Numerical study of heat pump contact drying

Numerical study of heat pump contact drying
Numerical study of heat pump contact drying
William Robert Catton
A thesis submitted for the degree of
Doctor of Philosophy
at the
University of Otago
Dunedin
New Zealand
May 2011
i
Abstract
Drying is one of the most energy intensive industrial operations, and it is well established that heat pump driers (HPDs), by recycling waste heat, may provide significantly higher drying energy efficiencies and lower net greenhouse gas emissions
than conventional driers. In addition, however, in the design of the HPDs themselves, there remains significant further scope for energy-efficiency improvements.
Second-law analyses of HPDs have shown previously that losses associated with the
convective transfer of heat to the drying process are a significant limiting factor for
energy efficiency. This thesis uses numerical simulation to explore the possibility
of improving on the energy performance of HPD systems by employing conductive
heat transfer from the refrigerant condenser, through a heating plate and through the
product itself, to drive the drying process in an "isothermal contact" HPD (ICHPD).
The duct model that is developed combines a detailed air-flow model, which solves
the mass, momentum and energy balances within the drier ducts, with a detailed internal drying process model, incorporating a description of the transport phenomena
occurring within the porous product medium. The whole-system dynamical HPD
model, which results when the drier-duct model is integrated with a pre-existing heat
pump model, is capable of describing the evolution of non-steady batch drying.
It is established that for applicable products the ICHPD configuration may increase
the energy efficiency of heat pump drying by as much as a factor of three compared
with conventional adiabatic HPDs. This ICHPD energy efficiency gain (relative to
the adiabatic mode) is, however, demonstrated to be highly sensitive to the product thickness (δ). The energy efficiency gain of ICHPD is also shown to be sensitive
to any constraint on the temperature and the maximum allowable relative humidity
above the product. Isothermal HPD is thus likely to be most applicable in the drying
of those products, such as sludges and pastes, that can be spread into thin layers, in
particular those that also are least vulnerable to quality deterioration at high temperature and humidity. Product throughput is shown to be simultaneously maximised at
low δ, implying that ICHPD provides an opportunity to avoid the adiabatic mode’s
trade-off between drying rate and energy efficiency, by using a thin product layer. A
case-study is presented of the economics of ICHPD in an industrial sludge-drying application, showing that isothermal HPD provides an opportunity to lessen exposure
to risk associated with electricity-price uncertainty. System performance is found to
be quite sensitive to variation in the surface area available for drying and also to the
dimensioning of the evaporator. A second-law analysis of the whole system is employed to examine the reasons underlying the energy performance gain associated
with the isothermal mode; ICHPD is found to reduce irreversibility equally within
the refrigerant cycle and in heat transfer from the condenser to the product plus the
drying process itself – a demonstration of the synergy enabled by the ICHPD design.
ii
Acknowledgements
I’m glad to have an opportunity to thank some of the people without whose help
this thesis could never have been completed. First, thank you to my two supervisors,
Zhifa Sun and Gerry Carrington, who have been strong role models and guides into
the world of research, generous in their efforts to secure the funding that made this
project feasible, and who came up with the idea of the project in the first place. It
has been fun exploring some of the impressive work that they have produced during
their collaboration. My thanks also to the other members of the research group, for
what I’ve learned from them, and for the camaraderie – it’s been great. The support
of my family has made a huge difference to me throughout my studies. Thank you,
Mum, Dad, Floss and Ellie. Thank you, grandparents. And welcome to the family,
my nephews Sebastian and Alexander, and thanks, little fellas, for being so good
at reminding us all what’s truly important! Finally: Eloise. Your contribution to
the completion of this thesis is unquantifiable. It has been your steady patience and
encouragement that has made it all possible. Now – here’s to what comes next!
iii
Publications arising from this study
Catton, W., Carrington, G., and Sun, Z. (2011). Exergy analysis of an isothermal
heat pump dryer. Energy, vol. 36, pp. 4616-4624.
Catton, W., Carrington, G., and Sun, Z. (2011). Performance assessment of contact
heat pump drying. International Journal of Energy Research, vol. 35(6), pp. 489500.
Catton, W., Sun, Z., and Carrington, G. (2010), Exergy analysis of an isothermal
heat pump dryer. Chemical Engineering Transactions, vol. 21, pp. 139-144.
Catton, W., Sun, Z., and Carrington, G. (2011), Dynamical modelling of an isothermal contact heat pump dryer. Oral presentation, Fifth Nordic Drying Conference,
Helsinki, Finland.
Catton, W., Sun, Z., and Carrington, G. (2010), Analysis of the energy efficiency
gain obtainable by incorporating indirect heat transfer into heat pump drying
technology. Oral presentation, 13th Conference on Process Integration, Modelling
and Optimisation for Energy Saving and Pollution Reduction, Prague.
iv
Contents
1
2
3
Introduction.
1
1.1
Context: heat pump drying . . . . . . . . . . . . . . . . . . . . . . . . .
1
1.2
Motivation for the thesis project . . . . . . . . . . . . . . . . . . . . . . .
8
1.3
Heat pump drying research . . . . . . . . . . . . . . . . . . . . . . . . .
15
1.4
Modelling of drying . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
21
1.5
Internal transfer processes . . . . . . . . . . . . . . . . . . . . . . . . . .
27
1.6
Aims and outline of thesis . . . . . . . . . . . . . . . . . . . . . . . . . .
32
1.7
Nomenclature for chapter 1 . . . . . . . . . . . . . . . . . . . . . . . . .
36
Initial performance assessment.
39
2.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
39
2.2
System description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
41
2.3
Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
42
2.4
Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . .
47
2.5
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
61
2.6
Nomenclature for chapter 2 . . . . . . . . . . . . . . . . . . . . . . . . .
62
Detailed drier-duct model.
64
3.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
64
3.2
Balance equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
65
3.3
Discretised equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
71
v
4
5
6
3.4
Refrigerant heat transfer coefficient . . . . . . . . . . . . . . . . . . . . .
84
3.5
Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . .
89
3.6
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
3.7
Nomenclature for chapter 3 . . . . . . . . . . . . . . . . . . . . . . . . . 110
Steady-state HPD model.
114
4.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
4.2
Pressure drop correlations . . . . . . . . . . . . . . . . . . . . . . . . . . 116
4.3
Compressor model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
4.4
Whole-system HPD model structure . . . . . . . . . . . . . . . . . . . . 121
4.5
Exergy functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
4.6
Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
4.7
Economic case-study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
4.8
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
4.9
Nomenclature for chapter 4 . . . . . . . . . . . . . . . . . . . . . . . . . 137
Dynamical HPD model.
139
5.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
5.2
Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
5.3
Results and discussion: 1-D model . . . . . . . . . . . . . . . . . . . . . 150
5.4
Results and discussion: 2-D duct model . . . . . . . . . . . . . . . . . . 152
5.5
Results and discussion: Full dynamical model . . . . . . . . . . . . . . 167
5.6
Results and discussion: The CDC method . . . . . . . . . . . . . . . . . 171
5.7
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188
5.8
Nomenclature for chapter 5 . . . . . . . . . . . . . . . . . . . . . . . . . 193
Discussion and conclusion.
196
vi
6.1
Survey of the findings of this study . . . . . . . . . . . . . . . . . . . . . 196
6.2
Comparison with other work . . . . . . . . . . . . . . . . . . . . . . . . 202
6.3
Strengths, weaknesses, suggested further work . . . . . . . . . . . . . . 204
6.4
Key conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206
A Background theory
209
A.1 Finite volume methodology . . . . . . . . . . . . . . . . . . . . . . . . . 209
A.2 Continuum mechanics background . . . . . . . . . . . . . . . . . . . . . 212
A.3 Conservation laws for multicomponent systems . . . . . . . . . . . . . 218
A.4 Linear constitutive relations . . . . . . . . . . . . . . . . . . . . . . . . . 227
A.5 Second law analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231
A.6 Turbulence and time-averaging . . . . . . . . . . . . . . . . . . . . . . . 237
A.7 Boundary-layer theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238
A.8 Properties of moist air and refrigerant R134a . . . . . . . . . . . . . . . 250
A.9 Nomenclature for appendix A . . . . . . . . . . . . . . . . . . . . . . . . 258
B Derivation of drying equations.
262
B.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262
B.2 Modifications for hygroscopicity . . . . . . . . . . . . . . . . . . . . . . 268
B.3 Mass balance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272
B.4 Mass transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277
B.5 Energy balance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285
B.6 Constitutive equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298
B.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302
B.8 Nomenclature for appendix B . . . . . . . . . . . . . . . . . . . . . . . . 305
References
308
1
Chapter 1
Introduction.
This chapter explains the motivation for the research, distinguishing between “adiabatic” and “isothermal” heat pump drying. The existing literature on the modelling
of heat pump drying systems is surveyed. This review highlights the absence of any
previously reported heat pump dryer (HPD) model that is able to provide prodictions
of the behaviour of the novel “isothermal contact” HPD (ICHPD) system design. The
aims, scope, and structure of the thesis are then described.
1.1.
Context: heat pump drying
The explosive recent diversification of human industry has brought with it a rapidly
growing array of applications for the removal of moisture. In industrialised societies
today, drying plays an integral role in the minerals, agricultural, pharmaceutical, textiles, wood, paper, and chemical industries, among others. Combined with water’s
high latent heat of vaporisation (∆hv ) – which is more than five times the energy required to raise liquid water from 0◦ C to 100◦ C – this ubiquity of drying processes
demands a vast supply of energy. In particular, drying has been estimated to contribute approximately 27% of the final embodied energy of paper, 50% of that of textiles, and 70% in the case of wood [101]. It has been estimated that in 1990 drying
contributed 12% of global industrial energy consumption, corresponding to about
16 × 1018 J [115]. In industrialised countries the proportion has probably climbed
since then, as energy consumption in other processes has fallen – not limited, as conventional drying systems are, by a thermodynamic barrier [97].
A wide variety of existing drying techniques matches the diversity of products
being dried. Drying systems are classified in a number of ways, including by their
mode of operation: batch or continuous; by the form taken by the product: paste, pow-
2
der, chip, sheet, plank; by the mode of heat transfer: convective (hot gas, usually air),
contact (or indirect, conduction through a hot plate or drum), or radiative (often microwave) [7, 94]. The theoretical focus of the present research, which is led by an idea
for increasing the energy efficiency of heat pump drying, also effectively defines its
scope of applicability. The work presented in this thesis is a theoretical and computational study. The thesis is concerned (1) with convective drying, (2) where contact
heat transfer is employed, and (3) where heat provision is from a refrigerant cycle
whose cold side is used to dehumidify the dryer atmosphere. The mode of operation
could be either batch-wise or continuous. The range of products to which such a system could usefully be applied is restricted by the above specifications, together with
some of the particulars of the system performance. These details are discussed in this
and other chapters. The present chapter reviews the HPD modelling literature, and
shows that such a system can be expected to provide exceptional energy efficiency –
but that no previously reported HPD model is capable of predicting this efficiency.
A typical configuration for a basic conventional convective drying system is illustrated in Fig. 1.1. In the system depicted, both latent heat and sensible heat (associated
with the higher-than-ambient temperature of the exhaust air) are lost to the environment. This heat flow is often supplied by the transformation of high-grade forms of
energy directly into heat. In contrast, rather than simply converting high grade energy to heat, HPD systems employ a lesser amount of work to recycle much of the
waste heat they produce. A simple dehumidification HPD system is shown schematically in Fig 1.2. The standard HPD design depicted in Fig 1.2 is sometimes referred
to as adiabatic (in contrast to isothermal) heat pump drying, a terminology that will be
explained in the following section. Dehumidification HPDs recirculate the air used in
the drying process, simultaneously reclaiming some of its unused sensible and latent
heat. The thermodynamic cycle most commonly used to transfer this heat against its
natural flow direction is the mechanical vapour-compression cycle [72]. The system
depicted in Fig 1.2 can be described as follows. The moist air leaving the product is
cooled by the evaporator – so-called for the role it plays to the refrigerant – to below
the moist air mixture’s dew-point temperature, causing some of the water vapour
to condense and drain from the system. The latent heat thus liberated is recycled
by pumping the gasified refrigerant to the higher-pressure, higher-temperature condenser, where as it condenses it transfers heat back to the air stream now approaching
the product. The resulting net transfer of heat up a temperature gradient requires an
input of available energy, which is supplied as (a smaller quantity of) electrical work,
to compress and drive the refrigerant in its closed-loop cycle.
Fig. 1.3 depicts an idealised version of the thermodynamic cycle taken by the refrigerant in the dehumidifier dryer of Fig. 1.2, and the corresponding air psychrometric cycle is shown in Fig. 1.7 (on page 12). Neglecting external heat loss from the
3
Air
Inlet
Air
Outlet
Fan
Heater
Figure 1.1: Conventional heat-and-vent tray dryer, adapted from [93].
compressor, the heating COP of the heat pump – the heat delivered by the condenser
divided by the work required (not including fan power consumption) – is given by
COP =
h2 − h3
h2 − h1
(1.1.1)
As can be seen in Fig. 1.3, the idealised cycle involves an isenthalpic expansion process in the throttle (from 3 to 4), and an isentropic compression process in the compressor (from 1 to 2). Within actual compressors substantial irreversibility takes place,
shifting state 2 to the right in each of the diagrams of Fig. 1.3, and this is quantified in
terms of the isentropic efficiency ηC of the compressor [81, p.150]:
ηC =
∆hs12
<1
∆h12
(1.1.2)
where ∆hs12 is the change in refrigerant specific enthalpy (between states 1 and 2) that
would result in the theoretical case of isentropic compression from the compressor inlet state to the given outlet pressure. Actual heat pump cycles differ from that shown
in Fig. 1.3 in several other significant ways, which include the presence of subcooling (state 3 lying to the left of the vapour-liquid coexistence region), of superheating
(state 1 lying to the right of the saturation region), and of non-negligible pressure
differences between locations 2 and 3 and between locations 4 and 1.
The potential benefits of heat-pump drying systems have been summarised by
Carrington [26], Kudra and Mujumdar [102], Chua et al. [47] and Eikevik et al. [62]
in recent reviews of the technology. By recycling the heat of evaporation, heat pump
dryers can offer significant energy savings. While conventional heat-and-vent drying
systems usually operate at a specific moisture extraction rate (SMER)
SMER =
moisture extracted (kg water)
total energy consumed (kWh)
4
B
C
Product
3
Throttle
4
A
Fan
Evaporator
Condenser
2
Condensed
moisture
1
Compressor
2
3
h=c
4
s=c
1
SPECIFIC ENTROPY, s
PRESSURE, p
TEMPERATURE, T
Figure 1.2: Typical dehumidifier dryer, adapted from [26].
3
2
s=c
h=c
4
1
SPECIFIC ENTHALPY, h
Figure 1.3: Idealised vapour-compression cycle, adapted from [3].
within about 0.2-0.6 kg kWh−1 , HPDs have exhibited SMERs as high as 7.94 kg kWh−1
[11]. This can be compared with a maximum of ∼ 1.6 kg kWh−1 for conventional
driers that recoup no latent heat. It must be emphasized that while conventional
drying systems typically consume heat, HPDs require high grade energy, usually in
the form of electrical power. Nevertheless, their high energy efficiencies can yield
substantial reductions both in primary energy consumption and in greenhouse gas
emissions, even after allowing for losses in the thermal generation of electricity [11].
Perhaps more significantly, they offer the potential, if combined with renewable electricity sources, to eliminate any need for fuel combustion. This fuel-conservation and
emissions-mitigation potential has been appreciated for decades [83, 99].
For several pressing reasons, the global impetus to streamline energy consump-
5
tion is currently trending upward [84]. Nevertheless heat streams and potential heat
sources remain an under-utilized waste product of many processes. For instance offcuts and other wood waste provide an abundant, currently economical heat source
for timber drying, where the provision of heat is seldom accompanied by electricity
cogeneration [57]. Such energy streams appear unlikely to become a valued commodity until greenhouse-gas externalities become substantially internalized, shifting
demand away from CO2 -intense fossil fuels. Even then, the cost of electrical energy
relative to that of heat will depend on factors that impact the ratio of heat-producing
processes to heat-consuming processes in society as a whole, such as the overall mix
of primary energy sources. Currently, with potential heat streams in plentiful supply,
the impetus to adopt HPD technology remains weak except where it adds value to
the drying process in other ways. This is reflected in the emphasis frequently placed
on the non-energy benefits, discussed below, that HPD can offer [26, 153, 47].
In addition to their energy performance, HPD systems can offer practical advantages such as closed drying environments that can be controlled independently of
ambient conditions [89, 64, 153]. Mujumdar [7] has listed processes by which product
quality can be degraded during drying. These include chemical changes such as oxidation, browning and loss of volatile and heat labile compounds; physical changes
such as shrinkage and texture changes; and in the case of food, nutritional changes
including growth of pathogenic micro-organisms, degradation of vitamins and proteins, and formation of carcinogens. Carrington [26] cites three key advantages of
heat-pump drying technology which can aid in controlling quality. These are the
ability to select (1) drying temperature, (2) absolute humidity and (3) the drying atmosphere, all independently of the environment, from a broad range of possibilities.
While drying conditions in HPDs are typically relatively mild (e.g. air velocities below 10 ms−1 and temperatures below 70◦ C), HPD systems can be operated across a
wide range of temperatures, from −20◦ C to 100◦ C. This control can enhance product
value both directly and indirectly, for instance in food drying by reducing the need for
preservatives and anti-browning agents. Closed-cycle dryer operation (Fig. 1.2) can
also be used to reduce atmospheric pollution, or to avoid costly exhaust air treatment
otherwise required to meet environmental regulations.
Carrington [26] also cites savings associated with HPD systems. There is the potential for significantly reduced life-cycle cost compared with conventional dryers,
due to the higher SMERs of heat-pump dryers. For small drying operations, HPDs
may also offer capital cost savings, e.g. by eliminating the need for boiler systems,
which may constitute a high fraction of the capital cost of the drying system. Heat
pump drying systems are capable of continuous operation, allowing maximal product throughput and short payback times, even where the capital cost is higher than
for other systems [47]. Heat pump systems can also offer a solution in locations where
6
environmental restrictions prohibit the use of fuel-fired drying units.
A detailed analysis of the barriers to the adoption of HPD that may underly the
relatively slow up-take of the technology, to date, would necessarily be complex and
interdisciplinary, and certainly lies beyond the scope of the present study. This and
the next paragraph nevertheless contain some brief speculation on the topic. It has
been demonstrated that HPD has the potential to offer cost savings while simultaneously delivering higher-value end products in a range of applications. The relative
significance of these benefits will depend on the details of the particular drying application, and must of course be weighed against the disadvantages of HPD, which may
include higher capital costs, costs of ongoing maintenance, and the need for fluorocarbons in the refrigerant cycle. On top of the disadvantages (and perceived disadvantages) listed above, a key barrier to HPD uptake to date may have been the complexity of HPD systems and their sensitivity to the manner of their operation, which
were investigated in detail by Cox-Smith [51]. Obtaining good performance from a
heat pump drier requires its use to be consistent with a detailed understanding of the
system, since HPDs exhibit rapidly declining performance away from optimal drying
conditions. As a general rule the moisture extraction rate (MER), the SMER and the
heat pump coefficient of performance (COP) can not be simultaneously maximised
[129]. Performance sensitivity to resulting design tradeoffs may have contributed to
the slow adoption of heat pump technology in industrial drying [33], despite several
decades of interaction between modelling work [196, 27, 6, 88, 87, 46, 132, 130, 162]
and empirical investigation [166, 165, 48, 31, 10, 131].
It should be acknowledged that some of the barriers already discussed are not
unique to heat pumps. For instance boiler systems and heat-and-vent kilns themselves require ongoing maintenance. Nevertheless HPD technology may be more
sensitive to its areas of relative weakness than conventional drying technology as a
consequence of the HPD technology’s relative immaturity, and of the “coordination
problems” associated with any shift from one technological convention to another.
Conventional drying systems today have reached a mature state of development,
with many easy performance gains already having been achieved, and thus with relatively little room left for incremental performance improvements. This observation
may apply not only to the state of the technology itself, but also to systems of human organisation and cultural traditions (ranging from expertise to habit and brand
awareness), which have had a long co-evolution with conventional technology. The
situation is illustrated metaphorically in Fig. 1.4: even when new technologies offer
significant promise in performance gains, their adoption and development would in
many cases require an initial reduction in (some aggregate measure of) performance.
In the case of HPD, such a barrier may manifest itself as an aversion to perceived
risk, perception of complexity and of increased capital outlay, fear of maintenance
7
issues, and perhaps incomplete understanding (together with the attitude, “if you
don’t understand it, don’t buy it”). Since in addition to these barriers, the economic
viability of heat pumps is tied to the relative costs of provisioning electricity and heat,
it seems likely that external forces such as environmental legislation may prove key
to spurring a significant adoption of HPD by industry [102, 113].
Performance
Conventional drying systems
Heat pump drying systems
Figure 1.4: Performance evolution in drying technology
Regarding Fig. 1.4 it is worth focusing briefly on the poor environmental track
record held by heat pumping technologies in general [152]. First-generation refrigerants (CFCs) have done, and continue to do, great damage to the ozone layer. Nonchlorinated halocarbons (HFCs), which have come to replace CFCs after these were
banned under the 1987 Montreal Protocol, are known to be significant greenhouse
gases. For instance HFC-134a has a global warming potential GWP ≈ 1300 [111],
i.e. it contributes 1300 times more than CO2 to the greenhouse effect, per unit mass.
To the extent that refrigerant leakage is unavoidable, current heat pumping technology can be regarded, at best, as a bridge toward the technologies that may ultimately
eliminate the climate impact of our activities [152]: although heat pump driers based
on HFCs may reduce net GHG emissions, they cannot eliminate them. One hopeful
area of ongoing research is a return to the use of natural refrigerants, which, because
they occur in the natural world, may be assumed to be relatively environmentally benign. Promising possibilities include hydrocarbons, ammonia, and CO2 , which were
among the first refrigerants. Of course, the fact that these had been abandoned as
refrigerants in favour of the HFCs points to the fact that the natural refrigerants have
problems of their own, which include inflammability in the case of hydrocarbons,
and toxicity and corrosiveness in the case of ammonia [102, 152].
8
1.2.
Motivation for the thesis project
The present work centres on a simple idea for increasing the energy efficiency of heat
pump driers. The idea arises naturally from consideration of the Gouy-Stodola law
[15], which is expressed in local form (for a multicomponent fluid mixture) by the
exergy balance equation (A.5.28) derived in the appendix. This is the result that the
lost opportunity for work (or in the case of work-consuming processes, the minimum
work input) associated with a process is given by the environmental temperature
times the total rate of entropy generation in the process. In a heat pump drying system, recycled heat is typically returned to the drying process convectively, by heating the dehumidified airstream as it reapproaches the product (Fig. 1.2). But from a
second-law viewpoint, this method – using air for heat transfer – appears wasteful.
Heat transfer both into the air and from the air to the drying process is responsible
for a substantial part of the entropy creation in such a system [27, 176]. So may be
losses owing to the airflow resistance of the system and fan friction [33]. Taken together, these losses result not only from the low thermal conductivity of air, but also
from the large mass circulation rate necessitated by its relatively low specific heat capacity. The straightforward idea underlying this thesis is that these entropy creations
can be avoided, or at least mitigated, by instead providing heat directly to the drying
process, through a conductive plate separating the refrigerant and the product. The
idea is summarized in the abstract of NZ patent 526648 [32], which was written by
Carrington and Scharpf prior to the research reported in the present thesis:
Dehumidifier drier for pastes, liquors and aggregate materials.
Abstract: A process and apparatus for drying pastes, liquors and aggregate materials by means of a heat integrated and/or heat pumping process and/or apparatus is disclosed. This includes a heat pump or heat
integrated apparatus operable in a drying apparatus with the heat pump
evaporator (36) or cold heat exchanger in primary thermal contact with
the drying gas medium (33) after said drying gas medium (33) has taken
up moisture from the material being dried (35) and the heat pump condenser (14) or hot heat exchanger (36) in primary thermal contact with
the material being dried and with both the drying gas medium (33) and
any heat pump refrigerant in nominally closed loop circulation paths (22).
This process and apparatus may provide improved efficiency and reduced
costs by reducing the required flow of drying gas through the system since
that drying gas is no longer the primary means for supplying heat to the
material being dried.
9
A schematic depiction of such a system is shown in Fig. 1.5. The term ‘isothermal
contact HPD’ describing such a system is shortened to ‘ICHPD’ in this thesis.
B
C
Product+Condenser
3
Throttle
4
A
Fan
Evaporator
Condenser
2
Condensed
moisture
1
Compressor
Figure 1.5: Dehumidifier contact dryer, adapted from [26].
The steady-state stationary control volume illustrated in Fig. 1.6 can offer an alternative view of the benefit promised by contact HPD. The control volume, which
encloses the product and airstream lying between locations B and C in Fig. 1.5, is
shown in Fig. 1.6 bounded by a dashed line. The drying system is conceptualised in
Fig. 1.6 as a counter-flow dryer operating at steady state. In an ICHPD system, the
heating plate that supports the product would contain heat pump condenser coils
with high-pressure refrigerant condensing within them, liberating heat at rate Q̇. By
employing several rough assumptions to simplify our analysis of Fig. 1.6, one can
illustrate another aspect of the contrast between adiabatic and isothermal HPD.
Moist air is shown entering the volume through the air-inlet portion at the top of
the left-hand bounding surface (‘B’) of Fig. 1.6. The air is regarded as comprising two
components, water vapour (entering at rate ṁv,B ) and ‘dry air’ (entering at rate ṁa ).
Assuming that the moist air can be treated as an ideal gas mixture, the enthalpies of
the two components can be treated independently [58, p.129]. The water vapour has
specific enthalpy hv,B at the air inlet, and the dry air has specific enthalpy h a,B . There
is no significant mass exchange of dry air between the product and airstream. Since
the control volume is at steady state, so that dry air cannot be accumulating within
the volume, an identical mass flow-rate of dry air, ṁa , departs the volume through the
air-outlet bounding surface on the right. Similarly, moist product enters the system
10
through the product-inlet portion at the bottom of the bounding surface ‘C’. The drysolid component of the product departs the system on the left at the same rate as
it enters on the right, ṁs . For the sake of argument, we assume that the product is
nonhygroscopic and that all the moisture is evaporated from the product into the air
stream, so that the air-outlet mass flow-rate of water vapour exceeds that at the inlet,
and the product-outlet mass flow-rate of condensed moisture is zero. A water-mass
balance across the control volume then yields ṁv,C − ṁv,B = ṁw,C . Finally, we assume
that the (arbitrary) product inlet temperature is identical to the air outlet temperature,
i.e. Tprod,C = Tair,C . This assumption slightly simplifies the following argument.
Insulation
B
C
Tair,B
ha,B ṁa
hv,B ṁv,B
Air
Tprod,B
ṁs
hs,B
ṁv,C − ṁv,B
Product
Q̇
Tair,C
ha,C ṁa
hv,C ṁv,C
Tprod,C
ṁs
hs,C
ṁw,C hw,C
Heating plate
Figure 1.6: Steady state drying control volume
The control volume of Fig. 1.6 is now used to examine the two drying modes represented in Figs. 1.2 and 1.5, by considering the two idealised cases respectively called
“adiabatic” and “isothermal” humidification processes by Keey [94, p.101]. This will
clarify the reasons why isothermal HPD (an idealised version of the system illustrated in Fig 1.5) can be expected to offer an energy efficiency improvement over the
conventional, adiabatic, HPD mode (illustrated in Fig 1.2). The following is a sketch
which will employ a number of assumptions and simplifications that are justified by
arguments or results appearing in later chapters. Using (A.3.61) in the stationary control volume energy balance (A.3.51), and neglecting kinetic energy, body forces, and
work done against stress forces, yields:
ṁ a h a,B + ṁv,B hv,B + Q̇ + (ṁv,C − ṁv,B )hw,C + ṁs hs,C = ṁ a h a,C + ṁv,C hv,C + ṁs hs,B
(1.2.1)
In (1.2.1), hw,C is the specific enthalpy of the water arriving at the product inlet at the
bottom of ‘C’. In (1.2.1) longitudinal heat conduction has been neglected, and Q̇ rep-
11
resents the heat flow integrated over the bottom surface of the control volume. For
simplicity, the terms ṁs hs,C and ṁs hs,B describing the convective energy transport
with the dry solid will be neglected. These terms can be shown to have a relatively
small effect if the incoming product has a large moisture content; also, in the isothermal case, the terms cancel to zero since TC = TB . Rearranging, (1.2.1) then becomes:
ṁ a h a,B + ṁv,B hv,B + Q̇ = ṁ a h a,C + ṁv,B hv,C + (ṁv,C − ṁv,B )(hv,C − hw,C )
(1.2.2)
By examining (1.2.2), we can compare the two idealised drying modes.
The present work adopts the convention [184] that the reference temperature T0 ,
at which the enthalpy of liquid water is defined to be zero, is 0.01 degrees centigrade:
T0 = 273.16 K, with hw ( T0 ) = 0 J. With this convention, across the moderate temperature range that is of interest in the present work, the outlet vapour enthalpy hv,C
provides a good approximation to the term hv,C − hw,C appearing in (1.2.2), since
h
i h
i
hv − hw = hw ( T0 ) + ∆h0v + cv ( T − T0 ) − hw ( T0 ) + cw ( T − T0 )
(1.2.3)
≈ ∆h0v + cv ( T − T0 ) = hv
(1.2.4)
where (1.2.4) can be obtained from (1.2.3) by noting that ∆h0v cw ( T − T0 ) for relevant temperatures. Using (1.2.4) in (1.2.2) produces the following equation:
ṁ a h a,B + ṁv,B hv,B + Q̇ = ṁ a h a,C + ṁv,C hv,C
(1.2.5)
In passing from (1.2.2) to (1.2.5) we have effectively neglected the enthalpy delivered
to the control volume by the liquid water arriving with the product at C.
Adiabatic drying corresponds to the situation where there is no significant heat
flow from the plate, i.e. Q̇ = 0. In this situation the total enthalpy per unit mass of
dry air is a constant: we can drop Q̇ from (1.2.5) and rearrange, to obtain
ṁ a h a + ωhv
C
= ṁ a h a + ωhv
B
(1.2.6)
In (1.2.6) ω = mv /m a is the humidity ratio (A.8.34), and h a + ωhv is the specific enthalpy of the mixture, measured per unit mass of dry air. In obtaining equation (1.2.6)
from (1.2.5) we have neglected the diffusive flux of moisture along the air flow direction, which is reasonable since it is generally tiny compared with the convective flux.
In adiabatic drying, since no heat arrives from beneath the product surface, the
energy required to evaporate the liquid water is provided by the air itself. As a consequence, in accordance with equation (1.2.6), the air remains at constant specific
enthalpy as it passes over the product. The upshot of this is captured in the psychrometric chart displayed in Fig. 1.7: because total enthalpy (per unit mass of dry air) is
constant, the air temperature falls as the air acquires moisture (note that the diagonal
lines in the figure are lines of constant enthalpy). In Fig. 1.7, air that approaches the
12
product at 20% relative humidity at 44 degrees centigrade (point B) can be seen to be
limited to a maximum moisture content of around 20 grams of moisture per kg of dry
air, as shown by the dotted line extended from point C to the saturation curve.
0.04
0.03
0.025
0.02
φ=20%
C
0.015
φ=10%
A
B
0.01
0.005
TWB=20°C
−10
0
10
20
30
T,
°
40
50
Humidity ratio, kg water/kg dry air
0.035
0
60
C
Figure 1.7: Air cycle, adiabatic drying, adapted from [26]. Points A, B and
C correspond to those illustrated in Figure 1.2. The three subprocesses are:
A→B: heating. B→C: humidification (drying). C→A: dehumidification.
In idealised isothermal drying, a nonzero heat flow Q̇ balances the latent heat consumed by the evaporation process, leading to constant temperatures throughout the
system: TC = TB = Ts = T. Examining (1.2.2) we can evaluate the required heat flux.
Since we are assuming ideal gas behaviour, the enthalpies of the air and vapour are
functions of temperature only. Since the inlet and outlet temperatures are identical,
we can cancel the first two terms on both sides of (1.2.2), to obtain:
Q̇ = (ṁv,C − ṁv,B )(hv,C − hw,C )
(1.2.7)
As we have assumed that the moist product is arriving (at C) at the same temperature
as the outlet air (at C), and thus as the rest of the system, we can write hv,C − hw,C =
∆hv ( TC ). Thus we obtain the heat flow condition for isothermal drying:
Q̇ = ∆hv ( TC )(ṁv,C − ṁv,B )
(1.2.8)
In isothermal drying, because heat is entering the control volume from beneath,
allowing the air to maintain a constant temperature as it acquires moisture, the enthalpy per kg of dry air increases over the product. Thus the total moisture acquired
13
in one pass is not limited by the adiabatic saturation temperature of the inlet air. This
is illustrated in Fig. 1.8, which has the same drier inlet condition (state ‘B’) as Fig. 1.7.
Note that convective drying is described by equation (1.2.6)), so that, subject to the
assumptions stated above, “convective” HPD is “adiabatic” HPD. In contrast, equation (1.2.8) represents a further condition on heat flow, which must be met in order
for contact drying to meet the idealised “isothermal” definition. The validity of the
idealisations employed in the above sketch are considered in chapters 3 and 5.
0.04
0.035
0.03
0.025
0.02
φ=20%
0.015
φ=10%
A
B
0.01
TWB=20°C
−10
0
10
20
30
40
50
Humidity ratio, kg water/kg dry air
C
0.005
0
60
T, ° C
Figure 1.8: Air cycle, isothermal drying, adapted from [26]. Points A, B and C
correspond to those illustrated in Fig. 1.2. The three subprocesses are: A→B:
heating. B→C: humidification (drying). C→A: dehumidification.
We now briefly consider the implications of the above discussion for the relative
energy performance of the isothermal and adiabatic HPD modes. The heating coefficient of performance of an idealized Carnot cycle, COPC , provides the limiting
heating COP for a real heat pump system. We have:
COP = ηC COPC = ηC
Tco
Tco − Tev
(1.2.9)
where Tco and Tev are the condensor and evaporator temperatures, respectively, and
ηC is the ratio of the heat pump heating COP to the ideal Carnot heating COP, which
satisfies 0 < ηC < 1. The ratio ηC tends to vary slowly over a range of temperatures
[196], and as a consequence, the heat pump heating coefficient of performance generally increases as the pressure (and temperature) difference between the condensor
14
and evaporator is reduced [25]. Since the SMER is strongly affected by the heat pump
COP, the SMER is affected by the condensing and evaporating temperatures. Thus for
a standard HPD with convective heat transfer (Fig. 1.2), a tradeoff exists between fan
power and compressor power: a lower compressor power and heat pump temperature lift Tco − Tev would improve the heat pump COP, but would also increase the
required air circulation and thus the power requirements of the fans, to achieve the
same rate of heat transfer to the product. The small specific heat of air corresponds to
the small negative gradient of the diagonal isenthalps in Fig. 1.7. Fig. 1.8 shows how
the isothermal drying mode may allow a HPD to escape this tradeoff, by freeing the
heat delivered, at a given air flow rate, from its dependence on the air temperatures
TB and TA , which are connected with the HP temperatures Tco and Tev .
Figures 1.7 and 1.8 have been adapted from the review by Carrington [26], and are
similar to Figures 1 and 9 respectively of references [89] and [152], papers from the
Norwegian de-watering technology research group (SINTEF, NTNU): thus it is evident that a distinction between adiabatic and isothermal HPD is reasonably standard
in the HPD literature. However, there appears to have been no previous systematic assessment of the energy performance gain that may be obtained by combining
HPD with contact heat transfer, and such an assessment forms a key objective of the
present study. As discussed above, because this is a “thermodynamics-led” project,
the scope of applicability of the work has been determined at the outset, by the range
of application possibilities for contact HPD systems. Part of this work has been to delineate the impact of product constraints on the performance of a contact HPD. The
most important constraints on the system that have arisen are that the product must
be sufficiently thin to enable good heat transfer, and should be able to be dried under
fairly high-humidity conditions (see chapters 2 and 6). Consistent with the patent abstract quoted on page 8, these constraints suggest that such a system would be most
applicable to the drying of products such as sludges and pastes that (1) can be spread
into thin layers and that (2) are invulnerable to degradation during a relatively slow
drying process in a high-humidity drying environment.
An interesting potential application of isothermal HPD is in the drying of bioalgae pastes (after mechanical dewatering) prior to biodiesel production, since energy
considerations play a central role in biofuel viability, and drying may turn out to be a
crucial, energy-intensive process step [44, 59]. To a plant operator, the applicability of
HPD technology to any particular use would be an economic matter, dependent (for
instance) upon a favourable balance between operational savings or benefits and the
cost of amortizing capital outlay. In the near term, ICHPD appears most likely to find
applications among the medium-temperature drying processes in which conductive
heating is already currently employed, particularly those that demand or would benefit from high energy efficiency or closed air-cycle operation. For instance, contact
15
heat transfer is widely used in the food industry to treat heat-sensitive products. In
food drying applications, closed air-cycle HPD can prevent the emission of volatile
organic compounds such as solvents (or eliminate combustion products that would
be released by fuel-fired dryers), and may eliminate the need for product additives
such as sulfites by allowing the use of a modified drying atmosphere [26]. Probably
the most likely near-term candidate for ICHPD is the drying of industrial sludges,
such as filter-cake sludges or wastewater sludges: contact heat transfer is already
routinely employed in industrial sludge drying, where closed air-cycle operation is
also commonly required to meet local environmental standards, and where energy
efficiency is a significant determinant of costs, and thus a key factor in the ongoing
evolution of the drying technology [43]. What is lacking so far is a marriage between
dehumidification HPD and contact heat transfer to the drying process.
1.3.
Heat pump drying research
Unlike conventional drying systems, whose irreversible provision of heat dwarfs
other entropy-generation mechanisms, HPDs are highly susceptible to incidental entropy generation, which increases the heat-recovery load on the heat pump. Consequently the analysis of such systems requires the full exergy functions (A.5.20) and
(A.5.21) to be considered, rather than just their energy components u + 12 v2 + ψ and
h + 12 v2 + ψ respectively, as would be appropriate for systems not employing heat
pumps. An early application of second-law analysis to drying processes was conducted by Carrington and Baines [6, 27], who employed the process-optimisation philosophy described by Linnhof [104]. Linnhof had emphasized a need to focus engineering effort on those subprocesses most likely to yield significant energy-efficiency
improvements. To simplify this sharpening of focus, he recommended compiling the
rational (second-law) inefficiencies of an aggregate process, and then categorizing
each loss mechanism as being either “avoidable” or “inevitable”.
Applying this approach to drying processes, Carrington and Baines considered the
following loss mechanisms for a simple HPD as depicted in Fig. 1.2: (1) irreversible
diffusion of water vapour from the product surface into the drier air, (2) transfer of
heat across heat exchangers and down temperature gradients, (3) discharge of air
above the ambient temperature and humidity, (4) discharge of condensed moisture
above the ambient temperature, (5) losses associated with the heat pump compressor,
(6) losses associated with the heat pump throttle. More generally, one could include
losses associated with (7) fan motors, and (8) frictional resistances to fluid flows. Of
the losses listed above, the first was considered to be the least avoidable [27]. We will
consider the limiting energy efficiency of drying that arises due to irreversible vapour
16
diffusion, since the argument bears significantly upon our entire thesis project. Consider the control volume in Fig. 1.9, which is similar to that in Fig. 1.6, except that it
does not enclose the product. The figure shows a molar flux n a of dry air and nv of
moist air entering the steady-state control volume, admixing with a molar flux ∆nv of
saturated vapour, and exiting the control volume at the right. The partial pressures
of the components at the various locations are also labelled.
na pa
pa + ∆pa na
nv pv
in
pv + ∆pv nv + ∆nv
out
∆nv
pv,sat
Figure 1.9: Steady state drying control volume
To evaluate the irreversibility generated by the scenario depicted in the figure, we
use the general control volume entropy balance, which can be written [15]:
!
!
nq
n
n
dS
Qi
=∑
+ ∑ ni si
− ∑ ni si
+ σ̇
dt
T
i =1 i
i =1
i =1
in
(1.3.1)
out
In (1.3.1), nq represents the number of distinct temperatures at which heat is supplied
to the control volume, and n represents the number of species that are exchanged
through its bounding surface. σ̇ represents the rate of entropy generation within the
volume. The loss due to irreversible vapour diffusion can be evaluated by considering the idealised scenario in which the fluxes n a and nv are very large compared
with ∆nv , the system is adiabatic and p a + ∆p a ≈ p a and pv + ∆pv ≈ pv , so that
the vapour diffusion is the only source of irreversibility. For the steady-state control
volume shown in Fig. 1.9, the entropy balance (1.3.1) becomes
!
!
σ̇ =
∑
i = a,v
ni si
out
−
∑
ni si
i = a,v
(1.3.2)
in
In the ideal-gas limit, (A.8.22) yields the molar entropy si of species i as
si = xi s0i ( T ) − xi R ln pi
(1.3.3)
where s0i ( T ) is the entropy of species i at unit pressure, at the given temperature.
Using (1.3.3) in (1.3.2), with the fluxes as shown in Fig. 1.9, yields:
σ̇ = −∆nv R ln φ
(1.3.4)
17
where φ = pv /pv,sat is the relative humidity of the inlet mixture. According to the
Gouy-Stodola law, the rate at which exergy is destroyed in this process is T0 σ̇. Thus
the minimum work requirement for the evaporation of water, per mole, is
∆W = − RT0 ln φ
(1.3.5)
where φ is the relative humidity of the unsaturated drier atmosphere.
Equation (1.3.5) illustrates an important point about energy efficiency considerations in drying. Maintaining a closed drier in an unsaturated state requires a work
input, since in the presence of condensed moisture only a saturated drier atmosphere
is thermodynamically stable. But the maximum SMER for such a system is unlimited
in principle, since the system can be operated as close to reversibility as desired, by
approaching an infinitely slow drying process at 100% relative humidity. As a matter
of fact, since drying is spontaneous in an unsaturated environment, in theory, in such
an environment, it could be used to provide work. However, non-thermodynamic
constraints on the engineering of the system and on the drying process inevitably impact energy performance. For instance, moisture activity at the product surface often
must be held below some maximum acceptable level – such as 0.6 [121] – to guarantee
product quality (e.g. to prevent microbial contamination or enzymatic browning of
food), and to obtain reasonable drying rates. It follows that a comparison of any set of
drying system configurations can only be made subject to the particular requirements
of the drying process for which they are to be used [27].
Because the major irreversibilities in dehumidification drying generally occur in
the heat transfer processes, in the compressor, and in the fans, key design tradeoffs
arise between the competing demands of system pressure losses and heat transfer
irreversibilities [33] – a situation that is typical in the engineering of heat transfer
equipment [15]. As has been discussed above, the aim of the present thesis is to assess
the potential (SMER) benefit of mitigating this tradeoff through a new HPD configuration, in which HPD is combined with contact heat transfer to the drying process.
To situate the present research within the literature, we will review past work which
has focused on optimising HPDs for energy performance. One parameter whose optimisation involves a tradeoff between pressure irreversibilities and heat transfer irreversibilities is the evaporator airflow bypass. The option of using evaporator bypass
to optimise the condenser and evaporator airflows independently was described by
Geeraert [68], and has received extensive subsequent analysis [27, 31, 33, 35, 48, 87].
Geeraert [68] emphasized the benefit that could be obtained by increasing the airflow through the product and the condenser, while holding the evaporator airflow
constant by allowing some of the product air to bypass the evaporator. This tends
to reduce the condensing temperature, with heat delivery to the drying process occurring at a relatively lower air temperature [6]. A lower heat pump temperature
18
lift Tco − Tev is thus required for a given rate of moisture extraction from the product. The resulting improvement in heat pump COP must be traded off against an
increase in fan work per kg of moisture extracted. Often the system airflow is fixed
by a prior choice of drier fans and drier geometry, and in this case it is the evaporator airflow that may be independently varied. Increasing the evaporator airflow
tends to increase the heat pump evaporating temperature, again improving the heat
pump COP. However this increased evaporating temperature also tends to decrease
the latent cooling proportion at the evaporator, reducing the condensation rate and,
ultimately, the SMER. Carrington and Liu [31] later showed experimentally that this
effect is sensitive to the relative humidity of the air approaching the evaporator.
Here we insert a parenthetical remark about the nature of evaporator bypass: different methods may achieve the same result. The Norweigian group at SINTEF employs evaporators designed for refrigerators [23]. These heat exchangers have relatively low-density coil spacing and fin pitch, allowing air to exit them – under typical
HPD conditions, with 100% of the air-flow passing through the evaporator – at a condition that significantly departs from that at the surface of the evaporator coils. In
the language of the previous paragraph, we might say that this amounts to allowing
part of the air-flow to ‘bypass’ the evaporator despite passing through the evaporator.
In contrast, the Otago University research group has employed heat exchangers designed for the air conditioning industry. These components have relatively denselypacked heat transfer areas, and optimisation of the evaporator air-flow requires air to
externally bypass the evaporator. In effect, one can either modify evaporator air-flow,
or evaporator heat-exchange area, to optimize the system SMER.
Geeraert [68] termed the limiting case of very high air recirculation through the
condenser and product region isothermal drying, since although the plate heat transfer
(Q̇ of Fig. 1.6) is nonexistent, in this limit negligible cooling of the air takes place as it
acquires moisture. By adiabatic drying he referred to a drying process in which the air
cools (along a path of constant wet-bulb temperature) as it acquires moisture, such as
a typical convective HPD when operated without evaporator bypass. In general, nonadiabatic operation enables the heat pump to operate over a smaller temperature lift,
for a given MER and airflow, than adiabatic operation, and a bypass-optimised HPD
with high product airflow may obtain a significantly higher SMER than a low-airflow
HPD without bypass, subject to the fan power tradeoff described above [31].
Most of the HPD research literature focusses on systems that may include evaporator bypass, but otherwise do not include heat transfer from the condenser during
the drying process. A number of numerical simulations of HPDs have been reported.
Comprehensive models have been developed and validated by Jolly, Jia and Clements
[48, 87, 88], by Pendyalla et al. [119, 120], by Praserstan et al. [130, 131], Prasertsan
19
et al. [132], and by Theerakulpisut [167]. These models are each built up from detailed heat and mass balances over differential elements of the heat exchangers and
of the drying process, and employ heat and mass transfer correlations from the refrigeration and chemical engineering literature. Simpler HPD models that have been
described include a model developed by Chou et al. [46] based on simple, idealised
theoretical considerations, and models based on empirical data developed by AlvesFilho et al. [1], by Charters and Aye [42], and by Carrington and Bannister [28]. None
of the models in the literature can be applied to ICHPD without further development,
as the heat pump condensers are modelled separately from the drying process.
The University of Otago energy research group has done extensive modelling of
HPD systems applied to timber drying, as well as of the drying process itself. Carrington et al. [24, 30] and [10, 11, 31] obtained performance data for a HPD system
based on a scroll compressor, whose key specifications are listed in Table 2.2 of this
thesis, and used the results to produce an empirical heat pump model that uses
lumped-parameter expressions to describe the heat transfer properties of the heat
exchangers, the refrigerant pressure drops, and the compressor’s isentropic and volumetric efficiencies [28]. Sun et al. [155, 157, 159, 160, 162] developed a comprehensive
model of the timber stack, and used detailed CFD calculations to establish heat and
mass transfer correlations for the stack. Vaughan et al. [175, 176] performed an exergy
analysis of the timber stack, and demonstrated that heat transfer was the primary
mechanism of exergy destruction within the stack under most drying conditions, corresponding to a limiting SMER on the order of 100 kg kWh−1 under typical conditions
in the constant drying-rate period. (Note however that much of the exergy destruction that may be avoidable through the use of contact heat transfer is associated with
the compressor and the fans, as well as the condenser, so that 100 kg kWh−1 does not
set an upper bound on the potential SMER improvent associated with contact HPD.
In this situation a binary division of exergy losses as being either ‘avoidable’ or ’inevitable’ may not be appropriate, due to the synergistic nature of the system – see
chapter 4.) By combining the empirical HPD model and the detailed timber stack
model, Sun and Carrington [33, 156, 157] went on to produce integrated, dynamical
whole-system models. These models were subsequently used for an assessment of
the impact on performance of air leaks and heat losses [34] and an investigation of
the coupling between pressure losses and heat transfer losses [33, 35].
The other recent work whose theoretical motivation is most similar to the present
project has come from the Norwegian group mentioned on page 14. This group has
developed heat pump systems for food drying and successfully placed these systems
in industry [60, 61, 62, 63, 64]. A high-performance non-adiabatic HPD has been created by configuring the condenser to progressively transfer heat to the air before and
after multiple drying stages in a single air pass, and has been implemented [89, 153] in
20
a high-SMER fluidised-bed system in which the compressor power, the air-flow rate
and the fan power have all been reduced relative to an adiabatic dryer. The group
makes the following general recommendations for maximising HPD efficiency [152]:
(1) Use lengthwise (as opposed to cross-flow) operation to maximise air outlet relative
humidity. (2) Use counter-current flow to maximise relative humidity at the outlet. (3)
Use continuous processing where possible, in order to maintain optimal conditions.
(4) Maximise the air temperature at the product inlet. (5) Avoid excess refrigeration,
which consumes energy to reduce relative humidity unnecessarily. (6) Optimize the
condensing and evaporating temperatures for SMER. Clearly these recommendations
need to be considered alongside the requirements of the product being dried. For instance, counter-current flow is generally harder to control [7], with fluctuations in the
air-inlet flow properties impacting directly at the product outlet.
As discussed above, the possibility under consideration here would be for a HPD
to be configured for contact drying, with the condenser placed in direct thermal contact with the product [32]. This possibility appears to be unexplored by other research groups, including the Norwegian group. We begin by modelling the general
HPD system depicted in Fig. 2.1 (page 40), which incorporates both the adiabatic
and isothermal modes, as well as evaporator bypass and venting. With reference to
Fig. 2.1, the two idealised drying modes can be defined as follows:
1. A purely “isothermal” dryer, which is an idealisation of the dryer depicted in
Fig. 1.5, is one where Q̇ D matches the drying rate so that air in the drying control
volume acquires moisture at constant dry-bulb temperature. Thus TE = TD .
2. A purely “adiabatic” dryer, which is an idealisation of the dryer depicted in
Fig. 1.2, is one for which Q̇ D = 0, so that air in the drying control volume cools
as it acquires moisture, along a path of constant wet bulb temperature. Thus
Twb,E = Twb,D , and since ωE > ωD , the dry bulb temperature TE < TD .
Chapters 2 and 4 contain steady-state modelling treatments of both modes, both with
and without evaporator bypass. Chapters 3 and 4 contain an assessment of how
closely an ICHPD may approach the isothermal mode. In the high-bypass limit, the
adiabatic mode resembles the isothermal mode, except for a comparatively high ratio
of fan power to evaporator airflow. The adiabatic mode with evaporator bypass is
essentially identical to the HPD systems considered by previous investigators [e.g.
6, 87, 132], with the possible exception that the vent port is located after the evaporator to maximise the vented RH, consistent with the recommendation of Solczyk and
Chow [147]. Although typically beneficial to the SMER, this vent location may lead
to additional design complexity and to restrictions on the applicability of venting, as
discussed in reference [29]. From the perspective of energy efficiency, the distinction
21
between adiabatic and isothermal drying is unimportant to conventional heat-andvent drying, since most of the irreversibility in a conventional heat-and-vent dryer
occurs in the conversion of high-grade energy into heat, and in venting [27].
1.4.
Modelling of drying
Arguably the most complex part of an HPD system is the drying process itself. A
typical convective drying process can be regarded as two moisture flow resistances
in series [139]. The first resistance relates to the transfer of moisture from within
the product to the product surface; the second to the transfer of moisture from the
product surface into bulk flow of the drying air. The driving force for each of these
subprocesses is influenced by the temperatures, which are linked to the external heat
transfer coefficient and to the modes of heat transfer that are present. The drying
process is conventionally divided into the Constant Rate Period (CRP) and the Falling
Rate Period (FRP) [49]. In the CRP, the moisture activity at the surface remains close
to 1, despite the falling surface moisture content. The surface drying flux is thus
governed by the external flow resistance, and equals the drying flux over a free liquid
surface under the same conditions. At the onset of the FRP, the surface moisture
content has fallen sufficiently to cause a significant lowering of the surface moisture
activity. As a result the surface vapour mass flux (nv,s ) starts to decline, and the
drying rate becomes determined by the internal transport properties of the product.
The internal processes in a drying product are complex, involving highly nonlinear,
tightly coupled heat and mass transport mechanisms.
Coumans [49], Katekawa [92] and Zhang [195] have recently provided useful surveys of the drying literature. A large number of dynamic drying models are reported
by a large number of authors, based upon several different approaches to modelling
internal moisture, heat and mass transport. Because of the size of the literature, a
complete review will not be attempted here; a selection of representative papers will
be used to illustrate the main approaches the modelling of drying. For completeness, mention must be made of the early work of Luikov [108, 109], who applied the
basic equations of nonequilibrium thermodynamics directly to the aggregate porous
medium to obtain a system of PDEs. However the alternative theoretical basis developed in large part by Whitaker [188] appears to have carried the day, with a
clear majority of modelling work published post-1980s adopting Whitaker’s volumeaveraging approach or some close equivalent [e.g. 5, 52, 66, 70, 106, 112, 122, 124, 126,
143, 172, 173, 178]. For a strident criticism of Luikov’s approach, see Whitaker [192],
who argues that cross-terms resulting from first-order thermodynamic driving forces
are likely to dominate the second-order thermodynamic forces. Largely because mod-
22
els with a complete theoretical underpinning are unwieldy, a number of simplifying
assumptions are often made when applying theoretical models to real-world situations. The differences of approach mainly reflect varying degrees of completeness in
the description of the drying phenomena. Based on the simplifying assumptions, a
system of equations is obtained that captures the idealised drying behaviour of the
product. These equations may then be solved, numerically except in the very simplest
of cases, subject to boundary conditions determined by the exchange processes at the
product surface, and initial conditions which characterise the product at the start of
drying. An overview of the necessary background is sketched below; the relevant
equations are developed in more detail in the appendix and in subsequent chapters.
In hygroscopic materials such as most products of biological origin, water can exist
in ‘bound’ form, energetically retained by interaction between water molecules and
the material itself. As a result, the moisture’s chemical potential is lowered relative to
that of pure water at the same temperature, leading to a lowering of the equilibrium
vapour pressure. The magnitude of this effect depends upon material composition
and temperature, generally increasing with polymer content and decreasing with
temperature. Several mechanisms may contribute to this effect, including capillary
pressure in sub-micrometer pores, as well as molecular adsorption at polar sites [93,
p.23]. (Another conceivable cause of chemical potential lowering is the dilution of
liquid water by involatile solutes. For low solute concentrations, the vapour pressure
will be reduced in proportion to the solute mole fraction, in agreement with Raoult’s
law [58, p.224]. However due to the relatively high molecular weight of the common biological solutes, this effect is generally minimal [95].) Although the maximum
bound water content of most porous materials is typically an order of magnitude
lower than their saturated moisture content [95], the transport of bound moisture has
been found to significantly affect the overall drying process [106].
For simplicity the moist air is here treated as a perfect gas mixture when dealing
with the hygroscopic behaviour of non-ideal solids. The moisture activity am in a
nonideal solid solution is defined in terms of its chemical potential µm by [58, p.271]:
µm = µ•m + RT ln am
(1.4.1)
where the superscript ‘• ’ denotes “over the pure substance at the same T”, so that µ•m
is the chemical potential of pure liquid water at the same temperature. This implies
that am → 1 as the dry-mass-basis moisture content xm → ∞. Treating moist air as a
perfect gas mixture, the water vapour’s chemical potential µv satisfies [58, p.111]:
µv = µ0v ( T ) + RT ln pv
(1.4.2)
When a non-ideal moist solid phase is at equilibrium with a gas phase, the water
component will have the same chemical potential in both phases. Equating (1.4.1)
23
and (1.4.2) and considering the case of very high moisture content xm , it follows that
µ•m = µ0v ( T ) + RT ln p•v
(1.4.3)
Using (1.4.3) to eliminate µ•m from (1.4.1), and, since µv = µm at equilibrium, equating
the resulting expression with the right-hand side of (1.4.2), it follows that for a moist
hygroscopic solid in equilibrium with moist air, the moisture activity am satisfies:
pv
am =
=φ
(1.4.4)
p•v T
That is, if we treat the moist air as a perfect gas mixture, then we can simply equate
the moisture activity in the nonideal solid solution, am , with the relative humidity of
vapour, φ, at equilibrium over the solution.
The function that relates the equilibrium dry-basis moisture content, xm , to water
activity at a fixed temperature is called the moisture sorption isotherm. By neglecting
interaction between water molecules and assuming that there are a finite number of
binding sites, an ensemble treatment of binding-site occupation can be used to derive
Langmuir’s isortherm [58, p.438]:
x m = x1
bam
1 + bam
(1.4.5)
where x1 is the dry-mass-basis moisture content at 100% relative humidity, and
∆hs
s − s0
•
b = pv exp −
+
(1.4.6)
RT
R
The term ∆hs in (1.4.6) is the enthalpy of adsorption, equal to the sum of the latent
heat of vapourisation and the binding energy for free water:
∆hs = ∆hv + ∆hb
(1.4.7)
As a matter of fact, multiple layers of water molecules interact at sorption sites,
significantly distorting the observed isotherm. Differing treatments of these multiple layers has led to a proliferation of semi-empirical models. The GuggenheimAnderson-de Boer (GAB) model is based on the assumption that the binding energy
for the first layer of adsorbed molecules differs from a constant binding energy that is
assumed to apply to all subsequent layers [128]. The resulting GAB sorption isotherm
has been widely adopted as a versatile and theoretically satisfactory expression for
describing moisture sorption behaviour [75, p.193]:
xm =
k
a1 =
xM
1
−1
C
am
a1 a2m + a2 am + a3
1
2
1−
a2 =
xM
C
(1.4.8)
a3 =
1
x M Ck
where the parameters C ( T ), k( T ) and x M ( T ) are fitted to experimental sorption data
24
[163]. Usually these parameters are fit to isotherms measured at different temperatures, and are thus themselves established as functions of temperature [128].
According to the film theory model of mass transfer (summarized in section A.7),
external moisture transfer between product and airstream is driven by a concentration gradient within a laminar boundary sublayer. The surface normal vapour diffusion flux is given by:
jv,s · n̂s = h•m (ρv,s − ρv,∞ )
(1.4.9)
In (1.4.9), n̂s is the unit surface normal, directed out of the product and into the air
flow. jv,s is the mass diffusion flux of vapour at the exchange surface and h•m is the
(local) mass transfer coefficient, which is defined by this equation (section A.7). The
subscript ‘s’ refers to conditions at the product surface, while the subscript ‘∞’ refers
to bulk flow conditions in the gas side. It is standard to assume that a state of local thermodynamic equilibrium exists at the product surface [49], so that the surface
vapour density can be obtained from the surface moisture content using a sorption
isotherm such as (1.4.8). Assuming ideal behaviour, it follows that
p•v ( Ts )
pv,∞
• Mw
am ( xm,s )
−
jv,s · n̂s = hm
R
Ts
T∞
(1.4.10)
The surface moisture flux is the sum of its diffusion and barycentric fluxes [16]:
nv,s = [ρv vv ]s = [jv + ωv ρv ]s
(1.4.11)
The vapour mass fraction ωv is quite low; for instance, ωv ≈ 0.08 at 100% relative
humidity at 50◦ C. Since vapour at the drying surface is typically diffusing through a
stagnant air phase, the diffusion flux is generally significantly larger than the barycentric flux. Many drying models neglect the correction in (1.4.11) and use (1.4.10) instead to evaluate the total surface moisture flux: two instances are refs. [49, 195].
The pure heat flux at the surface is given by the following equation:
qsc · n̂s = h• ( Ts − T∞ )
(1.4.12)
In (1.4.12), h• is the local convective heat transfer coefficient, defined by this equation. A fully detailed expression for the energy balance at the product surface would
include thermal flows from beneath, the enthalpy and kinetic energy flux associated
with vapour and air flows at the surface, and the rate of work being done by surface
stresses, as given in Equation (A.3.51). The dominant terms are the drying enthalpy
flux, the heat flux from beneath, and the surface heat exchange term. Other terms are
generally neglected in models in the literature, e.g. in refs. [49, 86].
The transfer coefficients h•m and h• are functions of the mass transfer rates, as well
as of the external flow conditions and geometry (section A.7.5). In general an iterative
procedure is therefore required to evaluate the external transfer rates.
25
A majority of the drying models reported in the literature are one-dimensional (1D), although in recent years several 2-D and 3-D models have been produced by a
number of researchers [124, 126, 127, 172, 173, 194]. Notable among this work are
the contributions of Turner and Perré, who have played a pioneering role in developing and testing 3-D anisotropic wood drying models. Few internal-process drying
models that have been reported in the literature are linked to models that track variation in the air flow conditions due to the drying process itself, although this situation
has begun to change; recent examples of dryer models which have accounted for the
variation in dryer air conditions due to the drying process are given by the references
[65, 127]. In modelling an integrated HPD system, a detailed air-flow model may be
crucial. In the results of Sun and Carrington [157, 160] – in the absence of air flow reversals – the initial drying rate can be seen to vary by a factor of approximately four
between the inlet and outlet ends of a 5m wood stack under dehumidifier drying conditions, owing to the large difference in humidities and product surface temperatures
between the stack air inlet and outlet. In an isothermal dryer the temperature variation along the dryer can be expected to be less than that within an adiabatic dyer.
Nevertheless, in order to assess the impact of the temperature and humidity variation that may occur in the isothermal case, and in order to obtain an accurate picture
of the overall behaviour of such a system, it is important to track the variation of air
conditions above the product, and of drying rates along the product.
The humidity ratio of the drier outlet air is given by
ωout = ωin +
ZL
0
dω
dx
dx
(1.4.13)
where the humidity ratio gradient is [119]:
dω
h• ρ a (ωs − ω ) dAm
= m
dx
ṁ a
dx
In (1.4.14)
dAm
dx
(1.4.14)
is the moisture mass exchange surface area per unit distance along the
drier. Let W and L be the width and length of each duct, with each duct presenting a
mass-exchange surface W × L to the air, and let ND denote the number of dryer ducts
through which a total dry-air mass flow rate of ṁ a passes, as for the tray-drier geometry depicted in Fig. 2.2. Then
dAm
dx
is equal to ND W. The mass transfer coefficient
hm can be estimated using the Chilton-Colburn analogy [16] and the Dittus-Boelter
equation for turbulent duct flow [135] (section A.7):
D2va
hm =
ρc p k2
k
Nu
h=
DH
1/3
h
Nu = 0.023Re4/5 Pr2/5
(1.4.15)
(1.4.16)
(1.4.17)
26
In (1.4.15), hm denotes the mass-transfer coefficient that applies at low mass transfer
rates, as is reflected in the absence of the superscript bullet symbol ‘•’. In both the
idealised isothermal and adiabatic drying modes, during the constant-rate drying
period, the product surface temperature is approximately constant through the duct.
In the adiabatic case the surface temperature equals the wet-bulb temperature of the
air, Ts = Twb,D [110]. In the isothermal case, product thermal conduction holds the
surface temperature near the dry-bulb temperature of the inlet air, Ts = Tin . In either
case, neglecting variation of transfer coefficients and air density within the duct, the
outlet humidity ratio can be obtained by integration of (1.4.14) as follows:
hm ρ a ND W
ωout = ωsat ( Ts ) + ωin − ωsat ( Ts ) exp −
L
ṁ a
120
(1.4.18)
Tin=55°C
80
60
Temperature
Humidity ratio ω, g moisture/kg dry air
100
40
Twb,in=36.0°C
20
0
Isothermal ω
Adiabatic ω
Adiabatic T
0
1
2
3
4
Position x along kiln airflow direction, m
5
Figure 1.10: Humidity and temperature in adiabatic and isothermal driers.
When combined with equations (1.4.15)–(1.4.17), (1.4.18) can be used to estimate
the air state profiles over an ideal (nonhygroscopic) solid undergoing drying. Varying
L in (1.4.18) allows the variation of air state within the duct to be evaluated. Figure
1.10 depicts the humidity ratio and temperature profiles in a dryer with geometry as
specified in Table 2.1 (page 43), and drier airflow ṁ a = 1 kg s−1 . The figure illustrates
27
how conductive heat transfer elevates the MER of an isothermal dryer, in comparison
with an adiabatic dryer with the same inlet condition. Fig. 1.10 is somewhat unusual
and so will be explained in detail here. Air enters the drier at dry-bulb temperature Tin = 55◦ C and at 30% relative humidity, i.e. ωin = 31 g kg−1 . The heat transfer coefficient h estimated from the Dittus-Boelter equation is 22.8 W/(m2 K, within
the range of values obtained by Salin [139] in a review article that surveyed literature on heat and mass transfer systems of similar dimensions and air mass flow
rates. The ratio of the heat and mass transfer coefficients is near to the ratio of the
film thermal conductivity k f and diffusivity Dva, f , as required by the analogy between heat and mass transfer (page 243). In the figure, the dashed and dash-dot
lines, which represent humidity ratios, are referred to the scale on the left. The
dotted and solid lines, which represent temperatures, are referred to the temperature scale on the right. The temperature scale has been set so that the inlet temperature, Tin = 55◦ C, is horizontally aligned with the saturation humidity ratio at
this temperature, ωsat ( Tin ) = 115 g kg−1 , and that the inlet wet-bulb temperature,
Twb, in = 36◦ C, is aligned with the saturation humidity ratio at this temperature,
ωsat ( Twb, in ) = 39 g kg−1 . This allows the connection between heat and mass transfer
to be seen – with all the adiabatic curves asymptotically converging on one another,
and likewise for the isothermal curves. However caution needs to be exercised since
the relationship between the temperature and humidity scales is not straightforward,
due to the nonlinear relationship between the temperature and the saturation humidity ratio. In the adiabatic drier the product surface remains near the inlet wet-bulb
temperature [94]. This results in a relatively small drying rate (humidity shown by
the bottom dash-dot line), and in rapid air cooling within the adiabatic drier (solid
line). In the isothermal contact dryer the air and product surface remain near the inlet
dry bulb temperature, with the vaporization heat provided by conduction through
the plate and the product layer, and drying proceeds more rapidly (drier humidity
ratio shown by the dashed line).
1.5.
Internal transfer processes
This section presents a brief sketch of internal-process drying models described in the
literature, organised by reference to their simplifying assumptions. This is followed
by a discussion of the situation being modelled in the present thesis project. Drying models often correspond to one of two idealisations of the material being dried:
continuum or porous. Continuum materials (such as carbohydrate solutions) shrink
ideally – that is, by an amount equal to the free-liquid volume of the moisture lost
– and allow evaporation at the surface only. Shrinkage is often modelled as unidimensional, which is reasonable for the drying of thin slabs, in conjunction with a
28
simple Fickian model of moisture transport [18, 49, 86, 92]. By contrast, porous materials such as timber contain voids which remain when the drying process is complete,
into which air must flow to replace the exiting moisture. Porous media are generally
modelled as rigid; that is, as undergoing no volume change during drying. These
two idealisations highlight the difficulty involved in the modelling of drying: actual
drying behaviour tends to lie somewhere between the extremes set by the continuum
and porous conceptual models. Since biomaterials generally contain on the order of
100% moisture (dry-mass basis), their volume change on drying can be considerable.
A majority of porous drying models in the literature do not take shrinkage into account, despite frequent acknowledgement that shrinkage is often an integral part of
the drying process, and ought eventually to be incorporated into the models [92].
Sn
Sw
d
T∞
Se
AIR
Ts
Sm
PRODUCT
δ
Tp
xp
Tw
D
PLATE
Tr
REFRIGERANT
l
Figure 1.11: Temperature profile in contact heat pump drying.
If we consider only the constant drying-rate period (or use the continuum material
model) then we can assume that evaporation takes place entirely at the product surface. A simple model of heat transfer within the product, similar to that outlined by
Keey [94, p.222], can then be used to calculate the drying rate of a product undergoing
conductive heat transfer on a heat pump condenser plate. Fig. 1.11 is an adaptation
of Fig. 6.15 of [94]. We will evaluate the energy balance here in detail. We take the
plate bottom surface to be adiabatic, and use the conduction equation on page 3-121
29
of [135] to evaluate the heat transfer from the tube wall to the plate top surface. We
assume that the refrigerant tubes are oriented transversely to the heating plate, and
that the heating plate is of width W, and neglect edge effects. The rate of heat transfer
Q̇tube from any one tube within the heating plate then becomes
Q̇tube = αr πDW ( Tr − Tw )
Q̇tube =
2π
ln
h
2l
πD
sinh
2πx p
l
(1.5.1a)
i k p W ( Tw − Tp )
Q̇tube = k δ Wl ( Tp − Ts )/δ
(1.5.1b)
(1.5.1c)
where heat transfer from the refrigerant has been equated to heat transfer through
that part of the plate associated with the given tube, and to heat transfer through the
corresponding part of the product. In (1.5.1) αr is the refrigerant condensation heat
transfer coefficient, k p is the plate thermal conductivity, and k δ is the thermal conductivity of the product. The mean product heat flux q can be evaluated by dividing the
heat transfer rate Qtube per pipe by the product area Wl per pipe, to obtain
πD
( Tr − Tw ) = α1 ∆T1
l
2πk p /l
i ( Tw − Tp ) = α2 ∆T2
q= h
2πx p
2l
ln πD sinh
l
q = αr
q=
kδ
( Tp − Ts ) = α3 ∆T3
δ
(1.5.2a)
(1.5.2b)
(1.5.2c)
where the meanings of ∆Ti and αi (i = 1, 2, 3) are clear from (1.5.2). Now by definition
3
3
i =1
i =1
of the ∆Ti we have Tr − Ts = ∑ ∆Ti = ∑ q/αi from (1.5.2), so that
q = αt ( Tr − Ts )
αt =
1
1/α1 + 1/α2 + 1/α3
(1.5.3)
(1.5.4)
This is the heat flux through the product, i.e. the heat flux at the product surface from
beneath. From a rough energy balance at the surface, we have [94, p.222]:
αt ( Tr − Ts ) = nv ∆hv + h( Ts − T∞ )
(1.5.5)
where h is the convective heat transfer coefficient at the product surface.
A steady-state model that incorporates equations (1.4.15)–(1.4.18) and (1.5.5) is
used in chapter 2 in a preliminary analysis of the energy-efficiency potential of ICHPD.
The model is based on the assumption that the conductive heat transfer coefficient αt
is sufficiently large to hold the product surface temperature near the temperature of
the inlet air, which has already been heated by the part of the condenser in thermal
contact with the air. The drying process is thus modelled as an isothermal vapour
diffusion process from the product into the airstream. The model is illustrated by the
30
example output presented in Fig. 1.10 on page 26. It does not incorporate the effects
of the drying process on the air flow parameters, variation of the local heat and mass
transfer coefficients, the pressure drop along the air flow direction, or deviation of the
surface temperatures from their idealised values due to variation in drying rate along
the product. Furthermore the equations apply only to an ideal (nonhygroscopic) solid
or a nonideal solid during the constant-rate drying period. A large part of the work
presented in chapters 3 and 4 has been in extending the drying model to incorporate
these effects, and to assess the validity of these idealisations.
A full description of the internal process within a porous medium involves the
simultaneous, linked transport of bound water (if the medium is hygroscopic), unbound (‘free’ liquid) water, water vapour and air, in addition to heat transfer. The
physical properties underlying these phenomena are generally composition and temperature dependent. Local equilibrium relationships can be used to evaluate the local
amount of water in each phase, and to close the model equations. An obvious difficulty with this approach is that model behaviour is potentially sensitive to a large
number of physical parameters, some of which may not be well known. This difficulty is compounded by the inherent variability of many materials, often even within
a single sample [47, 55, 163, 195]. Many researchers have therefore opted for simpler
drying models, in which evaporation is assumed to take place only at the surface,
and internal moisture transport is governed by a simple diffusion equation, e.g. simplified model 2 of [195]. In searching through the drying literature, several simple
plate-drier models based on Fick’s diffusion law were found [e.g. 105, 86], but no
plate-dryer models incorporating Whitaker’s [188] detailed drying theory was found.
Several novel features of ICHPD systems argue for a detailed analysis of the drying process in the falling-rate period. In convective drying situations such as timber
drying, the heat-transfer Biot number Bi =
hδ
k ,
which is a measure of the relative
ease of external heat transfer and internal heat flow, is often small. As a result, the
temperature within the product may often, in convective drying, be assumed to be
relatively homogeneous. Some modelers [e.g. 49, 103] have employed the assumption of uniform temperature within the product (allowing however that the product
temperature may vary with time), though other researchers have found that temperature variation can significantly affect moisture transport [e.g. 5, 178, 181, 195]. For
the drying situation of interest to us here (product layer heated from beneath by a hot
plate, and being dried by a cooler air stream flowing above it), there is no such reason
to assume that heat conduction from beneath the product cannot impose a significant temperature gradient between the bottom of the product layer and the drying
process occurring within the material and at its surface. Indeed, one key motivation for employing contact heat transfer is to circumvent the small air-product heat
transfer coefficient h that contributes to the second-law losses in convective HPD.
31
Furthermore, compared with a conventional dryer employing contact heat transfer,
the energy performance of an ICHPD system is likely to be relatively sensitive to factors which impact the drying kinetics, such as the temperature and moisture profile
within the product. Thus an assumption of isothermal conditions within the product
is not justified, and there is a need, in order to assess ICHPD performance, to develop
an understanding of the processes that take place within the product layer, to consider the impact temperature and moisture gradients within the product may have
on the drying process and on overall system performance.
Another simplification often employed in drying models is the assumption of an
internal drying front. In the falling rate drying period in convective drying, liquid
moisture transport to the drying surface cannot keep pace with the drying flux, and
a region of large gradient in moisture content, termed the drying front, develops and
recedes into the product. A number of models [e.g. 55, 76] have employed the simplifying assumption that above the drying front, moisture transport is due entirely to
the diffusion of water vapour, and below the front moisture transport is due entirely
to the capillary transport of liquid water. Again the boundary conditions relevant to
contact drying are different from a typical convective drying situation, and the unknown temperature profile within the product precludes the assumption of a drying
front. The imposed plate heat flux may lead to a significant temperature gradient
within the product. One goal of the modelling work described in this thesis was to
use the theory of drying (as results from Whitaker’s approach [188, 193] to modelling
transport in porous media) to assess the impact that plate heat transfer may have on
the evaporation and transport profiles within the product.
The simplest method for coping with the complexity of internal drying processes
is to avoid it by using a (usually empirically-determined) characteristic drying curve
[94, p.154]. The characteristic drying curve (CDC) is a function that maps the product moisture content to the ratio between the (current) drying rate and that during the
constant drying rate period. Using such a curve, the drying rate at any given time can
be estimated from the moisture remaining in the product and from the drying rate
from a water surface under the same external conditions. This approach has been
employed in the timber drying model that was developed by Sun and Carrington
[155, 159, 160, 161, 162] and combined with the previously-developed empirical heat
pump model to produce a whole-kiln dynamic timber HPD model [28, 33, 156, 157].
The justification for the CDC approach comes largely from past successes in reproducing experimentally obtained convective drying curves from a single drying curve
[94]. This approach was not initially pursued here as it is unable to provide the detailed (and new) information about temperature profiles within the product, and their
impact on the drying process, that the present investigation calls for. Numerical results that are presented in chapter 5 suggest that the approach is unlikely to yield
32
satisfactory results in the presence of contact heat transfer.
1.6.
Aims and outline of thesis
The aim of this project was to construct a numerical model of a contact HPD system.
The key goal of the project was to develop the capacity to assess the potential of such a
system to improve on the energy efficiencies currently attained in heat pump drying,
and also to develop an ability to optimise such a system, e.g. so that the demands
of the heat pump are matched to the properties and demands of the product being
dried. In order to obtain a model of the widest applicability and to take into account
the recommendations of the Norwegian group (listed on p.20), the goal established
at the outset was to develop a dynamic dryer model, incorporating a drying process
model based on the general drying equations developed by Whitaker [188].
The modelling task undertaken here has been to develop a model of a stack of drier
ducts in which heating takes place from condensing refrigerant flowing in condenser
tubes embedded within the heating plates. In each drier duct, a product layer is sandwiched between air (passing over the product) and a heating plate, through which
the hot condensing refrigerant passes in counterflow to the air. Once this drier duct
model was developed, it could be integrated with a comprehensive pre-existing heat
pump model. The duct model that is developed combines a detailed air-flow model,
which solves the mass, momentum and energy balances within the drier ducts, with
a detailed internal drying process model, incorporating a description of the transport phenomena occurring within the porous product medium. The whole-system
dynamical HPD model, which results when the drier-duct model is integrated with a
pre-existing heat pump model, is capable of describing the evolution of a non-steady
batch drying operation, for instance predicting the decline in energy efficiency that
occurs during the falling-rate period of drying. While this thesis presents no new
experimental work in support of the model that is developed, each component of
the model is tested against previously-established numerical models, against existing
empirical data, or against ideal scenarios to which theoretical solutions apply.
The model that has been developed can be used to investigate the effects of a variety of parameters, for instance the effects of product thickness, air velocity, temperature set-point, etc., on the system behaviour. The model can be applied to a wide
variety of porous materials, each of which is characterised by a different set of consitutive relations. In addition, although the model has only so far been used to examine
the dynamics of batch-mode HPD operation, it could readily be extended to make
predictions of system performance for continuous-operation driers. As discussed in
the previous section, one key goal has been to assess the impact that the product
33
temperature gradient may have on the evaporation and transport profiles within the
product, and that the drying kinetics ultimately have on the energy performance of
the drier. Further questions addressed include the impacts of variation in the transfer coefficients, of the high-mass-transfer enhancement factors and of user-controlled
parameters: for instance, what are the impacts of changing the the drier dimensions,
or the face area of the air-side condenser, or at the evaporator?
The key difficulty that is associated with developing a model of a contact HPD
is the tight linkage between the drying process and working-fluid condensation on
the refrigerant-cycle side. The novel feature of the current research is essentially
that it deals with this new linkage between condenser and drying process, which
has prevented previously-developed models from being applied to contact HPD, and
which has added considerable complexity to the simulation. The detailed timber
stack model described by Sun et al. [160] has provided a basis for developing the airside dryer model, while for simulating the remaining system components, the empirical HPD model developed by Carrington and Bannister [28] has provided a large
number of functions that were directly applicable in the present work. The internal
drying-process model is similar to a model described by Stanish et al. [151] although
based on the constitutive relations employed by Wang and Chen [178].
Although the analysis that is conducted in this thesis remains fairly general, modelling an HPD system and its components has required explicit assumptions to be
made on parameters such as component sizes and system configuration. This is
somewhat unfortunate, as it is clear that to actually develop a commercial system
would require expertise distributed amongst multiple specialists, would involve practical considerations that are beyond the scope of this thesis, and could even be an
application-specific process. Since the research described here is a theoretical investigation of an untried design possibility for HPD, it seems appropriate for the analysis
to remain fairly high-level. However, the value of this work would undoubtedly be
strengthened if it could be linked into a practical research programme toward a working system design, for a specified application. In the absence of such a programme,
the appropriate path has seemed to lie in producing a working model of a likely system design, with parameters selected to maximise the energy performance predicted
by the models. A few thoughts on system designs are presented in chapter 6.
The remainder of the thesis is structured as follows.
Chapter 2 presents a comparative analysis of the performance of adiabatic and
isothermal HPDs under constant drying-rate conditions, based on a simple conduction heat transfer model that combines equations (1.4.15)–(1.4.18) and (1.5.5). The
chapter’s system-level analysis combines this model of the drying process with empirical equations for evaporator and condenser heat transfer, obtained by Carrington
34
and Bannister [28], to obtain the system steady-state. The heat pump is assumed to
operate at a constant (50%) percentage of the Carnot heating COP. This analysis has
demonstrated that the isothermal contact drying mode has the potential to increase
the SMER obtainable from a heat pump dryer by a factor of 2-3.
Chapter 3 contains a development of a more comprehensive plug-flow model of
the air flow through the drying system. This air-side model is similar to the wood
stack model developed by Sun and Carrington [159, 160]. As discussed above, the
basic airflow model summarized as equations (1.4.15)–(1.4.18) does not incorporate
effects of the drying process on the flow parameters, variation of the local heat and
mass transfer coefficients, the pressure drop along the flow direction, or deviation of
the surface temperatures from their idealised values. Furthermore the equations only
apply to a nonhygroscopic solid or a nonideal solid during the constant-rate drying
period. The impacts of these idealisations are assessed in chapters 3 and 4.
Chapter 4 describes the extension of the heat pump model of chapter 2, to include
a detailed model of a scroll compressor used with R134a, as based on the correlations
of [30]. In addition correlations are incorporated for pressure drops along the refrigerant lines, for suction superheat, and for pressure drops on the air side across all
system components. The detailed duct model from chapter 3 is combined with these
functions to produce a steady-state whole-system ICHPD model. An exergy analysis
of the system is performed for both the isothermal and adiabatic modes, which shows
that the isothermal mode derives about half of its improved efficiency from a reduction in the irreversibility associated with the transfer of heat to the drying process. By
modifying the specification of a number of key system parameters, we determine that
the system performance is most sensitive to variation in the exposed product surface
area and the dimensioning of the (refrigerant) evaporator.
Chapter 5 describes the implementation of (a simplified version of) the set of drying equations whose derivation is discussed in appendix B, first into a one-dimensional
(1-D) model, which is tested by comparison of its outputs with those of Wang and
Chen [178], and then tested under the isothermal boundary condition relevant to the
heat pump model. This one-dimensional model is then implemented within a 2-D
model which is linked to the detailed air-side plug flow model developed in chapter
3. Results of this integrated two-dimensional model are presented and discussed. The
2-D drying model presented in chapter 6 is then linked into the full heat pump model
described in chapter 4, producing a dynamical whole-system model. This combined
model is used to investigate several of the questions raised above. In particular, the
effects of drying mode and of product thickness, and the impact of the drying process
dynamics on second-law efficiencies within the drier are considered.
Chapter 6 summarizes the key findings of this thesis, describes difficulties encoun-
35
tered during the thesis work, and discusses possibilities for further work.
The final two chapters, the appendices, form a logical ‘prequel’ to the remainder
of the thesis. Appendix A summarizes general theoretical background in fluid mechanics, thermodynamics, and heat and mass transfer (and not specific to the field
of drying), in an integrated format. The purpose of this chapter is to gather all of
the relevant background material, and present it together in a coherent, logically selfcontained format, employing consistent notation. As a consequence the chapter contains some elementary material that would not belong in the main body of a PhD
thesis, but provides the reader with all of the background required to understand the
thesis. Appendix B presents a derivation of the governing equations for the drying
of a hygroscopic porous medium. The equations extend Whitaker’s model equations
[188] to include the transport of bound moisture. The derivation ultimately arrives
at a set of equations that are similar (with minor differences) to those used by several
other researchers [69, 172]. However the path taken in the derivation presented here
appears to be novel, and the differences in approach from those of previous authors
are highlighted. The derivation also highlights the many assumptions that are involved in obtaining the drying equations using the volume-averaging method. The
derivation is included as appendix B even though the full set of equations derived,
(B.7.1)–(B.7.15), are not used in the dynamical model of chapter 5, which instead proceeds from the simplified set employed by Wang and Chen [178].
In summary, there are four components which form the basic building blocks that
are constructed, tested and assembled together in this thesis. These are: (1) a model of
the air flow through the drier, (2) a dynamical model of the transport processes occurring within the product, (3) a model of the refrigerant flow through the product heating plate, and (4) a model of the remaining heat pump components, which is based
on correlations taken from the empirical heat pump model developed by Carrington
and Bannister [28]. Taken together, components (1)–(3) form a detailed dynamical
model of the contact dryer, subject to the inlet refrigerant and air flow boundary conditions. In terms of the components (1)–(4), the thesis can be summarized as follows.
Chapter 2 entirely bypasses (1)–(3), using (4) only to conduct an initial assessment of
the potential energy performance benefit of the isothermal mode. Chapter 3 develops model (1) and (3). Chapter 4 integrates the resulting drier-duct model with model
(4). Appendix B and chapter 5 develop model (2). Finally in chapter 5, model (2) is
integrated into chapter 4’s model, to produce a dynamical whole-system model.
36
1.7.
Nomenclature for chapter 1
Roman and Greek symbols
Symbol
Definition [units]
Am
Mass-exchange area [m2 ]
ai
Activity of species i [–]
a1 , a2 , a3
Parameters appearing in GAB equation [–]
b
Constant appearing in Langmuir isotherm [–]
Bi
Biot number [–]
cp
Specific heat capacity [J/kg-K]
COP
Coefficient of performance [–]
C ( T ), k ( T ), x M ( T )
Parameters in GAB sorption isotherm [–]
D
Diffusivity [m2 /s]
d
Air duct depth [m]
D
Heating plate refrigerant tube internal diameter [m]
DH
Duct hydraulic diameter [m]
∆hv
Latent heat of vaporization [J/kg]
∆hb
Binding energy for free water [J/kg]
∆hs
Enthalpy of adsorption [J/kg]
∆hs12
Isentropic change in enthalpy [J/kg]
∆h12
Actual change in enthalpy [J/kg]
h
Specific enthalpy [J/kg], Specific enthalpy of moist air [J/kg-dry], Product surface heat transfer coefficient [W/m2 -K]
hm
Product surface mass transfer coefficient [m/s]
j
Mass diffusion flux vector [kg/m2 -s]
k
Thermal conductivity [W/K-m]
l
Heating plate condenser tube spacing [m]
L
Heating plate length [m]
ṁ
Mass flow rate [kg/s]
MER
Moisture extraction rate [kg/s]
Mw
Molar mass of water [kg/mol]
n̂
Unit normal vector
n
Mass flux [kg/m2 -s]
n
Number of moles [–], Molar flux [s−1 ]
ND
Number of ducts [–]
Nu
Nusselt number [–]
p
Pressure [Pa], number of passes through plate per circuit [–]
37
Pr
Prandtl number [–]
Q̇
Heat flow rate [W]
qc
Pure heat flow vector [W/m2 ]
R
Molar gas constant [= 8.314472 J/K-mol]
Re
Reynolds number (based on D H ) [–]
S
Entropy [J/K]
s
Specific entropy [J/K-kg]
s̄
Molar entropy [J/K-mol]
s0i
Ideal-gas molar entropy of pure species i at 1 atm [J/K-mol]
SMER
Specific moisture extraction rate [kg/kWh]
T
Temperature [K]
t
Time [s]
v, v
Speed, velocity [m/s]
W
Heating plate width [m], Work [J]
Ẇ
Power input [W]
x
Distance through kiln [m], Moisture content [kg/kg-dry], Mole fraction
[–]
x1
Hygroscopic moisture content at saturation [kg/kg-dry]
xp
Heating plate refrigerant tube centerline depth [m]
α
Heat exchange coefficient [W/m2 -K]
δ
Product thickness [m]
ηC
Ratio of heating COP to Carnot heating COP [–]
µi
Chemical potential of species i [J/mole]
ρ
Density [kg/m3 ]
σ̇
Rate of entropy generation [J/K − s]
φ
Relative humidity [–]
ψ
Potential [J/kg]
ω
Humidity ratio [kg vapour/kg-dry]
Subscripts and superscripts
Symbol
Definition
0
Environment or reference
1, 2,
20 ,
3, 4
Locations on refrigerant cycle
A, B, C, D, E, F, G
Locations on air cycle
co, ev
Condenser, evaporator
in, out
Inlet, Outlet
k, a, v, m, w
Species-k, Dry-air, Water-vapour, Moisture, Liquid-water
n, w, s, e
North, West, South, East (control-volume boundaries)
38
m
Mass-exchange
p
Heating plate
prod
Product
r
Refrigerant
S, s
Surface
sat
Saturation condition
t
Total, effective
wb
Wet bulb
•
Modified for high mass transfer rates; Value over the pure substance at
δ
Product
∞
Bulk-flow (‘at infinity’)
the same T
Mean
39
Chapter 2
Initial performance assessment.
This chapter presents a comparative performance analysis of simple HPD systems
operating at steady state in the constant drying-rate period. A contrast is established
between purely isothermal operation (which is an idealisation of ICHPD) and purely
adiabatic operation (which is a good description of convective HPD) in terms of the
heat transfer rate into the product, labelled Q̇ D , in accordance with the definitions
of p.20. The results of this chapter suggest that ICHPD can provide a substantial
energy efficiency improvement (by a factor of 3), subject to the assumption that the
idealised “isothermal” model successfully describes the behaviour of a contact HPD.
This motivates a detailed investigation of ICHPD, to assess this and the other model
assumptions, and to examine which parameters most impact system performance.
2.1.
Introduction
This chapter examines the impact of two operational constraints on the energy performance of the HPD depicted in Figure 2.1, under the adiabatic and isothermal drying
modes. We assume that venting is controlled to maintain a fixed temperature at location D. The constraints considered are (1) dry bulb temperature at drier entry, and
(2) maximum acceptable relative humidity of air passing over the product. These
constraints establish a relationship between the dehumidifier refrigeration capacity
and the airflow and geometry of the tray drier. For instance, for a given drier temperature, geometry and airflow, the dehumidifier capacity must be sufficient if the
relative humidity constraint is to be met. The independent parameters of the modelling procedure are the HP characteristics, the drier geometry, and the air flow rates.
From these parameters, conditions within the system are evaluated. The HP capacity or drier size required to meet the humidity constraint can then be established by
40
numerical solution. The drying process is modelled using simple generic models.
A standard tray drier configuration is assumed throughout, as depicted in Figure 2.2, and further key system parameters selected for the baseline scenario are listed
in Table 2.1. The purpose of the work presented in this chapter is to obtain an indicative assessment of the energy benefit associated with contact heat pump drying. Following [27], the environmental (dead state) air is taken to be saturated at 10◦ C. As is
described below, the model described here has suggested that the SMER of a contact
HPD system may be substantially greater than that of an adiabatic HPD, in part because it reduces the impact of the fan power tradeoff described in chapter 1. A related
potential benefit of isothermal HPD (where contact drying is applicable) is that a high
SMER may be compatible with a relatively high MER. However, as the results show,
the energy performance benefit is highly sensitive to the relative humidity constraint,
and also to the thickness (and thermal resistivity) of the product layer.
ṁa,con
F
evaporator bypass air
evaporator air
ṁa,ev
b · ṁa,con
x · ṁa,ev
cooling
f · ṁw
f · ṁw
A
drying
B
venting
Tout
C
mixing
Q̇E
TEV
ẆF
D
Q̇H
expansion
valve
ṁw
E
G
heating
Q̇D
Q̇W
TCO
compressor
evaporator
condenser
ẆP
Figure 2.1: Schematic of vented dehumidification HPD with evaporator air bypass. The heat pump is located in the lower half of the figure. Locations around
the air cycle are denoted A, B, C, D, E, F. Incoming moisture is heated in control
volume G before it enters the drying chamber, as described in the text. Variables
displayed in the figure are: dry air mass flow rates ṁ a,ev and ṁ a,co , bypass ratio
b, condensation fraction f , evaporation rate ṁw , heat transfer rates Q̇ E , Q̇ H , Q̇ D
and Q̇W , work inputs ẆP and ẆF , temperatures T0 , Tout , TEV and TCO .
ṁw
T0
41
adiabatic wall
w,m
(A)
d,m
air duct
product layer δ,m
supporting plate (isothermal mode: heating plate)
refrigerant flow
(isothermal mode)
air duct
(×10)
air flow
(B)
xp
l
(×10)
D
L,m
Figure 2.2: Modelled tray drier. (A) front view, (B) side view. The drier is a
stack of ND = 10 subunits as depicted. Condenser tubes pass through the
heating plate in the “isothermal”, but not the “adiabatic”, drying mode.
2.2.
System description
The idealised system depicted in Figure 2.1 consists of two linked cycles: a heat pump
cycle (depicted in the bottom half of Figure 2.1), and an air cycle (in the top half).
The system is considered at steady state, so that mass and energy flows into the system, and into each of its subsystems, are balanced by outward flows. The system
exchanges mass and energy with its environment in five ways, four of which are depicted in the figure as arrows pointing into and out of the system. In addition, a fraction x of the evaporator airflow is exchanged for air at the environmental dead state,
as depicted in the “venting” volume in the center of the figure. Venting is usually
used as a control variable for maintaining balanced conditions within a HPD system.
Where venting is not possible, energy must be shed to the environment through pure
heat transfer, typically resulting in a somewhat lower overall SMER [33].
The heat pump is depicted operating between the saturated evaporating and condensing temperatures, Tev and Tco . Shaft work rate ẆP drives a heat flow Q̇ E from
the evaporator. The condenser receives a heat flow Q̇ E + ẆP = Q̇ H + Q̇ D + Q̇W . Heat
is assumed to be transferred out of the system exclusively by venting and with the
condensate ( f ṁw ). The dotted line in the top half of Figure 2.1 indicates the air circulation stream. Under normal operating conditions, water vapour condenses in the
air ‘cooling’ volume as hot moist air encounters the evaporator cold surface. Heat
is transferred to the evaporator at rate Q̇ E . Condensate is drained from the system
at the rate f ṁw , where f represents the fraction of evaporated moisture that is not
eliminated from the system through venting. This condensate fraction f is given by
f = ṁ a,ev (ωF − ωA )/ṁw
(2.2.1)
42
where
ṁw = ṁ a,co (ωE − ωD )
(2.2.2)
Cool dehumidified air exits the cold heat exchanger, and passes through location A.
Here we assume that this air is saturated. (This assumption is only valid for finned
tube heat exchangers with sufficient fin density, fin depth, and with low enough face
velocity. The assumption is reasonable for the evaporator modelled here, which was
characterised by Carrington and Bannister [28], as can be verified by examining the
wet-bulb depression at the evaporator, which is typically less than 1.5◦ C. However
the assumption would not be valid for the low-contact-factor heat exchangers that
are employed as evaporators by the SINTEF research group, which effectively incorporate the bypass air into the air flow through the evaporator [23].)
After a fraction x of evaporator air is exchanged for air at the environmental dead
state, and the resulting mixed stream at point B is further mixed with evaporator bypass air, the airstream is heated at rate Q̇ H by the condenser. Thus TD > TC . In the
‘drying’ volume, water evaporates from the product into the airstream at the rate ṁw .
Water enters the drier from outside the system at the same rate, ṁw . This flow represents the water delivered by a product passing through a continuous-operation dryer,
carrying less moisture when it leaves. For simplicity this entering water is assumed,
in control volume G, to be brought to the adiabatic saturation temperature Tas,D of the
air at inlet D of the drying control volume, by a heat flow Q̇W from the hot reservoir,
although the effect of this assumption is small. The (sensible) heat Q̇W depicted in
Figure 2.1 is always much smaller than the heat Q̇ H + Q̇ D (which includes the latent
heat of vaporization for the drying process), its only significance being that it allows
us to state, in the adiabatic case, that the hot air cools along a path of constant wet bulb
temperature [110]. Heat removal by the departing solids is neglected in this chapter,
as is any interaction between solid matrix and moisture, such as hygroscopicity. Note
that in both modes, the greatest relative humidity above the product occurs at the
product air outlet, point E. A constraint on maximum product relative humidity is
thus equivalent to a constraint on the relative humidity φE at location E.
2.3.
Analysis
The venting fraction x is assumed to be regulated to maintain energy balance with TD
fixed. For each mode, given a fixed drier geometry and airflow ṁ a,co = 1 kg s−1 , the
constraint on the drier air-outlet relative humidity becomes equivalent to a constraint
on the humidity at the drier inlet. This in turn determines the required refrigeration
capacity of the heat pump. The modelling strategy (described in more detail below)
is to fix ẆP , together with ẆF , ṁ a,co , ṁ a,ev , TD , ηC , and to determine the conditions
43
Parameter (unit)
Baseline value
Shaft power ẆP (W)
5.0 × 103
Condenser face area Aco (m2 )
1.0
Evaporator face area Aev (m2 )
1.0
Number of ducts ND
10
Product air flow ṁ a,co
(kg s−1 )
1.0
System air pressure drop ∆PF (Pa)
200.0
Tray drier length L (m)
5.0
Tray drier width w (m)
1.0
Air duct depth d (m)
20.0 × 10−3
Product thickness δ (m)
Heating plate condenser tube separation l (m)
Heating plate condenser tube diameter D (m)
Heating plate condenser tube midline depth x p (m)
Drier maximum temperature TD (◦ C)
1.0 × 10−3
60.0 × 10−3
10.0 × 10−3
6.0 × 10−3
55.0
Table 2.1: Baseline values of key system parameters.
throughout the system, including the temperatures, venting fraction x, and drier outlet relative humidity φE . The ASHRAE psychrometric routines are used [184], incorporating terms up to the third order in the virial equation of state for mixtures.
The following further idealisations are used:
1. The system is in steady state.
2. Kinetic and gravitational potential energy effects are neglected.
3. For psychrometrics the air pressure is constant at p0 = 1 atm.
4. The evaporator air venting fraction x is assumed to be controlled by the system
operator to maintain energy balance, with air temperature at location D consistent with the temperature constraint. All component heat losses are neglected.
5. Under ordinary conditions in which dehumidification takes place, moist air at
A is saturated at the temperature TA , i.e.
ωA = ωsat ( TA )
(2.3.1)
The outlet condensate is the same temperature, i.e. Tout = TA .
6. Evaporator and condenser heat exchanger characteristics are estimated using
44
the empirical correlations established by Carrington and Liu [28]:
Q̇ E = Aev [ g( Twb,F ) f ev (vev )( Twb,F − Tev ) − 2.69] × 103
Q̇ H = Aco [ f co (vco )( Tco − TC ) − 2.50] × 103
(2.3.2)
(2.3.3)
where Aev and Aco are the face areas of the evaporator and condenser heat
exchangers, respectively, and the polynomials g, f E and FC are as follows:
g( x ) = 0.212 + 0.1283x − 0.001181x2
(2.3.4a)
f ev ( x ) = −0.017 + 1.486x − 0.5145x2
(2.3.4b)
f co ( x ) = −0.0065 + 1.1535x − 0.11079x
2
(2.3.4c)
Twb,F is expressed in centigrade, and vco and vev are expressed in units of m/s.
Equations (2.3.2) and (2.3.3) are valid for air velocities in the range 0.3-1.4 ms−1
at the evaporator and 1.5-3.0 ms−1 at the condenser, respectively. The components to which equations (2.3.2) and (2.3.3) apply are described in Table 2.2.
Compressor
Copeland scroll ZR61K2-TED
Refrigerant
HFC134a
Lubricant
Evaporator
Condenser
ICI Emkarate RL68S
fin type
aluminium ripple
fin pitch
2.1mm
tube rows
4
fin type
aluminium ripple
fin pitch
1.8mm
tube rows
3
Table 2.2: Description of system components, from [24].
In the adiabatic case, Q̇ D = 0, and all heat for drying is provided to the air
as Q̇ H . In this case (2.3.3) is used to relate the condenser heat transfer rate,
the condensing temperature and the air temperature. In the isothermal case,
all three heat transfer rates are nonzero, and the plate heat transfer Q̇ D must
also be included in the model. The plate heat flow rate Q̇ D , the condensing
temperature and the product surface temperature TD are described by equation
(1.5.3) (p.29), which here becomes:
Q̇ D = wLND αt ( Tco − TD )
αt =
1
1/α1 + 1/α2 + 1/α3
(2.3.5)
(2.3.6)
where α1 is the effective refrigerant convective heat transfer coefficient, α2 is the
effective heat transfer coefficient associated with the row of condenser tubes
45
embedded in the heating plate [135, p.3-121], and α3 is an effective heat transfer
coefficient associated with the product layer, as follows:
α1 = πDαr /l
α2 =
(2.3.7a)
2πk p /l
i
2πx p
2l
ln πD
sinh
l
h
α3 = k δ /δ
(2.3.7b)
(2.3.7c)
The dimensions D, l, x p and δ that appear in (2.3.7) are illustrated in Figure 2.2.
The quality-averaged flow condensation heat transfer coefficient αr for refrigerant R134a in a plain tube is near 2.5 kWm−2 K−1 over a wide range of mass
flow rates [90]. The thermal conductivities k p and k δ of the heating plate and
the product are assumed to be the thermal conductivity of copper and of water.
7. The condenser and product dry air mass flow rate ṁ a,co is fixed. An air pressure
change ∆PF = 200 Pa from E to F is used to estimate fan power, e.g. of 372 W in
the baseline scenario, ṁ a,co = 1 kg s−1 , TD = 55 ◦ C, using [54]:
ẆF =
1
ṁ a,co v a,D ∆PF
eF
(2.3.8)
When the duct length is modified from its baseline value of 5 m the pressure
change is rescaled proportionately; 75% of the system pressure drop is assumed
to take place within the tray drier air ducts. These estimates reflect the dynamic
loss coefficients and air flow velocities of a typical HPD configuration [35, 162].
A (typical) constant fan efficiency e F = 50% has been assumed [54].
8. The heat pump heating COP is ηC = 50% of the Carnot coefficient of performance, COPC , corresponding to Tev and Tco . This is consistent with the observation that heat pumps often operate at an approximately constant fraction
of COPC [196]. For instance, the reported COP data of [31] lie within 52 ± 2%
of COPC across a wide range of evaporator airflows and temperatures. The
value 50% for ηC is conservative: a reciprocating compressor with ammonia
may exhibit (cooling) COPs as high as 70% of the Carnot COP, at evaporating
temperatures approaching 40◦ C and a condensing temperature of 60◦ C [26].
Assumption 8 yields the following expression:
Q̇ H + Q̇ D + Q̇W = ηC
Tco
ẆP
Tco − Tev
(2.3.9)
where ηC is the ratio of the heat pump heating COP to the Carnot heating COP. Energy
balances over several system control volumes yield the following equations:
Q̇ E = ṁ a,ev (hF − hA ) − f ṁw h f ( TA )
(2.3.10)
Q̇ H = ṁ a,co (hD − hC )
(2.3.11)
Q̇ H + Q̇ D + Q̇W = ṁ a,co (hE − hC ) − ṁw h f ( T0 )
(2.3.12)
46
Equations (2.3.10) and (2.3.11) can be used to eliminate the heat flow terms Q̇ E and
Q̇ H from Equations (2.3.2) and (2.3.3). Similarly, combining (2.3.9) and (2.3.12) yields:
ṁ a,co (hE − hC ) − ṁw h f ( T0 ) = ηC
Tco
ẆP
Tco − Tev
(2.3.13)
By the conservation of energy across the whole system, at steady state, we have
ẆF + ẆP + ṁw h f ( T0 ) = x ṁ a,ev hA − h0 + f ṁw h f ( TA )
(2.3.14)
The terms on the right hand side of equation (2.3.14) represent the the net energy
shed from the system by venting, and the heat removed by the condensed moisture.
These energy flows represent heat that cannot be recycled by the heat pump: “waste”
heat. The system at steady state is further characterised by the following two sets of
equations, which express moisture and energy balance over its components.
Moisture balance, in terms of the humidity ratio ω (kg-vapour/kg-dry-air):
ωB = (1 − x )ωA + xω0
(2.3.15a)
ωC = (1 − b)ωB + bωF
(2.3.15b)
ωD = ωC
(2.3.15c)
ωE = g1 (ωD , TD )
(2.3.15d)
ωF = ωE
(2.3.15e)
Energy balance, in terms of h, where h = h a + ωhv :
hB = (1 − x )hA + xh0
(2.3.16a)
hC = (1 − b)hB + bhF
(2.3.16b)
hD = h( TD , ωD )
(2.3.16c)
hE = g2 (ωD , TD )
(2.3.16d)
hF = hE + ẆF /ṁ a,co
(2.3.16e)
The bypass fraction b appearing in equations (2.3.15b) and (2.3.16b) is defined:
b=
ṁ a,co − ṁ a,ev
ṁ a,co
(2.3.17)
The functions g1 (ωD , TD ) and g2 (ωD , TD ) that appear in Equations (2.3.15d) and (2.3.16d)
represent the drier model that was described in chapter 1 in the discussion around
Equation (1.4.18) (p.26). Equation (1.4.18) here becomes:
hm ρ a ND W
ωE = ωsat ( Ts ) + ωD − ωsat ( Ts ) exp −
L
ṁ a,co
(2.3.18)
In the adiabatic case the drier outlet air temperature is obtained from the inlet wetbulb temperature and the outlet humidity; in the isothermal case the outlet temperature is the inlet temperature. From the drier outlet specific humidity ωE and temperature TE , the drier outlet specific enthalpy hE can be obtained.
47
The state of the drying system is specified by the following set of 15 variables:
{hA , hB , hC , hD , hE , hF , ωA , ωB , ωC , ωD , ωE , ωF , x, Tco , Tev }
Given values for ẆF , ẆP , TD , ṁ a,co , ṁ a,ev , T0 , ηC , using definitions (2.2.1), (2.2.2) and
(2.3.17), and using the psychrometric relations of [184] to relate T, Twb , h, ω and φ for
the moist air, Equations (2.3.13)–(4.4.6) together with Equations (2.3.1)–(2.3.3) constitute a set of 15 equations in the above 15 unknowns, and specify the system steady
state. The solution to this set of equations is obtained numerically by combining
equations (2.3.1), (2.3.14) and (4.4.5) to obtain the required condensation rate f ṁw (by
numerical solution) as a function of TA . TA is then obtained by seeking the value that
satisfies (2.3.13), where the terms of Equation (2.3.13) are evaluated using equations
(2.3.1)–(2.3.3) and (2.3.14)–(4.4.6). The resulting solution has been found to satisfy all
component mass and energy balance requirements, as well as the second law requirement expressed in terms of a net exergy destruction over each component.
2.4.
Results and discussion
To test the model, its outputs have been inspected in the adiabatic mode under the
three conditions investigated experimentally by Carrington and Liu [31]. These correspond to the following conditions at location F: (a) 31◦ C, 93% RH, (b) 40◦ C, 48%
RH, (c) 45◦ C, 25% RH. The conditions have been recreated in the modelled system depicted in Figure 2.1 by fixing the shaft work rate ẆP = 5 kW, the condenser
and evaporator areas and airflows Aco = 1.2 m × 0.91 m, ṁ a,co = 1.7 kg s−1 , Aev =
0.38 m × 0.9 m, ṁ a,ev = 0.6 kg s−1 , and then appropriately modifying the drier length,
L, and the maximum air temperature TD . The evaporator airflow is then varied in
the range 0.2 < ṁ a,ev < 0.6 kg s−1 to produce the plots shown in Figure 2.3, which
illustrate the effect of evaporator airflow on MER and SMER. The MER and SMER
are shown both with reference to just the condensed moisture (subscript ‘c’) and also
including that moisture which is shed from the system by venting. Experimental
results from [31] are also plotted in the figure, appearing as points. “Standard volume” flow rates have been converted to dry-air mass flow rates by use of the fact that
the standard atmosphere to which these refer (dry air at 21.1◦ C) has specific volume
1.2 m3 /kg [31, p.653].
The situation investigated by Carrington and Liu [31] was somewhat different
from that modelled here. Since the evaporator and condenser were placed in separate
calorimeter chambers, air off the evaporator did not mix with the air approaching the
condenser, and air approaching both of these components was unaffected by changes
in the action of the dehumidifier, since chamber conditions were held stable externally. The present model also does not take into consideration the feedback between
48
25
MER
MERc
(a)
MER (kg/h)
20
15
(b)
10
5
(c)
0
SMER
SMERc
SMER (kg/kWh)
4
(a)
3
(b)
2
1
(c)
0
0.2
0.25
0.3
0.35
0.4
0.45
ma,ev (kg/s)
0.5
0.55
0.6
Figure 2.3: Effect of evaporator airflow on MER and SMER, adiabatic mode,
approximate conditions at location F as follows: (a) 31◦ C, 93% RH, (b) 40◦ C,
48% RH, (c) 45◦ C, 25% RH. MERc and SMERc refer to the rate of moisture
condensation on the evaporator coils. MER and SMER refer to the total drying rate, including net moisture vented from the system. Points represent
measurements of Carrington and Liu [31] in a related experiment employing
the heat exchangers modelled here, with matching evaporator inlet air states.
49
a change in the evaporator cooling power and the compressor shaft power, which the
model assumes to be fixed at 5 kW. Nevertheless the SMERc and MERc curves of Figure 2.3 are consistent (within about 20%) with the experimental data of [31], and are
fully consistent with the conclusions of that investigation, illustrating the significant
effect that humidity has on the optimum evaporator airflow. The range of evaporator air flow rates modelled here (0.20-0.60 kg s−1 ) is smaller than that investigated by
Carrington and Liu [31] (about 0.12-0.60 kg s−1 ). In the system modelled here, the
drier energy balance requirement imposes a lower bound on the evaporator airflow,
since it is used to maintain energy balance through venting.
Before inspecting the effects of the selected constraints on the dryer energy performance and MER, we consider the following baseline drying scenario: ṁ a,co = 1 kg/s;
∆P = 200 Pa; ẆP = 5 kW; TD = 55 ◦ C; T0 = 10◦ C; condenser and evaporator face
areas Aco = Aev = 1 m2 ; tray drier dimensions as specified in Figure 2.2. These
parameters have been selected to represent a small-scale food dryer. The baseline
product thickness is a thin film of 1 mm; the effect of product thickness is examined
subsequently. The system behaviour in the baseline scenario is summarized in Table 2.3 for the isothermal case, the adiabatic case with bypass set to optimise SMER,
and the adiabatic case without bypass. As Table 2.3 shows, the isothermal system
behaviour is characterised by high humidity and by high MER (approximately three
times greater than for the adiabatic mode). Thus the baseline isothermal fan and HP
work requirement per kg moisture extracted is approximately three times lower than
its adiabatic counterpart. The isothermal mode yields its greatest SMER with zero
evaporator bypass, as is suggested by its high relative humidity φE .
Adiabatic
Parameter (unit)
Isothermal
Tev (◦ C)
40.8961
24.9045
27.5179
Tco
( ◦ C)
57.4184
58.6157
58.7158
TA
( ◦ C)
50.8320
29.3987
31.8813
TD
( ◦ C)
55.0
55.0
55.0
b
0
0.3800
0
x
0.0098
0.1192
0.0612
φE
0.9438
0.6937
0.7010
MER (kg/h)
62.9570
22.7421
22.2734
SMER (kg/kWh)
11.8044
4.2642
4.1763
COP
9.0037
3.9207
4.3187
ẆP /MER (kWh/kg)
79.4193 × 10−3
219.8562 × 10−3
224.4834 × 10−3
ẆF /MER (kWh/kg)
5.2946 × 10−3
(with bypass)
14.6571 × 10−3
Adiabatic
14.9656 × 10−3
Table 2.3: Baseline behaviour of isothermal and adiabatic modes.
50
Adiabatic
Isothermal
Tev
E Tco
A
B
D
80
60
Tev
E
A
B
Tco
40
D
Humidity ratio, g moisture / kg dry air
100
20
φ=20%
φ=10%
10
20
30
40
TWB=20°C
50
60
0
T,°C
Figure 2.4: Psychrometric chart showing air property paths in baseline scenario for adiabatic and isothermal dryers with zero evaporator bypass. The
evaporating and condensing temperatures Tev and Tco are represented by the
vertical bars to the left and right of the cycles. The MER is given by the vertical displacement DE for each cycle (also the height of the vertical bars), multiplied by the condenser airflow ṁ a,co . The isothermal cycle corresponds to
ṁw = 63 kg h−1 = 0.0175 kg s−1 , while the adiabatic cycle corresponds to
ṁw = 22 kg h−1 = 0.0061 kg s−1 . The adiabatic cycle nevertheless requires a
larger temperature difference TD − TB and thus a larger heat pump tempera-
ture lift Tco − Tev , resulting in lower COP and SMER.
51
Adiabatic
Isothermal
Tev
E Tco
A
B
D
80
60
40
Tev
E
A
B
Tco
D
C
Humidity ratio, g moisture / kg dry air
100
20
φ=20%
φ=10%
10
20
30
40
TWB=20°C
50
60
0
T,°C
Figure 2.5: Psychrometric chart showing air property paths in baseline scenario for isothermal and adiabatic dryers, with evaporator bypass set to optimise SMER. The isothermal cycle is optimised at zero bypass and is identical to that depicted in Figure 2.4. The adiabatic cycle is SMER-optimised at
38% evaporator airflow bypass. Bypass leads to an increased temperature lift
Tco − Tev , and a reduced heat pump COP (Table 2.3), but this is compensated
by an increased MER due to enhanced condensation on the evaporator coils.
52
The psychrometric cycles experienced by the air in the baseline scenario, without evaporator bypass and with bypass optimised for SMER, respectively, are represented in Figs. 2.4 and 2.5. The isothermal cycles are represented by dashed lines and
the adiabatic cycles by solid lines. In each case, state B, resulting from the mixing of
saturated air at two different temperatures, lies slightly above the line of 100% relative humidity i.e. corresponds to a cloud state [110]. (As was stated at the start of this
chapter, the environment is assumed to be saturated at 10◦ C.) The vertical heights of
the cycles, when multiplied by the evaporator dry air mass flow rate ṁ a,ev , give their
respective drying rates. As the figures illustrate, the isothermal mode yields a significantly greater drying rate than the adiabatic mode, subject to the other constraints,
with a lower HP temperature lift TD − TB . This smaller temperature lift leads to a
significantly higher isothermal COP (as shown in Table 2.3), which contributes to the
improved isothermal energy performance. The lower temperature lift arises mainly
because heat is provided progressively throughout the drying process, rather than to
the air entirely before the process. The heat provided in the isothermal dryer thus
effectively drives the moisture directly into the airstream.
Figure 2.6 illustrates the effect that the air temperature constraint TD has on the
MER and SMER for the isothermal and adiabatic modes, with all other parameters as
in the baseline scenario described above. For simplicity bypass has not been considered. This has no effect on the isothermal mode, which is in any case optimised at
zero bypass, and it has little adverse effect on the performance of the adiabatic mode,
since the humidities of both modes in the baseline scenario are fairly high (e.g. those
in Table 2.3). Over the range 30◦ C < TD < 70◦ C, the temperature constraint has a significant effect on the MER and SMER. Figure 2.6 shows that both MER and SMER are
maximised at high temperature for both modes, as expected. The marginal increase
in the benefit associated with the isothermal mode with an increase in TD is seen to
fall with TD , and becomes negative above about 60◦ C. The performance benefit associated with contact heat transfer is thus at a maximimum at a temperature of around
60◦ C for the baseline model.
Figs. 2.7 and 2.8 illustrate the effect that varying the capacity of the heat pump, as
determined by the shaft work ẆP , has on the relative humidity φE , the MER and the
SMER. Once again, for simplicity, evaporator air bypass is not considered, although at
low humidities this may adversely affect the performance of the systems by as much
as 50%, as Figure 2.3 indicates. The relative energy consumption (per kg moisture extracted) of the heat pump and fan are also displayed. Since the plots span a very wide
range of values for the shaft power ẆP , they are not expected to predict the behaviour
of any particular system. Rather they provide a guide to the obtainable performance
of a system employing a heat pump that operates at 50% of COPC , over a range of
capacities, in the context of the baseline scenario described above for the two drying
53
70
60
WP=1kW
WP=5kW
WP=10kW
MER (kg/h)
50
40
Isothermal
30
20
10
Adiabatic
0
SMER (kg/kWh)
20
15
10
Isothermal
5
Adiabatic
0
30
35
40
45
50
55
60
65
70
TD, °C
Figure 2.6: Effect of TD and ẆP on performance of adiabatic and isothermal
HPD in the baseline configuration described in the text.
54
modes. Points on each plot that correspond to shaft powers of 1 kW, 2 kW, 5 kW,
10 kW and 20 kW are indicated, although the range of shaft powers that have been
used to generate the plot includes values that lie outside the range 1 kW to 20 kW.
Figure 2.7 shows the variation in system performance with compressor power for the
adiabatic system. The relative variation of the MER and SMER (near the point corresponding to ẆP = 5 kW) appears consistent with previously published results [87,
Figure 4]. As Figure 2.7 illustrates, varying the shaft work in the range 1-20 kW has
a significant effect on the drier outlet relative humidity φE of the adiabatic dryer. In
contrast, the relative humidity φE at the outlet of the isothermal drier remains above
0.85 over this entire range of ẆP (Figure 2.8). Both modes exhibit a peak in SMER
below a shaft power of about 1kW. At this low shaft power the MER is relatively
low for both modes. The isothermal MER is generally 2-3 times higher than that of
the adiabatic mode, and at low ẆP the isothermal MER is comparable with the highest MERs achieved by the adiabatic dryer over the range considered. Note that the
isothermal-mode baseline behaviour depicted in Figure 2.8 is for a thin-film product
(δ = 1mm) which has little thermal resistance.
Another way of meeting a constraint on the relative humidity in the drier, given a
fixed HP capacity, would be to appropriately modify the exposed product area, and
this possibility is investigated here by varying the drier length, L. Figs. 2.9 and 2.10
show the variation in relative humidity, MER and SMER under varying drier length
L for the adiabatic and isothermal modes, respectively. The rest of the system parameters are fixed at their baseline values; in particular, ẆP = 5 kW. At very high
relative humidities, corresponding to very long drier ducts, the impact of increasing drier length on fan power becomes increasingly important, as can be seen in the
downward SMER curvature. The figures indicate (1) that a HPD in the isothermal
mode may have significantly lower space requirements per unit drying capacity than
the adiabatic mode, and (2) that the SMER and MER benefits of the isothermal mode
are highly sensitive to the relative humidity constraint. Nevertheless, at any given
relative humidity, the isothermal SMER is approximately twice the adiabatic SMER.
Thus the high isothermal SMER can be seen to result partly from high system relative
humidities due to the enhanced drying rate, but also partly from the increased heat
pump COP that is obtained by avoiding the use of air as the primary route for heat
transfer to the drying process. As well as increasing product throughput, a reduced
product residence time due to high MER (compared with the adiabatic mode) could
relax the relative humidity constraints that arise from product degradation, and thus
allow energy performance to be maximised.
Figure 2.11 shows the effect of product thickness on the performance of the system. As the figure illustrates, the effect of product thickness δ on the SMER may be
significant, particularly at low heat pump capacities and product thicknesses. Thus
55
50
WP=20kW
WP=10kW
MER (kg/h)
40
WP=5kW
WP=2kW
30
WP=1kW
20
10
0
Power/MER (kJ/kg)
Compressor
Fan
600
400
200
0
SMER (kg/kWh)
8
6
4
2
0
0.4
0.5
0.6
0.7
φE
0.8
0.9
1
Figure 2.7: Performance of adiabatic HPD with φE ; varying HP capacity.
56
200
WP=20kW
WP=10kW
MER (kg/h)
150
WP=5kW
WP=2kW
WP=1kW
100
50
0
Power/MER (kJ/kg)
Compressor
Fan
150
100
50
0
SMER (kg/kWh)
20
15
10
5
0
0.85
0.875
0.9
0.925
φE
0.95
0.975
1
Figure 2.8: Performance of isothermal HPD with φE ; varying HP capacity.
57
50
MER (kg/h)
40
L=10m
L=5m
L=2m
L=1m
30
20
10
Power/MER (kJ/kg)
0
Compressor
Fan
800
600
400
200
SMER (kg/kWh)
0
5
2.5
0
0.3
0.4
0.5
0.6
φE
0.7
0.8
0.9
1
Figure 2.9: Performance of adiabatic HPD with φE ; varying drier length L.
58
100
MER (kg/h)
75
L=5m
L=2m
L=1m
L=0.4m
L=0.2m
50
25
0
Compressor
Fan
Power/MER (kJ/kg)
500
400
300
200
100
0
SMER (kg/kWh)
12.5
10
7.5
5
2.5
0
0.3
0.4
0.5
0.6
φE
0.7
0.8
0.9
1
Figure 2.10: Performance of isothermal HPD with φE ; varying drier length L.
59
except in the case of thin product layers, second-law losses that result from heat transfer through the product may limit the SMER obtainable from an isothermal contact
HPD. Note however that the model presented here only takes account – by way of a
simple conduction model – of heat delivered through the product itself. In practice,
in a stack of ducts, heat conduction through the duct top surface could play a role
in heat transfer that is not accounted for by the present model. Investigating such
possibilities, and examining the system-level impacts of irreversible mass transfer
processes (and linkages between heat and mass transfer) within the product, require
the drying processes to be modelled in substantially greater detail than it is in the
present chapter.
Our analysis has not assessed the impact of various system modifications that
could further enhance performance. For instance an evaporator economiser, by improving the latent-to-total-cooling ratio of the evaporator, might further improve the
SMER of a heat pump contact dryer. An investigation of such enhancing features
would be appropriate in a more detailed investigation of a particular system, and
goes beyond the scope of the fairly crude, system-level analysis presented here. In
any case, since isothermal drying generally involves high humidities, the benefit of
an economiser is likely to be small [10]. The present analysis also has not taken into
account any limitations on the operating conditions of the compressor, such as restrictions on the suction and discharge pressures. For example, the manufacturer of
the scroll compressor used in the dehumidifier model, a Copeland ZR61K2-TFD, recommends that the saturated suction temperature should not exceed 25 ◦ C, due to the
risk of excessive oil dilution [30]. This restriction has not been applied in the modelling work. There are now new developments in compressor technology, such as the
Danfoss Turbocor oil-free compressor, for which such restrictions do not apply. Such
developments may contribute to the feasibility of ICHPD technology.
In passing, it is intriguing to note that at the high SMERs predicted in this chapter,
which correspond to a specific work consumption (per kg moisture extracted) of less
than one tenth of water’s latent heat, an ICHPD, if clad in solar photovoltaic panels
of 20% conversion efficiency, could in principle extract moisture at more than twice
the rate of a solar thermal drier of the same area – even if the solar thermal drier were
able to operate at 100% efficiency. At typical solar drier efficiencies of around 5-20%
[138], the photovoltaic-ICHPD system could, in principle, extract ten to forty times
more moisture, per exposed surface area, than a solar thermal drier. Another observation which may be of interest is that the high energy efficiency of ICHPD allows
system energy balance to be maintained easily, requiring much less heat-shedding
than its adiabatic HPD counterpart. This makes venting relatively avoidable in the
case of ICHPD, which thus could lend itself to drying applications which require
zero-emission operation, such as sewage-sludge treatment.
60
150
WP=1kW
WP=2kW
WP=5kW
WP=10kW
MER (kg/h)
100
50
0
SMER (kg/kWh)
25
20
15
10
5
0
0
1
2
3
4
5
6
7
Product thickness, δ (mm)
8
9
10
Figure 2.11: SMER variation of isothermal HPD with product thickness.
61
2.5.
Conclusions
In this chapter we have investigated a typical dehumidification dryer configuration,
to determine the energy benefit that may be associated with the (idealised) isothermal contact drying mode [37]. Isothermal contact drying has been found to have the
potential to increase both the energy efficiency and the moisture extraction rate by as
much as 2-3 times compared with conventional, adiabatic dehumidification drying.
The energy efficiency and MER gain has been found to peak at around 60◦ C for the
baseline system configuration that has been used in the present investigation. The
isothermal energy performance is highest at high temperature, and falls rapidly with
decreasing maximum acceptible relative humidity over the product and with product
thickness. These constraint impacts suggest that isothermal HPD may be most applicable in the drying of those products, such as sludges and pastes or other products (1)
that can be spread into thin layers, that (2) are least vulnerable to quality deterioration
at high temperature and relative humidity. For such products, isothermal HPD has
the potential to substantially increase the SMER, effectively allowing the latent heat
of vaporization to be recycled on the order of 10 times. This high SMER may have
significant economic implications in the drying of products that can be spread onto
a surface in thermal contact with the condenser of a dehumidification HPD [32], and
may, in the longer term, increase the viability of some biofuel options [45]. The above
findings are subject to the assumption that the idealised “isothermal” model successfully describes the behaviour of a contact HPD. In order to assess this assumption we
turn to a more detailed duct model in the following chapter.
62
2.6.
Nomenclature for chapter 2
Roman and Greek symbols
Symbol
Definition [units]
A
Area [m2 ]
b
Bypass ratio (ṁ a,co − ṁ a,ev )/ṁ a,co [–]
cp
Specific heat capacity [J/kg-K]
COP
Coefficient of performance [–]
D
Diffusivity [m2 /s]
D
Heating plate refrigerant tube internal diameter [m]
d
Duct air gap [m]
DH
Duct hydraulic diameter [m]
f
Moisture condensation fraction [–]
f E , fC , g
Polynomials characterising evaporator and condenser
h
Specific enthalpy of moist air [J/kg dry air]
h
Product heat transfer coefficient [W/m2 -K]
hf
Specific enthalpy of condensed water phase [J/kg]
hm
Product mass transfer coefficient [m/s]
k
Thermal conductivity [W/K-m]
l
Heating plate condenser tube spacing [m]
L
Heating plate length [m]
ṁ
Mass flow rate [kg/s]
MER
Moisture Extraction Rate [kg/s]
Nu
Nusselt number [–]
ND
Number of ducts [–]
p
Pressure [Pa]
Pr
Prandtl number [–]
Q̇
Heat flow rate [W]
SMER
Specific Moisture Extraction Rate [kg/kWh]
T
Temperature [K]
v
Velocity [m/s], Specific volume [m3 /kg]
W
Heating plate width [m]
Ẇ
Power input [W]
x
Venting fraction, distance through kiln [m]
xp
Heating plate refrigerant tube centerline depth [m]
α
Heat exchange coefficient [W/m2 -K]
δ
Product thickness [m]
eF
Fan efficiency [–]
63
ηC
Ratio of heat pump heating COP to Carnot heating COP [–]
ρ
Density [kg/m3 ]
φ
Relative humidity [–]
ω
Humidity ratio [kg vapour/kg-dry]
Subscripts and superscripts
Symbol
Definition
a
Dry air
ev
Evaporator
co
Condenser
sat
Saturation condition
wb
Wet bulb
as
Adiabatic saturation
A, B, C, D, E, F
Locations on air cycle
G
Incoming water heating control volume
C, C
Condenser, Carnot
D
Drying, Ducts
E
Evaporator
F
Fan
H
Heating
out
Outlet
P
Heat pump, compressor
W
Inlet moisture heating
w
Water
m
Mass exchange
r
Refrigerant
p
Heating plate
v
Vapour
0
Environment
δ
Product
64
Chapter 3
Detailed drier-duct model.
In this chapter the equations for a detailed air-side model are stated, simplified and
implemented numerically. (Their derivation can be found in the appendix.) The numerical recipe that has been employed is the finite volume method, and the momentum balance equation is solved using the Semi-Implicit Method for Pressure-Linked
Equations, Revised (SIMPLER) [117]. The model is similar to the models of Vaughan
et al. [175, 176] and of Sun and Carrington [159, 160] for stack air-flow in the drying
of timber. Preliminary results, comparing the detailed model with the simple air-side
model for the constant drying-rate period, are presented and discussed.
3.1.
Introduction
It was noted in Chapter 1 and in the discussion of Chapter 2 that the simple gasside model captured in equations (1.4.15)–(1.4.18) (on page 25) involves considerable
idealisation. In particular the model neglects any deviation of the product surface
temperatures from their idealised values, and cannot describe the impact of the drying process on the air-flow parameters, the pressure drop along the flow direction,
or local variation of the heat, mass and momentum transfer coefficients. It was also
noted that the equations apply only to an ideal solid or to a nonideal solid during the
constant-rate drying period. Finally, the model is unable to deal with incidental processes such as duct ceiling heat transfer, and (small) corrections related, for example,
to the kinetic energy of the air stream. In the present chapter a more flexible air-side
model is developed and is used to assess the impacts of these effects. In later chapters
this detailed air-side model is linked with models of the internal drying processes and
of the heat pump components, to produce a dynamical whole-system model.
65
3.2.
Balance equations
The appendix (p.212) contains a systematic presentation of the theory underlying the
material contained in this chapter, and is referred to repeatedly. In particular, section
A.3 contains a discussion of the relationship between the local and integral forms of
the conservation equations, and the underlying postulates for body motion.
Time-averaged control volume balance equations
Equations (A.6.2a)–(A.6.2c) can be used to convert the control volume balances (A.3.17),
(A.3.34) and (A.3.51) of the appendix into time-averaged form. The moist-air gas system is regarded as a binary mixture of ‘air’ (a) and water vapour (v). No chemical
reactions take place, so that the source term for production of species k by chemiR n
cal reactions may be dropped from (A.3.17). The term
∑k=1 (ρk v + jk ) · fk dV in
R( t )
(A.3.51), which represents work due to external body forces, corresponds to gravity
in the present case of the air system. Changes in gravitational potential in a typical
HPD system will be on the order of 10 J kg−1 , which corresponds to tiny changes in
the air temperature, on the order of only 0.01 K. The external body force term can thus
be dropped safely from (A.3.51). The resulting equations are as follows (k = a, v):
d
dt
d
dt
d
dt
Z
R
Z
R( t )
Z
ρk dV =
R( t )
ρvdV =
Z
S( t )
Z S( t )
ρk (vk − w ) · (−n̂) dS
ρv (v − w ) − (σ + p0 δ ) · (−n̂) dS
(3.2.1)
(3.2.2)
Z
1 2 p0
1 2
ρ u+ v +
dV =
ρ h + v (v − w ) · (−n̂) dS
2
ρ
2
S( t )
−
+
Z S( t )
Z
S( t )
w ( p0 − p) + v · τ · (−n̂) dS
qc +
∑
!
h k jk
k = a,v
· (−n̂) dS
(3.2.3)
where the variables are as defined in Appendix A; in particular, w is the velocity of
the bounding surface of the control volume, u is the specific internal energy of the
moist-air gas mixture, and q c is the pure heat flow vector.
Equations (3.2.2) and (3.2.3) can be simplified by approximating the flow as incompressible. This assumption is largely justified due to the fact that relative pressure
variations in the system will be small, although there will be small density variations
66
associated with humidity and temperature variations across the flow system. Assuming incompressible flow, i.e. assuming that ρ is a constant in (3.2.2) and (3.2.3), leads
to:
d
dt
d
dt
Z
R( t )
Z
ρvdV =
R( t )
Z S( t )
ρv (v − w ) − (σ + p0 δ + σ (t) ) · (−n̂) dS
(3.2.4)
Z 1
p0
1
ρ u + v2 +
dV =
ρ h + v2 (v − w ) · (−n̂) dS
2
ρ
2
S( t )
Z −
S( t )
w ( p0 − p) + v · τ · (−n̂) dS
Z
+
qc
+
∑
!
· (−n̂) dS
h k jk
k = a,v
S( t )
(3.2.5)
In (3.2.4), σ (t) = ρv v − ρvv is the Reynolds stress tensor for incompressible flow
[145, p.185]. In investigating a similar flow situation (within a stack of timber under-
going drying), Vaughan [175] established that this term had a negligible effect on the
momentum balance, assuming isotropic turbulence with an intensity of up to 10%.
Dropping σ (t) amounts to neglecting the difference between the self-outer-product
of the time-averaged velocity, and the time average of the self-outer-product of the
instantaneous velocity. In order to obtain equations that are suitable for discretisation, we make a number of similar assumptions allowing us to use products of time
averages to estimate time averages of products. The equations that result are:
d
dt
d
dt
d
dt
Z
R( t )
Z
Z
Z
ρk (vk − w ) · (−n̂) dS
(3.2.6)
ρv (v − w ) − τ + ( p − p0 )δ ) · (−n̂) dS
(3.2.7)
ρk dV =
R( t )
ρvdV =
R( t )
1
p0
ρ u + v2 +
2
ρ
S( t )
Z S( t )
dV =
1 2
ρ h + v (v − w ) · (−n̂) dS
2
Z
S( t )
−
Z S( t )
+
w ( p0 − p) + v · τ · (−n̂) dS
Z
S( t )
qc
+
∑
k = a,v
!
h k jk
· (−n̂) dS
(3.2.8)
In order to obtain (3.2.6)–(3.2.8), we have dropped σ (t) and several similar correlation cross-terms from the balance equations. This move is consistent with Slattery’s
recommendation [146, p.225] that on entry and exit portions of a turbulent control
volume, the Reynolds stress tensor can generally be neglected from the momentum
balance equation, and that the work done by turbulent forces at the entry and exit
67
portions of a turbulent control volume can similarly be neglected [146, p.234]. These
assumptions also correspond to the ‘plug-flow’ model for a flow system discussed
by Wen and Fan [183], which assumes zero longitudinal dispersion, and full mixing
in the transverse direction, for all flow properties. (Dispersion arises from nonzero
products of perturbation terms e.g. in velocity and concentration.) In addition, we
have noted that σ = τ − pδ (see p. 229). Equations (3.2.6)–(3.2.8) have the same
form as the equations (A.3.17), (A.3.34) and (A.3.51) prior to time-averaging, and for
convenience the overbars will be dropped from subsequent expressions.
Partitioning of surface integrals
Sn
Sw
Se
Moist Air
Sm
Product
Figure 3.1: Control volume for air flow system
Fig. 3.1 illustrates a typical control volume for the air flow system in the dryer.
Slattery partitions the surface of a control volume into an entry and exit portion Sent ex
and a remaining impermeable portion [146]. The air flow system control volume illustrated in Fig. 3.1 is naturally partitioned into four portions, three of which are
permeable: entry portion Sw , exit portion Se , mass exchange portion Sm and impermeable portion Sn . Of these four surfaces, Sw , Se and Sm have zero normal velocity,
i.e. w · n̂ = 0, while Sm may have a (tiny) nonzero normal velocity due to shrinkage
and movement of the product. The barycentric velocity v is in general nonzero on
Sw , Se and Sm , but is zero at the top surface Sn . Dividing the surface integrals into the
four parts depicted in Fig. 3.1, we rewrite the balance equations using the constitutive
equations (A.4.2), (A.4.22) and (A.4.24), and noting that dωv = −dωa .
68
Control volume mass balance for species k (k = a, v):
d
dt
Z
Z
ρk dV =
R
(m)
(ρk v − ρDva ∇ωk ) · (−n̂)dS + mk
Sw, e
(3.2.9)
(m)
In (3.2.9) the following definition is used for the surface mass transfer mk
of species
k (where k = a, v):
(m)
mk
=
Z
Sm
ρk (vk − w ) · (−n̂)dS
(3.2.10)
We proceed by assuming that w v, i.e. that shrinkage leads to a surface motion
that is much smaller than the barycentric velocity due to evaporation at and beneath
the product surface. This can be justified by considering the relative densities of condensed moisture and water vapour. Neglecting w and using (A.7.3), we can write
(m)
mk
=
Z Sm
ωk ρv2 + h•m (ρk,m − ρk ) dS
(3.2.11)
Here the y-component of the barycentric velocity is denoted v2 . (Similarly, chapter
also refers to the unit vectors in the x and y directions as eˆ1 and eˆ2 .)
Control volume momentum balance:
d
dt
Z
R
ρv dV =
Z Sw, e
ρvv + ( p − p0 )δ − τ · (−n̂) dS − F f − F (m)
(3.2.12)
In (3.2.12), the momentum source terms F f and F (m) are defined as follows:
Ff =
Z Sm, n
F (m) =
( p0 − p)δ + τ · (−n̂)dS
Z
Sm
−ρv (v − w ) · (−n̂)dS
(3.2.13)
(3.2.14)
Control volume energy balance:
Z Z d
1
p0
1
ρ u + v2 +
dV =
ρ h + v2 v − v · τ · (−n̂)dS
dt
2
ρ
2
R
Sw, e
Z
+
Sw, e
[−k∇ T − ρ (hv − h a ) Dva ∇ωv ] · (−n̂)dS
− W + Q(m) + Q
(3.2.15)
In (3.2.15), the surface heat transfer Q, the energy transfer associated with the transfer
of mass Q(m) , and the work rate at the CV surface, W, are defined as follows:
Q=
Z
Sm, n
q c · (−n̂) dS
(3.2.16)
69
Q
(m)
=
Z
Sm
#
" 1 2
ρ h + v (v − w ) + ∑ hk jk · (−n̂)dS
2
k = a,v
W=
Z Sm
(3.2.17)
w ( p0 − p) + v · τ · (−n̂) dS
(3.2.18)
Simplification of air balance equations
The species-k mass, momentum and energy balance equations (3.2.10), (3.2.12) and
(3.2.15) are simplified by the considerations summarized in this section.
On Sw we have −n̂ = ê1 , while on Se we have −n̂ = −ê1 . It follows that the
species-k mass balance can be rewritten as follows:
Z
Z ∂ω
d
(m)
ρk dV =
ρk v1 − ρDva k ê1 · (−n̂)dS + mk
dt
∂x1
R
(3.2.19)
Sw, e
In (3.2.19), v1 represents the x-component of the time-averaged free-stream velocity, which in the unidimensional flow model is taken to prevail over the control
k
volume. The diffusion term −ρDva ∂ω
∂x is expected to be negligible compared with
1
the convection term ρk v1 . For instance the steepest longitudinal humidity gradient for the isothermal case depicted in Fig. 1.10 corresponds to
∂ωv
∂x
≈ 0.02 m−1 ,
which can be taken as an upper bound since the entrance relative humidity is very
low. From this value and a mass flow rate on the order of 1 kg/s we can estimate
jv /ρv v = O(10−5 ). We retain the diffusion term only because it enables us to
operate entirely within Patankar’s convection-diffusion framework [117].
In accordance with the unidimensional plug-flow model, we consider only the xcomponent of the momentum balance. On Sn we have −n̂ = −ê2 , while on Sm we
have −n̂ ≈ ê2 . Taking the inner product of the unit vector ê1 in the x-direction with
the momentum balance (3.2.12), we note the following:
ê1 · F (m) ≈
Z
Sm
−ρv1 (v2 − w2 )dS ≈ 0
(3.2.20)
since v1 ≈ 0 on Sm . Similarly, we have:
ê1 · F f =
Z
Sm, n
ê1 · τ · (−n̂)dS
(3.2.21)
Assuming ∇ · v = 0, we can use (A.4.22) to obtain τ · (−n̂) on Sm , as follows:
∂v1
∂v2
∂v2
τ · (−n̂) = µ
+
ê1 + 2
ê2
(3.2.22)
∂x2
∂x1
∂x2
∂v
≈ µ 1 ê1
(3.2.23)
∂x2
70
using (A.7.15). Thus on Sm , using (A.7.4) with τ · (−n̂) = τ · n̂s , we have
1
ê1 · τ · (−n̂) = τ · n̂s = ρv21 C •f
(3.2.24)
2
Using an analogous argument, we can establish that (3.2.24) holds also on Sn .
On Sw we have −n̂ = ê1 , while on Se we have −n̂ = −ê1 . Assuming again that
the (time-averaged) flow is incompressible, i.e. that ∇ · v = 0 on these surfaces, we
can evaluate ê1 · τ · (−n̂) using (A.4.22), and find that on Sw, e :
ê1 · τ · (−n̂) = 2µ
∂v1
ê · (−n̂)
∂x 1
(3.2.25)
Strictly speaking, because of shrinkage of the product, the area of Sw may differ
slightly from that of Se . We neglect this small difference, and get:
Z
Sw, e
− p0 δ · (−n̂)dS = 0
(3.2.26)
Using (3.2.20)–(3.2.26), the momentum balance (3.2.12) x-component becomes:
Z
Z
d
∂v ρv1 dV =
(3.2.27)
ρv21 + p − 2µ 1 ê1 · (−n̂) dS − F f
dt
∂x1
R
Sw, e
where the magnitude F f of the friction force is given by
Ff =
Z
Sm, n
1 2 •
ρv C dS
2 1 f
(3.2.28)
This includes the contribution to the friction force of the top-solid surface Sm .
Turning to the control volume energy balance (3.2.15), we can safely neglect the
work W due to compression and viscous forces at the product surface, and also the
work due to viscous forces on Sw, e [146, p.234]. Also the transport of energy by diffusion, ρ (hv − h a ) Dva ∇ωv on Sw, e , is expected to be negligible compared with the
convective transport. The conductive term −k∇ T is expected to be small too, but we
retain it in order to obtain a discretised equation compatible with Patankar’s formal-
ism [117]. Similarly we evaluate the term −v · τ · (−n̂) which is integrated on the
interfaces Sw,e in Equation (3.2.15) by using (3.2.25) to obtain
∂ 1 2
−v · τ · (−n̂) = −v1 eˆ1 · τ · (−n̂) = −2µ
v1 eˆ1 · (−n̂)
∂x 2
(3.2.29)
on Sw,e . Finally we neglect changes in the stored energy associated with atmospheric
pressure, caused by small changes in the volume due to product shrinkage, which
R p0 d
is represented in (3.2.15) by the term dt
ρ ρ dV. Dropping the above terms, the
R
energy balance can be simplified to the following:
Z Z d
1 2
1 2
∂T
d 1 2
ρ u + v1 dV =
ρ h + v1 v1 − k
− 2µ
v
ê1 · (−n̂)dS
dt
2
2
∂x1
dx 2 1
R
Sw, e
+ Q(m) + Q
(3.2.30)
71
Applying the no-slip condition on Sm , we approximate Q(m) as follows:
Q(m) =
Z
∑
ρk hk (vk − w ) · (−n̂)dS
k = a,v S
(3.2.31)
m
Using (A.7.2), we have
Z
Q=
Sm, n
h• ( Ts − T ) dS
(3.2.32)
Evaluating the dot products on Sm, n , we summarize the simplified mass, momentum and energy control volume balance equations for the air flow system as follows:
Control volume mass balance for species k (k = a, v):
Z
Z ∂ω
d
ρωk dV =
ρk v1 − ρDva k dS
dt
∂x1
R
Sw
Z ∂ω
(m)
−
ρk v1 − ρDva k dS + mk
∂x1
(3.2.33)
Se
Control volume momentum balance:
d
dt
Z
ρv1 dV =
R
Z
ρv21 + p − 2µ
Sw
−
Z
Se
∂v1 dS
∂x1
ρv21 + p − 2µ
∂v1 dS − F f
∂x1
(3.2.34)
Control volume energy balance:
Z Z d
1
1
∂T
∂ 1 2
ρ u + v21 dV =
− 2µ
v1 dS
ρ h + v21 v1 − k
dt
2
2
∂x1
∂x 2
R
Sw
Z
∂T
∂ 1 2
1 2
−
− 2µ
v
dS
ρ h + v1 v1 − k
2
∂x1
∂x 2 1
Se
+ Q(m) + Q
(3.2.35)
In the following section Equations (3.2.33)–(3.2.35) are discretised. In doing so, it
may appear as though variations in the conserved variables in the coordinate direction perpendicular to the average flow direction are neglected. Note however that
the definitions of the transfer coefficients (A.7.1)-(A.7.3) mean that empirical transfer
coefficient correlations effectively incorporate information about cross-flow variation.
3.3.
Discretised equations
Here the discretised balance equations for the moist air side are obtained from (3.2.33)–
(3.2.35). Throughout this chapter the coordinate system depicted on page 213 is used.
72
The air flow system is subdivided into Nx control volumes. With reference to the tray
drier configuration depicted on page 41, we have, for each air control volume:
∆x = L/Nx
(3.3.1)
∆y = d
(3.3.2)
∆z = w
(3.3.3)
The symbols ∆x, ∆y and ∆z will be used preferentially in what follows. Other spatial
variables that will be used are the air control volume face area A, the control volume
mass exchange area Am , and the control volume size V:
A = ∆y∆z
(3.3.4)
Am = ∆x∆z
(3.3.5)
V = ∆x∆y∆z
(3.3.6)
Patankar’s staggered grid approach has been adopted: the grid upon which we discretise the balance equations appears in Fig. 3.2, for which Nx = 3. The state of the
system at the main grid points is represented by a set of vectors containing Nx values (e.g. [ωW , ωP , ωE ]0 ), while the velocity field is represented by a vector containing
Nx + 1 values ([vin , vw , ve , vf ]0 ), of which vin is fixed as a boundary condition.
vin
pW
ρW TW
(W)
{j − 1}
vw
{i + 1}
(f)
{i}
(e)
{i − 1}
(w)
pP
ρP TP
(P)
{j}
ve
pE
ρE TE
p0
vf
(E)
{j + 1}
Figure 3.2: Staggered momentum balance control volumes.
73
Mass balance
The overall mass balance equation is obtained by summing (3.2.33) over k = a, v, and
noting that dωv = −dωa . The resulting equation can be discretised to
"
#
ρ1 − ρ0 V
(m)
(m)
= Fw − Fe + m a + mv
∆t
(3.3.7)
P
In (3.3.7), F = ρvA is the discretised convective mass flow at the entrance and exit
surfaces of the gas control volume.
The SIMPLER algorithm recommended by Patankar (described below) proceeds
by recalculating the density at each control volume by way of an equation of state:
ρP = f ( T, p, ω ) . We use the ideal gas state equation to evaluate this function. On
P
convergence, the algorithm produces a flow pattern which satisfies (3.3.7).
The vapour mass balance equation (3.2.33), with k = v, can be discretised to
"
#
ρ1 ωv1 − ρ0 ωv0 V
(m)
= Jv,w − Jv,e + mv
(3.3.8)
∆t
P
where the total vapour mass flow at the entrance and exit surfaces is given by
(m) ∂ωv
Jv = ρvωv − Γ
A,
(3.3.9)
∂x
with
Γ(m) = Dva ρ.
(m)
The discretised vapour mass transfer term mv
(m)
mv
(3.3.10)
appearing in (3.3.8) is given by
(m)
= h•m (ρv,m − ρv ) Am + ωv,m ∑ mk Am
(3.3.11)
k
where the expression for h•m = hm θ AB , and θ AB is given on page 249. When considering the impact of dry-air transport into the product, Equation (3.3.11) is most
easily solved by an iterative procedure. Alternatively, since the motion of the dryair component is typically very small (as it is flowing into the product to replace the
volume vacated by liquid water, which is three orders of magnitude denser), one can
(m)
reasonably set m a
= 0 and rearrange (3.3.11) to obtain
(m)
mv
=
1
h• (ρv,m − ρv ) Am
1 − ωv,m m
1 and subtracting from (3.3.8) yields:
Multiplying (3.3.7) by ωv,P
"
#
ωv1 − ωv0 ρ0 V
1
1
= Jv,w − Fw ωv,P
− Jv,e − Fe ωv,P
∆t
P
(m)
(m)
(m)
1
+ mv − ωv,P
m a + mv
.
(3.3.12)
(3.3.13)
74
Using Patankar’s fully-implicit convection-diffusion recipe, (3.3.13) becomes
#
"
ωv − ωv0 ρ0 V
(m)
(m)
= aW (ωv,W − ωv,P ) − aE (ωv,P − ωv,E )
∆t
P
(m)
(m)
(m)
(3.3.14)
+ mv − ωv,P m a + mv
(m) (m)
(m)
aW = Dw A( Pw ) + [[ Fw , 0 ]]
(m) (m)
(m)
aE = De A( Pe ) + [[− Fe , 0 ]]
(3.3.15)
(3.3.16)
in (3.3.14)–(3.3.16) the superscript ‘1’s denoting the updated values have been dropped
for convenience, and the following definitions have been used:
A( P(m) ) = [[0, (1 − 0.1 P(m) )5 ]]
Γ(m)
A
δx
F
= (m)
D
(3.3.17)
D (m) =
(3.3.18)
P(m)
(3.3.19)
Equation (3.3.14) can be rearranged to the following discretisation equation:
(m)
(m)
(m)
(m)
aP ωv,P = aW ωv,W + aE ωv,E + bP ,
(m)
aP
(m)
bP
(3.3.20)
ρ0P V
(m)
(m)
(m)
(m)
+ aW + aE + m a + m v
∆t
ω 0 ρ0 V
(m)
= v,P P + mv
∆t
=
The discretised equations can be solved with minimal effort using the matrixmanipulation capabilities that are built into MATLAB. The method that has been
used is illustrated here for the case of the species mass balance equation; essentially
the same method has been applied to each of the discretisation equations presented
below. We use Nx = 3 for the illustration. The main grid points in Fig. 3.2 have been
labelled in terms of their relationship to the central grid point (in both labelling systems: ’W’, ’P’ and ’E’, and also j − 1, j and j + 1). When Nx = 3 we can take the central
grid point to be grid point number two, i.e. j = 2, and refer to the three main grid
points as being grid points 1, 2 and 3. Each grid point (j=1, 2, or 3) then receives a set
(m)
(m)
(m)
(m)
of discretisation coeffecients aP j , aW j , aE j and bP j , as used in (3.3.20). In applying
the boundary conditions, the upwind scheme is used and only convective transport
is considered. Thus the vapour flux at the inlet is given by ωin ρin vin A, and the vapour
flux at the outlet boundary is given by ω3 ρ3 vf A. The upwind scheme [117, p.95] cor
responds to A( P(m) ) = 1 in equations (3.3.15) and (3.3.16), and since vin > 0 and
vf > 0, and only convective transport is considered at the boundary interfaces, the
75
discretisation coefficients for the boundary grid points can be written
(m)
aW1 = [[ρin vin A, 0 ]] = ρin vin A
(3.3.21)
(m)
aE3
(3.3.22)
= [[ρ3 vf A, 0 ]] = 0
Together with the boundary condition specified above, the discretised vapour mass
balance (3.3.20) can be written for all three grid points in matrix form as follows:

 
 


(m)
(m)
(m)
bP1
aP1
− aE1
0
ωin ρin vin A
ωv,1

  (m)  

(m)
(m)  
 
 

 (3.3.23)
 − a(m)
0
aP2
− aE2 


  ωv,2  =  bP2  + 
W2
(m)
(m)
(m)
0
ωv,3
bP3
0
− aW3
aP3
Mω = b + bc
or
(3.3.24)
This equation is solved by inverting the matrix M:
ω = M−1 (b + bc)
(3.3.25)
Energy balance
The energy balance equation (3.2.35) becomes, in discretised form:
n 2 o 
ρ u + 12 v2 − ρ0 u0 + 12 v0
V

 =
∆t
P
n ρ h + 12 v2 − ρ0 h0 +
=
∆t
1
2
0 2 o
V
v

p − p0 V
 =
−
∆t
P
=
where
(m)
Ju,w − Ju,e + JH,w − JH,e + QP
+ QP
d 1 2
1 2
Ju = ρv
v − 2µ
v
A
2
dx 2
dT
JH = ρhv − k
A
dx
(m)
QP
=
∑
i = a,v
(m)
mi
hi ( Ts )
QP = Am h• ( Ts − TP ) + Am h• ( Tn − TP )
(3.3.26)
(3.3.27)
(3.3.28)
(3.3.29)
(3.3.30)
where h• = hθt , and the expression for θt is given on page 249.
Equation (3.3.27) is already in convection-diffusion form, and can be written
1 2
1 2
( E) d
Ju = ρv
v − Γv
v
A
(3.3.31)
2
dx 2
76
with
( E)
Γv
= 2µ
(3.3.32)
In contrast, (3.3.28) requires modification to be cast into Patankar’s standard form,
since the spatial gradient is of a different variable (T) to that which is convected (h).
By the chain rule, the differential enthalpy change for the gas system is given by
∂h
∂h
∂h
dh =
dT +
dp + ∑
dωi
∂T P,ωi
∂p T,ωi
∂ωi T,p,ω j
i
∂h
∂h
= c p dT +
dp + ∑
dωi
(3.3.33)
∂p T,ωi
∂ωi T,p,ω j
i
Equation (A.3.60) implies that
∂h
∂ωi
= hi
(3.3.34)
T,p,ω j
Substituting (3.3.34) into (3.3.33) and noting that dωa = −dωv leads to:
∂h
dp + (hv − h a ) dωv
dh = c p dT +
∂p T,ωi
The pure heat flux term can thus be written as [159]:
dT
k dh
k ∂h
dp
k
dωv
k
=
−
− ( hv − h a )
.
dx
c p dx c p ∂p T,ωi dx
cp
dx
(3.3.35)
(3.3.36)
Substituting (3.3.36) into (3.3.28), the energy flow terms become:
JH = JHT + JHω + JHP ,
(3.3.37)
k dh
JHT = ρvh −
A
(3.3.38)
c p dx
k
dωv
JHω =
A
(3.3.39)
( hv − h a )
cp
dx
" #
dp
k ∂h
JHP =
A.
(3.3.40)
c p ∂p T,ωi dx
∂h
In order to evaluate JHP , the partial derivative ∂p
is evaluated numerically usT,ωi
ing the routines of Wexler et al. [184]. JHP contributes negligibly to overall energy
transport compared with JHT , as this includes convection, which is dominant.
The enthalpy transport term JHT can be written in convection-diffusion form:
( E) dh
JHT = ρvh − Γ H
A
(3.3.41)
dx
( E)
ΓH =
k
cp
(3.3.42)
77
Multiplying (3.3.7) by hP + (1/2)(vP )2 and subtracting it from (3.3.26) gives
n 2 o 
V
ρ h + 12 v2 − ρ0 h0 + 21 v0
 +

(3.3.43)
∆t
P
"
"
#
#
0
0
p−p V
ρP − ρP V
=
−
hP + (1/2)v2P
−
∆t
∆t
P
n
2 o 0

0
h − h0 + 12 v2 − v0
ρP V
p−p V

 =
−
∆t
∆t
P
= ( JHT,w − Fw hP ) − ( JHT,e − Fe hP ) +
1
1
+ Ju,w − Fw v2P
+ Ju,e − Fe v2P
2
2
+ ( JHω,w − JHω,e ) + ( JHP,w − JHP,e )
(m)
m
2
+ QP + QP − ( m m
a + mv ) hP + (1/2)( vP )
(3.3.44)
Applying Patankar’s convection-diffusion recipe to Equation (3.3.44) yields the following equation in the neighbouring enthalpies and velocities:
n
2 o 0

ρP V
h − h0 + 12 v2 − v0
p − p0 V


−
∆t
∆t
P
( E)
( E)
= a HW (hW − hP ) − a HE (hP − hE ) +
1 2
1 2
1 2
1 2
( E)
( E)
+ avW
+ avE
vW −
vP
vP −
vE
2
2
2
2
+ ( JHω,w − JHω,e ) + ( JHP,w − JHP,e )
(m)
m
2
+ QP + QP − ( m m
a + mv ) hP + (1/2)( vP )
(3.3.45)
with
( E) ( E)
( E)
a HW = D Hw A( Pw ) + [[ Fw , 0 ]]
( E) ( E)
( E)
a HE = D He A( Pe ) + [[− Fe , 0 ]]
( E) ( E)
( E)
avW = Dvw A( Pw ) + [[ Fw , 0 ]]
( E) ( E)
( E)
avE = Dve A( Pe ) + [[− Fe , 0 ]]
ΓH A
δx
( E)
( E)
DH
F
=
PH =
Pv
(3.3.48)
(3.3.49)
(3.3.50)
( E)
Γv A
δx
F
( E)
(3.3.47)
( E)
( E)
DH =
Dv
(3.3.46)
=
( E)
( E)
Dv
(3.3.51)
(3.3.52)
(3.3.53)
78
Equations (3.3.45) – (3.3.53) are equivalent to the following energy discretisation:
( E)
( E)
( E)
( E)
a HP hP = a HW hW + a HE hE + b HP
( E)
a HP =
( E)
a HW
( E)
a HE
( E)
avW
( E)
avE
( E)
b HP
(3.3.54)
ρ0P V
( E)
( E)
m
+ a HW + a HE + mm
a + mv
δt
( E) ( E)
= D Hw A( PHw ) + [[ Fw , 0 ]]
( E) ( E)
= D He A( PHe ) + [[− Fe , 0 ]]
( E) ( E)
= Dvw A( Pvw ) + [[ Fw , 0 ]]
( E) ( E)
= Dve A( Pve ) + [[− Fe , 0 ]]
pP − p0P V
ρ0P V
1 0 2 1 2
0
=
( v ) − vP
hP +
+
δt
2 P
2
∆t
1
1
1
1 2
( E)
( E)
2
2
2
+ avW
v
− vP
+ avE
v −
v
2 W
2
2 P
2 E
+ ( JHω,w − JHω,e ) + ( JHP,w − JHP,e )
1 2
(m)
m
m
+ QP + QP − ( m a + m v ) vP .
2
Momentum balance
As Fig. 3.2 shows, the velocity is specified on the boundaries between the control volumes for the other variables. The ith velocity is specified at the right-hand interface of
the jth ordinary control volume (where i = j but the different indexing letter is used
to highlight which variable-type is being referred to). The ith velocity control volume
likewise has the jth pressure specified at its left-hand face. The boundary conditions
that are imposed on the momentum balance are the inlet air velocity (vin ) and the
outlet pressure (p0 ). Since density ρ is specified as ρ j (or ρP ) at the centre of the ordinary control volumes (equivalently, on the faces between velocity control volumes),
an interpolation must be applied to determine ρ at the velocity control volumes. This
interpolated density, evaluated at the ith velocity control volume, is denoted by ρi (or
ρw ). An analogous remark holds for variables required at the interfaces which are
specified at the control volume centres, and also for the velocity when it is used (for
instance) to estimate the transfer coefficients at the main grid points.
We can obtain the following discretised form for the momentum balance (3.2.34):
"
#
ρv − ρ0 v0 V
= J p,P − J p,E − Ff,e + ( pP − pE ) A
(3.3.55)
∆t
e
79
where
∂v
J p = ρvv − Γ( p)
A
∂x
(3.3.56)
Γ( p) = 2µ
(3.3.57)
in which
The term Ff,e in (3.3.55) is given by
Ff,e = 2∆xe ∆z
1 2
ρv
2
e
C •f ,e
(3.3.58)
where C •f = 2jloc θv , and the expression for θv is given on page 249. This term contains
the magnitude of the friction force experienced by the moving gas in the i + 1th control
volume; its (negative) orientation is represented by the sign with which it occurs in
the equation.
The mass balance equation for the velocity control volume e is obtained by summing (3.2.33) over k = a, v. The resulting equation can be discretised to
"
#
ρ − ρ0 V
(m)
(m)
= FP − FE + m a, e + mv, e
∆t
(3.3.59)
e
with F = ρvA the convective mass flow at the entrance and exit surfaces of the control
volume. The subscript ‘e’ on the mass source terms indicates that this term must be
obtained by interpolation of the mass source terms defined on the ordinary control
volume faces. Multiplying (3.3.59) by ve and subtracting it from (3.3.55) gives
"
#
v − v0 ρ0 V
= Jp,P − FP ve − Jp,E − FE ve
∆t
e
(m)
(m)
− Ff,e − m a, e + mv, e ve + ( pP − pE ) A
(3.3.60)
Once again using Patankar’s convection-diffusion method, this becomes (on linearisation of the source term, [−Ff,e ] = SC,e + Se ve ):
#
"
v − v0 ρ0 V
( p)
( p)
= aw ( vw − ve ) − af ( ve − vf ) +
∆t
e
( p)
( p)
(m)
(m)
+ SC,e + Se ve − m a + mv
ve
+ ( pP − pE ) A
(3.3.61)
with
( p) ( p)
( p)
aw = DP A( PP ) + [[ FP , 0 ]]
( p) ( p)
( p)
af = DE A( PE ) + [[− FE , 0 ]]
Γ( p) A
δx
F
= ( p) .
D
(3.3.62)
(3.3.63)
D ( p) =
(3.3.64)
P( p)
(3.3.65)
80
Using ‘∗’ to represent the estimate obtained on the last estimate of φ, the general
source term linearisation, when applied to the velocity grid point e, is given by:
dS ∗
∗
(3.3.66)
S=S +
(φe − φe∗ )
dφ
dS ∗
dS ∗
= S∗ − φe∗
+
φe
(3.3.67)
dφ
dφ
= SC + Se φe
(3.3.68)
The friction force experienced by the gas flowing in control volume e is
1 2
(−Ff,e ) = −2je θv,e
ρv
(2∆xe ∆z)
2
e
(3.3.69)
where one factor 2 is due to the existence of two surfaces (Sm and Sn ) that are involved
in the fluid flow resistance. We seek the source term linearisation given by
S = SC,e + Se ve
(3.3.70)
SC,e = (−Ff,e )∗ − ve∗
d(−Ff,e ) ∗
Se =
dv
d(−Ff,e )
dv
∗
(3.3.71)
(3.3.72)
The goal of source-term linearisation is to obtain convergence; stability requires that
Se < 0. It would be possible to obtain explicit expressions for SC and Se for any
particular expression for the friction factor C •f = 2jloc θv , such as that given by DittusBoelter equation (A.7.30). However to keep the approach general it was decided to
use numerical differentiation within MATLAB to obtain the source term linearisation.
Rearrangement of (3.3.61) leads to the following discretisation equation:
( p)
( p)
( p)
( p)
ae ve = aw vw + af vf + be
with
( p)
ae
=
+ ( pP − pE ) A
ρ0e V
( p)
( p)
( p)
m
+ aw + af + m m
a + m v − Se
δt
and
( p)
be
( p)
= SC,e +
ρ0e v0 V
.
δt
(3.3.73)
(3.3.74)
(3.3.75)
Implementing SIMPLER
The Semi-Implicit Method for Pressure-Linked Equations, Revised (SIMPLER) algorithm recommended by Patankar [117, p.133] and described by Versteeg and Malalasekera [177, p.192] proceeds by way of a series of estimated flow fields, each of which
satisfies the law of mass conservation. The procedure is as follows.
• Start with the previous estimate of the velocity field, v∗ .
81
• Use this velocity field to obtain an estimate, p∗ , of the pressure field by use of
the discretised mass balance equation.
• Use the estimated pressure field to obtain a new estimate v∗ of the velocity field.
• Correct v∗ to obtain an improved estimate of the velocity field, which satisfies
mass conservation, by way of a ‘pressure correction’ p0 .
The procedure is repeated until adequate convergence has been obtained, periodically recalculating fields such as T and ρ. The procedure is designed to lead at each
step to a better estimate of the velocity field (i.e. is designed to be a contraction mapping, whose fixed point itself would converge on the actual velocity field as the maximum grid spacing became small and the number of control volumes became large).
The procedure is applied in this particular case as follows. (3.3.73) can be rewritten
ve = v̂e + de ( pP − pE )
(3.3.76)
with the ‘pseudo-velocity’ v̂e at location e given by
1 v̂e ≡
( p)
ae
( p)
( p)
( p)
aw vw + af vf + be
(3.3.77)
and with de given by
de ≡
A
( p)
.
(3.3.78)
ae
Substituting (3.3.76) into the discretised mass balance (3.3.7) yields the following
discretisation equation for the estimated pressure field p∗ :
( p∗ ) ∗
pP
aP
( p∗ )
aW
( p∗ )
aE
( p∗ )
aP
( p∗ )
bP
( p∗ )
( p∗ ) ∗
pE
∗
+ aE
= aW pW
( p∗ )
+ bP
(3.3.79)
( p)
= dw ρw A
( p)
= de ρe A
( p∗ )
( p∗ )
= aW + aE
ρ0P − ρP V
(m)
(m)
=
+ (ρw v̂w − ρe v̂e ) A + m a + mv .
∆t
The boundary conditions that are applied to the pressure equation (3.3.79) are as
follows. At the inlet boundary CV interface the velocity and density are specified as
ρin and vin , while beyond the outlet boundary the pressure is set to the ambient environmental pressure, 1.01325 × 105 Pa. In calculating the pressure of the first control
82
volume, the coefficients therefore become
( p∗ )
aW
( p∗ )
aE
( p∗ )
aP
( p∗ )
bP
=0
( p)
= de ρe A
( p∗ )
( p∗ )
= aW + aE
ρ0P − ρP V
(m)
(m)
=
+ (ρin vin − ρe v̂e ) A + m a + mv ,
∆t
( p∗ )
while to the source term bP
( p∗ )
aE .
for the final control volume, we add the quantity p0 ×
( p∗ )
In the notation of (3.3.24), we have bc(1) = ρin vin A and bc( Nx ) = p0 × aE( N ) .
x
We now possess the old estimate of the discretised flow field v∗ , together with
the discretised pressure field p∗ that we obtained from this flow field estimate. The
SIMPLER algorithm uses the estimated pressure field p∗ to update the flow field v∗
using the discretisation equation (3.3.73). Having thus obtained an updated estimate
v∗ of the discretised velocity field, a ‘pressure correction’ p0 is estimated and used
to correct the new velocity estimate. More specifically, the difference between the
converged solutions v and p that we seek and the starred estimates are called the
‘velocity correction’ and the ‘pressure correction’, v0 and p0 respectively:
v = v∗ + v0
(3.3.80)
p = p∗ + p0 .
From the momentum discretisation equation (3.3.73), given the discretisation coef( p)
ficents ai , the following equations hold for the real and estimated velocity fields:
( p)
( p)
( p)
( p)
ae ve = aw vw + af vf + be
( p)
ae ve∗
=
( p) ∗
aw vw
( p)
+ af vf∗
( p)
+ be
+ ( pP − pE ) A
+ pP∗ − pE∗ A
(3.3.81)
It follows that the following equation holds for the velocity and pressure corrections:
( p)
( p) 0
( p)
+ af vf0 + pP0 − pE0 A
ae ve0 = aw vw
(3.3.82)
which leads on resubstitution into the first part of (3.3.80) to
ve = ve∗ + ∆e + de pP0 − pE0 ,
(3.3.83)
with de defined by (3.3.78) and with
∆e =
1 ( p)
ae
( p) 0
( p)
aw vw
+ af vf0 .
(3.3.84)
Our purpose is to obtain an increasingly accurate estimate of the velocity field. The
SIMPLER algorithm proceeds by dropping the term ∆e , which contains contributions
from adjacent velocity corrections, thus introducing dependencies that we are not
equipped to cope with. This does not affect the final converged solution since as v∗
83
approaches v, both v0 and thus ∆ will necessarily approach zero [117, p.127]. The
corrected velocity field estimate is thus given by the ‘velocity correction equation’:
ve = ve∗ + de pP0 − pE0 .
(3.3.85)
In order to determine the discretised pressure correction field, (3.3.85) is substituted into the discretised mass conservation equation (3.3.7). This is similar to the
previous substitution of (3.3.76) into (3.3.7), and leads to a similar result for p0 :
( p0 )
( p0 )
( p0 )
( p0 )
0
aP pP0 = aW pW
+ aE pE0 + bP
( p0 )
( p)
( p0 )
aE
( p0 )
aP
( p)
de ρe
(3.3.86)
aW = dw ρw A
( p0 )
bP
=
(3.3.87)
A
( p0 )
(3.3.88)
( p0 )
= aW + aE
ρ0P − ρP V
(m)
(m)
∗
+ ( ρw vw
− ρe ve∗ ) A + m a + mv
=
∆t
(3.3.89)
(3.3.90)
Once again, at the inlet boundary the velocity and density are specified as ρin and
vin , while beyond the outlet boundary the pressure correction is set to zero. In calculating the pressure of the first control volume, the coefficients therefore become
( p0 )
aW = 0
( p0 )
aE
( p0 )
aP
( p0 )
bP
( p)
= de ρe A
( p0 )
( p0 )
= aW + aE
ρ0P − ρP V
(m)
(m)
=
+ (ρin vin − ρe ve∗ ) A + m a + mv ,
∆t
( p0 )
while to the source term bP
( p0 )
aE ,
for the final control volume, we add the quantity 0 ×
i.e. 0. In the notation of (3.3.24), we have bc(1) = ρin vin A and bc( Nx ) = 0.
The estimated pressure correction thus obtained is substituted back into the velocity correction equation to determine the corrected discretised velocity estimate:
ve = ve∗ + de pP0 − pE0 .
(3.3.91)
( p0 )
Convergence of SIMPLER occurs as the mass source term bP
of (3.3.90) becomes
small. As a consequence the pressure correction p0 also becomes small everywhere.
Unsteady vs. steady-state description
Like the three equations (3.2.33)–(3.2.35) from which they were derived, the discretised equations that were developed in the previous section describe the unsteady
84
case. This level of description turns out to be more than is necessary for HPD modelling. Even in unsteady batch operation, HPD systems proceed through a series
of quasi-steady states, which are determined by the (current) drying kinetics of the
product evolving over a relatively long timescale. (This became apparent when the
detailed air-flow model of this chapter came to be integrated into the steady-state
HPD model [28].) In general in the finite volume method described by Patankar [117],
the discretised equation for a steady state situation can be obtained from its unsteady
counterpart by considering the limit as the time step ∆t → ∞ [117]. The equations
that appear in this chapter can thus be simplified to describe the steady-state case as
follows. Set the time derivatives on the left-hand sides of equations (3.2.33)–(3.2.35)
to zero; delete each term in the discretised equations (below) containing ∆t in its denominator. The plots in the results section of this chapter have been produced using
a large timestep ∆t to eliminate transient behaviour. A steady-state version of the
discretised equation set presented in this chapter is used in subsequent chapters.
3.4.
Refrigerant heat transfer coefficient
The simple refrigerant flow model developed in this section is used to estimate the average refrigerant heat transfer coefficient αr,eff , for use in determining the local product temperature, by way of the conductive model described in Chapter 1. Our purpose is not to establish a detailed model of the refrigerant flow within the plate, but
to obtain an estimate for the mean plate refrigerant heat transfer coefficient. In developing the refrigerant flow model we encounter the conflict, discussed in Chapter 1,
that arises between the need to build a concrete model and a desire to avoid undue
specificity. The details of the refrigerant flow through the heating plates is a specific
design question which it would not seem appropriate to try to answer at this stage,
since factors that are currently unknown will likely arise in the design of such a system. On the other hand, it is clearly useful at this stage to gain an indicative picture
of the impact of refrigerant-side losses. It has thus seemed appropriate to perform
a preliminary analysis of the situation, employing the simple refrigerant-flow layout
illustrated schematically in Fig. 3.3. As depicted in the diagram, several parallel circuits of refrigerant flow pass through each plate, each running back and forth through
the length of the plate multiple times. Note that this orientation of refrigerant flow
through the heating plate is at right angles to the flow direction assumed in Chapter 2.
This configuration has the effect of averaging the wall temperature experienced near
a given location z along each refrigerant circuit, and also of averaging the heat transfer coefficient associated with a given location x on the plate. This allows us to assume an averaged wall temperature in the enthalpy balance at all locations along the
refrigerant flow, and then to calculate an effective (average) heat transfer coefficient
85
which applies at all locations on the plate. Any pressure drop within the condenser
plate is unlikely to lead to significant variation in the saturation temperature, since,
for example, for R134a at a saturation temperature of 55◦ C, even a pressure drop of
100 kPa would lead to a saturation temperature drop of under 3◦ C. We thus assume
a constant refrigerant saturation temperature throughout the plate, which we evalutate using the pressure at the condenser outlet. Using an average value αr,eff and a
constant saturation temperature is a standard approach in HPD modelling – see (for
instance) the modelling work of Fischer and Rice [67, p.51].
2’
2
CD1
LR
CD2
3
T
4
FD
x
z
Figure 3.3: Conceptual schematic of multi-pass condensing plate with three
loops (view from above). FD=refrigerant flow distributor. LR=liquid receiver.
Directions along the dimensions x and z are shown.
Traviss et al. [168] provided an early analysis of the condensation of refrigerant
within horizontal refrigerant condenser tubes – a complex heat and mass transfer
problem. Under high-vapour-quality, low-mass-flow conditions, condensed refrigerant runs to the bottom of the tube, and heat transfer occurs mainly across the thin
liquid film at the top and sides of the tube, since the largest thermal resistance through
the tube walls is in thermal conduction through the liquid refrigerant. Under lowervapour-quality, low-flow conditions the flow may contain waves of liquid refrigerant
spanning the entire tube cross-section (slug flow) or consist mainly of liquid enclosing bubbles of vapour (plug flow). With higher mass flow-rates the liquid takes the
form of a film, distributed approximately symmetrically on the inside of the tube
walls (annular flow). Using the analogy between momentum and heat transfer, and
assuming that the liquid film on the pipe inner wall obeys the von Karman universal
velocity distribution, Traviss et al. [168] obtained general correlations for heat transfer
86
and for the refrigerant pressure drop in the condenser in the annular-flow case. Cavallini et al. [39] have gathered several such correlations, developed by them and other
authors, to estimate the heat transfer coefficients under the various flow regimes that
may occur, compiled into a single general function. More recently, Cavallini et al. [40]
have consolidated the resulting equations to produce a simple model that predicts
heat transfer coefficients robustly across the range of refrigerants and flow regimes.
We employ the simple heat transfer model of Cavallini et al. [40]. In what follows, the
variable z will be used to represent the distance along a given refrigerant flow path
through the plate. We use lb to denote the length of each circuit, so that 0 < z < lb
for points along the refrigerant flow in each circuit. Assuming an effectively constant
temperature difference between the refrigerant and the pipe inner walls along the
length of refrigerant flow, the effective heat transfer coefficient between the refrigerant and the tube walls is the mean along each circuit [67, p.51]:
αr,eff
1
=
lb
Zlb
α(z)dz
(3.4.1)
0
z=0
4
3
5
12
2
6
11
1
7
10
8
9
z = lb
12 11 10 9 8 7 6 5 4 3 2 1
z=0
z = lb
Figure 3.4: Refrigerant flow subdivisions. Here n x = 4, p = 3, and nz = 12.
We use nb to represent the number of refrigerant flow circuits passing through
a single plate (e.g. in Fig. 3.3, nb = 3). If each circuit passes through the plate p
times (e.g. in Fig. 3.3, p = 5), then the total number Nb of refrigerant tubes in the
plate is given by Nb = nb p. The number of control volumes along the direction x
of air flow is denoted n x . Fig. 3.4 shows the adopted layout of gridpoint locations
zi distributed along a single circuit of refrigerant flow. The circuit is divided into nz
(here 12) gridpoints. The direction of increasing gridpoint index (i = 1, . . . , 12) is the
same as the direction of refrigerant flow. The refrigerant flow ṁr in each circuit is
ṁr =
ṁr,tot
Nd nb
(3.4.2)
87
In (3.4.2), ṁr,tot is the whole-system refrigerant mass flow rate, Nd is the number of
heating plates, and nb is the number of refrigerant flow circuits in each plate.
Q̇
z + ∆z
z
ṁr
D
∆z
Figure 3.5: Element of refrigerant tube.
For the purposes of establishing the variation of the refrigerant quality along the
flow direction, we neglect any small variation in the saturation temperature. Thus,
since the wall temperature is assumed effectively constant, the temperature difference
∆T = Tr − Tw between the refrigerant and the inside tube walls is constant. Fig. 3.5
depicts a small refrigerant flow control volume within the heating plate, between
locations z and z + ∆z. The steady-state mass and energy balances are:
ṁr (z) − ṁr (z + ∆z) =0
(ṁr hr )|z − (ṁr hr )|z+∆z = Q̇
(3.4.3)
(3.4.4)
By the definition of the heat transfer coefficient α, we have
Q̇ = (πD∆z) · ∆T · α
(3.4.5)
Using (3.4.3) and (3.4.5) in (3.4.4), we obtain in the limit as ∆z → 0
lim
z →0
hr (z) − hr (z + ∆z)
dh
πD∆T
=−
=
α
∆z
dz
ṁr
(3.4.6)
The heat transfer coefficient α in (3.4.6) is a function of the local quality of the saturated two-phase refrigerant, which is a function of its local enthalpy and of the pressure. Thus we can write (3.4.6) as follows:
dh
πD∆T
=−
α(h)
dz
ṁr
(3.4.7)
Equation (3.4.7) can be used in (3.4.1) to obtain:
αr,eff
1
=
lb
Zlb
0
1
α(z)dz =
lb
hZout
hin
α z(h)
dz
ṁr ∆h
dh = −
dh
πD∆Tlb
(3.4.8)
We use the Runge-Kutta method to integrate the initial value problem given by
(3.4.7) together with the initial condition h(z = 0) = h0 .
88
The classical (fourth-order) Runge-Kutta method integrates the equation
dh
= f (z, h),
dz
h ( z0 ) = h0
(3.4.9)
using
1
hn+1 =hn + ∆z(k1 + 2k2 + 2k3 + k4 )
6
(3.4.10a)
zn+1 =zn + ∆z
(3.4.10b)
where [134, p.970]:
k1 = f (zn , hn )
(3.4.11a)
1
k2 = f (zn + ∆z, hn +
2
1
k3 = f (zn + ∆z, hn +
2
1
∆zk1 )
2
1
∆zk2 )
2
k4 = f (zn + ∆z, hn + ∆zk3 )
(3.4.11b)
(3.4.11c)
(3.4.11d)
Since the derivative that appears in (3.4.7) does not explicitly depend on z, the ks in
(3.4.11) can be simplified (to Simpon’s rule). Using (3.4.7), we obtain the following:
πD∆T
α( hn )
ṁr
πD∆T
1
k2 = −
α(hn + ∆zk1 )
2
ṁr
πD∆T
1
k3 = −
α(hn + ∆zk2 )
ṁr
2
πD∆T
k4 = −
α(hn + ∆zk3 )
ṁr
k1 = −
(3.4.12a)
(3.4.12b)
(3.4.12c)
(3.4.12d)
The local heat transfer coefficient can be calculated as a function of the specific
enthalpy as follows. The enthalpy and pressure are used to establish the vapour mass
quality x and refrigerant physical properties (see the appendix). In the single-phase
regions (both desuperheating and subcooling), the heat transfer coefficient is given
by the following equations used by Jolly et al. [88]:
G = 4ṁr /(πD2 )
(3.4.13)
α = 0.023(λ/D )Pr0.4 Re0.8
(3.4.14)
In the the two-phase region the heat transfer coefficient is evaluated using the method
of Cavallini et al. [40], summarized in the following equations:
χtt =
µl
µv
0.1 1−x
x
0.9 ρv
ρl
Relo = GD/µl
αlo = 0.023
λl 0.4 0.8
Pr Relo
D l
0.5
(3.4.15)
(3.4.16)
(3.4.17)
89
h
1 − x 0.3321 i−1 h λ3 ρ (ρ − ρ ) gh i0.25
v
lv
l l l
+ (1 − x0.087 )αlo
αstrat = 0.725 1 + 0.741
x
µl D∆T
(3.4.18)
Jv = xG/[ gDρv (ρl − ρv )]0.5
(3.4.19)
The transition value of the dimensionless vapour velocity Jv is estimated using the
following empirical correlation, where for HFCs such as R134a, CT = 2.6:
JvT =
h
7.5/(4.3χ1.111
+ 1)
tt
i −3
+ CT−3
−1/3
(3.4.20)
When the dimensionless vapour velocity Jv > JvT (∆T-independent flow):
h
ρ 0.3685 µ 0.2363 µ − µ 2.144 µ c −0.100 i
l p,l
v
l
l
α A = αlo 1 + 1.128x0.8170 l
ρv
µv
µl
λl
(3.4.21)
When the dimensionless vapour velocity Jv ≤ JvT (∆T-dependent flow):
h
i
α D = α A ( JvT /Jv )0.8 − αstrat ( Jv /JvT ) + αstrat
(3.4.22)
Equations (3.4.21) and (3.4.22) give the heat transfer coefficient, in the cases of ∆Tindependent and ∆T-dependent flow, respectively. The file that implements the above
functions has been tested by generating Fig. 3.6, which agrees with Fig. 11 of Cavallini
et al. [40], showing the relationship between refrigerant quality and heat transfer coefficient at different mass flow rates, in an 8mm internal-diameter tube.
3.5.
Results and discussion
This section discusses results obtained from the flexible air flow model, when the
model is used to describe steady-state drying behaviour in the constant drying-rate
period. The model takes as input parameters the drier dimensions, the air inlet conditions, the saturated condensing temperature of the refrigerant, and the pressure
at the drier air outlet. For each control volume, the simple heat conduction model
equation (1.5.5) is solved numerically to estimate the product surface temperature.
The adiabatic mode can be modelled straightforwardly, by simply setting the plate
thermal conductivity effectively to zero, and thus eliminating plate heat transfer. The
dry-air mass flow rate will be near zero at the surface of a porous product undergoing
drying, since dry air flows into the product to fill the volume previously occupied by
liquid water, whose density is roughly 103 times greater than that of dry air. Thus
the mass flux of water vapour at the product surface can be expected to be O(103 )
times larger than that of dry air. The mass flux of dry air at the product surface has
therefore been set to zero, and the total vapour mass flux (convective plus diffusive)
at the product surface is evaluated from (1.4.11) through iteration of SIMPLER. A
90
9000
R134a
Ts=40°C
D=8mm
∆T=4°C
−2
7000 1: G=800 kg m s−1
2: G=400 kg m−2s−1
3: G=200 kg m−2s−1
6000 4: G=100 kg m−2s−1
5: G=50 kg m−2s−1
Heat transfer coefficient [W m−2 K−1]
8000
1
5000
4000
2
3000
3
4
2000
5
1000
0
0
0.2
0.4
0.6
Vapour quality
0.8
1
Figure 3.6: R134a heat transfer coefficients, by method of Cavallini et al. [40].
maximum value of 10−4 in any control volume for the relative source magnitude (relative to the quantity in the given control volume, as defined by Patankar [117]) of the
three conserved quantities has been adopted as an acceptable balance of convenience
and accuracy. Thus, convergence of the model implies that the discretised mass, momentum and energy conservation equations have been satisfied to within this convergence criterion. The plots in this chapter have been produced using Nx = 50 control
volumes. No significant change occurs when Nx is increased above this number. To
allow meaningful comparison with the model of Chapter 1, the plots discussed in
this section have been produced under the assumption of an adiabatic duct ceiling
Sn (see Fig. 3.1). That is, QP has been evaluated using only the first term of (3.3.30).
Convergence of the detailed model takes on the order of one minute elapsed time.
For the purpose of comparison with the simple model of Chapter 1, Fig. 3.7 has
been produced representing the flow situation that was previously modelled using
the simple model (Fig. 1.10 on page 26). Once again, the modelled situation involves
the kiln geometry specified in Table 2.1 (page 43), with inlet air at 55◦ C and 30% relative humidity, and with a dry-air mass flow rate of 1 kg s−1 . The refrigerant saturated
condensing temperature has been set 5◦ C greater than that of the inlet air. The most
striking feature when Figs. 1.10 and 3.7 are compared is their broad similarity. In
particular, agreement in the case of the adiabatic mode is very good. As expected
91
120
Tin=55°C
1
2
3
80
Temperature
Humidity ratio ω, g moisture/kg dry air
100
4
60
Twb,in=36°C
5
40
6
20
Temperatures
Humidity ratio ω
0
0
1
2
3
4
Position x along airflow direction, m
5
Figure 3.7: Humidity and temperature in isothermal (ISO) and adiabatic (ADI)
driers. The scales are identical to Fig. 1.10. 1: Air temperature (ISO). 2: Product
surface temperature (ISO). 3: Air temperature (ADI). 4: Humidity ratio (ISO).
5: Product surface temperature (ADI). 6: Humidity ratio (ADI).
the inlet conditions, at x = 0, are identical for both models. However the detailed
model shows a dip in the air temperature (line 1) due to a relatively low product surface temperature at the inlet (line 2). This effect is unsurprising: evaporative cooling
can be expected to be greatest near the air inlet, where the airstream is least humid.
The product surface temperature increases with position in the drier, and between
3 and 4 metres into the drier the surface temperature can be seen to exceed the air
temperature. Beyond this location the bulk air temperature increases with position.
An interesting effect visible in Fig. 3.7 is a flattening of the humidity ratio curve (line
4) compared with its counterpart in Fig. 1.10. Since the gradient of the humidity ratio
is proportional to the drying rate, this implies that the drying rate is more uniform
throughout the drier than predicted by the simple model of Chapter 1. This effect
can be understood as follows. The driving force for evaporation in the isothermal
mode is approximately proportional to the vertical separation between lines 2 and 4
in Fig. 3.7. This is because the figure is scaled in such a way that line 2, which repre-
92
sents the product surface temperature, also provides a first-approximation measure
of the product surface vapour density (see the discussion on page 27). The positive
gradient of the product surface temperature (line 2) thus has the consequence of reducing the variation in the driving force for drying along the length of the drier, and
this is reflected in the reduced curvature of line 4.
It should be emphasized that Figs. 1.10 and 3.7 result from local heat, mass and
momentum transfer coefficients that have been evaluated using a different correlation
from those employed by Sun [155], Sun et al. [160] and by Vaughan et al. [176] for the
somewhat similar (timber) drying situation that they modelled. The Dittus-Boelter
correlation (page 246) used for Figs. 1.10 and 3.7 applies to fully-developed duct flow.
The central section of a wood stack forms a set of ducts between layers of rows of
adjacent boards, but small (≈ 1mm) gaps exist between adjacent boards along the
flow direction, due to board imperfections and shrinkage, and these gaps tend to
disrupt the boundary layer, enhancing transfer [98]. The correlations that have been
used in the works listed above described local variations in the transfer coefficients
due both to development of the boundary layer in the duct between the wood boards
and to boundary layer detachment and subsequent re-attachment that occur as the
air flow encounters the trucated slabs presented by the boards [see 148, 139]. These
effects result in a net enhancement of the transfer processeses near the leading edge
of each board [139], with local maxima both at the leading edge and at a location
typically one or two cm from the leading edge [155, 154], as depicted in Fig. 3.8.
While it would be straightforward to include correlations to deal with boundarylayer separation and reattachment at the drier entry region in the present model, this
level of detail does not appear sensible here. The corrections relating to boundary
layer detachment are specific to the details of the inlet geometry; the detailed design
of an isothermal contact HPD would be application-specific, and would likely require
multidisciplinary input from process engineers, perhaps materials handling experts,
and so on. The present work should therefore remain generic, rather than attempting
to describe any particular system. On the other hand, development of the boundary
layer within a duct typically requires an entry length of 20-30 hydraulic diameters
[139, 135], on the order of 1m for the duct geometry of interest here, and the transfer
coefficients are typically significantly greater in the entry region than they are in the
fully-developed flow region. Fig. 3.9 shows j-factors (defined on page 244) obtained
from four different correlations: (a) from the Dittus-Boelter equation for duct flow;
(b) from the following correlation for turbulent flow past a flat plate [155]:
0.2
jloc = 0.0288Re−
x
(3.5.1)
where Rex is the local Reynolds number defined by location x; (c) from the simple
timber-stack transfer correlation used by Sun et al. [160]; and (d) from correlations
93
e
b
d
jD
a
x
c
b
Figure 3.8: Boundary layer for flow past a truncated slab. (a) leading edge. (b)
eddy. (c) reattachment point. (d) viscous sublayer. (e) turbulent region. The
curve for jD represents the correlations obtained by Sun [155].
obtained from theoretical considerations by Sun [155] for boundary layer detachment
and reattachment. Curve (d) has been produced by assuming a slab thickness of
0.025m. The correlation (c) is a multiple η of Equation (3.5.1), where the enhancement
factor (η = 1.54) was selected [160] to match mass transfer rates previously measured
in a wood stack by Kho [98]. The correlation (d) is defined using the following:
α0 = 9.0408 × 10−5 ReD 0.6017
(3.5.2a)
β 0 = 2.1443 × 10−4 ReD 0.6787
(3.5.2b)
α = 2.1513 × 10−6 ReD 1.1106 ReS 0.6501
β = 0.1542ReD 0.1410 ReS 0.0437
γ = 8.5197 × 10
−5
ReD
1.1172
ReS
(3.5.2c)
(3.5.2d)
0.5240
− β0
j1 = α0 Rex
(3.5.2e)
(3.5.2f)
−β
−β
(−0.2+γRex )
j2 = (0.0288 − αRex )Rex
(3.5.2g)
In (3.5.2) ReD is the Reynolds number defined by slab thickness, ReS is the Reynolds
number defined by the duct height, and Rex is the Reynolds number defined by position x. The correlation proposed by Sun [155] sets jloc equal to j1 from x = 0 up to the
smallest x such that j2 = j1 , and sets jloc equal to j2 beyond this point. As x → ∞, j2
tends toward the value obtained from (3.5.1). The j-factor correlations (b), (c) and (d)
can be seen to tend toward zero as x increases, reflecting the boundary layer thick-
94
ness for external flow over an isolated surface, which increases without bound. Salin
[139] has observed that in internal duct flow the limiting j-factor value should in fact
be that given by the Dittus-Boelter equation, which suggests that the curves (b)-(d)
underestimate the j-factors for large x. The correlation (c) used by Sun et al. [160],
with η selected to match average mass-transfer coefficients measured within a wood
stack, produces an average j-factor, through the 5m stack, that is close to that given
by the Dittus-Boelter equation. Salin calculates that small gaps between boards increase the transfer factors by about 6%. We therefore adjust η down 6% to 1.45, and
0.2
use η × 0.0288Re−
to estimate the local j-factors in subsequent work. Fig. 3.10 illusx
trates the impact that this new estimate for the transfer coefficents has on the model.
Compared with Fig. 3.7, heat and mass transfer is more intense at low values of x, and
weaker at larger values of x. In order to allow ongoing comparison with the simple
model we will use the Dittus-Boelter equation for the remainder of this chapter, but
in later chapters the modified version of Equation (3.5.1) is used.
Fig. 3.11 illustrates the impacts of several additional physical effects, not in the
simple model, that have been incorporated into the flexible drying model. One of
these effects is the modification of the transfer coefficients due to boundary layer distortion as a result of high mass transfer rates, which is developed in the theoretical
appendix on page 247. This effect is quantified by the enhancement factors θv , θ AB
and θt , which express the ratio of the high-mass-transfer to the low-mass-transfer
momentum-transfer, mass-transfer and heat-transfer coefficients, respectively. The
enhancement factors are found to be all everywhere less than 1, as shown in the first
subfigure, consistent with the direction of mass transfer, into the flow stream [16,
p.663]. In the drying scenario that we are considering here, which is reasonably intense, due to its low inlet relative humidity, the heat transfer coefficient is reduced
by as much as 6% by boundary layer distortion due to mass transfer. The second
subfigure in Fig. 3.11 shows the gas temperature, the product surface temperature,
and the estimated film temperature, which by definition is the average of the other
two temperatures [16] (see page 244). For comparison, these are all assumed everywhere identical to the air inlet temperature in the simple model. The final subfigure
of Fig. 3.11 shows the impact of convection on the total vapour mass flux at the product surface, by way of the ratio nv /jv of total local vapour flux to diffusion vapour
flux at the product surface. We are considering the constant drying-rate period, so
the product surface is saturated and the vapour mass fraction is a function only of
temperature. As the product surface temperature increases (with location, along the
airflow direction) the vapour mass fraction also increases. The barycentric mass flux,
which is the resultant of the vapour and dry air mass fluxes, thus also increases along
the airflow direction, and so does the evaporation enhancement due to convection.
The moisture extraction rates and air outlet temperatures estimated by the detailed
95
0.01
jD
(a)
(b)
(c)
(d)
0.005
0
0
1
2
3
Position x along flow direction, m
4
5
0.02
jD
(a)
(b)
(c)
(d)
0.01
0
0
0.02
0.04
0.06
Position x along flow direction, m
0.08
0.1
Figure 3.9: Variation of transfer j-factors evaluated using four different cor1/5 1/15
relations. (a) jloc = 0.023ReH −1/5 Pr1/15 . (b) jloc = 0.0288Re−
Pr
. (c)
x
1/5 1/15
Pr
, η = 1.54. (d) correlation proposed by Sun [155]
jloc = η × 0.0288Re−
x
as defined in the text. Top plot: whole duct. Bottom plot: close-up of first ten
centimetres of duct.
96
120
Tin=55°C
1
3
80
Temperature
Humidity ratio ω, g moisture/kg dry air
2
100
60
4
5
40
Twb,in=36°C
6
20
Temperatures
Humidity ratio ω
0
0
1
2
3
4
Position x along airflow direction, m
5
Figure 3.10: Humidity and temperature in isothermal and adiabatic driers,
for which the local transfer j-factors are evaluated using the relation jloc =
0.2
η × 0.0288Re−
, η = 1.45. 1: Air temperature (ISO). 2: Product surface temx
perature (ISO). 3: Air temperature (ADI). 4: Humidity ratio (ISO). 5: Product
surface temperature (ADI). 6: Humidity ratio (ADI).
and simple models are summarized in Table 3.5 for a range of inlet air temperatures
and relative humidities. In the isothermal mode the outlet air temperatures do not
deviate markedly from the inlet temperatures, but the moisture extraction rates do in
some cases vary significantly from those predicted by the simple model. The detailed
model sometimes predicts a higher MER than the simple model; sometimes a lower
MER. The key reason for these variations again appears to be that the product surface
temperature may deviate significantly from the air inlet temperature. This hypothesis has been tested by forcing the product surface temperatures to equal the air inlet
temperature in the detailed model. The observed result is near-agreement (within 4%
for all the scenarios tabulated here) with the MER predicted by the simple model. In
addition, in the isothermal case the curvature of the bulk relative humidity matches
the idealised model, confirming the discussion above (page 91). The relative surface
temperature deviation is determined by the intensity of the drying process, as is il-
97
Enhancement factors
1
0.98
0.96
0.94
θv
θAB
0.92
θt
0.9
Temperatures, °C
56
55
54
53
52
Tgas
51
Tfilm
50
Tsurface
49
1.08
nv/jv
1.06
1.04
1.02
1
1
2
3
Position x along flow direction, m
4
5
Figure 3.11: Variation of enhancement factors, temperatures and relative effect
of convection with x. Tin = 55◦ C, Tr sat = 60◦ C, φin = 30%, ṁa,in = 1.
98
lustrated by the last two scenarios detailed in Table 3.5. In the case of inlet condition
Tin = 55◦ C, φin = 90%, a low drying intensity leads to a product surface temperature
that exceeds the air inlet temperature by an average of several degrees (see Fig. 3.19
below). Since the airstream is close to saturation, this temperature difference has a
significant relative effect on the driving force for mass transer. In the case of inlet
condition Tin = 70◦ C, φin = 60%, a high drying intensity leads to a product surface
temperature that is several degrees less than the air inlet temperature, depressing the
driving force for drying. This initial analysis has shown that deviation of the product surface temperature from its idealised ‘isothermal’ value, as used in the simple
model of chapter 1, may have a significant effect on the drying rate. Under highhumidity HPD conditions, this could significantly increase the capacity of a contact
HPD compared with the prediction of the model used in chapter 2. However since
this effect depends on the difference between the refrigerant condensing temperature
and the air inlet temperature, an assessment will require the use of an integrated system model. Under moderate drying conditions the simple model appears to reflect
the behaviour of the detailed model reasonably well. The analysis has also provided
reassurance that the other additional physical effects that have been incorporated into
the detailed model can appropriately be regarded as corrections, rather than as primary aspects of the situation being modelled.
ISO
ADI
ISO
ISO
ISO
ISO
Idealised model
Detailed model
Detailed with Tideal
φin = 30%
MER = 199.3 kg h−1
MER = 179.4 kg h−1
MER = 200.3 kg h−1
Tin = 55◦ C
Tout = 55.0 ◦ C
Tout = 53.9 ◦ C
Tout = 54.4 ◦ C
φin = 30%
MER = 21.2 kg h−1
MER = 19.3 kg h−1
MER = 20.0 kg h−1
Tin = 55◦ C
Tout = 41.8 ◦ C
Tout = 42.4 ◦ C
Tout = 42.4 ◦ C
φin = 60%
MER = 50.4 kg h−1
MER = 73.8 kg h−1
MER = 50.4 kg h−1
Tin = 40◦ C
Tout = 40.0 ◦ C
Tout = 42.2 ◦ C
Tout = 41.8 ◦ C
φin = 60%
MER = 122.6 kg h−1
MER = 131.9 kg h−1
MER = 120.6 kg h−1
Tin = 55◦ C
Tout = 55.0 ◦ C
Tout = 55.4 ◦ C
Tout = 55.4 ◦ C
φin = 90%
MER = 33.1 kg h−1
MER = 80.2 kg h−1
MER = 30.7 kg h−1
Tin = 55◦ C
Tout = 55.0 ◦ C
Tout = 56.9 ◦ C
Tout = 56.5 ◦ C
φin = 60%
MER = 314.7 kg h−1
MER = 206.6 kg h−1
MER = 315.4 kg h−1
Tin = 70◦ C
Tout = 70.0 ◦ C
Tout = 68.0 ◦ C
Tout = 68.6 ◦ C
Table 3.1: Estimates of system behaviour. ṁa = 1.
Fig. 3.12 illustrates the effects (1) of the difference between the refrigerant condensing temperature and the inlet air temperature, Tr sat − Tin , and (2) of the product
thickness, δ, on the temperatures within the drier. The first subfigure corresponds to
the scenario already illustrated in Figs. 3.7 and 3.11. As seen previously, the resulting MER = 178 kg h−1 . The second subfigure shows the temperatures in the drier if
99
56
Temperatures, °C
55
54
53
52
Tgas
51
Tfilm
50
Tsurface
56
Temperatures, °C
54
52
50
Tgas
48
Tfilm
Tsurface
46
54
Temperatures, °C
52
50
48
46
44
42
Tgas
40
Tfilm
38
Tsurface
36
1
2
3
Position x along flow direction, m
4
5
Figure 3.12: Variation of temperatures with x. Tin = 55◦ C, φin = 30%, ṁa,in =
1. In the first subplot Tr sat = 60◦ C, and δ = 1 mm. In the second, Tr sat = 55◦ C,
and δ = 1 mm. In the third, Tr sat = 60◦ C, and δ = 10 mm.
100
Adiabatic
Isothermal
100
80
70
60
50
40
30
Humidity ratio, g moisture / kg dry air
90
20
φ=20%
φ=10%
10
20
30
TWB=20°C
40
50
10
0
60
T,°C
Figure 3.13: Psychrometric paths predicted by the detailed model. Tin = 55◦ C,
φin = 30%, ṁa,in = 0.4 kgs−1 (circles) and 1.6 kgs−1 (diamonds).
the refrigerant saturated condensing temperature is only equal to the air inlet temperature i.e. Tr sat = 55◦ C, rather than 5◦ C greater as before. This 5◦ C reduction in
the refrigerant condensing temperature leads to a 2◦ C lowering of the outlet air temperature, and to a 3◦ C lowering of the product surface temperature at the air outlet.
The resulting MER = 137 kg h−1 , 77% of its previous value. The third subfigure of
Fig. 3.12 shows the drier temperatures in the case δ = 10 mm. As can be seen, the
additional thermal resistance contributes to a large temperature decrease (as much as
10◦ C cooler than when δ = 1 mm) both at the product surface and in the bulk air flow.
The resulting MER = 71 kg h−1 , only 40% of that when δ = 1 mm. The 5◦ C temperature difference has been selected as representative of typical conditions in an HPD.
The results summarized in this paragraph show that the system behaviour is much
less sensitive to a fairly small (5◦ C) decrease in the temperature difference Tr sat − Tin
than it is to a fairly small (9 mm) increase in the product thickness.
Figure 3.13 shows psychrometric paths obtained from the detailed model under
the inlet condition Tin = 55◦ C, φin = 30%. Two inlet air mass flow rates are considered: ṁa,in = 0.4 kgs−1 , and ṁa,in = 1.6 kgs−1 . Locations 0, 1, 2, 3, 4 and 5 m into the
duct are indicated using circles for the low flow-rate scenarios and diamonds for the
high flow-rate scenarios. In the adiabatic case the humidity ratio and the temperature
101
are related to one another by the (inlet) specific enthalpy of the air, and the psychrometric path traced by the air is unaffected by the air flow rate, although the distance
into the duct at which a given state is attained depends upon the air speed. In the
isothermal case, a lower air flow rate enables the air to receive more heat as it passes
over the product, resulting in a greater air temperature at the air outlet. The increase
in ω is significantly less in the high flow rate scenarios, which nevertheless have a
greater MER than for the low flow rate scenarios, as MER = ṁa ∆ω.
Figures 3.14–3.17 offer a more detailed view of the variation of conditions along
the airflow direction. This detailed picture both helps to provide confidence in the
soundness of the model and reveals a number of behaviours that are of interest in the
context of the overall thesis project. Two inlet conditions have been selected: 55◦ C at
30% relative humidity and 55◦ C at 90% relative humidity. The refrigerant saturated
condensing temperature is again set 5◦ C greater than the inlet air temperature, i.e.
Tr sat = 60◦ C. Each inlet psychrometric condition is considered at dry air mass flow
rates of 0.4 kgs−1 , 0.8 kgs−1 , 1.2 kgs−1 and 1.6 kgs−1 . The case of ṁa = 1.0 kgs−1
considered in Figs. 1.10 and 3.7–3.12 would lie between the dotted and dash-dot lines
of Figures 3.14–3.17. Since the inlet airflow is specified in terms of the dry air mass
flow rate, the inlet velocities and mass flow rates are dependent on the inlet relative
humidities. This explains the difference between the inlet air velocities in Figs. 3.14
and 3.18 and between the inlet mass flow rates in Figs. 3.16 and 3.17.
Figures 3.14 and 3.18 show the air velocity, gauge pressure and humidity ratio of
the bulk air flow in the drier under the two inlet conditions. The system pressure
drop |∆p| is between 10 and 100 Pa, this latter pressure drop being the pressure pro-
duced by a 10 kg cat with a 1 m2 footprint. The observed range of pressure drops
spans the pressure drop of 20-25 Pa found by Sun et al. [160] for a timber kiln with
similar duct dimensions (5.76 m × 0.02 m as seen from the side) and an inlet air velocity of 4 ms−1 . In particular, the case represented by a dash-dot line in Figure 3.14 has
an inlet velocity near 4 m s−1 and exhibits a pressure drop of 33 Pa. In the adiabatic
mode with the same inlet condition, the pressure drop is 28 Pa. The small remaining
disagreement between the pressure-drop predictions of the present model in the adiabatic mode and Sun et al.’s model can be attributed to the absence of evaporation
from the duct ceiling (Sm in Fig. 3.1) in the present model and to the different transfer correlations employed in the two models. Notable features of Figures 3.14 and
3.18 are: (1) A slight (respectively ≈ 8% and ≈ 3%) increase in the air flow velocity
– equivalently, given the uniform duct cross-section, in the volume flow rate – along
the air flow direction in the drier. This can be attributed both to a falling air density
(see below) and an increasing mass flow rate as the airstream acquires moisture in the
drier. In contrast, in the adiabatic drying model with inlet conditions as considered in
Table 3.5, the air density increases along the air flow direction due to cooling, and the
102
volume flow rate decreases by 3% along the air flow direction. This latter behaviour
is consistent with that shown in Figure 3 of [160]. (2) An approximately linear pressure profile through the system, and a nonlinear pressure response to the air flow,
seen as the increasing vertical separation of the pressure lines while ṁa increases by
equal steps. (3) An inverse relationship between air flow and the total humidity ratio
increase ∆ω – although the drying rate, given by ∆ω × ṁa , is greatest at high air flow
rates. The MER in the adiabatic mode is consistent with the humidity change shown
in Fig. 2 of [160], to within a margin that is similar to that for the pressure drop. With
an inlet velocity of 4.0 m s−1 , the present detailed duct model predicts a change in the
vapour mass fraction of 0.0041 kg kg−1 , which is in fair agreement with the value of
0.0045 kg kg−1 that can be read from Fig. 2 of Sun et al. [160].
Figs. 3.15 and 3.19 show the system temperatures (air and product surface) and
the density of the moist air binary mixture. The mixture density is a function of
pressure, of temperature and of the humidity ratio. Both the falling pressure and the
increasing humidity ratio contribute to a decrease in the mixture density along the air
flow direction. The temperature has a more complex effect, since in the case of high
air flow rates the air temperature initially decreases, as discussed above. This feature
depends on the inlet relative humidity, as can be seen by comparing Figures 3.15 and
3.19: When φin = 0.9 (Fig. 3.19), evaporative cooling is insufficient to cause the air
temperature to decrease under any of the air flow rates considered. In the two cases
shown in Figures (3.14)–(3.17), the air density falls along the airflow direction.
Figs. 3.16 and 3.17 show the total moist air mass flow rate through the system, the
(evaporation) vapour flux at the product surface, and the relative humidity of the air
as it passes through the drier. The product surface vapour flux varies by as much as
70% from the air inlet to the air outlet, highlighting the need to model the variation of
the drying process along the length of a drier of this size. The proportional increase in
the air mass flow rate through the duct (respectively ≈ 5% and ≈ 2%) is less than the
proportional increase in the volume flow rate, as a result of the lower air density at the
air outlet. Fig. 3.16, in particular, highlights that the 5 m-long tray drier specified in
Table 2.1 provides enough evaporation surface for there to be a significant single-pass
increase in relative humidity in the isothermal mode.
3.6.
Conclusions
This chapter has described in detail the development of a flexible finite-volume airside model, which has been used to examine the drying process in quasi-isothermal
and adiabatic drying processes in detail. There are two standard ways in which numerical models are validated. First, a model can be used to predict the behaviour of a
Air velocity v, ms−1
103
8
6
4
2
Gauge pressure p− p0, Pa
0
100
50
Humidity ratio ω, g/kg
0
100
ma,in=1.6kgs−1
50
ma,in=1.2kgs−1
ma,in=0.8kgs−1
ma,in=0.4kgs−1
0
1
2
3
Position x along flow direction, m
4
5
Figure 3.14: Variation of air velocity, gauge pressure and humidity ratio. Tin =
55◦ C, Tr sat = 60◦ C, φin = 30%. ṁa,in as shown in legend.
104
Surface temperature T, °C
58
56
54
52
50
48
Air temperature T, °C
57
56
55
54
53
Air density ρ, kgm−3
52
1
ma,in=1.6kgs−1
0.8
ma,in=1.2kgs−1
ma,in=0.8kgs−1
0.6
ma,in=0.4kgs−1
1
2
3
Position x along flow direction, m
4
5
Figure 3.15: Variation of surface temperature, air temperature and air density.
Tin = 55◦ C, Tr sat = 60◦ C, φin = 30%. ṁa,in as shown in legend.
105
Mass flow rate, kgs−1
9
8
7
6
5
4
3
2
Vapour flux nv(m), gs−1m−2
1
2
1.5
1
0.5
0
Relative humidity φ
0.9
0.8
0.7
0.6
ma,in=1.6kgs−1
0.5
ma,in=1.2kgs−1
0.4
ma,in=0.8kgs−1
0.3
ma,in=0.4kgs−1
0.2
1
2
3
Position x along flow direction, m
4
5
Figure 3.16: Variation of air mass flow rate, surface vapour flux and relative
humidity. Tin = 55◦ C, Tr sat = 60◦ C, φin = 30%. ṁa,in as shown in legend.
106
Mass flow rate, kgs−1
9
8
7
6
5
4
3
2
1
Vapour flux nv(m), gs−1m−2
0.7
0.6
0.5
0.4
0.3
0.2
0.1
Relative humidity φ
0
0.98
0.96
ma,in=1.6kgs−1
0.94
ma,in=1.2kgs−1
ma,in=0.8kgs−1
0.92
ma,in=0.4kgs−1
0.9
1
2
3
Position x along flow direction, m
4
5
Figure 3.17: Variation of air mass flow rate, surface vapour flux and relative
humidity. Tin = 55◦ C, Tr sat = 60◦ C, φin = 0.9. ṁa,in as shown in legend.
107
Air velocity v, ms−1
10
8
6
4
2
Gauge pressure p− p0, Pa
0
100
50
Humidity ratio ω, g/kg
0
100
ma,in=1.6kgs−1
ma,in=1.2kgs−1
50
ma,in=0.8kgs−1
ma,in=0.4kgs−1
0
1
2
3
Position x along flow direction, m
4
5
Figure 3.18: Variation of air velocity, gauge pressure and humidity ratio. Tin =
55◦ C, Tr sat = 60◦ C, φin = 0.9. ṁa,in as shown in legend.
108
Surface temperature T, °C
60
59
58
57
56
Air temperature T, °C
55
58
57
56
55
Air density ρ, kgm−3
54
1
ma,in=1.6kgs−1
0.8
ma,in=1.2kgs−1
ma,in=0.8kgs−1
0.6
ma,in=0.4kgs−1
1
2
3
Position x along flow direction, m
4
5
Figure 3.19: Variation of surface temperature, air temperature and air density.
Tin = 55◦ C, Tr sat = 60◦ C, φin = 0.9. ṁa,in as shown in legend.
109
simple system which admits of analytical solution techniques. Second, in modelling
more complex situations, model predictions can be compared directly with experimental observations of the system being modelled. In effect, we have applied both
methods in the development of the duct flow model. By comparing the isothermal
drying behaviour with the idealised model described in chapter 1, we have employed
the first method. By moving into the adiabatic mode and comparing the model outputs with those of the model developed by [159], which was itself tested against measured data, we have indirectly employed the second technique. This justifies a fair
confidence that the model robustly represents drying within a duct.
The analysis has shown that deviation of the product surface temperature from its
idealised isothermal value, as used in the simple model, may have a significant effect
on the drying rate. At moderate drying rates the idealised and detailed models are in
good agreement; however at very high and very low drying rates, correspondingly
depressed and elevated surface temperatures (respectively) may have a substantial
effect on evaporation within the dryer, as is shown by Table 3.5. Taken all together,
these results indicate the following: (1) the idealised model provides a reasonable
first-order approximation; (2) the detailed model may be required in order to obtain
accurate predictions of HPD performance, because of significant temperature effects
that occur at low and high drying rates; and (3) the other additional physical effects
that have been included in the detailed model can be regarded as minor corrections,
rather than as primary aspects of the situation being modelled.
Our investigation has thus confirmed that under moderate drying conditions the
simple constant-parameter model described in chapter 1 provides a satisfactory firstapproximation model of the drying process, corroborating the results of chapter 2.
However it has also provided insight into several ways in which a contact drier operated under HPD conditions would deviate from the ideal isothermal case assumed in
chapter 2, and has shown how these deviations depend on the inlet flow conditions,
product thickness, and refrigerant condensing temperature. Of particular relevance,
the model has shown that at high relative humidities the rate of evaporation may be
significantly greater than predicted by the simple model. In order to assess the impact that this may have on the performance of isothermal HPD, the detailed air-side
model must be incorporated into a whole-system model. In the form described in
the present chapter, the air-side model is only able to describe the constant dryingrate period. However this airflow model has the potential to be combined with an
internal-process model, to produce a dynamical model of the whole drying process.
This flexibility, and the greater detail of the model, does come at a significant computational cost: the detailed model takes about fifty times longer than the simple model
of chapter 1, requiring typically about 30 iterations to converge from an initial guess
of the air inlet conditions prevailing throughout the flow system.
110
3.7.
Nomenclature for chapter 3
Roman and Greek symbols
Symbol
Definition [units]
Am
Mass-exchange area of Control Volume [m2 ]
a, b, d
Discretisation equation coefficients
A
Area [m2 ]
Cf
Friction factor [1/m2 ]
COP
Coefficient of performance [–]
cp
Specific heat capacity [J/kg-K]
D
Diffusivity [m2 /s]
d
Air duct depth [m]
D
Heating plate refrigerant tube internal diameter [m]
D (m)
Discretised humidity diffusion conductance [kg/s]
( E)
DH
( E)
Dv
D ( p)
Discretised enthalpy diffusion conductance [kg/s]
Discretised kinetic energy diffusion conductance [kg/s]
Discretised momentum diffusion conductance [kg/s]
ê1 , ê2
Unit vector parallel to x axis, y axis
F
Discretised convective mass flow [kg/s]
f
Specific body force [N/kg]
Ff
Net friction force acting on control volume boundaries [N]
F (m)
Momentum source associated with mass transfer [N]
Ff
Friction force [N]
Fm
Momentum transfer associated with mass transfer [N]
G
Mass velocity of refrigerant kg/m2 -s
∆hvap
Latent heat of vaporization [J/kg]
hk
Partial mass enthalpy of species k [J/kg]
h
Specific enthalpy of moist air [J/kg dry air]
h
Product surface heat transfer coefficient [W/m2 -K]
hm
Product surface mass transfer coefficient [m/s]
j
Mass diffusion flux vector [kg/m2 -s]
jloc
Local j-factor [–]
J
Discretised total mass flow, convective + diffusive, [kg/s]
Jp
Discretised total momentum flow, convective + diffusive, [kg-m/s2 ]
Jv
Discretised vapour mass flux [kg/m2 -s]
Ju , JH , JHT , JHω , JHP Discretised energy flux terms [J/m2 -s]
Jp
Discretised momentum flux [kg/m-s]
111
k
Thermal conductivity [W/K-m]
l
Heating plate condenser tube spacing [m]
lb
Heating plate condenser tube branch length [m]
L
Heating plate length [m]
ṁ
Mass flow rate [kg/s]
(m)
mk
Control volume mass-transfer rate for species k [kg/s]
M, b, bc
Terms in matrix representation of discretisation equations
ṁr,tot
System refrigerant mass flow rate [kg/s]
n
Mass flux [kg/m2 -s]
nb
Refrigerant circuits per plate [–]
n̂
Unit normal vector [–]
ND
Number of ducts [–]
Nx
Number of duct control volumes [–]
p
Pressure [Pa], passes through plate per circuit [–]
P(m]
Discretised humidity transport Peclet number [–]
( E)
PH
P( p)
( E)
Pv
p∗
Discretised enthalpy transport Peclet number [–]
Discretised momentum transport Peclet number [–]
Discretised kinetic energy transport Peclet number [–]
Previous discretised pressure field estimate [Pa]
p0
Pressure correction estimate [Pa]
Pr
Prandtl number [–]
Q̇
Heat flow rate [W]
qc
Pure heat flow vector [W/m2 ]
Q(m)
Discretised energy transfer associated with mass transfer [J/s]
Q
Discretised control volume heat transfer [J/s]
Re
Reynolds number [–]
R
Region occupied by control volume
S
Bounding surface of control volume
SC , S
Source terms in discretised balance equations
T
Temperature [K]
t
Time [s]
∆T
Mean difference Tr,sat − Tw between refrigerant saturated condensing
temperature and tube wall temperature [K]
u
Specific internal energy [J/kg]
v, v
Speed, velocity [m/s]
v1 , v2
x-component, y-component of velocity
V
Volume [m3 ]
v̂
Pseudo-velocity [m/s]
112
v∗
Previous discretised velocity field estimate [m/s]
w
Velocity of control-volume bounding surface [m/s]
w
Heating plate width [m]
Ẇ
Power input [W]
W
Work rate at the air control volume surface [W]
x, y
Spatial variables in air control volume [m]
xp
Heating plate refrigerant tube centerline depth [m]
x
Vapour mass quality [–]
z
Distance along refrigerant flow in CD2 [m]
α
Heat exchange coefficient [W/m2 -K]
Γ(m)
Discretised ‘diffusivity’ of humidity [kg/m-s]
( E)
ΓH
( E)
Γv
Γ( p)
Discretised enthalpy ‘diffusivity’ [kg/m-s]
Discretised kinetic energy ‘diffusivity’ [N-s/m2 ]
Discretised momentum ‘diffusivity’ [N-s/m2 ]
δ
Isotropic tensor [–]
δ
Product thickness [m]
θv , θ AB , θt
Velocity, mass-transfer and thermal transfer enhancement factors
λ
Refrigerant thermal conductivity [W/K-m]
µ
Dynamic viscosity of fluid [N-s/m2 ]
ρ
Density [kg/m3 ]
σ
Total stress tensor [Pa]
σ (t)
Reynolds stress tensor [Pa]
τ
Viscous stress tensor [Pa]
φ
Relative humidity [–]
χtt
Lockhart Martinelli parameter [–]
ω
Humidity ratio [kg-vapour/kg-dry-air]
ωi
Mass fraction of species i [kg-i/kg-mixture]
Subscripts and superscripts
Symbol
Definition
0, 1
Last value, Updated value
A
∆T-independent flow regime
D
∆T-dependent flow regime
( E)
Energy-equation term
g
Gas-phase
in, out
Inlet, outlet
k, a, v, w
Species-k, dry-air, water-vapour, liquid-water
l, v
Liquid, vapour phases of refrigerant (in condenser tubes)
113
lo
Liquid-phase with total flow
(m)
Mass-equation term
( p)
Momentum-equation term
( p∗)
Pressure-equation term
( p0 )
Pressure-correction equation term
p
Heating plate
r
Refrigerant
S, s
Surface
sat
Saturation condition
strat
Fully-stratified flow regime
t
Total, effective
w, e, n, m
West, East, North, Mass-exchange (control-volume boundaries)
W, P, E
West, Point, East (Control-volume nodes)
wb
Wet bulb
•
Modified for high mass transfer rates
x
Time average of x
∗
Last estimate
114
Chapter 4
Steady-state HPD model.
The preliminary isothermal performance assessment of chapter 2 indicated a SMER
benefit of 2-3 times compared with an adiabatic HPD. However, this analysis was
based on a very idealised system model. In particular, the heat pump heating COP
was assumed to be a constant percentage (50%) of the Carnot COP, and in the isothermal (contact) case the model assumed that the temperatures of the product surface
(Ts ) and of the bulk air (Tb ) were everywhere equal to that at the drier inlet. Thus
although the analysis was able to satisfactorily match measured data in the adiabatic
mode, uncertainty remained in its predictions for the isothermal mode. In this chapter the detailed air flow model of chapter 3 is linked into whole-system model based
on models of the remaining sytem components, which were established by Carrington and Bannister [28]. Deviation from purely isothermal behaviour is found to influence the predicted MER and SMER by only a few per cent. An exergy analysis shows
that the isothermal mode derives about half of its energy efficiency from a reduction
in the irreversibility associated with the transfer of heat to the drying process. The
remaining half is associated with the avoidance of air cooling during drying. System
performance is shown to be most sensitive to variation in the moisture evaporation
surface area and the dimensioning of the refrigerant evaporator. See also [36, 38].
4.1.
Introduction
Fig. 4.1 schematically illustrates the system considered in this chapter, which combines elements of the systems illustrated in Figs. 1.5 and 2.1. The geometry of the
dryer plate stack, of the condenser and of the evaporator remains as specified in chapter 2 (see Table 2.1 on p.43). The face areas of all air ducts are taken to be identical to
the condenser face area, 1 m2 . Again, the novel feature of the isothermal case is the
115
conductive plate thermally linking hot refrigerant with the drying process (labelled
CD2 in Fig. 4.1). The system depicted in Fig. 4.1 becomes equivalent to an adiabatic
HPD if the refrigerant bypasses the condenser plates (dotted line 2-20 ).
The detailed heat pump model that is developed in the present chapter provides an
opportunity to assess an aspect of contact HPD that could not be addressed in chapter
2. Running the refrigerant through the heating plates (labelled CD2) can be expected
to produce an additional pressure drop in the refrigerant cycle. The impact that this
additional pressure drop will have on overall system performance will depend on
the balance that must be sought with the heat transfer rate. The system geometry
that was assumed in chapter 2 would involve a total of (on the order of) 900 m of refrigerant tube running through the heating plates, which would lead to a prohibitive
pressure drop if the refrigerant passed through the tubes in series. Separating the
heating-plate refrigerant flow into many parallel circuits will reduce the length of
each circuit, as well as the local flowrate and friction throughout each circuit, reducing the pressure drop. However the reduced mass flow through each circuit will also
impact negatively on the refrigerant heat transfer coefficient. There are strong reasons
to believe, at the outset, that this trade-off will be much less severe than the air-side
trade-off that isothermal drying avoids: refrigerant-side heat transfer irreversibilities
are much lower (by about two orders of magnitude) than those on the air side, and
large pressure drops may be small in relative terms at the high-pressure end of the
heat pump. Nevertheless, it seems clear that a full, fair comparison between isothermal and adiabatic modes requires the additional pressure vs. heat transfer trade-off
to be taken into account, in order to test these intuitions. In the present chapter we
extend the drier model of chapter 3 to include a refrigerant-side model incorporating correlations for two-phase refrigerant flow within the condenser plates, in order
to estimate the plate refrigerant heat transfer coefficient αr and the pressure drop.
The refrigerant-flow model is based on functions obtained by Cavallini et al. [39]
and Traviss et al. [168]. Other pressure drops are evaluated using the correlations of
Carrington and Bannister [28]. To estimate key air-side pressure drops, general heatexchanger pressure drop correlations developed by Turaga et al. [171] are used for
the evaporator and condenser.
Having established models for the remaining ICHPD components, we link the detailed drier-duct model of chapter 3 into a whole-system HPD model that incorporates all of these component functions. The modelling approach that was used in
chapter 2, which involved numerically solving nested functions, is neither computationally efficient enough, for reasons discussed by Goldstein [73], nor flexible enough
for easy use with the detailed airflow model described chapter 3. (For example, minor system modifications would require an entirely new model structure.) The more
flexible and efficient approach of Goldstein [73] for finding the steady-state operating
116
venting
C
ẆF
B
C’
F
ṁa
CD1
CD2
CD2 E
D
EV=evaporator
CP=compressor
bṁa
CD=condenser
(1−b)ṁa T=throttle
F=fan
ẆF =fan power
EV
ẆP =shaft power
A
b=bypass ratio
CD2
3
20
2
T
CP
ẆP
1
condensate
4
Figure 4.1: Schematic of heat pump drying system being modelled.
condition, described below, is adopted instead. The accuracy of the preliminary analysis of chapter 2 is assessed. An exergy audit is used to explain the energy benefit
of the isothermal mode. An assessment is made of the sensitivity of system energy
performance to a set of parameters characterising the system geometry. Finally, a
comparative economic analysis of adiabatic and isothermal HPD is presented.
The following sections present: (1) refrigerant- and air-side pressure drop correlations (2) the compressor model, and (3) the whole-system HP model. Exergy functions are also developed, for use in second-law system analysis.
4.2.
Pressure drop correlations
The refrigerant flow model developed in chapter 3 is used to evaluate the refrigerant
pressure drop in the heating plate. The total pressure drop in the plate is evaluated by
numerically integrating the pressure derivative from z = lb to z = 0. Cavallini et al.
[39] suggest using the annular-flow pressure drop correlation to evaluate pressure
drops in all the flow regimes. We follow this approach, employing the pressure drop
relation of Traviss et al. [168]. Neglecting the external force of gravity, since the tubes
are horizontal, the pressure gradient along the tubes is given by [168]:
dp
dp
dp
=
+
dz
dz f
dz m
(4.2.1)
117
In (4.2.1) the pressure gradient is decomposed into a friction part, subscript f , and
a part due to momentum change, subscript m, which is negligible in the one-phase
region. In the one-phase region the friction part is given by the correlations [88]:
f G2
dp
=2
(4.2.2)
dz f
Dρ
where f =

0.3164Re−0.25
(Re ≤ 105 )
0.0032 + 0.221Re0.237
(4.2.3)
(105 < Re < 3 × 106 )
The Reynolds number in (4.2.3) is given by its usual definition Re = GD/µ. In the
two-phase region, the friction part of the pressure gradient is evaluated using [168]:
dp
dz
=
f
G2
GD −0.2 2
−0.09
φv
Di ρ v
µv
(4.2.4)
φv = 1 + 2.85χtt
(4.2.5)
the pressure gradient due to momentum changes is [168]:
dp
G2 dx
=
f (x)
dz m
ρv dz
f ( x ) = 2x + (1 − 2x )
ρv
ρl
1/3
+ (1 − 2x )
We evaluate the length of each circuit as
1
nb
ρv
ρl
2/3
(4.2.6)
− 2(1 − x )
ρv
ρl
(4.2.7)
times the total tube length in the plate,
where nb (as in the previous chapter) represents the number of refrigerant branches
in each heating plate. Examining the plate geometry in Fig. 3.4, we find:
1
W
lb = pL + ( p − 1) π
2 nb p + 1
(4.2.8)
where the variables are as defined in the discussion around Fig. 3.4, on page 86. We
neglect any extra pressure drops due to the curvature of the return bends.
For the remaining system components, refrigerant pressure drops are evaluated
using empirical correlations that have been inherited from the model first described
by Carrington and Bannister [28]. These are as follows:
Condenser pressure drop:
∆pCO = pr,20 − pr,3 = 0.025(0.0106 + 1.267ṁr + 2.91ṁ2r ) × 106
(4.2.9)
Liquid-line pressure drop:
∆pL = pr,3 − pr,4 = 0.025(−0.0054 + 0.00225ṁr + 11.049ṁ2r ) × 106
(4.2.10)
In (4.2.9) and (4.2.10), ṁr is the refrigerant mass flow rate in kg/s, and the locations
20 ,
3 and 4 are shown in Fig. 4.1.
118
Air-side pressure drops in the evaporator and condenser are evaluated using general heat exchanger pressure-drop correlations developed by Turaga et al. [171]. The
pressure drop across each heat exchanger is given by the following equation:
∆p = f a ·
2
2Ld Gm,
f
(4.2.11)
Dh ρ a
In (4.2.11), Gm, f = ṁ a /Am, f is the mass velocity evaluated using the minimum flow
cross-section within the heat exchanger, Am, f . The coil depth Ld is shown in Fig. 4.2.
The air-side hydraulic diameter of the heat exchanger is defined as follows:
Dh =
4Am, f Ld
A0
(4.2.12)
where A0 is the total airside heat exchange area. Assuming that the entire evaporator
face area is wet, we use the following correlations for the condenser and evaporator
friction factors, due to Turaga et al. [171]:
−0.28
A0
0.27
Re−
= 0.589
a
Ap
0.01 0.4
Sf
A0
0.41
= 0.325
Re−
a
Ap
Yf
f a,CO
f a,EV
(4.2.13)
(4.2.14)
In (4.2.13) and (4.2.14), Rea is the air-side Reynolds number calculated using the hydraulic diameter of the heat exchanger:
Rea =
A0
Ap
Gm, f Dh
µa
(4.2.15)
represents the ratio of the total airside heat
S exchange area to the primary (tube) area, and the ratio Yf represents the ratio of
In (4.2.13) and (4.2.14) the ratio
f
the fin spacing to the fin thickness, as shown in Fig. 4.2. From Table 4.1, the finpitch (1.8 mm) and number of rows (3) of the condenser are the same as those of
coil number 3 investigated by Turaga et al. [171], while the fin-pitch (2.1 mm) of the
evaporator is the same as that of coil number 2 of [171], and its number of rows (4)
is identical to coil number 4 of [171]. We assume that Turaga’s constant parameter
values ST = 38.1 mm, D0 = 13.4 mm, y f = 0.16 mm (these parameters are depicted
in Fig. 4.2) adequately describe the evaporator and condenser coils. With this assumption, we are able to adopt the parameter values presented in Table 4.1, which
are independent of the coil face area, directly from Table 2 of [171]. To estimate the
minimum flow cross-section of the heat exchanger coils, we use:
Am, f
Do
= 1−
= 0.648
Af
ST
(4.2.16)
Using (4.2.16) and the parameter values presented in Table 4.1 allows us to estimate
the air-side pressure drops across the condenser and evaporator coils, from the incoming air flow properties, using equations (4.2.11), (4.2.13) and (4.2.14).
119
Ld
A0 /A p
Condenser
Evaporator
11.4 cm
37.2
Dh
0.22 cm
Ld
15.2 cm
A0 /A p
31.8
S f /Y f
14.4
Dh
0.27 cm
Table 4.1: System component parameters, from [171].
Minor air-side pressure drops are estimated using the dynamic loss coefficients
(k-factors) obtained by Carrington et al. [35] for a timber stack, using:
1
∆p = k · ρ a v2a
2
(4.2.17)
The coefficients are listed in Table 4.2.
fan exit duct
0.8
90◦
turn
1.2
air entering and leaving stack
0.3
upper
Table 4.2: Dynamic loss coefficients (k-factors) for timber dryer [35].
4.3.
Compressor model
In addition to the detailed dryer model, the plate refrigerant model and the pressure
drop correlations described above, the full-system model incorporates correlations
for the isentropic and volumetric efficiencies of the ZR61K2-TFD scroll compressor
with R134a characterised by Carrington et al. [30]. The usual definition of the isentropic efficiency, ηi , is given by Haywood [81, p.150] as follows:
ηi =
h s − h1
h2 − h1
(4.3.1)
In Equation (4.3.1), which is an expanded version of Equation (1.1.2) on page 3, the
subscript ‘s’ refers to the state that would result from isentropic compression from
the suction state h1 , p1 to the outlet pressure p2 . For reasons of measurement accuracy, Carrington et al. [30] have employed the following alternative definition, which
incorporates the mechanical efficiency of the compressor:
ηi =
h s − h1
Pco /ṁr
(4.3.2)
120
SL
tubes
Lf
ST
CH
fins
CW
Ld
(a)
(b)
Sf
LC
(c)
Di Do
Yf
Figure 4.2: Geometrical parameters of heat exchangers, adapted from [171]:
(a) side view; (b) front view; (c) top view detail of tube and fins.
Comparing the electrical input power Pco with the refrigerant mass flow rate and
enthalpy change resulted in a significant energy imbalance ∆Eco , which was assumed
to take the form of heat rejected from the compressor [30]:
Pco = ṁr (h2 − h1 ) + ∆Eco
(4.3.3)
The compressor heat rejection ∆Eco is estimated using the following linear fit [30]:
∆Eco = 0.067Pco
(4.3.4)
Using (4.3.4) in (4.3.3) yields:
Pco =
1
ṁr (h2 − h1 )
1 − 0.067
(4.3.5)
Using (4.3.5) in (4.3.2) and rearranging, yields:
h2 = h1 + 0.933
h s − h1
ηi
(4.3.6)
The refrigerant mass flow rate is given by the equation
ṁr = ηv V̇ρ1
(4.3.7)
121
where the swept volume V̇ is 4.12 × 10−3 m3 s−1 [30]. The isentropic and volumetric
efficiencies ηi and ηv are evaluated using the following equations:
p2
x =ln
p1
y =ln ( p2 )
2
ηi = ∑
(4.3.8)
(4.3.9)
2
∑ a jk x j yk
(4.3.10)
j =0 k =0
2
ηv = ∑
2
∑ bjk x j yk
(4.3.11)
j =0 k =0
The coefficients used in (4.3.10) and (4.3.11) are listed in Table 4.3. The pressures in
a jk
b jk
↓k,j→
0
1
2
0
-1.755898
3.233047
-1.599213
1
0.9372472
-1.077541
0.6645025
2
-0.04633881
0.02790755
-0.06585382
0
3.165012
-4.447835
1.276352
1
-1.522796
3.037622
-0.8965961
2
0.2456160
-0.4839152
0.1366940
Table 4.3: Scroll compressor performance with R134a: model coefficients.
(4.3.8) and (4.3.9) are expressed in units of bar (= 105 Pa). Together with Equations
(4.3.10) and (4.3.11), Equations (4.3.6) and (4.3.7) allow the compressor outlet state and
the refrigerant mass flow rate to be calculated given the inlet state and the discharge
pressure. Using back-substitution into (4.3.5) and (4.3.4), the compressor electrical
input power Pco and heat rejection ∆Eco can then be evaluated.
4.4.
Whole-system HPD model structure
As discussed above, we adopt the general modelling procedure described by Goldstein [73], applying the Newton-Raphson method to a state-vector for the system.
Goldstein distinguishes between refrigerant systems employing a simple capillary
tube and those employing a thermostatic expansion valve for throttling. In the former case the degree of superheating at the evaporator outlet is variable, depending
on the refrigerant flow rate. In the latter case the evaporator outlet state is determined
by the saturated condition in the evaporator and by the superheat setpoint. In addition, systems that employ a liquid receiver after the condenser are insensitive to the
refrigerant charge. The system developed and investigated by Carrington and Bannister [28], which provides the basis for the system modelled here, employed both
122
a liquid receiver and a thermostatic expansion valve. Thus we can assume that the
system is charge-insensitive and maintains a constant evaporator superheat.
The whole-system model has a very similar structure to the model described by
Carrington and Bannister [28]. However it also incorporates the drier model of chapter 3, with the humidity ratio at E, ωE , appearing as a dependent variable. Fig. 4.3
shows the model’s flow chart. The Newton-Raphson method is applied to the state
vector x = ( x1 , x2 , x3 ), where, in terms of the locations shown in Fig. 4.1:
x1 = Trsat,1
(4.4.1)
x2 = Trsat,3
(4.4.2)
x 3 = ωD
(4.4.3)
Together with the compressor model, the fixed air temperature at location D, and
the heat pump component correlations described above, a unique value of this state
vector can consistently describe the system state, and the three values shown are
sufficient to deduce the condition at all system locations. (This is similar to the situation encountered in the modelling work of chapter 2.) Pressure drops in the liquid
and suction lines are evaluated using the condenser plate model described above,
together with the empirical correlations of Carrington and Bannister [28]. Using the
compressor model and assuming isenthalpic throttling, the two saturated refrigerant
states Trsat,1 and Trsat,3 are then used to update the estimated refrigerant mass flow
rate ṁr and thermodynamic cycle (where no subcooling and 10◦ C superheating is assumed). The refrigerant state at location 20 is calculated using the duct model, using
hr20 = h2 − Q̇ D /ṁr
(4.4.4)
where Q̇ D is the total heat transfer rate from the refrigerant in the drier ducts.
On the air-side, a (typical) constant fan efficiency of 50% is assumed Daly [54],
and the air pressure at location E is set equal to ambient: pE = 101325 Pa. In order
to evaluate conditions within the duct, the duct model is simultaneously iterated toward a solution by way of the SIMPLER algorithm as described in chapter 2. The air
pressure drop within the drier is thus evaluated using the SIMPLER algorithm; other
air pressure drops are estimated using the functions described above. As venting is
controlled to maintain TD fixed, the air state at location D is specified by ωD and pD .
Air states around the system are obtained using moisture and energy balances:
123
Moisture balance, in terms of the humidity ratio ω:
ωA = ωsat (hA , pA )
(4.4.5a)
ωB = bωE + (1 − b)ωA
(4.4.5b)
ωC = (1 − x )ωB + xω0
(4.4.5c)
0
ωC
= ωC
(4.4.5d)
0
ωD = ωC
(4.4.5e)
In (4.4.5), ωE is estimated from the state-vector x using the duct model.
Energy balance, in terms of h, where h = h a + ωhv :
hA = hE + − Q̇ev + f ṁw hw ( Twb,A ) /(bṁ a )
(4.4.6a)
hB = bhE + (1 − b)hA
(4.4.6b)
0
hC
= hB + ẆF /ṁ a
(4.4.6c)
0
hC = (1 − x ) hC
+ xh0
(4.4.6d)
hD = hC + Q̇co /ṁ a
(4.4.6e)
In (4.4.6), hE is estimated from the state-vector x using the duct model, and the condensation rate f ṁw and condensate enthalpy hw ( Twb,A ) are evaluated from the previous estimates of the air state at locations A and E. The fan power ẆF is estimated
using the same equation as was used in chapter 2:
ẆF =
1
ṁ a v a,B ∆PF
eF
(4.4.7)
where a constant fan efficiency e F of 50% has again been assumed.
The condenser and evaporator heat transfer rates Q̇co and Q̇ev in (4.4.6) are:
Q̇co = ṁr (h20 − h3 )
(4.4.8a)
Q̇ev = ṁr (h1 − h4 )
(4.4.8b)
These heat transfer rates are updated using estimates of h1 , h20 , h3 , h4 that are evaluated from the state-vector x, by way of the current estimates of the refrigerant pressure drops, assuming isenthalpic throttling, and employing the compressor model
and the duct model which have been described, above.
The error vector ∆ that is used in the Newton-Raphson method is specified:
∆1 = Qev − f ev (vev , Twb,E , T4 )
∆2 = Q̇co − f co (vco , TC , Trsat,3 )
∆ 3 = ωD − ωC
(4.4.9a)
(4.4.9b)
(4.4.9c)
124
In (4.4.9a)–(4.4.9c), the polynomials f ev and f co are specified by equations (2.3.2) and
(2.3.3). These are presented again here for convenience:
f ev = − 0.017 + 1.486x − 0.5145x2
f co = − 0.0065 + 1.1535x − 0.11079x2
(4.4.10)
(4.4.11)
Each iteration of the Newton-Raphson method occurs as follows. From the current
estimate of the state vector, x, an updated estimate of the states throughout the system
is formed, and the error vector ∆ is evaluated. An estimated value J of the Jacobian
matrix for the system is then used to update the state-vector estimate using
x1 = x0 − r · J −1 ∆
(4.4.12)
where r represents the relaxation that is applied, and x1 is the new estimate of the
system state, which is obtained from the previous estimate x0 . In practice, it has
been found that a constant, diagonal value of J leads to satisfactory convergence behaviour. The convergence criterion for the whole-system model is that the maximum
error function, across all submodels of the whole-system model, must be less than a
threshold, which is typically set to 10−5 . Each updated variable is used to generate an
error function for that variable. In the duct model these are the control-volume source
terms that arise from the SIMPLER algorithm. For the remaining system variables,
these are the relative change in the variables in the latest iteration. The magnitudes
of the error-vector components ∆1 , ∆2 , ∆3 are also included. Since 4.4.10 and 4.4.11
represent the heat transfer from the condenser and evaporator that was measured as
a function of the incoming refrigerant and air flow conditions [28], the components
of the error vector ∆ approach zero only if the heat transfer at these heat exchangers
is consistent with these empirical correlations. In addition, the global convergence
criterion requires that the mass, momentum and energy balances within the drier
ducts be satisfied, and that the product surface temperature estimate be satisfied, to
within this convergence criterion. Thus on convergence the prediction is stable and
is guaranteed to satisfy mass and energy balances across all system components.
4.5.
Exergy functions
In the introduction, on page 15, it was argued that since HPDs are highly susceptible
to incidental entropy generation mechanisms, a full performance analysis requires
the consideration of the second-law system losses. Once the whole-system model has
converged, the exergy functions are used to evaluate the exergy destruction in each
component. The environment has (as in previous chapters, and consistent with reference [27]) been assumed to be at 100% relative humidity at 10◦ C. We represent the
125
Start
Get input parameters: dimensions of drier, heating plate, evaporator
and condenser; TD set point, ṁa , b; environmental state.
Set initial estimate of refrigerant, plate, product and air
temperatures, air flow parameters, and all system pressures.
Using Trsat,3 , Trsat,1 and pressure drop estimates, update estimate of
ṁr and states around refrigerant cycle, starting with compressor
model. Using ṁr update refrigerant pressure drops.
Determine corrections
to Trsat,3 , Trsat,1 , ωD
by way of NewtonRaphson method.
Using CD2 plate tube inside wall temperature, refrigerant specific
enthalpy at CD2 entry, Trsat,3 and ṁr , update αr,p and ∆pr,p .
One iteration of dryer model (thin-layer steady state drying model
+ air-side SIMPLER algorithm), given ωD , TD , ṁa , Trsat,3 , αr , ṁr .
This updates estimates of conditions throughout the drier.
Evaluate remaining air states around air cycle using energy and
mass balances over each component together with estimated air
pressure drops. Update estimates of air pressure drops.
Evaluate error functions for evaporator, condenser and humidity
ratio at location D, maximum error from last SIMPLER iteration
(maximum local relative energy, mass, momentum source), and
maximum relative change in αr and in all system pressure drops.
All within convergence criteria?
No
Yes
Finish
Figure 4.3: Flow chart for whole-system steady-state model.
steady-flow exergy here using the letter ‘e’. Changes in gravitational potential energy
can reasonably be neglected in a HPD (see the discussion on p. 65). For simplicity we
also neglect changes in kinetic energy, which correspond to temperature differences
less than the order of 0.1 K. This allows us to drop the final two terms of Equation
(A.5.21). It follows that the steady-flow exergy can be written:
e = (h − T0 s) − ∑ µk,0 ωk
(4.5.1)
k
In the absence of chemical reactions, the mass fractions of the chemical species are
unaffected by the mixture state, and we can write ωk = ωk,0 . It follows from Equation
(A.5.5) and from the definition of the Gibbs free energy, G = H − TS, that ∑ µk,0 ωk,0 =
g0 = h0 − T0 s0 , so that we obtain the following expression:
e = (h − T0 s) − (h0 − T0 s0 )
(4.5.2)
126
This expression is standard in the absence of changes in potential energy, and of
chemical reactions [e.g. 22, p.153]. Equation (4.5.2) can be applied directly to the heat
pump working fluid, using the functions described on page 250.
Bejan [15, pp.220-231] uses (4.5.1) to evaluate the moist-air flow exergy per kg dry
air, which we represent here using the symbol ed , and obtains the equation
T
T
R
p
ed =(c p,a + ωc p,v ) T0
− 1 − ln
+ (1 + ω̃ )
T0 ln
T0
T0
Ma
p0
R
1 + ω̃0
ω̃
+
T0 (1 + ω̃ ) ln
+ ω̃ ln
Ma
1 + ω̃
ω̃0
(4.5.3)
In (4.5.3), the symbol ω̃ represents the number of moles of water vapour per mole of
dry air in the moist-air mixture, and is given by [15]:
ω̃ = 1.608ω
(4.5.4)
For simplicity in evaluating the moist-air exergy we use equation (4.5.3) together
with the constant values: R = 8.314472 J/K-mol, Ma = 0.0289645 kg/mol, Mw =
0.01801528 kg/mol, c p,a = 1003 J/(kg K), c p,v = 1872 J/(kg K) [15, p.227].
The condensate that forms on the heat pump evaporator and drains from the system carries exergy since in general its temperature differs from that of the environment. This specific flow exergy ew can be estimated using the equation [27]:
ew = cw T − T0 − T0 ln( T/T0 )
J/(kg H2 O)
(4.5.5)
In (4.5.5), cw = 4186 J/(kg K) is the specific heat capacity of liquid water.
The functions that implement the above equations for the exergy of moist air, refrigerant and condensed water have been tested by comparing their outputs with
results from the literature. Fig. 4.4 was produced from the air exergy function using
the same method as was used to produce Figure 1 of Carrington and Baines [27], as
a test of the function. The air exergy function has been used to calculate the ideal
SMER of an adiabatic drier in the high air-flow (isothermal) limit, in which the only
irreversibility is that associated with vapour diffusion into the unsaturated air stream.
This is evaluated as the derivative of the humidity ratio with respect to exergy at constant wet-bulb temperature. The curves represent contours of constant SMER.
Table 4.5 contains the expressions used to evaluate the exergy destruction in key
system processes and components, where the system locations are as depicted in
Fig. 4.1. The expressions total to the work input of the system, as required, since
no work is recovered from the system and the make-up water carries zero exergy into
the system because it enters the system at the dead-state condition.
127
70
500
100
50
50
40
30
30
20
20
15
Humidity ratio, g moisture / kg dry air
60
10
10
8
10
20
30
40
50
60
70
0
T,°C
Figure 4.4: Limiting SMER where the only thermodynamic loss is due to the
diffusion of water vapour into the unsaturated drier air, adapted from [27].
Plot show contours of constant SMER (as labelled, in kg/kWh) in the limiting
case based on psychrometric state at the drier outlet.
128
Component or process
Expression for exergy destruction rate
Condenser and product
(e2 − e3 )ṁr + (eC − eE )ṁa
ẆP + (e1 − e2 )ṁr
Compressor
Evaporator
(eE − eA )(1 − b)ṁa + (e4 − e1 )ṁr − ew, A f ṁw
(e3 − e4 )ṁr
Throttle
Fan friction
Mixing
Venting and condensate
Sum of the above
ẆF + (eB − eC0 )ṁa
eE b + eA (1 − b) − eB ṁa
ew, A f ṁw + (eC0 − eC )ṁa
ẆF + ẆP
Table 4.4: Expressions for exergy destruction by component.
4.6.
Results and discussion
We now consider an illustrative output from the detailed whole-system model. In
the (baseline) case that we will consider, the two modes again have identical specifications aside from plate heat transfer. In particular, TD = 55◦ C; evaporator bypass
ratio b = 0. The dimensions of the system are identical to those in the baseline scenario described in chapter 2, except for the tubes embedded in the condenser plate,
whose orientation are as schematically illustrated in Fig. 3.4. The adopted number nb
of refrigerant circuits per plate is 3, and the number p of passes through the plate for
each circuit is 5. These values lead to a tube separation l of 6.7 cm in the plate, in the
baseline case. Fig. 4.5 shows the refrigerant thermodynamic state-cycle under typical
isothermal and adiabatic HPD conditions. The isothermal condenser refrigerant pressure drop (between the locations labelled 2 and 3) is somewhat greater than that for
the adiabatic cycle, owing to the pressure drop within the condenser plates. Despite
this (relatively small) additional pressure drop, the isothermal mode can be seen to
enable the system to operate over a significantly smaller pressure range. The present
chapter’s condenser-plate pressure drop model thus leads to the prediction that the
refrigerant-side tradeoff between heat- and momentum-transfer irreversibilities does
not significantly impact on performance in the isothermal mode.
Table 4.6 shows the exergy destroyed per kg moisture removed from the product in
key system processes and components. Noteworthy features of the data are: (1) In the
adiabatic mode, condenser and product irreversibilities contribute 43% of the overall work requirement; (2) The isothermal mode greatly reduces this irreversibility, by
about six-fold per kg moisture removed; (3) A significant part of the overall reduction
of irreversibility nevertheless takes place in the compressor and the throttle. This last
result highlights the synergistic nature of a heat pump dryer, and also implies that
129
R−134a
3
2
2
Pressure (MPa)
3
4
ISOTHERMAL
1
0.9
1
0.8
0.7
4
0.6
ADIABATIC
1
0.5
0.4
140 160 180 200 220 240 260 280 300 320 340 360 380 400 420 440 460 480 500
Enthalpy (kJ/kg)
Figure 4.5: R134a state-cycle in the isothermal and adiabatic modes.
the exergy destruction in the condenser and product of an adiabatic HPD does not
set an upper bound on the SMER gain associated with isothermal HPD. (Indeed if it
did, then from Table 4.6 the isothermal SMER could not exceed 7 kg kWh−1 .) About
half of the energy efficiency gain (reduction in irreversibility) is seen to be associated
with the condenser and the drying process. Since most of the exergy destruction in
the condenser and product is associated with the transfer of heat [27, 176], this portion of the avoided irreversibility can be attributed chiefly to the isothermal mode’s
avoidance of the poor heat transfer coefficient of air (the thermal insulator). Most of
the rest of the energy efficiency gain is at the compressor and throttle, and can be attributed to the narrower temperature and pressure range of the refrigerant cycle for
the isothermal mode. Since the majority of the narrowing of the temperature range
(in the isothermal mode) is due to its avoidance of air cooling in the drier ducts, we
can associate most of this latter improvement with the fact that the isothermal mode
avoids using air (with its small specific heat capacity) as a heat carrier.
The system MERs and SMERs are summarized at the bottom of Table 4.5. Consistent with chapter 2, the isothermal mode is found to yield a relative SMER gain of
around 3 times. The psychrometric state-cycles taken by the air are shown in Fig. 4.6.
The nominal power rating of the compressor is 5kW, as was used in the baseline
scenario of chapter 2. However the actual compressor power varies with operating
condition, as reflected in the model of the present chapter, so that direct comparison
130
120
Tco
100
80
60
40
Tev
Tco
Humidity ratio, g moisture / kg dry air
Tev
20
20
30
40
50
60
0
T,°C
Figure 4.6: Psychrometric chart showing air property paths in baseline scenario for adiabatic and isothermal dryers with zero evaporator bypass. The
evaporating and condensing temperatures Tev and Tco are represented by the
vertical bars to the left and right of the cycles. The MER is given by the vertical
displacement DE for each cycle (also the height of the vertical bars), multiplied
by the condenser airflow ṁ a,co . Compare with Fig. 2.4 (page 50).
131
(×10−3 ), kWh kg−1
Adiabatic
Isothermal
∆
Condenser and product
106.9
17.8
-89.1
Compressor
64.7
21.3
-43.4
Evaporator
33.0
22.5
-10.5
Throttle
27.5
4.4
-23.1
Fan friction
3.2
1.5
-1.7
Venting and condensate
14.6
5.9
-8.7
MER, kg h−1
20.5
55.2
34.7
kg kWh−1
4.0
13.6
9.6
SMER,
Table 4.5: Specific exergy destruction by component.
between the outputs of the two models requires caution. Nevertheless the psychrometric cycles shown in Figs. 2.4 and 4.6 are very similar. Since the modelling techniques used to obtain the two figures are quite different, the agreement supports our
confidence in the validity of the two models, and in particular support the simplifying assumptions that were employed in chapter 2.
A question was raised on page 98 in regard to the accuracy of the idealised model
of chapter 2 in the isothermal mode. The question was whether deviation of the
product surface temperature from its idealised value, which had been found using
the detailed duct model to significantly affect the rate of evaporation from the product surface (under some inlet air and refrigerant conditions), could thereby impact
significantly on the predicted isothermal HPD performance. The integrated system
model now in our posession enables us to address this question. In chapter 3 we
found that the detailed drier duct model can be made to agree with the idealised
duct model, in which the surface temperature is assumed constant, to within a few
percent by artificially forcing the surface temperature equal to its idealised value (the
inlet air dry-bulb temperature). When this is done in the integrated whole-system
model for the baseline case, the isothermal mode yields the following performance
data: SMER = 13.4 kg kWh−1 MER = 54.0 kg h−1 . As can be seen, the ultimate impact of surface temperature variation on predicted system performance is quite small,
only a few per cent. (A similarly small impact has been observed when testing the
system in the other configurations discussed below.) On reflection this small sensitivity to the temperature idealisation is not a surprise. The product temperatures that
are employed in the drier duct models can have a strong impact on the predicted drying rates. But also associated with these drying rates is a variation in the rate of heat
delivery to the product. In the integrated HPD system, the rate of heat delivery from
the refrigerant is ultimately limited by the compressor specifications and the broad
temperature range across which the heat pump operates. This result (supported by
132
the agreement between Figs. 2.4 and 4.6) shows that the model outputs presented in
chapter 3 are compatible with chapter 2’s analysis of the isothermal mode.
Sensitivity to system parameters
The sensitivity of system performance to a number of parameters is briefly examined
in this subsection by varying the parameters, one at a time, from their baseline values.
We consider the set of parameters listed in the first column of Table 4.6. The baseline
value of each parameter is shown in the second column of the table. In each case, the
MER and SMER can be compared with the baseline values (presented on page 131).
Parameter
Baseline
δ
1.0 mm
l
6.7 cm
D
1 cm
w
1m
ND
10
Aco
1 m2
Aev
1 m2
MER, kg h−1
SMER, kg kWh−1
(55.2)
(13.6)
1.5 mm
54.2
13.2
0.5 mm
56.2
14.1
11.1 cm
54.1
13.2
4.8 cm
55.6
13.8
1.5 cm
55.4
13.8
0.5 cm
55.6
12.3
1.5 m
56.1
14.2
0.5 m
52.8
11.9
15
56.5
14.5
5
51.8
11.4
1.5 m2
56.4
13.8
0.5 m2
54.5
13.3
1.5 m2
56.5
14.4
0.5 m2
46.7
9.9
Modified
Table 4.6: Sensitivity of MER and SMER to key system parameters.
It is clear that there is scope for improvement in the system performance if the
system dimensions are appropriately modified: as Table 4.6 shows, the baseline scenario is not optimized with respect to any of the parameters that have been considered. On the other hand, the system performance is not very sensitive to any of
these parameters, in the sense that a fairly large change (50% of the baseline value)
in any one of the parameters typically causes a much smaller change in system performance. The performance appears to be most sensitive to variations in the product mass exchange area (through changes either in the width w or the number ND
of ducts) and to variations in the evaporator face area. This latter sensitivity is not
133
surprising, since in the isothermal mode the effective heat exchange area at the condenser is very large, and in the baseline case the evaporator is the component with
the largest entropy creation (Table 4.5). It appears that the evaporator face area assumed in the baseline case is undersized. A further substantial increase in the evaporator face area, to 2.5m2 , yields a smaller improvement, resulting in the following
performance: MER = 56.9 kg h−1 , SMER = 14.8 kg kWh−1 . Simultaneously adjusing several parameters with the aim of increasing system performance, we obtain
MER = 60.6 kg h−1 , SMER = 17.4 kg kWh−1 when l = 4.4 cm, D = 1.5 cm, w = 2 m,
ND = 20, Aev = 2.5 m2 , with other system parameters taking their baseline values.
We briefly return here to the impact on system performance of the product thickness. Fig. 4.7 shows the effect that product thickness has on the energy performance
of the isothermal and adiabatic modes. Since we are considering the constant dryingrate period, and convective heat transfer to the product is not affected by its thickness, the adiabatic performance is unaffected by product thickness. In contrast, a
thick product layer presents a significant thermal resistance to conductive heat transfer, and rapidly nullifies the benefit of the isothermal mode. The curve shown in
Fig. 4.7 is similar to the ẆP = 5kW of Fig. 2.11, as expected. As has already been
discussed, the requirement that the product must be able to be spread thin (together
with the typically very high humidities in the isothermal mode) limits the products
for which isothermal drying will be appropriate; a glance at Fig. 4.7(a) suggests that
the contact HPD system being modelled would yield a significant performance advantage only in the drying of products that can be spread into layers less than about
1cm thick. However, Fig. 4.7(b) shows a significant potential benefit of the isothermal
HPD mode, in cases where it may be applicable. The figure has been produced by
varying the product thickness, and plotting the resulting MER and SMER against one
another. As the figure indicates, isothermal HPD may enable energy performance
and MER to be maximised simultaneously (by ensuring a thin product layer). This
absence of a trade-off between the SMER and the product throughput contrasts (for
instance) with adiabatic timber HPD systems, which must be operated at relatively
low drying rates to obtain good energy performance.
4.7.
Economic case-study
Finally, we use the system performance values that we have obtained for ICHPD to
conduct a tentative analysis of the relative economics of ICHPD. We consider an operation that produces 1000 kg of waste sludge daily, with an initial moisture content
of 0.65 kg/kg (dry-mass basis). A waste-sludge drying operation has been selected
for the following reasons: (1) Adiabatic HPD of filter-cake sludge is common practice
134
15
15
10
SMER (kg/kWh)
SMER (kg/kWh)
Isothermal
Adiabatic
5
0
0
10
5
0
20 25 30 35 40 45 50 55 60
MER (kg/h)
10
20
30
40
50
Product thickness, δ (mm)
Figure 4.7: MER and SMER relationship with varying product thickness.
today. (2) The value that is added to the product by drying is typically not large compared with the energy cost involved. (3) Waste sludges could be dried under ICHPD
conditions that are set to optimize energy performance. In order to reduce transport
and landfilling costs, this sludge is to be dried to a final moisture content of 0.1 kg/kg,
corresponding to a required drying capacity of 25.625 kg/h. Using the rule of thumb
that the capital cost of adiabatic HPD is between $1 and $2 per watt of heat provision
at the condenser, we conservatively estimate a capital cost of $30, 750 for the adiabatic
case. We assume that this cost is financed at a 7% annualized interest rate.
Table 4.7 shows the value added per day (after electricity costs have been met),
the payback time, and the net present value (NPV) of installation, for adiabatic and
isothermal HPD systems, over a range of scenarios. We do not consider any costs associated with depreciation, servicing or labour, and in evaluating the NPV we assume
a long project lifetime. The equation for NPV is [19]:
∞
NPV =
Rt
∑ (1 + i ) t
(4.7.1)
t =0
where i is the future discount associated with an increment of time, and Rt is the net
revenue at time t. Evaluating the geometric series, we obtain for the present case:
NPV = −CC + Rw
i
1+i
(4.7.2)
135
In (4.7.2), Rw represents the net value added by the HPD system (net of electricity
costs) per time increment, and CC represents the capital cost. The parameters that
are varied to produce Table 4.7 are the cost per unit of electricity, the cost of sludge
transport and landfilling, and the relative capital cost of an ICHPD system, compared
with adiabatic HPD. In each scenario each independent parameter is either ‘high’ or
‘low’. Scenarios 1-4 reflect current electricity prices of roughly 10c kWh−1 . Scenarios 5-8 correspond to a significantly higher electricity price of 30c kWh−1 , and could
represent a future scenario characterised by energy shortages and/or a strong CO2 emission price signal. Scenario 6 (in which electricity is expensive, sludge disposal is
cheap, and ICHPD is costly to install) appears to be the only scenario in which neither
adiabatic HPD nor ICHPD is economically feasible. ICHPD appears to be a sensible
investment in scenarios 1,3,5,7, and 8. However, of the low-electricity-price scenarios considered in Table 4.7, ICHPD is a significantly better investment than adiabatic
HPD only in scenario 1, in which the capital cost associated with ICHPD is equivalent
to that of adiabatic HPD, while disposal costs are low. Both scenarios 3 and 4 have
high disposal costs. The economics favour adiabatic HPD somewhat in 3, and isothermal HPD somewhat in 4. In all of the high-electricity-cost scenarios 5-8, ICHPD is
strongly favoured. In particular, in scenario 5 ICHPD is a sensible investment while
adiabatic HPD yields a negative net value, and in scenario 8 ICPHD is preferable to
adiabatic HPD despite its much higher up-front cost. Electricity and waste disposal
costs vary with region, while the relative capital cost of ICHPD is currently unknown.
Our results indicate that at present the economic viability of ICHPD strongly depends
on its capital cost being less than three times that of adiabatic HPD, but that this dependency could be lessened or reversed by increases in the costs of sludge disposal or
of electricity. Viewed alternatively, Table 4.7 shows that the isothermal mode’s high
energy efficiency makes its economics relatively insensitive to the price of electricity,
a result which may be significant in a time of energy-price uncertainty.
Table 4.7: Economics of adiabatic HPD and ICHPD.
Scenario
1
2
3
4
5
6
7
8
Electricity cost, $/kWh
0.1
0.1
0.1
0.1
0.3
0.3
0.3
0.3
Disposal costs, $/kg
0.05
0.05
0.15
0.15
0.05
0.05
0.15
0.15
1
3
1
3
1
3
1
3
ADI
18.5
18.5
80.0
80.0
-6.2
-6.2
55.4
55.4
ISO
26.7
26.7
88.2
88.2
18.5
18.5
80.0
80.0
ADI
5.5
5.5
1.1
1.1
1.6
3.6
15.2
1.0
3.2
5.5
−
1.6
ISO
−
39.3
1.1
3.6
ADI
68.6
68.6
399.7
399.7
-63.9
-63.9
267.2
267.2
ISO
112.7
51.2
443.8
382.3
68.6
7.1
399.7
338.2
Relative capital cost (ISO)
Value added, $/day
Payback time, years
Net present value, k$
136
4.8.
Conclusions
A whole-system HPD model has been produced by combining the detailed air flow
duct model of chapter 3 with models of the remaining heat pump components. The
system model has been used to confirm the previous finding that isothermal contact
dehumidification drying may increase drying energy efficiency threefold compared
with conventional adiabatic dehumidification HPDs. In particular this finding has
been strengthened by accounting for the tradeoff between pressure and heat-transfer
irreversibilites on the refrigerant side. Deviation from purely isothermal behaviour
has been found to influence the MER and SMER that is predicted by the whole-system
model by only a few per cent. An exergy analysis of the system has been performed
for both the isothermal and adiabatic modes. This has shown that the isothermal
mode derives about half of its energy efficiency from a reduction in the irreversibility associated with the transfer of heat to the drying process. The remaining half is
associated with the avoidance of air cooling during the drying process. By modifying the specification of a number of key system parameters, we have determined that
the system performance is most sensitive to variation in the surface area of moisture
evaporation and the dimensioning of the refrigerant evaporator. By simultaneously
modifying five key parameters from their baseline values, we have obtained a significant improvement in predicted system performance, from a SMER of 13.6 kg kWh−1
to a SMER of 17.4 kg kWh−1 in the system, operating at 55◦ C. We have also examined
the relationship between the SMER, the product thickness δ, and the product throughput as indicated by the MER, showing that ICHPD may allow MER and SMER to be
optimised simultaneously through the use of a thin product layer.
Finally, we have examined a case-study of the economics of ICHPD. Since we are
not yet in a position to assess the likely capital cost of an effective ICHPD system,
we are unable to assess its current viability further. However, we have shown that
the viability of ICHPD would become substantially less sensitive to capital costs if
either waste-disposal or electricity prices increase significantly, and that isothermal
HPD provides an opportunity to minimize risk due to uncertain electricity prices.
At this point we have obtained a model that describes our baseline contact HPD
system in detail, in the constant drying-rate period. However, one of the key motivations for developing the flexible air-flow model was to enable a dynamical process
model to be constructed. In a batch drier the conditions at the start of system operation provide a poor guide to the conditions that will prevail later in the process.
To create a dynamical dryer model, enabling overall system performance to be predicted, the present chapter’s HPD model must be linked with a dynamic model of the
internal drying processes, to produce a detailed dynamical model of the entire drying
process. Such an effort makes up the remainder of the work reported in this thesis.
137
4.9.
Nomenclature for chapter 4
Roman and Greek symbols
Symbol
Definition [units]
A0 /A p
Ratio of total airside heat exchange area to the primary area [–]
a jk b jk
Scroll compressor model coefficients [–]
Af
Face area [m2 ]
Am, f
Minimum flow cross-section within heat exchanger [m]
b
Bypass ratio [ṁ a,co − ṁ a,ev ]/ṁ a,co [–]
CD1, CD2, EV
Air-side condenser, plate, evaporator
CP, F, FD, LR, T
Compressor, Fan, Flow distributor, Liquid receiver, Throttle
cw
Specific heat capacity of water [J/kg-K]
D
Heating plate refrigerant tube internal diameter [m]
Dh
Air-side hydraulic diameter [m]
Do
Tube outer diameter [m]
e
Steady flow exergy [J/kg]
∆Eco
Compressor heat rejection rate [W]
f
Moisture condensation fraction [–]
fa
Friction factor [–]
G
Mass velocity [kg/m2 -s]
G, g
Gibbs free energy [J], Specific Gibbs energy [J/kg]
H, h
Enthalpy [J], Specific enthalpy [J/kg]
k
Dynamic loss coefficient [–]
S, s
Entropy [J/K], Specific entropy [J/kg-K]
ST
Vertical tube separation in heat exchanger [m]
l
Heating plate condenser tube spacing [m]
L
Heating plate length [m]
lb
Length of each flow branch within CD2 [m]
Ld
Coil depth [m]
ṁ
Mass flow rate [kg/s]
MER
Moisture Extraction Rate [kg/s]
nb
Number of flow branches in each CD2 plate [–]
ND
Number of ducts [–]
nx
Number of lateral CVs in duct model [–]
p
Pressure [Pa]
P
Power input [W]
Q̇
Rate of heat transfer [W]
138
Re
Reynolds number [–]
S f /Y f
Ratio of fin spacing to fin thickness [–]
SMER
Specific Moisture Extraction Rate [kg/kWh]
V̇
Swept volume [m3 /s]
w
Width of heating plate [m]
W
Work input [J]
Ẇ
Power input [W]
x
Distance through kiln [m], Vapour mass quality [–]
z
Distance along refrigerant flow in CD2 [m]
δ
Paste thickness [m]
ηi
Isentropic efficiency [–]
ηv
Volumetric efficiency [–]
µ
Dynamic viscosity of fluid [N-s/m2 ]
µi
Chemical potential of species i [J/mole]
ρ
Density [kg/m3 ]
φ
Relative humidity [–]
χtt
Lockhart Martinelli parameter [–]
ω
Humidity ratio [kg vapour/kg-dry]
Subscripts and superscripts
Symbol
Definition
0
Environment
1, 2,
20 ,
A, B, C,
3, 4
C0 ,
Locations on refrigerant cycle
D, E
Locations on air cycle
co, ev
Condenser, evaporator
D, F, P
Ducts, fan, compressor
f
Friction
i
ith -control-volume
k, a, v, w
Species-k, dry-air, vapour, liquid
L
Liquid-line
n, w, s, e
North, west, south, east (control-volume boundaries)
m
Mass-exchange, momentum-change
p
Heating-plate
r
Refrigerant
S, s
Surface
sat
Saturation-condition
139
Chapter 5
Dynamical HPD model.
The equations presented at the end of Appendix B are a detailed representation of
much of what occurs within a porous hygroscopic medium undergoing drying. In
practice, in order to produce a workable whole-system model, much of the complexity of these equations has had to be stripped away. In order to make this achievable,
we have built upon an approach to simplifying the volume-averaged drying equations that has been developed by Wang and Chen [178]. Wang and Chen’s method
allows the effects of temperature gradients to be incorporated into a simple diffusion
drying model. A straightforward generalisation of the method, which is described
below, allows a sufficiently detailed representation of the porous product layer to capture plate heat transfer and temperature-gradient effects, while at the same time being relatively simple and easy to implement within the already-complex HPD model.
The material in the present chapter proceeds from the results of Appendix B.
5.1.
Introduction
A key motivation for developing the flexible framework of chapter 3 was to enable the
phenomena that occur in the falling-rate period of drying to be modelled. As was outlined in chapter 1, the majority of numerical drying simulations reported in the past
two decades have been based either on the volume averaging method introduced by
Whitaker [188] [e.g. 112, 53, 106]; on a simpler “diffusion” model of moisture transport [18, 86]; or on the highly simplified assumption of a discrete, receding, drying
front [95]. The trade-offs between comprehensiveness, computational tractability and
ease of implementation influence the correct choice of model for a particular context.
Detailed models are significantly more demanding to produce and run than diffusion models [86], which can be used when low drying intensity allows the governing
140
equations to be reduced to the form of the diffusion equation [193, 178, 123].
The somewhat unusual boundary conditions that arise from contact heat transfer
suggest that the dynamical internal-process drying model, if it is to successfully represent the drying process, will need to be consistent with temperature-gradient effects
that are predicted by the detailed theory of drying. Previous researchers have found
that temperature gradients may significantly influence internal moisture transport
in convective drying. In the case of an adiabatic slab as depicted in Fig. 5.1, Wang
and Chen [178] have shown how temperature effects can be incorporated into the
diffusion equation. The resulting effective moisture diffusivity De , which varies to
incorporate the link between heat and mass transport, can be predicted from local
conditions in the porous medium using the volume-averaging theory of Whitaker
[188, 193]. Such a diffusion model thus has the potential to combine comprehensiveness under appropriate conditions with simplicity and computational speed.
Strictly speaking, Whitaker’s drying equations [188, 193] are restricted to describing the drying of porous media whose rigid solid phase contains no bound moisture.
However Wang and Chen [178] have obtained satisfactory agreement between their
diffusion model and experimental results for the drying of banana pieces, by selecting an intrinsic permeability K D of 10−19 m2 . Furthermore, they have shown that
when the intrinsic permeability K D of the medium is 10−19 m2 or less, their diffusion
model predicts behaviour that deviates markedly from a discrete drying front. To
illustrate the linkage between temperature gradients and moisture transport, Fig. 5.2
has been produced using the method described in reference [178]. The figure shows
the relative contributions of vapour flux ( f v ) and liquid flux ( f β ), and of saturation
gradients ( f s ) and temperature gradients ( f T ), to moisture transport in the porous
medium in question (banana). (The product saturation s is defined as the ratio of the
local moisture content to its maximum value, which occurs when the pores are filled
with condensed moisture. Assuming constant ρ β , this can be written:
s=
eβ
e β ρ β + eγ h ρ v i γ
≈
eρ β
e
(5.1.1)
where the terms in (5.1.1) are defined in Appendix B.) Note that f v + f β = 1 and
f s + f T = 1 in Fig. 5.2: in the model of Wang and Chen [178] all moisture transport
is with vapour (v) and the liquid phase (β), and the driving force for moisture transport is created entirely by temperature (T) and saturation (s) gradients. The theory
underlying Fig. 5.2 is developed below, in the present chapter; the point here is that
under adiabatic drying conditions the contribution to moisture transport of temperature gradients (−1× line 4) is negative and, at low saturations, is significant in mag-
nitude relative to the contribution due to saturation gradients (line 3). The negative
sign indicates that temperature gradients retard moisture transport under adiabatic
drying conditions (i.e. in the absence of plate heat transfer).
141
T∞, φ∞, v∞
y
δ
top surface
product
plate interface
0
adiabatic plate
Figure 5.1: Schematic of 1-D slab undergoing convective drying.
Under adiabatic drying conditions, during the drying process, heat is transferred
from the product surface to the region within the product where the drying process is
occurring, and the surface temperature therefore exceeds the temperature within the
product. In contrast, in contact drying the temperature gradient within the product
generally takes the opposite sign (Fig. 1.11). Since the bulk of the heat transferred
within the product is converted to latent heat [178], a rough measure of the magnitude of the temperature gradient in the isothermal case is provided by the ratio of
the isothermal MER to the adiabatic MER. We have already, in chapter 4, established
this ratio to be a factor of approximately three under typical HPD conditions in the
constant drying-rate period. For a rough estimate, this can be taken to indicate (by extension) that the product temperature gradient in the isothermal mode may be on the
order of three times (3×) that in the adiabatic mode, and opposite in sign. This result,
together with the considerations in the previous paragraph, suggests that the temper-
ature gradient within a porous product in a contact drier may play a significant role
in driving moisture from the product during the falling drying rate period.
In this chapter a dynamical drying model is developed, using Patankar’s discretisation method. The model employs the constitutive relations that were used by Wang
and Chen [178], although the model is more general than theirs, in that it includes a
nonzero heat flux q0 at the base of the product, and the energy balance equation retains the accumulation term hρTot iC p
∂h T i
∂t
(see the discussion below). For simplicity,
the dynamical model is first developed as a 1D model. A 2D model, comprising
a series of side-by-side, non-interacting 1D models, each with its own independent
gas-side boundary condition, is a straightforward extension of the 1D model. In other
words, we use the idealisation of a thin product layer within which lateral moisture
transport can be neglected. This 2D model is then combined with the duct air-flow
model of chapter 3, to provide the gas-side boundary conditions experienced by the
product at different locations within the duct. This produces a dynamical duct model,
142
2
1.8
Fractional contribution to diffusivity
1.6
1.4
3
1.2
1
1
0.8
2
0.6
4
0.4
0.2
0
0
0.2
0.4
0.6
Saturation
0.8
1
Figure 5.2: Relative contributions to the effective moisture diffusivity,
calculated using the method of Wang and Chen [178]. (1) f v , due to
vapour diffusion; (2) f β , due to liquid capillarity; (3) f s , due to saturation gradients; (4) −1 × f T , due to temperature gradients.
whose outputs are presented and discussed. Finally, this duct model is incorporated
into the whole-system HPD model to produce a dynamical whole-system model. This
is used to investigate the time variation of system behaviour and performance, and
to assess the application of the characteristic drying curve method to ICHPD.
5.2.
Theory
In this section the discretised equations are derived for the 1-D product model that
is employed in this chapter. First a derivation is presented of an extension of the
equations of Wang and Chen [178], generalised to include a plate heat flux q0 . The
resulting set of equations has not been implemented within the whole-system model.
However, by modifying the derivation we obtain the discretised equations that do
underly the model that has been incorporated into the ICHPD model.
143
Modifying the drying equations into diffusion form
In their paper “Heat and mass transfer during low intensity convection drying”,
Wang and Chen [179] start from the classic drying equations, specifically equations
(IV.A-10), (VI-3), and (VI-1) of Whitaker [188]. In order to keep the present development self-contained, we proceed from equations (B.7.2), (B.7.3), (B.7.10) and (B.7.15)
of the present thesis. Whitaker’s equations result when the terms that describe hygroscopic behaviour are dropped from these equations, to obtain:
∂ ρ + ∇ · hnβ i = − ṁ βγ
∂t β
∂
hρv i + ∇ · hnv i = ṁ βγ
∂t
∂h T i
hρTot iC p
+ ∇h T i · ∑hni ic p,i + hṁ βγ i∆hv = ∇ · KTeff · ∇hTi
∂t
i
(5.2.1)
(5.2.2)
(5.2.3)
The three equations (5.2.1)–(5.2.3) are used as the basis of the dynamical drying model
that is developed in this chapter. Note that since these are simply Whitaker’s classic
drying equations, the material in appendix B, which derives a more general set of
equations, is not crucial to the work in the present chapter.
Wang and Chen [179] use a number of arguments to simplify Whitaker’s equations. We replicate their argument in terms of equations (5.2.1)–(5.2.3), as follows:
1. Assuming heat transfer associated with convection mass transfer can be ne
glected, the term ∇h T i · ∑hni ic p,i is dropped from (5.2.3).
i
2.
∂
∂t h ρv i
is dropped from (5.2.2), due to the following. Since hρv i hρ β i for
almost all of the drying process, comparing (5.2.2) and (5.2.1) we may write:
∂
∂
hρ β i + hργ i
hρ β i ≈
∂t
∂t
hρv iγ γ
γ
= − ∇ · hnβ i + hρv i hvγ i − hργ i Deff · ∇
hργ iγ
(5.2.4)
(5.2.5)
i.e., using (5.2.1) again,
hρv iγ = ṁ βγ
∇ · hρv iγ hvγ i − hργ iγ Deff · ∇
γ
hργ i
3. The term hρTot i C p
∂h T i
∂t
(5.2.6)
is dropped from (5.2.3), due to the following. The to-
tal energy change associated with the heating of the drying product is much
smaller than the energy consumed by the conversion of liquid water into vapour;
that is, expressed in terms of the Kossovich number K0 :
(1 − xm,0 )c p,s + xm,0 c p,β ( Tair − Twb )
hρTot i C p ∆T
−1
K0 =
≤
1
∆hv · ∆hρM i
∆hv · xm,0
(5.2.7)
144
In (5.2.7), ∆T is the difference between the final product temperature, which
(under constant drying conditions) is the same as the ambient temperature Tair ,
and the initial temperature, which can be estimated using the wet-bulb air temperature, Twb , as a lower bound. Finally xm,0 represents the initial moisture
content (on a wet-mass basis). The inequalities in (5.2.7) allow us to write:
hρTot i C p
∂ hT i
∆hv ṁ βγ
∂t
(5.2.8)
After making the assumptions and dropping the terms listed above, the governing
equations for the 1-D case become:
usat
∂s ∂hn β i
+
= −hṁ βγ i
∂t
∂y
∂hnv i
= hṁ βγ i
∂y
∂ T ∂h T i k
∆hv · hṁ βγ i =
∂y eff ∂y
(5.2.9)
(5.2.10)
(5.2.11)
In Eqs (5.2.9)–(5.2.11), the porous medium is assumed rigid and of constant porosity
e ≡ e β + eγ , the symbol s ≡ e β /e represents the saturation of its pores, and usat ≡ ρ β e
is the liquid density (per unit volume of the porous medium) when s = 1. We assume
that variations in the liquid density ρ β may be neglected in the temperature range
under consideration, so that ρ β ≡ hρ β i β and usat is constant, with:
usat s = e β ρ β = e β hρ β i β = hρ β i
(5.2.12)
We follow Wang and Chen [179] with the assumption Deff = eγ Dva = e(1 − s)Dva .
The convection-diffusion equations (B.7.9) and (B.7.10) then become as follows:
hρ a iγ
hna i = hρ a iγ hvγ i − hργ iγ e(1 − s)Dva ∇
(5.2.13a)
hργ iγ
hρv iγ
hnv i = hρv iγ hvγ i − hργ iγ e(1 − s)Dva ∇
(5.2.13b)
hργ iγ
Since the density of liquid water typically exceeds that of air at atmospheric pressure
by a factor greater than 800, the net mass flow of dry air into the product (throughout
the drying process) is much smaller than the vapour mass flow out. Throughout the
drying process, the inward dry-air mass flux can therefore be assumed much smaller
than the outward vapour mass flux, as discussed by Whitaker [189]. This low dry-air
mass flux is created when vaporization of water inside the pores produces an excess
internal pressure, i.e. ∂h pγ iγ /∂y < 0, that results in an outward gas flux according to
Darcy’s law, that matches the diffusion mass fluxes. Setting hna i = 0 in (5.2.13a) and
noting that hρ a iγ + hρv iγ = hργ iγ , we can sum (5.2.13a) and (5.2.13b) to obtain
hnv i = h ρ γ i γ hvγ i =
hργ iγ
h ρ a i γ hvγ i
hρ a iγ
(5.2.14)
145
Substituting (5.2.13a) with hna i = 0 into (5.2.14) yields
hργ iγ
hρv iγ
γ
h
ρ
i
e
(
1
−
s
)
D
∇
hnv i = −
γ
va
hρ a iγ
hργ iγ
(5.2.15)
Assuming a rigid (stationary) solid phase which contains no moisture, and neglecting the impact of gravity, Darcy’s law (B.7.7) and (B.7.11) for the β phase becomes:
hn β i = − h ρ β i
β
krβ (s)k0β U
µβ
· ∇[h pγ iγ − pc ]
(5.2.16)
Wang and Chen [179] assume that the gas pressure is approximately constant throughout the porous medium. This can be deduced from Darcy’s law on the assumption
that the evaporative flux is small compared to kγ ργ p0 /(µγ δ), where p0 is the ambient pressure and δ is the product thickness. Assuming constant pressure and density
within the gas phase, equations (5.2.16) and (5.2.15) reduce for the 1-D case to
krβ k0β ∂pc
hn β i =ρ β
µ β ∂y
hnv i = −
(5.2.17a)
hργ iγ
∂ hρv iγ
e
(
1
−
s
)
D
va
hρ a iγ
∂y
(5.2.17b)
The diffusion vapour flux has been estimated by Wang and Chen [179] using
hnv i = −e(1 − s)Dva
∂hρv i
∂y
Note that Equation (5.2.17b) differs from (5.2.18), by a factor of
hργ iγ , this additional factor will have a fairly small effect.
(5.2.18)
hργ iγ
.
hρ a iγ
Since hρ a iγ ≈
Neglecting variations in ∆hv , Equations (5.2.10) and (5.2.11) combine to yield:
T
∂ keff
∂h T i ∂hnv i
=
∂y
∂y ∆hv ∂y
(5.2.19)
Integrating (5.2.19) from 0 to y, we find that:
k T ∂h T i k T ∂h T i hnv i − hnv i = eff
− eff
∆hv ∂y y ∆hv ∂y 0
y
0
(5.2.20)
In the case considered by Wang and Chen [179], of a product sitting
on an adiabatic
k T ∂h T i plate (Figure 5.1) the midline y = 0 is symmetric and both hnv i and ∆heffv ∂y can
0
0
be dropped. In the case of a plate drying system, there is a nonzero heat flux at y = 0
and the second term must be retained. In this case we obtain:
hnv i =
T
In (5.2.21), q0 = −keff
∂h T i ∂y 0
T
keff
∂h T i
q
+ 0
∆hv ∂y
∆hv
(5.2.21)
is the heat flux through the plate. We will proceed using
(5.2.21) rather than the simpler equation that results when q0 = 0. To obtain the
146
equations of Wang and Chen [179], set q0 = 0 everywhere in what follows. Using the
chain rule for partial derivatives, characterising all physical properties in terms of the
independent parameters s and T, we can use (5.2.17b) to rewrite (5.2.21) as follows:
T
h ∂hρ i ∂h T i ∂hρ i ∂s i
keff
hργ iγ
q
∂h T i
v
v
=(−e)
(
1
−
s
)
D
+
− 0
va
γ
∆hv ∂y
hρ a i
∂h T i ∂y
∂s ∂y
∆hv
⇒
hρ iγ
∂hρ i
(5.2.22)
q
γ
v ∂s
0
∂h T i (−e) hρa iγ (1 − s)Dva ∂s ∂y − ∆hv
= hρ iγ
∂hρ i
∂y
T /∆h
e hργiγ (1 − s)Dva ∂hTvi + keff
v
(5.2.23)
a
=−K
∂s
− K q q0
∂y
(5.2.24)
where
hρ iγ
K=
and
Kq =
e hργiγ (1 − s)Dva
a
e(1 − s)Dva
∂hρv i
∂h T i
∂hρv i
∂s
T /∆h
+ keff
v
1/∆hv
hργ iγ
∂hρ i
e hρ iγ (1 − s)Dva ∂hTvi
a
T /∆h
+ keff
v
(5.2.25)
(5.2.26)
Combining (5.2.9) and (5.2.10) we obtain
usat
∂hn β i ∂hnv i
∂s
=−
−
∂t
∂y
∂y
(5.2.27)
Wang and Chen [178] proceed by noting that the intrinsic-averaged vapour density
hρv iγ and the capillary pressure pc can both be expressed as functions of h T i and s.
The following chain rules for partial derivatives then apply:
∂hρv iγ ∂hρv iγ ∂h T i ∂hρv iγ ∂hsi
=
+
∂y
∂h T i ∂y
∂s
∂y
∂pc
∂pc ∂h T i ∂pc ∂s
=
+
∂y
∂h T i ∂y
∂s ∂y
Using (5.2.17) and (5.2.28) in (5.2.27), we can obtain
∂
∂h T i
∂s
∂h T i
∂s
∂s
usat
=
DβT
+ Dβs + DvT
+ Dvs
∂t
∂y
∂y
∂y
∂y
∂y
(5.2.28a)
(5.2.28b)
(5.2.29)
where
DβT = −
Dβs = −
krβ k0β
µβ
krβ k0β
µβ
ρβ
∂pc
∂h T i
(5.2.30)
ρβ
∂pc
∂s
(5.2.31)
hργ iγ
∂hρv iγ
(1 − s)Dva
γ
hρ a i
∂h T i
γ
hργ i
∂hρv iγ
=e
(1 − s)Dva
γ
hρ a i
∂s
DvT = e
(5.2.32)
Dvs
(5.2.33)
147
Finally substituting (5.2.24) into (5.2.29) we obtain:
∂
∂s
∂s
∂s
=
[ Dβs + Dvs ] + [ DβT + DvT ](−K − Kq q0 )
usat
∂t
∂y
∂y
∂y
∂
∂s
∂
=
Cs
De
− q0
∂y
∂y
∂y
(5.2.34)
(5.2.35)
where the effective diffusivity De and heat flux correction coefficient Cs are given by
De = Dβs + Dvs − K ( DβT + DvT )
(5.2.36)
Cs = [ DβT + DvT ]Kq
(5.2.37)
Gathering together the governing equations (5.2.24) and (5.2.35), we have:
∂s
∂T
= − K − K q q0
∂y
∂y
∂Cs
∂s
∂
∂s
usat
=
De
− q0
∂t
∂y
∂y
∂y
(5.2.38)
(5.2.39)
In the case where q0 = 0, (5.2.39) becomes the pure diffusion equation. With nonzero
s
q0 , (5.2.39) is the diffusion equation with local source term −q0 ∂C
∂y .
We briefly define the quantities f v , f β , f s and f T that appeared at the start of the
chapter, in terms of the quantities defined above, as follows:
f v =( Dvs − KDvT )/De
(5.2.40)
f β =( Dβs − KDβT )/De
(5.2.41)
f s =( Dvs + Dβs )/De
(5.2.42)
f T = − K DvT + DβT /De
(5.2.43)
In the adiabatic case, in which q0 = 0, the total moisture flux can be expressed as
∂s
∂s
( f v + f β ) De ∂y
, or as ( f s + f T ) De ∂y
. Thus f v + f β and f s + f T represent alternative
decompositions of the total moisture transport. The first decomposition expresses
the total moisture transport in terms of the vapour transport and the liquid-phase
transport, and the second decomposition expresses the total moisture transport in
terms of the saturation driving-force and the temperature driving-force.
Drying model equations
The equations derived in the previous subsection represent a generalisation of those
of Wang and Chen [179], since they incorporate the effects of the plate heat flux q0 . In
the present section an alternative set of discretised equations is developed, based on
the definitions of the previous section, and employing the finite volume method described by Patankar [117]. Rather than specifying the heat flux at the base of the product, the boundary condition representing heat transfer from the plate is expressed in
148
terms of a heat transfer coefficient at the bottom of the product. This has been done
in order to be able to combine the model easily with those of the previous chapters.
The moisture balance, as expressed by Equation (5.2.29), can be written as follows:
∂T
∂s
∂
∂s
DT
(5.2.44)
usat
=
+ Ds
∂t
∂y
∂y
∂y
In (5.2.44), the following definitions have been used:
DT = DβT + DγT
(5.2.45)
Ds = Dβs + Dγs
(5.2.46)
This becomes, upon integration and discretisation using the fully-implicit scheme:
[usat V ]
T 1 − TS1
s1P − s0P
T 1 − TP1
=[ DT A]n N
− [ D T A ]s P
∆t
∆yn
∆ys
+ [ Ds A ] n
s1 − s1S
s1N − s1P
− [ Ds A ] s P
∆yn
∆ys
(5.2.47)
Rearrangement of (5.2.47) yields
s 1
aPs s1P = bPs + aN
sN + aSs s1S
(5.2.48)
where the discretisation coefficients aP , bP , aN and aS are as follows:
Ds A
s
aN
=
∆y n
Ds A
aSs =
∆y s
usat V
a0s
=
P
∆t P
s
s
aPs = a0s
P + aN + aS
0
bPs = a0s
P sP + [ D T A ]n
(5.2.49)
(5.2.50)
(5.2.51)
(5.2.52)
1
TN
TP1
−
∆yn
− [ D T A ]s
TP1
TS1
−
∆yn
Dropping the convective heat transfer term from (5.2.3) yields:
∂T
∂hnv i
∂
T ∂T
hρTot iC p
=−
∆h +
keff
∂t
∂y
∂y
∂y
(5.2.53)
(5.2.54)
Or, upon integration and discretisation:
hρTot iC p V 1
T 1 − TS1
T 1 − TP1
T
T
TP − TP0 =[keff
A ]n N
− [keff
A ]s P
∆t
∆y
∆y
+ hnv i A s − hnv i A n ∆h
(5.2.55)
Rearrangement of Equation (5.2.55) yields the following discretisation:
T 1
aPT TP1 = bPT + aN
TN + aST TS1
(5.2.56)
149
where the discretisation coefficients aP , bP , aN and aS are as follows:
#
"
T A
keff
T
aN =
∆y
"
#n
T
k A
aST = eff
∆y
(5.2.57)
(5.2.58)
s
hρTot iC p V
∆t
T
0T
T
aP = aP + aN
+ aST
0
bPT = a0T
hnv i A s − hnv i A n ∆h
P TP +
a0T
P =
(5.2.59)
(5.2.60)
(5.2.61)
The 1-D model has been implemented with a relaxation factor which is applied
to both the temperature and the saturation discretisation [117, p.67]. Typically this
relaxation factor has been set to a value of 0.5, although smaller values have been
applied in the 2D duct model, which has been found to be more prone to numerical
instability. The convergence criterion that has been adopted is a maximum absolute
relative change of less than 10−4 in the updated estimate of the temperature or saturation in any control volume. Since the 2D duct model is obtained straightforwardly
by replacing the time-invariant drying model of chapter 1, which has been used in
the duct model described in chapter 3, with the dynamical model described above,
we do not describe it further here, simply noting that the resulting model has as its
convergence criterion that the convergence criteria of both submodels must be simultaneously met. A global relaxation factor has been applied to all variables that are
updated with each iteration, and this has been set to gradually decrease, so that the
model gradually mixes in less and less of its new estimate with each iteration. This
allows the system to quickly converge toward the solution when it is in a numerically
stable region, but to avoid oscillating indefinitely in regions of numerical instability.
Boundary and initial conditions
The boundary conditions for the internal-process drying model are as follows:
∂T
∂s + Ds
=0
∂y
∂y y=0
(5.2.62)
∂T
∂s + Ds
= h•m hρv iγy=δ − ρv,∞
y
=
δ
∂y
∂y
(5.2.63)
DT
T
−keff
DT
∂h T i
+ ∆hv n β y=δ = h• h T iy=δ − T∞
∂y
(5.2.64)
∂h T i = α p ( Tr − Tp )
∂y y=0
(5.2.65)
T
−keff
150
where, from an argument entirely analogous to that on page 29:
1
1/α1 + 1/α2
πD
α1 = αr
l
2πk p /l
i
α2 = h
2πx p
2l
ln πD
sinh
l
αp =
(5.2.66)
(5.2.67)
(5.2.68)
where α p denotes the total plate heat transfer coefficient (for heat transfer from the
refrigerant to the base of the product). The adiabatic mode is straightforwardly simulated by simply setting α p to a very small value, such as 10−8 W/(m2 · K).
The initial condition that is adopted in this chapter is that the initial product temperature is equal to the dry-bulb temperature of the inlet air, and the product saturation s = 0.99. (This initial condition would correspond to an experimental protocol
of allowing the system to heat up before switching on the fans for air flow.)
5.3.
Results and discussion: 1-D model
In this section, results from the 1D model are presented and discussed. Fig. 5.3 shows
the half-hour saturation profiles within the product, and Fig. 5.4 shows the drying
rate and surface temperatures, in the adiabatic and isothermal cases, under the drying conditions investigated by Wang and Chen [178], obtained by setting T∞ , ρv,∞ , h•m ,
h• to the values employed in that reference (Table 5.7). In addition, for the isothermal
case, a refrigerant saturated condensing temperature 5◦ C greater than the air inlet
temperature, i.e. Trsat = 65◦ C, has been assumed. In each case the top subfigure represents the adiabatic case, and the bottom subfigure represents the isothermal case.
The adiabatic results show good agreement (in terms of the saturation profiles and
overall drying time) with those that have been presented by Wang and Chen [178].
In the isothermal case, a surprising dip in the product surface temperature occurs
at the end of the drying process – Fig. 5.4(b), dashed line. This late dip in the surface
temperature contrasts with the adiabatic case, in which the surface temperature, after
the initial cool-down at the start of the drying process, only increases as the drying
proceeds. A survey of the drying literature has failed to show up any discussion of
this phenomenon in the context of plate drying. However, the author is aware of
some (unpublished) experimental results that suggest that such a temperature dip
may be a general feature of plate drying. Experimental work conducted in the Otago
heat pump drying research group consistently revealed a late drop in surface and air
temperature, for a variety of product mediums, within a plate drier [140].
Fig. 5.5 shows four snapshots of the isothermal drying process, in order to illus-
151
trate the processes that contribute to this dip in the surface temperature. In order
to more clearly illustrate the effects that underlie the temperature ’dip’, an increased
refrigerant saturated condensing temperature, of 70◦ C, has been used. The figure
shows scaled representations of the volumetric drying intensity, vapour and liquid
fluxes, and temperature, as well as the product saturation. In the first panel, the drying intensity is a maximum at y = 4 mm. Liquid flux is significant below this value of
y, but negligible above. The vapour flux increases significantly around y = 4 mm, to
a maximum at the product surface at y = 5 mm. The circles at the right-hand side of
the figure represent the product surface temperature, which increases from the first
panel to the second, decreases again in the third panel, and then increases again. (All
variables except the local saturation have been scaled to produce this figure, with the
same scaling factor, for each of the variables, applied to all four panels.) The chief reason for this late temperature dip appears to be that, in contrast to the adiabatic case,
the region of maximum drying intensity (the point at which the dashed line peaks)
does not simply recede into the product, but moves back toward the product surface
near the end of the drying process. Thus heat transfer from the plate to the region
of maximum drying intensity must take place, in this final stage of drying, through
an increasing product region, of decreasing saturation and thus thermal conductivity. An increasingly negative temperature gradient within the product thus extends
increasingly far from the heating plate, as can be seen by comparing panel 3 with
panel 2, and the surface temperature falls. This part of the drying process is the only
period during which heat transfer at the product surface is downward, as indicated
by the positive temperature gradient near y = 5 mm. Finally, when drying ceases
altogether, the negative temperature source term associated with drying goes to zero
and the surface temperature approaches its steady-state value, intermediate between
the air temperature (60◦ C) and the refrigerant temperature (65◦ C).
We must in turn explain why the region of maximum drying intensity moves upward during the late stages of drying. This can be understood by again examining
Figs. 5.3 and 5.5, which show that unlike in the adiabatic case, in which an extended
drying front moves progressively deeper into the product, the isothermal saturation profile decreases in a complicated manner, where the saturation at y = 0 falls
faster than the saturation at intermediate values of y. As a consequence, in the late
stages of drying, most remaining moisture is not immediately adjacent to the heating
plate. This complicated saturation profile, seen most clearly in Fig. 5.3(b), appears to
arise because of the temperature gradient within the product, due to plate heat transfer, which drives vapour upward through the product. Examining the first panel of
Fig. 5.5 closely, one can see that the liquid flux at the interface with the heating plate
(y = 0) is negative, matching the positive vapour flux. This negative liquid flux is
driven by a slight positive saturation gradient that initially arises from the unmatched
vapour flux removing moisture from the product at y = 0. Interestingly, this small
152
initial saturation gradient appears to be self-reinforcing, since falling saturation (at
the plate interface y = 0) tends to enhance vapour flux and to inhibit liquid flux. As a
consequence the saturation at y = 0 falls away relative to that immediately above it,
until finally the effect is nullified by declining vapour pressure due to low saturation.
5.4.
Results and discussion: 2-D duct model
We now turn to the outputs of the 2D model that is produced when a row of the
1D drying models described in the previous section replace the steady-state drying
model that had been used in chapters 3–4. The structure and convergence criteria for
the resulting dynamical duct model have been described above. Figure 5.6 shows the
drying rate profiles observed at different locations within the duct, when Tin = 60◦ C,
Tr sat = 65◦ C, φin = 10%, ṁa,in = 1.8 kg/s. This air mass flow rate has been selected
here because, for the baseline geometry under consideration (which is summarized
in Table 2.1 on page 43), this air mass flow rate yields a mean heat transfer coefficient
h = 43.3 W/(m2 · K), which very closely resembles the heat transfer coefficient in
the situation modelled by Wang and Chen [178], h = 43.3 W/(m2 · K). (However,
the mass-transfer coefficient assumed by Wang and Chen [178], hm = 0.0172 m/s, is
not compatible with their heat transfer coefficient and the Chilton-Colburn analogy,
which yields a mean mass-transfer coefficient of 0.045 m/s when ṁa,in = 1.8 kg/s.
Thus the mass-transfer boundary condition employed in this section is numerically
different from that which has been employed in the previous section.)
The top panel of Fig. 5.6 shows the drying rates observed in the adiabatic case,
and the bottom panel shows the drying rates for the isothermal case, with the air and
refrigerant inlet conditions as described in the previous paragraph. At first sight, at
this air mass flow rate, the drying profiles at different locations within the duct do not
appear markedly different from one another. (Lower air flow rates lead to wider variation in local conditions and drying rates.) Nevertheless there is a significant spread
in the times taken to reach completion of the drying process: in the adiabatic case,
product at the duct inlet takes under 450 minutes to essentially reach its final moisture content, while product near the duct air outlet takes about 600 minutes. In the
isothermal case, for which drying times are significantly shorter, the corresponding
spread is from a drying time of about 170 minutes at the inlet to about 190 minutes
at the air outlet. It accords with intuition that the adiabatic mode shows the greater
percentage spread in drying times. In the isothermal mode, refrigerant heat transfer tends to maintain relatively high temperatures (and drying rates) throughout the
duct. The drying rate profiles thus should depend on the product thickness δ, which
determines the thermal resistance presented by the product layer, and, thus, plate
153
1
Saturation
0.8
0.6
0.4
0.2
0
0
1
2
3
y, mm
4
5
4
5
(a) Adiabatic case.
1
Saturation
0.8
0.6
0.4
0.2
0
0
1
2
3
y, mm
(b) Isothermal case.
Figure 5.3: 1D Porous drying model. Half-hour saturation profiles within the product
layer. Initial saturation is 0.99. In the adiabatic case (a) a drying front can be seen
retreating into the product. The situation is more complex in the isothermal case (b).
0.8
70
0.6
60
0.4
50
0.2
40
0
0
Temperature, °C
Drying rate, g/(m2 s)
154
200
400
t, min
30
600
1.6
70
1.2
60
0.8
50
0.4
40
0
0
Temperature, °C
Drying rate, g/(m2 s)
(a) Adiabatic case.
200
400
t, min
30
600
(b) Isothermal case.
Figure 5.4: 1D model. Drying rate (solid) and surface temperature (dashed).
155
1
t=30min
0.5
0
1
t=100min
0.5
0
1
t=125min
0.5
0
1
t=150min
0.5
0
0
1
2
3
4
5
y, mm
Figure 5.5: Four snapshots of internal processes in the isothermal drying
mode, illustrating the reasons for temperature dip and end of drying process.
Solid lines: saturation. Dash-dot line: scaled temperature. Dotted lines: scaled
vapour and liquid fluxes. Dashed line: scaled drying intensity.
156
heat transfer.
Fig. 5.7 shows the path taken by the air during the drying process in the adiabatic mode. The dotted lines trace out instantaneous psychrometric profiles within
the ducts, and their endpoints are marked by diamonds. The solid line connecting
the diamonds is the psychrometric state-path traced by the duct outlet air. The times
that are illustrated (by dotted lines and diamonds) are: t = 0 s (the rightmost path),
120 s, 570 s, and every subsequent 30 minutes. Clearly in the first few minutes there
is a rapid cool-down phase in which the psychrometric path gradually acquires the
adiabatic profile that is expected at steady state (see e.g. Fig. 3.13). Once this adiabatic, constant-enthalpy profile is attained, the path remains constant. The amount of
drying that occurs within the duct (and thus the amount of evaporative cooling of the
airstream) then gradually declines as the saturation, and thus the drying rate, falls.
Fig. 5.8 shows how the surface temperatures and product saturations along the
duct vary with time in the adiabatic mode. The simulation that has produced Figs. 5.7–
5.9 has been conducted using 16 lateral control volumes (in the x-direction), corresponding to each of the 16 lines shown in each panel of Fig. 5.8. The surface temperatures show essentially the same behaviour as has been observed in the 1-D model
case (Fig. 5.4), with minor deviations arising in the duct model, due to the varying
conditions experienced by the product at different locations within the duct. The saturations that are shown in the bottom panel are each the mean value at the given
x-location, i.e. y-averaged values. The bottom panel clearly shows the spread in
drying times that arises due to the differing conditions at either end of the duct. The
direction of air flow is indicated in each sub-figure. During the constant-rate period of
drying, one would expect a constant surface temperature throughout the duct, equal
to the wet-bulb temperature. Fig. 5.8 (a) illustrates that this is not the case throughout the drying process, with surface temperature variation exceeding 5◦ C throughout
much of the drying process. The variation can be explained as follows. Drying is most
intense at the air inlet, so the saturation falls fastest at this end of the duct. As a consequence, the temperature climbs near the duct inlet, since evaporative cooling lessens
as the saturation, and thus the drying rate, falls. Thus (during the falling-rate period)
the surface temperature is highest at the duct inlet, and falls through the duct.
Fig. 5.9 illustrates the variation in the air temperature and humidity ratio, at various duct locations, with time, in the adiabatic mode. As before the path taken by each
control volume is shown. At the air inlet, the air temperature takes a markedly different path from that taken by the surface temperature, as was shown in the first panel
of Fig. 5.8. Unlike the product surface temperature, the air temperature near the duct
inlet remains very close to the inlet dry-bulb temperature (60◦ C). This makes intuitive sense: convective heat transfer maintains a higher duct air temperature despite
157
Drying rate, g/(m2 s)
0.8
0.6
0.4
0.2
0
0
200
400
t, min
600
(a) Adiabatic case.
Drying rate, g/(m2 s)
0.8
0.6
0.4
0.2
0
0
200
400
t, min
600
(b) Isothermal case.
Figure 5.6: 2D duct porous drying model. Plots show time variation of drying rates at
different duct locations: each curve shows the local drying rate at a different control
volume along the flow direction within the duct.
50
40
30
20
10
10
20
30
40
T,°C
50
60
Humidity ratio, g moisture / kg dry air
158
0
Figure 5.7: Psychrometric chart showing snapshots of psychrometric path
traced by air within duct (dotted lines) and the path traced by the outlet air
state (solid line). Adiabatic case. Dashed lines and diamonds correspond to
times t = 0 s, t = 120 s, t = 570 s, and then to every subsequent 30 minutes.
Here ṁ a = 1.8 kg/s, Tin = 60◦ C, φin = 0.1, Trsat = 65◦ C, δ = 1 mm.
159
Surface temperature, °C
60
50
Air flow direction
40
0.9
0.8
saturation, −
0.7
0.6
0.5
Air flow direction
0.4
0.3
0.2
0.1
0
0
100
200
300
400
500
600
700
Time, minutes
Figure 5.8: Variation of product surface temperature and mean product saturation (at each control-volume stack in the x-direction) with time. Each curve
shows the time variation of surface temperature or saturation at a given control volume location within the duct. The direction of air flow is indicated.
Adiabatic case, Tin = 55◦ C, φin = 0.1, ṁa,in = 2 kg/s, δ = 5 × 10−3 m.
160
a lower product surface temperature. At the end of the drying process, as the air
encounters no evaporative cooling, both the temperature and the humidity ratio become constant within the ducts, and all of the temperature lines and the humidity
lines converge. This contrasts with the isothermal case (described below), where as
the drying process comes to an end, heat transfer from the refrigerant drives the air
temperature upward as the air passes through the duct, as can be seen in Fig. 5.9 (a).
Fig. 5.10 shows the variation of gauge pressure and air flow velocity within the
duct in the adiabatic mode. As can be seen, during the drying process, evaporative
cooling lowers the air flow velocity as the air passes through the duct, despite its
acquisition of moisture. This effect has been discussed in chapter 3. As the drying
process comes to an end, evaporative cooling ends, and the only reason for a variation
in the air flow velocity through the duct becomes the slightly greater pressure at the
duct inlet. Associated with this greater pressure is a marginally greater air density at
the duct inlet, and a tiny increase in the air flow velocity through the duct. Since the
air is no longer decelerating within the duct, as it did throughout the drying process
due to cooling, a slightly greater driving force for air flow is required, and the gauge
pressure at the duct entrance increases toward the end of the drying process.
Figs. 5.11–5.14 are the isothermal-case analogues of Figs. 5.7–5.10 described above.
The psychrometric path taken by the air at the duct outlet in the isothermal case,
which is shown as the solid line in Fig. 5.11, initially appears more complicated than
the corresponding adiabatic path. However, the humidity ratio can be seen to decrease monotonically with time, with only the temperature behaving in a complicated
manner. After the initial cool-down phase, the outlet air can be seen to gradually
increase in temperature, with this increase interrupted by a dip in the temperature
toward the end of the drying process. The reasons for the dip in the product surface
temperature, which leads to a dip in the air outlet temperature, have been discussed
above in relation to the 1-D product mode. The dip is clearly seen in Fig. 5.12, and
the consequent effect on air temperature is shown in Fig. 5.12. The corresponding air
temperatures and humidities are shown in Fig. 5.13. The velocities and pressures in
the duct shown in Fig. 5.14 again illustrate a behaviour that was observed in chapter 3: in the isothermal mode the air tends to accelerate within the duct, with a greater
pressure drop across the duct than in the adiabatic mode.
Figs. 5.15 and 5.16 illustrate the air-stream properties when the product thickness
δ = 1 mm, the inlet relative humidity φin = 0.3 and the air mass flow rate ṁa =
1 kg/s, with all other conditions unchanged from before. No qualitative change is
observed from the drying behaviour that has already been discussed. The thinner
product layer, compared with previously-discussed simulations, lowers the overall
drying time. Comparing Figs. 5.15 and 5.16 with Figs. 5.13, we see that the lower air
161
Air temperature, °C
60
55
Air flow direction
50
Humidity ratio, (g vapour)/(kg dry air)
45
0.045
0.04
0.035
0.03
Air flow direction
0.025
0.02
0.015
0.01
0
100
200
300
400
500
600
700
Time, minutes
Figure 5.9: Variation of air temperature and humidity ratio. Each curve shows
the time variation of air temperature or humidity ratio at a given control volume location within the duct. The direction of air flow is indicated. Adiabatic
case. Tin = 55◦ C, Tr sat = 60◦ C, φin = 0.9, ṁa,in = 1.8 kg/s.
Gauge pressure, Pa
162
100
Air flow direction
50
0.224
0.222
Air flow direction
Air flow velocity
0.22
0.218
0.216
0.214
0.212
0.21
0.208
0.206
0
100
200
300
400
500
600
700
Time, minutes
Figure 5.10: Variation of gauge pressure and air flow velocity within the ducts.
Each curve shows the time variation of gauge pressure or air flow velocity at
a given control volume location within the duct. The direction of air flow
is indicated. Adiabatic case. Tin = 55◦ C, Tr sat = 60◦ C, φin = 0.9, ṁa,in =
1.8 kg/s.
50
40
30
20
10
10
20
30
40
T,°C
50
60
Humidity ratio, g moisture / kg dry air
163
0
Figure 5.11: Psychrometric chart showing snapshots of psychrometric path
traced by air within duct (dotted lines) and the path traced by the outlet air
state (solid line). Isothermal case. Dashed lines and circles correspond to times
t = 0 s, t = 120 s, t = 570 s, and then to every subsequent 30 minutes. Here
ṁ a = 1.8 kg/s, Tin = 60◦ C, φin = 0.1, Trsat = 65◦ C, δ = 1 mm.
164
Surface temperature, °C
60
55
50
Air flow direction
45
0.9
0.8
saturation, −
0.7
0.6
0.5
Air flow direction
0.4
0.3
0.2
0.1
0
0
20
40
60
80
100
120
140
160
180
200
Time, minutes
Figure 5.12: Variation of product surface temperature and mean product saturation (at each control-volume stack in the x-direction) with time. Each curve
shows the time variation of product surface temperature or mean product saturation at a given control volume location within the duct. The direction of
air flow is indicated. Isothermal case, Tin = 55◦ C, Tr sat = 60◦ C, φin = 0.1,
ṁa,in = 2 kg/s, δ = 5 × 10−3 m.
165
62
Air temperature, °C
60
58
56
54
Air flow direction
Humidity ratio, (g vapour)/(kg dry air)
52
0.045
0.04
0.035
0.03
Air flow direction
0.025
0.02
0.015
0.01
0
20
40
60
80
100
120
140
160
180
200
Time, minutes
Figure 5.13: Variation of air temperature and humidity. Each curve shows
the time variation of air temperature or humidity at a given control volume
location within the duct. The direction of air flow is indicated. Tin = 55◦ C,
Tr sat = 60◦ C, φin = 0.9. ṁa,in as shown in legend.
Gauge pressure, Pa
166
100
Air flow direction
50
0.225
0.224
Air flow velocity
0.223
0.222
0.221
Air flow direction
0.22
0.219
0.218
0.217
0.216
0
20
40
60
80
100
120
140
160
180
200
Time, minutes
Figure 5.14: Variation of gauge pressure and air flow velocity within the ducts.
Each curve shows the time variation of gauge pressure or air flow velocity at
a given control volume location within the duct. The direction of air flow is
indicated. Isothermal case. Tin = 55◦ C, Tr sat = 60◦ C, φin = 0.9, ṁa,in =
1.8 kg/s.
167
mass flow rate leads to wider relative state variation at different duct locations.
5.5.
Results and discussion: Full dynamical model
In this section, illustrative results from the dynamical whole-system ICHPD model
are presented and discussed. The simulation is run until the overall drying rate has
fallen to 10−8 kg s−1 . Fig. 5.17 shows the psychrometric state profiles within the system when operated in the isothermal mode. A 1 mm product layer has been used
here, in contrast to the 5 mm product layer considered above. Each cycle, in which
the air stream passes the drier, evaporator and condenser, in sequence, represents a
snapshot of the system. The snapshots are separated (in this figure) by 15-minute
intervals. Initially the humidity in the system is high, and the humidity increase that
occurs within the drier duct is appreciable (curved right-hand side of state-cycle triangles). As the drying process proceeds, the humidity in the system falls. The solid
line connecting the diamonds in the figure represents the time-evolution of the air at
the drier-duct air outlet. The temperature at the air inlet is constrained to equal 55◦ C,
with the venting controlled in the model to maintain this constant duct inlet temperature (see chapter 4). Thus the system tends to maintain a constant high temperature,
but the low temperature (at the evaporator) falls as the system humidity drops. After
about 75 minutes, moisture condensation ceases in the system as humidity falls and
the evaporator surface is no longer below the dew point temperature. Thus the final
state-cycle shown in Fig. 5.17 does not reach the saturation line. At this point the
ICHPD system effectively becomes a conventional drying system, with all moisture
removed from the product being vented to the environment as vapour.
Fig. 5.18 shows the time-evolution of the system refrigerant state-cycle. The snapshots are again separated by 15-minute intervals. The condensing temperature can be
seen to be fairly stable through the drying process, as a result of venting which maintains constant TD . In contrast the evaporating temperature decreases dramatically
through the drying process, as the relative humidity falls (Fig. 5.17).
The impact of the evolution of the psychrometric and refrigerant state-cycles on
the system performance is shown in Fig. 5.19. This figure shows the time evolution
of the product saturation (top panel), the breakdown of irreversibility within the system (centre panel), and the system SMER (bottom panel). In the first panel, the time
evolution of the product saturation throughout the duct is plotted as a function of
time. The saturation falls most rapidly at the air inlet of the duct. After about 75
minutes, moisture condensation on the surface of the evaporator ceases and the system behaviour enters a new regime in which the system is acting as a conventional
dryer. This time is indicated in Fig. 5.19 by the vertical dotted line that cuts through
168
Air temperature, °C
60
55
Air flow direction
Humidity ratio, (g vapour)/(kg dry air)
50
0.055
0.05
Air flow direction
0.045
0.04
0
50
100
150
200
250
Time, minutes
Figure 5.15: Variation of air temperature and humidity ratio. Each curve
shows the time variation of air temperature or humidity ratio at a given
control volume location within the duct. The direction of air flow is indicated. Adiabatic case, Tin = 60◦ C, Tr sat = 65◦ C, φin = 0.3, ṁa,in = 1 kg/s,
δ = 1 × 10−3 m.
169
Air temperature, °C
63
62
61
Humidity ratio, (g vapour)/(kg dry air)
60
Air flow direction
0.08
0.07
0.06
Air flow direction
0.05
0.04
0.03
0
10
20
30
40
50
60
70
Time, minutes
Figure 5.16: Variation of air temperature and humidity ratio. Each curve
shows the time variation of air temperature or humidity ratio at a given
control volume location within the duct. The direction of air flow is indicated. Isothermal case, Tin = 60◦ C, Tr sat = 65◦ C, φin = 0.3, ṁa,in = 1 kg/s,
δ = 1 × 10−3 m.
170
the figure. By the time this regime is reached, the drying process has nearly reached
completion and saturation is fairly uniform throughout the duct. The second panel
shows that exergy destroyed by the compressor, by the draining of condensed moisture, in throttling, and in venting (to maintain system energy balance), do not change
appreciably throughout the drying process. Irreversibility in the evaporator climbs
noticeably at the end of the drying process. Irreversibility occurring at the condenser
and product decreases at the end of the drying process as the product temperature
increases. The net effect is an approximately 20% increase in total system irreversibility from the start of drying to the end of the process. Combined with the decrease in
evaporation rates (top panel), this leads to the evolution of the system SMER shown
in the bottom panel of Fig. 5.19. Clearly the SMER at the start of drying provides a
poor indication of the mean system SMER evaluated over a whole drying process.
Fig. 5.20 shows the time evolution of the product surface temperature, duct humidity ratio, and product surface mass flux. The product surface temperature (first
panel) shows complicated dynamics which include the temperature dip and a change
of behaviour after moisture condensation at the evaporator ceases (dotted line). The
humidity ratio within the ducts exhibits a straightforward decline throughout the
drying process (second panel). The evolution of the drying intensity occurring at each
of 20 duct control volumes (third panel) shows an interesting behaviour. Initially the
drying rate is a maximum near the duct entrance. Later, the drying rate near the duct
inlet collapses as the product near the inlet reaches completion, and the drying rate
near the duct air outlet actually increases. By the time moisture condensation ceases,
the evaporation intensity has collapsed throughout the duct.
Figs. 5.21 and 5.22 illustrate the psychrometric paths and the evolution of the refrigerant state-cycle of the system when it is operated in the adiabatic mode. The impact of the variation on the system performance is shown in Fig. 5.23, which depicts
the product saturation, exergy destruction rates and SMER. Fig. 5.24 shows the time
evolution of the surface temperature, humidity ratio and drying intensity within the
duct. The drying process takes about twice as long as in the isothermal mode, with an
approximate factor of two difference in average SMER. The most striking differences,
when comparing the adiabatic model outputs with those of the isothermal model,
are the path taken by the air at the duct outlet (solid line connecting the diamonds in
Fig. 5.21) and the spread of drying times in the duct. While the drying process is seen
to end fairly abruptly in the isothermal mode, with product at all locations within the
duct reaching completion simultaneously, this does not occur in the adiabatic mode.
Instead a drying front is seen to advance through the duct, with a substantial delay
separating completion of drying at the two ends of the duct. The behaviour seen in
Fig 5.23 appears qualitatively similar to that produced by the integrated timber-stack
HPD model of [159]. A significantly smaller variation in drying times in the isother-
171
mal case would be expected: one might anticipate that plate heat transfer would
reduce the variation in drying rate with location. Nevertheless the starkness of the
difference in behaviour is striking, and perhaps surprising.
Figs. 5.25–5.32 show analogous results for the case in which the product thickness δ is 2.5mm. The mean SMERs (prior to condensation halting) in the isothermal
cases are 6.41 kg kWh−1 and 5.48 kg kWh−1 , or 36% and 27% less than the SMERs
predicted in Fig. 4.7. At δ = 5 mm the isothermal mean SMER is 4.48 kg kWh−1 , or
25% less than that predicted by Fig. 4.7. (For comparison, the adiabatic mean SMERS
for δ=1mm, δ=2.5mm, δ=5mm are 2.66 kg kWh−1 , 2.49 kg kWh−1 and 2.00 kg kWh−1 .)
These results suggest that the SMERs obtained from the steady-state HPD model of
chapter 4 overestimate the energy performance of the system by approximately 30%.
This highlights the substantial penalty that results in batch operation from the system
operating away from optimal conditions. Nevertheless one interesting feature of the
model outputs is the fact that in the isothermal mode, the decline in drying rate and
energy performance at the end of the process is more sudden than in the adiabatic
mode. This leads to a smaller performance penalty due to batch operation than in the
adiabatic mode. Another finding is that the time-averaged performance appears to
be less sensitive to product thickness than predicted by the steady-state model.
5.6.
Results and discussion: The CDC method
In this section we briefly consider the characteristic drying curve (CDC) method in
the context of ICHPD. The use of the CDC method has been described, for instance,
by Keey and Suzuki [96]. The method assumes that a functional relationship exists
between the remaining moisture content of the product and the rate of drying, relative to its maximum which would occur over a saturated product surface, under the
given temperature and humidity conditions. We assess this method in the context of
ICHPD using the data generated by the simulation runs described above. We use the
fixed ideal surface temperature (here 55◦ C) with the bulk flow conditions and modified transfer coefficients predicted by the detailed model, to evaluate the maximum
drying rate, and then use the drying rates actually observed to calculate the relative
drying rates. Thus we are testing the ability of a very simple model to replicate the
drying behaviour predicted by the comprehensive model. Fig. 5.33 illustrates the relative drying rate as a function of saturation, in the isothermal mode. The 20 lines
in the figure correspond to the 20 lateral control volumes in the duct. As can be
seen, there is a spread of approximately 20-30% in the relative drying rates occurring
at a given saturation. This provides an estimate of the uncertainty associated with
the CDC method in the context of ICHPD. Fig. 5.34 shows the drying curves in the
172
80
70
60
50
40
30
20
10
10
20
30
40
50
Humidity ratio, g moisture / kg dry air
90
0
60
T,°C
Figure 5.17: Psychrometric path traced by air cycle with time. Isothermal
mode, δ = 1 mm. Cycles are snapshots of the system, separated by 15 minutes.
0.4
140 160 180 200 220 240 260 280 300 320 340 360 380 400 420 440 460 480 500
Enthalpy (kJ/kg)
0.5
0.6
0.7
0.8
1
0.9
2
3
3
4
ISOTHERMAL
1
2
R−134a
173
Pressure (MPa)
Figure 5.18: Evolution of R134a state-cycle. Isothermal mode, δ = 1 mm. Cycles are snapshots of the system, separated by 30 minutes.
174
1
Saturation
0.8
0.6
0.4
Cumulative Ex. Dest, kW
0.2
0
4
4
3
2
2
1
SMER kg/kWh
0
10
8
6
4
2
0
0
20
40
60
Time, minutes
80
Figure 5.19: Isothermal mode, δ = 1 mm. First panel: saturation. Second
panel: Cumulative exergy destruction plot. 1: compressor. 2: throttle. 3:
condenser and product. 4: evaporator. 5: fan. 6: venting. 7: condensate (very
thin band). Third panel: system SMER.
Surface temperature, °C
175
59
58
57
56
55
54
Air humidity ratio, g/kg
53
50
Drying intensity, g/m2
0
0.3
0.2
0.1
0
0
20
40
60
Time, minutes
80
Figure 5.20: Isothermal mode, δ = 1 mm. First panel: surface temperature.
Second panel: duct air humidity. Third panel: drying intensity.
176
70
60
50
40
30
20
10
10
20
30
40
50
Humidity ratio, g moisture / kg dry air
80
0
60
T,°C
Figure 5.21: Psychrometric path traced by air cycle with time. Adiabatic mode,
δ = 1 mm. Cycles are snapshots of the system, separated by 75 minutes.
0.4
140 160 180 200 220 240 260 280 300 320 340 360 380 400 420 440 460 480 500
Enthalpy (kJ/kg)
0.5
0.6
0.7
0.8
1
0.9
2
3
3
4
ADIABATIC
1
2
R−134a
177
Pressure (MPa)
Figure 5.22: Evolution of R134a state-cycle. Adiabatic mode, δ = 1 mm. Cycles are snapshots of the system, separated by 30 minutes.
178
1
Saturation
0.8
0.6
0.4
Cumulative Ex. Dest, kW
0.2
0
4
2
SMER kg/kWh
0
10
8
6
4
2
0
0
50
100
150
Time, minutes
200
Figure 5.23: Adiabatic mode, δ = 1 mm. First panel: saturation. Second panel:
Cumulative exergy destruction plot. 1: compressor. 2: throttle. 3: condenser
and product. 4: evaporator. 5: fan. 6: venting. 7: condensate (very thin band).
Third panel: system SMER.
Air humidity ratio, g/kg
Surface temperature, °C
179
55
50
45
40
60
40
20
Drying intensity, g/m2
0
0.3
0.2
0.1
0
0
50
100
150
Time, minutes
200
Figure 5.24: Adiabatic mode, δ = 1 mm. First panel: surface temperature.
Second panel: duct air humidity. Third panel: drying intensity.
180
80
70
60
50
40
30
20
10
10
20
30
40
50
Humidity ratio, g moisture / kg dry air
90
0
60
T,°C
Figure 5.25: Psychrometric path traced by air cycle. Isothermal mode, δ =
2.5 mm. Cycles are snapshots of the system, separated by 15 minutes.
0.4
140 160 180 200 220 240 260 280 300 320 340 360 380 400 420 440 460 480 500
Enthalpy (kJ/kg)
0.5
0.6
0.7
0.8
1
0.9
2
3
3
4
ISOTHERMAL
1
2
R−134a
181
Pressure (MPa)
Figure 5.26: Evolution of R134a state-cycle. Isothermal mode, δ = 2.5 mm.
Cycles are snapshots of the system, separated by 30 minutes.
182
1
Saturation
0.8
0.6
0.4
Cumulative Ex. Dest, kW
0.2
0
4
4
3
2
2
1
SMER kg/kWh
0
10
8
6
4
2
0
0
50
100
150
Time, minutes
200
250
Figure 5.27: Isothermal mode, δ = 2.5 mm. First panel: product saturation
within the duct, both spatially and with time. Second panel: Cumulative exergy destruction plot. 1: compressor. 2: throttle. 3: condenser and product. 4:
evaporator. 5: fan. 6: venting. 7: condensate (very thin band). Third panel:
system SMER.
Air humidity ratio, g/kg
Surface temperature, °C
183
58
56
54
52
50
Drying intensity, g/m2
0
0.3
0.2
0.1
0
0
50
100
150
Time, minutes
200
250
Figure 5.28: Isothermal mode, δ = 2.5 mm. First panel: surface temperature.
Second panel: duct air humidity. Third panel: drying intensity.
184
80
70
60
50
40
30
20
10
10
20
30
40
50
Humidity ratio, g moisture / kg dry air
90
0
60
T,°C
Figure 5.29: Psychrometric path traced by air cycle. Adiabatic mode, δ =
2.5 mm. Cycles are snapshots of the system, separated by 15 minutes.
0.4
140 160 180 200 220 240 260 280 300 320 340 360 380 400 420 440 460 480 500
Enthalpy (kJ/kg)
0.5
0.6
0.7
0.8
1
0.9
2
3
3
4
ADIABATIC
1
2
R−134a
185
Pressure (MPa)
Figure 5.30: Evolution of R134a state-cycle. Adiabatic mode, δ = 2.5 mm.
Cycles are snapshots of the system, separated by 30 minutes.
186
1
Saturation
0.8
0.6
0.4
0.2
Cumulative Ex. Dest, kW
0
4
2
SMER kg/kWh
0
10
8
6
4
2
0
0
100
200
300
400
Time, minutes
500
600
Figure 5.31: Adiabatic mode, δ = 2.5 mm. First panel: drying intensity within
the duct, both spatially and with time. Second panel: Cumulative exergy destruction plot. 1: compressor. 2: throttle. 3: condenser and product. 4: evaporator. 5: fan. 6: venting. 7: condensate (very thin band). Third panel: system
SMER.
Air humidity ratio, g/kg
Surface temperature, °C
187
55
50
45
40
50
Drying intensity, g/m2
0
0.3
0.2
0.1
0
0
100
200
300
400
Time, minutes
500
600
Figure 5.32: Adiabatic mode, δ = 2.5 mm. First panel: surface temperature.
Second panel: duct air humidity. Third panel: drying intensity.
188
adiabatic mode. Interestingly, the spread is much larger, which implies that in the
isothermal mode, variation in relative drying rates is substantially less than in the
adiabatic mode. This suggests that the CDC method may be at least as applicable
in the ICHPD context as in the adiabatic context. This is significant since the CDC
method has been used successfully to model adiabatic HPD [159, 156, 160].
5.7.
Conclusions
At the start of this chapter we have examined the drying equations and have argued
that in order to satisfactorily predict the dynamical behaviour of an isothermal drier,
the interaction between internal heat and internal mass transfer will need to be considered, requiring a model based on the volume-averaged drying equations. We have
therefore extended the equations of Wang and Chen [178] to account for plate heat
flux. The results of this chapter support our assessment that temperature effects may
play a complex role in the drying dynamics. The internal-process dynamical drying
model has exhibited a counterintuitive behaviour (the “late temperature dip”) which
qualitatively agrees with the experimental evidence that we have available. Although
we have insufficient evidence to reach firm conclusions regarding this phenomenon,
this chapter’s 1-D model has provided a (provisional) explanation for the reasons for
the dip in the product surface temperature measurements. This explanation involves
the temperature gradient imposed by the plate boundary condition, and it appears
plausible that the dip may be a general feature of plate drying.
As expected, the 2-D dynamical duct model has exhibited behaviour which is consistent with the 1-D drying model discussed above. The duct model’s observed behaviour is also consistent with that seen from the static duct model of chapter 3. However a number of additional features in the modelled behaviours have been seen arising from the variation of flow conditions throughout the duct. The broadest feature
of the model output is an illustration of how the linked drying processes at different
locations within the duct can lead to a significant spread in the drying times, with a
drying front progressing through the duct as well as into the product. In addition,
the dynamical duct model’s results illustrate how the drying dynamics can interact
with flow parameters such as the pressure drop and air flow velocity.
We have re-combined the dynamical duct model with the whole-system HPD model
described in chapter 4. This has produced a whole-system dynamical HPD model
which we have used for several indicative simulation runs. The results have highlighted the importance of using dynamical modelling for batch processes: the discrepancy between the initial steady-state behaviour and the averaged behaviour is
substantial. We have compared the dynamical behaviour of the system in the adi-
189
0.5
Relative drying rate
0.4
0.3
0.2
0.1
0
0
0.2
0.4
0.6
0.8
1
0.6
0.8
1
Saturation
0.5
Relative drying rate
0.4
0.3
0.2
0.1
0
0
0.2
0.4
Saturation
Figure 5.33: Local drying-rate curves. Isothermal mode. Top panel: δ = 1 mm.
Bottom panel: δ = 2.5 mm.
190
0.5
Relative drying rate
0.4
0.3
0.2
0.1
0
0
0.2
0.4
0.6
0.8
1
0.6
0.8
1
Saturation
0.5
Relative drying rate
0.4
0.3
0.2
0.1
0
0
0.2
0.4
Saturation
Figure 5.34: Local drying-rate curves. Adiabatic mode. Top panel: δ = 1 mm.
Bottom panel: δ = 2.5 mm.
191
abatic and isothermal modes. The results have shown how the drying process in
the isothermal case interacts with the refrigerant cycle, allowing the heat pump to
operate more efficiently than in the adiabatic mode. They have also shown that the
isothermal mode may allow the drying process to occur more rapidly, reducing the
energy-performance penalty associated with batch drying operation. Since it is based
on the general framework provided by the volume-averaging theory, the model developed in the present chapter could in principle be used to model the drying, in an
ICHPD system, of a variety of products. On the other hand, we have investigated the
drying curves that occur at various parts of the duct. According to the present model,
the CDC method is likely to be at least as valid in the isothermal mode as in the adiabatic mode. Thus, in fact, a comprehensive product model does not appear to be
required to capture behaviours arising from contact heat transfer, and the CDC may
provide a useful practical tool for modelling dynamical ICHPD system behaviour.
192
Quantity
Value
Unit
Dva
0.256 × 10−4
m2 /s
1 × 10−19
m2
∆hv
2.44309 × 106
J/kg
0.44
–
usat
758
kg/m3
e
0.758
–
kγ
0.025
kβ
0.65
J/(s · m · K)
kσ
0.0629
ργ
1.29
ρβ
1000
kg/m3
ργ
183
kg/m3
µβ
0.458509 − 5.30474 × 10−3 T + 2.31231 × 10−5 T 2
kβ
scr
σ(T )
ρv (s, T )
J/(s · m · K)
J/(s · m · K)
kg/m3
−4.49161 × 10−8 T 3 + 3.27681 × 10−11 T 4
0.121978 − 0.0001683 × T
p( T ) exp − pc (s, T )/(ρ β · R · T ) /(R · T )
kg/(m · s)
kg/s2
kg/m3
0.020023/s + 0.009547/(s − 1.028) − 0.12s + 0.4415
q
J (s) × σ ( T ) × e/k β

[(s − scr )/(1 − scr )]3 , s > scr
0,
s≤s
–
T∞
333.15
K
p∞
1.01325 × 105
Pa
10%
–
s0
0.99
–
δ
5 × 10−3
m
1.5
m/s
h
43.3
hm
0.0172
W/(m2 · K)
J (s)
pc (s, T )
krβ (s)
Pa
–
cr
RH
u∞
Quantities, relations and operating conditions adopted from [178]
m/s
193
5.8.
Nomenclature for chapter 5
Symbols (units)
Symbol
Definition [units]
A
Area [m2 ]
ADI, ISO
Adiabatic, isothernal
Cs
Plate-heat-flux moisture source term coefficient [kg/J]
c p,i
Constant-pressure specific heat capacity of pure species i [J/kg-K]
D
Diffusivity [m2 /s]
d
Air duct depth [m]
D
Heating plate refrigerant tube internal diameter [m]
D
Diffusivity [m2 /s]
Dvs
Mass “diffusivity” of gas phase caused by the saturation distribution
[kg/m-s]
DvT
Mass “diffusivity” of gas phase caused by the temperature distribution
[kg/m-K-s]
Dβs
Mass “diffusivity” of liquid phase caused by the saturation distribution
[kg/m-s]
DβT
Mass “diffusivity” of liquid phase caused by the temperature distribution [kg/m-K-s]
De
Effective diffusivity [kg/m-s]
DT
Mass “diffusivity” caused by the temperature distribution [kg/m-K-s]
Ds
Mass “diffusivity” caused by the saturation distribution [kg/m-s]
fv
Fraction of moisture transport due to vapour transport [–]
fβ
Fraction of moisture transport due to liquid transport [–]
fs
Fraction of moisture transport due to saturation gradient [–]
fT
Fraction of moisture transport due to temperature gradient [–]
∆hvap
Latent heat of vaporization [J/kg]
h
Product surface heat transfer coefficient [W/m2 -K]
hm
Product surface mass transfer coefficient [m/s]
hi
Specific enthalpy of pure species i [J/kg]
∆hv
Latent heat of vaporization [J/kg]
k
Thermal conductivity [W/K-m]
K
Diffusion model variable temperature coefficient [K]
K0
Kossovich number [–]
KD
Intrinsic permeability of medium [m2 ]
Kq
Diffusion model variable temperature coefficient [s-m-K/J]
KTeff
Overall effective thermal conductivity tensor [J/s-m-K]
194
T
keff
Effective thermal conductivity of porous medium (scalar) [J/s-m-K]
krj
Relative permeability of medium to phase j [–]
k0j
Intrinsic permeability of medium to phase j [m2 ]
l
Heating plate condenser tube spacing [m]
L
Heating plate length [m]
hṁ βγ i
Volumetric evaporation rate [kg/s-m3 ]
Mi
Molar mass of species i [kg/mol]
MER
Moisture extraction rate [kg/s]
n
Refrigerant circuits per plate [–]
ND
Number of ducts [–]
hni
Volume-averaged mass flux [kg/m2 ]
n̂
Unit normal vector [–]
p
Pressure [Pa], passes through plate per circuit
pc
Capillary pressure [N/m2 ]
s
Saturation [–]
scr
Critical saturation [–]
SMER
specific moisture extraction rate [kg/kWh]
s̄v
Molar entropy of water vapor [J/K-mol]
T
Temperature [K]
T∞
Gas-side temperature [K]
t
Time [s]
U
Unit isotropic tensor [–]
usat
Saturated liquid density of porous medium [kg/m3 ]
v
Speed [m/s]
x, y
Spatial variables [m]
xp
Heating plate refrigerant tube centerline depth [m]
z
Distance along refrigerant flow in CD2 [m]
α
Heat exchange coefficient [W/m2 -K]
Γγ
Discretised vapour “diffusivity” [m-kg/s]
δ
Product thickness [m]
∆
Change
e
Volume fraction of a phase; porosity [–]
µ
Dynamic viscosity of fluid [N-s/m2 ]
ρ
Density [kg/m3 ]
hρTot i
Density of porous medium [kg/m3 ]
h ρM i
Moisture density of porous medium [kg/m3 ]
σ
Surface tension [kg-m/s2 ]
φ
Relative humidity [–]
195
ψ
Generalised quantity
ω
Humidity ratio [kg-vapour/kg-dry-air]
Subscripts and superscripts
Symbol
Definition
0
Environment, Initial
1, 2,
20 ,
3, 4
Locations on refrigerant cycle
A, B, C, D, E, F
Locations on air cycle
b
Bulk
b, s, v, a, β
Bound-moisture, dry-solid, vapour, dry-air, free-liquid
co, ev
Condenser, evaporator
D, F, P
Ducts, fan, compressor
eff
Effective (porous-medium)
in, out
Inlet, outlet
i, j
Species-i, Phase-j
k, a, v, w
Species-k, dry-air, water-vapour, liquid-water
m
Mass-exchange
n, w, s, e
North, west, south, east (control-volume boundaries)
p
Heating plate
r
Refrigerant
S, s
Surface
S, P, N
South, “Point” under consideration, North
sat
Saturation condition
t
Total, effective
wb
Wet bulb
•
Modified for high mass transfer rates
δ
Product
β, γ, σ
Free water phase, Gas phase, Solid phase
∞
Free-stream condition
196
Chapter 6
Discussion and conclusion.
6.1.
Survey of the findings of this study
Chapter 1 and appendix A together represent a literature survey, and a summary of
the theoretical framework in a coherent form using a consistent notation. The closedform analytic duct model, which allows the temperature and humidity variation to
be predicted within idealised “adiabatic” and “isothermal” drying ducts, may be a
new result of chapter 1. In the present section we briefly consider the findings of the
theoretical and simulation work of appendix B and chapters 2-5.
In chapter 2 a simple whole-system HPD model was implemented employing the
purely-isothermal or purely-adiabatic drier model corresponding to Equation (1.4.18).
This idealised duct model has been combined with polynomial heat-transfer correlations (2.3.2) and (2.3.3) characterising the evaporator and condenser heat-exchangers
that were investigated by Carrington and Liu [31], and with a highly simplified compressor model that uses the assumption that on the refrigerant side, the heat pump
operates at a fixed percentage (50%) of the Carnot heating COP. In developing the
model, we have found that given values for ẆF , ẆP , TD , ṁ a,co , ṁ a,ev , T0 , ηC , the system steady state is fully specified by equations expressing the whole-system energy
balance, the heat pump COP, heat transfer models for the condenser and evaporator,
and energy and moisture balances across all system components.
The model has been validated in the adiabatic mode by comparison with measured data published by Carrington and Liu [31]. Agreement with experiment (with
a typical error of 20%) is rough, but is more than adequate to support the chapter’s
conclusions, given the size of the predicted difference between isothermal and adiabatic HPD performance. In validating the HPD model, we have also seen that the
model predicts variation of optimal evaporator air flow with incoming air humidity
197
in a manner that is qualitatively consistent with the experimental investigation.
A baseline scenario representing a small-scale food dryer has been selected. The
model has illustrated (Table 2.3) that the isothermal mode tends to produce highhumidity conditions within the dryer, accompanied by high MER and SMER (approximately three times those in the adiabatic mode in the baseline case). This is
reflected in the shapes of the psychrometric cycles plotted in Fig. 2.5. The isothermal
cycles are taller (implying a greater MER), and narrower (implying a smaller system
∆T, which leads to improved COP, contributing to the increased MER and SMER). We
have found that the adiabatic SMER is optimised at a bypass ratio of b = 38%, while
in contrast, the isothermal SMER is optimised at a bypass ratio of b = 0%, consistent
with the much higher humidity that prevails in the case of the isothermal mode.
We have found that (although the isothermal SMER increases with T) the difference between isothermal SMER and adiabatic SMER has a maximum with respect
to T, implying that at higher values of T, isothermal SMER increases more slowly
with T than the adiabatic SMER. By plotting φE against MER and SMER as ẆP is
varied, we have considered the simultaneous effect of capacity on duct-outlet relative humidity and performance. Adiabatic humidity is very sensitive to ẆP , and is
substantially lower than the isothermal humidity at the same ẆP . This contributes to
the higher isothermal SMER, but by comparing the isothermal and adiabatic SMERs
at a given φE we have found that the increased system humidity is not responsible
for all of the SMER gain associated with the isothermal mode. By plotting φE against
MER and SMER as L is varied, we have found that the isothermal mode may be compatible with modest space requirements compared with those of the adiabatic mode.
We have also found that the isothermal mode, just like the adiabatic mode, remains
very sensitive to the constraint on φE . We have found that the product thickness δ has
a significant impact on both MER and SMER as δ deviates significantly from a thin
(1mm) film. We have found that ICHPD, combined with photovoltaics, provides the
possibility (albeit a capital-intensive one) of leveraging sunshine to obtain 10-40 times
the drying per solar irradiance that is obtainable using conventional solar driers.
In chapter 3 discretised versions of the control-volume balance equations for a detailed duct model have been developed and implemented, which uses the SIMPLER
algorithm for the air side [117], and which also incorporates a refrigerant-side model
that calculates the mean refrigerant heat transfer coefficient. The reasons for developing this detailed model were threefold. By evaluating the refrigerant heat transfer
coefficient, and allowing the refrigerant-side and air-side pressure drops to be evaluated within the tray drier, the detailed duct model allows the trade-off between heatand momentum-transfer irreversibilities to be directly assessed once the model is incorporated into a whole-system ICHPD model (in the following chapter). Secondly,
198
its greater flexibility allows several other additional details to be built into the model:
(1) a detailed model of refrigerant flow within the plate, accounting for variation in
heat transfer coefficient and pressure; and (2) details of the air flow, including fluid
friction; energy transport associated with mass exchange; heat transfer through the
duct bottom surface; variation of the transfer coefficients along the duct; enhancement of the transfer coefficients due to mass transfer. Furthermore, the model is compatible with simulation of hygroscopic porous solids throughout the drying process,
allowing the performance of an ICHPD system to be evaluated away from the initial
steady state. In light of the results of chapter 5, which show significant difference
between steady-state system performance and average dynamical behaviour, this is
a crucial addition. In order to compare the detailed model with the idealised model
developed in chapter 1, we have examined its outputs with the same drier geometry
under the same inlet conditions: Tin = 55◦ C, φin = 30%, ṁa = 1kg/s. The saturated
condensing temperature of the refrigerant has been set to Tr,sat = 60◦ C. This has enabled us both to validate the detailed model, and to assess the impact of the various
corrections included in the detailed model.
We have found that the detailed model predicts variation in T and ω within the
drier ducts that is in broad agreement with the idealised model. The most significant
additional feature in the detailed model’s predictions is a depressed product surface
temperature at the air inlet, and elevated product surface temperature at the air outlet, which leads to the air temperature dipping and then climbing again as it passes
through the duct. This temperature variation causes a flattening of the drying-rate
profile through the duct, and also relatively higher drying rates at high humidities
than those predicted by the idealised model. We have considered (Fig. 3.10) the impact that variation in the local transfer coefficients will have on the drying profile
within the duct, due to boundary-layer development within the duct.
We have considered (Fig. 3.11) the variation of enhancement factors (due to masstransfer), temperature, and convective enhancement of mass transfer through the
ducts. Transfer-coefficient enhancement factors vary between 0.94 and 1.0. Vapour
flux enhancement due to convection mass transfer at the product surface increases
through the duct, to a maximum of 1.08. We have explored the variation in outlet air
temperature and duct MER over a range of inlet air conditions {Tin ,φin } (Table 3.5).
Under all inlet conditions that have been tested, forcing the product surface temperature equal to its idealised value yields very good agreement with the drying rate
predicted by the idealised model (within 4%). This has been taken to indicate that
the detailed model is functioning correctly, and that the additional physical effects
accounted for in this model result in relatively minor corrections to the results.
We have examined (Fig. 3.12) the effects on temperatures through the duct of varia-
199
tions in the product thickness δ and the refrigerant saturated condensing temperature
Tr,sat . We have considered (Fig. 3.13) illustrative psychometric paths taken with two
different inlet air velocities in the isothermal and adiabatic modes, showing how the
psychrometric path that is followed is a function of air flow rate only in the isothermal
case. Finally, we have provided a detailed depiction (Figs. 3.14–3.17) of air flow conditions as they vary throughout the drier duct. The results shown in these figures are
all in good agreement with the results from Sun et al. [160] modelling a timber-stack,
using a model which itself had been tested against measured data [8, 9].
One interesting phenomenon that shows up in Figs. 3.14–3.17 is an increase in
the air flow velocity in the drier in the isothermal mode – which stands in contrast
to the adiabatic drying mode, for which the air flow velocity decreases along the
air flow direction. Associated with this acceleration, an approximately 20 greater
duct pressure drop has been observed in the isothermal mode. The product surface
vapour flux varies by as much as 70% from the air inlet to the air outlet, highlighting
the need to model the variation of the drying process along the length of a drier of this
size. In summary, the results of chapter 3 are broadly in line with expectations, with
temperature effects playing the most important role in causing divergence between
the idealised model of chapters 1 and 2 and the detailed model of chapter 3.
In chapter 4 the detailed air flow model of chapter 3 has been linked into a detailed
whole-system model. Illustrative results have been presented, including a comparative second-law analysis of the adiabatic and isothermal modes. We have considered
system performance sensitivity to a number of parameters, paying particular attention to the tradeoff between heat- and momentum transfer irreversibilities.
After presenting correlations from the literature describing the remaining heat
pump components, and presenting expressions for the exergy destruction rate at each
system component, the whole-system model has been developed, which implements
the procedure of Goldstein [73]. The comprehensive model agrees well with the preliminary model under the baseline geometry that was previously specified in Chapter 2. We have examined a pressure-enthalpy plot (Fig. 4.5) illustrating the R134a
state-cycle in the baseline scenario (in both the isothermal and the adiabatic modes),
and have considered the breakdown of rates of exergy destruction in both modes.
About half of the reduction in irreversibility (in the isothermal mode, compared with
the adiabatic mode) was seen to be associated with the avoidance of the poor air-side
convective heat transfer coefficient. The other half of the irreversibility avoidance is
associated with the avoidance of air cooling within the drier ducts.
We have found that in the context of the whole-system HPD model, the deviation of the drier duct from idealised “isothermal” conditions, which was observed in
chapter 3, makes very little difference to the predicted HPD performance. This has
200
been attributed to the fact that the compressor specifications provide the practical
limit on the rate of heat delivery to the drying process. We have examined the sensitivity of system performance to a number of parameters, of which the most influential
were the evaporator face area and the product mass exchange area, as well as product thickness. By simultaneously adjusting several parameters we have increased the
predicted ICHPD SMER to over 25% more than its baseline value, illustrating the potential that exists to obtain significant further performance gains through appropriate
system modifications. By plotting the HPD SMER against δ and MER while varying
δ, we have examined the potential offered by ICHPD to avoid a trade-off that applies
to the adiabatic mode, between energy performance and product throughput.
We have evaluated the NPV of installation of isothermal and adiabatic HPD in a
hypothetical sludge drying case-study under a range of economic scenarios. The results indicate that the viability of ICHPD (as an alternative to adiabatic HPD) is currently highly dependent on its relative capital cost, but that this dependency would
be reduced under scenarios of high energy costs or high sludge-disposal costs. Our
results also show that ICHPD has the potential to lessen energy-price risk exposure.
We now jump to appendix B, which develops the dynamical heat and mass transport processes that occur within a porous medium undergoing drying. As discussed
on p.143, appendix B derives a more general set of equations than are employed in
the main body of the thesis, in chapter 5. The derivation of the full set of equations
for hygroscopic porous media has nevertheless been included, as an appendix, because of the information that it provides on the derivation of drying equations in
general: (1) by providing a novel path connecting several past derivations of the
equations, and (2) by emphasizing the large number of unqualified assumptions that
are required to obtain the equations by the use of the volume-averaging theory. After
reviewing the volume-averaging approach, which allows local balance equations to
be applied in averaged form to the aggregate porous medium, an attempt has been
made to rigorously develop the drying equations for deformable hygroscopic porous
media. In doing so, we have seen (in the case of the conservative form of the energy
balance) how Whitaker’s method allows the intuitions of several early researchers,
whose equations have formed the basis of much subsequent work, to be justified
formally (page 291). However, at the same time, this has shown that the standard
equations become valid only in the case where a large number of terms (listed on
page 304), some of which are poorly understood, can be neglected. This result has no
practical bearing on our subsequent work, since it provides no basis for evaluating
the impact of neglecting these terms. However, it highlights the difficulty involved in
capturing the full complexity that arises, even in so biologically simple a process as
drying, and forces us to view results from the detailed dynamical drying model with
some degree of caution. In light of these results (together with results from chapter
201
5 which provide support for the CDC method) it has been recommended that in follow up work the much more straightforward CDC method be used to simulate the
time evolution of the drying process. In terms of the logical structure of the thesis,
the main result of appendix B is the set of equations (B.7.1)–(B.7.15), which describe
the drying process in detail, subject to the considerations just discussed. The classic drying equations of Whitaker [188] result from these equations when the terms
describing hygroscopic behaviours are dropped from them.
In chapter 5, the internal drying processes have been used to develop a dynamical ICHPD model, based on Whitaker’s classic drying equations, and generalizing
the diffusion model of Wang and Chen [178] to include plate heat transfer. The
development of this model proceeds from Whitaker’s classic drying equations – or
equivalently, from the set of equations presented at the end of Appendix B, but neglecting the terms arising from the presence of a distinct bound moisture species. As
expected, the unusual boundary conditions that occur in ICHPD are predicted by
the model to cause accelerated moisture transport, and unusual saturation profiles
within the product. The 1D model output is in good agreement with the results of
Wang and Chen [178] in the adiabatic mode. In the isothermal mode, the model has
shown a temperature-dip phenomenon which has been accounted for in terms of the
transfer mechanisms present in the product. A 2-D model has been produced by incorporating the 1D dynamical model into the detailed duct model of chapter 3. This
model has been used to plot the time evolution of conditions at different duct locations, with constant air inlet condition and refrigerant temperature. The results of
this work are consistent with the results of previous chapters. The adiabatic mode
shows the greater percentage spread in drying times, since in the isothermal mode
plate heating maintains a more uniform drying rate through the duct. Once the adiabatic, constant-enthalpy profile is attained, the adiabatic psychrometric path remains
constant, while the isothermal psychrometric path evolves in a complicated manner
reflecting the variation in plate heat transfer as the drying process evolves. Finally, as
the drying process comes to an end, evaporative cooling ends, and air temperature,
velocity, and pressure profiles in the two modes behave differently and in ways that
are consistent with the presence of plate heat transfer only in the isothermal mode.
Finally this 2D model has been integrated into the whole-system HPD model,
and have plotted the evolving psychrometric state-cycles in the isothermal mode
(Figs. 5.17 and 5.25) and the adiabatic mode (Figs. 5.21 and 5.29). Plots have been
produced of the time-evolution of the product saturation, the breakdown of exergy
destruction (by plotting the specific exergy destruction at each component in a cumulative plot of specific exergy destruction) and the time-evolution of the SMER
(Fig. 5.19, 5.23, 5.27, 5.31). The results have highlighted the importance of using dynamical modelling for batch processes: the discrepancy between the initial behaviour
202
and the averaged behaviour is substantial. The results have illustrated the connection
between the drying process and the refrigerant cycle, which allows the heat pump to
operate in the isothermal case within a narrower temperature-band than in the adiabatic mode. However the mean difference seen in the results of chapter 5 is less than
was predicted by the steady-state model of chapter 4, since it decreases as the drying
rate falls. The model results have also shown that the isothermal mode may allow
the drying process to occur more steadily until the end, reducing the batch-operation
SMER penalty. In addition, they have shown that the impact of product thickness on
SMER increases slightly less rapidly than was suggested by the steady-state model.
The mean isothermal SMERs seen in the results of chapter 5 are significantly lower
than the SMERs predicted by the steady-state model. On the other hand, the mean
adiabatic SMERs are themselves significantly lower than SMERs of up to 5 kg/kWh
that have been obtained from real HPD systems (based on technology closely related
to that modelled in the present study) when operating in batch timber dryers [23].
The predicted ratio between the isothermal mode SMER and the adiabatic SMER remains unchanged (2-3). An important open question for follow-up research is how
much of the steady-state performance can be recovered by operating an ICHPD in
steady-state continuous mode, rather than in batch operation.
A comparison has been made between the dynamical behaviour of the system in
the adiabatic and isothermal modes by also plotting surface temperature, air humidity and drying intensity as a function of time (Fig. 5.20, 5.24, 5.28, 5.32). The results
have shown that in the context of the varying conditions occurring within the modelled HPD, the spread in drying times in the adiabatic mode is very different from the
isothermal mode, in which the drying process is essentially complete by the time dehumidification at the evaporator coils ceases. We have considered the drying curves
that occur at various parts of the duct, and have found that the CDC method is likely
to be at least as valid in the isothermal mode as in the adiabatic mode. This result
suggests that a comprehensive product model may not be required to capture behaviours arising from contact heat transfer, and that the CDC method may provide a
useful practical tool for future dynamical modelling of ICHPD system behaviour.
6.2.
Comparison with other work
This thesis is dependent on comparisons that have been made with the previouslypublished work of others, particularly as it contains no new experimental data in
support of the model that has been developed. In chapter 1 we have surveyed the
literature into which this research fits, highlighting the relationship between ICHPD
and other approaches to non-adiabatic HPD. The fairly extensive body of published
203
HPD research cited in chapter 1, which is centered around the observation that nonadiabatic HPD operation can improve SMERs, provides some indirect corroboration
of our key result. More importantly, however, the components of the present model
have been tested in a number of more direct ways throughout the thesis. In this
section we summarize key points of contact between this thesis and the literature.
On page 27 we have considered the heat transfer coefficient obtained using the
Dittus-Boelter equation (for the case of the example flow situation that has been examined in chapter 2), and have confirmed that the value falls within the range of
the heat-transfer coefficients that were measured [139] in flow situations of approximately the same geometry and flow characteristics. The ratio of the heat- and masstransfer coefficients produced by the numerical routines has also been confirmed to
be very nearly identical to the ratio of the thermal conductivity to the binary diffusivity of the moist-air mixture, as is required by the analogy between heat and mass
transfer, expressed in Equations (A.7.22)–(A.7.27). The detailed duct model developed in chapter 3 is based on control-volume equations that are structurally the same
as those previously developed by Sun and Carrington [159] and Vaughan [175]. This
correspondence has provided a guide in developing the equations, and an opportunity to check the equations; the equations used here agree with those presented by
Sun and Carrington [159]. Furthermore, agreement between the detailed model and
the idealised model (in the adiabatic mode) is very good, as is discussed on page 90.
This effectively provides a test of the detailed model against the analytical solution
that applies, given the idealised boundary condition. Finally, outputs from the detailed model (in the adiabatic mode) have been compared with those of the model of
Sun et al. [160] on page 102. In effect, this stands as a comparison between the output
of the present model and the measured data reported by Bannister et al. [8, 9].
The comparisons detailed in the previous paragraph (together with the detailed
examination of model outputs described on pages 101–106) provide fairly robust evidence that the detailed model is functioning correctly in the adiabatic mode. When
the detailed model is used in the isothermal mode, the term αt ( Tr − Ts ) appearing in
(1.5.5), which represents the additional heat transfer from the refrigerant, also comes
into play. Our confidence in the outputs in the isothermal mode depends upon the
observation that the conduction model is fairly elementary and that the model outputs accord with physical intuition as embodied in the idealised isothermal model.
The preliminary whole-system model used in chapter 2 has also been tested in the
adiabatic mode under the three conditions investigated experimentally by Carrington and Liu [31]. Because of the way in which the test-condition humidities have been
recreated in the modelled system (by modifying the drier length, L), this test is effectively a test of the remainder of the HPD, not including the drier ducts. As discussed
204
on pages 49 and 196, this test has provided validation that is adequate to support the
conclusions of the chapter. Since the model developed in chapter 2 has been wholly
tested (in two parts: duct model and rest of system), the agreement that is observed
between the predictions of the preliminary model and the detailed model developed
in chapter 4 also provides support for the whole-system model of chapters 4 and 5.
The 1-D dynamical model has been tested successfully (p.153) in the adiabatic
mode under the adiabatic conditions investigated numerically and experimentally
by Wang and Chen [178]. As has been discussed at length at the end of appendix B,
there are strong reasons to be cautious regarding the results of such a model; given
the results of chapter 5 the author’s recommendation would be to employ the simpler CDC approach in future ICHPD modelling work. Nevertheless the model has
provided interesting insights including providing a qualitative account for the temperature ‘dip’ that has been observed in plate drying [140], which represents another
point of contact between the modelling work reported here and empirical data.
6.3.
Strengths, weaknesses, suggested further work
In this section we consider some of the strengths and weaknesses of the work reported in this thesis, and indicate areas where further work could produce interesting
results. Considering first the strengths of the modelling work, a comprehensive physical model of a complex system has been produced, whose outputs (wherever tested)
have satisfied both qualitative and quantitative assessments of their accuracy. The
model development has involved contributions to the understanding of the physics
of contact drying, in the idealised models of chapter 1, the detailed duct model of
chapter 3, and the porous modelling work in chapter 5 and appendix B. The results
from the models accord with, and have enabled further development of, physical
intuition regarding how such systems could work, and have indicated that ICHPD
shows significant promise as an energy-efficient drying technology for appropriate
products. The model that has been developed has the potential to be a useful tool for
ongoing research, and to enable further development of the ICHPD idea.
One interesting question that has not been addressed in the thesis is what performance can be obtained from the ICHPD system that has been modelled when it is
operated in the continuous mode. With some modification, the model developed in
chapter 5 could be used to evaluate the system steady state in a drier in which product is transported continuously through the system as it dries (e.g. in the screw drier
that is depicted, below, in Fig. 6.1). The energy performance of this system can be
expected to lie somewhere in between the time-averaged performance of the batch
drier modelled in chapter 5, and the steady-state performance predicted in chapter
205
4 – that is, somewhere between 6.41 kg kWh−1 and 13.6 kg kWh−1 . Where the SMER
falls in this range would be interesting to find out.
The most significant weakness of the work reported in this thesis is undoubtedly
the limited connection to new empirical results. While a great deal of effort has been
made to connect the model outputs both to theory and to experiment, the isolation
of this work from a practical apparatus, which would allow the model to be put to
a direct test, is a significant limitation. Also, in the absence of a real physical system
explicitly being modelled, some of the choices of system parameters have been somewhat arbitrary. Clearly this is something that could be remedied: it is hoped that
the simulation that has been developed in the course of this thesis may be adapted
usefully to future research which could include an experimental programme.
Other problems encountered in the research resulted from the modelling approach
that was chosen. As was discussed at the end of appendix B, the comprehensive
approach that was adopted for modelling the falling-rate period of drying is inherently problematic from the standpoint of theoretical justification of the equations.
Although the model’s generality means that it could in principle be used for a variety of porous products, by using appropriate constitutive relations for the given
product, it is not clear that any accuracy benefits provided by the elaborate detail of
the model justify the effort involved in implementing it. Adopting a simpler, more
empirically-based approach might be more fruitful. The theoretical work presented
in appendix B did not produce any result with any practical implication for the subsequent modelling work, although it did clarify the relationship between several existing approaches to modelling drying. The internal-process model that has been
implemented in chapter 5 is significantly simpler than the equations developed in appendix B, not explicitly including hygroscopically-bound moisture, and instead using
the constitutive relations employed by Wang and Chen [179] to link product saturation to hygroscopicity. Thus the model’s accuracy is subject to even more doubt than
the equations developed in appendix B. In any case it seems that there is always more
physical detail that could be built into the model: for instance, product warp could
reduce the thermal contact between the product and the heating plate, with effects unaccounted for by the model as it stands, and very difficult to account for theoretically.
Regarding the whole-system model, on the other hand, additional elementary detail
could be straightforwardly and meaningfully incorporated. The most obvious example is radiative heat transfer from the heating plates above the product, which has not
been considered even though it may have a non-negligible effect [94]. Again, including such details would perhaps make more sense when modelling a specific system,
rather than for collecting general performance indications. It should be emphasized
that none of the effects just described is likely to significantly impact the validity of
the key modelling results of the thesis, since these affects result from second-order
206
correction terms that are unlikely to qualitatively alter system behaviour.
6.4.
Key conclusions
We close this chapter by linking the key conclusions with the aims and objectives of
the study as outlined in chapter 1. As stated on p.32, the fundamental goal of this
project was to develop a numerical model of an ICHPD system, in order to develop
the capacity to assess the potential of such a system to improve on the energy efficiencies currently attained in heat pump drying, and also to develop the capacity to
optimise such a system, e.g. so that the demands of the heat pump are matched to the
properties and demands of the product being dried. Both of these goals have been
realised. The model that has been developed can be used to investigate the effects
of a variety of parameters, for instance the effects of product thickness, air velocity,
temperature set-point, etc., on the system behaviour. The model could be applied to
a wide variety of porous materials, each characterised by a different set of consitutive
relations. However, on the basis of the results of chapter 5, we have concluded that
an empirical approach based on the CDC method would be a satisfactory, and more
straightforward approach, if further using the simulation as a design tool.
The model has established that for applicable products the ICHPD configuration
may substantially increase the energy efficiency of heat pump drying, by a factor
of three compared with conventional adiabatic HPDs. This ICHPD energy efficiency
gain is, however, highly sensitive to the product thickness. The energy efficiency gain
of ICHPD is also sensitive to any constraint on the temperature and the maximum
allowable relative humidity above the product. Further questions that have been
addressed include the impacts of variation in the transfer coefficients, of the highmass-transfer enhancement factors (both slight) and of user-controlled parameters
(where SMER has been found to be most sensitive to variation in the surface area
available for drying and also to the dimensioning of the evaporator).
Isothermal HPD is thus likely to be most applicable in the drying of those products,
such as sludges and pastes, that can be spread into thin layers, in particular those that
also are least vulnerable to quality deterioration at high temperature and humidity.
Product throughput has been shown to be simultaneously maximised at low product thickness, implying that ICHPD provides an opportunity to avoid the adiabatic
mode’s trade-off between drying rate and energy efficiency, by using a thin product
layer. A second-law analysis of the whole system has shown the reasons underlying
the energy performance gain associated with the isothermal mode; ICHPD is found
to reduce irreversibility occurring within the refrigerant cycle by roughly the same
amount as that occurring in heat transfer from the condenser to the drying process
Discussion and conclusion.
207
and in the drying process itself, highlighting the synergistic nature of HPD systems.
One key theoretical goal has been to assess the impact that the product temperature gradient may have on the evaporation and transport profiles within the product,
and that the drying kinetics ultimately have on the energy performance of the drier.
It has been found that the temperature gradient contributes significantly to moisture
transport within the product, and that as a result in batch ICHPD the drying process
occurs comparatively steadily until the end, so that the mode is affected less by the
performance penalty that results from operation away from the initial steady state.
Although the model has only been used so far to examine the dynamics of batchmode HPD operation, it could be readily extended to make predictions of system performance for continuous-operation driers. This represents an important opportunity
for further work, as the modelled performance of ICHPD in the initial steady state of
the constant drying rate period was significantly greater than the time-averaged performance in batch operation, and the steady-state operation of a continuous drier is
likely to provide the limiting case for energy efficiency of a practical ICHPD system.
The work discussed in this thesis has suggested that ICHPD has significant merit
as an avenue for increasing drying energy efficiency for appropriate products. We
close with an idea for how a practical ICHPD might potentially be configured – even
though this is only loosely tied to the work contained in the thesis. Fig. 6.1 shows a
schematic illustration of how a continuous-operation ICHPD could be configured. In
the figure a rotary spreader is used to maintain contact between product in the drier
and a stationary outer cylindrical shell, in which the refrigerant condenser tubes can
be embedded. This approach is commonly used by pharmaceuticals processors, with
heat typically provided at the outer jacket by steam or hot oil. Combining such a
system with refrigerative HPD technology would be a relatively small technological
step, which, as the results of this thesis suggest, could offer significant energy returns.
Discussion and conclusion.
208
ẆP
8
14
Condensate
ẆF
15
7
10
6
3
9
13
12
11
1
5
2
4
8
Dry product
Top view
(whole system)
Wet product
ẆA
Rotary spreader
Outer heating shell
Condenser tube
Blade
Air
Product
End view
(rotary spreader + contact condenser)
Figure 6.1: Example design for a continuous-operation thin-film ICHPD. 1:
dry product outlet. 2: wet product inlet. 3: condensate drain. 4: rotary drive.
5: rotary spreader + contact condenser. 6: evaporator. 7: subcooler. 8: air-side
condenser. 9: compressor. 10: liquid receiver. 11: suction-line accumulator.
12: thermostatic expansion valve. 13: fan. 14: outlet vent. 15: inlet vent.
209
Appendix A
Background theory
This chapter comprises a self-contained development of the basic governing equations that are used throughout the thesis. Since much of the material presented here
is conventional, although it has been assembled from a number of sources, this chapter is placed as an appendix. There are a variety of notational conventions scattered
throughout the literature, and a key purpose of this development is to bring together
the relevant theory in a convenient, self-contained form that employs consistent notation. For instance the notation of [141] and [12] has been adopted in the representation
of the stress tensors σ and τ , under which the stress tensor components σij and τij are
considered positive for tension, and Table A.1 below summarizes the relationship
between the definitions used here and the notation used by other key references.
A.1.
Finite volume methodology
The result of any numerical simulation of a physical system is akin to the outcome
of an experiment, in that it is necessarily limited to a finite set of values, from which
an approximate representation, Ψ, of the system can be reconstructed. For instance,
the values might represent intensive physical properties at a set of N locations {xi }
called grid points, at instants separated by a fixed interval ∆t. If our representation
Ψ is to reflect the physical system’s real behaviour, then as N becomes large and ∆t
becomes small, Ψ must tend to the true solution of the system’s governing equations
and boundary conditions. In order to obtain Ψ, we seek to create an iterative process
yielding successive approximations that are guaranteed to get progressively closer to
Ψ. Such a procedure is called a contraction map for which Ψ is a fixed point [20].
Several different approaches are used to obtain grid point values; a common feature is that the set of points involved in determining the value at any particular grid
Background theory
210
point is small, and the variation of the parameters between grid points is approximated by simple (often piecewise-linear) functions of position. The first of these simplifications is an expression of the principle of local action, while the second can also be
justified, since as the number of grid points increases, and their separation decreases,
linear interpolation between the grid point values will better and better approximate
the actual variation of the physical parameters ([146], p.38); [117]. In general, we can
form an approximation Ψ(x) of a physical quantity throughout space using a linear
combination of N basis functions φi (x):
Ψ (x) =
N
∑ ai φi (x)
(A.1.1)
i =1
If we know the value of Ψ at the gridpoints {xi }, linear interpolation of Ψ corresponds
to taking piecewise-linear basis functions φi which satisfy

i=j
1,
φi x j =
0,
i 6= j
(A.1.2)
with the coefficients
a i = Ψ ( xi )
(A.1.3)
The general differential equation which expresses the local rate of change of a
quantity Ψ = ρφ that can be created, conducted and convected may be written:
∂Ψ
+ ∇ · JΨ = S
∂t
(A.1.4)
In (A.1.4) JΨ = ρuφ − Γ∇φ is the total (convection + conduction) flux of Ψ [117].
Multiplying (A.1.4) by an arbitrary weight function wi (x) and integrating over the
solution domain leads to the integral form of the differential equation:
Z ∂Ψ
+ ∇ · Jψ − S wi (x) dV = 0
∂t
(A.1.5)
R
If all terms appearing in (A.1.5) are continuous, the integral form is equivalent to
(A.1.4) if it is satisfied for arbitrary wi (x). By applying the divergence theorem (A.2.5),
the integral form may be recast into so-called weak integral form [118]:
Z Z
∂Ψ
− S wi − JΨ · ∇wi dV + JΨ wi · n̂dA = 0
∂t
R
(A.1.6)
S
By using N different weighting functions wi (x), enough equations can be obtained to
fix the N coefficients ai : this is the so-called weighted residual technique. Different formulations result from substituting different weight functions into (A.1.5) and (A.1.6).
For instance, the Finite Element Method (FEM) formulation developed by Galerkin
takes the N basis functions themselves (often piecewise-linear, each taking the value 1
at one grid point and 0 at all others) as weight functions. In the Finite Volume Method
Background theory
211
(FVM) formulation recommended by Patankar [117] which is used in this work, the
solution domain is divided into N subdomains (‘control volumes’) surrounding the
N grid-points, and each weight function is set as the characteristic function of a given
control volume (equal to 1 within the control volume, 0 elsewhere). In more detail,
the approach may be described as follows:
1. The calculation domain is partitioned into non-overlapping regions (the ‘controlvolumes’), with one region surrounding every grid-point.
2. Setting wi (x) as the characteristic function of the ith control volume, the conservation equation is integrated over the entire domain (equivalently, the conservation equation is integrated over each control volume). Applying the divergence theorem (A.2.5) and the generalized transport theorem (A.2.35) to (A.1.5)
yields, for a control volume fixed in space:
d
dt
Z
cvi
ΨdV +
Z
∂cvi
Jψ · n̂dA =
Z
SdV
(A.1.7)
cvi
3. Equation (A.1.7) is discretised. Integrands are assumed constant throughout
the domain of integration. Surface conduction fluxes are calculated by using
piecewise profiles to evaluate the gradients of the physical parameters at the
control-volume interfaces, and combined (conduction + convection) fluxes are
estimated using Patankar’s conduction-convection recipe.
4. The fluxes thus calculated are associated with the interfaces themselves and are
therefore identical for neighbouring control volumes.
By this method a set of discretised equations is produced, which can be solved using an iterative process. An important feature of the FVM is that the profiles used to
generate the discretised equations are subsequently discarded, and are not regarded
as part of the solution. This allows different profiles for the same variable to be employed in developing the discretised equations. Thus in point 3 above, Ψ is assumed
constant throughout a given control volume for the purpose of evalutating the volume integral, while it is assumed to vary linearly with position for the purpose of
evaluating the surface flux.
The key attraction of the FVM is that by its construction (because of point 4 above),
solutions are guaranteed to satisfy conservation principles precisely, even when the
number of grid points is small. The conservative balances will therefore be exact
for any group of control volumes (including both single control volumes and the
entire solution domain), which is a significant advantage for troubleshooting code.
The link between the finite volume method and the underlying conservation law is
summarized as follows:
Background theory
212
General Conservative Flow Equation:
Rate of
Net convective
increase of φ
of volume
+
element
flow of φ
out of volume
element
Rate of
=
Rate of
increase of φ
due to
diffusion
+
increase of φ
due to
sources
Finite Volume Formulation:
Rate of
Net rate of flow of
increase of φ
of control
volume
A.2.
=
φ into control volume
due to combined
convection and diffusion
Rate of
+
increase of φ
due to
sources
Continuum mechanics background
This section states the divergence theorem [149, p.236], introduces the basic kinematic
concepts of continuum mechanics such as the material derivative [136], and presents a
straightforward derivation of the generalized transport theorem [145, 146].
Tensors and the divergence theorem
The physical quantities of continuum mechanics are entities whose mathematical description must be consistent, regardless of the reference frame within which they are
considered [134, p.674]. For such entities in Euclidean 3-space, the language of Cartesian tensors is the simplest for their description. When a coordinate system has been
adopted, these tensors can be represented in terms of their components, as referred
to the fixed coordinate directions. In this thesis, for the modelling of the belt dryer,
the coordinate system depicted in Figure A.1 is used. In order for the coordinate system to be right-handed, the z-direction must point upwards out of the page. Velocity
components in the x, y and z directions will sometimes be referred to respectively
using the letters u, v and w (up out of the page), as indicated in the figure.
In any Cartesian coordinate system the isotropic tensor δ has coordinates δij given
by the Kronecker delta function, which is 1 if i = j and 0 otherwise. Similarly the LeviCivita symbol eijk is used to represent the Cartesian components of the permutation
tensor, taking the value 1 when (i, j, k) is an even permutation of the 3-tuple (1, 2, 3),
the value −1 when (i, j, k) is an odd permutation of (1, 2, 3), and the value 0 if any
index is repeated [134, p.688]. Summation over repeated indices will be assumed
throughout this chapter, unless explicitly stated. Following the convention adopted
by Slattery [146], the gradient ∇v of a vector field v is defined to be the following
Background theory
213
air flow
y
product
x
heating plate
Figure A.1: Coordinate system within dryer.
second-order tensor. In dyadic notation [146]:
∇v ≡
∂vi
ee
∂x j i j
(A.2.1)
The divergence ∇ · v of the vector field v is defined as the trace of its gradient:
∇ · v = tr(∇v ) = ∇v : δ =
∂vi
∂xi
(A.2.2)
Similarly for a second-order tensor field τ , we have [146]:
∂τij
eee
∂xk i j k
∂τij
∇ · τ = ∇τ : δ =
e
∂x j i
∇τ =
(A.2.3)
(A.2.4)
Care must be taken when comparing the equations presented in this thesis with the
equations for instance of references [16] and [17], in which the definitions ∇v ≡
∂vi
∂x j e j ei
and ∇τ ≡
∂τij
∂xk ek ei e j
are used instead of the above – see [146, p.658-670].
Throughout the following overview of the theoretical background, frequent use is
made of the divergence theorem which relates volume integrals to surface integrals, in
any of the following three forms [17, p.824]:
Z
(∇s) dV =
R
Z
R
Z
R
(∇ · v ) dV =
(∇ · τ ) dV =
Z
(sn̂) dS
(A.2.5a)
(v · n̂) dS
(A.2.5b)
(τ · n̂) dS
(A.2.5c)
S
Z
S
Z
S
In (A.2.5), S is the closed bounding surface of the enclosed region R, n̂ is the unit
outward normal on S, s is any scalar (rank zero tensor), v is any vector (tensor of rank
one) and τ is any tensor of second or higher rank.
Background theory
214
Kinematics
A material body is formally defined as a set satisfying a number of properties [170,
p.6], each of whose elements, ζ, is called a material particle. The set is such that there
exists a continuous bijection χ, called a configuration, mapping it into the Euclidean
space of our everyday experience. Thus the objects we encounter may be idealised as
configurations of (unspecified) bodies.
A motion is a family of configurations smoothly parametrized by time:
x = χ (ζ, t)
(A.2.6)
where x = ( x1 , x2 , x3 ) is the vector location of particle ζ at time t.
A particular configuration ℵ(ζ ), called the reference configuration (often the system’s starting configuration), is used to specify particles. The particle at the location
x0 = (ξ 1 , ξ 2 , ξ 3 ) in the configuration ℵ is given by ζ = ℵ−1 (x0 ). The motion of this
particle is given by
x = χ (ζ, t) = χℵ (x0 , t)
(A.2.7)
χℵ (x0 , t) = χ ℵ−1 [x0 ] , t
(A.2.8)
where χℵ is related to χ by
Time derivatives
This section follows [12, p.43-71], [170, p.4-30], and [145, p.1-66]. We will call a timepath through space any family of spatial locations smoothly parametrized by time:
x = x(t) = ( x1 [t], x2 [t], x3 [t])
(A.2.9)
For any locally specified quantity Ψ(x, t), associated with every time-path through
space is a function Ψ p which takes the local value of Ψ at each t:
Ψ p ( t ) = Ψ (x [ t ] , t )
(A.2.10)
The total time derivative of Ψ p along the time-path through space is given by the chain
rule for partial derivatives as follows:
dΨ p
∂Ψ dx1
∂Ψ dx2
∂Ψ dx3
∂Ψ
=
+
+
+
dt
∂x1 dt
∂x2 dt
∂x3 dt
∂t
∂Ψ
= ẋ · ∇Ψ +
∂t
where ẋ is the time rate of change of location along the path.
derivative of Ψ.
∂Ψ
∂t
(A.2.11)
(A.2.12)
is called the local time
Background theory
215
From (A.2.6), associated with every material particle for any given motion is a
time-path through space x(t), and thus a function Ψ (x [t] , t) which takes the value
of Ψ experienced by the particle at every t. We may define a function Ψℵ which gives
Ψ as a function of location in the reference configuration, and of time:
Ψ ℵ ( ξ 1 , ξ 2 , ξ 3 , t ) = Ψ ( x1 [ t ], x2 [ t ], x3 [ t ], t )
The material time derivative of a material quantity Ψ, written
(A.2.13)
dm
dt [ Ψ ]
(and which will
also sometimes be written Ψ̇), is the time rate of change of Ψ associated with a particular material particle. It is given by the partial time derivative of Ψℵ at the particle’s
fixed position in the reference configuration:
dm Ψ
∂Ψℵ =
dt
∂t (ξ 1 ,ξ 2 ,ξ 3 ,t)
(A.2.14)
Using the chain rule for partial derivatives on the right side of (A.2.13), the material
time derivative is obtained as the total time derivative of Ψ along the path taken by
that particle:
dm Ψ
∂Ψ
= v · ∇Ψ +
dt
∂t
(A.2.15)
where v is the velocity of the given material particle. In particular taking Ψ = r
(the displacement vector field), since ∂∂tr = 0, the material derivative of the particle’s
position is just the particle’s velocity itself:
dm r
= v · ∇r = v · δ = v = ( ẋ1 [t], ẋ2 [t], ẋ3 [t])
dt
(A.2.16)
Time rate of change of body integrals
The body integral of any quantity Ψ is the integral of Ψ over a given material body. In
order to emphasise that the region over which Ψ is being integrated is that occupied
by the body, a subscript ‘m’ will be used in denoting the region of integration. The
domain of integration Rm is in general a function of time. Using a standard result
from calculus, e.g. [149, p.211], the integral can be transformed into one over the
body in its reference configuration Rmℵ :
Z
Rm
where
Ψ dV =
Z
ΨJ dV
(A.2.17)
Rm ℵ
∂ ( x1 , x2 , x3 ) J=
∂(ξ 1 , ξ 2 , ξ 3 ) (A.2.18)
Consider the time rate of change of the body integral of any quantity Ψ. If we use
(A.2.17) to transform the integral into one over the body in its reference configuration,
Background theory
216
in which the region occupied by the body is time-invariant, we may exchange the
order of integration and differentiation [145, p.19]:
d
dt
Z
Ψ dV =
Rm
=
d
dt
Z
Rm ℵ
Z Rm ℵ
=
ΨJ dV
Z Rm
∂Ψ Ψ ∂J
+
∂t
J ∂t
(A.2.19)
J dV
dm Ψ Ψ dm J
+
dt
J dt
(A.2.20)
dV
We now turn to the substantial time derivative of the Jacobian,
(A.2.21)
dm J
dt ,
on the right side
of (A.2.21). The Jacobian is given in full by [145, p.611]
J (ξ 1 , ξ 2 , ξ 3 , t) =
∂x1
∂ξ 1
∂x2
∂ξ 1
∂x3
∂ξ 1
= eijk
∂x1
∂ξ 2
∂x2
∂ξ 2
∂x3
∂ξ 2
∂x1
∂ξ 3
∂x2
∂ξ 3
∂x3
∂ξ 3
(A.2.22)
∂x1 ∂x2 ∂x3
∂ξ i ∂ξ j ∂ξ k
(A.2.23)
mJ
It follows that the material derivative J̇ = ddt
is given by [136, p.84]:
∂J J̇ =
∂t (ξ 1 ,ξ 2 ,ξ 3 ,t)
∂ ∂x1 ∂x2 ∂x3
+...
= eijk
∂t ∂ξ i ∂ξ j ∂ξ k
(A.2.24)
(A.2.25)
where for simplicity two other terms resulting from the product rule have not been
shown. Using Equations (A.2.16) and (A.2.14) and some calculus yields:
∂v1 ∂x2 ∂x3
J̇ = eijk
+...
∂ξ i ∂ξ j ∂ξ k
∂v1 ∂xl ∂x2 ∂x3
= eijk
+...
∂xl ∂ξ i ∂ξ j ∂ξ k
=
∂v1 ∂xl
∂xl ∂ξ 1
∂x2
∂ξ 1
∂x3
∂ξ 1
∂v1 ∂xl
∂xl ∂ξ 2
∂x2
∂ξ 2
∂x3
∂ξ 2
∂v1 ∂xl
∂xl ∂ξ 3
∂x2
∂ξ 3
∂x3
∂ξ 3
+...
(A.2.26)
(A.2.27)
(A.2.28)
Expanding this determinant along the first row, only the terms for which l = 1 contribute, since the other two terms contain a multiple of one of the other two rows.
This determinant therefore reduces to
∂v1
∂x1
J. Considering the other two determinants
needing to be evaluated, the material derivative of J can be seen to be
∂v1
∂v
∂v
J̇ =
+ 2+ 3 J
∂x1
∂x2
∂x3
= J ∇·v
(A.2.29)
(A.2.30)
Background theory
217
This allows us to write (A.2.21) as
d
dt
Z
dm Ψ
+ Ψ ∇ · v dV
dt
Rm
Z
∂Ψ
=
+ ∇ · (Ψv ) dV
∂t
Z Ψ dV =
Rm
(A.2.31)
(A.2.32)
Rm
after using relation (A.2.15) to substitute for the substantial derivative in (A.2.31).
Using the divergence theorem, this equation may be expressed as follows:
Z
Z Z
d
∂Ψ
Ψ dV =
dV + Ψv · n̂ dS
dt
∂t
Rm
Rm
(A.2.33)
Sm
This is the transport theorem for material bodies [145, p.19]. A nonrigorous intuitive
derivation is given in [187, p.193-197]. Note that the integrals are over the body’s
interior Rm , and its bounding surface Sm , which are functions of time.
Let Ψ be any locally specified quantity (e.g. a tensor of unspecified rank, which
may be associated with the particles of a body). Consider the integral of the function
Ψ over any body consisting of ‘fictitious particles’. From (A.2.32) it follows that:
Z
Z d
∂Ψ
Ψ dV =
+ ∇ · (Ψw ) dV
(A.2.34)
dt
∂t
R( t )
R( t )
=
Z R( t )
∂Ψ
∂t
dV +
Z
S( t )
Ψw · n̂ dS
(A.2.35)
In (A.2.34) and (A.2.35) R(t) is used to denote the system (the region occupied by
the fictitious body), and w · n̂ is the normal velocity of the [particles at its] bound-
ing surface S(t). This result is called the generalized transport theorem [145], and is a
generalization of Leibniz’s rule for the derivative of a definite integral [149, p.186].
Consider a control volume which is fixed in space. The volume can be regarded
as being occupied by a body of fictitious particles, all of which are stationary. The
region occupied by this body is constant, its bounding surface is constant, and the
surface velocity w is identically zero. Thus for such a body the generalized transport
theorem (A.2.35) reduces to the following relation:
Z
Z d
∂Ψ
Ψ dV =
dV
dt
∂t
R
(A.2.36)
R
In other words, swapping the order of differentiation and integration for a stationary
control volume, i.e. using the relation (A.2.36), is equivalent to using the generalized
transport theorem in the case of a stationary control volume.
Background theory
A.3.
218
Conservation laws for multicomponent systems
In fluid mechanics, multicomponent systems are regarded as the superposition of
multiple interpenetrating bodies, each representing one species. The present section develops the multicomponent balance equations over a single phase. After the
conservation of mass is formulated for each separate species, the principles of momentum and energy conservation are postulated for a fictitious body whose particles
move everywhere at the local mass-average velocity [146, p.431]. This approach leads
to the governing differential equations of nonequilibrium thermodynamics [56].
Conservation of mass
Consider the body that represents the species k. Let Φk be the total local mass production rate of species k in chemical reactions (per unit volume). According to the
principle of mass conservation, in the presence of chemical reactions, the rate of change
of the total mass of species k is given as:
d
dt
Z
ρk dV =
Rk
Z
Φk dV
(A.3.1)
Rk
where Rk denotes the region occupied by species k. Equation (A.3.1) can be regarded
as the defining equation for Φk . Applying the transport theorem in the form of
(A.2.32) to the left side of (A.3.1), we obtain:
Z
Z ∂ρk
+ ∇ · (ρk vk ) dV = Φk dV
∂t
(A.3.2)
Rk
Rk
As this integral relationship holds for arbitrary material regions, since any portion
of a body is a body [145, p.666], it must hold identically throughout the body. Thus
everywhere
∂ρk
= −∇ · (ρk vk ) + Φk
∂t
(A.3.3)
Taken together, the k separate bodies comprise our entire system. Summing (A.3.3)
over k leads to the following relation:
∂ρ
= −∇ · (ρv ) + ∑ Φk
∂t
k
(A.3.4)
In (A.3.4) the following definitions have been used:
n
ρ=
∑ ρk
(A.3.5)
n
ρ k vk
= ∑ ω k vk
ρ
k =1
k =1
(A.3.6)
k =1
n
v=
∑
Background theory
219
ρ is called the total density and v is called the barcyentric velocity; the mass fraction
ωk of species k is given by ωk = ρk /ρ. We further define the diffusion velocity uk of
species k as its excess over the barycentric velocity, and the diffusion flux jk as the flux
of species k associated with its diffusion velocity:
uk = vk − v
(A.3.7)
jk = ρ k (vk − v )
(A.3.8)
Now define a body consisting of fictitious system particles which initially occupy
the region R0 , and dictate that they take the mass-average (barycentric) velocity everywhere. We denote the region occupied by this system at time t by Rv(t), bounded
by the surface Sv(t). Integrating ρk over Rv , we find from (A.2.35) and (A.3.3), and an
application of the divergence theorem:
Z
Z
Z d
∂ρk
dV + ρk v · n̂ dS
ρk dV =
dt
∂t
Rv
Rv
=
Z
Rv
=
Z
(A.3.9)
Sv
[−∇ · (ρk vk ) + Φk ] dV +
Φk dV +
Rv
Z
Sv
Z
Sv
ρk v · n̂ dS
jk · (−n̂) dS
(A.3.10)
(A.3.11)
The surface integral on the right of (A.3.11) gives the rate of diffusion of species k into
the system through its bounding surface. Equation (A.3.11) reduces to (A.3.1) in the
case where only one species is present, since in that case jk = 0. Summing (A.3.11)
over k, noting definitions (A.3.5) – (A.3.8), we obtain:
d
dt
Z
Z
ρ dV =
Rv
Rv
Clearly if ∑ Φk = 0 everywhere, then
k
d
dt
R
∑ Φk dV
(A.3.12)
k
ρ dV = 0. Assuming that all physical
Rv
parameters are described by well-behaved functions (in particular, all functions are
assumed to be everywhere differentiable, thus continuous), the converse is also true,
because if ∑ Φk 6= 0 anywhere, then there must necessarily be a finite region within
k
which ∑ Φk 6= 0 and takes the same sign, so that (since any portion of a body is
k
R
d
a body) it would be possible to define a body for which dt
ρ dV 6= 0. We have
Rv
therefore demonstrated the equivalence of two different expressions of the principle of
mass conservation for a multicomponent body:
• Overall mass conservation for a multicomponent body:
d
dt
Z
Rv
ρ dV = 0
(A.3.13)
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220
• Mass conservation in chemical reactions:
∑ Φk = 0
(A.3.14)
k
We take the principle of mass conservation for a multicomponent body as our first
postulate. In terms of the R independent chemical reactions that may take place
R
among the mixture’s components, Φk = ∑ νkj Jj where νkj is the specific stoichioj =1
metric coefficient of species k in the jth reaction, which locally is proceeding at volumetric rate Jj [158]. Subsituting (A.3.14) into (A.3.4), we obtain the total differential
mass balance:
∂ρ
+ ∇ · (ρv ) = 0
∂t
(A.3.15)
Equation (A.3.15) is the local form of the law of conservation of mass. Noting expression (A.2.15), we can obtain the substantial form:
dm ρ
+ ρ (∇ · v ) = 0
dt
(A.3.16)
Integrating (A.3.3) and (A.3.15) over an arbitrary control volume R(t), we can use
(A.2.35) to swap the order of differentiation and integration to obtain:
Control Volume Mass Balance For kth Component:
d
dt
Z
ρk dV =
R( t )
Z
S( t )
ρk (vk − w ) · (−n̂) dS +
Z
Φk dV
(A.3.17)
R( t )
Control Volume Overall Mass Balance:
d
dt
Z
ρdV =
R( t )
Z
S( t )
ρ (v − w ) · (−n̂) dS
(A.3.18)
In Equations (A.3.17) and (A.3.18), w · n̂ is the normal velocity of the bounding sur-
face S(t) of the control volume R(t). In the absence of chemical reactions, for instance
in a moist-air system, the final term in Equation (A.3.17) can be dropped.
Combining (A.3.15) and the form of the substantial time derivative (A.2.15) leads
to the following relation for any locally specified property a:
ρ
∂( aρ)
dm a
=
+ ∇ · ( aρv )
dt
∂t
(A.3.19)
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221
Equation (A.3.19) relates the substantial time derivative on the left to the local time
derivative on the right. Substituting (A.3.19) into (A.2.32), the transport theorem for
Ψ = ρa for a multicomponent body can be written as follows:
d
dt
Z
Z
ρa dV =
Rv
ρ
Rv
dm a
dV
dt
(A.3.20)
Conservation of momentum
Our second and third postulates, respectively the principle of the conservation of linear
momentum and the principle of the conservation of angular momentum, are expressed for
a multicomponent system as follows [145, p.433-435]:
d
dt
d
dt
Z
Rv
Z
Rv
∑ ρi vi dV =
i
t dS +
Z
Sv
Z
∑ ρi fi dV
Rv
Sv
∑ ρi (r × vi ) dV =
i
Z
(r × t) dS +
(A.3.21)
i
Z
Rv
∑ ρi (r × fi ) dV
(A.3.22)
i
Once again the region of integration is defined by a set of fictitious system particles,
whose velocity everywhere is the barycentric velocity.
The quantity on the left-hand side of (A.3.21) is the rate of change of body momentum. The term t which is integrated on the right over the body’s surface represents
the contact force per unit area exerted on the body by the material outside Sv , and fi
denotes any external body force per unit mass of species i. The specific body force fi
acting on species i is assumed to result from a time-independent conservative scalar
potential field ψi [56, p.15]:
∂
ψ =0
∂t i
fi = ∇ψi ,
(A.3.23)
We define the specific force f and potential ψ as follows:
f =
∑
ψ=
∑
ρ i fi
ρ
ρi ψi
ρ
(A.3.24)
(A.3.25)
The total force experienced by the body is given on the right hand side of (A.3.21)
by the resultant of the total force exerted on the body’s surface and the total force
exerted on its interior. In equation (A.3.22), the rate of change of angular momentum
is similarly equated to the sum of the moments due to the surface and body forces.
By considering a limiting process involving progressively smaller tetrahedral fluid
elements [145, p.32], t can be shown to be given by
t = σ · n̂
(A.3.26)
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222
where n̂ is the outward unit normal and σ is a second-rank tensor called the total
stress tensor, which is a function of the fluid motion.
Substituting equations (A.3.6) and (A.4.20) into (A.3.21), and using (A.3.20) on the
left-hand side and the divergence theorem on the right, we obtain:
Z
ρ
Rv
dm v
dV =
dt
Z
(∇ · σ + ρf ) dV
(A.3.27)
Rv
Continuity of the integrand leads once again to a differential expression, called Cauchy’s
first law (alternatively, the differential momentum balance) [136, p.102]:
ρ
dm v
= ∇ · σ + ρf
dt
(A.3.28)
A similar rearrangement of (A.3.22) employing (A.2.16) leads to the equation:
Z
Z
(A.3.29)
(r × ρv̇ ) dV = r × ∇ · σ + eijk σkj ei + ρf dV
Rv
Rv
Substituting (A.3.28) into (A.3.29) and invoking continuity leads to
eijk σkj ei = 0
(A.3.30)
which is equivalent [136, p.103] to the equation:
σ = σT
(A.3.31)
Equation (A.3.31) is known as the angular momentum principle.
Setting a = v in (A.3.19) yields
ρ
dm v
∂(ρv )
=
+ ∇ · (ρvv )
dt
∂t
(A.3.32)
where the term vv is the dyadic product of v with itself: expressed in terms of components, [vv ]ij = [v ]i [v ] j . Comparing (A.3.28) and (A.3.32) and noting that ∇ · p0 δ = 0
where p0 is the ambient pressure, we obtain the local form of the momentum balance:
∂(ρv )
= −∇ · ρvv − (σ + p0 δ ) + ρf
∂t
(A.3.33)
Integrating (A.3.33) over an arbitrary region R(t), and applying the divergence theorem, we can use (A.2.35) to swap to order of differentiation and integration to obtain
the control volume form for the momentum balance:
Control Volume Momentum Balance:
d
dt
Z
R( t )
ρvdV =
Z S( t )
ρv (v − w ) − (σ + p0 δ ) · (−n̂) dS + F
(A.3.34)
Background theory
223
where the total body force F on the matter within the region is given by
F =
Z
ρf dV
(A.3.35)
R( t )
Equation (A.3.34) is equivalent to Equation (4.4.3-8) of reference [146].
Conservation of energy
According to the first law of thermodynamics, or the principle of conservation of energy, the total energy change of a system must equal the total energy flow into that
system. This postulate is expressed for a multicomponent body as follows:
!
Z
Z
Z
d
1
2
ρ u + ∑ ωk vk + ψ dV = v · (σ · n̂) dS + jq · (−n̂) dS
dt
2
k
Rv
Sv
Sv
+
Z
Sv
∑ ψk jk · (−n̂) dS
k
(A.3.36)
The term u appearing on the left-hand side of (A.3.36) represents the specific internal
energy of the multicomponent mixture, including energy contained in the vibrational
and rotational modes. The term ∑ 21 ωk v2k represents the specific kinetic energy, with
k
v2k = vk · vk . The term ψ represents the specific potential energy defined by (A.3.25).
Thus the left-hand side tracks the rate of change of the sum of the internal, kinetic
and potential energies. The first term on the right-hand side of (A.3.36) is the rate at
which work is done on the multicomponent system by stress forces at its surface. The
second term represents the inward diffusion flow of energy through the surface: jq
is the total heat flux vector, which is given in terms of the pure heat flux vector q c as
follows [145, p.474]:
n
H̄k
1 2
jq =q + ∑
+ u k jk
Mk
2
k =1
n 1 2
c
=q + ∑ h k + u k jk
2
k =1
c
where H̄k =
∂H
∂nk
T,p,n j
(A.3.37)
(A.3.38)
is the partial molar enthalpy of species k, and hk =
H̄k
Mk
is
called the partial mass enthalpy (or partial specific enthalpy) [188, 158]. The total heat
flux vector jq thus comprises the energy flux – internal, compressional and kinetic –
associated with the diffusion of the k distinct species relative to the barycentric motion
of the fictitious system particles, and thus to the system boundary, in addition to pure
heat transfer. Similarly, the third term on the right hand side of (A.3.36) represents
the inward flow of potential energy associated with the diffusion of the k distinct
species across the system boundary. Thus the rate of change of the body’s total energy
Background theory
224
(internal, kinetic and potential) on the left is equated with the total rate at which
energy is flowing into the system.
Using the Reynolds transport theorem (A.3.20) on the left hand side of (A.3.36),
and the divergence theorem on the right, we can invoke the familiar continuity argument to obtain the following differential balance:
!
!
dm
1
2
ρ
u + ∑ ωk vk + ψ = ∇ · σ · v − jq − ∑ ψk jk
dt
2
k
k
(A.3.39)
Since the diffusion fluxes ρωk uk sum to zero, and the mass fractions ωk sum to 1,
ρ
dm
dt
1
dm
1
∑ 2 ωk v2k = ρ dt ∑ 2 ωk
=ρ
dm
dt
1 2
v
2
v2 + 2v · uk + u2k
+ρ
dm
dt
1
∑ 2 ωk u2k
(A.3.40)
(A.3.41)
The second term on the right side of (A.3.41) represents the material time derivative
of the kinetic energy associated with diffusion. It is generally disregarded in the literature [e.g. 158], leading to the following expression:
dm
1
dm 1 2
2
ρ
ω v =ρ
v
dt ∑ 2 k k
dt 2
(A.3.42)
Substituting (A.3.42) into (A.3.39) yields the following equation [158, p.35]:
!
1 2
dm
u + v + ψ = ∇ · σ · v − jq − ∑ ψk jk
(A.3.43)
ρ
dt
2
k
Multiplying the mass balance for species k, Equation (A.3.3), by the kth scalar potential ψk , and noting that ψk is time-invariant, leads to the following equation:
∂ρk ψk
= −∇ · (ρk vk ψk ) + ρk vk · ∇ψk + Φk ψk
∂t
(A.3.44)
Summing (A.3.44) over k, noting that ∑ Φk ψk = 0 when potential energy is conserved
k
in chemical reactions [56, p.16], using (A.3.23) and rearranging yields [158]:
!
∂ρψ
= −∇ · ρψv + ∑ ψk jk − ∑ ρk vk · fk
(A.3.45)
∂t
k
k
In substantial form (A.3.45) becomes:
dm ψ
ρ
= −∇ ·
dt
∑ ψk jk
k
!
− ∑ ρ k vk · fk
(A.3.46)
k
Subtracting (A.3.46) from (A.3.43) yields
dm
1 2
u + v = ∇ · σ · v − jq + ∑ ρ k vk · fk
ρ
dt
2
k
(A.3.47)
Equation (A.3.47) is equivalent to the second equation of [146, Table 8.5.2-1, p.480],
with no heat generation term ρQ. It can also be obtained from the second equation
Background theory
225
of [17, Table 19.2-4, p589]. The total stress tensor σ can be expressed in terms of the
thermodynamic pressure p and the viscous stress tensor τ as in Equation (A.4.7), below,
as follows: σ = − p δ + τ . With this definition of τ , it follows that:
∇ · (σ · v ) = −∇ · ( pv ) + ∇ · (τ · v )
(A.3.48)
Substituting (A.3.42), (A.3.48) and (A.3.38) into (A.3.39), and using (A.3.19) to convert the resulting expression into local form, the following differential energy balance
can be obtained:
∂
1 2
p 1 2
ρ u+ v
= − ∇ · ρ u + + v v + ∇ · (τ · v )
∂t
2
ρ
2
"
#
n
1 2
c
− ∇ · q + ∑ h k + u k jk
2
k =1
+ ∑ ρ k vk · fk
(A.3.49)
k
As an expression of the first thermodynamic law, (A.3.49) may be understood as the
definition of the pure heat conduction flux vector q c [56, p.18]. In the context of this
project, in which the heat and mass flux are modeled using simple constitutive laws,
the equation serves as a means of tracking the internal energy of the system. For
simplicity, the kinetic energy associated with the diffusion flux, 12 u2k , which in engineering applications is generally insignificant next to the partial mass enthalpy term
hk , will be dropped from subsequent expressions. Introducing the specific enthalpy
p
with the definition h = u + ρ , we note that
∂p0
∂t
= 0, and write:
∂
1 2 p0
1 2
ρ u+ v +
= − ∇ · ρ h + v v + ∇ · (τ · v )
∂t
2
ρ
2
"
#
− ∇ · qc +
n
∑ h k jk
k =1
+ ∑ ρ k vk · fk
(A.3.50)
k
Integrating (A.3.50) over an arbitrary region R(t), we can use (A.2.35) to obtain the
following control volume energy balance equation:
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226
Control Volume Energy Balance:
Z
1 2 p0
d
ρ u+ v +
dV
dt
2
ρ
R( t )
=
S( t )
−
1 2
ρ h + v (v − w ) · (−n̂) dS
2
Z
Z S( t )
w ( p0 − p) + v · τ · (−n̂) dS
+
Z
c
q +
∑ h k jk
!
k =1
S( t )
+
n
Z
R( t )
· (−n̂) dS
n
∑ (ρk v + jk ) · fk dV
(A.3.51)
k =1
Equation (A.3.51) can be obtained from [146, (7.4.1-9), p.401]. On impermeable portions of S, we have w = v and the second integral on the right-hand side of (A.3.51),
R
evaluated on these portions, becomes equal to − v · [σ + p0 δ ] · (−n̂) dS, as σ =
R
− p δ + τ . The integral W = v · [σ + p0 δ ] · (−n̂) dS represents the work done by
the system on the surroundings at these impermeable surfaces [146, p.401].
Another form of the energy balance for multicomponent systems is used in chapter 5 in developing the theory of drying. Taking the dot product of Cauchy’s first law
(A.3.28) with the barycentric velocity yields:
dm v
ρ
· v = (∇ · σ ) · v + ρ(f · v )
dt
(A.3.52)
Subtracting (A.3.52) from (A.3.47) yields the following equation [56, p.18]:
ρ
Now,
ρ
dm
( u ) = σ : ∇v − ∇ · jq + ∑ jk · fk
dt
k
dm
dm
(h) = ρ
dt
dt
u+
p
ρ
=ρ
dm
dm
(u) + ρ
dt
dt
(A.3.53)
p
ρ
(A.3.54)
And
dm
ρ
dt
h ṗρ − pρ̇ i
p
=ρ
ρ
ρ2
ρ̇
= ṗ − p
ρ
= ṗ + p(∇ · v )
(A.3.55)
Background theory
227
using (A.3.15). Since p(∇ · v ) = pδ : ∇v and since by definition σ = τ − pδ, adding
(A.3.53) and (A.3.55) yields:
ρ
dm
(h) = τ : ∇v − ∇ · jq + ∑ jk · fk + ṗ
dt
k
(A.3.56)
Substituting (A.3.38) into (A.3.56), we recast the equation into local form:
n
∂
(ρh) + ∇ · (ρhv ) = τ : ∇v − ∇ · qc − ∇ · ∑ hk jk + ∑ jk · fk + ṗ (A.3.57)
∂t
k =1
k
Under typical moist-air conditions within a porous medium undergoing drying, the
work done against viscous stress forces τ : ∇v, the potential energy diffusion work
term ∑ jk · fk and the pressure material time-derivative ṗ are all negligible [188] and
k
are generally dropped from equation (A.3.57), to obtain:
n
∂
(ρh) + ∇ · ρhv + ∑ hk jk = −∇ · qc
∂t
k =1
(A.3.58)
It is possible to simplify the total enthalpy flux due to combined convection and
n
diffusion, ρhv + ∑ hk jk , which appears in (A.3.49), (A.3.51) and (A.3.58). The mok =1
lar enthalpy of a mixture, h̄ = H/n, can be expressed in terms of the partial molar
∂H
as follows [58, p.101], [81, p.333]:
enthalpies H̄i = ∂n
i
T,p,n j
h̄ =
∑ H̄i xi
(A.3.59)
i
Multiplying (A.3.59) by the number of moles per unit mass of the mixture, we obtain:
h=
∑ h i ωi
(A.3.60)
i
where hi = H̄i /Mi is the partial mass (or partial specific) enthalpy of species i.
Equation (A.3.60) can be used to evaluate the total enthalpy flux due to combined
convection and diffusion. Also using (A.3.8) we have
ρhv + ∑ hk jk = ∑(ρk vhk + ρk uk hk )
k
k
= ∑ ρ k vk h k
(A.3.61)
k
where ρk vk = ρk (v + uk ) is the total mass flux of species i, as given in (A.3.7).
A.4.
Linear constitutive relations
Before the balance equations can be used, the mass, momentum and heat fluxes must
be calculated. In this work linear relations are used and only the primary flux driving
Background theory
228
forces are modeled. Most authors concerned with engineering applications consider
the so-called Dufour and Soret cross-effects (which emerge from nonequilibrium thermodynamics, and by which mechanical driving forces can drive energy flux, and
temperature gradients can drive mass flux, respectively) to be negligible [16, 145].
We do not consider these effects in this thesis. Indirect linkages between heat and
mass transfer within the porous product medium do arise, which are unrelated to
these effects. For instance, the vapour transport associated with temperature gradients within a porous medium mentioned on p. 262 arises as a result of Fickian diffusion down the vapour mass concentration gradient in the γ phase. This is only an
indirect effect of the temperature gradient, and is not attributable to the Soret effect.
Mass transfer: Fick’s law of diffusion
For a binary system, Fick’s law states that the molar diffusion flux of each species,
relative to the molar-average velocity v ∗ = xA vA + xB vB , is negatively proportional
to the gradient of its molar fraction:
JA∗ = cA (vA − v ∗ ) = −cDAB ∇ xA
(A.4.1)
In a binary mixture, species A’s mass diffusion flux (relative to the mass-average
velocity) can be shown from (A.4.1) to be negatively proportional to its mass fraction
gradient [16, 16.2-1]:
jA = ρA (vA − v ) = −ρDAB ∇ωA
(A.4.2)
Equation (A.4.2) is the form of Fick’s law that is used throughout this work. Some
key steps in a proof that (A.4.2) follows from (A.4.1) are as follows:
nA = MA cA v ∗ − cDAB ∇ xA
(A.4.3)
h
ρ i
ρ =ωA ρv ∗ − cDAB ∇
− ρDAB ∇ A
(A.4.4)
c
ρ
h
i
=ωA ρv ∗ − ( MA − MB )cDAB ∇ xA − ρDAB ∇ωA
(A.4.5)
h
i
=ωA ρA vA + ρB vB − ρDAB ∇ωA
(A.4.6)
ρ ρ
ρ
Equation (A.4.4) can be obtained by using ∇ MA xA = ∇ cA = ∇ ρA c , and
ρ
(A.4.5) can be obtained by noting that ∇ c = ∇ xA MA + xB MB = MA − MB ∇ xA .
Finally, equation (A.4.6) can be obtained by substituting cA vA − cA v ∗ in place of
−cDAB ∇ xA in (A.4.5) and rearranging. Equation (A.4.2) follows as jA = nA − ρA v.
Momentum transfer: stress σ in a Newtonian fluid
The total stress tensor σ is expressed as a sum of the isotropic thermodynamic pressure, whose magnitude is given, assuming local thermodynamic equilibrium, by
Background theory
p=−
∂u
∂v
s,ωi
229
(see section A.5.1 below), and a viscous part τ :
σ = −p δ + τ
(A.4.7)
On contraction with δ, (A.4.7) gives [12, p.93]:
1
p = − tr (σ − τ )
3
(A.4.8)
In (A.4.7), p is considered positive for compression. The following table summarizes
how the definitions for the stress tensors that are used in this work relate to those of
several key references:
[12, 141]
[16, 17, 158, 56]
[145, 146]
σ
−P
T
τ
−P v
S
Table A.1: Summary of stress tensor notations.
In section (A.3) it was shown that σ, and hence τ , must be symmetric: expressed in
terms of its Cartesian components, τij = τji . It is generally assumed that the viscous
stress tensor τ is a function of the gradient of the velocity, ∇v. However, ∇v can be
decomposed into a symmetric part e and an antisymmetric part Ω:
∇v = e + Ω
(A.4.9)
It can be shown that the antisymmetric tensor Ω corresponds to a rigid rotation. According to the postulate of material indifference, the stress response of a material cannot
depend on rigid body motion [136, p.191]. Thus the stress must be a function of the
symmetric strain rate tensor e, whose components are
!
∂v j
1 ∂vi
eij =
+
2 ∂x j
∂xi
(A.4.10)
For sufficiently slow spatial velocity variation, this relationship may be approximated as linear [12, p.93]. For a statistically isotropic fluid such as air, the relationship
must be of the form
τij = Cijkl ekl
(A.4.11)
where Cijkl is the most general fourth-rank isotropic tensor [136, p.111]:
Cijkl = a1 δij δkl + a2 δik δjl + a3 δil δjk
(A.4.12)
This approximation of fluid behaviour is called the Newtonian viscous fluid model. By
the symmetry of ekl , we may disregard the antisymmetric part of Cijkl in k and l, since
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230
s e + C a e = C s e , on decomposing C into parts symmetric and
Cijkl ekl = Cijkl
kl
ijkl kl
ijkl kl
antisymmetric in k and l (C s and C a respectively). Thus
0 = Cijkl − Cijlk = ( a2 − a3 ) δik δjl + ( a2 − a3 ) δil δjk
(A.4.13)
Considering the components Cijlk that satisfy i = j = k = l leads to the equation:
a2 = a3
(A.4.14)
Cijkl = a1 δij δkl + a2 δik δjl + δil δjk
(A.4.15)
Thus we may write [12, p.94]
and, on substituting (A.4.15) into (A.4.11):
i
1h
τij = a1 δij δkl + a2 δik δjl + δil δjk
2
∂v
= a1 k δij + 2a2 eij
∂xk
∂vk
∂v
+ l
∂xl
∂xk
= a1 ekk δij + 2a2 eij
(A.4.16)
It follows from (A.4.8) and (A.4.16) that
1
2
p = − σii + a1 + a2 ekk
3
3
(A.4.17)
The quantity a1 + 23 a2 is called the fluid’s coefficient of bulk viscosity, represented by
the symbol ζ, and the quantity a2 is called the fluid’s coefficient of shear viscosity,
represented by the symbol µ [145, p.49].
In tensor notation, (A.4.16) can be written
2
τ = ζ − µ (∇ · v ) δ + 2µe
3
=ζ (∇ · v ) δ + 2µe0
(A.4.18)
(A.4.19)
and, using (A.4.7):
σ = − pδ + ζ (∇ · v ) δ + 2µe0
(A.4.20)
where e0 is the deviator (traceless tensor) formed from the symmetric rate of deformation tensor, with the definition
1
e = e − (∇ · v ) δ
3
0
(A.4.21)
The Navier-Stokes equations result when (A.4.20) is substituted into the equation of
motion for a fluid element (A.3.28).
It is conventional to disregard the second term on the right-hand side of (A.4.20),
on the basis that ∇ · v = 0 for incompressible fluids, and ζ is negligible for dilute
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231
gases [158]. The total stress tensor and viscous stress tensor thus take the final form:
σ = − pδ + 2µe0
τ =2µe0
(A.4.22)
(A.4.23)
These equations evidently give rise to symmetric σ and τ . Since tr e = ∇ · v, it
follows from (A.4.8) that for incompressible flow, p = −tr σ.
Heat transfer: Fourier’s law of heat conduction
According to Fourier’s law of heat conduction, the pure heat flux is (negatively) proportional to the temperature gradient [16, 18.4-2]:
q c = −k∇ T
(A.4.24)
where k is the local instantaneous thermal conductivity of the mixture, assumed here
to be scalar. Radiative heat transfer is not considered in this thesis, on the basis that
there are no high-temperature objects in a heat pump dryer [160].
A.5.
Second law analysis
The total exergy of a system is defined as the maximum work that can be obtained
during a process in which the system is brought into physical and chemical equilibrium with its environment. Thus a system’s exergy is a function both of the state of
the system and of the state of its environment [164]. The second law analysis conducted in chapter 4 of this thesis is an audit of the rates of exergy destruction in the
various system components. The present section summarizes the theory that underlies the second law analysis used in this thesis. The presentation includes a brief
summary of the thermodynamics required to derive the exergy balance equations for
nonequilibrium thermodynamics, which were obtained by Sun and Carrington [158].
Because of their generality, these equations are a convenient starting point for the
exergy analyses conducted in this thesis. The equations of nonequilibrium thermodynamics encompass the physics of heat transfer, mass transfer and hydrodynamics
[56, p.42], and so the general exergy balance equation (A.5.28), below, can be applied
in integral form to each of the phases present in the system under consideration.
Condition for membrane equilibrium
The present section contains a brief derivation of the equilibrium condition that is required to develop the concept of exergy. The development starts from the second law
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232
of thermodynamics, in the form of the statement that the entropy S ≡ S0 +
R
dQrev
T
of
a system is a state function (where S0 , the entropy at some reference state, is a matter
of convention), such that δS ≥ 0 during a transformation of any isolated system, with
equality holding for reversible transformations [58].
For a multicomponent system containing k chemical species at equilibrium, it is
postulated [58, p.77] that we can express the internal energy U of the system in terms
of S, V and mi , e.g. U = U (S, V, m1 , . . . , mk ). It follows that dU = ∂U
∂S V,m dS +
i
k ∂U
∂U
dmi . For closed systems we have that dU = TdS −
∂V S,m dV + ∑
∂mi S,V,m
i
j 6 =i
i =1
pdV from energy conservation in a reversible differential change, yielding ∂U
∂S V,m =
i
∂U
T and ∂U
=
−
p.
We
also
use
the
definition
µ
≡
where
µ
i
i is
∂V
∂m
i
S,mi
S,V,m j6=i
called the chemical potential of species i in the mixture [58, p.78]. Thus the differential change in U can be written:
k
dU = TdS − pdV + ∑ µi dmi
(A.5.1)
i =1
The Gibbs function G is defined G ≡ H − TS ≡ U + pV − TS, and by (A.5.1) must
satisfy dG = VdP − SdT for a closed system. From a line of reasoning analogous
to that leading to equation (A.5.1), the differential change in G, when expressed as
G = G ( T, p, mi ), is:
k
dG = −SdT + Vdp + ∑
i =1
∂G
∂mi
dmi
(A.5.2)
T,p,m j6=i
Subtracting (A.5.1) from (A.5.2) yields, since dmi is arbitrary, µi =
∂G
∂mi
T,p,m j6=i
. Thus
k
dG = −SdT + Vdp + ∑ µi dmi
(A.5.3)
i =1
Comparing (A.5.3) with the differential change in G for a closed system, we have
k
∑ µi dmi = 0
(A.5.4)
i =1
From this result (and the conservation of mass), it follows that at equilibrium, the
chemical potential µ is constant across phase boundaries and across semi-permeable
membranes. The following expression for the Gibbs function of a phase can be found
by integrating (A.5.3) at constant pressure and temperature [58, p.93]:
k
G=
∑ µi mi
(A.5.5)
i =1
From (A.5.5) it can be seen that the Gibbs free energy per unit mass of a singlecomponent phase is identical to the chemical potential of its only component.
Background theory
233
Exergy balance
This section presents a derivation of the maximum work obtainable from a steady
process that has inlet stream (h1 , ṁ1 , ωi 1 ) and outlet stream (h2 , ṁ2 , ωi 2 ). To do so
we consider the work Ẇout obtained from a completely reversible process in which
the ith -species mass flow shortfall ṁ2 ωi 2 − ṁ1 ωi 1 is made up by extraction from the
environmental mixture. Since we are considering a completely reversible process,
we know [81, p.101] that we must have Ẇout = Ẇout, max . Figure A.2 depicts the
conceptual flow situation that we consider. The diagram consolidates and streamlines
the proofs presented by Haywood [78, 79, 80, 81]. The diagram depicts a flow system
situated in an environment containing a mixture of z chemical species, at temperature
T0 , pressure p0 . A steady flow of matter into the system (bottom left) entering at
pressure p1 carries an energy flow due to specific enthalpy h1 into the system. The
flow out of the system (bottom right) is at pressure p2 , specific enthalpy h2 . Between
these locations the flow system evolves work and exchanges heat and mass with its
environment. The exchange of mass involves a subset numbering k < z of the species
that occur in the environmental mixture. The work rate Ẇout represents the maximum
net power obtainable from the system.
The exchange of mass is depicted for typical species i by way of two regions
bounded by semi-permeable membranes (impervious to all species except species
i). In the first region, species i is in equilibrium through the first membrane with the
environmental mixture. Within control volume A the dead-state mixture is reversibly
combined with the inlet flow at state 1, and reversibly transformed into the outlet
stream at state 2, requiring provision of heat at temperature T. The flow system in
control volume A receives heat at rate Q̇ + Q̇mix and produces work at a total rate
k
Ẇout + ∑ Ẇi + ẆP . The energy balance for control volume A can be written
i =1
k
k
i =1
i =1
0
Ẇout + ∑ Ẇi + ẆP = h1 ṁ1 − h2 ṁ2 + ∑ hi0 ṁi + Q̇ + Q̇mix
(A.5.6)
Now, for an ideal Carnot heat pump,
ẆP = Q̇ − T0
Q̇
T
(A.5.7)
where an entropy balance for (reversible) control volume A yields
k
0
Q̇
Q̇
= − mix + ṁ2 s2 − ṁ1 s1 − ∑ ṁi si0
T
T0
i =1
(A.5.8)
Equation (A.5.8) can be seen to hold by considering the equation Sgen = 0 which
applies to the reversible closed system defined as consisting of the matter enclosed in
the volume at time t, together with the small amount of matter that is about to enter
the system in the subsequent interval to t + ∆t, as shown by [15, p.23 & p.66].
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234
Using (A.5.8) in (A.5.7):
k
ẆP = Q̇ + Q̇mix − T0
0
ṁ2 s2 − ṁ1 s1 − ∑ ṁi si0
!
(A.5.9)
i =1
Combining (A.5.9) and (A.5.6) yields:
0
k
k
0
Ẇout + ∑ Ẇi = ṁ1 (h1 − T0 s1 ) − ṁ2 (h2 − T0 s2 ) + ∑ ṁi hi0 − T0 si0
(A.5.10)
i =1
i =1
The energy and entropy balances over the ith -species extraction module give:
0
0
0
0
ṁi hi0 =ṁi hi00 + Ẇi + Q̇i
ṁi si0 =ṁi si00 +
(A.5.11)
Q̇i
T0
(A.5.12)
k
⇒
0
0
k
k
0
0
ṁ
h
−
T
s
=
ṁ
h
−
T
s
+
∑ i i0 0 i0 ∑ i i00 0 i00 ∑ Ẇi
i =1
i =1
(A.5.13)
i =1
Using (A.5.13) in (A.5.10) yields
0
k
0
Ẇout = ṁ1 (h1 − T0 s1 ) − ṁ2 (h2 − T0 s2 ) + ∑ ṁi hi00 − T0 si00
(A.5.14)
i =1
0
0
0
Now by definition, hi00 − T0 si00 = gi00 , the specific Gibbs function for the ‘mixture’
containing pure species i in membrane equilibrium with the environmental mixture.
0
From (A.5.5), this specific Gibbs function gi00 is identical to µi00 , the chemical potential of pure species i in equilibrium with the environmental mixture. Also we have
ṁi = ωi, 2 ṁ2 − ωi, 1 ṁ1 . Equation (A.5.14) can therefore be rewritten:
h
i
i
h
k
k
Ẇout = ṁ1 h1 − T0 s1 − ∑ ωi,1 µi00 − ṁ2 h2 − T0 s2 − ∑ ωi,2 µi00
i =1
(A.5.15)
i =1
h
k
i
That is, Ẇout = −∆ ṁ h − T0 s − ∑ ωi µi00 = −∆ [ṁbs ], where
i =1
k
bs = h − T0 s − ∑ ωi µi00
(A.5.16)
i =1
is called the specific steady-flow exergy function. Haywood [81, p.196] shows that
bs ≥ 0 for all T, on the condition that the mixture satisfies p = p0 .
Equation (A.5.15) can also be used to deduce the maximum energy obtainable from
a non-flow system, by considering the flow system depicted in Figure A.3. Here
ṁ2 = 0, and the environmental pressure p0 does work on the system as its volume
decreases to zero. We have Ẇout = bs 1 (−ṁ) + ( p0 − p)(−V̇ ), and, integrating
Wout = bs 1 m + ( p0 − p)V
(A.5.17)
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235
Environment at T0 , p0 , ωi . z chemical species.
ṁi
T0 ,
0
Q̇i 0
pi 00 ,
0
hi 00 .
Q̇0
Q̇mix
T0 ,
0
pi 0 ,
0
hi 0 .
Ẇi
ṁi
Ẇp
Ẇout
Q̇
T 0 , p0 .
k species.
T
ṁ2 − ṁ1
h2
h1
p1
ωi 1
ṁ1
control volume A.
ṁ2
p2
ωi 2
Figure A.2: Diagram for steady-flow exergy.
where m and V are the initial mass and volume of the system. Thus
Wout
= bs 1 + ( p 0 − p ) v = bn
m
(A.5.18)
where bn is called the non-flow exergy function, satisfying
k
bn = u + p0 v − T0 s − ∑ ωi µi00
(A.5.19)
i =1
In the general case in which there may be significant changes in the kinetic and potential energy of a system or stream during a process, the total specific exergy contains contributions from these terms, as arises when the h terms are replaced by
h0 = h + 12 v2 + ψ in (A.5.6), yielding:
1
bTn =u + p0 v − T0 s − ∑ µk,0 ωk + v2 + ψ
2
1 2
bTs =h − T0 s − ∑ µk,0 ωk + v + ψ
2
(A.5.20)
(A.5.21)
For a stationary mixture at T0 , p0 and ψ = 0, with the same composition ωi as the
environmental mixture, the exergy functions satisfy bTs = 0 and bTn = 0.
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236
Environment at T0 , p0 , ωi . z chemical species.
−ṁi
Q̇
Ẇout
p0
p, V ,
m
bs 1
−ṁ
Figure A.3: Diagram for non-flow exergy.
Local equilibrium and exergy
The equations of classical thermodynamics cannot be applied across nonequilibrium
systems, whose intensive parameters vary with location. However in such systems
it is often possible to apply the results of classical thermodynamics to small fluid
elements that locally are in equilibrium. A general application of this approach is in
the equations of non-equilibrium thermodynamics [56], in which the Gibbs equation
(A.5.1) is assumed to hold for the fictitious system particles, moving at the barycentric
velocity, that were referred to in section A.3.1. The present section summarizes the
results of a paper by Sun and Carrington [158], in which exergy balance equations are
developed for a nonequilibrium fluid mixture using the equations of nonequilibrium
thermodynamics [56]. Since the Gibbs equation (A.5.1) holds for the system particles,
it follows that [56, p.23]; [158, p.36]:
ρT ṡ = ρu̇ −
p
ρ̇ − ∑ µk ρω̇k
ρ
k
(A.5.22)
De Groot and Mazur [56] have shown how (A.5.22) can be re-expressed in the form of
a local entropy balance equation. Regarding ρk as a property of the particles moving
at the barycentric velocity, Equation (A.2.15) can be applied to obtain
ρ̇k − v · ∇ρk =
∂ρk
∂t
(A.5.23)
Using (A.5.23) in the species-k mass balance (A.3.3) and rearranging yields:
ρ̇k + ρk ∇ · v + ∇ · jk = Φk =
R
∑ νkj Jj
(A.5.24)
j =1
Using (A.5.24), (A.3.16), (A.4.7) and (A.3.53) in (A.5.22), one can obtain the local entropy balance [56, 158]:
∂
(ρs) + ∇ · (ρsv + js ) = σ
∂t
(A.5.25)
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237
where the entropy flux density js is defined
1
jq − ∑ µ k jk
js =
T
(A.5.26)
and where the local volumetric entropy production σ is defined
µ 1
fk
∇v
µ
σ = jq · ∇
+ ∑ jk ·
+τ :
−∇ k
+ ∑ Jj − ∑ k νkj
(A.5.27)
T
T
T
T
T
In (A.5.26) and (A.5.27), in the absence of radiative heat transfer, the heat flux jq
is given by (A.3.38). The term σ defined in (A.5.27) is seen to consist of a sum of
products of thermodynamic “flows” and “forces” [56, 158].
Sun and Carrington [158] have used the substantial-form version of (A.5.25) to establish the general exergy balance for a differential fluid element, that is, for each
of the multicomponent system particles described in section A.3. An alternative approach which leads to identical results is to determine the local time derivative of the
specific nonflow exergy. The equation that results when (A.5.25) and (A.3.43) – after
conversion to local form using (A.3.19) – are used to evaluate the local time derivative
of the volumetric nonflow exergy obtained from (A.5.20) is as follows [158]:
∂
(ρbTn ) + ∇ · (ρbTs v + jbs ) = − T0 σ
∂t
(A.5.28)
In (A.5.28) σ is again the local volumetric entropy generation rate given by (A.5.27),
and the exergy flux density jbs is given as [158]:
T0
jbs = −τ · v + 1 −
q c + ∑ bsk jk + ∑ ψk jk
T
(A.5.29)
In (A.5.29), the specific exergy bsk of species k is
bsk =hk − T0 sk − µk,0
∂Bs
=
∂mk T,p,m j6=k
(A.5.30)
(A.5.31)
where Bs = mbs is the total flow exergy that would be associated with a mass m of
the mixture at the local equilibrium state.
As a balance equation for local exergy containing the negative source term − T0 σ,
Equation (A.5.28) is an example of the Gouy-Stodola law. This is the name for the
general result that the opportunity to obtain useful work is destroyed at a rate equal
to T0 σ, the environmental temperature times the rate of entropy generation [14].
A.6.
Turbulence and time-averaging
In order to cope with the turbulence that is encountered in typical engineering flow
systems, a time-average of the control volume balances can be formed by integrating each balance equation over a finite duration ∆t (which is long compared with a
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238
characteristic time of the fluctuations, but short compared with a characteristic time
of the process under consideration) and then dividing by ∆t. The time average of any
quantity B is defined using an equation such as [187, 146]:
B=
1
∆t
Z t+ 1 ∆t
2
t− 12 ∆t
B(t0 )dt0
(A.6.1)
For any quantity Ψ, provided the appropriate timescale ∆t exists, the single-phase
control volume time averages satisfy the following relationships [146, p.221]:
d
dt
Z
ΨdV =
R
Z
ΨdV =
R
Z
S
d
dt
Z
Z
ΨdV
ΨdV
(A.6.2b)
ΨdS
(A.6.2c)
R
ΨdS =
Z
S
w · n̂ =w · n̂
A.7.
(A.6.2a)
R
(A.6.2d)
Boundary-layer theory
Drying involves external moisture transfer to a moving airstream, and in order to
model the drying process, a brief foray into boundary-layer theory is required. In
this section the transfer coefficients are defined, and the Chilton-Colburn analogy,
which links the mass, momentum and heat transfer coefficients, is presented. To
relate this empirical result to theory, the boundary-layer equations for laminar flow
past a flat plate are presented, following the developments of [85] and [141] – though
employing a more general definition, Equation (A.7.3) below, of the mass transfer
coefficient than [85] as discussed by Bird et al. [16, p.639]. This section highlights that
the derivation of the Reynolds analogy, of which the Chilton-Colburn analogy is an
extension, requires the assumptions (1) that the enthalpy flux associated with mass
diffusion can be neglected, and (2) that the flow is incompressible.
As is discussed below, it has been convenient in this work to define the mass transfer coefficient in terms of the species-A mass diffusion flux jA . By applying the assumption of constant boundary-layer density to the film-theory approach described
by Bird et al. [16], correction factors for high mass transfer rates are obtained (below)
in terms of the mass-average velocities, rather than their molar counterparts. Strictly
speaking, the assumption that this approach requires, of constant density within the
boundary layer, is incommensurable with the assumption of constant molar concentration employed by Bird et al. [16, p.660]. Since according to Equation (A.7.16e)
below, the pressure is constant in the transverse direction within the boundary layer,
Background theory
239
the total concentration would be expected to be constant for an ideal gas mixture in
the isothermal case. In contrast, in a binary mixture containing species of differing
molar masses, the density would vary with the molar fractions of the species. As
the boundary layer is non-isothermal, and is of varying molar composition, neither
the assumption of constant concentration nor the assumption of constant density is
entirely justified. However the error introduced by either assumption should be low
for the concentration and density variations across the boundary layer (less than x%
and y%, respectively) that turn out to be typical for the system being modeled. And
in any case the corrections for high mass transfer turn out to be less than 10%.
Transfer coefficients
When a nonideal fluid flows over a stationary surface, the zero-slip condition requires
that particles coinciding with the surface become stationary. These particles slow the
neighbours shearing past them in the layer above, which slow their upstairs neighbours in turn. The distance from the surface at which this effect becomes negligible
is called the velocity boundary-layer thickness δ, sometimes defined as the thickness at
which the velocity equals 99% of the free-stream velocity. This definition is necessarily arbitrary because variation within the boundary layer is continuous. The stress at
the surface is used to define the dimensionless parameter C f called the friction coefficient, defined as the ratio of surface stress to free-stream kinetic energy density [85]:
Cf ≡
|τs · n̂s |
1
2
2 ρu∞
(A.7.1)
A thermal boundary layer also develops when a fluid flows over a surface whose
temperature differs from the free-stream temperature. The thermal boundary-layer
thickness δt is defined as the distance from the surface at which the deviation from
the free stream temperature falls below 1% of the difference between plate and free
stream temperatures. The heat transfer at the surface is used to define the convection
heat transfer coefficient h as follows [85]:
h≡
qsc · n̂s
Ts − T∞
(A.7.2)
A concentration boundary layer, analogous to the velocity and thermal boundary
layers, also develops when mass exchange occurs across the surface. The concentration
boundary-layer thickness δc for species A in a binary mixture is the distance at which
the deviation from the free stream concentration of A falls to 1% of the difference
between surface and free stream concentrations of A. The diffusion flux of species A
at the surface can be used to define the mass-transfer coefficient hm as follows [16, 85]:
hm ≡
jA,s · n̂s
ρA,s − ρA,∞
(A.7.3)
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240
Here the approach of [16] is adopted, in defining the mass-transfer coefficient in terms
of the diffusion flux rather than the total flux as elsewhere. This allows the film theory corrections for high mass transfer rates (described below) to be employed. However for convenience the mass-transfer coefficient is expressed here in terms of the
mass diffusion flux jA = ρA (vA − v ), rather than in terms of the molar diffusion flux
JA∗ = NA − xA N as in the reference [16, p.639]. The mass transfer coefficient de-
fined by [16] has nonstandard units (molar flux rather than velocity). Furthermore,
mass transfer is expressed throughout this thesis (as a matter of convenience) in terms
of the mass-average flux relative to the mass-average velocity. The use of definition
(A.7.3) has implications for the precise form taken by high mass-transfer corrections
which are introduced later. A derivation of the film-theory corrections that apply
to the transfer coefficients defined by equations (A.7.1)–(A.7.3) is presented in section A.7.5 below. The development runs parallel to that of Bird et al. [16, p.656-662],
and leads to results that can be shown to be equivalent (given the boundary-layer assumptions summarized below) to those presented in that reference. The rationale for
identifying the surface mass flux with a multiple of ρA,s − ρA,∞ lies in the assumptions
(1) that the gradient ∇ρA |s at the surface of the density of species A is approximately
proportional to ρA,s − ρA,∞ , and (2) that total density ρ is approximately constant
within the boundary layer, so that (A.7.3) follows from Fick’s law (A.4.2).
The transfer coefficients defined in (A.7.1)-(A.7.3), and the corresponding boundarylayer thicknesses, are defined locally and are generally position-dependent.
Boundary-layer equations
In this section the boundary-layer equations for laminar flow over a flat plate are
obtained from the governing equations derived in the previous sections. Low masstransfer rates are assumed throughout this section; a subsequent section deals with
film-theory adjustments that account for the boundary-layer distortion that is caused
by high mass-transfer rates [16]. In the flat plate geometry, the transfer coefficients
defined in expressions (A.7.1)–(A.7.3) may be evaluated as follows:
µ ∂v Cf = 1
2 ∂y y =0
2 ρu∞
−k f ∂T h=
Ts − T∞ ∂y y=0
−ρD AB ∂ωA hm =
ρA,s − ρA,∞ ∂y y=0
(A.7.4)
(A.7.5)
(A.7.6)
The conservation equations in differential form (A.3.3), (A.3.15), (A.3.28), (A.3.49),
together with the phenomenological relations (A.4.22), (A.4.2), (A.4.24), allow the behaviour of fluid flowing past a flat plate to be described in detail. In order to proceed,
Background theory
241
a number of simplifying assumptions are made [85, p.278], [141, p.107]:
1. The situation is essentially two-dimensional.
2. The situation is steady (time-independent).
3. Moist air behaves as an ideal mixture of constant-heat-capacity components.
4. Body forces are negligible.
5. Chemical reactions are disregarded.
6. The enthalpy flux associated with mass diffusion is negligible – referring to
(A.3.38), this assumption may be expressed by the relation jq ≈ q c .
7. Density ρ is independent of position within the boundary layer.
8. Fluid properties (DAB , µ, k) are constant throughout the layer.
Assumptions 1 and 2 may be expressed as follows:
∂(•)
∂(•)
=
= 0,
∂z
∂t
w=0
(A.7.7)
With assumptions 2 and 7, the equation of continuity (A.3.15) takes the form:
∂u ∂v
+
=0
∂x
∂y
(A.7.8)
Together with assumptions 7 and 8, substituting Fick’s law (A.4.2) and the equation
of continuity (A.7.8) into the differential mass balance (A.3.3) for species A gives:
2
∂ρA
∂ρA
∂ ρA
∂2 ρA
u
+v
= DAB
+
(A.7.9)
∂x
∂y
∂x2
∂y2
The viscous stress tensor may be evaluated using (A.4.22), (A.7.7) and (A.7.8) as:
∂u
∂x
∂v
=2µ
∂y
τ11 =2µ
τ22
(A.7.10)
τ12 =τ21 = µ
∂u
∂v
+
∂y
∂x
Together with assumption 2, (A.7.10) and (A.7.8) may be substituted into the momentum balance equation (A.3.28) to obtain the boundary-layer momentum balance:
2
∂u
∂u
∂ u ∂2 u
∂p
ρ u
=µ
+
−
+v
(A.7.11)
∂x
∂y
∂x
∂x2
∂y2
2
∂v
∂v
∂ v
∂2 v
∂p
ρ u
+v
=µ
+ 2 −
(A.7.12)
∂x
∂y
∂y
∂x2
∂y
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242
The differential energy balance (A.3.57) can be expressed in material form as [16]:
ρ
dm h
dm p
= −∇ · q c +
dt
dt
(A.7.13)
Using equations (A.2.15), (A.3.38), (A.4.2) and assumptions 1, 2, 3, 6 and 8, Equation
(A.7.13) may be evaluated as:
2
∂2 T
∂T
∂p
∂T
∂ T
∂p
+
ρc p u
+
u
+v
=k
+
v
∂x
∂y
∂x
∂y
∂x2
∂y2
(A.7.14)
By utilising the so-called boundary-layer approximations, these expressions may be
further simplified. An order-of-magnitude analysis can be applied [141, p.107], using
the fact that the boundary-layer thicknesses are typically very small compared with
a characteristic length of the system. The following results follow:
u v
(A.7.15a)
∂T
∂T
∂y
∂x
∂u
∂u ∂v ∂v
, ,
∂y
∂x ∂y ∂x
∂ρv
∂ρv
∂y
∂x
(A.7.15b)
(A.7.15c)
(A.7.15d)
Using the boundary-layer approximations to simplify the boundary-layer equations leads to the following system [85]:
∂u ∂v
+
=0
∂x
∂y
∂ρ
∂ρ
∂2 ρA
u A + v A = DAB
∂x
∂y
∂y2
∂T
∂T
k ∂2 T
u
+v
=
∂x
∂y
ρc p ∂y2
∂u
∂u
µ ∂2 u 1 ∂p
+v
=
u
−
∂x
∂y
ρ ∂y2
ρ ∂x
∂p
=0
∂y




u
=
v
=
0
u = u∞






y = 0 : T = Ts
y=∞:
T = T∞








ρA = ρA,s
ρA = ρA,∞
(A.7.16a)
(A.7.16b)
(A.7.16c)
(A.7.16d)
(A.7.16e)
(A.7.16f)
In particular, the pressure in the outer flow is ‘impressed’ upon the boundary layer
(A.7.16e). Together with assumption 2, (A.7.16e) implies that p is a function only of
x, and the partial derivative
this fact.
∂p
∂x
in (A.7.16d) is sometimes written as
dp
dx
to emphasise
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243
The Reynolds analogy
The Reynolds analogy may be understood by examining a dimensionless form of the
equations (A.7.16a)-(A.7.16d). Normalising the variables in terms of a characteristic
length L, the free stream velocity u∞ , and the plate and free stream temperatures and
concentrations (Ts , T∞ , ρ A,s and ρ A,∞ ), the boundary-layer equations may be recast in
the following dimensionless form:
∂v∗
∂u∗
+ ∗
∗
∂x
∂y
∗
∗ ∗ ∂ρA
∗ ∂ρA
u
+v
∂x ∗
∂y∗
∗
∂T
∂T ∗
u∗ ∗ + v∗ ∗
∂x
∂y
∗
∂u∗
∂u
u∗ ∗ + v∗ ∗
∂x
∂y


u∗ = v∗ = 0



y∗ = 0 : T ∗ = 0



 ∗
ρA = 0
=0
(A.7.17a)
=
∗
1 ∂2 ρA
ReL Sc ∂y∗2
(A.7.17b)
=
1 ∂2 T ∗
ReL Pr ∂y∗2
(A.7.17c)
=
1 ∂2 u ∗
dp∗
− ∗
∗
2
ReL ∂y
dx
(A.7.17d)
y=∞:









u∗ = 1
T∗ = 1
(A.7.18)
∗ =1
ρA
The dimensionless quantites in (A.7.17a)-(A.7.17d) are defined as follows:
ρu∞ L
2 ∂u∗ ReL =
=
µ
C f ∂y∗ y∗ =0
µc
Pr = P
k
µ
Sc =
ρDAB
(A.7.19)
(A.7.20)
(A.7.21)
The Reynolds number ReL is a characteristic of the particular flow situation, while
(assuming constant fluid properties) the Prandtl (Pr) and Schmidt (Sc) numbers depend only on the fluid itself.
The Reynolds analogy follows from the observation that when Pr ≈ 1, Sc ≈ 1 and
dp∗
dx ∗
≈ 0, the dimensionless pressure, velocity and density profiles are all subject to
the same PDE and boundary conditions. They therefore must be the same function of
location. From (A.7.19) it follows that for laminar flat-plate flow:
Cf
ReL
= Nu = Sh
2
(A.7.22)
where the Nusselt and Sherwood numbers (Nu and Sh) are defined as follows:
∗ ∂ρA
hm L
Sh =
=
(A.7.23)
∂y∗ y∗ =0
DAB
∂T ∗ hL
Nu =
=
(A.7.24)
∗
∂y y∗ =0
k
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244
The Chilton-Colburn analogy
The Reynolds analogy may be extended to situations where the conditions Pr ≈ 1,
Sc ≈ 1 do not hold, by use of an empirical relationship known as the Chilton-Colburn
analogy, which is valid in the ranges 0.6 < Sc < 2500 and 0.6 < Pr < 100 [182]:
Cf
Nu
=
2
ReL
1
Pr
1/3
Sh
=
ReL
1
Sc
1/3
(A.7.25)
The second and third quantities in (A.7.25) are known as the ‘j-factors’:
Nu
jH =
ReL
Sh
ReL
jD =
1
Pr
1/3
1
Sc
1/3
(A.7.26)
(A.7.27)
These equations, which reduce to the Reynolds analogy when Pr = Sc = 1 (and
which can be shown to hold precisely for laminar flow past a flat plate), have been
validated for a wide range of flow conditions and geometries, including turbulent
flow situations not meeting the requirement that
dp∗
dx ∗
≈ 0 [16, 17].
In the absence of experimental data, the full Chilton-Colburn analogy is assumed
to hold for all flow situations encountered throughout this work. The recommendation of [16] is to evaluate the j-factors using parameters evaluated at the local
“film conditions” given by the average of the surface and free-stream conditions, e.g.
T f = 12 ( Ts + T∞ ). The friction part of the analogy expressed by (A.7.25) is known to
be rough: in situations where form drag or surface roughness plays a significant role,
the friction factor can significantly exceed the other transfer coefficients, and only the
limited analogy jH = jD may hold [16, 17]. This caveat is not considered exceptional,
because the more general analogy characterises those transfer coefficients – mass, and
heat – which most strongly influence drying rates in a heat pump dryer.
Turbulent flow
The boundary-layer equations (A.7.16a)-(A.7.16d), describing laminar flow past a flat
plate, were solved by Blasius (for the case of constant fluid properties) by using a
similarity variable to reduce the system to a single ODE [141, 146]. The resulting
expressions are well matched by experimental data for the laminar region [17].
For sufficiently large Reynolds numbers (e.g. Rex = ρux/µ), amplification of flow
instabilities causes a transition to turbulent flow in the boundary layer. The transition location depends on the turbulent intensity of the upstream flow, as well as
on properties of the particular situation, including geometry, pressure gradient and
surface roughness [85, 169]. Turbulent flow is characterised by random fluctuations
Background theory
245
in the flow properties, and is inherently difficult to describe analytically, although
time-averaging techniques may be used to develop governing equations which resemble equations (A.7.16a)-(A.7.16d) but include additional correlation cross-terms
[141]. These terms account for the additional convective transport that takes place as
a result of the turbulent fluctuations. Turbulent mixing has the effect of extending
free-stream conditions toward the surface, decreasing effective film thicknesses and
enhancing transfer rates. Inside the turbulent boundary layer, near to the stationary surface, there remains a thin region within which flow is approximately laminar.
Prandtl assumed the existence of such a laminar sublayer, and showed that for a fluid
whose Prandtl number is approximately 1 (e.g. air, for which Pr ≈ 0.7), the Reynold’s
analogy holds also in the case of turbulent boundary-layer flow [141, p.495].
For turbulent flow over a flat plate, data characterising heat transfer is readily
available. Mass transfer coefficients obtained from heat transfer coefficients using
the Chilton-Colburn analogy have been found to approximate measured values reasonably well, although there is a fair amount of scatter in the mass-transfer data [169,
p.75]. A generic transition Reynolds number of 5 × 105 has been recommended for
flat plate flow [85], together with the relations:
Transfer for flow parallel to flat plates [85]
Rex < 5 × 105
5 × 105 < Rex < 107
jH,x = 0.332Rex−1/2
δ = 5xRex−1/5
jH,x = 0.0296Rex−1/5
δ = 0.37xRex−1/2
The situation being modeled in this work is one of internal flow within a rectangular duct (Figure A.4). In duct flow, the growing boundary layers eventually merge
over the entire flow cross-section, so that flow everywhere throughout the duct is affected by the walls. The distance from entrance to this point is called the hydrodynamic
entry length. In the absence of heat and mass transfer within the duct, flow conditions
beyond the entry region become independent of position. The resulting flow may
be turbulent or laminar: this is determined by a Reynolds number Re for the flow
situation. In [135], Kays and Perkins present tabulated results relating to flow in a
rectangular duct. For flow in a rectangular duct, the Reynolds number Re is defined
using the hydraulic diameter Dh :
Dh = 4Ac /P
(A.7.28)
Re = ρuDh /µ
(A.7.29)
where Ac and P are the cross-sectional area and wetted perimeter of the flow, respectively. The critical Re for turbulence depends on the aspect ratio, or relative linear dimensions of the rectangular cross-section (as well as on the particular investigation),
but is in the region 2, 000 − 6, 000, with typical values near 2, 800 [135, p.7-116]. For
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246
turbulent flow in ducts with aspect ratios ranging from 1 : 1 to 40 : 1, various investigators have found that the Dittus-Boelter equation, originally developed to describe
flow in a circular tube, holds satisfactorally:
Nu = 0.023Re4/5 Pr2/5
(A.7.30)
In (A.7.30), Nu is defined by equation (A.7.24). Typical entry lengths range from 20
diameters (for Re > 4, 000) to about 40. As an order-of-magnitude estimate of the
quantities involved, consider the following table containing some properties of dry
air near standard temperature and pressure (STP):
Approximate properties of air [182]
P (Pa)
Dva (m2 /s)
ρ (kg/m3 )
1.0133 × 105
2.5994 × 10−5
1.225
µ (Pa · s)
1.789 × 10−5
k (W/m · K)
2.4671 × 10−2
c p (J/kg · K)
1.0057 × 103
These constant values are not used in the model (where the correlations described in
section A.8.1 are instead used), but are stated here for reference and illustration.
Figure A.4: Development of Flow Within a Duct
The moist-air plug-flow model which is used in this work assumes that flow is
fully mixed laterally, with no longitudinal mixing and no dispersion due to correlation of fluctuations [160, 169, 183]. Thus time-averaged physical properties are all
Background theory
247
assumed to be uniform across the entire flow cross-section. This approach is justified
partially by the fact that the viscous flow region can be estimated, using the methods
described in reference [177, p.57–59], to be relatively thin (≈ 0.5mm) compared with
the duct thickness. The effects of the thin laminar sub-layer adjacent to the bounding
surface of the duct are captured using heat, mass and momentum transfer correlations established for time-averaged turbulent flow.
In general, as a consequence of surface mass exchange, the normal component of
velocity (here represented as the velocity y-component, v) cannot be assumed zero
at the surface. The transfer coefficients consequently depend upon the mass transfer rates themselves. In a drying system, where the mass-transfer rates are high, this
dependence can significantly alter the transfer properties [85]. Additionally, section
(A.7) made use of the assumption that the enthalpy flux associated with mass diffusion could be neglected. This assumption breaks down in a drying system, where
heat transfer associated with diffusion fluxes is a significant component of the overall
energy equation. The film theory corrections [16] are therefore incorporated into the
model. A derivation of versions of the correction factors that are appropriate for the
transfer coefficients defined in equations (A.7.1)–(A.7.3) follows.
The assumptions listed at the start of section (A.7.2) are again applied, with the
exception of assumption 6. In order to solve the boundary-layer equations under
conditions of high mass-transfer, [16] make the additional assumption that all variables vary significantly only in the y-direction:
∂(•)
=0
∂x
(A.7.31)
Apart from the energy equation (A.7.16c), equations (A.7.16a)–(A.7.16e) again result,
but with the x-derivatives absent from all expressions. Because this development
tracks diffusion relative to the mass-average velocity, the equations diverge here from
those appearing in the reference [16], running in parallel. Denoting the diffusion
velocities with a superscript ‘r’, the divergence of j q for the binary mixture can be
obtained from the assumptions together with equations (A.3.38) and (A.4.24) as:
∇ · (j q ) = − k
d2 T
d
+
(ρ vr h + ρB vBr hB )
2
dy A A A
dy
(A.7.32)
By substituting (A.7.32) into (A.7.13) and making use of the various assumptions
listed above, the enthalpy balance may be evaluated as:
d
d2 T
( ρA vA hA + ρB vB hB ) − k 2 = 0
dy
dy
(A.7.33)
Background theory
248
The boundary-layer equations therefore further simplify as follows:
v
dv
=0
dy
⇒
d2 ρA
dρA
− DAB
=0
dy
dy2
⇒
v · ρA − DAB
du
d2 u
−µ 2 = 0
dy
dy
⇒
ρv · u − µ
ρv
ρv = ρA vA + ρB vB = const.
(A.7.34)
dρA
= ρA vA = const.
dy
(A.7.35)
du
= const.
dy
(A.7.36)
and
d
dy
∑ ρi vi hi
i
!
−k
d2 T
=0
dy2
⇒
dT
∑ ρi vi c p,i T − k dy
= const.
(A.7.37)
i
As stated earlier, the boundary-layer equations used in this development proceed
from the assumption that density ρ is a constant within the boundary layer, and this
assumption is implicit in (A.7.35). In contrast, constant concentration is assumed in
reference [16], in which the molar form of Fick’s law is used (A.4.1).
The integration constants in these ODEs can be expressed in terms of the (as yet
unknown) surface fluxes τs , qs and jA,s = ρA,s (vA,s − v). Equations (A.7.35)–(A.7.37)
then become:
d
(ρ − ρA,s )
dy A
d
ρv (u − us ) − µ
(u − us )
dy
d
∑ ρi vi c p,i (T − Ts ) − k dy (T − Ts )
i
v (ρA − ρA,s ) − DAB
=
jA,s
(A.7.38)
= τs
(A.7.39)
= qs
(A.7.40)
Equations (A.7.38)–(A.7.40) may be solved subject to the boundary conditions


u = us = 0
y = δ : u = u∞



y = 0 : ρA = ρA,s
(A.7.41)
y = δc : ρA = ρA,∞




T = Ts
y = δt : T = T∞
to obtain the following solutions:
τs
ρv
u−0 =
1 − exp
y
ρv
µ
jA,s
v
ρA − ρA,s =
1 − exp
y
v
DAB
∑ ρi vi c p,i
qs
T − Ts =
y
1 − exp
k
∑ ρi vi c p,i
(A.7.42)
(A.7.43)
(A.7.44)
The surface fluxes τs , jA,s and qs may be related to the boundary-layer thicknesses δ,
δc and δt by substituting the values in the second column of (A.7.41) into (A.7.42)–
(A.7.44). Introducing a superscript bullet ‘• ’ to label coefficients corrected for high
Background theory
249
mass-transfer rates, we may substitute these surface fluxes into the transfer coefficient
definitions (A.7.1)–(A.7.3), to express the corrected transfer coefficients in terms of the
boundary-layer thicknesses and surface mass fluxes:
C •f =
h•m =
1
2 u∞
n
exp
v
ρv
µ δv
v
exp
exp
v
DAB δc
−1
o
(A.7.46)
−1
∑ ρi vi c p,i
h• =
∑ ρi vi c p,i
δt
k
(A.7.45)
(A.7.47)
−1
Bird et al. [16] proceed by assuming that the film thicknesses are not affected
by mass-transfer rates. The thicknesses may then be evaluated from the measured
mass-transfer coefficients at low mass transfer rates by taking the limit of expressions
(A.7.45)–(A.7.47) as vA → 0 and vB → 0 (and thus v → 0):
1
δv = µ /
ρu∞ C f
2
(A.7.48)
δc = DAB / hm
(A.7.49)
δt = k / h
(A.7.50)
Finally dividing the corrected exchange coefficients in (A.7.45)–(A.7.47) by their
low mass-transfer limits, which are obtained by rearrangement of equations (A.7.48)–
(A.7.50), the ‘correction factors’ θv , θAB and θt can be obtained:
C •f = C f θv
(A.7.51)
h•m = hm θAB
(A.7.52)
h• = hθt
(A.7.53)
where
φ
−1
1
φv = ρv /
ρu∞ C f
2
θ=
(A.7.54)
eφ
(A.7.55)
φAB = v / hm
(A.7.56)
φt = ρA vA c p,A + ρB vB c p,B / h
(A.7.57)
Equations (A.7.55) and (A.7.57) are equivalent to (21.5-37) and (21.5-38) of [16, p.661]
by definition of the terms appearing in these expressions. When the assumption of
identical molecular masses holds (as would be implied if both total molar concentration and density were constant throughout the boundary layer), Equation (A.7.56)
becomes equivalent to (21.5-39) of [16]. For duct flow as encountered in this work,
for the values ‘at infinity’ of the variables u, ρ A and T, we substitute the bulk stream
conditions within the duct, obtained from the plug-flow model [16, p.665].
Background theory
A.8.
250
Properties of moist air and refrigerant R134a
In this section the correlations used to evaluate the properties of moist air are presented. The thermodynamic properties have been evaluated using the routines developed for the American Society of Heating, Refrigerating and Air-Conditioning Engineers (ASHRAE) by Wexler et al. [184], which were translated (with fellow PhD
student Sam Lowrey) from FORTRAN, and implemented in MATLAB. The physical properties have been evaluated using methods described by Reid and Sherwood
[133] and from several correlations described in the PhD thesis of Ashworth [4].
Physical properties of moist air
Semi-empirical approximation methods described by Reid and Sherwood [133] have
been used throughout this work to determine the viscosity, thermal conductivity and
binary diffusivity of the vapour-air mixture. The equations used are summarized as
follows:
The binary diffusion coefficient Dva of moist air is estimated using [4]:
Dva = 2.20757 × 10−5 + 1.32048 × 10−7 TC + 3.02105 × 10−10 TC2
(A.8.1)
The viscosities µi , and thermal conductivities k i , at p0 = 101325 Pa for the pure
components of moist air are estimated using [4, 175]:
µv =8.12095 × 10−6 + 3.89574 × 10−8 TC + 6.70819 × 10−12 TC2
µ a =1.71687 × 10−5 + 4.96315 × 10−8 TC − 3.85243 × 10−11 TC2
(A.8.2)
(A.8.3)
k v =1.74995 × 10−2 + 6.56938 × 10−5 TC + 6.37385 × 10−8 TC2
(A.8.4)
k a =2.41240 × 10
(A.8.5)
−2
+ 7.17753 × 10
−5
TC
These equations were originally obtained by Ashworth [4, p.51] by using a leastsquares method to fit third-order polynomials to data obtained from several sources.
Wilke’s equation for the viscosity of a binary gas mixture is [133, p.200]:
µmix =
µ1
µ2
+
1 + ( x2 /x1 )φ12
1 + ( x1 /x2 )φ21
1
(A.8.6)
1
[1 + (µi /µ j ) 2 ( M j /Mi ) 4 ]2
φij =
√
1
8(1 + Mi /M j ) 2
(A.8.7)
According to Reid, Wilke reported that Equation A.8.6 yielded a deviation from the
measured viscosity, averaged over 17 systems, of less than 1 per cent [133, p.200].
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251
Wassiljewa’s equation for the thermal conductivity of a binary mixture is similar
in form to Wilke’s equation (A.8.6) for the viscosity of a binary mixture [133, p.240]:
k1
k2
+
1 + ( x2 /x1 ) A12
1 + ( x1 /x2 ) A21
(
" # 12 )2
34
1 + Sij /T
µi M j
1 + Si /T
1
Aij =
1+
4
µ j Mi
1 + S j /T
1 + Si /T
km =
(A.8.8a)
(A.8.8b)
In (A.8.8) the Sutherland constant Si for species i is estimated as S = 1.5Tb where Tb
is the boiling temperature in degrees Kelvin. The values used are S1 = Sa = 119 K
for air [107] and S2 = Sv = 560 K for water vapour. The Sutherland constant for the
mixture, S12 = Sav , is estimated using S12 = 0.735(Sa Sv )0.5 ≈ 190 K since water is a
polar species [133, p.241]. Although these Sutherland constant estimates may not be
particularly accurate, (A.8.8) is quite insensitive to errors in S, and can be expected to
yield a value that is accurate to within about 2 per cent [133, p.241].
Thermodynamic properties of moist air
The equation of state of a pure real gas is written as follows:
pv
=Z
RT
(A.8.9)
As a result of the molecular theory of gases, the compressibility factor Z that appears in
(A.8.9) can be expressed as a power series of the pressure p, or more commonly, as a
power series of the molar volume v [82, p.131]:
Z = 1+
B
C
+ 2 +...
v
v
(A.8.10)
The virial coefficients B and C, etc., which are functions of temperature, can be regarded as corresponding to interactions respectively involving up to two molecules,
up to three molecules, and so on [184, p.156]. An ideal gas, whose molecules are
non-interacting, satisfies (A.8.9) together with the truncated expansion Z = 1.
The equation of state for mixtures [82, p.153] is given by Equation (A.8.10), where
the second and third virial coefficients for the mixture are given by:
Bm = ∑ ∑ Bij xi x j
(A.8.11)
Cm = ∑ ∑ ∑ Cijk xi x j xk
(A.8.12)
i
i
j
j
k
In (A.8.11) and (A.8.12), xi is the mole fraction of species i, and Bii and Ciii are the
second and third virial coefficients for pure species i. The mixture coefficients Bij
(i 6= j) and Cijk (where either i 6= j or i 6= k), which correspond to the interactions of
molecules of different species, are called the called cross-virial coefficients.
Background theory
252
From (A.8.10) it can be seen that at a large molar volume v∗ the equation of state of
a real gas approaches that of an ideal gas. This fact allows the thermodynamic state
of a real gas mixture to be calculated by considering a two-step path from a reference
state at temperature T0 and (very large) specific volume v0∗ . For instance, the molar
enthalpy of a real gas is h( T, v) = u( T, v) + pv, and it follows that, for a process
involving constant mole numbers:
h( T, v) = u( T0 , v0∗ ) +
ZT
v0∗
Zv cv dT +
T
v0∗
T0
=
u( T0 , v0∗ ) +
∂u
∂v
ZT
v0∗
dv + pv
Zv p∗ v0∗ T
cv dT +
− RT + T
{z
}
∗
|
T0
(A.8.13)
T
T
v0
=0
∂p
∂T
v
− p dv + pv
(A.8.14)
= h( T ) + pv − RT +
where h( T ) = u( T0 , v0∗ ) +
RT
v0∗
T0
Z∞ T
v
p−T
∂p
∂T
dv
(A.8.15)
v
cv dT + p∗ v0∗ T is the molar enthalpy of the ideal (zero-
density) gas at temperature T, and the ideal-gas equation of state has been used to
obtain (A.8.14) from (A.8.13). The two integrals in (A.8.13) correspond to the second
and third steps in the paths described by Beattie [13, p.162]. In Equation (A.8.14), the
∂p
partial derivative ∂u
= T ∂T
− p is evaluated using
∂v
T
v
du = T
where
∂s
∂v
T
=
∂p
∂T
v
∂s
∂T
dT +
v
∂s
∂v
T
dv − pdv
(A.8.16)
, by a Maxwell relation equating cross second derivatives of
the Helmholtz function F = U − TS [81, p.278].
∂s
∂u
Equation (A.8.16) also yields ∂T
= T1 ∂T
=
v
v
∂p
, the molar entropy of a real gas can be written:
∂T
cv
T.
Again using
∂s
∂v
T
=
v
s( T, v)
ZT
=s( T0 , v0∗ ) +
v0∗
T0
=s( T, v0∗ ) +
Zv T
v0∗
In (A.8.18), s( T, v0∗ ) = s( T0 , v0∗ ) +
cv
dT +
T
RT
v0∗
T0
∂p
∂T
cv
T dT
Zv T
v0∗
∂p
∂T
dv
(A.8.17)
v
R
−
dv + [ R ln v]vv∗
0
v
v
(A.8.18)
is the molar entropy of the ideal gas at
Background theory
253
state ( T, v0∗ ). The corresponding ideal-gas molar entropy at unit pressure is given by
0
s (T )
=s( T, v0∗ ) −
Z1 T
p∗
∂v
∂T
dp
(A.8.19)
p
=s( T, v0∗ ) + R ln p∗ T
In (A.8.19) the Maxwell relation
∂s
∂p
= −
T
(A.8.20)
∂v
∂T
p
(which equates cross second
derivatives of the Gibbs function G = U + pV − TS) has been used, and (A.8.20) fol ∂v
= Rp for an ideal gas. Combining (A.8.20) and (A.8.18)
lows from the equation ∂T
p
yields
Tv
s( T, v) =s ( T ) + R ln ∗ ∗ v0 p T T
0
=s0 ( T ) + R ln
v
RT
+
+
Z∞ T
v
Zv T
∞
R
−
v
∂p
∂T
R
−
dv
v
v
∂p
∂T
dv
(A.8.21)
(A.8.22)
v
where the ideal gas equation of state has been applied to the state ( T, v0∗ ).
Equations (A.8.15) and (A.8.22) apply both to single-component real gases, and to
multicomponent real gas mixtures. The equations can be further developed in the
case of mixtures by applying the Gibbs-Dalton law [81, p.329] to the low-pressure
state corresponding to ( T, v̄∗ ), and noting [81, p.348] that:
h̄( T, xi ) = ∑ xi h̄i0 ( T )
(A.8.23)
s̄( T, v̄∗ , xi ) = ∑ xi s̄i0 ( T, v̄i0∗ )
(A.8.24)
i
v̄i∗ =
i
v̄∗
xi
= v̄i0∗
(A.8.25)
Variables with a prime (0 ) in (A.8.23)–(A.8.25) represent properties of pure species
i in equilibrium with the mixture through a semi-permeable membrane. The unitpressure molar entropy of an ideal gas mixture (at low molar density v̄∗ ) can be evaluated by combining (A.8.24) and (A.8.20) to obtain:
s0 ( T, xi ) = ∑ xi s̄i0 ( T, v̄i0∗ ) + R ln p∗
(A.8.26)
i
∗ v̄
= ∑ xi s̄0i ( T ) + R ln
+ R ln p∗
x
RT
i
i
0
= ∑ xi s̄i ( T ) − R ln xi
(A.8.27)
(A.8.28)
i
Equation (A.8.27) is obtained by use of (A.8.22) together with (A.8.25), since by using
(A.8.9) and (A.8.10) the integral term in (A.8.22) can be shown to vanish as v̄ → ∞.
Using equations (A.8.23) and (A.8.28) in (A.8.15) and (A.8.22) yields the expres-
Background theory
254
sions [184, p.159] for the molar enthalpy and entropy of a multicomponent gas:
h( T, v, xi ) =
Z∞ s( T, v, xi ) =
Z∞ v
∑ xi s0i (T ) + ∑ xi R ln
i
p−T
∂p
∂T
∑ xi hi (T ) + pv − RT + T
i
i
v
xi RT
+
T
v
R
−
v
dv
(A.8.29)
v
∂p
∂T
dv
(A.8.30)
v
Substituting the power-series expansion of Z, up to the third term as shown in (A.8.9),
into (A.8.29) and (A.8.30), yields approximations of the real-gas enthalpy and entropy. Wexler et al. [184] have compiled correlations for the virial coefficients for
water vapour and air and cross-virial coefficients for the vapour-air mixture, as well
as expressions obtained from statistical mechanical computations based on spectroscopic data for the ideal-gas specific enthalpy and unit-pressure (1 bar) entropy of
dry air and water vapour. With these correlations, the specific enthalpy and entropy
of a moist-air mixture of a given composition and state can be evaluated.
The composition of a moist-air system may be specified in a number of ways. The
ASHRAE psychometric routines [184, p.317-353] use standard numerical solution
techniques to allow the state and composition of a moist-air system to be specified
by its pressure and any of the following pairs: (ω, h); (ω, Twb ); (ω, v); ( T, h); ( T, ω );
( T, φ); ( T, Twb ); ( T, v). These terms are defined in the following paragraphs.
The relative humidity φ of a moist-air system is the ratio of the water vapour mole
fraction to its mole fraction at saturation (in equilibrium with a flat liquid surface),
xw
φ=
(A.8.31)
xws p,T
The saturated water vapour mole fraction xws that appears in (A.8.31) satisfies:
xws =
f pw
p
(A.8.32)
where p is the total mixture pressure, and f = f ( p, T ) is a dimensionless enhancement
factor which incorporates the effects of molecular interactions between the various
species present as well as the Poynting effect and the Henry’s effect [184, p.162]. The
enhancement factor f is related [184, p.188] to the saturated mole fractions, as well as
to the virial coefficients for air and for water, the cross-virial coefficients, and several
physical constants. As the mole fractions at saturation are themselves determined
by the enhancement factor, f is obtained by an iterative procedure [184, p.193]. The
Background theory
255
saturated vapour pressure pw (in the absence of dry air) is given by [184, p.183]:
3
ln pw =
∑
gi T i + g4 ln T
(A.8.33)
i =−1
g−1 = − 5.8002206 × 103
g0 =1.3914993
g1 = − 4.8640239 × 10−2
g2 =4.1764768 × 10−5
g3 = − 1.4452093 × 10−8
g4 =6.5459673
The humidity ratio ω of moist air is the ratio of water vapour mass to dry air mass
in a given volume, and is given by [184, p.194]:
ω=
Mw x w
Mw φ f p w
=
Ma x a
Ma ( p − φ f pw )
(A.8.34)
The wet-bulb temperature Twb is the temperature of liquid water whose evaporation into the air will saturate the air at the same temperature. Consideration of the
energy balance for this scenario shows that for a mixture at T, ω, the wet-bulb temperature Twb is the solution to the following equation [184, p.312]:
h( T, ω ) + ωs ( Twb ) − ω hw ( Twb ) = h Twb , ωs ( Twb )
(A.8.35)
The dew-point temperature Tdp is the temperature of a saturated moist-air system
with the same humidity ratio as that of the system under consideration. The molar
volume v̄ of moist air is obtained by rearrangement of (A.8.9). The specific volume
vm , specific entropy s, and specific enthalpy h, per unit mass dry air, are given by:
vm =v̄/( x a Ma )
(A.8.36)
h =h̄/( x a Ma )
(A.8.37)
s =s̄/( x a Ma )
(A.8.38)
Adopting the constant heat capacities listed in [151] for the s component of the σ
phase, the pure β phase, and the dry air component in the γ phase, we may write:
hs ( T ) = c p,s ( T − Tref )
(A.8.39a)
h β ( T ) = c p,β ( T − Tref )
(A.8.39b)
h a = c p,a ( T − Tref )
(A.8.39c)
where c p,β = 4, 180 J/kg K and c p,a = 1, 000 J/kg K. The following additive expression for the specific heat of food is recommended by Sweat in [163]:
c p = 4, 180xw + 1, 711x p + 1, 928x f + 1, 547xc + 908x a
(A.8.40)
Background theory
256
where the subscripts stand for food constituents as follows: w = water, p = protein, f
= fat, c = carbohydrate, a = ash. Equation (A.8.40) is considered reliable over a wide
range of temperatures. In the present case this leads to a bone-dry solid specific heat
capacity of c p,s = 1, 604 J/kg K (refer to p 301 for more details).
The vapour enthalpy is given by the enthalpy of liquid water at the dew point temperature, plus the heat of vaporization at this temperature, plus the heat required to
raise the resulting vapour from the dew point temperature up to its final temperature.
This is evaluated in [151] as follows:
hv =1, 950( T ) + 1.65 × 106 + 2, 070( Tdp ) − 3.43( Tdp )2
Tdp =230.9 + 2.10 × 10
−4
( pv ) − 0.639( pv )
1/2
+ 6.95( pv )
(A.8.41)
1/3
(A.8.42)
Noting that in the present work the reference temperature Tref is set to 273.15K, we
set c p,v = 1, 950 and recast equation (A.8.41) into the following form:
hv = c p,v ( T − Tref ) + ∆hv
∆hv = 2.1826 × 106 + 2, 070( Tdp ) − 3.43( Tdp )2
(A.8.43a)
(A.8.43b)
Thermodynamic properties of R134a
Thermodynamic properties of R134a are evaluated using an inherited set of routines
originally based on equations (1)–(5) in the 1992 International Institute of Refrigeration booklet “Tables and diagrams for the refrigeration industry: R134a” [100]. These
functions are based on essentially the same theory as the non-ideal psychrometric
state equations described above. The functions produce the thermodynamic properties T, p, s, h, v of saturated vapour, saturated liquid, and the mixture, from any
of the following pairs of knowns: 1) T-p, 2) T-Q, 3) p-Q, 4) p-s, 5) p-h, 6) T-v. The
routines have been used in producing the pressure-enthalpy diagram shown e.g. in
Figure 4.5. The routines have been checked by comparing their outputs with the
pressure-enthalpy diagram that comes with the IIR documentation [100].
The thermal conductivity, viscosity and surface tension of R134a are evaluated using the equations presented on pages 14-15 of reference [100]. To rapidly evaluate
the constant-pressure specific heat capacity of liquid and vapour, data presented in
the table on page 26 of [100] have been fit using fourth-order polynomials in the saturated temperature TC (centigrade). The specific heat capacity data for the temperature
range 0◦ C < TC < 80◦ C yields the following polynomials and corresponding R2 val-
Background theory
257
ues:
4
c p,v =103 × ∑ ai TCi
(R2 = 0.9996510)
(A.8.44)
i =0
a0 =9.073187 × 10−1
(A.8.45)
a1 =1.034608 × 10−3
(A.8.46)
a2 =2.762619 × 10−4
(A.8.47)
a3 = − 6.500676 × 10−6
(A.8.48)
a4 =6.257613 × 10
(A.8.49)
−8
4
c p,l =103 × ∑ bi TCi
(R2 = 0.9993238)
(A.8.50)
i =0
b0 =1.347049
(A.8.51)
b1 =8.071823 × 10−4
(A.8.52)
b2 =1.745927 × 10
(A.8.53)
−4
b3 = − 3.858910 × 10−6
b4 =3.668025 × 10−8
where the specific heat capacities c p,v and c p,l are expressed in J/(kg K).
(A.8.54)
(A.8.55)
Background theory
A.9.
258
Nomenclature for appendix A
Roman and Greek symbols
Symbol
Definition [units]
Ac
Contact area [m2 ]
A
Area [–]
ai
Weighting coefficient for basis function i
am
Water activity [–]
bw
Steady-flow exergy function [J/kg]
bn
Non-flow exergy function [J/kg]
bTw
Total steady-flow exergy function [J/kg]
bTn
Total non-flow exergy function [J/kg]
c
Molar concentration [mol/m3 ]
c p,i
Specific heat capacity of pure species i [J/kg-K]
Cf
Friction factor [1/m2 ]
Dh
Hydraulic diameter [m]
D
Diffusivity [m2 /s]
e
Symmetric strain rate tensor [s−1 ]
e0
Traceless part of symmetric strain rate tensor [s−1 ]
f
Enhancement factor [–]
f
Specific body force [N/kg]
g − 1 , . . . , g4
Coefficients in eqation for pw [–]
G
Gibbs free energy of mixture [J/kg]
g
Specific Gibbs function [J/kg]
Hk
Partial molar enthalpy [J/mol]
h
Heat transfer coefficient [W/m2 -K]
hm
Mass transfer coefficient [m/s]
∆hv
Latent heat of vaporization [J/kg]
J
Jacobian [–]
Jψ
Flux of quantity ψ [ψ/m2 -s]
Jj
Volumetric rate of reaction j [kg/s-m3 ]
jq
Total heat flux vector [J/s-m2 ]
jk
Diffusion flux of species k [kg-m/s]
∗
JA
Molar diffusion flux relative to molar average velocity [mol/m2 -s]
jD , jH
Mass-transfer, Heat-transfer j-factors [–]
k
Thermal conductivity [J/s-m-K]
k
Number of chemical species in equilibrium mixture [–]
Mk
Molar mass of species k [kg/mol]
Background theory
259
mi
Mass of species i in mixture [kg]
ṁi
Mass transfer of species i [kg/s]
n̂
Unit normal vector [–]
n̂ · w
Control volume interface normal velocity [m/s]
Nu
Nusselt number [–]
p
Pressure [N/m2 ]
P
Wetted perimeter [m]
Pr
Prandtl number [–]
pw
Saturated vapour pressure [Pa]
p0
Ambient pressure [N/m2 ]
qc
Pure heat flux vector [J/s-m2 ]
dqrev
Heat transfer from system in a reversible transformation [J]
Q̇mix
(−1×) heat of mixing [J/s]
Q̇
Heat input [J/s]
R
Region
Rm
Region occupied by material body
Rv
Region occupied by control volume
Re
Reynolds number [–]
Sv
Bounding surface of control volume
S
Volumetric source of quantity ψ [ψ/m3 -s]
S
Surface
s
General scalar quantity
S
Entropy of system [J/K]
S
Sutherland constant [K]
Sc
Schmidt number [–]
t
Contact force per unit area [N/m2 ]
T
Temperature [K]
TC
Temperature in Celcius [◦ C]
∆t
Averaging time for turbulent flow [s]
u
Specific internal energy [J/kg]
U
Internal energy of system [J]
u, v
x-component, y-component of velocity [m/s]
v
General vector quantity
V
Volume [m3 ]
vk
Velocity of species k [m/s]
vk
Speed of species k [m/s]
V
Volume occupied by multicomponent system [m3 ]
v∗
Molar average velocity [m/s]
Background theory
260
v
Molar volume [m3 /mol]
wi
Weighting-function i
W
Work rate on surroundings at impermeable surfaces [J/s]
Ẇout
Work-rate obtainable from reversible steady-flow process [J/s]
Ẇi , Q̇i0
Work and heat input to drive species i isothermally up to equilibrium
with mixture entering control volume A [J/s]
ẆP
Work input to Carnot heat pump [J/s]
x
Location [m]
x0
Position in reference configuration
xi
Position of gridpoint i [m]
xi
Mole fraction of species i [–]
x, y, z
Spatial coordinates [m]
xm
Moisture content [–]
z
Number of chemical species present in environment [–]
Z
Compressibility factor [–]
ℵ
Reference configuration of body
δij
Kronecker delta function
δ
Momentum boundary layer thickness [m]
δc
Concentration boundary layer thickness [m]
δt
Thermal boundary layer thickness [m]
eijk
Antisymmetric tensor
ζ
Particle
θ
Correction factors [–]
µ
Dynamic viscosity of fluid [N-s/m2 ]
µi
Chemical potential of species i [J/kg]
νkj
Stoichiometric coefficient of species k in reaction j [–]
ρ
Density [kg-m]
σ
Total stress tensor [Pa]
τ
Viscous stress tensor [Pa], General tensor quantity
σ
Local volumetric entropy generation rate [J/K-m3 ]
φ
Relative humidity [–], Dimensionless quantity in high-mass transfer correction equations [–]
φi
Basis-functions in weighted-residual approach
Φ
Local volumetric mass production of species k [kg/m3 -s]
χ
Configuration of body
Ψ
Approximation of physical quantity
Ψp
Local value of ψ associated with time-path at time t
dm Ψ
dt
Material time derivative of Ψ [Ψ/t]
ω
Humidity ratio [–]
Background theory
261
Ω
Antisymmetric part of velocity gradient [s−1 ]
ωk
Mass-fraction of species k [–]
Subscripts
Symbol
Definition
0
Initial, environment
1, 2
System entry, system exit
i00
Species i at equilibrium with environment
i0
Species i at equilibrium with make-up mixture
∗
At low density, dimensionless
Molar, Time-averaged
dp
Dew-point
wb
Wet-bulb
S
Surface
β, γ, σ
Free water phase, Gas phase, Solid phase
∞
Free-stream condition
b, s, v, a, β
Bound-moisture, dry-solid, vapour, dry-air, free-liquid
S, P, N
South, “Point” under consideration, North
n, s
North interface, South interface
i, j, k
Species-i, Phase-j, Species-k
eff
Effective (porous-medium)
•
Modified for high mass transfer rates
262
Appendix B
Derivation of drying equations.
This appendix develops the theory that underlies the internal-process porous drying model that is subsequently implemented, and integrated with the HPD model, in
chapter 5. In particular, a critical assessment is made of the connection between the
standard drying equations and the detailed drying theory elaborated by Whitaker
[188, 193]. This highlights many assumptions that are required to obtain the drying
theory using the volume-averaging approach. Particular attention is paid to the relationship between various forms of the volume-averaged energy balance equation for
the porous medium, which is explained using a novel derivation.
B.1.
Introduction
In modelling the drying of a hygroscopic porous medium in detail, the separate transport of four different phases and species (free water, bound water, vapour and air)
must together be analysed. A theoretical basis for the study of multiphase transport was developed, chiefly by Whitaker, Slattery and Gray, during the 1960s and
1970s [186, 74, 188]. To avoid the difficulty of describing pore structure in detail on
the micro-scale, so-called volume averaging techniques are used in order to treat the
medium as a locally uniform continuum. The porous structure is ‘smoothed out’, allowing the local microscopic balance equations to be recast into averaged equations
that hold equally for all points within the medium, regardless of which phase they
happen to lie within. This approach, first applied to drying by Whitaker [188], has
become standard in modelling the drying of porous media [e.g. 50, 91, 173]. Whitaker
[193] has recently revisited the drying problem using new techniques in the solution
of so-called coupled closure problems. In the resulting equations, additional linkages
appear between the heat and mass transport processes (see below).
Derivation of drying equations.
263
In Whitaker’s work, three separate phases are recognised: gas, liquid and solid,
each exhibiting its own characteristic properties. The local behaviour within each
phase is expressed by versions of the balance equations which are presented in the
appendix as (A.3.3), (A.3.15), (A.3.28), (A.3.31) and (A.3.49). Whitaker has restricted
his attention to non-hygroscopic granular media, within which water may exist only
in two forms, namely as a liquid and as a vapour. Whitaker assumes the solid phase
to be rigid, impenetrable, and stationary. More generally, in a material which may
exhibit hygroscopic behaviour, moisture can exist within each of the three phases:
within the gas as vapour, as free liquid water, and associated with the solid phase as
hygroscopically bound water. The adsorption of moisture may also lead to volume
changes in the solid phase, further complicating the drying phenomena.
A straightforward extension of Whitaker’s equations, which can describe the drying of rigid hygroscopic media, has been used by many authors. However there appear to have been few attempts, and no decisive success, at deriving these extended
drying equations from first principles using Whitaker’s original volume-averaging
method. The most detailed attempt appears to be in an unpublished report by Gibson
[69], in which equations are derived describing transport within a deformable hygroscopic porous medium. Unfortunately Gibson’s development starts from a restrictive,
and physically incorrect, expression for the enthalpy of bound moisture (discussed
below). While the impact of this restriction on typical model predictions seems likely
to be minimal, since the heat of wetting can often in any case be neglected [93, p.34],
the restriction is unfortunate in the context of an attempt to produce a rigorous theory
of heat and mass transfer in deformable hygroscopic media. A different derivation
of a similar set of equations was presented by Turner [172]. The resulting equation
set had also, earlier, been stated without derivation and used by Harmathy [77] and
by Stanish et al. [151]. Turner’s equations, which have formed the basis of much subsequent work [e.g. 106, 116, 125, 127, 173], involve volume-averaging only implicitly.
In short, the drying equations for nonhygroscopic porous media are well-known and
have been derived from first principles; a set of equations that describes the drying of
hygroscopic porous media is well-established, widely known and commonly used,
but does not appear to have received a thorough theoretical treatment.
One difference between the equations obtained by Gibson [69] and by Turner [172]
is that the energy balance equation obtained in [69] is expressed in terms of the local
temperature change-rate, like that of Whitaker [188], rather than in an explicitly conservative form. (Variants of both forms are common.) In the present appendix a set
of transport equations is developed for a deformable hygroscopic porous medium.
In particular, an alternative derivation of the energy equation is presented, which is
based on Whitaker’s rigorous approach but takes a path that has not been seen in
the drying literature. The derivation highlights the relationship between Whitaker’s
Derivation of drying equations.
264
volume-averaging theory and the energy equation of Turner [172], and explicitly connects the latter to Gibson’s [69] formulation of the energy balance, while highlighting
the restrictive assumption, mentioned above, that is built into this equation.
At the outset, it would probably be wise to attempt to manage the reader’s expectations. While every effort has been made to justify the equations that are adopted
at the end of this appendix, by connecting them to the rigorous framework offered
by the fundamental continuum mechanics equations and by the volume-averaging
theory, it has often felt as though the net result was to open new questions faster
than old questions were being answered. The theoretical framework that is developed here, and used in chapter 5, is consistent with a large body of modelling work
reported in the drying literature. However, as is discussed in the concluding section
of the present appendix, this appendix’s detailed examination of the derivation of the
drying equations has highlighted a large amount of theoretical uncertainty surrounding the derivation of the standard drying equations. An upshot of this appendix is
that the equations cannot be derived from first principles without first making an
astonishing number of assumptions. This observation holds even in the “simple”
case first analysed in depth by Whitaker [188], of a nonhygroscopic granular porous
medium. As a consequence, the modelling work presented in chapter 5, which is
based upon the theory presented in this appendix, must be accompanied by some degree of doubt, even aside from issues surrounding the determination, for any given
product medium, of the many parameters that arise in the detailed theory. A more
effective approach to developing a kinetic model of the HPD system developed in
chapters 1–4 might have been to use empirical drying-curve relationships. We return
to a discussion of this and related points in chapter 6.
To the sharp assessment above, it should perhaps be added that the equations
themselves, whatever the status of their derivation, are elegant, intuitively plausible,
and widely accepted. From a pragmatic standpoint, the absence of a rigorous derivation of the drying equations in the hygroscopic case may not be a significant problem.
The rigour of Whitaker’s approach is in any case sacrificed as soon as it comes to implementing the drying equations in a numerical model: for instance, tensorial quantities with complex definitions are approximated using simple, often scalar, functions.
And a fairly broad consensus that the method works has been built on agreement
between modelled and measured drying behaviour. In the absence of a satisfactory
derivation from first principles, the equations can be accepted as phenomenological
description that adequately describes what has been observed in experiment.
Derivation of drying equations.
265
The volume averaging approach
In the case of a porous medium undergoing drying, three separate phases together
comprise the medium: the solid phase, designated the greek letter σ; the gas phase,
designated γ; and the (free) liquid water phase, designated β. When the medium
is hygroscopic, water may be associated with the solid (σ) phase. For phase i, the
volume distribution function αi (r ) is defined as follows:

1 if r lies in the region occupied by phase i
α i (r ) ≡
0 otherwise
(B.1.1)
In (B.1.1) the placeholder i can take any of the three values σ, γ, β. Thus, for instance,
if the location at displacement r0 from the origin lies within the gas γ phase (and not
in the solid σ or liquid β phase), then αγ (r0 ) = 1, while ασ (r0 ) = α β (r0 ) = 0.
Let V be any region occupied by the porous medium, bounded by the closed surface A. The volume in V occupied by phase i is denoted Vi (t), i = σ, γ, or β:
Vi (t) ≡
Z
αi (t) dV
(B.1.2)
V
The three phases are assumed to partition V, so that αi = 1 implies (j 6= i) ⇒ (α j =
0), and so that the three volumes Vi sum to the volume occupied by V, which we
(somewhat loosely) identify by the same symbol, V. We can thus write:
⇒
Vσ (t) + Vγ (t) + Vβ (t) = V
(B.1.3)
eσ ( t ) + eγ ( t ) + e β ( t ) = 1
(B.1.4)
where ei (t) ≡ Vi (t) /V is the volume fraction occupied by phase i, equal to the phase
average of the volume distribution function for phase i.
Consider any quantity ψi which is associated with phase i. By convention, the
value of ψi is set as zero outside of phase i, so that, expressed in terms of the volume
distribution function αi (r ), the following identity holds for all r:
ψi (r ) = ψi (r ) · αi (r )
(B.1.5)
The phase average hψi i of ψi in V and the intrinsic phase average hψi ii of ψi in V are
defined respectively as follows:
hψi i ≡
1
V
hψi ii ≡
1
Vi
Z
ψi dV
(B.1.6)
ψi dV
(B.1.7)
V
Z
V
Dividing (B.1.6) by (B.1.7) yields the following relationship between the volume fraction, the phase average and the intrinsic phase average:
hψi i = ei hψi ii
(B.1.8)
Derivation of drying equations.
266
In order to produce a well-behaved average of the property under consideration
(whose value does not fluctuate, depending upon which phase encloses the precise
location begin considered), an averaging volume V must have a characteristic length
l significantly greater than the characteristic length d of the pores. In [186], Whitaker
further demonstrates that in order for the following key relationship to hold,
hhψi ii = hψi i
(B.1.9)
the characteristic length l of the averaging volume must also be significantly less than
the characteristic length L of the system as a whole. Thus the averaging approach
assumes the existence of an intermediate scale, small compared with the large-scale
anisotropies but large compared with the scale of porosity:
dlL
(B.1.10)
A cross-section through a spherical averaging volume is shown schematically in
Fig B.1.1. The solid and bound water are treated together as the single phase σ. The
Averaging volume
n̂σγ
n̂γσ
Solid (‘σ’) phase
n̂σβ
n̂βσ
n̂γβ
Liquid (‘β’) phase
Gas (‘α’) phase
n̂βγ
Figure B.1: Cross-section of an averaging volume
porosity, which is denoted by the symbol e, is defined as follows:
e ≡ eγ + e β = 1 − eσ
(B.1.11)
The region occupied by phase i in V is bounded by a closed surface Ai . This surface
may be decomposed into its intersection with the exterior bounding surface A of V,
Derivation of drying equations.
267
written Aei , and its interior portion Aii :
Ai = Aei + Aii
(B.1.12)
Since there are three phases present within the medium, the internal bounding surface Aii of the i phase can be decomposed into two parts, corresponding to the interface between the i phase and each of the two other phases. Thus, for instance,
the bounding surface Aσ of the σ phase contained in an averaging volume can be
expressed as the following sum:
Aσ = Aeσ + Aiσ
(B.1.13)
Aiσ = Aσγ + Aσβ
(B.1.14)
where Aσγ = Aγσ is the interface between the σ and γ phases, and Aσβ = A βσ is
the interface between the σ and β phases, both interfaces being contained within the
averaging volume. The notation of Whitaker [188, II.B-5] is followed here: thus the
unit normal n̂σγ points out of the σ phase and into the γ phase, and
n̂σγ = −n̂γσ on Aσγ = Aγσ
(B.1.15)
and similarly for the other interfaces.
The volume-averaged governing equations can be obtained from the microscopic
balances using Slattery’s theorem for the volume average of a gradient [146, p.202]:
h∇ψi i = ∇ hψi i +
1
V
Z
(B.1.16)
ψi n̂dA
Aii
Using an expression such as (B.1.14) the final term in (B.1.16) can be expressed as a
sum of two integrals, over each of the two internal interfaces which together comprise the internal bounding surface Aii . When ψi represents the flux of a conserved
physical quantity, the final term of (B.1.16) tracks internal transfer between the different phases, and can be evaluated by use of the jump balance conditions for a phase
interface [e.g. 146, p.427], [144]. Equation (B.1.16) effectively converts the average of
a gradient on the left side into a gradient of an average on the right. This allows the
local balances within each phase to be converted into volume averaged forms which
hold identically throughout the porous medium. Applying (B.1.16), for example, to
the σ phase, Slattery’s averaging theorem can be expressed as follows:
h∇ψσ i = ∇ hψσ i +
1
V
Z
ψσ n̂σγ dA +
Aσγ
1
V
Z
ψσ n̂σβ dA
(B.1.17)
Aσβ
The physical significance of the term ∇ hψi i can be seen by applying the divergence
theorem (A.2.5) to the left side of (B.1.16), which yields the following equation:
1
∇ hψi i =
V
Z
ψi n̂dA
Aei
(B.1.18)
Derivation of drying equations.
268
In words, Equation (B.1.18) implies that the gradient of the volume average of ψi
(where ψi is a quantity that is nonzero within phase i, and zero in the other phases)
is equal to the integral of ψi n̂ over the exterior of the averaging volume. This physical
interpretation allows many of the drying equations to be interpreted intuitively.
In the next five sections, equations describing the drying of a hygroscopic porous
medium are developed. Frequent reference is made to the derivations of Whitaker
[188, 193] to show the relationship between the equations presented here and Whitaker’s
drying theory. The porous medium comprises three phases: the gas (γ) phase, the liquid (β) phase, and the solid (σ) phase. The γ phase is treated as a dry-air and water
vapour binary continuum. The basic continuum-mechanics equations are integrated
over each phase within an averaging volume. Product-averages can be decomposed
into sums of average-products and dispersion terms [74], so that for instance, for a
scalar field a j and a vector field w j in phase j (j = γ, σ or β):
h a j w j i = h a j i j hw j i + h ã j w̃ j i
(B.1.19)
Terms marked by tildes in equation (B.1.19) represent the local deviation of a quantity within phase j from its intrinsic phase average (defined using the averaging volume associated with the given point). The first term on the right-hand side can be
rewritten h a j i j hw j i = e j h a j i j hw j i j = h a j ihw j i j , i.e. the intrinsic phase average can be
applied to either variable. For the derivation of Equation (B.1.19), see Gray [74].
B.2.
Modifications for hygroscopicity
Several modifications need to be made to Whitaker’s derivation, in order to cope with
phenomena that arise due to the presence of bound moisture.
First, in Whitaker’s derivation local equilibrium assumptions are used to evaluate
thermodynamic properties of matter in the various phases. Thus the local-volumeaveraged temperatures in each phase are assumed to be equal (Whitaker’s assumption of local thermal equilibrium), and the vapour pressure is assumed to be related
to the saturated vapour pressure at this local temperature by the capillary pressure
exerted on the β phase, by way of the Kelvin equation (the local mass equilibrium
assumption) [193]. Applying this approach to the case of a hygroscopic material, the
local amount of moisture in each phase can be evaluated as follows [151, 173]:
1. By inverting the sorption isotherm for a given temperature, the equilibrium
bound moisture content at a given relative humidity can be obtained.
2. The saturation point, defined for a given temperature, is the moisture content
at which all free water has been removed, and only bound water remains in
Derivation of drying equations.
269
equilibrium with the water vapour. If the local moisture content exceeds the
saturation point, then free water must be present, and phase equilibrium is assumed between the liquid and moisture in the σ- and γ-phases. That is, the
water activity am is evaluated using the Kelvin equation [193].
3. In the absence of free water the vapour pressure is assumed saturated with
respect to the bound water, and is evaluated using the sorption isotherm.
In general, mass transfer between the σ and β phases may take place. However,
as a general rule if the temperature is roughly constant, above the saturation point
there will be little net transfer of bound moisture into or out of the σ phase, because
everywhere in this regime the relative humidity φ ≈ 1, and the σ phase is saturated
with bound water at the local temperature; meanwhile, below the saturation point
no β phase remains. As a consequence mass transfer between the σ phase and the β
phase can be expected to be small compared with mass transfer between the σ phase
and the γ phase.
A more detailed discussion of the assumptions of local mass and local thermal
equilibrium appears below, after the derivation of the energy equation.
Second, a choice needs to be made between allocating a separate ‘bound water’
phase, or instead associating the bound water with the solid phase. Both approaches
have previously been adopted: e.g. the former approach was taken by Turner [172],
the latter by Gibson [69]. In the present work, the bound water is associated with the
σ phase. That is, the σ-phase is considered to be a multicomponent system incorporating the ‘bone-dry’ solid, denoted ‘s’, and the bound water, denoted ‘b’. Thus the
density of the σ phase and the energy per unit volume of the σ phase are given by
ρσ =
ms + mb
= ρs + ρb
Vσ
ρσ hσ = ρs hs + ρb hb
(B.2.1)
(B.2.2)
where hb is given by (B.2.13a), and no effective distinction needs to be made between
specific internal energy u and specific enthalpy h for the solid and bound moisture
components, because of their high respective densities [85].
Figure B.2 displays some sorption data for porridge, obtained as part of this work,
together with sorption curves produced by fitting the GAB equation to this data.
The differential heat of wetting can be evaluated from the sorption isotherm. According to the Clausius-Clapeyron equation, the specific latent heat associated with a
Derivation of drying equations.
270
0.25
0.2
T=20C
xm
T=35C
0.15
T=50C
0.1
0.05
0
0
0.2
0.4
φ
0.6
0.8
1
Figure B.2: Typical sorption isotherm. Experimental data (circles, 20◦ C; triangles, 35◦ C; squares, 50◦ C); Best-fit GAB correlation (dash-dot lines).
condensate-vapour phase change is given by [58, p.198]:
T∆v dpv
Mw dT
RT 2 d ln pv
≈
Mw dT
∆hv =
(B.2.3)
(B.2.4)
where the molar volume change ∆v has been approximated as the volume occupied
by one mole of vapour, since vapour is much less dense (has a much greater specific
volume) than the condensed phase. Equation (B.2.4) also applies to non-ideal solid
solutions [58, p.218], and can be applied to bound water as follows. Figure B.3 depicts a hypothetical apparatus with which to imagine adsorption of moisture on a
hygroscopic solid. After the two isothermal chambers are connected, the liquid water
is gradually evaporated as water vapour is adsorbed onto the solid. If the process
takes place at constant temperature in a quasi-equilibrium process where the vapour
pressure in the gas phase γ1 over the liquid water remains near p•v , then if the moist
air system is regarded as ideal,
dQvap = (hv − h β )dmw
(B.2.5)
dQ ads = (hv − hb )dmw
(B.2.6)
where hv , h β and hb are partial mass enthalpies, of which hv and h β can be regarded
as functions only of temperature [58, p.117].
Derivation of drying equations.
271
dmw
plug
T , p0 , p•v
T , p0 , pv
γ2
γ1
σ, ms
β, mw
T
dQads
dQvap
Figure B.3: Isothermal sorption process
Thus the net heat evolved per mass increment of adsorbed moisture satisfies
dQ ads − dQvap
dQnet
=
= ( hv − hb ) − ( hv − h β )
dmw
dmw
(B.2.7)
Using (B.2.4), (B.2.7) becomes
RT 2 ∂ ln pv
d ln p•v dQnet
=
−
dmw
Mw
dT
dT
2
RT
∂ ln am
=
Mw
∂T
(B.2.8)
(B.2.9)
The net heat evolved per increment of the moisture content xm of the solid is
dQnet
dQnet dmw
RT 2
=
=
dxm
dmw dxm
Mw
∂ ln am
∂T
ms
(B.2.10)
Dividing through by the volume of the moist hygroscopic solid at the end of the
process, the specific enthalpy of the moist hygroscopic solid satisfies
(ρs + ρb )hσ = ρs hs + ρb h β − Qnet
= ρs hs + ρb hb
where hb = h β −
1
mw
R
dQnet
dxm dxm
(B.2.11)
(B.2.12)
is the specific enthalpy of the bound water. Using
Eq. (B.2.9), the specific enthalpy of the bound water can be written
hb = h β − ∆hb
∆hb =
RT 2 1
Mw x m
Zxm 0
∂ ln am
dxm
∂T
(B.2.13a)
(B.2.13b)
Derivation of drying equations.
272
Equations (B.2.13) are consistent with the equations of Keey [94, p.33] when Keey’s
equations are expressed in mass rather than molar units.
Third, bound water may force a modification to Whitaker’s theory by causing motion in the σ-phase. In treating the σ-phase as a combination of two components (s
and b), the velocity of the s-component can be taken as the reference velocity to which
the diffusion velocity of the b-component is referred [16, page]. For the region Vσ (t)
allocated to the σ phase to coincide with the region occupied by the s component, the
interfacial normal velocities n̂σβ · wσβ and n̂σγ · wσγ on the surfaces Aσβ and Aσγ are
defined equal to the normal velocity n̂ · vs of the solid (s) component on these sur-
faces. Moisture transfer occurs between the σ-phase and the other phases when the
normal velocity of the moisture at the interface differs from the interface velocity.
B.3.
Mass balance
Each species present within each phase satisfies the local mass balance equation [16]
∂ρi
+ ∇ · ( ρ i vi ) = 0 :
∂t
(B.3.1)
where in the σ phase, i = s, b, in the β phase, i = β, and in the γ phase, i = a, v.
The volume-averaged mass equations are obtained by integrating (B.3.1) over an averaging volume, and then using the volume-averaging theorem and the generalized
transport theorem.
Interfacial mass jump conditions are required in order to use the volume averaging approach to obtain the volume-averaged mass equations for the porous medium.
Some of the jump conditions obtained in [188] carry over to the hygroscopic case,
while others need to be modified to account for the presence of bound moisture.
Internal species mass jump conditions
The jump conditions for the water species (β, v) and the dry air species (a) on the
interfacial surface A βγ = Aγβ are unaffected by the presence of bound water in the σ
phase, and are given by equations IIB-24 and IIB-25 of Whitaker [188]:
ρ β (vβ − wγβ ) · n̂βγ = ρv (vv − wγβ ) · n̂βγ
(B.3.2)
ρ a (va − wγβ ) · n̂βγ = 0
(B.3.3)
Since n̂βγ · ρ β (vβ − wγβ ) is the normal flux of the β phase relative to the interface A βγ ,
which moves with velocity wγβ , the left hand side of (B.3.2) is the rate, per unit area,
Derivation of drying equations.
273
at which water passes through the interface Aγβ on the β side. Similarly, the right
hand side is the rate, per unit area, at which water passes though the interface Aγβ
on the γ side. Evidently (B.3.2) states that mass is not created, destroyed, stored, or
transported away, within the plane of the interfacial surface Aγβ . Using the methods
of [144], the species jump conditions on Aσγ = Aγσ can be shown to be
ρb (vb − wγσ ) · n̂σγ = ρv (vv − wγσ ) · n̂σγ
(B.3.4)
ρ a (va − wγσ ) · n̂σγ = ρs (vs − wγσ ) · n̂σγ = 0
(B.3.5)
Finally, the species jump conditions on the surface Aσβ = A βσ can be shown to be
ρb (vb − wβσ ) · n̂σγ = ρ β (vβ − wβσ ) · n̂σβ
(B.3.6)
ρs (vs − wβσ ) · n̂σβ = 0
(B.3.7)
Evidently (B.3.4) and (B.3.5) assert – as may reasonably be assumed of the phase interfaces within a biological material [52] – that moisture flow is preserved as the moisture passes through the σ-γ interface, and that dry air does not penetrate this surface.
Equations (B.3.6) and (B.3.7) assert similar statements for the β-σ interface.
Solid phase b-component mass balance
In modelling the solid phase σ as two interpenetrating components, each component
obeys its own conservation principle. The local mass balance for the b-component in
the σ phase is obtained in the appendix as (A.3.3), and is given as:
∂ρb
+ ∇ · ( ρ b vb ) = 0
∂t
(B.3.8)
Consider integrating the bound water balance (B.3.8) over the volume Vσ (t) as depicted in Fig. B.4. The whole sphere (including γ, β and σ phases) has constant volume V. As the drying process proceeds, the volume Vσ (t) will tend to retreat as the
hygroscopically bound moisture departs from the σ phase. This shrinkage will be
overlaid on top of motion due to bulk shrinkage of the porous medium. The velocity of the σ-γ interface is denoted wσγ or wγσ (where wσγ = wγσ ), and the outward
velocity of the σ-β interface is denoted wσβ or wβσ (where wσβ = wβσ ).
Integrating (B.3.8) over Vσ (t) and dividing by V yields the following equation:
1
V
Z
Vσ (t)
∂ρb
1
dV +
∂t
V
Z
Vσ (t)
∇ · (ρb vb )dV = 0
(B.3.9)
Noting the definition of the phase average (B.1.6), we apply the generalized transport
Derivation of drying equations.
274
Vγ (t+∆t)
Vγ (t)
Vσ (t)
Vσ (t+∆t)
Vβ (t)
Vβ (t+∆t)
Figure B.4: Changing averaging volume partition
theorem (A.2.35) to the first term in (B.3.9) to obtain
1
V
Z
Vσ (t)
∂ρb
d
1
dV = hρb i −
∂t
dt
V
−
Since
d
dt h ρb i
1
V
Z
Aσβ (t)
Z
Aσγ (t)
(ρb wσβ ) · n̂σβ dA
(ρb wσγ ) · n̂σγ dA
(B.3.10)
in (B.3.10) is associated with a given point in space, it is identical to
the partial time derivative of the function hρb i(x, t), which returns the local-volume-
averaged gas density as a function of location and time. Since the argument (x, t) of
this last function is unambiguous, it can be dropped. Thus we can write [188]:
∂
d
∂
hρb i =
hρb i(x, t) = hρb i
dt
∂t
∂t
(B.3.11)
On the right-hand side of (B.3.11), hρb i is implicitly understood to be a function both
of time and of location. Equation (B.3.11) allows us to rewrite (B.3.10) as follows:
1
V
Z
Vσ (t)
∂ρb
∂
1
dV = hρb i −
∂t
∂t
V
−
1
V
Z
Aσβ
Z
Aσγ
(ρb wσβ ) · n̂σβ dA
(ρb wσγ ) · n̂σγ dA
(B.3.12)
Applying Slattery’s theorem for the average of a gradient (B.1.16) to the second
Derivation of drying equations.
275
term in (B.3.9) yields the following equation:
1
V
Z
Vσ (t)
∇ · (ρb vb )dV =∇ · hρb vb i +
+
1
V
Z
Aσγ
1
V
Z
Aσβ
(ρb vb ) · n̂σβ dA
(ρb vb ) · n̂σγ dA
(B.3.13)
Substituting (B.3.12) and (B.3.13) into (B.3.9) yields the following equation:
∂
1
h ρ i + ∇ · h ρ b vb i +
∂t b
V
+
1
V
Z
Aσγ
Z
Aγβ
ρb (vb − wσβ ) · n̂σβ dA
ρb (vb − wσγ ) · n̂σγ dA = 0
(B.3.14)
Equation (B.3.14) can be written
∂
hρ i + ∇ · hρb vb i + hṁσβ i + hṁσγ i = 0
∂t b
(B.3.15)
In (B.3.15) the rate of liberation of bound water per unit volume hṁσβ i, and the rate
of evaporation of bound water hṁσγ i per unit volume, are defined as follows [69]:
hṁσβ i =
1
V
hṁσγ i =
1
V
Z
Aσβ
Z
Aσγ
ρb (vb − wσβ ) · n̂σβ dA
(B.3.16)
ρb (vb − wσγ ) · n̂σγ dA
(B.3.17)
The rate of evaporation of free water hṁ βγ i per unit volume is defined similarly:
hṁ βγ i =
1
V
Z
A βγ
ρ β (vβ − wβγ ) · n̂βγ dA
(B.3.18)
Solid phase s-component mass balance
We now turn to the mass balance for the s-component of the σ phase. The local mass
balance for the s-component can be written as follows:
∂ρs
+ ∇ · ( ρ s vs ) = 0
∂t
(B.3.19)
Since equation (B.3.19) is of exactly the same form as (B.3.8), integrating over the same
region and dividing by V leads to the following equation, analogous to (B.3.14):
∂
1
h ρ s i + ∇ · h ρ s vs i +
∂t
V
+
1
V
Z
Aσβ
Z
Aσγ
ρs (vs − wσβ ) · n̂σβ dA
ρs (vs − wσγ ) · n̂σγ dA = 0
(B.3.20)
Derivation of drying equations.
276
Using (B.3.5) and (B.3.7), Equation (B.3.20) becomes
∂
h ρ s i + ∇ · h ρ s vs i = 0
∂t
(B.3.21)
By evaluating hns i = hρs vs i using a decomposition similar to that in Equation (B.1.19)
and neglecting the small deviation term hρ̃s ṽs i, we can rewrite (B.3.21) as follows:
∂
hρs i + ∇ · hρs ihvs iσ = 0
(B.3.22)
∂t
Gas species mass balance
The local mass balance for species i in the gas phase is given as follows:
∂ρi
+ ∇ · ( ρ i vi ) = 0
∂t
(B.3.23)
Consider integrating this local balance over the volume Vγ (t) as depicted in Fig. B.4.
Again the whole sphere (including γ, β and σ phases) has constant volume V. As
the drying process progresses, the volume Vγ (t) will tend to expand, replacing the
retreating β and σ phases. The external bounding surface of the volume is stationary.
Integrating (B.3.23) over Vγ (t) and dividing by V yields the following equation:
1
V
Z
Vγ (t)
∂ρi
1
dV +
∂t
V
Z
Vγ (t)
∇ · (ρi vi )dV = 0
(B.3.24)
We apply the transport theorem (A.2.35) to the first term in (B.3.24), again writing
∂
∂t h ρi i
rather than
d
dt h ρi i
1
V
to represent the time derivative of this local volume average:
Z
Vγ (t)
∂ρi
∂
1
dV = hρi i −
∂t
∂t
V
−
1
V
Z
Aγσ
Z
Aγβ
(ρi wγβ ) · n̂γβ dA
(ρi wγσ ) · n̂γσ dA
(B.3.25)
Applying Slattery’s theorem for the average of a gradient (B.1.16) to the second
term in (B.3.24) yields the following equation:
1
V
Z
Vγ (t)
∇ · (ρi vi )dV =∇ · hρi vi i +
1
+
V
Z
Aγσ
1
V
Z
Aγβ
(ρi vi ) · n̂γβ dA
(ρi vi ) · n̂γσ dA
(B.3.26)
Substituting (B.3.25) and (B.3.26) into (B.3.24) yields the following equation:
∂
1
h ρ i + ∇ · h ρ i vi i +
∂t i
V
+
1
V
Z
Aγσ
Z
Aγβ
ρi (vi − wγβ ) · n̂γβ dA
ρi (vi − wγσ ) · n̂γσ dA = 0
(B.3.27)
Derivation of drying equations.
277
Setting i = v in (B.3.27) and using Equations (B.3.2) and (B.3.4) yields:
∂
hρv i + ∇ · hρv vv i − hṁ βγ i − hṁσγ i = 0
∂t
(B.3.28)
where the volumetric evaporation rate definitions (B.3.18) and (B.3.17) have been
used. Setting i = a in (B.3.27) and substituting (B.3.3) and (B.3.5) into the resulting
expression yields the following equation:
∂
hρ a i + ∇ · hρ a va i = 0
∂t
(B.3.29)
Substituting Fick’s law (A.4.2) and (A.3.8) into equations (B.3.28) and (B.3.29) yields:
∂
hρv i + ∇ · hρv vγ i − hṁ βγ i − hṁσγ i − ∇ · hργ Dva ∇ωv i = 0
∂t
∂
hρ a i + ∇ · hρ a vγ i − ∇ · hργ Dva ∇ωa i = 0
∂t
(B.3.30)
(B.3.31)
These equations are analogous to (II.C-65) in [188], with the additional mass source
term owing to bound water evaporation. Summing (B.3.28) and (B.3.29), we obtain:
∂
hργ i + ∇ · hργ vγ i − hṁ βγ i − hṁσγ i = 0
∂t
(B.3.32)
In summing (B.3.28) and (B.3.29), the linearity of volume-averaging has been invoked.
Equation (B.3.32) expresses the volume-averaged mass balance for the gas phase.
Liquid phase mass balance
The derivation of the liquid phase mass balance is affected by the existence of bound
moisture in exactly the same way that the gas phase mass balance is. The resulting
equation is identical to that of Whitaker [188], except that that it includes the source
term hṁσβ i accounting for liberation of bound water from the σ phase into the β
phase. As has already been commented, this term is expected to be small. For reasons
of space, the equation is presented here without its derivation:
∂
hρ β i + ∇ · ρ β vβ = ṁσβ − ṁ βγ
∂t
B.4.
(B.3.33)
Mass transport
Governing equations for the bulk motion of all three phases remain to be presented.
Because biological materials may undergo bulk deformation, due to shrinkage, changes
need to be made to the equations presented in papers such as [188], which assume a
rigid σ phase. The changes that have been adopted are outlined in the present section.
Derivation of drying equations.
278
Gas species diffusion equations
Applying the volume averaging theorem to the continuity equations in the previous section yielded the following mass balance equations for the five species [193]:
∂hρ a i
∂t
∂hρv i
∂t
∂hρ β i
∂t
∂hρb i
∂t
∂hρs i
∂t
= −∇ · hna i
(B.4.1)
= −∇ · hnv i + hṁ βγ i + hṁσγ i
(B.4.2)
= −∇ · hnβ i − hṁ βγ i + hṁσβ i
(B.4.3)
= −∇ · hnb i − hṁσγ i − hṁσβ i
(B.4.4)
= −∇ · hns i
(B.4.5)
In equations (B.4.1)–(B.4.5), hna i ≡ hρ a va i is the phase-averaged air flux (kg s−1 m−2 ),
hnv i ≡ hρv vv i is the phase-averaged water vapour flux, hnβ i ≡ hρ β vβ i is the phase-
averaged liquid water flux, hnb i ≡ hρb vb i is the phase-averaged bound moisture flux,
and hns i ≡ hρs vs i is the phase-averaged solid-species flux. Using Equation (B.1.19)
the gas species flux hni i ≡ hρi vi i (where i = a or v) becomes:
hρi vi i = hρi vγ i + hji i = hρi iγ hvγ i + hρ̃i ṽγ i + hji i
(B.4.6)
where ji = ργ Dva ∇ωi is the local diffusion flux of species i relative to the gas phase
velocity vγ . When convective mass transport is significant in the γ-phase, the dispersion term hρ̃v ṽγ i in (B.4.6) may contribute significantly to overall moisture transfer. In
the absence of mass transfer at the bounding surfaces of the γ-phase, this term would
lead to a modified effective diffusivity [185, 21]: for instance, according to the TaylorAris result, in a bundle of capillary tubes the effective diffusivity D is increased by a
factor of 1 +
hvz i2 r02
48D2
[2]. Since mass transfer does occur at the γ-β and γ-σ interfaces,
the situation is more complex. Paine et al. [114] have demonstrated theoretically that
a pulse of solute released into a bundle of capillary tubes, undergoing irreversible adsorption at the tube walls, can move up to 57% faster than the average fluid velocity
hvz i, as slow-moving solute adjacent to the tube walls is removed from the stream.
In drying processes where vapour transfer is diffusion-limited (e.g. in the absence
of forced air convection through the product), especially at low temperatures where
ωv 1, the contribution of convective (and thus dispersive) mass transport to overall
vapour transport is likely to be be fairly small [189]. Adopting a diffusive model [185]
for the dispersion, one can write [188]:
hρ̃v ṽγ i =
(v)
DD
hρv iγ
·∇
hργ iγ
(v)
(B.4.7)
where DD , which depends on the gas velocity hvγ i, is expected to be fairly small
compared with the effective diffusivity of vapour in the medium.
Derivation of drying equations.
279
In Whitaker’s early work [e.g. 188, 193], the quantity hρ̃v ṽγ i − hρi iγ hvγ i is anal-
ysed into a form that resembles Fick’s law of diffusion at the volume-averaged level.
However, in its most comprehensive form [193], Whitaker’s drying theory predicts
that the active liquid-gas interface within an averaging volume can affect internal
heat and mass transfer in drying processes. A linkage arises between the spatial deviation vapour density ρ̃v and the spatial deviation temperature T̃γ , due to the local
vapour density at the β-γ interface being determined by the local temperature by way
of the Clausius-Clapeyron equation. There is also a linkage between the spatial deviation vapour density gradient ∇ρ̃v and the spatial deviation temperature gradient
∇ T̃γ , due to the local evaporation rate at the β-γ interface being linked to the local
heat flux on either side of the interface. These linkages result in couplings between
macroscopic heat and mass transfer that Whitaker has called the “multiphase Dufour
effect” and “multiphase thermal diffusion” [193]. However, these influences appear
to be poorly understood, and have not been acknowledged in any of the numerical
drying simulation work referenced by this thesis.
Substituting (B.4.7) into (B.4.6) and following Whitaker’s analysis leads to the vapour
transport equation – refer to Equations (II.C-65)–(II.C-72) and (IV.A-10) of Whitaker
[188] and to Equations (524)–(533) of Whitaker [193]:
hnv i = hρv iγ hvγ i + hjv i + hρ̃i ṽγ i
(v)
= hρv iγ hvγ i − hργ iγ Deff · ∇
(B.4.8)
γ
hρv i
− eγ hρv iγ h T i−1 DT · ∇h T i
hργ iγ
(B.4.9)
(v)
The effective diffusivity Deff appearing in (B.4.9) is defined by equation (IV.A-9) of
(v)
Whitaker [188], and includes the dispersion tensor DD :
(v)
(v)
(v)
Deff = eγ Dva δ + DD + DD
(B.4.10)
where δ is the unit tensor. In (B.4.10), Dva is the mutual diffusivity of the binary moist
(v)
air mixture, given in the appendix by Equation (A.8.1). The tortuosity DD appearing
in (B.4.10) can be decomposed into a passive and active part [193]:
(v)
0( v )
∗(v)
DD = DD + DD
∗(v)
The term DD
(B.4.11)
appearing in (B.4.11) represents the enhanced effective diffusivity of
vapour due to the effect of the β-γ interface. In establishing the coupled closure
problem for ρ̃v and T̃ in order to obtain this result, Whitaker [193] assumes that the
barycentric velocity vγ is small compared to the vapour species velocity vv . Equations
(B.4.9)–(B.4.11) show that five separate physical effects contribute to the vapour flux
nv : (1) Darcy flow in the γ-phase, through the term hρv iγ hvγ i, (2) passive diffusion
0( v )
by way of the effective diffusivity tensor eγ Dva δ + DD , (3) enhanced diffusion by
∗(v)
(v)
way of the tensor DD , (4) dispersion by way of the tensor DD , and (5) multiphase
thermal diffusion by way of the tensor eγ hρv iγ h T i−1 DT . Equations (B.4.9)–(B.4.11)
Derivation of drying equations.
280
become equivalent to those derived in Whitaker’s earlier work [e.g. 188, 193] if the
∗(v)
new coupling terms DT and DD
are set to zero.
The extra contributions of these terms cannot be evaluated without a detailed numerical solution of the associated coupled closure problem. Their contribution has
been neglected from previous numerical models of drying; however this important
assumption does not appear to have been acknowledged in the literature. Most researchers have effectively assumed that all of the correction terms, including the dispersion term, may be neglected or subsumed into a simple (generally scalar) correlation for the effective diffusivity, and have dropped the coupling term involving DT
from (B.4.9). For instance Wang and Chen [178] have simplified (B.4.10) to:
(v)
Deff ≈ eγ Dva δ
(B.4.12)
Like previous modellers, we have no choice but to neglect the enhancement of the
effective diffusivity and the multiphase thermal diffusion effect. The classic species
flux, expressed in terms of the passive diffusivity tensor, is expressed [189, p.32]:
h ρi iγ
(i )
hni i = hρi iγ hvγ i − hργ iγ Deff · ∇
(B.4.13)
hργ iγ
(i )
In Equation (B.4.13) the effective diffusivity Deff is the classic passive diffusivity ten(i )
0( i )
(i )
sor, including the dispersion coefficient: Deff = eγ Dva δ + DD + DD .
Bulk γ and β phase velocities
The theoretical basis for Darcy’s law in a porous medium with one fluid phase is
outlined in [190], where the final general form is presented as follows:
hv β i = −
1
Kβ · ∇h p β i β − hρ β i β g
µβ
(B.4.14)
In [191] equation (B.4.14) is extended to the case where there are two fluid phases,
resulting in the following expressions:
1
Kβ · ∇h p β i β − hρ β i β g + Kβγ · hvγ i
µβ
1
hvγ i = − Kγ · ∇h pγ iγ − hργ iγ g + Kγβ · hvβ i
µγ
hv β i = −
(B.4.15)
(B.4.16)
The coupling tensors Kβγ and Kγβ appearing in (B.4.15) and (B.4.16) have little effect
in drying situations, and are generally neglected [193, p.40]. The effect of gravity on
the gas phase, given by the term hρ β i β g in (B.4.15), can generally be neglected [189].
Dropping the final terms of (B.4.15) and (B.4.16), we are left with standard Darcytype equations of the form (B.4.14). Stanish et al. [151] among others have successfully
Derivation of drying equations.
281
applied Darcy’s law in this form to both gas and free liquid flow in the modelling of
drying by evaluating the free liquid pressure h p β i β , as follows:
h p β i β = h pγ iγ − pc
(B.4.17)
The capillary pressure pc appearing in (B.4.17) is generally evaluated as a function of
the saturation (void fraction occupied by the β phase) [151].
The development of equations (B.4.15) and (B.4.16) involves the boundary condition vβ = 0 on A βσ . But in the drying of biological materials, shrinkage and deformation may significantly alter this boundary condition. In order to consider what effect
shrinkage will have, consider the following two idealisations of shrinkage behaviour,
described by Gibson [69]. (These two idealisations are depicted in Fig. B.5.)
(a)
(b)
Figure B.5: Solid distribution in two idealisations of shrinkage [69].
• In idealisation (a), the bulk thickness of the medium is unaffected by the shrink-
age of the σ phase, and as the volume occupied by the σ phase decreases, the
pore space within the medium increases by precisely the same amount. Clearly
this will lead to variable porosity e within the sample, as drying proceeds at different rates at different locations; another implication is that the local velocity
of the σ phase will remain negligible.
• In idealisation (b) there is no change in porosity (that is, in the volume fraction
occupied by the β and γ phases), and instead bulk shrinkage occurs, leading to a
local σ-phase velocity which can be determined from the overall shrinkage rate.
In the current work, this second idealisation is adopted, although the actual
Derivation of drying equations.
282
shrinkage behaviour of a product undergoing drying is likely to lie somewhere
between these two extremes.
Several papers which deal with shrinkage, including [69, 143], simply apply Darcy’s
law (B.4.14) to the γ and β phases, without modifying the equation to account for the
motion of its σ phase. This approach may be valid in the first idealisation of the previous paragraph, in which shrinkage produces only local changes in porosity (and
hence only, in turn, to changes in the permeability tensor Kβ ). But when bulk shrinkage causes noticeable deformation of the product, and noticeable motion of the σ
phase, it is reasonable to expect that the other two phases will be dragged along by
this motion. In the present work, the following intuitive, nonrigorous argument is
used to modify (B.4.14) into a form acceptable for the second idealisation:
1. The effect of mass exchange hṁσβ i is neglected, since the associated mass flux
is expected in general to be much lower than the bulk mass flux in the β phase.
2. The effect of shrinkage on the boundary condition vβ = 0 on A βσ is broken into
two parts. It is assumed that the first, associated with local shrinkage of the σ
phase, can be neglected. The second, associated with bulk deformation of the
porous medium, cannot. Inertial effects are neglected.
3. The σ phase is defined as the region occupied by the s-component. Neglecting
the exchange of moisture hṁσβ i through the interface A βσ , it follows from the
no-slip condition that the velocity of the β phase at the surface A βσ is vs . To
deal with the effect of bulk deformation, the situation is therefore considered
from the point of view of a frame of reference which is being dragged along
with the dry solid, that is, moving at velocity hvs iσ . In this frame of reference,
the boundary condition vβ0 = 0 on A βσ is met within the averaging volume,
and the derivation of (B.4.14), for instance by Whitaker [191], is valid. In particular, substituting (B.1.8) into (B.4.14), we see that the intrinsic phase averaged
velocity hvβ0 i β measured in this reference frame will satisfy
hvβ0 i β = −
Kβ
· ∇h p β i β − hρ β i β g
eβ µ β
(B.4.18)
Transforming back to the stationary frame of reference, the following intrinsic-averaged
velocity will be observed:
hv β i β − hvs i σ = −
Kβ
· ∇h p β i β − hρ β i β g
eβ µ β
(B.4.19)
Combining (B.4.19) with (B.4.17), and following an identical procedure for the γ
phase momentum balance, yields the following equations, which are used in the
Derivation of drying equations.
283
present work to evaluate the intrinsic phase averaged velocities:
Kβ · ∇[h pγ iγ − pc ] − hρ β i β g
eβ µ β
Kγ hvγ i γ − hvs i σ = −
· ∇h pγ iγ
eγ µ γ
hv β i β − hvs i σ = −
(B.4.20)
(B.4.21)
Bulk dry solid σ phase s-component velocity
We follow the analysis of shrinkage outlined by Crapiste et al. [52]. One-dimensional
shrinkage in the thickness direction is assumed. Thus the phase-averaged velocity
hvs i = hvs iê2 of the solid phase s-component is assumed to be purely in the y-
direction (refer to the axes depicted in Figure A.1 on page 213), with zero shrinkage
velocity at the interface between the porous medium and the plate. This assumption
allows us to rewrite (B.3.22) as:
∂
d
hρs ihvs iσ ) = − hρs i
dy
∂t
⇒
hρs ihvs i = −
σ
Zy
0
0 ≤ y ≤ δ(t)
∂hρs i
dy
∂t
(B.4.22)
(B.4.23)
In integrating (B.4.22) to obtain (B.4.23) the integration constant is set equal to zero
because hvs iσ = 0 on the surface y = 0 (which corresponds to the interface between
the heating plate and the product, in the coordinate system depicted on page 213).
We introduce the local shrinkage, s∗ [52]:
s ∗ = h ρ s i0 / h ρ s i
(B.4.24)
Here the subscript ‘0’ means ‘at time t = 0’. Dividing by hρs i, equation (B.4.23) can
be written:
h ρ s i0
hvs i = −
hρs i
σ
= −s
∗
Zy
0
Zy
0
∂
hρs i/hρs i0 dy
∂t
∂
1/s∗ dy
∂t
(B.4.25)
(B.4.26)
Bound moisture migration
Two different methods for predicting bound moisture flux are in widespread use. In
a number of papers [e.g. 18, 49, 65, 86, 92, 124] the flux is assumed to be driven by
a Fickian diffusion process, and (on the further assumption of isotropy) is evaluated
as a simple multiple of the bound water concentration gradient. Transcribed into the
Derivation of drying equations.
local notation, the expression is usually of the form
hρb i
σ
hρb ihvb i = −Db eσ ∇
hρs i
284
(B.4.27)
It is a well-known result of classical thermodynamics that under isothermal conditions the chemical potential µ provides the driving force for diffusion (across semipermeable membranes). Haywood [81, p.311] shows how this result follows from the
fact that ∑ µi dmi = 0 at equilibrium, and that p increases with µ at constant temperi
ature. Stanish et al. [151], and several more recent authors including Lu et al. [106]
have proceeded from the assumption that under drying conditions the bound moisture driving force can be evaluated from the gradient of the chemical potential of the
bound water. Expressed in terms of the volume-averaging which lurks implicitly in
the work of [151], the relationship that they obtain can be written as follows:
hs̄v iγ
1
σ
γ
hρb ihvb i = −Db eσ −
∇h pv i
(B.4.28)
∇h T i +
Mw
hρv iγ
where the bound water diffusivity Db is considered to be an unknown, which must
be matched to empirical data. The molar entropy term s̄v which appears in (B.4.28)
may be estimated as a function of the volume-averaged temperature and pressure:
hT i
h pv iγ
R
γ
0
hs̄v i ≈ s̄v + cp,v ln
ln
+
(B.4.29)
Mw
T0
p0
where the constants are listed by Stanish et al. [151] as follows: s̄0v = 187 JK−1 , cp,v =
35.1 JK−1 , T 0 = 298.15 K, p0 = 101325 Pa, Db = 1.8 × 10−14 kg s−1 m−3 .
Since neither (B.4.27) nor (B.4.28) have been derived by the volume-averaging approach from more basic (and generally accepted) postulates, it is somewhat ironic to
use either of these equations within the otherwise rigorous volume-averaging framework. However, in applying Whitaker’s formalism to any real modelling application,
it is inevitable that one will run into limits in how far the analytical approach can go.
As a matter of fact, the use of quasi-theoretical, quasi-empirical equations like (B.4.27)
and (B.4.28) appears, at least in the case of bound moisture migration, to have been
accepted by the wider drying research community. A significant body of empirical
data provides justification for using equation (B.4.28): see [106], [150], and [151]; each
of these papers reports a reasonable agreement between measured mass fluxes and
fluxes predicted by (B.4.28). We adopt Stanish’s flux expression (B.4.28) in the present
work. Note that (B.4.28) is obtained from the assumption of a stationary dry solid. As
in the cases of the γ and β phases, solid motion must be taken into account. In order to recast the equation into a form applicable to a deformable biological material,
we rearrange (B.4.28) to obtain the intrinsic phase average velocity, from which we
subtract the intrinsic phase averaged s-component velocity:
Db e σ
hs̄v iγ
1
σ
σ
γ
hvb i − hvs i = −
−
∇h T i +
∇h pv i
hρb i
Mw
hρv iγ
(B.4.30)
Derivation of drying equations.
B.5.
285
Energy balance
The local energy equations for a multicomponent system, which are to be integrated
over each of the phases contained within the averaging volume, are derived in the
appendix as Equation (A.3.57). The appropriate versions for the three phases are
stated here for convenience. Neglecting the kinetic energy of diffusion, gravitational
potential energy and the work done against viscous stress forces, the local energy
balance equation (A.3.57) within the γ phase can be written as follows:
∂
(ργ hγ ) + ∇ · ∑ ρi vi hi = −∇ · qγ + Φγ
∂t
i = a,v
(B.5.1)
The local energy balance within the single-component β phase is as follows:
∂
(ρ h ) + ∇ · (ρ β vβ h β ) = −∇ · qβ + Φ β
∂t β β
(B.5.2)
The local energy balance within the σ phase is:
∂
(ρσ hσ ) + ∇ · ∑ ρk vk hk = −∇ · qσ + Φσ
∂t
k =s,b
(B.5.3)
Equations (B.5.1)–(B.5.3) include local heat source terms Φγ , Φ β , Φσ which represent
radiative heating of the medium, for instance by the absorption of microwaves.
Interfacial energy jump conditions are required in order to use the volume averaging approach to obtain the energy equations for the porous medium. Some of the
jump conditions obtained by Whitaker [188] carry over to the hygroscopic case, while
others need to be modified to account for the bound moisture and σ-phase motion.
Internal energy jump conditions
The energy jump condition for the β-γ interface is unaffected by the presence of
bound water in the σ phase. Thus, on A βγ [188, II.B-21]:
h
qβ + ρ β h β (vβ − wγβ ) − qγ − ργ hγ (vγ − wγβ ) −
∑
i = a,v
i
ρi ui hi · n̂βγ = 0
(B.5.4)
The convection and diffusion enthalpy fluxes can be combined in (B.5.4) using Equation (A.3.61), resulting in the following equation:
h
qβ + ρ β h β (vβ − wγβ ) − qγ −
∑
i = a,v
i
ρi hi (vi − wγβ ) · n̂βγ = 0
(B.5.5)
Derivation of drying equations.
286
The jump conditions at the σ-γ and σ-β interfaces are affected by the presence of
bound water in the σ phase. These are analogous to (B.5.5) and are [69]:
h
qγ +
∑
i = a,v
h
ρi hi (vi − wσγ ) − qσ −
qβ + ρ β h β (vβ − wσβ ) − qσ −
∑
(B.5.6)
k =s,b
i
ρk hk (vk − wσγ ) · n̂γσ = 0
∑
i
ρk hk (vk − wσβ ) · n̂βσ = 0
(B.5.7)
k =s,b
on Aγσ and A βσ respectively. Since on these interfaces we have n̂γσ · wσγ = n̂γσ · vs
and n̂βσ · wβσ = n̂βσ · vs , the terms involving k = s drop out of these equations.
Deriving the conservative form of the energy equation
The overall volume-averaged energy equation for a deformable hygroscopic material can be derived by integrating the energy balance for each phase over that phase
within the averaging volume, and adding together the resulting expressions. The
resulting expression is then transformed into a useable form by the use of several
assumptions which are adopted from the work of Whitaker. At each stage, the assumptions and their counterparts in Whitaker [188] and [193] are highlighted.
Integrating the energy equation (B.5.3) for the σ phase over Vσ (t), and dividing by
V, yields the following equation:
1
V
Z
Vσ
∂
(ρσ hσ )dV + h∇ · (qσ + ∑ ρk vk hk )i = hΦσ i
∂t
k =s,b
(B.5.8)
The first term of (B.5.8) can be evaluated by the use of the generalized transport
theorem (A.2.35) as follows:
1
V
Z
Vσ (t)
∂
∂
(ρσ hσ )dV = hρσ hσ i −
∂t
∂t
1
−
V
Z
Aσγ (t)
Z
Aσβ (t)
(ρσ hσ wσβ ) · n̂σβ dA
(ρσ hσ wσγ ) · n̂σγ dA
(B.5.9)
The gradient-averaging theorem, Equation (B.1.16), can be applied to the second
term of (B.5.8), to obtain:
h∇ · (qσ +
∑
k =s,b
ρk vk hk )i =∇ · hqσ +
+
1
V
Z
Aσγ
∑
k =s,b
(qσ +
ρ k vk h k i +
∑
k =s,b
1
V
Z
(qσ +
Aσβ
ρk vk hk ) · n̂σγ dA
∑
k =s,b
ρk vk hk ) · n̂σβ dA
(B.5.10)
Derivation of drying equations.
287
Substituting (B.5.9) and (B.5.10) into (B.5.8) yields the σ phase energy balance:
∂
1
hρσ hσ i+∇ · hqσ + ∑ ρk vk hk i +
∂t
V
k =s,b
+
Z h
1
V
qσ +
∑
k =s,b
Aσγ
Z h
qσ +
∑
k =s,b
Aσβ
i
ρk hk (vk − wσβ ) · n̂σβ dA
i
ρk hk (vk − wσγ ) · n̂σγ dA = hΦσ i
(B.5.11)
Integrating the β-phase energy equation (B.5.2) over Vβ (t), and dividing by V,
yields the following equation:
1
V
Z
Vβ (t)
∂
(ρ h )dV + h∇ · (qβ + ρ β vβ h β )i = hΦ β i
∂t β β
(B.5.12)
Applying the generalized transport theorem (A.2.35) to the first term of (B.5.12), and
the theorem for the average of a gradient (B.1.16) to the second term, produces the
following β phase energy balance:
1
∂
hρ h i+∇ · hqβ + ρ β vβ h β i +
∂t β β
V
+
Z 1
V
A βγ
Z A βσ
qβ + ρ β h β (vβ − wβσ ) · n̂βσ dA
qβ + ρ β h β (vβ − wβγ ) · n̂βγ dA = hΦ β i
(B.5.13)
Integrating the energy equation (B.5.1) for the γ phase over Vγ , and dividing by V,
yields the following equation:
1
V
Z
Vγ (t)
∂
(ργ hγ )dV + h∇ · (qγ + ∑ ρi vi hi )i = hΦγ i
∂t
i = a,v
(B.5.14)
Applying the generalized transport theorem (A.2.35) to the first term of (B.5.14), and
the theorem for the average of a gradient (B.1.16) to the second term, produces the
following γ phase energy balance:
1
∂
hργ hγ i+∇ · hqγ + ∑ ρi vi hi i +
∂t
V
i = a,v
+
Z 1
V
Aγβ
qγ +
∑
i = a,v
Z
Aγσ
[qγ +
∑
i = a,v
ρi hi (vi − wγσ )] · n̂γσ dA
ρi hi (vi − wγβ ) · n̂γβ dA = hΦγ i
(B.5.15)
The overall volume averaged energy equation can be obtained by summing equa-
Derivation of drying equations.
288
tions (B.5.11), (B.5.13) and (B.5.15):
∂
∂
∂
hρσ hσ i + hρ β h β i + hργ hγ i
∂t
∂t
∂t
+∇·h
∑
k =s,b
ρ k vk h k i + ∇ · h ρ β v β h β i + ∇ · h
∑
i = a,v
ρ i vi h i i
+ ∇ · hqσ i + ∇ · hq β i + ∇ · hqγ i
Z h
i
1
+
qβ + ρ β h β (vβ − wβγ ) − qγ − ∑ ρi hi (vi − wβγ ) · n̂βγ dA
V
i = a,v
A βγ
+
+
1
V
Z h
Aσβ
qσ +
ρk hk (vk − wσγ ) − qγ −
∑
i
ρk hk (vk − wσβ ) − qβ − ρ β h β (vβ − wσβ ) · n̂σβ dA
k =s,b
∑
i
ρi hi (vi − wγσ ) · n̂σγ dA
∑
k =s,b
Aσγ
Z h
1
V
qσ +
i = a,v
= hΦσ i + hΦ β i + hΦγ i
(B.5.16)
Inspecting the final three terms on the left of (B.5.16), we see that each of these
equals zero, by the jump conditions (B.5.5), (B.5.6) and (B.5.7) respectively. We can
therefore write the overall energy balance equation as follows:
∂
hρσ hσ i + hρ β h β i + hργ hγ i
∂t
+∇· h
∑
k =s,b
ρ k vk h k i + h ρ β v β h β i + h
+ ∇ · hqσ i + hq β i + hqγ i
∑
i = a,v
ρ i vi h i i
= hΦσ i + hΦ β i + hΦγ i
(B.5.17)
Referring to (B.1.18), the physical interpretation of equation (B.5.17) is clear: the total
energy content of the averaging volume changes at a rate equal to the total inward
flow of energy through its bounding surface, plus the rate at which energy is supplied
internally through radiative heating. Our use of the energy jump conditions to pass
from (B.5.16) to (B.5.17) accords with the intuitive notion that the energy lost from the
β phase through its interface with the γ phase (within the averaging volume) should
equal the energy gained by the γ phase through that same interface, and similarly for
the other two interfaces. It should be emphasized that up to this point in the derivation of the energy equation, no use of scale arguments has been made. Equation
(B.5.17) holds for all volumes within the medium, regardless of their size. However
the terms appearing in (B.5.17) are the averages of products, and in order to transform
these terms into more useful products of averages, several assumptions and restrictions must be modified from Whitaker [188]. We proceed by expressing the mass flux
ρi vi , the enthalpy hi , the density ρi and the velocity vi of gas species i in terms of their
Derivation of drying equations.
289
deviations from the intrinsic phase averages, with the following definitions:
γ
ρg
i vi = ρ i vi − h ρ i vi i
in the γ phase
(B.5.18a)
h̃i = hi − hhi iγ
in the γ phase
(B.5.18b)
ρ̃i = ρi − hρi i
γ
in the γ phase
(B.5.18c)
vei = vi − hvi iγ
in the γ phase
(B.5.18d)
where all of these quantites are 0 by definition outside the γ phase. The following
equations, which are analogous to (II.C-78) and (II.C-49) of Whitaker [188], can be
derived by the method of Gray [74], from the definitions (B.5.18):
hργ hγ i = hργ ihhγ iγ + hρ̃γ h̃γ i
(B.5.19a)
hρi vi hi i = hρi vi ihhi iγ + hρg
i vi h̃i i
(B.5.19b)
Similarly, for the β phase:
hρ β h β i = hρ β ihh β i β + hρ̃ β h̃ β i
(B.5.20a)
hρ β vβ h β i = hρ β vβ ihh β i + h]
ρ β vβ h̃ β i
β
(B.5.20b)
Similarly, for the σ phase:
hρσ hσ i = hρσ ihhσ iγ + hρ̃σ h̃σ i
(B.5.21a)
hρk vk hk i = hρk vk ihhk i + hρg
k vk h̃k i
γ
(B.5.21b)
Substituting (B.5.19), (B.5.20) and (B.5.21) into (B.5.17) yields:
∂
j
σ
β
γ
h
ρ
ih
h
i
+
∇
·
h
ρ
v
ih
h
i
+
h
ρ
v
ih
h
i
+
h
ρ
v
ih
h
i
j
j
β β
β
∑ k k k
∑ i i i
∂t j=∑
σ,β,γ
i = a,v
k=s,b
= −∇ · hqσ i + hqβ i + hqγ i − ∇ · hξ i + hΦi
(B.5.22)
where the combined liquid, gas and solid dispersion is represented, by analogy with
(III.A-4) of [188], in terms of ∇ · hξ i where
∇ · hξ i = ∇ · h
∑
k =s,b
h]
ρk uk h̃k i + hρ]
β u β h̃ β i +
i = a,v
∑
hρg
i ui h̃i i
∑
hρ̃i h̃i i
(B.5.23)
and where
hΦi = hΦσ i + hΦ β i + hΦγ i +
∂
∂t
i =σ,γ,β
(B.5.24)
In the present modelling work, internal radiative (e.g. microwave) heating of the drying product is not considered. Thus the terms hΦσ i, hΦ β i and hΦγ i can be set equal to
zero (these terms represent just the local heat source due to radiative absorption – see
p.285). Further, the final source term appearing in (B.5.24) is the time-derivative of a
Derivation of drying equations.
290
product of deviations and can in general be neglected [188, p.151]. We can therefore
drop the final term of (B.5.22), resulting in the following expression:
∂
j
σ
β
γ
hρ j ihh j i + ∇ · ∑ hρk vk ihhk i + hρ β vβ ihh β i + ∑ hρi vi ihhi i
∂t j=∑
σ,β,γ
i = a,v
k=s,b
= −∇ · hqσ i + hqβ i + hqγ i − ∇ · hξ i
(B.5.25)
Turning to the volume-averaged heat flux terms hqσ i, hqβ i and hqγ i appearing on
the right hand side of (B.5.25), one can invoke Fourier’s law (A.4.24), neglect variation in the thermal conductivities of the three phases, and use the averaging theorem
(B.1.16) to write
Z
h
i
1 Z
1
hqσ i = −k σ ∇ eσ h Tσ iσ +
Tσ n̂σβ dA +
Tσ n̂σγ dA
V
V
(B.5.26a)
Z
h
i
1 Z
1
hqβ i = −k β ∇ eβ h Tβ i β +
Tβ n̂βσ dA +
Tβ n̂βγ dA
V
V
(B.5.26b)
Z
i
h
1 Z
1
Tγ n̂γσ dA
Tγ n̂γβ dA +
hqγ i = −k γ ∇ eγ h Tγ iγ +
V
V
(B.5.26c)
Aσβ
A βσ
Aγβ
Aσγ
A βγ
Aγσ
Equation (B.5.26a), for instance, is identical to (II.C-18) of [188]. Note that variations
in thermal conductivity within each phase have been assumed to take place over
length scales significantly greater than the averaging volume, allowing the thermal
conductivity to be pulled out of the volume averaging integral. The assumption of
local thermal equilibrium results in equation (III-3) of Whitaker [188]:
h Tσ iσ = h Tβ i β = h Tγ iγ = h T i
(B.5.27)
Using (B.5.27) to replace the intrinsic temperature averages in (B.5.26), we can
therefore write the right hand side of (B.5.25) as follows:
− ∇ · hqσ i + hq β i + hqγ i − ∇ · hξ i =
Z
1
= ∇ · ∇ (k σ eσ + k β eβ + k γ eγ )h T i + (k σ − k β )
Tσ n̂σβ dA
V
Aσβ
+ (k β − k γ )
− ∇ · hξ i
1
V
Z
A βγ
Tβ n̂βγ dA + (k γ − k σ )
1
V
Z
Tγ n̂βγ dA
Aγσ
(B.5.28)
In [188] Whitaker presents an extensive argument, running from equation (III.A-17)
through to equation (III.B-18), that the terms whose divergence appear on the right
side of equation (B.5.28) can be expressed as the product of an effective thermal conductivity tensor KTeff and the temperature gradient ∇h T i. Tracing this derivation, we
Derivation of drying equations.
291
discover that we can rewrite (B.5.28) as follows:
−∇ · hqσ i + hqβ i + hqγ i − ∇ · hξ i = ∇ · KTeff · ∇h T i
(B.5.29)
where KTeff is defined by equation (III.B-18) of Whitaker [188]:
KTeff = Keff + KD
(B.5.30)
The tensor KD in (B.5.30) is a dispersion coefficient [185] such that hξ i = KD · ∇h T i.
Substituting (B.5.29) into (B.5.22), and applying the definition hnk i ≡ hρk vk i, we
obtain the energy balance equation in its final conservative form:
∂
j
σ
β
γ
h
ρ
ih
h
i
+
∇
·
h
n
ih
h
i
+
h
n
ih
h
i
+
h
n
ih
h
i
j
j
β
β
∑ k k
∑ i i
∂t j=∑
σ,β,γ
i = a,v
k =s,b
(B.5.31)
= ∇ · KTeff · ∇h T i
Equation (B.5.31) can be compared with equations (B.5.1)–(B.5.3), to which it is analogous in form. Referring to his energy balance equation, Whitaker [188, p.161] notes,
This form of the total thermal energy equation is quite appealing for all of
our difficulties have been incorporated into a single second order tensor
KTeff which must be determined experimentally or on the basis of further
theoretical developments.
It should be stressed that unlike to Equation (B.5.31), Whitaker’s equation (III.B-17),
to which the above quote refers, explicitly tracks the rate of change of the volumeaveraged temperature – that is, it is analogous to Equation (B.5.49) below. The above
quote nevertheless applies equally to Equation (B.5.31).
Equation (B.5.31) can be compared with equation (3) of Stanish et al. [151], which
was applied to a hygroscopic medium (pinewood) whose dry-solid component was
assumed rigid. For convenience, that equation is reproduced here:
∂
ρσ hσ + ρ β h β + ργ hγ = −∇ · nb hb + nβ h β + ∑ ni hi − k ∇ T
∂t
i = a,v
(B.5.32)
Volume-averaged terms do not appear in (B.5.32) since volume averaging is not explicitly used in [151]. However the terms in (B.5.31) do correspond directly with those
in (B.5.32), since the density (ρ) terms appearing in (B.5.32) are defined as mass per
unit volume of the porous medium, and the mass flux (n) terms in (B.5.32) are defined as the normal mass flow-rate per unit area of the medium. Equation (3) of [151]
is presented as a generalisation of equation (31) of Harmathy [77], which is stated as
a heuristically reasonable consequence of Assumption 2 of [77]: that
the phases of the porous system are so finely distributed that from the
macroscopic standpoint the system is a quasi-one-phase system. (It is not
Derivation of drying equations.
292
implied that the phases are distributed into macroscopically small units.
In fact, each phase may form a macroscopically large, continuous network.)
The present section’s derivation of Equation (B.5.31) using results borrowed from the
volume-averaging approach of Whitaker can be understood as a formal version of
the route that was intuitively taken from Assumption 2 of Harmathy [77] to the energy balances (31) of Harmathy [77] and (3) of Stanish et al. [151]. Similarly, since the
energy balance equations employed by Turner [172] and subsequent workers, [e.g.
106, 116, 125], are stated in a form equivalent to (B.5.31), the above derivation corresponds to a formal version of Turner’s intuition that an equation of the same general
form as the single-phase energy balance equations (B.5.1)–(B.5.3) can be applied at
the aggregate porous-medium level. Of course, since the tensor KTeff contains “all of
our difficulties”, this correspondence must be taken with a grain of salt.
It is interesting, as it was for the vapour transport equation, to compare the standard expression for the thermal energy flux, as presented above in Equation (B.5.29),
with that obtained by Whitaker [193] in a more recent investigation into the drying of granular porous media. Employing a diffusion model for thermal dispersion,
Whitaker again expresses the thermal dispersion appearing as the term ∇ · hξ i of
equation (B.5.22) as the product of a tensor KD with the temperature gradient ∇h T i.
The term −∇ · hqσc i + hqβc i + hqγc i that appears in (B.5.28) is equivalent to the right
hand side of Equation (550) of Whitaker [193]. Evaluating the coupling between heat
and mass transfer, Whitaker [193] ultimately expresses this in terms of a passive con∗ and a multiphase Dufour effect [193,
ductivity K0eff , an enhanced conductivity Keff
p.91]. In this case (B.5.28) becomes (for a nonhygroscopic porous medium):
− hqσc i + hqβc i + hqγc i − ∇ · hξ i = KTeff · ∇h T i + K A · ∇hρv iγ
(B.5.33)
|
{z
}
| {z }
effective conduction
multiphase Dufour
where KTeff is defined by Equation (555) of [193]:
∗
KTeff = K0eff + Keff
(B.5.34)
∗
Here K0eff is the passive thermal conductivity defined by Equation (B.5.30), and Keff
represents a thermal conductivity enhancement that is analogous to the enhanced dif∗(v)
fusivity tensor DD . Again, an order-of-magnitude analysis conducted by Whitaker
[193] indicates that the enhanced conduction and multiphase Dufour effects may contribute significantly to overall heat transfer, as may thermal dispersion when convective heat transfer is significant. None of the numerical drying simulation papers that
have been encountered have acknowledged the existence of these terms, though several of them appeared after 1998. Whitaker’s analysis provides no additional basis
for evaluating these terms, and again, for lack of a better option, we neglect them.
Derivation of drying equations.
293
Converting the energy equation to temperature form
This section proceeds from (B.5.31) to the energy balance obtained by Gibson [69]. In
general, the integral heat of wetting, ∆hb = h β − hb , will be a function of moisture
content xm and of temperature T. This function can be obtained by integrating the
differential heat of wetting, ∆h0b , and is expressed by (B.2.13), derived in the appendix.
By assuming that the partial mass enthalpy of bound moisture in the σ phase can be
written c p,b T + const, Gibson [69] has implicitly assumed that the integral heat of
wetting ∆hb can be represented by a linear function of temperature only:
∆hb = ∆h0b − a( T − T0 )
(B.5.35)
In (B.5.35), ∆h0b is the heat of wetting at the reference temperature T0 , and must be
independent of moisture content xm for Gibson’s assumed form for hb to hold. In
this case the differential and integral wetting heats become identical, since the heat of
wetting is independent of bound moisture content: ∆hb = ∆h0b . From (B.5.35) it also
follows that
hb = h β − ∆hb = c p,β ( T − T0 ) − ∆hb
= c p,b ( T −
T0 ) − ∆h0b
(B.5.36)
(B.5.37)
where c p,b = c p,β + a; see Equations (2.3.15)–(2.3.17) of Gibson [69]. To show that
(B.5.35) must hold, given the assumption employed in [69], one can use the definition of ∆hb to move backward through the above derivation from (B.5.37) to (B.5.35).
This section will use (B.5.35) and (B.5.37), which will allow easy comparison with the
results of Gibson [69], but by subsequently redefining the terms employed we will
incorporate the dependence of the bound water enthalpy on both T and xm .
At the reference temperature T0 , the enthalpy of liquid water is zero by definition.
Considering a process in which water is evaporated at the reference temperature and
then heated to the temperature T, neglecting variations in the specific heat capacity
of the vapour, the equation for the enthalpy of water vapour is analogous to (B.5.37):
hv = c p,v ( T − T0 ) + ∆h0v
(B.5.38)
Using (B.5.37) and (B.5.38), the latent heat and the differential heat of wetting at the
local volume averaged temperature h T i can be written:
∆hv = ∆h0v + h T i − T0
∆h0b = ∆hb00 + h T i − T0
c p,v − c p,β
(B.5.39a)
(B.5.39b)
c p,β − c p,b
Derivation of drying equations.
294
We can expand the terms in the energy balance (B.5.31) as follows:




hna ic p,a h T − T0 i
hρ a ic p,a h T − T0 i


! 
! 




c p,v h T − T0 i
c p,v h T − T0 i




+h
nv i



 +hρv i
0
0




+∆hv
+∆hv




∂ 




 + ∇ ·  +hnβ ic p,β h T − T0 i
 +hρ β ic p,β h T − T0 i

∂t 
! 
! 




c p,b h T − T0 i
c p,b h T − T0 i





 +hnb i

 +hρb i
0
0
0




−∆hb
−∆hb




+hns ic p,s h T − T0 i
+hρs ic p,s h T − T0 i
= ∇ · KTeff · ∇h T − T0 i
(B.5.40)
In (B.5.40) the volume-averaged enthalpies are evaluated in terms of the specific
heat capacities c p,i and the relative temperature T − T0 within each phase, averaged
over that phase. We assume that the c p,i are effectively constant over the relevant
range of conditions. The assumption of local thermal equilibrium [193, p.54] has allowed us to equate the intrinsic-volume-averaged temperatures in the three phases.
In obtaining (B.5.40) from (B.5.31) we have used the following approximations for the
moist-solid enthalpy: hρσ ihhσ iσ = hρσ ih
ρs hs +ρb hb σ
i
ρσ
≈ hρσ ihρs hs + ρb hb iσ /hρσ iσ ≈
hρs ihhs iσ + hρb ihhb iσ , where hb can be evaluated using (B.2.13a) and (B.2.13b). The
first approximation can be justified by an argument analogous to that made by Whitaker
ρ
[193] to justify Equation (187) of that reference, h ργv iγ =
hρv iγ
.
hργ iγ
The second approxima-
tion is equivalent to our dropping the term Φ appearing in (B.5.22). The ∆h0b appearing under the time derivative in Equation (B.5.40) is the integral heat of wetting at the
local temperature and moisture content, while the ∆h0b0 appearing under the diver-
gence operator is the differential heat of wetting, which arises from the partial mass
enthalpy of the bound moisture component that appears under the divergence operator in the second term of Equation (B.5.31). Under the assumptions employed by
Gibson [69], these two heats are equal. Analogous approximations have been used to
evaluate the enthalpy of the moist air. According to Equations (B.5.37) and (B.5.38),
at the reference temperature, the vapour and bound moisture enthalpies equal the
reference heats of vaporization and wetting.
Multiplying (B.4.2) by ∆h0v and (B.4.4) by ∆h0b0 yields, after rearrangement:
∂
hρv i∆h0v + ∇ · hnv i∆h0v = hṁ βγ i + hṁσγ i ∆h0v
∂t
∂
hρb i∆h0b0 + ∇ · hnb i∆h0b0 = − hṁσγ i + hṁσβ i ∆h0b0
∂t
(B.5.41)
(B.5.42)
Derivation of drying equations.
295
Substituting equations (B.5.41) and (B.5.42) into (B.5.40) yields:
∂
hρTot iC p h T i − T0 + ∇ · ∑hni ic p,i h T i − T0
∂t
i
00
+ hṁσγ i + hṁσβ i ∆hb + hṁσγ i + hṁ βγ i ∆h0v
= ∇ · KTeff · ∇ h T i − T0
(B.5.43)
In (B.5.43) the following definitions have been used:
hρTot i = ∑hρi i
(B.5.44)
i
Cp =
1
hρi ic p,i
hρTot i ∑
i
(B.5.45)
Equations (B.5.39) allow us to rewrite (B.5.43) as follows:
"
#
∂
hρTot iC p h T i − T0 + ∇ · ∑hni ic p,i h T i − T0
∂t
i
+ h T i − T0 c p,b − c p,β hṁσγ i + hṁσβ i + ∆h0b hṁσγ i + hṁσβ i
+ h T i − T0 c p,β − c p,v hṁσγ i + hṁ βγ i + ∆hv hṁσγ i + hṁ βγ i
= ∇ · KTeff · ∇ h T i − T0
(B.5.46)
i.e., expanding the first divergence term and rearranging slightly,
∂
hρTot iC p h T i − T0
∂t
i
h
+∇ h T i − T0 · ∑hni ic p,i + h T i − T0 ∑ c p,i ∇ · hni i
i
i
h
+ h T i − T0
c p,b − c p,β hṁσγ i + hṁσβ i + c p,β − c p,v
+∆h0b hṁσγ i + hṁσβ i + ∆hv hṁσγ i + hṁ βγ i
= ∇ · KTeff · ∇ h T i − T0
hṁσγ i + hṁ βγ i
i
(B.5.47)
By multiplying equations (B.4.1)–(B.4.5) each by the appropriate specific heat capacity, summing the equations that result, and then multiplying through by −(h T i −
T0 ), we can evaluate the terms contained in the 2nd to 4th lines of (B.5.47), obtaining
h T i − T0 ∑ c p,i ∇ · hni i
i
+ h T i − T0
h
c p,b − c p,β
= − h T i − T0
i
hṁσγ i + hṁσβ i + c p,β − c p,v hṁσγ i + hṁ βγ i
∂
hρTot iC p
∂t
(B.5.48)
Using (B.5.48) in (B.5.47) and dropping derivatives of the constant T0 , (B.5.47) be-
Derivation of drying equations.
296
comes:
∂h T i
+ ∇h T i · ∑hni ic p,i
∂t
i
0
0
+hṁσβ i∆hb + hṁσγ i ∆hb + ∆hv + hṁ βγ i∆hv
= ∇ · KTeff · ∇hTi
hρTot iC p
(B.5.49)
This is the energy balance derived in reference [69], by a route that much more
closely resembled Whitaker’s original derivation [188]. In the absence of hygroscopicity and solid motion, (B.5.49) reduces to (III.B-17) of Whitaker [188], since hṁσβ i = 0,
hnb i = hns i = 0 and ∆hb = 0. Equation (B.5.49) can be written in the following form:
∂h T i
+ ∇h T i · ∑hni ic p,i
∂t
i
0
+ hṁσγ i + hṁσβ i ∆hb + ∆hv + hṁ βγ i − hṁσβ i ∆hv
= ∇ · KTeff · ∇hTi
hρTot iC p
(B.5.50)
Here hṁσγ i + hṁσβ i represents the volumetric rate of transfer of bound water from
the solid phase into the other two phases within the averaging volume. Similarly
hṁ βγ i − hṁσβ i represents the volumetric rate of transfer of free liquid water from the
β phase into the other two phases within the averaging volume.
We have seen (on p.269) that the source term hṁσβ i for moisture mass exchange
between the solid and liquid phase is likely to be much smaller than the other two
mass exchange terms. Dropping hṁσβ i from (B.5.49) or from (B.7.15) yields
∂h T i
+ ∇h T i ·
∂t
= ∇ · KTeff · ∇hTi
hρTot iC p
∑hni ic p,i
i
+ hṁσγ i ∆h0b + ∆hv + hṁ βγ i∆hv
(B.5.51)
Intuitively, (B.5.51) can be understood as follows. Local sensible cooling occurs as a
result of three processes: (1) migration of species up a thermal gradient, since in coming to equilibrium with its new environment, the migrant mass must absorb heat. (2)
evaporation (hṁσγ i below the saturation point, hṁ βγ i above). (3) net heat conduction
away from the location. These physical processes correspond to the terms of (B.5.51).
The derivation summarised in this section has shown that the energy balance
(B.5.31) is equivalent to (B.5.49), or in other words to the energy balance obtained
by Gibson [69], only in the case where Gibson’s assumption expressed by (B.5.35)
holds to a good approximation. The assumptions of constant c p,b = c pβ + a and
constant ∆h0b are crucial to the above derivation, in passing from (B.5.40) to (B.5.43).
Since, in contrast, the derivation of (B.5.31) did not depend upon the assumption of
Equation (B.5.35), Equation (B.5.31) is of greater generality than (B.5.51), and Equa-
Derivation of drying equations.
297
tion (B.5.31) has been used in the detailed model which is outlined in chapter 5. However, the work to be described in chapter 5 leads to a strong argument for adopting
an energy balance of a form similar to Equation (B.5.51). Here we will describe here
how a more general form can be derived in the case where the assumptions described
above do not hold.
The dependence of the bound water enthalpy on both T and xm can be accounted
for by allowing ∆hb to vary as an arbitrary function of these variables and (without
loss of generality) setting a = 0 in (B.5.35). Rather than going through the derivation
again, we will simply describe the modifications that need to be made to the above
derivation. Thus in the general case, which encompasses that examined by Gibson
[69] as a special case, we replace ∆h0b by the local heat of wetting ∆hb , a function of T
and xm , everywhere it appears in the above derivation, and we replace c p,b with c p,β .
The resulting derivation is otherwise identical to the above, except that the terms
b
∇ ∆h0b · hnb i + ∂h∂tρb i ∆hb − ∆h0b + hρb i ∂∆h
∂t now appear on the right-hand sides of
∂hρ i
(B.5.42), (B.5.43), and (B.5.46)–(B.5.51). The term ∂tb ∆hb − ∆h0b is second-order,
since ∆hb = O(∆h0b ). Meanwhile ∆hb ∆hv except at low moisture contents [93,
∂hρ i
p.34], so the term ∂tb ∆h0b − ∆hb can be expected to generally be much smaller
than the term hṁσγ i ∆h0b + ∆hv . Dropping the former, we obtain:
∂h T i
+ ∇h T i · ∑hni ic p,i + hṁσγ i ∆h0b + ∆hv + hṁ βγ i∆hv
∂t
i
∂∆hb
= ∇ · KTeff · ∇hTi + ∇ ∆h0b · hnb i + hρb i
(B.5.52)
∂t
hρTot iC p
In the case where the s-species is stationary, Equation (B.5.52) becomes equivalent
to the energy equation that is stated without derivation by Couture et al. [50] for a
b
rigid hygroscopic medium, except for the final term hρb i ∂∆h
∂t , which does not ap-
pear in Couture’s equation. One can argue that the effect of this term is likely to be
small, since the heat of wetting will vary rapidly only in the hygroscopic region as the
bound moisture density hρb i becomes small. Also, as the work presented in chapter 5
shows, the very large amount of heat consumed by evaporation leads to much greater
heat transfer by conduction than is carried convectively. This allows us to drop the
convection and correction terms, and the energy balance that has been used in the
product model incorporated into the HPD model in chapter 5 is:
hρTot iC p
∂h T i
+ hṁσγ i ∆h0b + ∆hv + hṁ βγ i∆hv = ∇ · KTeff · ∇hTi
∂t
(B.5.53)
The key results of the present section can be summarized as follows:
• Whitaker’s volume-averaging method has been used to derive the equations
stated without derivation by Harmathy [77] and Turner [172].
Derivation of drying equations.
298
• This derivation has required a number of terms to be dropped without satisfactory theoretical justification.
• The resulting energy balance has been shown to be equivalent to that of Gibson
[69] if several other (demonstrably small) terms are also neglected.
B.6.
Constitutive equations
Closure of the governing equations requires expressions for the following quantities:
β
• the capillary pressure pc ≡ h pγ iγ − p β ;
• the permeability of the porous medium to the liquid phase, Kβ ;
• the gas-phase transport coefficients (the effective diffusivity, Deff , and the permeability of the porous medium to the gaseous phase, Kγ );
• the effective thermal conductivity, KTeff ;
Expressions adopted to evaluate the above properties are summarised below.
Capillary pressure
Closed-form equations that were originally developed by Van Genuchten [174] for
predicting capillary pressures and permeabilities in unsaturated soils have been used
in a wide variety of settings with good results. For instance Celia [41] has successfully employed Van Genuchten’s model to the simulation of aquifer flow; a fullmorphology simulation conducted by Schulz et al. [142] has demonstrated a good fit
between detailed CFD simulations of multiphase fuel-cell systems and Van Genuchten’s
model. These equations have recently been adopted for modelling moisture transport
in porous foodstuffs – see references [137], [180] and [181].
Different definitions for the saturation have been used by different people. In one,
the saturation is the local proportion of void space occupied by free liquid:
s≡
eβ
eβ
=
e
e β + eγ
(B.6.1)
Inverting equation (3) presented by Van Genuchten [174] leads to the following equation for the capillary pressure pc :
pc =
1/n
1 n/(1−n)
s
−1
α
(B.6.2)
Derivation of drying equations.
299
In (B.6.2) α is a constant whose dimensions here are Pa−1 (the units used in [174] are
different), and n is dimensionless. For various soils to which these parameters have
been fit by Van Genuchten [174], α−1 ranges between 4.87 × 103 Pa and 64.06 × 103 Pa.
In the case of tea leaves, Weerts et al. [181] has fit values of α−1 between 4.426 × 105 Pa
and 8.852 × 105 Pa. Meanwhile for soil in [174], n takes values between 1.17 and 10.4;
while for tea leaves in [181] n ranges between 1.68 and 1.82. The capillary pressure
profile resulting from (B.6.2) is shown in Fig. B.6 for the case α = 0.1 kPa−1 , n = 4.
4
Capillary pressure (Pa)
4
x 10
3
α=0.1 kPa−1
n=4.0
2
1
0
0
0.2
0.4
0.6
Saturation
0.8
1
Figure B.6: Van Genuchten capillary pressure profile (B.6.2)
Permeabilities
In the present work, the permeabilities Kβ and Kγ are assumed to be dependent only
on the saturation s, and to be isotropic. The permeabilities can then be written
Kβ (s) = kβ (s)δ = krβ (s)k0β δ
(B.6.3)
Kγ (s) = kγ (s)δ = krγ (s)k0γ δ
(B.6.4)
where δ is the unit isotropic tensor. Following convention, in (B.6.3) and (B.6.4) the
permeability magnitudes k(s) have each been decomposed into two parts: a dimensionless relative permeability kr (s), which is a function of saturation s and ranges between zero and one, and an absolute permeability k0 , with dimensions m2 , which corresponds to the maximum value taken by k(s). Applying the Van Genuchten model
Derivation of drying equations.
300
[174], the relative permeabilities krβ and krγ can be evaluated as follows:
h
1/r i2
krβ (s) = s1/2 1 − 1 − sr
h
1/r i2
krγ (s) = [1 − s]1/2 1 − 1 − [1 − s]r
(B.6.5)
(B.6.6)
where r = n/(n − 1) and n is the same parameter as appears in (B.6.2). The absolute
permeabilities k0β and k0γ remain to be determined. The resulting relative permeability
profiles, in the case where n = 4.0, are shown in Fig. B.7.
Relative permeability kr
1
n=4.0
0.8
0.6
krγ
krβ
0.4
0.2
0
0
0.2
0.4
0.6
Saturation
0.8
1
Figure B.7: Van Genuchten kr profiles (B.6.5) and (B.6.6)
Phase equilibrium calculations
The vapour-air gas mixture is modeled as an ideal gas system, and local thermodynamic equilibrium is assumed, so that phase equilibrium relations may be used to
obtain the local amount of water in each phase. The water activity may be evaluated
from the capillary pressure using Kelvin’s equation as follows [12, p.348]:
!
!
2τ cos θMw
Mw
am = exp −
= exp −
pc
rρ β RT
ρ β RT
(B.6.7)
In (B.6.7), τ is the surface tension of the liquid-gas meniscus, r is the radius of curvature of the meniscus, and θ is the contact angle between the meniscus and the capillary surface; as usual, ρ β is the density of free liquid water, Mw is the molecular
weight of water, and R is the universal gas constant. Since the capillary pressure is
a function only of saturation s, so is the water activity in the presence of free water.
Over the range of paramters reported for the Van Genuchten model (on page 299),
Derivation of drying equations.
301
the moisture activity is essentially 1 across virtually the entire range of saturation
values. In the hygroscopic region (after the removal of all free water) the GAB equation (1.4.8) is used to evaluate the water activity as a function of the moisture content
of the σ phase. We summarise the resulting relations as follows [50, 188]:
h pv iγ = pv,sat h T i
h pv i = pv,sat h T i · am xm
γ
when hρ β i > 0
(B.6.8a)
when hρ β i = 0
(B.6.8b)
hρv iγ = h pv iγ /Rh T i
(B.6.8c)
hρ a iγ = h p a iγ /Rh T i
h pγ i = h pv i + h p a i
γ
γ
(B.6.8d)
γ
(B.6.8e)
hργ iγ = hρv iγ + hρ a iγ
(B.6.8f)
Effective diffusivity
The equations of Whitaker [188, 193] yield different expressions for the effective passive diffusivity of vapour and dry air in the porous product. Following Stanish et al.
[151] and others, we assume an identical passive diffusivity for vapour and air. The
diffusivity is evaluated using the following expression for random-pore diffusivity:
Deff = αγ Deff δ = αγ eγ2 Dva δ
(B.6.9)
where δ is the unit isotropic tensor, where αγ is an attenuation factor which depends
on pore structure, and where Dva may be estimated using Equation (A.8.1).
Thermal conductivity
Assuming isotropy, the thermal conductivity may be written as follows:
T
KTeff = keff
δ
(B.6.10)
Neglecting the transfer of heat due to convection and diffusion within pores, processes which are dealt with separately in this model, two idealisations of pore distribution are useful in considering the effective thermal conductivity of a porous body.
In the first idealisation, the different phases are aligned parallel to the direction of
heat flow, and heat transfer occurs in parallel through all of the different phases. In
our case this would lead to the following effective thermal conductivity [163]:
T
keff
= eσ k σ + e β k β + eγ k γ
(B.6.11)
where ei is the volume fraction, and k i is the local thermal conductivity, of phase i.
Derivation of drying equations.
302
In the second idealisation, the phases are stacked perpendicular to the direction
of heat flow, and the transfer of heat takes place in series through all of the different
phases. In our case, this would lead to the following effective thermal conductivity:
T
keff
=
he
σ
kσ
+
eβ
e γ i −1
+
kβ
kγ
(B.6.12)
Goedeken et al. [71] have found that a composite scheme incorporating the forms
of both (B.6.11) and (B.6.12) adequately describes the thermal conductivity of bread
over a range of moisture contents and temperatures. Goedeken’s scheme is as follows
[71, Equations (6) and (7), corrected for typo]:
"
T
keff
eσ / ( eσ + e β ) e β / ( eσ + e β )
= (1 − eγ )
+
kσ
kβ
# −1
+ eγ k γ
(B.6.13)
Adopting the thermal conductivities of air and water listed in [71], we can evaluate
T from (B.6.13) using the following parameters:
keff
k γ = 0.0241 + 7.3 × 10−5 ( TC )
k β = 0.56 + 0.002( TC ) − 6.8 × 10
(B.6.14)
−6
( TC )
2
k σ = 0.61( xw ) + k s (1 − xw )
(B.6.15)
(B.6.16)
where k s is the thermal conductivity of the bone-dry solid.
B.7.
Conclusions
The governing equations that we have obtained for a shrinkable hygroscopic porous
slab undergoing drying on a horizontal plate are summarised as follows:
Mass balance for dry air in the γ phase:
∂
h ρ a i + ∇ · hn a i = 0
∂t
(B.7.1)
Mass balance for vapour in the γ phase:
∂
hρv i + ∇ · hnv i = ṁ βγ + hṁσγ i
∂t
(B.7.2)
Mass balance for water in the β phase:
∂ ρ + ∇ · hnβ i = ṁσβ − ṁ βγ
∂t β
(B.7.3)
Mass balance for bound water in the σ phase:
∂
hρ i + ∇ · hnb i = −hṁσβ i − hṁσγ i
∂t b
(B.7.4)
Derivation of drying equations.
303
Mass balance for dry solid component in the σ phase:
∂
h ρ s i + ∇ · hns i = 0
∂t
(B.7.5)
Momentum balance (Darcy’s law) for the γ phase:
hvγ i γ − hvs i σ = −
krγ (s)k0γ δ
· ∇h pγ iγ
eγ µ γ
(B.7.6)
Momentum balance (Darcy’s law) for the β phase:
krβ (s)k0β δ · ∇[h pγ iγ − pc ] − hρ β i β g
hv β i − hvs i = −
eβ µ β
β
σ
(B.7.7)
1-D slab shrinkage model for velocity of the dry-solid component of the σ phase:
hvs i = − s
σ
∗
Zy
0
∂
∗
1/s dy ê2
∂t
Convection + diffusion equation for dry air in the γ phase:
hρ a iγ
hna i = hρ a ihvγ iγ − hργ iγ Deff · ∇
hργ iγ
Convection + diffusion equation for water vapour in the γ phase:
hρv iγ
hnv i = hρv ihvγ iγ − hργ iγ Deff · ∇
hργ iγ
(B.7.8)
(B.7.9)
(B.7.10)
Convection equation for the β phase:
hnβ i = hρ β ihvβ i β
Convection + diffusion equation for bound moisture of the σ phase:
1
hs̄v iγ
σ
γ
hnb i = hρb ihvs i − Db eσ −
∇h T i +
∇h pv i
Mw
hρv iγ
(B.7.11)
(B.7.12)
Convection equation for solid-component of the σ phase:
hns i = hρs ihvs iσ
(B.7.13)
Energy balance. (1) Energy conservation form:
∂
hρ j ihh j i j + ∇ · ∑ hnk ihhk iσ + hnβ ihh β i β + ∑ hni ihhi iγ
∑
∂t j=σ,β,γ
i = a,v
k =s,b
T
= ∇ · Keff · ∇h T i
(B.7.14)
Energy balance. (2) Temperature form:
∂h T i
+ ∇h T i · ∑hni ic p,i + hṁσγ i ∆h0b + ∆hv + hṁ βγ i∆hv
∂t
i
∂∆hb
T
= ∇ · Keff · ∇hTi + ∇ ∆h0b · hnb i + hρb i
(B.7.15)
∂t
hρTot iC p
Derivation of drying equations.
304
The system of equations (B.7.1)–(B.7.15) resemble the equations (VI-1)–(VI-6) of [188],
but include, in addition, the mass and energy transport associated with the moisture
which is hygroscopically bound to the solid matrix. These governing equations are
completed by the constitutive equations (B.6.2)–(B.6.9) and (B.6.13). The theory contains the following free parameters: α, n, e, k0β , k0γ , αγ , k s , Db together with a shrinkage
function s∗ (s), which characterizes the shrinkage of the aggregate porous medium as
a function of its saturation. Taken together, these parameters characterise the porous
medium. The permeabilities k0β and k0γ may differ, although these two quantities are
related by k0β = O(k0γ ), as can be seen by examining Equation (4.16) of Whitaker [191]
and comparing the magnitude of kβ as e β → 1 with kγ as eγ → 1.
As this appendix’s discussion has shown, even in the relatively straightforward
non-hygroscopic case, numerical drying simulations currently implement the comprehensive drying theory only to the extent that the dispersion terms hρ̃i ṽγ i and hξ i,
γ
−1
γ
the coupling terms
−eγ hρv i h T i DT · ∇h T i and K A · ∇hρv i and the enhancement
∗(v)
terms DD · ∇
hρv iγ
γ
hργ i
∗ · ∇h T i, all of which will vary with local conditions,
and Keff
can be neglected. The assumption that spatial deviations are small compared with
characteristic average values is only generally valid for variables not subject to a
Dirichlet boundary condition [193], which excludes the dispersion terms hρ̃i ṽγ i and
hξ i. Nevertheless in many drying contexts these dispersion terms are unlikely to sig-
nificantly alter drying behaviour: vapour dispersion will be small when the vapour
mass-fraction ωv is small and transport in the γ-phase is diffusion-limited. It may be
permissible to neglect convective heat transfer, except for that part associated with the
mass rate of evaporation, and in such a case the thermal dispersion may reasonably
be neglected. On the other hand, by way of an order-of-magnitude analysis, Whitaker
has provided reason to believe that the enhanced diffusion and the multiphase thermal diffusion may have a significant impact on diffusion-limited drying processes
[193]. Unfortunately, Whitaker’s method does not allow the impact of these effects to
be predicted in detail, and we have been forced to neglect them in our model.
Derivation of drying equations.
B.8.
305
Nomenclature for appendix B
Roman and Greek symbols
Symbol
Definition [units]
A
Area [m2 ]
ai
General scalar field in phase i
aγ
Attenuation factor for diffusion within the porous medium [–]
am
Moisture activity [–]
cp
Constant-pressure specific heat capacity [J/kg-K]
d
Characteristic length of pores [m]
D
Diffusivity [m2 /s]
Db
Bound-moisture diffusion coefficient [kg-s/m2 ]
(v)
DD
(v)
DD
0( v )
DD
∗(v)
DD
Dispersion coefficient [m2 /s]
DT
Multiphase thermal vapour diffusion flux coupling tensor [m2 /s]
Tortuosity tensor [m2 /s]
Passive part of tortuosity tensor [m2 /s]
Active part of tortuosity tensor [m2 /s]
h
Specific enthalpy [J/kg]
∆hv
Latent heat of vaporization [J/kg]
∆h0b
Differential moisture binding energy [J/kg]
k
Thermal conductivity [W/K-m]
Kβγ , Kγβ
Coupling tensors for multiphase Darcy flow [m2 ]
Kγ , K β
Permeability of porous medium to gas and liquid flow [m2 ]
T
keff
Effective thermal conductivity [W/K-m]
KTeff
Effective thermal conductivity tensor [W/K-m]
K0eff
∗
Keff
Passive thermal conductivity tensor [W/K-m]
KA
Multiphase Dufour effect coupling tensor [J-m2 /s-kg]
l
Characteristic length of averaging volume [m]
Thermal conductivity enhancement [W/K-m]
L
Characteristic length of porous medium [m]
hṁσγ i
Volumetric rate of evaporation of bound moisture [kg/m2 -s]
hṁ βγ i
Volumetric rate of evaporation of free water [kg/m2 -s]
hṁσβ i
Volumetric rate of liberation of bound moisture [kg/m2 -s]
Mw
Molar mass of water [g/mol]
n̂ij
Unit normal vector on interface pointing from phase i into phase j [–]
ni
Mass flux of species or phase i [kg/m2 -s]
n, r
Dimensionless parameters in the Van Genuchten model [–]
Derivation of drying equations.
p
Pressure [Pa]
pc
Capillary pressure = pγ − p β [Pa]
306
q
Pure heat flow vector [W/m2 ]
s∗
Local shrinkage [–]
T
Temperature [K]
U
Unit isotropic tensor [–]
vi
Velocity of phase i or species i [m/s]
V
Volume, volume of averaging-volume [m3 ]
wi
General vector field in phase i
wij · n̂ij
Normal velocity of interface ij
x, y
Spatial variables [m]
xw
Mass fraction of bound moisture in the σ phase [kg-water/kg-total]
α i (r )
Characteristic function for phase i [–]
δ(t)
Thickness of product layer as a function of time [m]
ei
Volume fraction of phase i [–]
θ
Contact angle between meniscus and capillary surface [–]
µ
Dynamic viscosity of fluid [N-s/m2 ]
hξ i
Thermal dispersion [J/m2 -s]
ρ
Density [kg/m3 ]
h ρ s i0
Initial volume-averaged s-density [kg/m3 ]
τ
Surface tensions of liquid-gas meniscus [N/m]
φ
Relative humidity [–]
Φ
Volumetric heat source due to radiative heating [W/m3 ]
ψi
Quantity associated with phase i
hψi i
Phase average of ψi
ω
Humidity ratio [kg-vapour/kg-dry]
ωi
Mass fraction of species i [kg-i/kg-total]
hψi ii
Intrinsic phase average of ψi
Subscripts and superscripts
Symbol
Definition
0
Environment
e
Exterior of averaging volume
i
Interior of averaging volume
k
Species-k
w
Liquid-water
v, a
Water-vapour, Dry-air
b, s
Bound-moisture, Dry-solid
S, s
Surface
Derivation of drying equations.
sat
Saturation condition
t
Total, effective
wb
Wet bulb
γ
Gas-phase
β
Liquid-phase
σ
Solid-phase
307
308
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