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) Background theory 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) Background theory 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) Background theory 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: Background theory 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 Background theory 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 Background theory 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 Background theory 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]. Background theory 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) Background theory 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. Background theory 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) Background theory 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 Background theory 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) Background theory 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 Background theory 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 Background theory 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 Background theory 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 Background theory 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]. Background theory 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 References [1] Alves-Filho, O., Strømmen, I., and Thorbergsen, E. 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