Modelling of Two-Phase Flows with
 ISSN 0280-5316
ISRN LUTFD2/TFRT--5629--SE
Modelling of Two-Phase Flows with
Modelica™
Olat Bauer
Department of Automatic Control
Lund Institute of Technology
November 1999
Document name
Department of Automatic Control | MASTER THESIS
Lund Institute of Technology Date of issue
Box 118 November 1999
Document Number
SE-221 00 Lund Sweden ISRN LUTFD2/TFRT--5629--SE
Author(s) Supervisor
Olaf Bauer H. Tummescheit, À. Jakobsen, J. Fredsted and N.
Pettit
Sponsoring organisation
Title and subtitle
Modelling of Two-Phase Flows with Modelica-TM. (Modellering av tvá-fas flóden med Modelica-TM.)
Abstract
Modelica-TM is an object-oriented language for modeling physical systems that was designed in the last
years with the goal to become a standardized multi-domain modeling language. This paper describes a
robust model for homogeneous and inhomogeneous two phase flows with dynamic or static slip correlation.
It was developed in the context of developing a Modelica base library for thermo-hydraulic applications.
The model describes the transient behavior of a fluid moving through a pipe during a phase change caused
by heat transfer or pressure changes. Measurements from a refrigeration cycle were used to validate the
model. Physical approaches were taken to model friction and momentum exchange between the phases.
Spezialized thermodynamic equations of state were developed in order to improve the simulation speed.
The model also includes the one-phase flow of liquid or vapor as limiting cases in order to make the
simulation of a complete phase-transition possible. The model is numerically robust in all flow regions.
Modelica's language features are used to structure the code for reusability in different contexts. These
features make the model well suited for a reusable model library.
Key words
Classification system and/or index terms (if any)
Supplementary bibliographical information
ISSN and key title ISBN
0280-5316
Language Number of pages Recipient's notes
English 100
Security classification
The report may be ordered from the Department of Automatic Control or borrowed through:
University Library 2, Box 3, SE-221 00 Lund, Sweden
Fax +46 46 2224422 E mail ub20ub2.1u.se
Contents
Nomenclature . . . . .. ss у
Greek Letters . . oo... ae aaa vi
Subscripts . .. .. ._ xx...) ae vi
Superscripts . . . . ee ee vii
Dimensionless Numbers . . . .... aaa 0 Vil
Operators . . . . .............. aaa vii
1. Introduction . . . . . . . . ee aaa aa 1
2. Balance Equations . . . . . . .... e ae aaa 2
2.1 Quantities of State . . . . . ... aaa 2
2.2 Balance Equations . . . . . . . . eee 2
3. State Variables for Two-Phase Flow . . . . . .......... 7
3.1 Extensive Quantities of State . . . . . . . . 0 7
3.2 Thermodynamic State . . . . . . aaa ea aaa 9
3.3 Hydrodynamic State . . . .. .. eee 11
4. One-Dimensional Two-Phase Flow . . . . . ........... 15
4.1 Mass Balance . . . . . . . 2... aa aa e aa 15
4.2 Momentum Balance . . . ..................... 17
4.3 Energy Balance. . . . . . . . e ea aa 21
5. Thermodynamic Model . . . . . . ... .............. 24
5.1 Differential Equations . . . . . ... . 2.2 ea. 24
5.2 Equation of State .... 22220101 44411 LL LL 1120 27
6. Hydrodynamic Model . . . . .......... В 34
6.1 Mass Flow Equation . . . . .. .................. 34
6.2 Static Slip-Flow Equation . . . . . . . .............. 35
6.3 Dynamic Approach ........................ 38
6.4 Dynamic Slip-Flow Equation . . . . ............... 44
7. Discretized Model Equations . . ... .............. 49
7.1 The Finite Volume Method . . . . . . .. ............ 49
7.2 Pipe Model . . . . . aaa aa aaa 50
8. Modelling of an Evaporator . . .... 54
8.1 Model Equations . . . . . . e. a 54
8.2 Implementation . .. .. ......... 2 ee e 58
9. Simulation of an Емарогабог. ............. aa 60
9.1 Measurement Data .. 1.112211 414 LL 2 ea ea 60
9.2 Steady-State Simulation . . 1.111114 2 144144 12440 62
9.3 Transient Simulation .. 11224 1111 LL LL LL 111110 66
10. Summary . 1 LL LL LL 4 LL LL LL 4 A LL 1240 71
A. Balance Equations for Kinetic and Internal Energy ..... 75
A.1 Mass Balance . . . 111 414111 LL LL LL LL aaa 75
A.2 Momentum Balance . . . . .................... 75
A.3 Energy Balance. . . . . . . aaa aa 76
B. Correlations for heat transfer coefficients and friction factors 79
B.1 Heat Transfer Coefficient . . . . ................. 79
B.2 Friction Factor . . . . . . . aaa aaa 80
C. Thermodynamic Properties and Derivatives. . . . . . .. .. 83
il
G.1 Fundamental Equations .. .. 122414414411 41420
C.2 Transformation of Partial Derivatives . . . . ..........
C.3 Derivatives in the Two-Phase Region . . . ...........
List of Figures
2.1
3.1
4.1
4.2
0.1
5.2
5.3
5.4
5.5
5.0
0.7
5.8
6.1
6.2
6.3
6.4
6.5
7.1
7.2
8.1
8.2
8.3
8.4
9.1
9.2
9.3
9.4
9.5
9.6
9.7
9.8
9.9
Velocities on a surface element . . . . . . . ..........
Stratified two-phase flow ... 2244244 44 4 ere.
Areas in a two-phase pipeflow . . . . .............
Velocities at the phase interface . . . . . . ..........
log p,h-diagram of R22 . . . . . . . . . .. ...........
Taylor expansions for 1" in the vapour region . . .......
Taylor expansions for » in the vapour region . . . ......
Taylor expansion for p in the liquid region. . . . . . . . ...
Absolute error of the approximation function for 7°. . . . . .
Relative error of the approximation function for p . . . . ..
Relative error of the approximation function for 0p/0p|;,
Relative error of the approximation function for Op/Oh],
Flow quality and quality as functions of the void fraction
Velocity ratios as functions of the void fraction . . . . . . ..
Vapour pressure force .. ... 4444444444 4411420
Relative mass flow rates . . . . 1144444444 4411440
Interfacial friction factor . . . . 1.124424 4 44144 4440
Thermal grid .. 1.112211 LL LL 40 ea aa ee
Flow grid .. 22144411 LL LL LL LL 4 1 LL 4 LL 44
Test evaporator . . . 1.224124 44 4 4 4 4 LL 4 14 14 20
Pipe dimensions . . . . e... e ee... a
Theoretical heat transfer coefficients for R22 .........
Evaporator model ... 1220110444 414 4 e aa.
Refrigeration test plant ... 1.444444 44 4 4 4 1420
Measured entering and leaving mass flow rates . . . . . . ..
Measured condenser and evaporator pressure . . .. . . . ..
Measured refrigerant and fluid temperatures . . . .. . . ..
Outlet volume flow rate .. . 1.212414 444444 4440
Simulated pressure drop in the evaporator . . . .. . .....
Simulated outlet pressure . . . ... . 2. rea
Simulated evaporator charge . .................
Simulated evaporator outlet mass flow rate . . . . . .....
9.10 Simulated outlet temperature
9.11 Simulated outlet pressure
9.12 Simulated outlet pressure
B.1 Two-phase heat transfer coefficient
B.2 Two-phase friction factor
C.1 Legendre transformation
«+ # ##‘## EEE AE AA
+8 #1 EC EE A EE AE AE EEE EE
LA 4 #2 A A A&A 4 4 A x x Aa Fw
8 & #4 "4 + ¥ = = w= a =
# #1 4 4 4 #1 x = E ® 4 = a 4 + »¥ = a =
4 = 4 ¥ a oF x = = x = » 4 8 3 w® = «x =
111
Nomenclature
Symbol Unit
s
e
a E о Но ВоВ оз К ЕВ Нем Нео 0) © © кр р по бо TASAS
m/s
m
kg/s
kg
kg /kmol
Pa
Ww
J/m*s
J/s
J/(kgK)
(m/s)/ (m/s)
K
N/m?
J/kg
J
m°/kg
m*
Physical Meaning
thermodynamic property
velocity of sound
surface area
coefficient
specific heat capacity
limiting value of c
constant
diameter
specific energy
unit vector
energy
specific free energy
axial force
force vector
specific free enthalpy
acceleration due to gravity vector
mass velocity
specific enthalpy
enthalpy flux
counter
axial momentum
axial momentum flux
momentum vector
Jacobian matrix
correction factor
length
mass flow rate
mass
molecular weight
number of cells
unit normal vector
pressure
power
specific heat flux vector
heat flux
radius of curvature
specific entropy
slip ratio
time
unit tangential vector
temperature
stress tensor
specific internal energy
internal energy
specific volume
volume
NOMENCLATURE
3 Е Е
3S
п че в
m/s
m/s
Nm
kg/kg
(kg/s)/(kg/s)
Greek Letters
*
CHE KE SK QU MITT >-xREaDBN ya MR RIDE
W/(m*K)
m*/m*
m
N/N
Pas
m*/m*
W/mK
kg/ (ms)
kg/(m*s)
kg/m?
N/m
N/m?
Subscripts
C2HgO
corr
evap
ext
vi
inlet
outlet
axial velocity
velocity vector
work
independent variables
quality
flow quality
axial position
position vector
heat transfer coefficient
thermodynamic property
void fraction
indicator function or weight factor
thickness
fraction of acceleration force shared by the vapour phase
interfacial friction factor
С Aint/ Ay
dynamic viscosity
specific surface area
thermal conductivity
mass velocity through a moving interface
Mint/ A
friction factor
density
surface tension
shear stress
two-phase multiplier
function
angle between g and e,
quantity of state
specific quantity of state
specific quantity of state between two control volumes
surface
ethanol
condenser
correction
evaporation
external
fluid
1510
in
int
kin
out
ref
sat
th
Superscripts
gravity
Water
counter, room coordinate
inlet
phase interface, internal
counter, room coordinate
pressure, isobaric
kinetic
outlet
reference
saturation
friction
theoretical
isochoric
wall
axial component
Subscripts are also used to denote partial derivatives
Superscripts
liquid
vapour
Dimensionless Numbers
Fr w?/(gD)
Nu aD/A
Pr nep/ À
Re pwD/n
We pw*D/o
Operators
A
d
д
det
div
Froude number
Nusselt number
Prandtl number
Reynolds number
Weber number
difference
total differential
partial differential
determinant
divergence
factorial function
vil
1. Introduction
A mathematical model for one-component two-phase flows is developed for
the simulation of evaporators and condensers in refrigeration cycles. Since the
refrigerant undergoes a complete phase transition, the model has to include
one-phase flows as a limiting case. Due to the on/off control strategy com-
monly applied to refrigerators, the resulting process is dominated by transient
phenomena. As simulations of refrigerant evaporators show [1], the accuracy
of a two-phase flow model in a transient process depends on the modelling
detail of the hydrodynamic behaviour of liquid and vapour.
The most simple approach is to define average properties and utilize the model
equations for one-phase flows. This approach is, however, restricted to homo-
geneous flows, which in any case is a simplifying assumption that is justified
when one phase is dispersed within the other, i.e. as tiny drops or bubbles.
Otherwise a spatial separation of the phases occurs, especially in horizontal
pipes. For a given pressure gradient, the velocity of the vapour is, due to its
low density and viscosity, higher than the velocity of the liquid.
The homogeneous model can be derived from the physical principles of the
conservation of mass, momentum and energy. Considering the actual hetero-
geneity requires an additional equation to determine the velocity difference.
Empirical correlations are available, but a heterogeneous model can also be ob-
tained from separate momentum balances for each phase. The latter approach
provides more physical relevance, but contains the difficulty of predicting fric-
tion and momentum exchange between the phases, which will be attempted
within this thesis
The model is implemented in the object-oriented programming language Mod-
elica and to become a part of a thermohydraulic model library. Refrigerant
properties can be computed from the NIST REFPROP library, but simple
equations of state will be developed to minimize the computation time. For
evaluation of the model, the simulation environment Dymola is used, which
contains algorithms for integration of ordinary differential equations over the
time. To account for the spatial distribution of the properties of the flow, the
model equations have to be discretized. Simulation results will be compared
with measurement data from a refrigeration test plant at Danfoss, Denmark.
2. Balance Equations
2.1 Quantities of State
The thermodynamic state of a fluid particle can be described by the infinites-
imal quantities of volume dV, mass dM, momentum dI and energy dE. lt is
more common to use specific variables, representing a quantity dW per unit
mass
aw |
ill 2.1
The specific values of volume, momentum and energy are therefore
dVv dl dE
= — —— _ ——— 2.2
UT aM “7 IM “7 IM 2.2)
where w is the velocity of the fluid. The reciprocal value of the specific volume
is known as the density
1 dM
= > = — 2.3
p=-=— (2.3)
The energy dE may be split up into internal and kinetic energy
dE — dU + dErin (2.4)
which gives two more specific values
dU lw |?
— ZZ но = —— 2.5
и ЯМ €kin 9 ( )
where u quantifies the mechanical energy of the molecules in a motionless fluid.
The overall specific energy becomes
е = и + erin (2.6)
A potential energy is not declared; the related quantity will be considered as
the work of the gravitational field.
2.2 Balance Equations
Integration over the infinitesimal values d¥ yields the overall quantity Y
v= [av= | нам = | ppav (2.7)
2.2 Balance Equations
The rate of change of a quantity is formally written as
dv d
Application of the Leibnitz rule [7] yields
dv PV) /
—— = dA
= / En dV + | pYw AT (2.9)
The first term on the right hand side represents the rate of change of a quantity
V for a control volume keeping its shape; the derivative operator can thus be
put under the integral. The second term accounts for the rate of change of
VW due to a displacement of the volumes’ surface A. Therein, w4 denotes the
local velocity of surface displacement, while n is the unit normal vector of
the surface (outward direction is positive). The scalar product wan yields the
component of w 4 normal to the surface, fig. 2.1.
Figure 2.1 Velocities on a surface element
The influence of the second term becomes obvious when inserting v» = v, which
gives the rate of change of volume
d
= | wanda (2.10)
de J,
The Leibnitz rule serves to switch between different approaches of balancing: In
an Fulerian approach, the control volume is considered as fixed, w4 = 0, and
the second term in eq. (2.9) disappears. In a Lagrangian approach, the surface
velocity equals the velocity of the fluid particles on the surface, w4 = w;
in that case no particle enters or leaves the control volume, which therefore
contains permanently the same particles. Besides these two approaches, w4
may be defined in any appropriate way.
The Leibnitz rule will now be applied to the quantities mass, momentum and
energy, leading to the basic balance equations.
Chapter 2. Balance Equations
Mass Balance
For % = 1 the Leibnitz rule eq. (2.9) yields the derivative of mass
dM др
= — A 2.11
FT | 5 av + [ pwnd (2.11)
In a Lagrangian approach, ws = w, the control volume contains a constant
mass, thus
Op av + / pwn dA = 0 (2.12)
y O A
which essentially describes the conservation of mass and is also known as the
continuity equation. Solving this equation for the first term and inserting it
into eq. (2.11) gives
Mo / plw4 — w)n dA (2.13)
dt A
Since no mass is created or destroyed, the term on the right hand side repre-
sents the flow of mass through the surface of the control volume, i.e. the mass
flow rate
т, := / plw4 — w)n dA (2.14)
A
which, in this definition, is positive for a flow of mass into the control volume.
Momentum Balance
The Leibnitz rule, eq. (2.9), evaluated for Y) = w, yields the derivative of
momentum
dl Ö(pw)
— = 2.1
Hi / у dV + | ww) dA (2.15)
According to Newton's second law, the momentum of a constant mass (wy =
w) increases due to the applied forces
O(pw) побит) dA =
/ d+ [o (wn) dA => F (2.16)
Ot
It is common to distinguish between body forces and surface forces. If gravity
is the only body force present, the force on a mass dM becomes dF, = gdM,
where g is the vector of the acceleration due to gravity, which is constant.
Integration yields the overall gravity force
F,= | eg dV = Mg (2.17)
2.2 Balance Equations
The surface forces are usually split up into the pressure force and the friction
force. The pressure force acts opposite to the unit normal vector, dF, =
—pndA, causing an overall force on the surface of an amount
F, = - | mdd (2.18)
A
The friction force of an infinitesimal small element, caused by viscous and
turbulent forces, is expressed by use of a stress tensor
711 721 731
Т = | m2 T2 73 (2.19)
713 723 1733
Where 1,2,3 are the room coordinates and 7;; denotes the shear stress in
direction of à on a surface 7 = const. Multiplication with the unit normal
vector n yields the stress force vector on a surface element, dF, = TndA.
Integration gives
Fo = / Tn dA = Tji(ne;)e; dA (2.20)
A A a
1,3
where e; is the unit vector in direction of i. Pressure and friction forces are
also present inside the volume, but cancel themselves out and thus have no
influence on the momentum of the control volume as a whole. Solving eq. (2.16)
for the first term and inserting it into eq. (2.15) yields
dl
pri | wa wn dA + Fo + Fy, + Е; (2.21)
The first term on the right hand side accounts for the convective transport of
momentum.
Energy Balance
Applying the Leibnitz rule eq. (2.9) with 7 = e yields the rate of change of
energy
dE — [ д(ре)
A =), Sn av + [ pewan dA (2.22)
The energy of a closed system (wa = w) is, according to the first law of
thermodynamics, increased only by an addition of heat and work. If ¢) denotes
the heat flux and P denotes the power, this gives
д
/ ие) av + / pewn dA = >. P+Q (2.23)
The power P is the integral of the local power dP, which is the local work
per unit time dP = dW/dt. The work results from the movement of a particle
Chapter 2. Balance Equations
along a line dz, caused by a force dF attacking on the particle. The work is
the component of the force in direction of the movement times the length, i.e.
the scalar product dW = dFdz. The quotient of dz and time dt is the flow
velocity vector w, thus
P=| dP = 1 an 15 dFdz - | wdF (2.24)
Inserting the forces introduced in the previous section yields
P, = | pau av | (2.25)
V
БВ, = — | puns (2.26)
A
P, = || тет дл (2.27)
А ,
When solving eq. (2.23) for the first term and inserting it into eq. (2.22) one
obtains
dE
A
where the first term on the right hand side quantifies the convective transport
of energy.
In chapter 4 the equations derived here will be applied to the two-phase flow
in a pipe. The variables to be used for that purpose will be presented in the
next chapter.
3. State Variables for
Two-Phase Flow
3.1 Extensive Quantities of State
Specific Quantities of State
Contrary to a mixture of gases, a liquid-vapour mixture does not form a con-
tinuum. The phases are separated from each other by an interface. A local
specific quantity of state Y = dU /dM, e.g.
=v, w,u,e,.. (3.1)
may therefore be expressed as
= 6¢" + (1 — I)" (3.2)
where ó is an indicator function, denoting the phase present
0 liquid
5 — | iqui (3.3)
1 vapour
while 4” and 1’ are the the specific quantities of vapour and liquid, respec-
tively.
Overall Quantities of State
The overall quantity of state Y of a control volume containing liquid and
vapour fractions is obtained from an integration similar to eq. (2.7)
y = / dM = / [64 + (1 — 6)W] dM (3.4)
The indicator function can be eliminated by integrating separately over each
phase and then summing up. With ö = 0 in the liquid phase and § = 1 in the
vapour phase we find
v/, ии ms | dM (3.5)
Replacing the mass with dM = pdV in each phase yields
U = / op" dV + / ply dV (3.6)
vi V!
The terms on the right hand side of eq. (3.5) and eq. (3.6) represent the overall
quantities of state of vapour Y” and liquid Y', thus
v=4"+y (3.7)
Chapter 3. State Variables for Two-Phase Flow
Average Quantities of State
The average specific quantity is defined analogously to the local specific quan-
tity. It is defined for each phase separately and for the control volume as a
whole
yy” y”
ap" = UN = M' (3.8)
\
Pom - м
This definition has the advantage that an overall quantity Y can be expressed
as a product of the average specific quantity and the average density, ¥ = pi)V.
Inserting the equations above into eq. (3.7) gives
DM = pM" + M (3.9)
Division by M = M" + M' yields
4 = 2" + (1-7 (3.10)
where x is the quality
M"
In eq. (3.9) the masses may be replaced by M = pV
UNA — Y + UY (3.12)
division by V = Y” + V’ yields
оф = 0 Y +(1—-)9U (3.13)
where the void fraction y has been used
\/”
If the control volume reduces to an area, the void fraction becomes
А’
> ig (3.15)
The quantities vp” and (1—+)p’ in eq. (3.13) are the partial densities of vapour
and liquid respectively. Comparison of eq. (3.13) and eq. (3.10) shows
vp" = xp (1—-P = (1 —2)p (3.16)
3.2 Thermodynamic State
From the two equivalent definitions eq. (3.10) and eq. (3.13) various average
variables may be derived. The ones needed in this thesis are in terms of the
void fraction
р = 70" + (1-7)? (3.17)
I #1
py = au +(1— pu (3.18)
and in terms of the quality
и то” + (1 — x) (3.19)
и ти" ++ (1 — x)” (3.20)
в = г" + (1-м (3.21)
и zw” + (1 — z)w' | (3.22)
where h denotes the enthalpy, defined for each phase (and as an overall average
quantity in case of equal pressures, p’ =p’) by
h:=u+ 5 (3.23)
and w denotes the axial component of the average velocity
w:= we, (3.24)
The averageing process described above is only applicable to extensive quan-
tities. Average intensive quantities, such as pressure and temperature, must
be defined in a different way.
3.2 Thermodynamic State
In a one-component one-phase fluid, two independent thermodynamic proper-
ties determine the thermodynamic state. If pressure p and temperature 1" are
used as independent variables, internal energy and density can be expressed
as follows
u =ulp, 7) (3.25)
These dependencies are usually obtained from measurements and then imple-
mented into an algebraic equation of state. Such an equation is only locally
valid, but will, for simplicity, also be applied to the average values of p and u.
Eq. (3.25) thus implicitly defines the average pressure and temperature.
In a one-component two-phase fluid, the average values of p and u may in each
phase be computed from an equation of state
p° = p'(p",T”) в’ = p'(p', T”) (3.26)
ul — u (nt, Т”) и — u (p', T) (3.27)
Chapter 3. State Variables for Two-Phase Flow
which implies, that vapour and liquid may have different (average) pressures
and temperatures. The difference in each case is, at least close to the interface,
negligibly small. The interface itself is in state of equilibrium.
Equilibrium Properties
In the exceptional case of thermal, mechanical and chemical equilibrium, tem-
perature, pressure and specific free enthalpy of both phase are in each case
1
equal
T — T — T
/ И
р = Р=р
9 = 9=9 (3.28)
Since q itself is determined by 7 and p, only one parameter among these three
can be varied independently. This implies, that the temperature is determined
by the pressure
T= Tsat(P) (3.29)
which is known as the saturation temperature. As a result, the specific quan-
tities of each phase are functions of the pressure only
p° = p°(p) p = p'(p) (3.30)
и” = и” (р) и’ = (р) (3.31)
The average overall density and internal energy can then be computed from
eq. (3.19) and eq. (3.20). Apart from the pressure, only the quality x remains
as the second independent parameter. Since x provides non-trivial information
only in the two-phase region, it is not applicable to the one-phase region. À
couple of independent variables that can be used in both regions of state is, for
example, pressure and enthalpy. In the two-phase region, the quality is then
obtained from eq. (3.21)
7 + 0)
h"(p) — W'(p)
(3.32)
Thus p and A determine the equilibrium thermodynamic state in both regions
of state. They will be used as independent variables in this thesis.
Non-equilibrium Properties
If the equilibrium conditions eq. (3.28) were skipped, each phase would main-
tain its own dependency on two independent variables and may achieve a
metastable state. The equilibrium conditions, however, would have to be im-
posed upon the system as tendencies, to make sure that equilibrium is reached
Mt should be noted, that mechanical equilibrium does not necessarily imply equal pres-
sures. As shown in section 4.2 this is only true for a planar phase interface and the absence
of mass transfer over the interface.
10
3.8 Hydrodynamic State
when external influences are removed. This leads to the complex task of de-
termination of the heat and mass transfer between the phases. The related
transfer coefficients as well as the shape and movement of the interface would
have to be described. The resulting equations then replace the ideal assump-
tions made in eq. (3.28).
In small control volumes with relatively slow changes of pressure any deviation
from equilibrium is almost instantly achieved, compared to the transport of
heat and mass over its surface. For a control volume less than 1 m* the as-
sumption of equilibrium is justified for pressure changes slower than 10 bar/s
[10]. The processes studied in this thesis are restricted to control volumes
of less then 107° m”, and the maximum speed of pressure changes is about
0.5 bar/s, which is clearly within the limits. Hence there is no need to include
non-equilibrium effects. |
3.3 Hydrodynamic State
The mass flow rate through a surface has, most generally, been defined in
eq. (3.18). In a cross-section of a pipe the unit normal vector n is in the
direction of the axial unit vector e,. The mass flow rate will be defined positive
in direction of e,. For a fixed cross-section (w4 = 0) this gives
mo = / pwe, dA = / pw dA = pwA (3.33)
A A
where pw inside the integral is the local axial mass velocity G, while in the
last term it stands for the average value of G, which, due to the definition of
average values made in section 3.1, equals the product of average density and
average velocity.
The mass flow rate can be resolved into the vapour and liquid fractions by
multiplication of eq. (3.18) with A, which gives
m=m"+m (3.34)
where the vapour and liquid mass flow rates are
т" = wa (3.35)
m = (1-9wWaA (3.36)
and y is the void fraction in a cross-sectional area, eg. (3.15). In a stratified
flow, the above relations are also obtained from a geometrical consideration,
fig. 3.1.
The stratified flow is a special case among several flow regimes that occur in a
horizontal heated pipe [2]. We will not consider these in detail, but distinguish
only between dispersed flow (bubble and spray flow) and separated flow (strat-
ified and annular flow). This distinction is roughly related to the distinction
between homogeneous and heterogeneous flow.
11
w > "
w
Pg.
Nm
Figure 3.1 Stratified two-phase flow
Homogeneous Flow
Following the pattern of the conditions of thermodynamic equilibrium, which
gave 7” = T' and p” = y', one might assume an equality of vapour and liquid
velocities w = w” = w'. Eq. (3.35) and eq. (3.36) then become
H
mm” = yp'wA (3.37)
m = (1—")p'wA (3.38)
Substitution of the partial densities with the quality from eq. (3.16) gives
m' = zm (3.39)
m = (1-2)m (3.40)
Thus, the quality z determines the partition of the mass flow rate into its
vapour and liquid fractions. This simple relation between thermodynamic and
hydrodynamic properties is, however, only applicable to homogeneous flows,
e.g. In dispersed flows, where a particle of the distributed phase is entirely
surrounded by particles of the continuous phase. In that case, the interfacial
friction will almost equalise the velocities.
Heterogeneous Flow
In a separated flow, the interfacial friction equalises the velocities at the in-
terface, but barely affects the average velocity of each phase, fig. 3.1. A con-
siderable difference of the average velocities occurs, which is often expressed
in terms of the slip ratio S
H
S = (3.41)
a)!
€
Due to the low density and viscosity of the vapour, its average velocity is
usually higher than the average liquid velocity. In technical applications S is
in the range [10]
1 < 5< |5 (3.42)
12
3.8 Hydrodynamic State
Since (for pressures below the critical point) 9” > p” the velocity difference
cannot be neglected. The mass flow rate of each phase must therefore be
computed from eq. (3.35) and eq. (3.36) and summed up to obtain the overall
mass flow rate. Similar to the quality eq. (3.11) a vapour mass flow ratio can
be defined
AH
B= (3.43)
mn + m'
which will be called the flow quality. Insertion of eq. (3.35) and eq. (3.36) gives
Ной
— 70" 10
эра” (= р 344
For w” = w' the velocities can be cancelled down and eq. (3.44) turns to
eq. (3.16), thus & = x. Using the flow quality in the case when vapour and
liquid have the same mass flow rates, but different flow directions, results in
a division by zero singularity. The slip ratio S causes a similar problem for a
vapour flow over motionless liquid. For that reason, the velocity difference
Aw = w" — w (3.45)
will be used as a characteristic variable in this thesis, when computational
problems are likely to occur for S and à. From eq. (3.22) the velocity of each
phase can be written in terms of the velocity difference and the average velocity
w = w+(1-1)Aw (3.46)
/
w = w—zAw (3.47)
Inserting this into eq. (3.35) and eq. (3.36) and using eq. (3.16) gives
mM = zm + Meorr (3.48)
Mm = (1-2) — Тсот (3.49)
with a correction mass flow rate
Meorr = (1 — TY) PAWA (3.50)
Division of eq. (3.48) by m yields a relation between the flow quality and the
quality
A
t=z+x(l— x) = (3.51)
10
When compared to the homogeneous model, eq. (3.48) and eq. (3.49) reveal
the influence of the velocity difference on the mass flow distribution: For a
given quality, the vapour mass flow rate is increased by an amount of corr
on the cost of the liquid mass flow rate.
13
Chapter 3. State Variables for Two-Phase Flow
A second interpretation of Meorr is obtained from the point of view of an
observer moving at average velocity w: The relative velocities are then w"—w =
(1—x)Aw for the vapour and w'—w = —xAw for the liquid. The related mass
flow rates are yp"(1 — x)AwA and —(1 — y)p'xAwA. Application of eq. (3.16)
shows that both phases have the same mass flow rate Mcorr, but opposite flow
directions (see also fig. 6.4 on page 43).
14
4. One-Dimensional
Two-Phase Flow
The balance equations derived in chapter 2 will now be applied to a fixed seg-
ment of a pipe with constant diameter containing liquid and vapour fractions
of a pure fluid, fig. 4.1. The surface integrals are resolved into the ones for inlet
A1, outlet As and wall A,,. Due to friction, the fluid particles adhere at the
wall; the fluid velocity is thus w = 0 in the integrals over the wall area. Since
the pipe does not move, the velocity of the surface displacement is w 4 = 0.
Figure 4.1 Areas in a two-phase pipe flow
The balance equations for mass and momentum will also be applied to each
phase separately. The surface of the vapour phase is formed by fractions of
inlet AY, outlet A] and wall A] plus the entire interfacial area between the
phases Aju. Similarly, the liquid contacts the areas A1, A5, A, and A¿nt. At
the interface both phases may have different velocities 0, wy, ,. The interface
itself moves with the velocity Win.
4.1 Mass Balance
Overall Mass Balance
With w = 0 at the wall and 24 = 0 eq. (2.13) simplifies to
dM
— = — wn dA 4.1
= I (4.1)
where the right hand side represents the entering and leaving mass. Since
ni = —e, and ny = e, we obtain, considering the definition of the mass flow
rate in eq. (3.33),
dM
- — ) — 4.9
E m1 — Mo (4.2)
15
Chapter 4. One-Dimensional Two-Phase Flow
or, since m = т" т,
dM
SE = ri] — rif + ri, = ri (4.3)
Separate Mass Balances
The general mass balance equation eq. (2.13), evaluated for vapour and liquid
separately, yields
ам” 1 41 H H и
a -/ pwun dA + | p (Wint — Win) int dA (4.4)
dt Af Aint
dM’
a 7 7 / pwn dA + / PlWint — Wing) Tin; dA (4.5)
Alo Aint
In each equation the first term on the right hand side represents the entering
and leaving mass flow rate of the phase considered. The second term quantifies
the mass flow over the interface due to phase change. Therein, nn; ап тир
are the unit normal vectors of the interfacial area, which are directed opposite
to each other, fig. 4.2,
Mint = —Nint (4.6)
By use of the (relative) interfacial mass velocities,
=P (Wint — Win)Nin (4.7)
/ /
и о :== p'(Wint — Wint)int (4.8)
the two mass balances can also be written as follows
ам”
= = + u dA (4.9)
Aint
dM’
= a = ma | w dA (4.10)
dt Aint
Interfacial Mass Balance
Adding up the separate mass balances and subtracting the result from the
overall mass balance eq. (4.3) yields
/ и" аА + / и’ аА = 0 (4.11)
Aint A
int
which can be regarded as the mass balance for the interface: Since it contains
(arguably) no mass, just as much mass enters as leaves the interface. This
must also hold for a single point on the interface, thus
и’ + и’ = 0 (4.12)
16
4.2 Momentum Balance
Figure 4.2 Velocities at the phase interface
+
To show the influence of the interfacial mass flow on the velocities at the
interface, eq. (4.7) and eq. (4.8) are multiplied with the unit normal vectors
and then solved for win:
И
Wing = Wine + 7 nll, (4.13)
HW.
Wint = Wing + Pins (4.14)
Subtraction of eq. (4.14) from eq. (4.13) gives, together with eq. (4.12), and
eq. (4.6)
1 1
(wily — why) = E _ я 7 A (4.15)
which proves that the vapour and liquid velocities normal to the interface must
differ locally, in case a phase transition takes place. Multiplication of eq. (4.15)
and eq. (4.13) with the tangential unit vector t gives, since nt = 0,
the equality of tangential velocities. If the mass flow over the interface is
primarily perpendicular to the flow direction, which can be assumed in a duct
with small diameter, the tangential velocities are approximately equal to the
axial velocities, fig. 4.2. Eq. (4.16) then leads to
Wing ~ Wint ~ Wint (4.17)
4.2 Momentum Balance
Overall Momentum Balance
Three-Dimensional Equation Considering the adhesion condition w = 0
17
Chapter 4. One-Dimensional Two-Phase Flow
at the wall, the general momentum balance eq. (2.21) evaluated for a fixed
segment of a pipe, Wa = 0, gives
dl
— = — / pulwn) dA + Ко + Ру + Е, (4.18)
at A1,2 |
The first term represents the momentum transfer due to convection through
in- and outlet. The forces are
F, = es dV = Mg (4.19)
| У
F, = -/ pn aa— | pn dA (4.20)
| A1,2 Aw .
F, = / — TndA+ / Tn dA (4.21)
A1,2 Aw
One-Dimensional Equation The axial component of the equation is ob-
tained from multiplication with the axial unit vector e,. Using w = we, we
get
al
— = — / шрит dA + Fy + Fp, + Fr (4.22)
dt A1,2 |
where I,w, F are the axial components of I,w, F. The axial gravity force is
F, = / pge, dV = Mge, (4.23)
JV
Since nije, = —1, nie, = 1 and nyez = 0, the axial pressure force becomes
B= pdA— | pdA = (p1 —p2) A (4.24)
A1 А» | |
where p; and po are the average pressures at in- and outlet. In the friction
force term, the axial component of Tn at in- and outlet equals the normal
shear stress 722 which is usually negligible compared to the pressure. The only
considerable axial contribution is the wall friction
Fx Ру = / (Try) e, = / 7, dA = Ty, Ay (4.25)
where 7, is the local axial shear stress and Ty denotes its average value.
Eq. (4.22) can be expressed as
ddl. : |
= = | — Io + Fo + Fp + Fr (4.26)
where I is the flow of momentum in direction of the axial position vector
I = wpwe, dA | (4.27)
A
18
4.2 Momentum Balance
If liquid and vapour have different velocities, the integral must be split up to
be evaluated
i=) w''p"'w"e, a+ | w'p'w'e, dA (4.28)
!
These terms will be approximated as follows
1 = m'w” + m'w' (4.29)
Replacing m” and m' with eq. (3.48) and eq. (3.49) vields
I = rhw + Teper (w” — и’) (4.30)
where w and Meorr are defined in eq. (3.22) and eq. (3.50), respectively.
Separate Momentum Balances
Three-Dimensional Equation Applied to the vapour phase separately,
the momentum balance takes the form
dr”
Ef
where the interfacial mass velocity 4”, defined in eq. (4.7) has been employed.
The forces are
(wn) dA + / yw, dA + F + F/ + F/ (431)
H
1,2 Aint
J. / 09 dV (4.32)
И” -
Е, = -/ pn de — | рп. га - | р’ пли dA (4.33)
Ато AN, Aint -
FY = / Tr dA + / Tn, dA (4.34)
Au Aint
The normal shear stresses in F" have been neglected.
T ©
One-Dimensional Equation Multiplication of eq. (4.31) with ez yields
the axial component of the momentum balance
ar’. :
TRH [wuld E+ B+ FL (4.35)
int
where wi; = Win:éz denotes the axial vapour velocity at the interface. The
axial vapour momentum flow will be approximated by
И = / wW'p'w"e, dA = m'w" (4.36)
19
Chapter 4. One-Dimensional Two-Phase Flow
The gravity and pressure forces are
F/ = M'ge, (4.37)
= A [mie da — (4:38)
int
The friction force is resolved into wall friction and interfacial friction
F"=F"4 Fa (4.39)
with
Po / (Tru) e: dA = / 7" dA (4.40)
1, = / (Tn ne) е, da = | пы А (4.41)
Aint Aint
where 7, and 7//, denote the axial components of the wall shear stress and
interfacial shear stress respectively. The axial vapour momentum balance now
takes the form
ат” |
— = Mi — 90 + / Ww, dA+ Fl + F) + Fi + Fl, (442)
int
The liquid momentum balance is obtained by replacing the superscripts ” with
/
Interfacial Momentum Balance
When forming the three-dimensional liquid momentum balance similar to
eq. (4.31), then adding up both separate momentum balances and finally sub-
tracting the result from the overall momentum balance eq. (4.18), we arrive at
the momentum balance for the interface
| Wr] dA [wl hs + Tl + Tin] de
int int
(4.43)
Evaluated locally with y” = —u' and ni, = -,, it gets
(p —P "т; Mint = И "(wins — Wint) + Tn я ni + Tn Mint (4.44)
Normal Component The component of eq. (4.44) normal to the interface
is obtained from a multiplication with nn} ,
/ H H
р — р = E (Win: —_ Wins Mins + (Tn int) Mint + (Tn ne) TY Tint (4.45)
Shear stresses normal to the interface are caused by the surface tension a.
Applying the Laplace law yields [4]
1 1
p =p" =p (wi, — win )ni, to + — (4.46)
Ri UR,
20
4.3 Energy Balance
where Ry; and Hs are the radii of curvature of the interface. The positive sign
is valid, if the centre of curvature is located in the liquid phase. The equation
can further be simplified by replacing the velocity difference with eq. (4.15) [7]
/ И 11\ 2 1 1 | 1 1
— р" = HE to + — 4.47
ров = (4) 7 ‚| AE] “40
If the interface is planar, Rj, Rp—0o0, the mass transfer causes a higher pressure
in the liquid, regardless of the direction of the interfacial mass flow, evapora-
tion (u" > 0) or condensation (u” < 0). The influence of u is negligible for
small drops or bubbles Ri, Ry—0. In the case of bubbles, the negative sign
is valid; the pressure in the bubble is thus higher than in the surrounding
liquid. The determination of the average pressures from this equation will not
be attempted, instead, a zero pressure difference will be assumed.
Tangential Component A scalar multiplication of eq. (4.44) with the tan-
gential vector t leads us to the tangential component of the interfacial momen-
tum balance. Since nt = 0 the pressure term vanishes
О = (Wins — Wine) + (Tn) + (Tine) (4.48)
int 7
Due to the equality of tangential velocities, eq. (4.16), the first term on the
right hand side is zero and the equation simplifies to an equilibrium of tan-
gential stresses
0 = (T'mine)t + (T'nine)t (4.49)
int
As fig. 4.2 illustrates, in a stratified or annular flow the axial unit vector e; is
approximately in the direction of ¢, thus
0% Tins + Ting | (4.50)
4.3 Energy Balance
The overall energy balance eq. (2.28), evaluated for a fixed straight pipe with
w = 0 at the wall, yields
dt | A12 |
With dV = dAdz and dz = dze, the power due to gravity forces can be
written as follows
P, = / pgw dV = / pgw dAdze, = / / pwe, dAgdz = mgdz
V У. Az JA Az
(4.52)
21
Chapter 4. One-Dimensional Two-Phase Flow
Since the pipe wall is fixed, the surface forces which apply at the wall perform
no work. The remaining terms for in- and outlet are
P, = - | pwn dA (4.53)
A1,2
РВ; = / Twn dA = 0 (4.54)
A1,2
where Pr is the work of the normal shear stresses, which are negligible. P,
must not be mistaken for the power due to friction, which does not appear
in the overall energy balance. Friction does not affect the overall energy, but
causes a transformation of kinetic energy to internal energy within the fluid.
In eq. (4.51) these energy forms may be written explicitly
dU dE, 2 |
— + —— №9 -- pun dA— | p—wmn dA + P,+ P, + P, + 0
dt dt A12 Aa 2
| (4.55)
Resolving this equation into one equation for each energy derivative is difficult.
In appendix A the following equations are derived from integration of balance
equations for an infinitesimal small fluid particle (Navier-Stokes equations):
dEpin / w? Je Op fu OT
— == — — dA + Pa — — dV dV 4.56
dt Aj 2 p 2 wn + 9 dz; + V Oz;
ди;
“dv im dV 4.57
Po +/ Tjs Oz d (4.57)
T--|/ pun dA + Ó — |
dt Ars
The last two terms in eq. (4.56) quantify the rate of change of kinetic energy
due to surface forces. The related terms in eq. (4.57) cause a deformation of the
fluid particles. The deformation work cannot be stored as potential energy, but
is irreversibly transformed into heat, and thus increases the internal energy
6]. Eq. (4.57) is equivalent to
alu
: | др ‚дих
ar =>], rendd+0+5+ | wav + | UTA dv (4.58)
The last term therein is positive — otherwise internal energy would be trans-
formed into kinetic energy, which violates the second law of thermodynamics.
The pressure integral is negative, since a negative pressure gradient is required
to overcome the friction. It will be assumed that both integrals sum up to ap-
proximately zero |
Op Ow;
— == 4,
Le гр dV + | = ig, AV = 0 (4.59)
22
4.8 Energy Balance
In the remaining internal energy balance the pressure work and the convective
term can be combined
an
dt
Since h = u + p/p the integral represents the flow of enthalpy
f= | phn dA (4.61)
A
The integral may be resolved into the flow of enthalpy of each phase
H = p'h'w"n dA + | p'h'w'n dA ‚ (4.62)
A! А’
which will be approximated by
H ~ mh + mK (4.63)
The right hand side can, in analogy to eq. (4.30), also be expressed in terms
of the correction mass flow rate, eq. (3.50)
H = mh+ Thcorr (h” — A) (4.64)
where h is the average enthalpy, defined in eq. (3.21). The energy equation to
be used reads
du. | ,
— =H —H+Q (4.65)
23
= — / [оч + plwn dA + © (4.60).
A1,2
5. Thermodynamic Model
Гео thermodynamic variables must be known to determine the thermody-
namic state of a pure fluid. The objective of this chapter is to provide appro-
priate differential equations from the balance equations derived in the previous
chapter. The convective terms appearing therein will be modelled in the next
chapter.
5.1 Differential Equations
Differential equations for p and u are obtained from balance equations for mass
and internal energy. The resulting primary equations can be transformed into
secondary forms providing improved stability and performance in dynamic
simulations.
Primary Equations
A differentiation of M = pV and U = uM for a constant volume yields
dp dM
dp _ dM 1
de dt (5-1)
du du dM
ou _ ay am 9
dt de “de (5.2)
Equations for the rate of change of mass and internal energy have been derived
in chapter 4
dM
de = m1 — Mo (5.3)
dU , , .
The flow of enthalpy has been expressed as follows
H = mh+hcorr (h" — h”) (5.5)
with a correction mass flow rate
Meorr = (1 — 2) pAwA (5.6)
The heat flux is commonly written in terms of a heat transfer coefficient a
© — о А, (Tw — 7) (5.7)
where A, and T,, denote the wall area and its average temperature. Empirical
correlations for the heat transfer coefficient « in the one- and two-phase region
are listed in appendix B.
24
5.1 Differential Equations
(Numerical) integration of eq. (5.1) and eq. (5.2) yields values for p and u.
Experience shows, that the differential equation for the density eq. (5.1) causes
numerical stiffness in the liquid area, where, due to the low compressibility,
minimal changes of the density, caused by the numerical algorithm, lead to
large, unrealistic changes of the pressure. It is thus desirable to transform the
system of differential equations into a numerically more stable secondary form.
Secondary Equations
From eq. (5.1) and eq. (5.2) differential equations for any combination of two
state variables can easily be derived. In general, the independent parameters
of the available equation of state should be chosen. The equation of state used
within the NIST REFPROP library is, however, a (fundamental) equation for
the free enthalpy f(7, 0) containing the density, which cannot be used for the
reasons described above.
The NIST REFPROP library contains external functions for other combina-
tions of independent parameters, i.e. (7',p), (p, h) and (p, s), which, however,
apply time-consuming numerical methods to solve the fundamental equation
for its independent parameters. Moreover, all combinations, including (T, p),
use numerical methods to compute the saturation properties from the equilib-
rium conditions eq. (3.28).
lo keep the computation time low, the NIST REFPROP library will not be
used directly during the simulation. Instead, fast approximation functions will
be created, whose independent variables can then be chosen as appropriate.
In this thesis, pressure and enthalpy, common variables for a description of
refrigeration cycles, will be used.
To transform eq. (5.1) and eq. (5.2) into differential equations for p and h, the
general correlations p = p(p, h) and u = u(p, h) are differentiated by the time.
Application of the chain rule yields
de _ Op) do Op) dh 58)
de ôp|, dt ôh|, dt
du Ou| dp ди| ай
@ Op|, dl” On, de (5.9)
These equations can also be written in vector notation
d 9 а | р
(0) +5 (0) (5:10)
where J is the Jacobian matrix, containing the partial derivatives
3e 2]
д; oh
J= | Pir Tp (5.11)
dul Ou
To obtain differential equations for pressure and enthalpy eq. (5.10) must be
25
Chapter 5. Thermodynamic Model
solved for the derivative of (p, h)
d | p +d p
(je) 0
The inverse of the Jacobian is computed as follows [3]
1 5 - 5)
~1 _ В Л
det J | _ óu Op (5.13)
Op h др h
with the determinant
Op| dul _— др| ди |
— 5.14
t= Bp, aml, ón, ap), (5.14)
The partial derivatives of u can be reduced to the ones of p. From u = h—p/p
we obtain
ди р др ou 1 pop
Oh|, p? Oh op |p, p p* Opl,
Therefore
Op 1 др 1
= -—| = — 5.16
det J = ap), + PET , 72 ( )
where a is, as shown in appendix C.2, the velocity of sound. The inverse of
the Jacobian becomes
д д
au} 8
JT! = a Pp Pp (5.17)
1— P Op др
P p* Op h др h
and the new system of differential equations, multiplied with the mass M = pV
reads
p dp p Op dp Op du
— M— 0.18
a pps! Oh Ne Oh|, “di (5.18)
p dh p Op dp op du
= (1-=— JV M— 5.19
a? dt ( p Op ) de ôp|, dt (5.19)
Inserting eq. (5.1) and eq. (5.2) and substituting the internal energy with u =
h— p/p gives
p dp др dM Op| dU
—— = h—| | — — —| — 5.20
za ( Mm ) de Ohl, di (5:20)
р ай др dM Odp| dU
LE = (1-1 22 ZR 22 5.21
a dt Op ) dt Op|, dt (5:21)
26
5.2 Equation of State
Integration of these equations yields values for p and h. The next task is to
provide equations of state based on these two variables, that enable a fast
computation of other properties during a simulation run.
5.2 Equation of State
The properties to be computed from (p, h) are the temperature 7', which de-
termines the heat flux in eq. (5.7), and the density p, as well as its derivatives
, and dp/ Ohl, which appear in the equations above. The medium used
др/ др
in the refrigeration test plant is R22. Fig. 5.1 shows the log p, h-diagram of R22
including lines of constant temperatures and densities. It was created on the
basis of subroutines from the NIST REFPROP library. The library was ex-
tended by C-routines computing certain first and second partial derivatives of
properties from the derivatives of the fundamental equation, using correlations
derived in appendix C.
1
Ц I
|
1 ! |
! ala i
1
1 a !
I q
! i !
! 1 |
! 1
t | !
! 3 |
! I I
f
! 1 i
10 - | и
L ! 7
J dS
I У
| 7
+
1 + + и”
1 / +
F / и
I 4; и
! и #
— ! # 7
+ 7 7
E
< ! ’ # #
их
0 Y JS и”
— ¿ à „и
/ и
= dt / ‚”
© + ñ *
wn os ‘ x
© 7 7 „”
= UF 7 и 2°
A и
#; ‘ #
Ой # -
Pros =
+ / ’ #
A 4
МУ, A £
7 7
ATA 7
(PD 7 -
Ho ñ „”
7 / + Pa
7 Fa
Lar и #
Ia ñ 5 „”
/ 7 >
Li и
500
300 350
enthalpy [kJ/kg]
Figure 5.1 log p,h-diagram of R22
The determination of the region of state of a given point (p, h) is obtained from
a comparison of the enthalpy with the equilibrium liquid and vapour enthalpy
h < h'(p) liquid
h'(p)<h<h"(p) two-phase
h"(p) <h vapour
(5.22)
To obtain algebraic equations of states for theses regions, Taylor expansions
on the dew and boiling point are used. This approach has the advantage that
a continuous transition of the temperature and density functions on the phase
boundaries is achieved. The density derivatives are not continuous, as can be
seen from fig. 5.1.
27
Chapter 5. Thermodynamic Model
Properties and derivatives on the saturation curves are calculated from the
NIST REFPROP library and then approximated by polynomials (least squares
fit) using the pressure as independent variable. For numerical reasons the
pressure and/or the property was in some cases replaced by a function. The
order of the polynomial was not adapted to the individual case, but chosen as
constant, n = 5. For h'(p) a different approach was taken (see below).
The relative error for Tsat, A”, A’ and p’ is less than 0.03% in a pressure range
1..20 bar of R22. For p” the maximum error is 0.5%, which is due to its low
values at low pressures. The relative error of the other polynomial functions is
of secondary interest, since they are only used within the Taylor expansions.
The error of the Taylor expansions for 7, p, d0/0p|, and 0p/0h|,, is presented
in section 5.2.
Vapour Region
As fig. 5.1 shows, a wide region of state can be covered with a Taylor expansion
for isobars starting at h = h"(p). The related Taylor series is
Ap! (h— h'(p)}
Using a subscript notation for partial derivatives
ду ду
и, = X => (5.24)
Pop h oh |,
and the abbreviation
Ah = h — №" (р) (5.25)
eq. (5.23) becomes
H H 1 Ah”
bp, h) = Y (p) + 4 WD)AR + Urn(D) —— +. (5.26)
For the temperature in the vapour region a linear approach is made
T = Tsat(p) + Ty (p) AR (5.27)
Taylor expansions of second or third order provide an increased accuracy close
to the dew point. However, as fig. 5.2 shows, they yield qualitatively wrong ap-
proximations for superheated vapour. This is due to a change of the curvature
of an isobar close to the dew point.
Instead of the density, the specific volume is written in expansion form, be-
cause, compared to an isobar in a p(h) diagram, the same line is almost linear
in a v(h) diagram, fig. 5.3. A second order expansion is chosen
1
о = (р) + vh(p) AR + Sun (p) AR’ (5.28)
28
5.2 Equation of State
Taylor expansion for the temperature in the vapor region at p = 5bar
650
foal I
Sen inear —
A quadratic ee
RN qubic A
550 =
X, 500
v
5
® 450
D
ca
5 400
350
300
250
400 450 500 550 600
enthalpy [kJ/kg]
Figure 5.2 Taylor expansions for 7" in the vapour region
Taylor expansion for the specific volume in the vapor region at p = 5bar
0.12
real
———— inear
01 eee quadratic
2 0.1
m
£, 0.09
Е
= 0.08
o
>
£ 007
o
©
co
2 0.06 e
0.05
0.04
400 450 500 550 600
enthalpy [kJ/kg]
Figure 5.3 Taylor expansions for v in the vapour region
Consequently, the derivatives of v were used for a linear expansion
/ и
Un = Up(p) + Upp(P)Ah (5.29)
__ / и
Up = Up(p) + (P)AR (5.30)
The density and its derivatives are then obtained from
— —1
p = y (5.31)
dp
— — —2 — 2
Ph = — Un =-U “Un =—P Uh (5.32)
du
dp
— — —-2, — 2
Pp = dy, PU =—- AP Up (5.33)
The properties on the dew point, denoted with a superscript ”, as well as
Tsat(p) are approximated by polynomial functions, as explained in the previous
subsection.
29
Chapter 5. Thermodynamic Model
Liquid Region
As fig. 5.1 shows, temperature and density in the liquid region are almost
independent of the pressure. Therefore, extrapolations along lines of constant
enthalpies yield highly accurate approximations. The Taylor series is evolved
as follows
bon) = Te 00)? (5.34)
i
where p'(h) is the inverse function of h'(p), which is needed for identification
of the phase region. Both functions should be completely consistent to avoid
discontinuities. The following function was used
p'(h) = [bo + bih + bah?]” (5.35)
For a given pressure this function can be solved for h = h'(p)
b1 db Y SB bo
'D)=—5-—A5-) + — 5.36
UP) 2b (3) , ba (5.36)
As mentioned above, the relative error of h'(p) is less than 0.03% in a range
p = 1..20 bar of R22.
Using the abbreviation
Ар = p— p'(h) (5.37)
and subscripts to denote partial derivatives, eq. (5.34) gets
rr 1 Гу бор Ap?
Yip, h) = Y (D) + bp (0) Ap + ap PP) +» (5.38)
Linear expansions are completely sufficient. The relative error of 7" and p in
the region of interest is less than 0.1%.
Т = Tsat(p) + T,(p)Ap (5.39)
р = p(p)+p,(p)Ap (5.40)
Ph = PnlP)+ PholP)Ap (5.41)
Pp = Pp(P)+Ppp(P)Ap (5.42)
The functions %(p') are polynomials, except for the second derivatives of the
densities, which were not suitable for an approximating polynomial. Instead,
the related derivatives of the specific volume, Uhp and v,, were used. From
these, the second derivatives of the density are computed as follows
2 2 2
Php = (PUR) = —20'vhp, — pup, = Php Phy (5-43)
2 2 12 2
bop = (PU) =-2U0p PU =P) Pm (544)
30
5.2 Equation of State
Fig. 5.4 demonstrates the accuracy of the Taylor expansion. The deviation on
the extrapolation point at p = 5bar is due to the error of the polynomial
function p'(p). It does not cause a discontinuity, because the same function
p'(p) is used in the two-phase region.
Taylor expansion for the density in the liquid region at h = 200kJ/kg
1292
real
=== linear N
1288 —
7
1286 di
density [kg/m*]
1284
1282 |--
1280
5
10
15
20
25
pressure [bar]
Figure 5.4 “Taylor expansion for p in the liquid region
Two-Phase Region
In the two-phase region, the properties can be computed from the properties
on the dew and boiling point. Taylor expansions are not needed. For a known
enthalpy the quality is obtained from eq. (3.21), which gives
h — h'(p)
* = WW) 45)
The enthalpy on the boiling point A'(p) is computed from eq. (5.36). On the
dew point a polynomial function is used. Since the specific volume is calculated
from eq. (3.19) the two-phase density gets
1 1
Poy Tw) + (1—2(p) 2/0) +0 -x)/7)
(5.46)
where the functions for the liquid and vapour densities have already been used
within the Taylor series for liquid and vapour. Now the void fraction can be
computed. Eq. (3.17) gives
/
о — P (pP)
y = 5.47)
"070 |
The temperature equals the saturation temperature
T = Tsat(p) (5.48)
31
Chapter 5. Thermodynamic Model
which is a polynomial fit. As shown in appendix C.3, the density derivatives
in the two-phase region can be expressed as follows
op ре? ат
= HE (5.49)
Oh р T dp
Op —_ cp ° al - par (5.50)
OP |, ST ap T dp |
where the symbolic derivative of the function Ts (p) Was used to obtain a
function for its derivative dT/dp.* The specific isochoric heat capacity in the
two-phase region can be written as
Cy = TE, + (1 — 7)6, (5.51)
The limiting isochoric heat capacities ¢, (which must not be mistaken for ¢,
of boiling liquid or dewing vapour) are written as polynomials. The formula
for ¢, is derived in appendix C.3
Accuracy of the Equation of State
The accuracy achieved by the approximation functions is documented in the
diagrams below. In the vapour region the relative error increases with the
distance from the dew line, which is due to the extrapolation form chosen. The
relative error of 0p/0p|, and 0p/0h|, is larger by a factor up to 5 compared
to the one of the density, which is a result of the linear approach and the
low values of the derivatives at low pressures. The objective to provide good
approximations close to the two-phase region is obviously fulfilled.
absolute error of the temperature
T 1 ” 1 I | 1 X \ k
15K и fo \ ``
———— 5K и” TE \
[RR 1 K Pa - 1} N
——— 0.1K i ``
10 }- Ч so
= € ! ^^
| si
TT" ;
[ue 1 —
© орет
< E o
— ое
D i
3 San? Lo
2 ; e 7
" à Mi
o. 7 ñ рой
hi /
; и + 7
A ’
, гу” ñ
1 i и ñ
’ ! ’
1 I И
i 21 ñ
+ ! /
ei /
i I i i ! Li / !
150 200 250 400 450 500
300 350
enthalpy [kJ/kg]
Figure 5.5 Absolute error of the approximation function for T'
This appears natural, but the derivative of a polynomial is often not usable in this way.
In this case the result was, however, particularly accurate.
32
pressure [bar]
5.2 Equation of State
relative error of the density
T Fm = T I ; — :
| -———— 5% - ) Poo
ee 1% a : J (
ere 0.5% „КР ; !
——— 0.1% a |
10 Г Ka t } / =
| FT, ;‘ / } Ч
L A > / / J
fr 2 | /
| ; ya т „т ; \ /
woe г \ !
Ena mn \ 1 =
Pas Ni ? Вы \ A
Le : > ! !
r J oe : \ ! } +
te o ¢ i ! i
A Су» , Pa /
CE ' и, /
A ! ий И
1 e E / D ba В —
| } 1 | | L } 1 /
150 200 250 300 350 400 450 500
enthalpy [kd/kg]
Figure 5.6 Relative error of the approximation function for p
relative error of dp/oh|,
T 7 T I > “” 1 1 | : 4 4 À N
| {— — 10% RO o
po 5% „” ; | i
| росе 1% +” ; ! !
i TTT 0.5% . ? i ! _
10 Г , и” ; / / M
Г I A ! 1
i a ‘ и / 7
= | A2 /
=, i «о
© Г Pg
= I a
о - Ne
A wo
© ASS
a Por unan
Гу
Ри
FLE .
1 и a em TTT TT
| | 1 | | i
150 200 250 300 350 400 450 500
enthalpy [kJ/kg]
Figure 5.7 Relative error of the approximation function for 0p/0p|,
relative error of dp/ap|,
1 } F I
- 10% d
-—————- 5%
neon nea 1% и”
mmm 0.5%
10 | - 7
= Г - E 7
a т 7
— . Lo
o = 4
= SA
5 И -
N ‘
o TT
a | > -
pa
i
1 zl LS “7
| I 1 } | : A |
150 200 250 300 350 400 450 500
enthalpy [kJ/ka]
Figure 5.8 Relative error of the approximation function for dp/0h|,
33
1ydrodynamic
The convective terms in the equations eq. (5.3) and eq. (5.4), i.e. the mass flow
rate and the flow of enthalpy, are still unknown. The thermodynamic properties
at in- and outlet will later be approximated by the average value of the control
volume located upstream. The remaining properties to be determined in this
chapter are the mass flow rate m and the correction mass flow rate corr.
The latter is, for known thermodynamic properties, a function of the velocity
difference Aw.
6.1 Mass Flow Equation
The mass flow rate in a cross-section is related to the average axial momentum
I of a pipe segment
г= | гоу = | | pu dad: = таг (6.1)
У Az JA Az
For an axial range becoming infinitesimal small we obtain
lim I =mAz (6.2)
Az—dz
while for a finite length Az the quotient I/Az can be regarded as an average
mass flow rate through a pipe segment. The time derivative of this average
mass flow rate is then
dm di!
Substitution of the right hand side of eq. (6.3) with the momentum balance
eq. (4.26) gives a differential equation for the average mass flow rate
dm
AZ =I) — Is + pge:V + (p1 — p2)A + Fu (6.4)
where the gravity and pressure force terms from section 4.2 have been inserted.
The momentum flow was found to be
I = mw + Meorr Aw (6.5)
where the second term on the right hand side accounts for the effect of slip-
flow.
34
6.2 Static Ship-Flow Equation
Wall Friction Force
The wall friction force can be expressed as follows
Fo = / TdA = Tu Au = Тобой (6.6)
Aw
where 7, denotes the average sheer stress and k,, denotes the wall area per
cross-sectional area, which for a pipe segment of length Az is
A, 7wDAz 4Az
er = (6.7)
A mE D
Кар =
The average shear stress 7,, is usually expressed in terms of a friction factor
¢, which relates the shear stress to the dynamic pressure pw*/2. To make sure
that 7,, is directed inverse to the flow direction, it is written as follows
то = pull (6.8)
Inserting Æy and Ty into eq. (6.6) yields
Az
Fy =— > PululA (6.9)
Division by V = AAz yields the pressure drop due to wall friction
Ар, = —£22 p20 (6.10)
Correlations for £ in the one- and two-phase regions are found in appendix B.
The wall friction force in eq. (6.9) can also be written as follows
Fy = € mlul (6.11)
6.2 Static Slip-Flow Equation
In a homogeneous flow, the correction mass flow rate mq» appearing in the
convection terms for momentum and enthalpy equals zero, and eq. (6.4) pro-
vides sufficient information to close the system of equations. In a heterogeneous
flow, an additional equation is required to determine the mass flow rate of each
phase. Empirical approaches usually correlate two parameters among x, .S and
y. À correlation containing à or S as independent parameter would require a
numerical solution. For that reason a correlation by Levy, cited by Wang [19],
V1 29) + 9/01 -29)? + 24-01-99? + 101 —21)]
25 (1-7)? + 701 —27)
(6.12)
x =
Зо
Chapter 6. Hydrodynamic Model
is favourable, because + is known from the thermodynamic state via eq. (5.47).
Moreover the equation contains no empirical variables, and no flow regimes
have to be considered. The equation is therefore easier to handle than more
detailed approaches such as listed in [10]. In the root term of eq. (6.12) the
term (1 — +)? can be factored out
НИ 7(1 — 27) + 7(1 =v) Ja = 27) + 2757 (6.13)
2.6 (1 — 0)? + 0(1 — 27)
In Ав. 6.1 г Бот ед. (6.13) ала г = vp” /p are plotted against the void fraction
for R22 at p = 5 bar.
quality and flow quality at p = 5 bar
quality !
—————- flow quality /
0.8 /
0.6 7
ê
1
4 f
: La
: r
; ”
y r
: ”
„
rd
0.4 у
-
7
x
>
„”
#
и”
>
rn
Re
0.2 о
“ e
—
_—
—
7
mm
—
na
—_—
quality
void fraction
Figure 6.1 Flow quality and quality as functions of the void fraction
It can be seen that for a given void fraction the quality is always smaller than
the flow quality z < z. In a steady state the flow quality is determined by the
energy balance eq. (5.4)
H=m[ih" + (1 —2)h] = Hi + © (6.14)
For a given heat flux Q and inlet enthalpy flux Hy the equation can be solved
for x
: H+Q h
ESA wy WW (6:15)
Therefore, and thus m' and m” at the outlet are determined. Since m” =
vow” A a higher vapour velocity enforces a lower void fraction and thus a
lower quality. From eq. (3.51) follows that the difference between & and x is
proportional to Aw
A
2 — а = x(l — 2) — (6.16)
36
6.2 Static Shp-Flow Fquation
Eq. (6.16) can be solved for the velocity difference. Employing eq. (3.16) yields
Aw р
по — ла = пр ВР ne E
From this equation Aw can also be computed in the limiting cases y—-0 and
v—1: Inserting eq. (6.13) and using eq. (3.17) yields
Aw _ p py(L— a — 2) + 274 — (1 —P (618)
w yd — 797” 287(1— 4)? +41 = 25)
Now y(1 — y) can be cancelled down
A PY (1-27) + 275 — of
AL, ё (6.19)
wp 21— 7) +p (1-27)
From eq. (6.19) the related liquid and vapour velocities are obtained via eq. (3.46)
and eq. (3.47). To enable an evaluation independent of the average velocity
these equations are divided by w
" и? Aw
Aw
—=1 1-x— — =] — :—— ‚20
и + = и a (6.20)
Division of these terms yields the slip ratio S = w"/w'. The results obtained
for p = 5 bar are shown in fig. 6.2.
velocity ratios at p = 5 bar
velocity ratio
с
0 0.2 0.4 0.6 0.8 1
void fraction
Figure 6.2 Velocity ratios as functions of the void fraction
At y = 0 both velocities are equal, while at v = 1 the velocity of the last
evaporating drops is about 20% of the vapour velocity. The latter result is
arguable in a spray flow, where the liquid drops with a vanishing diameter are
more likely to be accelerated to the vapour velocity.
37
Chapter 6. Hydrodynamic Model
6.3 Dynamic Approach
The static slip-flow equation derived in the previous section does not account
for acceleration effects. Differential equations containing these effects can be
obtained from the separate momentum balances derived in section 4.2. One
might use these equations in a similar way as eq. (6.3) to compute the mass
How rates of each phase
dam” dI” dm’ dr
A — A = — 6.21
dt dt dt — dt (6:21)
‘The velocities, however, would then have to be computed from
AH 7
I m / т.
— — —— 6.22
PA PA ca,
which obviously causes computational problems, if y tends to one or zero,
because the velocity of the vanishing phase then tends to 0/0. À numerical
evaluation of this term usually leads to a division by zero singularity.
Differential equations for the velocities provide a better basis to avoid singu-
larities. From a differentiation of I” = и" М” хе ве!
y dw” dar" a " dM"
de de CU
(6.23)
Inserting the vapour mass balance eq. (4.9) and the axial component of the
vapour momentum balance eq. (4.42) yields
d H
MT = mj (w] — w") — mi (wh — w") | и” aa+ | uw dA+ > F"
Aint Aint
(6.24)
By replacing ” with '* we obtain the equation for the liquid velocity
d /
MT = (wf, —w) — sup — w') = | d+ | Win, dA+ > PF
Aint Aint
(6.25)
To solve these equations for the velocity derivatives, they have to be divided by
M” and M' respectively. As pointed out in [7], this division is also legitimate
for M"—0 and M'—0, because the right hand side of the equation for the
vanishing phase also tends to zero. However, the limiting value of the velocity
derivative cannot be computed numerically; a division by 0 singularity would
occur.
The problem can be solved by applying a number of simplifying assumptions
to the terms on the right hand side of eq. (6.24) and eq. (6.25). In the following
sections these terms will be rearranged to include M” or M' as a factor, so
that the masses can be eliminated from the equations.
"referring to the differential form of eq. (6.24) and eq. (6.25)
38
6.3 Dynamic Approach
Convective Momentum Transfer
In the convective term of eq. (6.24) the average velocity inside the control
volume w” and the average mass flow rate m” = yp"w”" A will be approximated
by the arithmetic average values of the related quantities at in- and outlet”:
H И fl И
wy + w wy + W
ИИ И 2 H( A И I 2 H( A 1 2 ИДИ 1 2
т) (0; — ww") — то(0о —w) = m( Lo )— 2 (Wy — ——— )
H И И И
1° n°01 — Wa . 1702 — Un
= mq ——— — Mo —_—
2 2
fi AH
my +m
— 1 2 И И
a 9 (wy or wy)
— HH и и
= yp w Aw; — wy
Inserting w” = (w{ + w5)/2 and rearranging yields
wl? — wi
«Hi И 1 Ni И HA LL И 1
mi (Wi — 4") — то(шо — и") =p A 9 (6.26)
In a similar manner we can rewrite the convection term in eq. (6.25)
и! * — wh?
/ / / ./ / IN / 1 7 2
т (401 — Ш) — то(шо — Ш) = (1 — ))р A——— (6.27)
Acceleration Due to Phase Change
The integral in the third terms on the right hand side of eq. (6.24) and eq. (6.25)
represent the mass flow rate over the interface. It will be denoted as 7h, and
defined positive for a flow towards the vapour. Since u” = — pu’ the following
definitions are equivalent
Mint := / и" аА = -/ Wu dA (6.28)
Aint int
Provided min: is equally distributed over the interface, then the interfacial
mass velocity does not vary spatially, thus
Mint = и” Аут — —H Aint (6.29)
Following the premise made, the mass velocity can also be extracted from the
integral in the fourth terms of eq. (6.24) and eq. (6.25). The remaining integrals
then represent average axial vapour and liquid velocities on the interface. For
the vapour we obtain
а I
/ ply dA = titi — / Ww! dA = rT, (6.30)
‘In a finite difference approach these manipulations are obtained directly from d (thw) —
wdm = mdw, which is, however, only true for an infinitesimal small difference. The control
volume approach reveals the assumptions actually made when applying this relation to a
finite difference.
39
Chapter 6. Hydrodynamic Model
The related liquid term can be rewritten in the same way. The third and fourth
terms in eq. (6.24) and eq. (6.25) finally take the form
| y dar | War dA = —(W — Wiy)Mint (6.31)
int int
-м | и ал+ | pw, dA = —(,— WJmine (6.32)
Aint Aint
For a flow where the interfacial mass flow is primarily perpendicular to the
flow direction eq. (4.17) can be used, which after averageing over the interface
turns to
7 и 7 mA.
Wint 7 Vint 7 Vint (6.33)
where Win: is the average interfacial velocity (the velocity of displacement of
the interfacial area). During an evaporation a particle passing through the
interface accelerates from the average liquid velocity w’ to the average vapour
velocity w”. The evaporating mass flow causes a reaction force
Foyay = —(и” — 0 int (6.34)
From eq. (6.31) and eq. (6.32) we can conclude that each phase is slowed down
by a fraction of this force, which depends on the value of Wint. Using a factor
€ to denote the fraction of the acceleration force that acts upon the vapour we
can write
Клар = —E(w"— wine = —(w" — Wine) Mine (6.35)
Fevap = —(1 — e)(w” — w')mint = — (Wine — W int (6.36)
In an isentropic process the force is equally shared by both phases [18], e then
takes the value 0.5. Another assumption is obtained from solving eq. (6.35) for
Wint
Dint = EU" + (1 — EJu” (6.37)
For € = 1 we get Wint = W', Which is the case in spray flows: The velocity on
the surface of a small liquid drop hardly differs from the average velocity of the
liquid inside the drop. Spray flow appears for high values of the void fraction,
v—1. Рог е = 00 the above equation turns to Win; = W, which can be assumed
in bubbly flows appearing at low void fractions y—0. These considerations
show, that € is related to y. Thus
EN (6.38)
will be used as a first approach that fulfils the conditions in the limiting cases
discussed above.
AO
6.3 Dynamic Approach
Figure 6.3 Vapour pressure force
Gravity and Pressure Force
The axial components of the gravitational forces, derived in section 4.2, require
no further manipulation
F, = yp"ge,V в = (1 - vp ge, V (6.39)
In the equation for the axial pressure force, eq. (4.38), the contribution of the
interfacial pressure is not yet modelled
EF) = pA] — pa Ay — / pn. .e, dA (6.40)
int
To evaluate the integral we consider the pressure p at the interface to be
constant. The resulting force acting from the interface towards the vapour
phase in axial direction is obtained from a multiplication of p with the axial
projection area of Aint- As shown in fig. 6.3 the forces cancel themselves out
at waves, bubbles, etc. The force not compensated is p(A§ — A’). Employing
the void fraction in a cross sectional area eq. (3.15) yields
F5 = (mp1 = орг) А + p72 — m) A (6.41)
The average pressure p will approximated by the arithmetic average of the
pressures at in- and outlet, which yields
+
FE) = (эр: — поро) А+ E (a — yı)A
2
ny
2
= (21 — P2)
Applying the same approximation to the void fraction yields
Fy = (pı — p2)YA (6.42)
Since the liquid and the vapour pressure force must sum up to F, = (p1 —p2)A,
the liquid pressure force is
Fl = (в — po) (1 — YA (6.43)
41
Chapter 6. Hydrodynamic Model
Friction Force
The friction force acting on a phase is the sum of the wall friction and the
interfacial friction.
Wall Friction The wall friction force acting upon each phase depends on
the fraction of the wall area in contact with the phase and the shear stress at
the wall
Po / 7" dA г = / 7! dA (6.44)
A A
1 /
w us
the sum of these yields the overall wall friction force
Fu = Fo + Fo (6.45)
Which in section 6.1 has been expressed as follows
Az
= — -— 4
Fu Ep PululA (6.46)
The fraction of F acting upon the liquid phase rises with the wetted fraction
of the pipe wall. In most flow regimes the wall can be considered as completely
wetted, and the entire friction force acts upon the liquid, F, # Fy. In hori-
zontal pipes, however, the wall gets partially dry as the void fraction increases,
and for y—1 the condition F, — F,, must be fulfilled in any case.
The vapour fraction of the wall friction should therefore rise from almost
zero up to one when the void fraction tends to one. This can qualitatively
be achieved by resolving the density p = yp" + (1 — y)p" into the sum of the
partial densities
Az
ЕЙ о = — ap 1 wjw|A (6.47)
Az
Fy = “Esp 4 Mpw[wjA (6.48)
With eq. (3.16) this is equivalent to
Ей = 2F, F! = (1-2)F, (6.49)
The quality thus serves as a parameter denoting the fraction of the wall force
acting upon the vapour. As fig. 6.1 shows, it provides a qualitatively correct
dependence on the void fraction.
Interfacial Friction Due to the condition of equilibrium of tangential
forces at the interface, eq. (4.50), the interfacial friction forces must sum up to
zero
+ Fa =0 (6.50)
int
42
6.3 Dynamic Approach
The axial vapour wall friction can be written as follows
Mo И =n on
int — / Tint dA = intAint — int int À (6.51)
int
where Æ;n: 1s the interfacial area divided by the cross-sectional area. Empirical
approaches for the interfacial shear stress 7, , usually have the same form as
the one for the shear stress at the wall eq. (6.8). The relevant velocity is the
velocity difference Aw = w” — w'. If it is positive, the vapour is decelerated,
thus
1
Tint =
pres AulAu) (6.52)
which gives
И
int — — Preg ul AulrincA (6.53)
To make a proper choice for the reference density eq. (6.53) is rewritten similar
to eq. (6.11)
Fl, = ites Ali (6.54)
with ref = pref AwA. No mass actually moves with the velocity Aw. Viewed
from an observer moving with the liquid, it is the velocity of the vapour and
vice versa, leaving it to the point of view, which density is to be considered as
relevant.
Figure 6.4 Relative mass flow rates
A mass flow that is related to both phases is the correction mass flow, defined
in eq. (3.50). As explained in section 3.3, for an observer moving at average
velocity w both phases have the same mass flow rate corr, but opposite flow
directions, fig. 6.4. Applying the wall friction force analogy for the vapour in
fig. 6.4 yields F, + Meorr AW. For the liquid we find F,, Ÿ —McorrÂw. The
ant 2
43
Chapter 6. Hydrodynamic Model
approach is in accord with the condition eq. (6.50). Moreover, since corr Y
Aw, the resulting force is proportional to the square of the velocity difference,
which is consistent with eq. (6.53), thus
Mref = Meorr = T(1 — x) pAwA (6.55)
will be used. Now eq. (6.54) gets
int
u Sl — x)pAwA|Aw|rint À (6.56)
Multiplication with 1 = k,,/k, and employing eq. (6.7) yields
Tint 4 (6.57)
К,
10
int ——
Fl, = al — 2) pAw| Aw]
Due to the difficulty of predicting the interfacial area, the term Kint/Kw Will
be included in the friction factor
Ета = ia — x) pAw|Aw|A (6.58)
nt —
where
Rint _ c Ant —¢ Aint
Kay Aw TDAz
с = С (6.59)
is to be determined empirically. Eq. (6.58) has the advantage that, as a first
approach, (* may be approximated by a constant value, since the condition,
that the interfacial area disappears for v—0 and v—1, is implicitly fulfilled by
the factor (1 — x) which after insertion of eq. (3.16) can also be expressed as
Ny =P до
x(1—x) a (1-2) 5 (6.60)
Contrary to eq. (6.53) with the common choice pre; = p” [18], eq. (6.58) also
allows explicit division by y and (1 — y).
6.4 Dynamic Slip-Flow Equation
Inserting the equations for the forces and convection terms derived in the
previous section into the differential equation for the vapour velocity eq. (6.24)
yields with M” = vp" AzA and eq. (6.60)
И 12 12
du Wi —u
1 AZA— = PA aw —wmin
+yP ge: AzA + (pi — pa) YA
5 оо А [ен + С*(1 — 2) Аман] | (6.6)
44
6.4 Dynamic Slip-Flow Equation
The same procedure carried out for the average liquid velocity yields with
int = — Font
dw’ Ww? — wh? ,
I-PAZA— = (=p A=—— — (1 —y)W” = win
+(1 — y)p'ge,AzA + (p1 — p9)(1 — y)A
A
— (1 =p Algwlw] - CedulAw]] (6.62)
These equations can now be divided by vp” A and (1 — «)p’ A respectively.
Using a specific interfacial mass flow rate v
int 1 / И
= = — dA 6.63
A ah (6.63)
we obtain
du” y — ay? 1
Az— — — + PAL — p9 — Awv) + ge,Âz
Az +
3D [Ew|w] + C* (1 — 2) Aw]Aw]] (6.64)
and
dw’ au! 2 _ и’ 2 1
Az = — + 7 — po — Awv) + ge, Az
Az ;
-5p ¡Ewlw| — C*zAw|Aw|| (6.65)
Numerical integration of these equations yields the velocities w and w'. The
mass flow rates m” and m' are then obtained from eq. (3.35) and eq. (3.36). In
that case the differential equation for the overall mass flow rate eq. (6.4) is not
needed. The equations for the velocity difference are, however, applicable only
in the two-phase region. In the one-phase region the equation for the mass
flow rate eq. (6.4) must be used instead.
To avoid a switching of differential equations, it is desirable to use the equation
for the mass flow rate eq. (6.4) also in the two-phase region. As a second
equation either eq. (6.64) or eq. (6.65) can be applied, but the choice is difficult.
Therefore, a combination of both will be used: Subtraction of eq. (6.65) from
eq. (6.64) yields a differential equation for the velocity difference, a dynamic
slip-flow equation
dA w Ww? — wl? — wl? wh? 1 1
Az == 1 2 5 1 2 =| pi = pa — Au
A
055 = Aw|Aw (6.66)
As can be seen, the gravitational force and the wall friction terms have van-
ished; the latter result follows from the modelling approach made in eq. (6.49).
45
Chapter 6. Hydrodynamic Model
The interfacial friction in eq. (6.66) appears in the form of wall friction term.
The pressure gradient is reduced by the acceleration force.
From the mass flow rate and the velocity difference the vapour and liquid
velocities can be computed from eq. (3.46) and eq. (3.47). Since the average
velocity w is defined via ™ = pwA this gives
w! = = +(1—-x)Aw (6.67)
w = 5 — zAw | (6.68)
Using an average velocity
ne wr (6.69)
eq. (6.66) can be rewritten in a more compact way
1 1 A
— 01 Awq — Wo Awe + = — 5) [P1 — ро — Awv] — CS Aw] Aw
о” o
(6.70)
dAw
A
“al
A similar equation for Aw has been derived by Kolev [10] on the basis of the
differential form of the balance equations. However, the terms for the interfacial
mass transfer and friction therein are not modelled and the equation cannot
be applied in the limiting cases у — 0 and y — 1. Contrary to this, the above
equation has the advantage that no singularities appear at the one/two-phase
boundary. The equation can easily be replaced by the static slip-flow equation
eq. (6.19) or the homogeneous model Aw = 0 to switch between different
modelling approaches.
Interfacial Mass Transfer
The specific interfacial mass flow rate v = Mint/A is not yet determined.
Under the assumption of two-phase equilibrium m;,; can be expressed as a
function of the rate of change of mass and pressure. The interfacial mass flow
rate appears as the last term in the vapour mass balance eq. (4.9)
ам”
ra my — Ms + Mint (6.71)
To compute M;n; from this equation, the derivative on the left hand side must
be determined from another condition: The volume of the control volume adds
up from the fractions of both phases.
M" МГ
V=V+V = 7 y (6.72)
Differentiating this equation by the time leads to
ar 1 dM’ М” а И M'd /
dv 1dM 1dM Map dp (6.73)
da 7a 7d 7d 7d
46
6.4 Dynamic Slip-Flow Equation
Inserting dM’ = dM — dM” and M" = yp"V and M" = (1 — y)p'V yields
(6.74)
dy 11 1] dM” там _ y dr 1— 1, 4e
dt 9" pl dt o dt p" dt 0 dt
At saturation equilibrium the densities are dependent on the pressure only,
thus
WV _ и = рам" лам [лай 1-yd/1dp — oe
de pp dt pdt р” ар 0’ dp | dt |
Solving for the vapour mass derivative yields with V = const
ам” ам do” dp' | dp
lA — — по V Pr 1— HN" | + 6.76
PP —P) Pt + eE (6.76)
where dM /dt and dp/dt are known from eq. (5.3) and eq. (5.20). The interfacial
mass flow rate can now be computed from the vapour mass balance
ам”
Interfacial Friction Factor
The modified interfacial friction factor * in eq. (6.70) is considered to be de-
termined empirically. It includes the influence of the interfacial friction friction
factor ¢ and the interfacial area Asnt
C= (6.78)
With the help of the static slip-flow equation eq. (6.19) it is possible to obtain
a first guess in case of a stationary adiabatic two-phase flow. In that case the
left hand side of eq. (6.70) equals zero. In an adiabatic process evaporation
occurs only due to the pressure loss. The related interfacial mass velocity v
is negligible. Moreover, the change of density is small and thus the velocities
remain almost constant. The dynamic slip-flow equation then reads
1 1 „Az
о = | _ 5) [21 — рэ] — © эр Aw|Aw| (6.79)
Applying the same simplifications to the mass flow equation eq. (6.4) gives
with eq. (6.9)
Az
О = pge;V + (pr — p2)A — Egy pwlw] A (6.80)
Eliminating the pressure difference from both equations yields
¢ op [Aw Aw — Е — 5 | орион — pge, Az (6.81)
47
Chapter 6. Hydrodynamic Model
In a horizontal pipe the gravity term vanishes
« _ ||P _ PL WW
С 6% 5) Au — (6.82)
From this equation it becomes obvious that a homogeneous model Aw = 0
is equivalent to an infinite interfacial friction. In a heterogeneous flow, the
velocity ratio can be computed from eq. (6.19) and is thus a function of the
pressure and the void fraction. Fig 6.5 shows the result obtained for C*/£ as a
function of the void fraction.
ratio of friction factors CE
1000 т
100 Е
о -
©
O
Su
5
5 10
5
‚©
©
ТЕ
0.1
0 0.2 0.4 0.6 0.8 1
void fraction
Figure 6.5 Interfacial friction factor
At low qualities, (* exceeds & by several orders of magnitude; this is due to
eq. (6.19), which yields Aw = 0 for x = 0. The result is in accord with the
large surface area appearing in bubble flow. At higher qualities we find
C* = (0.3..2) € for y > 0.8and p = 1..10 bar; (6.83)
Annular flow is most likely to occur in that region: The liquid forms a film
covering the entire pipe wall. As the liquid mass fraction decreases the film
becomes thinner and thus the interfacial area tends to the wall area A;,;— A.
The liquid is more and more restrained and the velocity difference becomes
Aw = w, while p>p” « p' and thus eq. (6.82) turns to
mé (6.84)
Depending on the pressure, C* is in times even lower than £, which reveals that,
due to the low viscosity of the vapour, the interfacial shear stress is actually
small. The interfacial friction is thus dominated by the size of the interfacial
area.
After passing through a minimum value at v ~ 0.95 the curves rise again
and arrive at C*=1.5€ for x = 1. This is in accord with the appearance of
spray flow, where the interfacial area is increased, but the actual value should
rise almost as rapid as for y— 0. The relatively low value of (* at v = 1 is
a result of the static slip flow equation, which produces a remaining velocity
difference. The above function for C* is thus restricted to a separated flow.
48
7. Discretized Model
Equations
The differential equations derived in the previous chapters are now used to cre-
ate a model that allows prediction of the outlet conditions of the flow through
a heated or cooled straight pipe. The general approach which is taken is known
as the finite volume method.
7.1 The Finite Volume Method
The main characteristic of the finite volume method, that distinguishes it
from the finite difference and the finite element method, is the use of balance
equations in integral form. In principle the entire pipe could be treated as one
single control volume, but the results would be inaccurate, since the influence
of the spatial distribution of the properties is neglected. Increased accuracy
is achieved by subdividing the pipe into a number of non-overlapping control
volumes, each of which is small enough to justify its representation by average
properties.
If the diameter of the pipe is small compared to the length, DL, an ax-
ial subdivision is sufficient; the control volumes are thus pipe segments. The
related one-dimensional model equations have been derived in chapter 5 and
6. The outlet conditions of each pipe segment equals the inlet condition of
its neighbour, but the actual interface condition is not known. Therefore, the
application of the control volume method requires two preparations:
e The determination of the size and location of the control volumes,
i.e. the grid structure
e A relation for the approximation of properties between the control vol-
umes,
i.e. the interface conditions
The technique chosen was first described by Patankar [13].
Grid Structure
Two different grid structures are applied: The basic grid structure serves to
determine the thermodynamic state. It will be referred to as the thermal grid.
The pipe is divided into n numbered cells of equal length Az = L/n as shown
in fig. 7.1. An average state variable inside cell number 7 is denoted as 1;, while
1; denotes the state at the inlet of the cell. Yo and Vn11 denote the state of
the flow before the inlet and behind the outlet of the pipe.
To compute the mass flow rates at the interface of the thermal cells, a staggered
grid is used, which will be called the flow grid. As fig. 7.2 illustrates, each
thermal cell overlaps two flow cells. The entering mass flow rate of the thermal
cell 7 is computed as the average mass flow rate of the flow cell i. The pressure
between the flow cells à? and à + 1 is approximated by the average pressure in
49
Chapter 7. Discretized Model Equations
inlet Y, Y. outlet
_ / — _
A Mi Vi Ming |
100 | | a
0 1 .. i .. n il
Figure 7.1 Thermal grid
the thermal cell ¢. The average thermodynamic properties of the flow cell i
equals the state at the inlet of the thermal cell 2.
inlet | | outlet
mie Pia DL P, Pia Po
: : т, i m +1 : :
1 ... L i+l ... n+l
Figure 7.2 Flow grid
Interface Conditions
The thermodynamic state at the interface Wb; 15 approximated by the state of
the thermal cell located upstream, thus
i = бир; + (1 — di) Via i=1.n+1 (7.1)
where д; 15 an indicator function denoting direction of the the mass flow
О m;>0
| hi = (7.2)
1 m; <0
The following interface variables are needed
db; — a, TP, Pas PY Po Bis, Da } (7.3)
The upwind method described here is recommended for convection-dominated
problems. Compared to other approaches it is more stable and yields reason-
able results even for a small number of cells n [13].
7.2 Pipe Model
Thermal Grid
Mass and energy balance are applied to each thermal cell. Using the numbering
shown in fig. 7.2 we obtain
d M;
dU; :
т Н; — Ныа + ©; (7.5)
50
7.2 Pipe Model
The computation of the heat flux Q; is part of the wall model described in the
next chapter. The enthalpy flow is
Hi = uh + Meorr Un — hi) (7.6)
with a correction mass flow rate
Meorr == Z; (1 — Ti) pi Aw; A (7.7)
The equations above are inserted into eq. (5.20) and eq. (5.21) to obtain dif-
ferential equations for enthalpy and pressure
Pi dpi _ 4 p Opi| \ dM: Opi| dU;
ade > CL Oh ) de — Óh|, de (7:8)
pi dh; Opi| \ dM; = Op;| dU;
Vs = |l-hi; — a
a? de ( " Op ) de ópl, dt (7.9)
where a; is the velocity of sound, which is a function of the density derivatives
1 opi
+7 à (7.10)
Pp
Numerical integration of eq. (7.8) and eq. (7.9) yields the values of p; and h;.
The related properties are computed from the equation of state described in
section 5.2
Op;
др;
Vi = (pi, hi) D = {ni on er
dpi de
y dp dp
(7.11)
The derivatives of the equilibrium liquid and vapour densities are needed to
compute the interfacial mass flow rate: The vapour mass balance eq. (6.77)
gives
A “A
Mint 1 a и
A Ald
+ Mis — UA (7.12)
The unsteady term therein is obtained from
dM! dM;
/ H Y И 2
dp; dp; | dpi
9 + (1-9 2—| — .13
+ |на do (1 — v)p; do | dí (7.13)
Flow Grid
The mass flow rates and velocities are calculated as average values of the flow
cells. The momentum balance in the form of eq. (6.4) applied to the flow cell
i in fig. 7.2 reads
dm;
dt
; Az.
Az—— = Al): + f:lg| cos pV + (pi_1 — р) А — “ори (7.14)
51
Chapter 7. Discretized Model Equations
Where y is the angle between the axial unit vector e; and g. The difference of
the momentum of the entering and leaving mass in a flow cell A(7); is approx-
imated by the average difference of the momentum flows of the overlapping
thermal celis
da — li) + (hi — ha) 17. |
A(T); 7 5 = 2 (da — ha) 1=2..n (7.15)
This central difference approach serves to avoid discontinuities in the momen-
tum balance. In the entering and leaving cross section, the momentum of the
cell located outside the pipe is approximated by the momentum of the first
and last flow cell respectively
A(T); m IL — I (7.16)
Ant = La — Laa (7.17)
The momentum flow is computed from
I; = Mi¿Wi + Mcorr AWi (7.18)
where w; is the average velocity
(7.19)
The velocity difference may either be neglected, which gives the homogeneous
model,
or be computed from the static slip-flow equation eq. (6.19)
AJA -20)+204—A
Pi
Aw; = Wi — - — ——— - (7.21)
104 21 = %)208 + %07(1 — 241)
or from integration of the dynamic slip-flow equation eg. (6.70)
dAw; _ 1 1 A y AZ
Az de b= A(WAw); + Е — | (Pi—1 — Pi — Аи) — € > vil Avil
(7.22)
where the convective term is modelled analogously to eq. (7.15)
_ I _ _
A(wAw); ~~ 2 (0; 1 Aw;—q — юн Аш) 1 = 2..т (7.23)
52
7.2 Pipe Model
The interfacial mass flow rate per cross-sectional area 7; is obtained from
application of the indicator function eq. (7.1), and @ is an arithmetic average
/
И
п (7.24)
W; =
In every case the velocities and mass How rates are computed as follows
wi Wi + (1 — £;)AW; (7.25)
Wi = w;— Aw; (7.26)
mi = Вити + Тот (7.27)
il = (1 — hi — Тот (7.28)
At in- and outlet the velocity difference will be set equal to zero
Aw1 — Али = () (7.29)
to avoid inconsistencies with other models, which are usually based on the
assumption of homogeneous flow.
Closure of the System of Equations
The model contains four differential equations, i.e. the ones for p, kh, 7 and Aw.
The remaining differentials appearing in the equations above are computed
from algebraic equations.
In the basic form of the model, the mass flow equations at in- and outlet are
not included. The following boundary conditions are then required to close the
system of equations
e pressure and enthalpy on the upstream side
e mass flow rate m; at inlet 1 = 1 and outlet t =n +1
e heat flux Q; for every thermal cell 1 = 1..n
e heat transfer coefficient a; for every thermal cell ¿ = 1..n
e friction factors &;, (; for every flow cell 7 =1..n +1
This basic model is extended by submodels containing the differential equa-
tions for mm at in- and/or outlet. The pressure on the related side of the pipe
is then a boundary condition, regardless of the flow direction.
The heat transfer coefficients are needed in case Q; is not known and must
thus be computed from the temperature difference between pipe wall and fluid.
For a known heat flux distribution, however, a may be used to compute the
wall temperature. The interfacial friction factor C* is only used in the dynamic
heterogeneous model. The computation of œ and € depends on the direction
of the heat flux and the incline of the pipe. The case of a horizontal heated
pipe is considered in the next chapter.
53
8. Modelling of an Evaporator
The two-phase flow model is applied to simulate an evaporator in a test refrig-
eration plant described in [1]. The evaporator pipe is placed inside a second
pipe, fig. 8.1, where a mixture of ethanol and water flows, which will be re-
ferred to as “the fluid” (subscript f). The refrigerant R22 in the inner pipe
and the fluid in the ring slot have opposite flow directions (counter-flow heat
exchanger). The pipe dimensions are shown in fig. 8.2
| Fluid
HS;
Roo_. SS ss
a EA
3500
Figure 8.1 Test evaporator
8.1 Model Equations
The effect of the pipe elbows will be neglected. Two concentric straight pipes
of length L = 7m form the basis of the model. Since the pipe diameter is
small compared to the length of the pipe, the one-dimensional two-phase flow
model from chapter 7 can be applied. Complementary models are required to
compute the distribution of heat fluxes towards the refrigerant, i.e. a model
for the fluid in the ring slot, which will be treated as incompressible, and a
model for the heat transfer through the wall of the inner pipe.
Figure 8.2 Pipe dimensions
D = 12.7mm д = 0.8mm D, = 22.2 mm
04
8.1 Model Equations
Wall Model
The pipe consists of copper; the related properties at 7 = 0°C are [17]
Pu = 8960 kg/m?
Cy = 381J/(kgK)
Aw = 401W/(mK)
The pipe wall is separated into n elements corresponding to the number of
thermal cells in the two-phase flow model, fig. 7.1. The internal energy balance
eq. (4.57) is applied: Since no movement takes place within the wall, w = 0,
and the pressure is spatially constant, it simplifies to
dU 4 :
ZUR Ov 8.1
with Un = UwPwVw. The specific internal energy is approximated by uy, =
Cwdw, Where cy is the heat capacity of the wall and T, is the average temper-
ature of a wall segment. The volume of the pipe wall segment is
(D + 268)? — D*
и, =
и — TT A
Az =m(D + 0)0Az (8.2)
The internal energy decreases due to the heat flux towards the refrigerant on
the inside
Qi = a; Aint (Twi — Ti) (8.3)
and the heat flux towards the fluid on the outside
О: — Of iÂext (Tai — Tri) (8.4)
where the wall surface temperature has been approximated by the average
temperature, which is justified due to the high ratio A,,/ó. The surface areas
are Aint = TDÂz and Aext = TÍD + 20)Az. The energy balance now reads
with Cuy = const
Hud = — (@ + Ori) (8.5)
Pw Cw Vw
The heat transfer coefficients are computed within the models for the refrig-
erant and the fluid.
Incompressible Fluid Model
The liquid flowing through the ring slot is ethanol with zoo = 10% mass
content water. The mixture will be treated as incompressible, the mass flow
rate is therefore spatially constant and the internal energy balance eq. (4.57)
reads
dei 1e» |
e = ms (Ая — hpi) + Q fi (8.6)
59
Chapter 8. Modelling of an Evaporator
The heat loss through the insulation is neglected. With ¢; = ¢, = cy % const
we obtain
dl т , A A +
PEI = Те) Tra - Tin + ©. (8.7)
with a volume
(Dj — D?)
У; = пд Ай — Vu (8.8)
The temperature on the boundary of a cell is determined by the flow direction
(upwind method):
Te; ne > 0
Pi=0 TS (8.9)
Tha mys < 0
dince fluid and refrigerant have opposite flow directions and the numbering of
cells refers to the thermal grid in the refrigerant model, the fluid mass flow
rate 1s negative.
The properties of the fluid will be approximated by constants. From the den-
sities at T = 0°C, px,o = 999.8kg/m* and pc.Hso = 807 kg/m* [17], the
mixture density is obtained in the same way as the density of a one-component
two-phase fluid, eq. (5.46)
1 kg
= = 823 —
FI TH20/ PH20 + (1 = TH,0)/PCyHeO m?
(8.10)
With cH,o = 4217 J/(kgK) and co.n,0 = 2232 J/ (kgK) [17] the mixture heat
capacity at 7 = 0°C gets
Cf = TH,O0CH,0 + (1 — 2H0)Cc,H,0 = 2430 keK (8.11)
Data of the thermal conductivity and dynamic viscosity of ethanol/water mix-
tures are listed in [15] as functions of the mole fraction of water. With the
molecular weights My,o = 18.02 kg/kmol and Mc.,H,0 = 46.07 kg/kmol the
mole fraction of water is obtained via [16]
Мс,нво/Мноо
NH,0 = TH,0 (8.12)
Using this value, linear interpolations of the data from [15] at p = 1 bar and
Т = 275 К give
Ay = 196-103 W/mK пу = 2566 - 107° Pas (8.13)
Application of eq. (B.4) yields the average fluid heat transfer coefficient. The
result obtained for a cell of length Az = 1.4m can be approximated by
ar = 4820 - m ; |kg/s] — 170 in W/m*K (8.14)
which is valid for hy = 0.22..0.5kg/s. The deviation from eq.(B.4) is less
than 1%.
50
8.1 Model Equations
Pipe Model
The model equations for a two-phase flow have been derived in chapter 7. For
computation of the properties of R22 the approximation functions described
in section 5.2 will be used. The transport properties are approximated by the
values of the saturated vapour and liquid, which are expressed as polynomial
functions.
The one-phase heat transfer coefficients can be computed from eq. (B.1). It
was evaluated for the range of mass flow rates and pressures appearing in the
measurement data. Fig. 8.3 shows the results for a cell length Az = 1.4m. The
actual value of the heat transfer coefficient often shows a large deviation from
a theoretical value. Therefore, the results from eq. (B.1) are multiplied with
correction factors ki, ko that will be determined empirically
о’ = К. о, a" = ko - ar, (8.15)
280
o atp=5 bar
260 Oh at p=3 bar
oy, at p= 5 bar
240 . —-—— a, at p= 3 bar
220
200
180
160
140
heat transfer coefficient [WIm*K]
120
100
80
0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8
R22 mass flow rate [kg/min]
Figure 8.3 Theoretical heat transfer coeflicients for R22
The one-phase friction factors obtained from the Blasius equation eq. (B.10)
barely varied, &, = 0.036 ..0.044, €, = 0.017 .. 0.02; thus constant values will
be used
E = 0.04. in = 0.02 (8.16)
These values are valid for a hydraulic smooth pipe, which is an ideal assump-
tion. Moreover an additional pressure loss is likely to occur due to 12 ther-
mometers, which are placed inside the pipe in a distance of 20 cm starting at
the outlet. To account for these installations and the actual roughness of the
pipe, € and £" will be adapted to the measured pressure loss.
In the two-phase region empirical correlations for œ and € are used, which are
based on the values for a one-phase flow with the same mass flow rate, i.e. the
values computed above. In appendix B these correlations are rearranged into
the following form for the heat transfer coefficient
fH
a= — (8.17)
(02 _ 0.014122 + (a)220-01 p72
ol
Chapter 8. Modelling of an Evaporator
and the friction factor
25
/
= 2° + (1 — 2)? +39 (1— 1) о; (8.18)
P
These equations provide a continuous transition into the one-phase values at
z = 0 and & = 1. The functions di, $2 and ¢3 are listed in section B.1 and
B.2, which also contain plots of the above equations.
In the dynamic heterogeneous model, the interfacial friction factor C* is an
additional parameter that will be approximated by a constant. As shown in
section 6.4, it can also be expressed as a function of £.
8.2 Implementation
The model described above is implemented in the object-oriented modelling
language Modelica [5]. Fig. 8.4 gives an overview over the code structure. The
arrows indicate where the equation is placed that provides the information
required to compute the related variable.
jet] “ hs] hs] fT [right
А fluid
fluid_in wall fluid _out
T+,
HQ
Ory f
, ext
ref_in wall ref_out
lhs int | rhs]
Ap Al. À
A TQ ml
ym Oly Y P
left| г wal right
1 | Min Q PIPE 4
o_O Ao Yin Pouy Neon ,
ED —] a E
, GA | y ' ! * i
htc | 3 therm ВН flo ¡EC fric
= о о — n — L p = о = xy am em a
p A op/ah | P ho о 7
y T (Pro Joep |" Ар | п Р
ah BA EA Eh A kd dom hE BAR AA RE ok kh Ake Ad kk dk hom hhh od hdd EAE bdrm Era er hr rE Ted Er ia EP Ara Eanes kasama al
DOTA D
©
=
о
----—- 5 ыы вы ан ME ded mek ен dee dem нк н= н= ен нк ее н== ее = == нк == == == == == == == === == == == жк жет тт ото сша EA Ea ны EA жи el dk нею ан зы ее dem dk ны ее == == = = = — od
Figure 8.4 Evaporator model
The evaporator model contains the models for the fluid, the wall and the pipe.
Additional models are used for the boundary conditions of the flows at in-
58
8.2 Implementation
and outlet. The naming refers to the design flow direction of the refrigerant;
a counter-flow heat exchanger is obtained by assigning a negative fluid mass
flow rate to fluid_out.
The pipe model contains the thermodynamic model therm (computed in the
thermal grid) and the hydrodynamic model flo (computed in the flow grid).
The analytical equations for friction factors and heat transfer coefficients are
placed within fric and htc. The model pro serves to compute thermodynamic
properties including inlet and outlet; the related vector has thus n + 2 com-
ponents, where n is the number of thermal cells.
59
9. Simulation of an
Evaporator
The simulation program Dymola is used to evaluate the evaporator model.
Version 4.0, released in July 1999, is the first that supports the Modelica
language. The simulation results will be compared with data obtained from
measurements carried out by Antonius [1] at Danfoss, Denmark. Fig. 9.1 shows
a simplified diagram of the refrigeration test plant and the data obtained from
measurements. |
Cooling Tower
A
| | Condenser
MN. q.
P. Te
R22 cycle
Expansion Valve | Ce oyo > O Compressor
A ,
£ Mou
inlet outlet O Pout , Tout
Evaporator
Tour O—— fluid cycle o My
fin
>"
Figure 9.1 Refrigeration test plant
3
Some of the measured variables will be selected to serve as boundary condi-
tions for the simulation. Steady state measurement results are used to identify
heat transfer coefficients and friction factors. The model is finally applied to
simulate a transient process. The measurement data will be presented at first
to give an overview over the thermohydraulic behaviour of the test plant.
9.1 Measurement Data
À part of a measurement recording of t = 400s duration was selected. During
this period the compressor speed is changed four times between certain levels.
It regulates the mass flow in the refrigerant cycle and thus the removal of heat
from the fluid cycle.
Hydraulic Behaviour
For a given compressor speed a certain volume of refrigerant per unit time is
removed from the evaporator. As the speed is reduced at t = 25s, the outlet
mass flow rate instantly drops, fig. 9.2. Since the entering mass now exceeds
the leaving mass, the amount of mass inside the evaporator increases and thus
the pressure rises, fig. 9.3.
While the evaporator pressure rises, the condenser pressure decreases due to
the reduced entering mass, fig. 9.3. As a result of the decreasing pressure dif-
60
9.1 Measurement Data
mass flow rate
1.4
, \
0.6
=, ret + —T Ma A
0.4 ¡DEIA Ao. I iA ©
0 50 100 150 200 250 300 350 400
Time [s]
mass flow rate [kg/min]
0.2
Figure 9.2 Measured entering and leaving mass flow rates
ference at the expansion valve, Тит, 15 reduced. At the same time mou: begins
to rise again, which is due to the increasing density at the evaporator outlet.
The mass flow rates finally agree on a reduced level at à = 60s. The evapora-
tor pressure has then reached a maximum value and begins to decrease again,
because of a lowering heat flux towards the refrigerant.
Condenser pressure (outlet)
16.5
16
15.5
15
Pressure p [bar]
14.5
0 50 100 150 200 250 300 350 400
Time [s]
Evaporator pressure (outlet)
5.5
5 en AS)
bf] [|
N || | — \—
О 50 100 150 200 250 300 350 400
Time [s]
Pressure p [bar]
IN
—
Hla
TT
Figure 9.3 Measured condenser and evaporator pressure
When the compressor speed is increased at t = 147s, the same process takes
place with reversed signs. Now the evaporator pressure drops due to the re-
moval of mass at the outlet, while the condenser pressure rises due to the
added mass. After adaption of the mass flow rates at t ~ 170s, the evapora-
tor pressure rises due to the increased heat flux towards the refrigerant.
61
Chapter 9. Simulation of an Evaporator
Thermal Behaviour
The driving force for the heat flux towards the refrigerant is the temperature
difference between the refrigerant and the fluid. Since the entering refrigerant
is a two-phase mixture with x = 0.2 mass content vapour, its temperature is
determined by the pressure 7 = Tsat(p). The saturation temperature related
to the outlet pressure is shown in fig. 9.4. It also includes the refrigerant outlet
temperature Tout and the fluid inlet temperature 1’ ;,.
15 т
— R22 outlet temperature
—————-- Fluid inlet temperature
>" E E ee TTT R22 saturation temperature
Temperature °C
o
0 50 100 150 200 250 300 350 400
Time [s]
Figure 9.4 Measured refrigerant and fluid temperatures
The heat flux towards the refrigerant rises with the difference between the
fluid temperature and the saturation temperature of the refrigerant AT =
Trin — Tsat(p). Therefore, a reduced evaporator pressure, caused by a high
compressor speed, leads to an increased removal of heat from the fluid cycle.
It also provides an increased refrigerant mass flow rate, which is necessary to
remove the heat from the evaporator.
After the refrigerant is evaporated, the vapour is superheated up to a temper-
ature 7, close to the fluid inlet temperature. Most of the time, the measured
outlet temperature exceeds the fluid temperature by 1K, which is actually
impossible and must be due to a warming of the refrigerant at the outlet or a
measurement error.
When the compressor speed is reduced at t = 245s, the saturation pressure
rises. Since Ty, has decreased, the temperature difference AT is now just
about sufficient to enable a complete evaporation of the refrigerant, and thus
the superheat begins to decrease. As the compressor speed is increased again at
$ = 305s, the superheat has finally reduced to 3 K. Now the refrigerant in the
pipe is almost completely in state of saturation and thus To. initially drops
even more due to the reduction of the pressure. As the temperature difference
gets larger, the heat flux is increased; the fraction of the pipe required for
evaporation gets smaller and Tout rises again.
9.2 Steady-State Simulation
A steady state simulation is used to identify the empirical parameters, i.e. the
one-phase friction factors &', €” and the correction factors for the one-phase
62
9.2 Steady-State Simulation
heat transfer coefficients k;,k9. The number of cells is n = 5; a higher number
of cells does not give a noticeable change of simulation results.
Boundary Conditions
The boundary conditions chosen are
e R22 inlet enthalpy hin
R22 inlet pressure Pin = Pout + Ap
R22 inlet mass flow rate Min = Me
R22 outlet volume flow rate Vout
e fluid inlet temperature 7 in
fluid mass flow rate m f
The inlet mass flow rate mn is set equal to the measured mass flow rate at
the outlet of the condenser, because the throttling process can be considered
as stationary. The inlet enthalpy h;n is obtained from an energy balance for
the expansion valve. For a stationary, adiabatic, homogeneous flow eq. (4.51)
gets, after division by m,
2 2
w Ww;
hin = he + > — > +g (Ze — Zin) (9.1)
where z is the position vector. The last term is negligible. During the throttling
process the refrigerant evaporates partially, which causes a decrease of the
density and thus an increase of the velocity. The reduction of enthalpy due to
this effect will, however, be neglected for simplicity, thus
hin = he = h(Te, Pe) (9.2)
The pressure at the inlet p;, is required to compute the saturation densities
and the void fraction of the entering refrigerant. It does not determine the
pressure inside the pipe, which is a simulation result and may differ from p;,.
In agreement with [1] the outlet volume flow rate is chosen as a boundary con-
dition, since the compressor removes a certain volume per unit time, regardless
of the density. The data are computed from
Mout
Voy = — out
out PlPout, Tout)
(9.3)
The following steady-state data [1] are used as boundary conditions
Table 9.2 shows the measured pressure drop and outlet conditions. lt also
contains results of the simulation program SINDA FLUINT [1] including the
charge M.
63
Chapter 9. Simulation of an Evaporator
No. | тт Vout iy Trin Te Pe Pin hin
kg/min m*/min kg/min °C °C bar bar kJ/kg
1 0.60 0.044 0.22 0.7 26.6 15.9 3.39 232.3
2 0.69 0.045 0.28 4.2 28.5 17.1 3.91 234.7
3 0.48 0.042 0.28 -1.8 25.9 15.0 2.88 231.4
4 0.72 0.046 0.33 9.4 30.0 18.0 4.12 236.6
Table 9.1 Boundary conditions for stationary simulation
No. source Pout Ар Tout = Trou M
bar bar °C °C g
1 measured 3.29 0.090 -2.580 -230 —
homogeneous | 3.20 0.045 -13.70 -2.28 17
heterogeneous | 3.20 0.050 -13.70 -2.29 61
2 measured 3.77 0.110 2.30 150 —
homogeneous | 3.66 0.055 -3.40 1.40 18
heterogeneous | 3.66 0.060 -4.10 1.37 62
3 measured 2.79 0.080 -140 -3.80 —
homogeneous | 2.75 0.035 -3.96 -3.84 12
heterogeneous | 2.75 0.038 -4.05 -3.87 32
4 measured 3.93 0.120 9.30 7.00 —
homogeneous | 3.95 0.058 7.74 6.86 16
heterogeneous | 3.96 0.063 7.70 6.84 41
Table 9.2 Simulation results of SINDA FLUINT [1]
Simulation Results
In table 9.3 the simulation results of the homogeneous and static heterogeneous
model are compared with measurement data. The theoretical one-phase fric-
tion factors and heat transfer coefficients were used, i.e. €, = 0.04, €, = 0.02
and оу, ay, from fig. 8.3. Both models yield almost identical results concern-
ing the intensive variables. Compared to the measurements, poy: and T' out
show reasonable good agreement. However, the refrigerant outlet temperature
is up to 8 K below the measured value and the pressure drop is too low by a
factor 4. The result is thus similar to the one of SINDA FLUINT, table 9.2,
where the pressure loss is too low by a factor 2 and the superheat is too low
by 1.5.. 11K.
The low simulated pressure loss is a result of the ideal assumption of a hy-
draulic smooth pipe and the neglect of the pressure loss due to the pipe elbows
and the thermometers placed inside the pipe. Further pressure losses may oc-
cur in the space between the measuring point and the in- and outlet of the pipe.
From the measured pressure drop the actual friction factors can be identified
as € = 0.1 and €” = 0.1.
64
9.2 Steady-State Simulation
No. source Dout Ар Tout Trou M
bar bar °C °C g
1 measured 3.29 0.090 -2.50 -2.30 —
homogeneous | 3.20 0.026 -10.29 -247 20.8
heterogeneous | 3.20 0.026 -10.30 -2.47 64.6
2 measured 3.77 0.110 2.30 1.50 —
homogeneous | 3.66 0.032 -3.66 1.31 22.9
heterogeneous | 3.66 0.032 -3.07 1.32 65.4
3 measured 2.79 0.080 -1.40 -3.80 —
homogeneous | 2.78 0.020 -4.07 -3.86 14.4
heterogeneous | 2.78 0.020 -4.07 -3.86 36.4
4 measured 3.98 0.120 9.30 7.00 —
homogeneous | 3.92 0.033 6.00 6.78 20.4
heterogeneous | 3.92 0.033 6.00 6.78 46.9
Table 9.3 Simulation results of Dymola
The low simulated superheat may be due to the application of eq. (8.17) in
the two-phase region, which is valid for convective evaporation and does not
account for nucleate boiling, which intensifies the heat transfer and may appear
at low void fractions. Moreover, liquid drops are still present when the vapour
is superheated and thus a higher heat transfer coefficient may be present in the
region where due to h > h”(p) the value for pure vapour is used. Measurements
on boiling nitrogen, cited in [14], show that for a value of # = 1, computed
from eq. (6.15), the actual flow quality is 0.6..0.8. From fig. B.1 it can be seen
that the heat transfer coefficient is then approximately six times as high as
the related value for pure vapour.
Considering the influences discussed above and the general uncertainty con-
cerning the prediction of heat transfer coefficients in boiling two-phase flows,
it appears to be justified to apply correction factors ki, kg, as proposed in
eq. (8.15), to achieve better agreement with the measured superheat. Table
9.4 shows the simulation results obtained for В) = 2 and ky = 2. The friction
factors are &’ = 0.1 and £” = 0.1.
The deviation from the measured outlet temperature is reduced to less than
2K. The error of the pressure loss is less than 10%. The above values for the
correction parameters are thus kept for further simulations.
From the tables it is already apparent that the mass inside the evaporator is
up to three times bigger in the heterogeneous model. This is due to the lower
void fraction y in a heterogeneous flow, which has been discussed in section
6.2. From p = vp" + (1 — y)? follows that for a given pressure the density is
higher in a heterogeneous flow. The charge has a huge effect on the transient
behaviour, as the following section will show.
05
Chapter 9. Simulation of an Evaporator
No. source Pout AD Tout Trou M
bar bar °C °C g
1 measured 3.29 0.090 -2.50 -2.30 —
homogeneous | 3.30 0.084 -3.59 -2.55 20.3
heterogeneous | 3.30 0.085 -3.60 -2.55 58.3
2 measured 3.77 0.110 2.30 1.50 —
homogeneous | 3.72 0.099 0.50 1.27 214
heterogeneous | 3.72 0.099 0.50 1.27 55.0
3 measured 2.79 0.080 -1.40 -3.80 —
homogeneous | 2.80 0.072 -2.61 -3.88 13.0
heterogeneous | 2.80 0.072 -2.61 -3.88 27.0
4 measured 3.98 0.120 9.30 7.00 —
homogeneous | 3.95 0.114 8.02 6.76 18.7
heterogeneous | 3.95 0.114 8.02 6.76 36.9
Table 9.4 Simulation results of Dymola - fitted parameters
9.3 Transient Simulation
Boundary Conditions
The measurement data have been presented in section 9.1. The data used as
boundary conditions, i.e. time-dependent data for the variables in table 9.1,
were computed from linear interpolation
t— ti
b(t) = Plti) + ——— Wti+1) — Y(ti)] for
tL <t < L,
‘1 — Li Pes
(9.4)
C-code was created to apply this interpolation automatically to data tables.
The above method was not used for the outlet volume flow rate. As fig. 9.5
shows, heavy oscillations appear, which are obviously due to statistical mea-
surement errors. To provide physically realistic boundary conditions, Vout is
approximated by piecewise constant functions, except at fast transients, where
third order polynomial functions are used. The result is shown in fig. 9.5.
The fluid mass flow rate is constant m; = 0.33kg/s. The inlet enthalpy is
almost constant, h;, = 233.234 kJ/kg. The number of pipe segments is n = 5.
Simulation Results
Fig. 9.6 compares the measured pressure loss with the simulation results of the
homogeneous and the heterogeneous model. À large pressure loss occurs when
the pressure is low, which is due to the low related vapour density causing a
high velocity at the outlet. The resulting reaction force is particularly high
due to the large mass flow rate present at the same time, which also increases
the friction force.
Both models yield good agreement with the measured pressure loss, except
when the compressor speed is increased at t = 147s and t = 305s, where
66
9.3 Transient Simulation
0.07
measured
polynomial fit
0.06
=
=
5 0.05
E,
©
©
> 0.04
©
QO
5 0.03
o
>
0.02
0.01
0 50 100 150 200 250 300 350 400
Time [s]
Figure 9.5 Outlet volume flow rate
0.25 T
measured
———==—- homogeneous
rns heterogeneous
— 0.2
о
==}
CL
<
8 0.15
c
©
2
> 0 |
2 ес et
A PET —
© |
- 0.05 Kad WN
А ALAA ahah a |
E Y + tas Tay v
0
0 50 100 150 200 250 300 350 400
Time [s]
Figure 9.6 Simulated pressure drop in the evaporator
the homogeneous model produces a too quick decrease from an initial peak.
The same effect is obvious from fig. 9.7, which shows the simulated outlet
pressure. The homogeneous model produces large overshoots as the pressure
changes. Application of the heterogeneous model yields a qualitatively better
agreement with the measured value. The error is less than 10% during the
whole measurement recording.
The improved accuracy of the heterogeneous model is caused by the higher
simulated charge, fig. 9.8: Due to the lower void fraction, more liquid mass
is present. As the compressor speed is increased, the pressure drops and the
liquid evaporates. As a result, the outlet mass flow rate suddenly rises and then
slowly decreases as the amount of liquid is reduced, fig. 9.9. In the homogeneous
model, the amount of liquid is lower and thus the outlet mass flow rate adapts
too quickly to the given inlet mass flow rate (fig. 9.2).
Due to the increased accuracy of the pressure, a qualitatively better result
is also obtained for the outlet temperature. At t = 40s and t = 250s the
homogeneous model produces too high pressures, causing a low temperature
G7
Chapter 9. Simulation of an Evaporator
55
RENEE TUTTI i 7 |
4.5 E
2, Ef \
a 4 À
D \
5 | \
a 35 4
0 |
a. —— | \
3 | X mE Sa AT
/ /
2.5 |- measured У y
nui homogeneous
TT heterogeneous,
0 50 100 150 200 250 300 350 400
Time [s]
Figure 9.7 Simulated outlet pressure
120
homogeneous
100 heterogeneous
© 80
D
2 60
©
<
© 40
20
0
0 50 100 150 200 250 300 350 400
Time {s]
Figure 9.8 Simulated evaporator charge
measured
homogeneous
Mass flow rate [kg/min]
0 50 100 150 200 250 300 350 400
Time [s]
Figure 9.9 Simulated evaporator outlet mass flow rate
difference AT between the saturated refrigerant and the fluid. The simulated
heat flux is therefore too low and the superheat drops. At t = 250s the su-
perheat reduces to zero in the homogeneous model, while the heterogeneous
model gives a better picture of the decreasing superheat.
08
92.3 Transient Simulation
Temperature °C
measured
homogeneous
0 50 100 150 200 250 300 350 400
Time [s]
Figure 9.10 Simulated outlet temperature
As explained in section 9.1, the sudden temperature drop at ¢ = 305s is
caused by the decrease of the pressure and the related saturation temperature.
The simulation produces a similar effect, but there is a large deviation from
the measurement recording. The deviation is partially due to the high time
constant of the thermometer, causing a delayed notice of sudden changes: At
t = 147s the measured value only changes its slope, while the simulations
show a temperature drop similar to the one at ¢ = 305s.
The above results of the heterogeneous model are based on the static slip-flow
equation eq. (7.21). To apply the dynamic slip-flow equation eg. (7.22), the
factor (* must be determined. From fig. 6.5 we can read off a first guess: Since
the void fraction in the measured data is y > 0.8 we obtain (* = €. Using this
relation the dynamic slip-flow model gives nearly the same results as the static
slip-flow model. In the scaling used in the plots above no difference would
be visible. Fig. 9.11 gives a closer look at the outlet pressure: The dynamic
equation yields only a slight improvement.
1
measured
—————- homogeneous
нее static heterogeneos
e dynamic heterogeneous —
Pressure p [bar]
2.5
145 150 155 160 165 170 175 180
Time [s]
Figure 9.11 Simulated outlet pressure
The interfacial friction factor was also approximated by a constant value
69
Chapter 9. Simulation of an Evaporator
¢* = 0.3..0.5, fig. 9.12. For С* > 0.5 the result tends to the one of the ho-
mogeneous model, which is due to the increased interfacial friction, causing an
equilibration of the velocities. For {* < 0.3 the simulation gets unstable. While
C* = 0.3 turns out to be the appropriate choice for fast transients, (* = 0.5
provides increased accuracy in a steady-state. Application of (* = 0.4 yields
the best overall result.
i
measured
------- a = 0.3
ен С = 0.4
С° = 0.5
=
2,
co
D
=
0
© 35
a
3
2.5
145 150 155 160 165 170 175 180
Time [s]
Figure 9.12 Simulated outlet pressure
The results show that a better agreement with measured data can be achieved
by variation of (* instead of using a constant value. However, the result de-
pends very much on the values of a’ and a” which have been adapted very
roughly. Before determination of a function for (*, the heat transfer coefficients
should be computed more precisely.
70
10. Summary
Among the variables applied to describe two-phase flows, thermodynamic and
hydrodynamic properties can be distinguished. For each category a distinct
model is created. The thermodynamic model is based on the integral form of
the balance equations for mass and energy, which yield differential equations
for density p and specific internal energy u. To increase the numerical stability,
these equations are transformed into differential equations for pressure p and
enthalpy À by symbolic inversion of the Jacobian matrix.
Simple equations of state are created using p and À as independent param-
eters. The properties in the two-phase region can be expressed as a linear
interpolation between dew- and boiling point. In the one-phase region Taylor
expansions at the dew- and boiling point are used, which provide a continuous
transition on the phase boundaries. The approach requires solely functions for
the properties and certain derivatives of saturated liquid and vapour, which
can be approximated by polynomial functions. The technique was applied to
the refrigerant R22 on the basis of data computed from the NIST REFPROP
database.
A hydrodynamic model is derived in order to determine the convective terms
appearing in the thermodynamic model. In the case of homogeneous flow as-
sumption, sufficient information is provided by a differential equation for the
mass flow rate which is derived from a momentum balance. In case of a hetero-
geneous flow an additional equation is required to account for the difference of
the average velocities of vapour and liquid Aw. An algebraic equation for Aw
is obtained from an empirical equation for the flow quality. As an alternative,
a differential equation for Aw is derived from separate momentum balances
for each phase.
The separate momentum balances contain terms quantifying momentum ex-
change and friction forces between the phases. Common approaches cause
singularities in the limiting case of a vanishing phase. To ensure a safe tran-
sition into the one-phase region, these terms are modelled under the premise
that in the limiting cases realistic results are obtained instead of singularities.
A relation for the mass flow through the interfacial area follows from a vapour
mass balance. The interfacial friction is expressed in terms of a factor C*, which
is defined in analogy to the wall friction factor © in a one-phase flow.
Thermo- and hydrodynamic model form a system of coupled differential equa-
tions. Following the principles of the finite volume method, the equations for
each model are discretized to account for the spatial distribution of properties.
Thermo- and hydrodynamic equations are applied to different grid structures.
The properties on the boundary of a pipe segment are approximated by the
average properties of the cell located upstream. Complementary models are
required to compute the heat transfer through a heated or cooled pipe wall.
For heat transfer coefficients and friction factors empirical correlations are
selected that provide a continuous transition on the phase boundaries.
The model is applied to simulate an evaporator in a refrigeration test plant.
Measured inlet conditions and outlet volume flow rate are used as boundary
71
Chapter 10. Summary
conditions. The simulated pressure loss and superheat temperature are too low
compared to the measured values; by increasing the one-phase friction factors
and heat transfer coefficients a better agreement is achieved. In steady-state
and slow transient simulations the homogeneous and heterogeneous model
yield almost identical results, except for the charge, which is several times
larger in the heterogeneous model compared to the homogeneous model.
The influence of the different charges becomes evident in fast transient pro-
cesses: As the evaporator outlet volume flow rate is abruptly increased, the
heterogeneous model produces a delayed pressure drop due to the time it takes
to evaporate the large amount of liquid mass in the evaporator. Contrary to
this, application of the homogeneous model causes overshoots of the pressure
that are not in agreement with the measurement recording. The maximum
error of the simulated pressure is 45% in the homogenous model compared to
10% in the heterogeneous model.
The simulated superheat temperature is only roughly in accord with mea-
surement data; the deviation is due to the high time-constant of the thermo-
element, which thus cannot detect the sudden temperature changes predicted
by the simulation. Moreover, the computation of heat transfer coefficients in
boiling two-phase flows is still afflicted with a huge inaccuracy, and a precise
determination of a function for a in the given evaporator has not been at-
tempted. The heterogeneous model provided only a slight improvement due
to the increased accuracy concerning the pressure.
The differential equation for Aw in the heterogeneous model gives almost the
same results as the related analytical equation, if the interfacial friction factor
C* is set equal to the wall friction factor E, which is justified in annular flows.
In some cases it is possible to obtain a better agreement between data and
simulation by adaption of {*. Due to the strong influence of the heat transfer
coefficient, which had been adapted very roughly, a more accurate determi-
nation of (* should not be attempted before the heat transfer coefficients are
computed more precisely.
The transition from the two-phase region into the vapour region did not cause
any numerical problems. The transition into the liquid region is not contained
in the measurement recording. Test runs using fictitious boundary conditions
show that numerical problems arise when the void fraction in a pipe segment
gets zero, which is due to the abruptly decreasing compressibility. This problem
can be solved by application of separate energy balances for each phase, but
this is beyond the scope of this thesis.
[1] ANTONIUS, JESPER
Distribuerede fordampermodeller pa flere detaljeringsniveauer
Danmarks Tekniske Universitet, 1998
12] BAEHR, HANS DIETER; STEPHAN, KARL
Warme und Stoffubertragung
3. Auflage, Springer-Verlag, Berlin, Heidelberg, 1998
[3] BRONSTEIN, I.N.; SEMENDJAJEW, K.A.
Taschenbuch der Mathematik
14. Auflage, Verlag Harri Deutsch, Thun und Frankfurt/Main, 1989
72
E
[12]
13]
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CAREY, VAN P.
Liquid- Vapour Phase-Change Phenomena
Hemisphere Publishing Corporation, 1992
ELMQVIST, H. ET AL.
Modelica Tutorial and Rationale
http://www.modelica.org
GUYON, ETIENNE; HULIN, JEAN-PIERRE; PETIT, LUC
Hydrodynamik
Verlag Vieweg, Braunschweig/ Wiesbaden, 1997
HETSRONI, GAD
Handbook of Multiphase Systems
Hemisphere Publishing Corporation, 1982
HEUSSER, PETER ANDREAS
Modelling and Simulation of Boiling Channels with a General Front
Tracking Approach
SCS - Society for Computer Simulation International, Ghent, 1996
KOHLER, JURGEN
Warme- und Stoffubertragung in Zweiphasenstromungen
Verlag Vieweg, Braunschweig/Wiesbaden, 1996
KOLEV, NIKOLAY IVANOV
Transiente Zweiphasenstromung
Springer-Verlag, Berlin, Heidelberg, 1986
MAYINGER, FRANZ
Strömung und Warmeubergang in Gas-Flissigkeits-Gemischen
Springer-Verlag, Wien, New York, 1982
NIST REFPROP DATABASE
National Institute of Standards and Technology, Version 6.0
PATANKAR, SUHAS, V.
Numerical Heat Transfer and Fluid Flow
Hemisphere Publishing Corporation, 1980
STEPHAN, KARL
Warmeubergang beim Kondensieren und beim Sieden
Springer-Verlag, Berlin, Heidelberg, 1988
STEPHAN, KARL; HECKENBERGER, THOMAS
Thermal Conductivity and Viscosity Data of Fluid Mixtures
Chemistry Data Series 10, Frankfurt am Main, 1988
STEPHAN, KARL; MAYINGER, FRANZ
Thermodynamik, Band 2: Mehrstoffsysteme und chemische Reaktionen
14. Auflage, Springer-Verlag, Berlin, Heidelberg, 1999
VDI-WARMEATLAS
8. Auflage, Springer-Verlag Berlin, Heidelberg, 1997
WALLIS, GRAHAM, B.
One-Dimensional Two-Phase Flow
McGraw-Hill, New York, 1969
73
Chapter 10. Summary
119] WANG, HONGWEI
Modelling of a Refrigerating System Coupled with a Refrigerated Room
Delft University of Technology, 1991
74
Balance Equations for
Kinetic and Internal
Energy
An infinitesimal small cubical fluid particle with lateral length dz is considered.
The edges of the cube are parallel to a system of coordinates, whose directions
are numbered 7 = 1, 2,3. The velocity is counted positive in direction of the
coordinates.
A.1 Mass Balance
The mass within the cube dM = pdV changes due to the entering and leaving
mass in all three coordinates. Using a Taylor series approximation this gives
Op O(pw;)
* — > "
where the subscript j implies summation over j = 1,2; 3 (Einsteins’ summation
convention). Division by dV yields
др ‚ O(pw;)
Е — 0 А.2
ot oz; (4.2)
Ed. (A.2) will be used to simplify the derivative of a quantity py, which can
be expanded as follows
Mo) Apwib) — OY Ou Op | O(pu;)
ll pen = pL о — + —— А.З
or Taz, Pa Pig TV aT az (4.3)
According to eq. (A.2) the last term equals zero, therefore
opp) Mowz) _ OY ду (A4)
a oz; “a “az;
A.2 Momentum Balance
The change of momentum in direction of a coordinate %
Ö(pwi)
ve (A.5)
equals the sum of the momentums in direction ¢ transfered with the entering
and leaving mass in all directions 7 = 1,2, 3
O( pu;
LY (as)
pwiw;dA — | рии; + Oz 9.
75
Chapter A. Balance Equations for Ei, and U
plus the gravity force
pgidV (A.7)
plus the pressure force
pdA — р + Pg | dA = ———dzdA (А.8)
Zi
minus the forces due to shear stresses
- | mida — [+ а ал} = лгал (A.9)
where 75; is the shear stress on a surface 7 = const in direction of i. Summing
up and dividing by а\ yields
Opus) | Opuiv;) __ 0p Ory
= ; A.10
7 02, да dz 9 (4.10)
According to eq. (A.4) this is equivalent to
— = — A.11
p A + PW; dz; dz; + Oz 2j * + pgi ( )
A.3 Energy Balance
The energy stored within the cube is
dE = pedV (А.12)
where e is the specific energy, i.e. the sum of internal and kinetic energy
2
e=u+ = (A.13)
The convective transport of energy into the volume is
pew;dA — | pew; + Nvews) y, dA = _Ppeus) 4, dA (A.14)
Oz; Oz;
The power due to the forces is obtained from multiplication of the forces
introduced in the previous section with the related velocity component in all
three directions
_ Op) y. dA + si ritos) y. dA + pg;w;dV (A.15)
OZ; Oz;
76
A.3 Energy Balance
The energy also increases due to convective transfer of heat into the volume
. 0g; да;
4 + 28| = _ 98 Ал
1 Ё + 2 OZ; 6)
Altogether this gives
O(pe) , O(pew;) Opw; O(mjiwi) да;
à "05 du | oz PIN Dm (AI
Kinetic Energy Balance
Multiplication of eg. (A.11) with w; and summation over i = 1, 2,3 yields
д и? д и? др OTij
Poy 7) + PU (3) = + Wi Oz + wi pg; (A.18)
which by application of eq. (A.4) turns into the kinetic energy balance
0 ( wf д и? др ÖTzi
Ea (o's) + 0 (ou =) = “Wig, + Way + Wi pg; (A.19)
The second term on the left side can be expressed with the divergence operator
divi) = Ov; /0z;, while the last term on the right side equals the scalar product
pgw. Integration over a control volume V gives
д и? a? Op От;
— | p— | dV di — | dV = dv — Wi — ‚| dV
[5 (03) + fair 05] = [00m [ugh sw
(A.20)
For a fixed control volume, the derivative operator in the first term can be
put before the integral. The second term can be transformed into a surface
integral by use of the Gaul theorem. This gives
d 2 2 y
pd | op wnda= f pgw vs) 2 ра TE dV
2 A 2 у V O Oz;
dt V 24
(А.21)
or
dPrin Op OT ji
_ inwn dA + P —Wi— + Wi 22
Ti KC wn dA + + [| Wig, TW | dv (A.22)
where FP, is the power due to gravity forces.
77
Chapter A. Balance Equations for Erin and U
Internal Energy Balance
Separating the internal and kinetic energy on the left side of eq. (A.17) yields
д(ри) Ölpuw;) Ölpe) Opew;) 0 ( w? 0 Wi
a общ дб оды > laz) 5 2
2
(A.23)
The terms on the right side can be replaced by eq. (A.19) and eq. (A.17)
Apu) Ölpuw;z) д(ри;) | O(Tjiwi) 0g;
— TT __—_—] ET ; ; — А.24
ot a dz Toy 1 PAS gl (424)
Op OT;
wig wig + Wing (A.25)
rearrangement yields
O(pu) O(puw;) (pw) Op ow; дд;
ANTI — ; — A.26
ot + Oz; OZ; tw "Oz; + Oz; Oz; ( )
Integration over a control volume gives
Öl(p . . |
| ze Dav | div(puw) dV = - || div(pw) av— divg dV
у V
др ди
; 2
+ |v i, + | a (A.27)
For a fixed control volume after applying the Gauf theorem we get
d д ow;
dv == [ тат аа — | от ау — | впал + | |w + + Tj =
dt A A A oz Oz;
(A.28)
Using P, and Q to denote the power due to pressure forces and the heat flux
we obtain
dU
de =>] runda+P+0+ /
e + rie] dv (A.29)
dt у
' Oz; Oz;
In eq. (A.26) the pressure terms may be simplified analogously to the terms
containing the shear stresses. This leads to an equivalent equation
dU ди ди
_ ‘то A.30
ET [ост ал + 9+ || |- Poe + ao | a ( )
78
B. Correlations for heat
transfer coefficients and
friction factors
B.1 Heat Transfer Coefficient
One-Phase Region
The heat transter coefficient for turbulent one-phase flow inside a horizontal
pipe of length Az is obtained from [2]
(5/8) (Ве — 1000)Pr D -
№ 1+ ( < В.
1+12.7,/€/8 (Prè/3 _ 1) | + (=) (B.1)
where € is the friction factor
& = (0.79 п Re — 1.64)” (B.2)
The equation is applicable for 2300 < Re<5 - 10°, 0.5 < Pr<2000, Az/D > 1.
The Nusselt, Prandtl and Reynolds number are defined as follows
D .
aD Pr — 1% ре = PWD_ MD
Nu =
6 A A 7 An
Therein A denotes the thermal conductivity, 7 denotes the dynamic viscosity
and c¢, denotes the specific isobaric heat capacity.
In an isolated ring slot the heat transfer coefficient for the heat transfer towards
the inner wall can be estimated from [2]
N on —0.16
— slot — 0.86 (5) (B.4)
Nu ext
where Nu is computed from eq. (B.1) with the diameter D replaced by a hy-
draulic diameter Dy, = Dext — Dint.
Two-Phase Region
The heat transfer coefficient in a horizontal pipe in the case of convective
evaporation can be expressed as a function of the related values a’ and a” for
79
Chapter B. Correlations for a and €
a liquid and vapour flow with the same mass velocity G = pw [17]
—2.2
a 0.01 . 04{0 037
—0.5
0.67) 17°
+ 4001 = | + 8(1 — 2)°7 (5) || (В.5)
о о” -
Eq. (B.5) can be written in a more compressed form
RA
а = ce (B.6)
(020 _ 30014722 + (a/)220-01 4
with
г \ 0.37
di = 1-i+1.24% 5) (B.7)
z\ 0.67
фо = 1+8(1— 3)” (5) (B.8)
Fig. B.1 shows the result obtained for R22 with o/ = 150 and a” = 120 W/m*K.
two-phase heat transfer coefficient a
1600
—— p=1bar
1400 HF O 6 bar
1200 En b= 100 еше
1000 те ene he A
800 ne TOO | |
600 oT ma Eee ae " |
TT A vam TT ел 3
fe TUTTO E me Sá
RM TT ETE e
ищет!
400 =f>- aaa еее у
7 зон X
pame
>
200 #
0 0.2 0.4 0.6 0.8 1
flow quality
heat transfer coefficient [W/m*K]
Figure B.1 Two-phase heat transfer coefficient
B.2 Friction Factor
One-Phase Region
In a one-phase flow, the frictional pressure drop in a pipe is written as
dp 1 pwjw|
a AE nn ——— B.
dz D 2 B-9)
80
B.2 Friction Factor
The friction factor € may be obtained from the Blasius equation
С
ал (В.10)
© =
where Re is the Reynolds number, see eq. (B.3). The values of C' and n depend
on the state of the flow and the roughness of the pipe. For laminar flows is
C = 64 and n = 1. For turbulent flows in smooth pipes the exponent gets
n = 0.2..0.25. The constant is C = 0.184 for n = 0.2 and C = 0.3164 for
n = 0.25 [9].
Two-Phase Region
The frictional pressure drop in a two-phase flow is usually expressed as a
multiple of the one-phase pressure drop, e.g.
dp 2 dp|
— = 95 = B.11
de dz, 8.1)
where Фо is a modified two-phase multiplier and the liquid pressure drop is
computed for a liquid flow with the same mass velocity G' = pw [2]
dz |, D 20 Dp 2
Inserting the last expression into eq. (B.11) yields
dp 1 pwlw|
== == ее ее В.13
dz D 2 ( )
which is identical with eq. (B.9), but £ gets
E
е = %%5, (B.14)
The factor ®3 can be obtained from an empirical correlation [11]
2 2.25 P 0.685 0.24 (PY
% = (1-9 +37 + 3433001 — a) (55)
0.22 0.89
, (5) (1 _ = Fr 0.047777, — 0.0334 (B.15)
7) 7
The Froude and Weber numbers are computed for a liquid flow with the same
mass velocity
G? G?D
— lal(9)?D We = op!
(B.16)
31
Chapter B. Correlations for a and €
where o is the surface tension. Inserting eq. (B.15) into eq. (B.14) gives
E e 92€ 0
5 — E + (1 — 1) + 089 (1 — à) 024 ba (B.17)
with
7x 0.8 nN 0.22 nN 0.89
da = 3.43 (55) (5) (1 — 5) Fr70.047yye70.0834 (В.18)
Pp 7) 7)
For & = 0 and à = 1 the above equation yields € = €" and € = €" respectively.
Fig. B.2 shows the two-phase friction factor as a function of the void fraction
y = xp/p" for the homogeneous model © = x and for the static heterogeneous
model with à = &(p,2) computed from eq. (6.12) for a pressure p = 5bar,
a mass velocity G = 60kg/m?s, a pipe diameter D = 12.7mm and constant
one-phase friction factors €” = 0.04, €” = 0.02.
two-phase friction factor § at p= 5 bar
0.2
homogeneous
0.18 heterogeneous
0.1
friction factor
0.08
0.06
0.04
0.02
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
void fraction
Figure B.2 Two-phase friction factor
¢ is higher in a heterogeneous flow, because for a given flow quality # the void
fraction is lower, fig. 6.1, and thus the density is higher. The maximum ratio
E/p is the same in both models.
82
C. Thermodynamic
Properties and Derivatives
The approximation functions described in section 5.2 are based on data from
the NIST REFPROP database, which applies fundamental equations to com-
pute thermodynamic properties and derivatives.
C.1 Fundamental Equations
This section gives a short overview over fundamental equations in general. The
purpose is the derivation of some equations, which will be used later.
Basic Form
Comparison of the first law of thermodynamics
du = Tds — pdv (C.1)
with the total differential of a function u = u(s, v)
ou Ou
du = ps | 094 Da | de (C.2)
shows
ou ди
т = < —— —
OS |, Р до | (C-3)
Therefore, an equation of state u = wu(s,v) does not only serve to compute
the internal energy, but also yields temperature end pressure when being dif-
ferentiated symbolically. The enthalpy is then obtained from h = u + pv. An
equation of state, that allows all thermodynamic functions to be computed
from two independent variables, is called a fundamental equation.
Secondary Forms
The independent variables in u(s,v) may be changed and still the thermody-
namic state is determined, but the related equation may not contain sufficient
information to allow computation of all properties by differentiation. Loss of
information is avoided, when applying the Legendre transformation [14].
The Legendre transformation serves to create secondary forms of a fundamen-
tal equation ¢(z,y). In its simplest form, one independent variable is replaced
by the partial derivative with respect to that variable.
= 5) (0.4)
ow) (2
The derivative represents a tangent on a curve y = const in a (x) diagram.
To determine that tangent, its crossing point with the ordinate is required,
83
Chapter C. Thermodynamic Properties and Derivatives
Figure C.1 Legendre transformation
which gives the definition of the new function y. This definition makes sure,
that the new function 9 contains the same information as the original ¢. This
transformation is possible as long as Ow/Ox + 0, in other words, ¥(z,y =
const) is convex or concave for all y.
dan) + = ф— 99) а (C5)
y
Free Energy The function obtained from replacing the entropy in u(s, v)
with the related derivative is called the free energy f. Together with eq. (C.3)
the Legendre transformation yields
wo) > (32| но) = 100) (C6)
And the definition of f is found to be
u(s,v) = f=u— ди
B| $7 Ts (C.7)
U
Differentiation of the last expression gives, after replacing du with eq. (C.1),
df = du — Tds — sdT = —pdv — sdT (C.8)
Comparison with the total differential of f(7,v), similar to eq. (C.2), shows
of _ O
P= ди |; 5 OT |, (C-9)
Internal energy and enthalpy are obtained from u = f + Ts and h= u + pv.
Often v is replaced by p = 1/v to give a fundamental function f(7, p). In that
case the pressure is obtained from
— dp of
_ _ 29
dv Op
C.10
T Op )
T
84
C.1 Fundamental Equations
Free Enthalpy The fundamental equation for the free enthalpy g is ob-
tained from replacing the specific volume in f(T’, v) with the related derivative
fro) 9 (17
) = (Тр) (C.11)
T
The definition of g is found to be
v=f+pu (C.12)
T
то) от
Differentiation of the last expression yields with df from eq. (C.8)
dg = df + pdv + vdp = —sdT + vdp (C.13)
Comparison with the total differential of g(T, p) shows
99
8 == —
OT |,
y — 09 (C.14)
op |r
Enthalpy and internal energy are obtained from and h = g+Ts and u = h—pv
Enthalpy If the temperature in g(7,p) is replaced by the derivative of q
with respect to 1" we obtain the fundamental equation for the enthalpy h
дд
q(T, p) — h Е
») = h(s, p) (C.15)
defined by
да
9(Т, р) — В = 9 — ==
ЭТ Т = 9+ 5! (С.16)
14
The last term is differentiated and rearranged by use of eq. (C.13)
dh = dg + sdT + Tds = vdp + Tds (C.17)
Comparison with the total differential of h(s, p) shows
_ on
-
Oh
т = <
Os |,
(C.18)
U
The internal energy is obtained from u = h — pv. Replacing the pressure in
h(s,p) with 0h/0p|, yields u(s, v), which brings us back to the starting point
of eq. (C.1).
85
Chapter C. Thermodynamic Properties and Derivatives
C.2 Transformation of Partial Derivatives
From a fundamental equation ¢(z,y) derivatives of thermodynamic functions
with respect to x and y can easily be obtained from symbolic differentiation,
but a derivative
да,
| (C.19)
where b and/or c differs from x and/or y appears to require numerical ap-
proaches. As will be shown in this section, it can be reduced to derivatives
with respect to x and y.
First Derivatives
Let a and another variable 0 be functions of b and c.
a = a(b,c)
BD, с)
D
||
The related Jacobian matrix contains the partial derivatives of a and 6
да д
= | as| os (С.20)
ob дс
b
Its determinant is computed as follows
да
‚ Oc
_ д(а, В) _ да
det J = A(b,c) ab
ap
op
% ce (С.21)
p Ob],
In the special case В=с we obtain, since dc/dc|, = 1 and dc/0b|, = 0,
да
— 2
5 (C.22)
с
which is the derivative to be determined. The right side is expanded by appli-
cation of the multiplication theorem for functional determinants [3]
da
Ob
_ Ola, c) AT, y) — Aa,c)/A(7,4)
с OT, y) O(p, с) Ш O(b, c)/0(x,y) (C.23)
which is equivalent to
da
Ob
_ 0a/02|, 9c/Oy|, — da/0y|, Oc/0x|,
— 24
¿ 0b/0x|, Oc/Oy|, — 0b/Oy|, Oc/Ox}, (0-24)
36
C.2 Transformation of Partial Derivatives
This equation allows the derivative on the left side to be written in terms of
derivatives with respect to x and y. Further simplification is possible, if a,b
and/or c agrees with x and/or y, because
km em
which gives
= = ET (C.26)
= > “unt (C.27)
| — Del del, a (C.28)
Example: Velocity of Sound As an example from this thesis, eq. (5.16)
will be proved: 'The velocity of sound a is
_ [Op
a = ap |, (C.29)
therefore
=> (C.30)
Application of eq. (C.28) to the right side yields with (z,y) = (p, h)
From eq. (C.27) and eq. (C.18) we find
Sa = 5. == (C.32)
Therefore
а > DA, (C.33)
which is eq. (5.16). More applications are found in section C.3
87
Chapter C. Thermodynamic Properties and Derivatives
Second Derivatives
The equations above can easily be used to form second derivatives, which may
be written as follows
OA _ да
— ith А = = 34
Bl, with 5. (C.34)
The derivative OA/0B|, can be reduced to derivatives of A, B,C with respect
to x or y. Therein, the derivatives OA/Ox|, and/or OA/Oy|, will appear. To
compute these, A is reduced to derivatives of a, b, c with respect to 7 or y. Then
A can be differentiated by x and/or y, which gives the required derivatives. In
the same way derivatives of any desired order can be formed.
The above relations were implemented in Maple. The resulting program deriv
reduces any first or second derivative of the thermodynamic functions Г, р, v,
h, u, s, f, 9, YT to derivatives of the fundamental equations f(7,v), g(7,p) or
h(s,p). The fundamental equation for the free energy is also included in the
form of f(T, p).
For every fundamental equation several basic properties and derivatives are
used as an option to substitute for the derivatives of the fundamental equation.
In the fundamental equation f(T’, p) these are
р = Pb
s = —Jr
pr = Pre
Po = 2pfp+ о” Гор
с, = — fa
prr = Pp frre
Pop = 2fp + 4pfop + P* Fopp
Prp = 20fro+P fro
cor = —Jrr—Tfrrr
where c, is the specific isochoric heat capacity. Except for this quantity, sub-
scripts denote derivatives, e.g. pr denotes dp/9T|, and pr, denotes dpr/ Oplr-
Examples for the output of deriv:
pl, Pope + Тру Ohl, — Poco +Tpr
2
Op _— p (cop + PT) др — P PT (C.35)
C.3 Derivatives in the Two-Phase Region
In the two-phase region the derivatives cannot be obtained from differentiation
of the fundamental equation. The two-phase equilibrium conditions have to be
considered.
88
C.3 Derivatives in the Two-Phase Region
Density Derivatives
With (x,y) = (v, T) eg. (C.28) gives
—1
ul _ on _ 0p/0T1, (C.36)
ml, | |, Oh/0vlp 0p/0T|, — 0h/0T|, p/0v),
aul _ fap] _ Oh [OT (C.37)
Op |, | Ov h a Op/0v|p Oh/OT|, — Op/AT!|,, OR/Ou|r
Since h = u + pv the enthalpy derivatives are
öh| oul Op
ar |, — ar,” EN (C.38)
oh ди др
до! = 9). TY Bu), TF (C.39)
where Ou/0T|, is the specific isochoric heat capacity
ou
G7 =o (C.40)
U
and Ou/Ov|p can be obtained from differentiation of u = f + Ts. Employing
eq. (C.9) yields
de TE Ta - + (C.41)
Therefore
| = + (C.42)
=| = Tp + (C.43)
Now eq. (C.36)and eq. (C.37) get
Bal, = TET — оу Le
% A TORTEN (C-45)
In case of two-phase equilibrium the pressure is a function of the temperature
only, thus
Op} — Op| dp
E — 0 — (C.46)
т or|, ат
v
89
Chapter C. Thermodynamic Properties and Derivatives
The gradient of the saturation pressure is obtained from the Clausius-Clapeyron
relation
dp $—s 1h" —FH
A 4
ar ov’ —o T= (CAT)
The equations now simplify to
ии
де 1dl_v uv (C.48)
Ohl, Tdp s!—8
Ou| otoldp/dl) ea (dT ° vdr (C.49)
opl, T(dp/dT)? — TXdp T dp |
And for the density derivatives we find
dp 9 OU р? ат
ЭВ — | РО C.50
Oh |, "д, Tp (6:50)
ay (dTY T
Op) _ 0 PO (%) pa (C.51)
OD | OP | T \ dp T dp
Heat Capacity
lo compute the isochoric heat capacity in the two-phase region eq. (C.28) is
applied with (7, x) as independent parameters
In the last term, x can be cancelled down:
= 55 7 Bel. (C.53)
From eq. (C.43) and eq. (C.46) we find
= i = ad —p (C.54)
Differentiation of 4 = 24 + (1—x)W and v = xv + (1—x) yields, since the
liquid and vapour properties are a function of 7” only,
ди du” du’
ду de” dv’
The total differentials of u and v can be written as
du ou ou| dp
— = — — | — (C.57)
dr OT D op |p dT
dv ov ov| dp
о — — | = ‚58
dT OT р op |p dT (C58)
90
C.3 Derivatives in the Two-Phase Region
Application of eq. (C.26) yields
OT |, Ju/du|- OT |, др/ ди;
and from eq. (C.28) we find
ou ди ди || др/дТ'
— | =—| EN 2H C.60
Oplp ult OTI, OpYoulr |
Therein Ou/0T|, and Ou/0Ov|, are known from eq. (C.40) and eq. (C.41). Al-
together we obtain
со = TE, + (1-7), (C.61)
where ©) and &, are the limiting isochoric heat capacities on the dew and
boiling point when approached from the two-phase region
2
= 7 1 dp Op ”
Cp = Cp—— | == — => (C.62)
др/ди|т (2 AT |,
2
- T dp op!
a = cd — ———— | == == (C.63)
Op/ Ov | Е oT )
91
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