a novel special distributed method for dynamic refrigeration

8/6/2019 A Novel Special Distributed Method for Dynamic Refrigeration

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A novel special distributed method for dynamic refrigerationsystem simulation

F.Q. Wanga,*, G.G. Maidmenta, J.F. Missendena, R.M. Tozerb

aLondon South Bank University, Faculty of Engineering, Science & the Built Environment, 103 Borough Road, SE1 0AA London, UKbWaterman Gore-Mechanical and Electrical Consulting Engineers, Versaillers Court, 3 Paris Garden, SE1 8ND London, UK

Received 20 August 2004; received in revised form 13 September 2006; accepted 5 October 2006

Abstract

A novel dynamic mathematical model based on spatially distributed approach has been developed and validated in this paper.

This model gives good agreement in predicting the system COP and other parameters. The validated model has been used to

enhance the prediction of the micro variations of superheat and sub-cooling. The novel spatial distributed model for the con-

denser and evaporator in refrigeration system, calculates the two-phase region in gas and liquid field separately since the gas

and liquid in the two-phase region have different velocities. Previous researchers have used a pre-defined function of the

void fraction in their spatially distributed model, based on experimental results. This approach results in the separate solution

of the mass and energy equations, and less calculation is required. However, it is recognized that the mass and energy equations

should be coupled during solving for more accurate solution. Based on the energy and mass balance, the spatial distribution

model constructed here solves the velocity, pressure, refrigerant temperature, and wall temperature functions in heat exchangers

simultaneously. A novel iteration method is developed and reduces the intensive calculations required. Furthermore, the con-

denser and evaporator models have shown a parametric distribution along the heat exchanger surface, therefore, the spatial dis-

tribution parameters in the two heat exchangers can be visualised numerically with a two-phase moving interface clearly shown.

2006 Elsevier Ltd and IIR. All rights reserved.

Keywords: Refrigeration; Condenser; Evaporator; Two-phase flow; Superheating; Subcooling; Modelling; Heat transfer; COP

Simulation dynamique dun systeme frigorifique a laide dunenouvelle methode distribuee

Mots cles : Refrigeration ; Condenseur ; Evaporateur ; Ecoulement diphasique ; Surchauffe ; Sous-refroidissement ; Modelisation ; Transfert

de chaleur ; COP

* Corresponding author. Present address: Fulcrum Consulting, 62e68 Rosebery Avenue, London EC1R 4RR, UK. Tel.: 44 0207 520 1305;fax: 44 0207 520 1355.

E-mail address: [email protected] (F.Q. Wang).

0140-7007/$35.00 2006 Elsevier Ltd and IIR. All rights reserved.

doi:10.1016/j.ijrefrig.2006.10.010

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Please cite this article in press as: F.Q. Wang et al., A novel special distributed method for dynamic refrigeration system simulation, Int. J.

Refrigeration (2006), doi:10.1016/j.ijrefrig.2006.10.010
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1. Introduction

Generally, in refrigeration system simulation, heat ex-

changer models dealing with unsteady compressible two-

phase flow fall into one of two categories: the lumped

parameter approach or the spatially distributed approach

[3]. The lumped parameter model (LPM) divides the heat

exchangers into mainly three regions, the superheated gas

region, the two-phase region, and the sub-cooling region.In each region, average values are used to represent the para-

meters of thewhole region. The spatial distributed parameter

method considers the parameters in each region to vary. The

complexity of the spatially distributed approach is much

greater than the lumped parameter model. The main merit

of the spatial model is that the parameter distribution over

distance is detailed and the moving two-phase interface is

clearly defined. Therefore, the heat exchange processes in

the refrigeration system can be displayed numerically but

also more precise results can be produced.

During transient operation, the refrigerant mass flow rate

is continuously changing, which causes spatial variations in

the refrigerant distribution in the system components as well

as variable refrigerant states at the inlet and outlet of each

component. It is seen that the mass distribution and other pa-

rameters within the heat exchangers are functions of time

and space. Hence the spatially distributed model has been

used by various researchers. The transient simulation of

the refrigeration system was reviewed by a number of re-

searchers and their key conclusions in relation to spatial

distribution modelling is described below:Chi and Didion [1] simulated an air-cooled system with

moving refrigerant phase boundaries, with each phase region

treated as a counter-flow heat exchanger. Refrigerant flow in

both heat exchangers is one-dimensional and homogenous.

Internal resistances of metallic components were neglected.

A direct expansion evaporator for water-cooling was simu-

lated by Yasuda et al.s [9] model. The refrigerant in each

phase in the two-phase region was treated using lumped pa-

rameters, while the superheated region was simulated using

a finite difference approach. The method considers the heat

transfer distribution within the evaporator tube material, sec-

ondary fluid and refrigerant phases. This approach was used

Nomenclature

A area (m2)

C specific heat (kJ kg1 K1)

D diameter (m)

F force (N)G refrigerant mass flow flux (kg m2 s1)

g gravity constant (m2 s1)

h enthalpy (J kg1)

i control variable in a loop

L length (m)

_m mass flow rate (kg s1)

Q heat flux (W m2)

P pressure (Pa)

R gas constant (J kg1 K1)

Re Reynolds number

r radius (m)

T temperature (K)

Tr reduced temperaturet temperature (C)

Tf heat transfer fluid temperature (K)

U overall heat transfer coefficient (W m2 K1)

VV volume (m3)

v specific volume (m3 kg1)

W velocity (m s1)

x coordinate (m)

X dryness fraction

Xdif the cell coordinate (m)

Xcv the cell interface coordinate (m)

Y the displacement of the spring in TEV

Greek symbols

a heat transfer coefficient (W m2 K1)

r density (kg m3)

l thermal conductivity (W m1 K1)

hf fin effectiveness

ttime (s)J void fraction

6 wetted perimeter (m)

Suffix

Cr refrigerant side

w wall

ca air side

tp two-phase region

cond condenser

tev thermal expansion valve

comp compressor

sub sub-cooling

evap evaporatorsat saturation or saturation point

f heat transfer fluid

s constant entropy process

g gas

r reduced parameters, refrigerant

i inside

r3 the compressor outlet

L liquid

r4 the condenser outlet or the TEV inlet

n the nth iteration

r6 the evaporator outlet

o outside

v vapour

p constant pressure

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in order to be able to predict theevaporator exit condition pre-

cisely, especially when observing the hunting phenomenon.

In the evaporative region a mean void fractionwas determined

from the Hughmark correlation, assuming uniform heat flux.

Macarthur [4] modelled refrigerant heat exchangers on

a finite-volume basis by discretizing the length of the heat

exchanger into phase-independent control volumes and ap-plying the mass and energy balances on the refrigerant,

tube material and secondary fluid in each control volume.

In contrast to the condenser, the evaporator is modelled as

a two-phase region, the liquid and vapour being handled sep-

arately while being in equilibrium. Heat from the secondary

fluid passes exclusively to the liquid refrigerant to cause

evaporation. Mass and energy balances on the vapour and

liquid phases are linked through the evaporation rate. The

single-phase region in the evaporator was modelled in the

same way as in the condenser. In Sami et al.s [6] model,

the total system is divided into a number of lumped param-

eter control volumes. The condenser is modelled as a com-bination of the three different phase regimes: the fully

superheated, the two-phase and the sub-cooled. The vapour

and liquid phases in each regime were modelled by a set of

coupled mass and energy balances.

Rossi and Braun [5] modelled the heat exchanger on

a fixed-boundary, finite difference basis. Elemental mass

and energy balances are written for each element of the

heat exchanger, including the heat transfer equations be-

tween the refrigerant, the tube-wall and the air. The momen-

tum equation does not appear as the pressure drops are

considered negligible. The heat exchanger model uses the

inlet and outlet refrigerant flow-rates and the inlet enthalpy

as boundary conditions. The air side heat transfer is mod-

elled differently for heat transfer in the condenser and evap-

orator. For the condenser the dry analysis was based on the

temperature difference between the wall and the air, while

for the evaporator the wet analysis is based on the enthalpy po-

tential between the refrigerant and the air. An effectiveness-

NTU formulation was employed in the analysis.

Svensson [7] modelled a condenser consisting of two

zones: the condensing and the sub-cooling, both of which

are assumedto have fixed volumes. Energy andmass balances

are obtained for each of these zones as a set of coupled ordi-

nary differential equations. These are then combined to give

an equation for the pressure difference in terms of: tube-walltemperature, incoming mass flow rate and saturated refriger-

antproperties. The tube-wall and thewater volume are discre-

tized into n volumes in thecondensing zone and1 volume in

thesub-cooling zone andfinitedifference forms of theenergy

balance equations areused. For each of these n volumes, the

refrigerant side conditions are considered identical.

Mostresearchers have previously useda pre-defined func-

tion of the void fraction in their spatially distributed model,

based on experimental results. This approach results in the

separate solution of the mass balance and energy equations,

and less calculations are required. However, it is recognized

that the mass and energy equations should be coupled during

solution. The spatial distribution model constructed here is

based on the principle of energy and mass balances, to solve

the velocity, pressure, temperature, and wall temperature

functions in heat exchangers simultaneously.

2. Condenser model

2.1. The assumptions of the spatial distributed model

The following assumptions are made in this study:

(1) Axial conduction in the refrigerant is negligible.

(2) Liquid and vapour refrigerant phases in the heat

exchangers are in thermal equilibrium.

(3) The effects of pressure wave dynamics are negligible.

(4) The pressure drops in the condenser, evaporator and

short connecting pipe work are neglected. (This has

a small, variable effect on COP.)

(5) Thermal resistances of metallic elements in the systemare negligible in comparison with other thermal resis-

tances, however, their capacitance is important.

(6) In the two-phase region, axial conduction in the tube is

neglected. However, this is considered in the single-

phase flow region.

(7) Annular flow in the condenser and evaporator.

2.2. Mass and energy balance equations

The condenser has three different heat-dissipating zones:

a cool down zone for the superheated gas, a condensation

zone and a sub-cooling zone for liquid refrigerant. Each re-

gion was divided into two control volumes. The condensermodel is shown in Fig. 1.

According to above assumption the refrigerant flow in the

heat exchanger can be simplified to one-dimensional flow.

The mass and energy equations can be expressed as follows:

vr

vtvrw

vx 0 1

vrh

vtvrwh

vxvP

vt

4

DaiTw T 2

The pipe wall energy equation can be written as:

CwrwAwvTw

vt aipDiT Tw a0AohfTa T

lwAwv

2Tw

vX23

The above three equations include four parameters: pres-

sure P, pipe wall temperature Tw, refrigerant temperature T

and velocity w. The refrigerant property state equation

must be included.

fP; v;T 0 4

Thus with four independent variables, the four equations

are closed. In the two-phase region, the pressure and the

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temperature are dependent parameters and can be easily de-

termined, however, the void fraction or the dryness fraction

need to be calculated. This group of equations is difficult to

solve, and numerical simulation is required. In order to

obtain convergent solutions speedily, numerical methods

are discussed below.

2.3. Equations in the sub-cooling liquid

region and superheat region

In both these regions, the fluid is in single phase, and Eqs.

(1e4) can be used directly. In this model, the liquid density

is a function of pressure and temperature, which is calcu-

lated by the thermodynamic properties model as describedelsewhere [8].

2.4. Equations in the two-phase region

In the two-phase region, the model assumes that the liq-

uid and gas are in equilibrium, the temperature and pres-

sure in the gas region are the same as that in the liquid

region. However, the liquid density is much greater than

that of gas and the condensed liquid will accumulate at

the bottom of the tube due to gravity. The two-phase

flow has its own particular features: the gas velocity and

the liquid velocity are not the same, with the liquid velocitybeing lower than the gas velocity due to its higher viscosity.

Although this phenomenon is neglected by most re-

searchers who assume the same velocity in the two-phase

region. In this model the individual liquid and gas veloci-

ties were needed to accurately predict performance. To

achieve this the void fraction was calculated in Eq. (5) to

split the liquid and gas phases.

j Vg

V5

where Vg is the gas volume, and Vis the total volume at the

control volume unit.

Macarthur [4] assumed heat from the heat transfer fluid is

transferred exclusively to the liquid refrigerant in the evap-

orator, (the wall being completely wet) a similar assumption

was made in this condenser model, hence the liquid and gas

equations can be written as:

Vapour region:

vjrg

vt

vjrgwg

vx

_m

V 0 6

vjrghg

vt

vjrgwghg

vx

vP

vt

_m

Vhfg 0 7

Liquid region:

vrl1 j

vtvrlwl1 j

vx

_m

V 0 8

vrlh1 j

vtvrlwlhl1 j

vxvP

vt

_m

Vhfg

4

DaiTw T 9

where _m is condensed liquid mass flow rate.

3. The energy and mass balance

equations in the evaporator

The model developed in the evaporator is based on the

following assumptions:

(1) Axial conduction in the refrigerant is negligible.

(2) The liquid and vapour refrigerant phases in the heat

exchangers are in thermal equilibrium.

(3) The effects of pressure wave dynamics are negligible.

(4) The pressure drop in the evaporator is neglected.

(5) The thermal resistances of metallic element in the sys-

tem are negligible in comparison with other thermal

resistances, however, their capacitance is important.

VV_

cond1

L_sub L_sat L_sup

m_r4

h_r4m_r3

h_r3

P,w

h, t,x

VV_condL

VV_condG

P,w

h, t,x

XQ_cond1Q_cond2Q_cond3

Metal tube

VV_cond3

Fig. 1. The conceptual model of the condenser.

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(6) In the two-phase region, the axial conduction of the

tube is neglected. However, this is considered in the

single-phase flow region.

(7) Refrigerant enters the evaporator directly from the TEV.

(8) Thefluidstateattheentrance to theevaporator is two-phase

already, some refrigerant has flashed into gas in the TEV.

As with the condenser model, the conceptual model of

the evaporator is similar. However, in the evaporator nor-

mally only two regions exist: the two-phase region and the

superheat region. The mass, energy, and pipe wall energy

equations are the same with the condenser; the only differ-

ences are their initial and boundary conditions.

As in the condenser, for the liquid continuously changing

phase, the heat from outside is assumed exclusively to be

transferred to the liquid. Then, the energy and mass balance

equations in the two-phase region are exactly the same as in

the condenser. Here the equations are omitted as they are

already defined in Section 2.As the refrigerant pressure drops due to the suction of

compressor, more liquid flashes into gas in the direction of

the evaporator, this phenomenon causes the liquid tempera-

ture to drop and the initial thermal balance to break down.

The heat from outside the tube then balances the energy

needed by the flashing process. This energy stored in the

gas is then removed away by the compressor.

4. The other parameters calculation

in two-phase region

4.1. Void fraction and dryness fraction

As defined in Eq. (5), the void fraction is the ratio of

vapour volume to the total volume in the two-phase region.

With the void fraction, the density of the fluid can be

expressed as:

r rl1 j rgj 10

Dryness fraction is defined as:

Xmg

mg ml11

The dryness fraction is used to determine the propertiesof the refrigerant mixture. For liquid and vapour mixture,

the thermodynamic parameters can be expressed as:

h 1 Xhl Xhg 12

v 1 Xvl Xvg 13

4.2. Relation between the void fraction

and dryness fraction

Dryness fraction and void fraction are not independent

parameters. From the void fraction, the dryness fraction

can be calculated as:

Xrgj

rgj 1 jrl14

Similarly, the void fraction can be calculated as:

j vgX

vgX 1 Xvl

15

Therefore calculation of variable dryness fraction can be

calculated from the variable void fraction.

5. Boundary conditions

5.1. Boundary conditions at inlet (x 0) and outlet (x L)of the condenser

In the model, the inlet of the condenser is set as the outlet

of thecompressor.This assumes there is no heat loss between

the compressor and condenser. Some extra heat exchanger

surface between the compressor and the condenser hasbeen added to the actual condenser surface to account for

the real thermal conditions (a similar addition was made to

evaporator area). At the outlet of the condenser, the mass

flow rate is assumed as that passing through the TEV. The

boundary conditions can be described mathematically as:

x 0; hr3 hcomp2 16

_mr3 _mcomp2 17

x L; _mr4 _mtev 18

5.2. The boundary conditions at inlet (x L) and

outlet (x 0) of the evaporator

At the outlet of the evaporator (x 0) the parameters areassumed the same as those at the inlet to the compressor. The

refrigerant mass stored in the compressor is neglected, and

the inlet mass flow rate is equal to the outlet mass flow rate

(storage only therefore occurs in the heat exchangers). At

the inlet to the compressor the mass flow rate is equal to

themass flow rate of theTEV, andthe parameters at this point

are equal to the outlet parameters of the TEV, therefore the

boundary conditions can be expressed as follows:

x

0;

hr6

hcomp1

19

_mr6 _mcomp1 20

x L; _mr4 _mtev 21

5.3. Boundary conditions on the heat exchanger surface

At the outside of the condenser or evaporator coil, the air

is drawn through the heat exchanger by a fan and heat is

taken away by convection. The boundary condition can be

expressed as:

q aotw tair 22

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where ao is the heat transfer coefficient which can be found

in Section 10, and is defined differently for both the evapo-

rator and condenser.

6. Initial conditions

6.1. Initial conditions in the condenser

In a dynamic model, all the parameters are considered as

time dependent, and the initial conditions affect system-

starting performance. The initial condition plays an important

role in theresults.The initialmass distribution andtemperature

also affect the solution. Unfortunately, it is very difficult to

track the initial mass distribution in the system. The initial

condition is only based on the assumption that the condenser

is in two-phase equilibrium or in a superheated state.

The mass distribution will initially have an effect on

system performance. If a quantity of mass is stored in the

condenser at the initial state, the rate at which the pressurein the condenser will rise will be affected by the mass pres-

ent. The initial transient period will be shorter with less mass

contained, and this has been demonstrated by a lumped

model investigation detailed elsewhere [8].

6.2. The initial conditions of evaporator

Theinitial condition in the evaporator is somewhat differ-

ent from the condenser. When a system starts, the evaporator

is normally always charged with refrigerant. This is because

the evaporator temperature and pressure are always less than

the condenser temperature and pressurewhen the compressoris stopped. The refrigerant therefore migrates to the evapora-

tor due to this pressure difference. However, the exact refrig-

erant charge in the evaporator is difficult to predict. The

initial conditions always assume the fluid exists in the two-

phase stateat a saturation temperature equivalentto the initial

temperature of the calorimetric chamber. For initial values,

the refrigerant was distributed 80% by mass in condenser,

20% in evaporator, residue being vapour of low density.

7. Discretization equations

7.1. The method of meshing

The heat exchanger is meshed along the refrigeration

flow direction using a method by which the first node is at

the entrance of the heat exchanger and the last node is at

the exit of the heat exchanger. The node coordinates value

can be calculated and stored as:

dxL

N 123

Xdifi i 1dx; i 1;2;.;N 24

The coordinates value of the interface of the control

volume can then be calculated:

Xcvi Xdifi 1 Xdifi

2; i 1;2;.;N 1 25

7.2. The method of discretization

The model uses a control volume based technique to con-

vert the governing equations to algebraic equations that canbe solved numerically. This control volume technique con-

sistsof integrating thegoverning equations about eachcontrol

volume, yielding discrete equations that conserve each quan-

tity on a control volume basis. For the time-based equations,

all the discrete equations will be expressed implicitly with

time and this method will enable the iterations to converge.

For the non-steady, one-dimensional problem, all

parameters in the equations vary with space and time. In a

discrete expression these parameters will be stored in a two-

dimensional array, the subscript represents space, and the

superscript represents time. For example, t12 presents the

second node temperature at present time, and t0

2 expressesthe second node temperature at the previous time. In

order to save memory, the maximum-element in each two-

dimensional array is N2. All the parameters will be sentto an external data file at each time step or selected time step.

7.3. The discretization of the equations

The finite difference method was employed in all equa-

tion discretizations. Mass and energy balance method was

used on all boundary conditions over the cells.

7.3.1. Condenser

Fig. 2 shows the two-phase region mesh, the dashed linesare the interfaces of the control volumes, the solid lines are

the nodal position for each control volume. There are three

regions in the condenser including superheated gas, two-

phase and liquid region. For each region, the energy and

mass balance equations in the discrete control volume terms

can be expressed as:

(1) Superheated gas:

r1i1w1i1 r

1i w

1i

Dxr1i r

0i

Dt26

r1i1w1i1Ah

1i1 arefpDDxt1wi t1i r1i w1i Ah1i

ADxr1i h

1i r

0i h

0i

DtADx

P1i P0i

Dt27

Dx Xcvi Xcvi 1 28

(2) Two-phase region:

In this region, the liquid and gas flow at different

velocities. The energy and mass equations will be

conducted separately.

For the gas:

j1i1r1g i1w

1g i1 j

1i r

1g iw

1g i

Dx

_m

ADxj1i

r1g i r

0g i

Dt

29

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j1i1r1g i1w

1g i1Ah

1g i1 j

1i r

1g iw

1g iAh

1g i _mhfg

j1i ADxr1g ih

1g i r

0g ih

0g i

Dtj1i ADx

P1i P0i

Dt30

For the liquid:1 j1i1

r1L i1w

1L i1

1 j1i

r1L iw

1L i

Dx

_m

ADx

1 j1i

r1L i r

0L i

Dt

311j1i1

r1L i1w

1L i1Ah

1L i1

1j1i

r1L iw

1L iAh

1L i

arefpDDx

t1w i t1i

_mhfg

1j1i

ADx

r1L ih1L i r

0L ih

0L i

Dt

1j1iADx

P1i P0i

Dt32

Dx Xcvi Xcvi 1 33

(3) Sub-cooled region:

r1i1w1i1 r

1i w

1i

Dxr1i r

0i

Dt34

r1i1w1i1Ah

1i1 arefpDDx

t1w i t

1i

r1i w

1i Ah

1i

ADxr1i h

1i r

0i h

0i

DtADx

P1i P0i

Dt35

Dx Xcvi Xcvi 1 36

(4) Wall energy equation:

arefpD

t1i t1w i

aoFohf

tair t

1w i

lwAw

t1w i1 t1w i

xdifi 1 xdifilwAw

t1w i1 t1w i

xdifi xdifi 1

cwrwAwxcvi xcvi 1t1w i t

0w i

Dt37

7.3.2. Evaporator

There are two regions in the evaporator, that is, the super-

heated gas and the two-phase region. For each region, the en-

ergy and mass balance equations in discrete control volume

terms can be defined as:

(1) Superheated gas:

r1i1w1i1 r

1i w

1i

Dxr1i r

0i

Dt38

r1i1w1i1Ah

1i1 arefpDDxt1w i t1i r1i w1i Ah1i

ADxr1i h

1i r

0i h

0i

DtADx

P1i P0i

Dt39

Dx Xcvi Xcvi 1 40

(2) Two-phase region:

In this region, the liquid and gas flow at different

velocities. The energy and mass equations will be

conducted separately.

For the gas:

j1i1r1g i1w

1g i1 j

1i r

1g iw

1g i

Dx

_m

ADxj1i

r1g i r

0g i

Dt

41

j1i1r1g i1w

1g i1Ah

1g i1 j

1i r

1g iw

1g iAh

1g i _mhfg

j1i ADxr1g ih

1g i r

0g ih

0g i

Dtj1i ADx

P1i P0i

Dt42

For liquid:1 j1i1

r1L i1w

1L i1

1 j1i

r1L iw

1L i

Dx

_m

ADx

1 j1i

r1L i r

0L i

Dt

43

1j1

i1r1

L i1w1

L i1Ah1

L i1 1

j1

ir1

L iw1

L iAh1

L i

arefpDDx

t1w i t1i

_mhfg

1j1i

ADx

r1L ih1L i r

0L ih

0L i

Dt

1j1iADx

P1i P0i

Dt

44(3) Wall energy equation:

arefpDi

t1i t1w i

aoFohf

tair t

1w i

lwAw

t1w i1 t1w i

xdifi 1 xdifi lwAw

t1w i1 t1w i

xdifi xdifi 1

cwrwAwxcvi xcvi 1t1w i t

0w i

Dt45

Control ControlVolume

I + 1

Xdif (i+1) Xcv (i) Xcv (i-1) Xdif (i-1)Xdif (i)

Volume I

Volume

I - 1

Control

Fig. 2. The two-phase region mesh in condenser.

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7.3.3. The inlet and outlet boundary conditions

discretization

7.3.3.1. Condenser entrance (node i 1). The refrigerantat the entrance of the condenser will be the superheated gas

from the compressor; then the energy and mass balance

equations can be expressed as follows:

Mass balance equation:

mr3 r11w

11A

A dxr11 r

01

Dt

46

Energy balance equation in the refrigerant:

mr3hr3 aipD dx

t1w 1 t11

r11w

11Ah

11

A dxr11h

11 r

01h

01

DtA dx

P11 P01

Dt47

Energy balance equations in the wall:

arefpDDx

t11 t1w 1

aoFhfDx

tair t

1w 1

lwAw

t1w 2 t1w 1

Xdif2 Xdif1 cwAwrw dx

t1w 1 t0w 1

Dt48

In above equations, Dx Xcv1 Xdif1.

7.3.3.2. Condenser exit (node i Nc)Mass balance equation:

r1n1w1n1A mr4

ADxr1n r

0n

Dt

49

Energy balance equation in the refrigerant:

mr4h1n arefpD dx

t1w n t

1n

r1n1w

1n1Ah

1n1

ADxr1nh

1n r

0nh

0n

DtADx

P1n P0n

Dt50

Energy balance equations in the wall:

arefpDDx

t1n t1w n

aoFhfdx

tair t

1w n

lwAw

t1w n1 t1w n

Xdifn Xdifn 1

cwAwrwDxt1w n t

0w n

Dt51

In the above equations, Dx Xdifn Xcvn 1.

7.3.3.3. Evaporator entrance (node i 1). The gas enter-ing the evaporator from the TEV will be wet, with dryness

fraction X, and then the energy and mass balance equations

can be expressed separately in the liquid and gas phases as

follows.

For the vapour, mass balance equation:

mr4Xj1nr

1g nw

1g nA _m

1n

j1nADxr1g n r

0g n

Dt

52

The energy balance equation in the refrigerant:

Xmr4h1g _m

1nh

1g r

1g nw

1g nAh

1g j

1nAdx

r1g nh1g n r

0g nh

0g n

Dt

Aj1n dxP1n P

0n

Dt53

For liquid, the mass balance equation:

mr41 X

1 j1nr1L nw

1L nA _m

1n

1 j1n

ADx

r1L n r

1L n

Dt

54


mr41 XhL 4 aipD dx

t1w n t1n

1 j1nr1nw

1nAh

1n _m

1nh

1L n

1 j1nA dx

r1nh1L n r

0nh

0L n

Dt

1 j1nA dx

P1n P0n

Dt

55

The energy balance equations in the wall:

arefpDDx

t1n t1w n

aoFhfDx

tair t

1w n

lwAw

t1w n1 t1w n

Xdifn Xdifn 1

cwAwrw dxt1

w n t0

w n

Dt56

In the above equations, Dx xdifn xcvn.

7.3.3.4. Evaporator exit (node i N)The mass balance equation:

r12w12A mr6

ADxr11 r

01

Dt

57


mr6h11 arefpD dx

t1w 1 t

11

r12w

12Ah

12

ADxr11h

11 r

01h

01

DtADx

P11 P01

Dt58

The energy balance equation in the wall:

arefpDDx

t11 t1w1

aoFhf dx

tair t

1w1

lwAw

t1w 1 t1w 2

Xdif2 Xdif1 cwAwrwDx

t1w 1 t0w 1

Dt59

In the above equations, Dx Xcv1 Xdif1.

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7.3.4. The phase transition point discrete equations

7.3.4.1. The transition point in condenser. Because of the

different heat transfer mechanisms, the rate of single-phase

heat transfer in a sub-cooled fluid is always less than the

heat transfer coefficient taking place in the condensing pro-

cess. This makes establishing the point of saturation impor-

tant in accurate model performance.During normal system operation this point always varies

with the condenser ambient air condition. When the system

starts, the delivery temperature and condenser pressure are

low, and the low-pressure ratio enables the compressor to

deliver a large mass flow rate. Under start up conditions the

condensing point will be somewhere just after the entrance

of the condenser. However, with further system running,

the delivery temperature and condenser pressure both rise.

With greater superheat, the condensing point would move

towards the exit of the condenser. The sub-cooling transition

point also moves with the running conditions. Initially, this

point moves because the surface area available for condens-ing decreases.

To establish the points, an enthalpy-comparison method

was developed and is used, where the gas enthalpy from

the energy equation is compared with the saturation vapour

enthalpy at the same condenser pressure. If it is less than the

saturated vapour enthalpy, the point is then marked as a start-

ing point for the condensing process. The recorded point will

be stored as a variable, and the moving boundary of the in-

terface is gradually revealed. The point of sub-cooling is

found in a similar way.

7.3.4.1.1. Gas condensing point. At the point of conden-

sation, the gas is condensing to liquid and this will be

attached to the wall in a very thin film. The liquid velocity

is assumed to be 0 at the initial condensing point. When

the void fraction at this point reaches 0.99, the liquid begins

to move. The equations of the initial condensing point are

a little different from those in the two-phase region and at

this point they are recorded and marked as I ktr1 in theprogram code. The mass and energy balance equations at

this point are given below.

For the gas:

j1i1r1g i1w

1g i1 j

1i r

1g iw

1g i

Dx _m

ADx

j1i r1g i r0g iDt 60

j1i1r1g i1w

1g i1Ah

1g i1 j

1i r

1g iw

1g iAh

1g i _mhfg

j1i ADxr1g ih

1g i r

0g ih

0g i

Dtj1i ADx

P1i P0i

Dt61

Dx Xcvi Xcvi 1

For liquid:

_m

1 j1i

r1L i

DtDx 62

arefpDDx

t1w i t1i

_mhfg

1 j1i

ADx

r1L ih1L i

Dt

1 j1iADx

P1i P0i

Dt63

Dx Xcvi Xcvi 1

7.3.4.1.2. The sub-cooling liquid point. The last cell in

the two-phase region, as marked I ktr2 in the modelcode, is recognized as a transition point from the two-phase

region to liquid region. The energy and mass balances for

this cell in gas region are:

j1i1r1g i1w

1g i1 j

1i r

1g iw

1g i

Dx

_m

ADxj1i

r1g i r

0g i

Dt

64

j1i1r1g i1w

1g i1Ah

1g i1 j

1i r

1g iw

1g iAh

1g i _mhfg

j1i ADxr1g ih

1g i r

0g ih

0g i

Dtj1i ADx

P1i P0i

Dt65

Dx Xcvi Xcvi 1

r1g i r1g i1 r

1g 66

For liquid:

1 j1i1

r1L i1w

1L i1

1 j1i

r1L iw

1L i

Dx

_m

ADx

1 j1ir

1L i r

0L i

Dt67

1 j1i1

r1L i1w

1L i1Ah

1L i1

1 j1i

r1L iw

1L iAh

1L i

arefpDDx

t1w i t1i

_mhfg

1 j1iADx

r1L ih1L i r

0L ih

0L i

Dt

1 j1iADx

P1i P0i

Dt

68

Dx Xcvi Xcvi 1 69

Some attention needs to be paid to the first cell in the

sub-cooled region. This cell on subsequent iterations can

become sub-cooled, just saturated or two-phase. The analy-

sis of this must be careful to ensure accurate prediction of its

subsequent state. The liquid and gas enter the cell simulta-

neously from the upper and lower positions of the cell.

The energy and mass balance equations in this cell can be

expressed as:

r1Lw1L i1

1 j1i1

w1g i1r

1gj

1i1 r

1Lw

1i

Dxr1i r

0i

Dt70

ARTICLE IN PRESS





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r1Lw1L i1Ah

1L

1 j1i1

Ar1gw

1g i1j

1i1h

1g

arefpD Dx

t1w i1 t1i1

r1i1w

1i1Ah

1L

ADxr1i1h

1i1 r

0i1h

0i1

DtADx

P1i1 P0i1

Dt71

Dx xcvi 1 xcvi 72

For this point, if the calculated enthalpy h1i1 is larger

than the saturated liquid enthalpy then the point will be in

the two-phase region, and this point will be marked as

I ktr2 and recognized as the last point of the two-phaseregion. If the enthalpy of the refrigerant is less than the satu-

rated liquid enthalpy, this point is still in the liquid region.

By recording this point, the moving interface can be traced

properly.

7.3.4.2. The transition point in evaporator. In a similar way

to the condenser model, this point was also traced in the

model. However, in the evaporator, there is only one phasetransition point, from the two-phase region to the super-

heated gas region The transition point is a balance point af-

fected by the refrigerant mass flow passing through the TEV

and the thermal load exerted on the evaporator. For a smaller

thermal load, less mass of refrigerant evaporates in the heat

exchanger, and this point will be adjacent to the outlet of the

evaporator, the heat transfer surface for the superheated gas

will be less and lower superheat will be obtained resulting in

a smaller TEV mass flow.

For a higher thermal load on the evaporator, more mass of

refrigerant will be evaporated and the balance point will be

further away from the outlet of the evaporator, resulting inmoreheat exchanger surfacefor thesuperheated gas.A higher

outlet temperature of the superheated gas is possible which

will increase the mass flow rate in the TEV in the system.

In the code, the control cell i represents this point. To

establish this point, the liquid mass is used as a criterion,

such that if the liquid mass is less than or equal to zero,

the cell will be in the dry region. However, it may be

more complex while the liquid is moving. This is because

the liquid can move forward because of gas friction or

backward because of high thermal load on the evaporator

or gravity (in this model the gravity force is neglected).

The energy and mass balance for this point can be

expressed as follows.

For the gas:

j1i1r1g i1w

1g i1 j

1i r

1g iw

1g i

Dx

_m

ADxj1i

r1g i r

0g i

Dt

73

j1i1r1g i1w

1g i1Ah

1g i1 j

1i r

1g iw

1g iAh

1g i _mhfg

j1i ADxr1g ih

1g i r

0g ih

0g i

Dtj1i ADx

P1i P0i

Dt74

Dx Xcvi Xcvi 1 75

For liquid:

A

1 j1i1r1L i1w

1L i1 _m

1 j1i

m1L i m

0L i

Dt

76

1 j1i1r1L i1w1L i1Ah1L i1 arefpDDxt1w i t1i _mhfg

1 j1iADx

r1L ih1L i r

0L ih

0L i

Dt

1 j1iADx

P1i P0i

Dt

77

Dx Xcvi Xcvi 1 78

uxli 0 if a1i > 0:998 x> 92% 79

With the above equations, a1i can be calculated. If a1i is

greater than 0.998, the dry fraction is 0.9 or more, and

the flow is recognized as fog flow in the evaporator. The liq-

uid forms very small particles and will move with the gas atthe same velocity. Ifa1i is less than 0.998, the liquid and gas

will separate with a defined interface and the liquid will

move at a speed different to that of the gas.

7.3.5. The conditions for iteration

In a set of differential equations with a close chain of

implicit parameters, the numerical solution is based on

initial assumptions. Under these initial values, the next

values can be calculated from the equations and then com-

pared with the assumed values. If the chosen tolerance is

not met, the initial assumptions can be modified, then cal-

culation should be carried out again, until the chosen tol-

erance is met. For example, in this model, at each timestep, the initial pressures of the condenser and the evapo-

rator are assumed, then the mass flow rate of the compres-

sor and TEV can be calculated, the condensing and

evaporator temperatures can also be calculated, then heat

transfer process can be analysed in the heat exchangers,

with known mass flow-rates. The condenser and the

evaporator model can be carried out with the results of pa-

rameters at each tube section, these parameters then affect

the TEV and compressor mass flows and parameters.

The assumed pressures in the condenser and evaporator

are correct when the required tolerance is met; otherwise

the simulation needs to be repeated until the convergencetolerance is met.

In the condenser and evaporator models, an iterative pro-

cedure is also required. The initial parameters such as refrig-

erant temperature, wall temperature, and gas void fraction

distributions are assumed, then the differential equations

are solved and these parameters may be then recalculated.

Comparing the calculated parameters with the assumptions,

the assumptions are then modified again till the chosen tol-

erances are met. A relative error is used to compare with the

chosen tolerance (usually 1.0 106). Once the iterationconditions are met, the solution can then be stored and the

solution for the next time step considered.

ARTICLE IN PRESS





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8. The novel triangle and tri-point

method developed for iteration

In order to compare the new method with general

methods for iteration, the general methods including Bisec-

tion, Secant, False Position, and NewtoneRaphson methods

are listed below as reviewed by EFUNDA [2].

8.1. Bisection method

The idea of the bisection method is based on the fact

that a function will change sign when it passes through

zero. The method can be expressed by the following

program:

To find a root of f(x) 0 in the interval of (a0, b0)With which f(a0) f(b0) < 0

Pick tolerance 3

Xk1 (bk ak)/2, k 0,1,2,3,.If (jf(xk1)j < 3) root found, stop iteration.

Else

If (f(xk1) f(bk) < 0) ak1 xk1; bk1 bkElse ak1 ak; bk1 xk1

End if

End if

When an interval contains a root, the bisection method is

the one that will not fail. However, it is amongst the slow-

est. When an interval contains more than one root, the

bisection method can find one of them. When an interval

contains a singularity, the bisection method converges tothat singularity.

8.2. Secant method

To improve the slow convergence of the bisection

method, the secant method assumes that the function is ap-

proximately linear in the local region of interest and uses

the zero-crossing of the line connecting the limits of the in-

terval as the new reference point. The next iteration starts

from evaluating the function at the new reference point

and then forms another line. The process is repeated until

the root is found.

To find a root of f(x) 0 in the interval of (x0, x1)With which f(x0) f(x1)< 0

xk1 xk xk xk1

fxk fxk1fxk; k 1;2;3;.

Mathematically, the secant method converges more rap-

idly near a root than the false position method (discussed be-

low). However, since the secant method does not always

bracket the root, the algorithm may not converge for func-

tions that are not sufficiently smooth.

8.3. False position method

Similar to the secant method, the false position method

also uses a straight line to approximate the function in the

local region of interest. The only difference between these

two methods is that the secant method keeps the two most

recent estimates, while the false position method retainsthe most recent estimate and the next recent one which has

an opposite sign in the function value.

To find a root of f(x) 0 in the interval of (a0, b0)With which f(a0) f(b0) < 0

Pick tolerance 3

xk1 bk bk ak

fbk fakfbk; k 1;2;3;.

If (jf(x k1)j < 3) root found, stop iteration.Else

If (f(xk1) f(bk) < 0) ak1 xk1;bk1 bkElse ak1 ak; bk1 xk1

End if

End if

The false position method, which sometimes keeps an

older reference point to maintain an opposite sign bracket

around the root, has a lower and uncertain convergence

rate compared to the secant method. The emphasis on brack-

eting the root may sometimes restrict the false position

method in difficult situations while solving highly non-

linear equations.

8.4. Newtone Raphson method

The NewtoneRaphson method finds the slope (the

tangent line) of the function at the current point and uses

the zero of the tangent line as the next reference point.

The process is repeated until the root is found.

To find a root of f(x) 0 with the initial guess x0,

xk1 xk fxk

f0xkwhere k 0;1;2;3.

The NewtoneRaphson method is much more efficient

than other simple methods such as the bisection method.

However, the NewtoneRaphson method requires the calcu-

lation of the derivative of a function at the reference point,

which is not always easy. Furthermore, the tangent line often

shoots wildly and might occasionally be trapped in a loop

and the solution does not converge. It is recommended to

monitor the step obtained by the NewtoneRaphson method.

When the step is too large or the value is oscillating, other

more conservative methods should take over the case. How-

ever, this method cannot be used in cycle iterations, because

the function is not known explicitly.

ARTICLE IN PRESS





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8.5. The novel method

The general iteration methods are listed above. The

Bisection method is too slow and the NewtoneRaphson

method is not effective when the function is explicit. The

other two methods are also not effective when tried by the

author repeatedly. This is because in either the condenseror evaporator, both the pressures are changing. When using

the Secant method, one heat exchanger reaches a converging

point, however, the other might be still far away from con-

vergence. When the other heat exchanger is near to conver-

gence, the previous converged point has drifted away. In

order to solve this problem, a new method of iteration was

developed. This method is based on the false position

method, but the minimum positive and negative errors of

each iteration are recorded, respectively. In the condenser it-

erations, this method was initially used, and in the evapora-

tor a simple constant factor is used initially, this makes the

iterations in the evaporator relatively slow. When the con-denser iteration reaches the tolerance then the same method

is applied to the evaporator, if the iterations in the condenser

diverge, the recorded minimum positive and negative errors

will be applied to the condenser iteration. This method en-

sures the iteration continues and makes the iteration three

times faster than the other methods mentioned here.

9. Simulation of TEV and compressor

The compressor generates a mass flow from the low-

pressure side of the cycle to the high-pressure side of the

cycle. The heat generated by the electric motor is absorbed

by the refrigerant rather than transferred to the ambient.Thus the compression is assumed to be adiabatic and the

isentropic efficiency can be found from the manufacturer,

which can be expressed as a function of pressure ratio:

hi h2 h1h02 h1

f

p2

p1

80

The refrigerant mass flow in the compressor is given by

the following equation:

_m r$VV$hv 81

where VV is the swept volume rate of the compressor, which

is calculated by:

VV p

4D2 S n=60 82

where S is stroke of piston, D is the piston diameter; n is the

rotational speed of the shaft (rev/min), hv is the volumetric

efficiency of the compressor, which was determined from

makers data as:

hv fp2

p1 83

The compressor power, pe, can be calculated:

pe _m

h02 h1

hm84

The TEV shown in Fig. 3 is a proportional controller with

the control signal from the evaporator outlet superheat. It re-sponds to the difference between the pressure of the refrig-

erant at the evaporator outlet and the pressure developed

in the temperature-sensing remote phial attached to the out-

let of the evaporator. The phial is normally charged with the

same refrigerant as the plant. A spring setting in the valve is

provided to adjust the required superheat of the vapour leav-

ing the evaporator.

The TEVoperation can be modelled by a set of algebraic

equations, which are given below. The forces acting on the

TEV spindle, when it is in equilibrium, are shown in Fig. 3:

Fs F2 F1 85

where F1 is the force exerted as a result of the pressure of the

liquid/vapour mixture in the remote phial, which is the sat-

urated pressure at the outlet temperature of the evaporator

calculated by the state equation. F2 is the outlet pressure

of the evaporator (or inlet pressure of the evaporator minus

the pressure drop in the evaporator). Fs is the closing force

exerted by the spring.

The spring pre-tension can be adjusted. The force exerted

when the valve starts to open is denoted byF0. Therefore, the

force exerted by the spring is given by:

Fs F0 KY 86

where Yis the displacement in millimeter and Kis the spring

stiffness in N/mm. Manipulating Eqs. (85) and (86) gives:

KY F2 F1 F0 87

or KY P2 P1A F0

whereA is the area of the diaphragm,P1 is the pressure in the

remote phial and P2 is the evaporator pressure at the position

of the pressure-sensing connection. When Y 0, the springpre-tension is known as the static pressure of the valve,

which is usually expressed as:

F0 DP0A 88

where Y is given by:

Y C0P2 P1 DP0 89

where C0 A/K; also Y is the adjustment of the TEV.The flow area A of the TEV is a function ofY. The actual

flow area of the TEV can be determined with the calculated

value of the displacement Y. The mass flow rate can then be

calculated:

M CdAffiffiffiffiffiffiffiffiffiffiffiffiffiffiffirDP

p90

A hp

4D Y Ctga2

p

4D2isina 91

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10. Heat transfer coefficient used in the model

In this model the Shah correlation and the Dobson and

Chato method will be used to predict the heat transfer behav-

iour in the condenser.

The Shah correlation for condensation is a two-phase

multiplier approach valid for annular flow. It is based on

the liquid heat transfer coefficient, which is shown in Eq.

(92). The liquid heat transfer coefficient is calculated fromthe DittuseBoelter correlation.

acond aL

"1 X

0:83:8X0:761 X

0:04

Pr0:38

#92

where

PrPsat

Pc; reduced pressure

aL l

D

0:023

ReL

1 X

0:8Pr0:4L

ReL G1 XD

mL> 350

PrL CpLmLlL

In the sub-cooling region, the heat transfer coefficient

on the refrigerant side can be calculated from the standard

DittuseBoelter equation:

Nu 0:023 Re0:8 Pr0:4 93

The Kandlikar correlation [Kandlikar 1987] is shown:

a aL

1:136C0:9o 25Fr0:3667:2B0:7o Ffl

94

where Ffl is fluid dependent parameter, for R22,

Ffl 2:2

aL 0:023 Re0:8L Pr

0:4L lL=d

Co 1 X

X!

0:8

rG

rL!

0:5

; X is dryness fraction

Fr G2

r2LgD; G is mass flux; kg=m2 s

Bo q

G$hfg

Gungor and Winterton correlations were found to pro-

duce mean deviations of 19% [Kandlikar [12]], therefore,

in this model, Kandlikar [11] and Gungor and Winterton

[10] correlations are used.

11. Model validation

The test plant is based on an IMI Marstair model C60/E,

direct air cooling split system. The indoor unit comprised of

an evaporator (IMI impact CU6E) and a fan. The outdoor

unit (IMI CUE60) comprised of a condenser, compressor

and propeller fan.

The evaporator was sited inside a calorimetric chamber

and was a twin circuit air-cooled coil with 26 tubes, two

of which were blanked. The tubes had external aluminium

alloy plate fins at a pitch of 551/m. The total external heat

transfer surface area was 11.432 m2, while the total internal

heat transfer area was 0.533 m2 with refrigerant tube diam-

eters of 9.52 mm, giving a total metal mass of 6.8 kg.

Remote bulb

To evaporator

Superheat adjustment

Liquid

refrigerant in

Fs

Fs

F2

F1

Pressure in bulb P2Connect to

evaporator outletPressure in

evaporator P1

Fig. 3. A typical TEV and Forces balance on TEV spindle.

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The air was driven over the evaporator by a fan having

three speeds. These gave 0.233, 0.283 and 0.312 kg/s of

air, respectively. The measurements were all taken with

the evaporator fan operating at the top speed (0.312 kg/s).

The condenser was an air-cooled coil having 63 tubes

(one blanked off), arranged in three parallel circuits. The

tubes had external aluminium plate fins at a pitch of 472/m.The total external heat transfer area was 17.955 m2 and the

total internal heat transfer area was 0.867 m2. The refriger-

ant tube diameter was 9.53 mm and the total metal mass

was 8.2 kg. The air was driven over the condenser by a

propeller fan having two speed settings. The air mass flow

rate at the low speed was 0.563 kg/s and 0.7 kg/s at the

high speed setting, at which the experimental work was

performed.

The compressor was a Tecumseh model, No. TAH

5527E, having a swept volume of 0.0023611 m3 /s, at

2900 rpm. The unit was a reciprocating, suction gas cooled

hermetic type.The thermostatic expansion valve is a Danfoss model

(with orifice 03), having a capacity of 5.2 kW. The outside

diameter of the suction line is 15.88 mm, this was insu-

lated to minimise the heat gain from the surroundings.

The outside diameter of the liquid line to the evaporator

pipe is 9.525 mm. All the valves are flare valves made

by Danfoss.

The spatial model was validated against the test results

for the system. The validation was carried on one group of

test data. Like the lumped parameters model, both the inlet

air temperature of the condenser and evaporator were fed

in as known parameters into the model. The calculated and

test parameters were represented on the same diagram

against time. The results are shown in Figs. 4e6.

Fig. 4 shows the calculated pressures with the test data

in the condenser and the evaporator. The calculated pres-

sures agreed with the test pressures as shown in Fig. 4.

Fig. 5 shows the temperature before the TEV for the model

and the test data. This shows there is 3K difference. This

error is caused by a number of factors. The boundary and

initial conditions influence the results. Unlike the lumped

parameters, the spatially distributed model is more depen-

dent on this condition, and includes detailed geometric

parameters as well. Fig. 6 shows the temperatures from

simulated and tested results in the evaporator. The

calculated evaporating temperature is 2 K less than the

test data. This error mainly comes from the heat transfer

coefficient, but also initial and boundary conditions do in-

fluence the results. For example if the mass flow rate

through the compressor (the boundary condition) is higher

than the actual rate, the calculated evaporating temperature

would be lower than the actual one.

12. The other parameters shown against space

and time in two heat exchangers

As mentioned in Section 1, one additional benefit of the

spatial model is that the spatial distributed parameters can be

visualized against time and space. If reasonable initial and

boundary conditions are given, a valid result will be obtained

by this model. Figs. 7e12 show some of these parameters

distributed in the heat exchanger. Fig. 7 shows the dryness

fraction distribution along the condenser tube. As time in-

creases, the dryness fraction distribution will be stabilizedas long as running conditions remain unchanged. At the en-

trance of the heat exchanger (x 0), it clearly shows that thelength of the superheated gas section is changing. As the out-

let temperature of the compressor increases, the gas (super-

heat) length increases, and the position of the two-phase

region interface is moving. Fig. 8 shows the refrigerant tem-

perature distribution in the condenser, it is a function of time

and space. The horizontal line represents the phase changing

temperature. The length of each region including gas,

0.0

0.3

0.6

0.9

1.2

1.5

1.8

0 200 400 600 800 1000

Time (sec)

Pressure(MPa)

Pc_test Pe_test

Pc_cal Pe_cal

Fig. 4. Pressure validation.

1520

25

30

35

40

45

0 200 400 600 800 1000

Time (sec)

T

emperature(C)

TEV inlet_Test

TEV inlet_Cal

Fig. 5. Validation of the temperature before TEV.

5

9

13

17

21

25

0 200 400 600 800 1000

Time (sec)

Temperature(C)

Test_Te Test_Evaporator outlet

Cal_T3 Cal_Evaporator outlet

Fig. 6. Validation of the temperature in the evaporator.

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two-phase, and sub-cooled can be clearly shown in this fig-

ure. Fig. 9 shows the wall temperature distribution along the

tube against time. These lines are similar to the refrigeranttemperature distribution. The only difference is that their rel-

ative values are lower than the refrigerant temperatures. In

these the refrigerant can be seen flowing from right to left,

the exit is at x 0 and the inlet is xLevap. From the dia-gram, the lowest temperature in the evaporator is around

the superheat and two-phase interface in the initial stage.

This is because a quantity of refrigerant vapour was drawn

away by the compressor at the initial stage, and liquid refrig-

erant is being flashed into gas intensively around this point,

which reduces the wall temperature.

13. The moving boundary

Fig. 11 shows the length of superheat section in the

evaporator. It can be seen that this length changes rapidly

at the beginning of the cycle. This is because a large quan-

tity of vapour is drawn into the compressor, and the

pressure in the evaporator drops quickly. This then results

in much liquid flashing into vapour and liquid flowing

backwards in the evaporator against vapour flow direction.

This results in an increase in the superheat section. Fig. 12

shows how the two-phase region varies with time. The

length of this region is important to the accuracy of the

pressure calculation in the condenser. The longer this sec-

tion is, the more gas that is condensed and the lower con-

densing pressure. The main parameter determining the

length is the heat transfer coefficient, which can be up to

15% different than the test data.

14. Conclusions

A spatial distribution model in two heat exchangers has

been presented in this paper. The complexity of the model

is much greater than the lumped parameter model. The

main merit of the spatial model is that the parameter distri-

bution over distance is detailed and the moving two-phase

0

0.2

0.4

0.6

0.8

1

1.2

0 10 20 30 40

Condenser length (m)

Drynessfraction

0.1 Sec

0.4 Sec

0.6 Sec

2 Sec

120 Sec

360 Sec

Fig. 7. Dryness fraction distribution ong the condenser tube.

280

290

300

310

320

330

340

350

360

370

380

0 10 20 30 40

Condenser length (m)

Temperature(K

)

0.1 sec

0.5 sec

1.5 sec

8.0 sec

31.9 sec120.4 sec

Fig. 8. The refrigerant temperature distribution in the condenser.

280

284

288

292

296

300

0 5 10 15 20 25

Evaporator length (m)

Temperature(K)

0.1 Sec 0.5 Sec 6.0 Sec

15.4 Sec 30.9 Sec 120.9 Sec

Fig. 9. The wall temperature distribution in the evaporator.

276

278

280

282

284

286

288

290

292

294

296

0 4 8 12 16 20 24

Evaporator length (m)

Temperature(K)

0.5 Sec 1.0 Sec 3.0 Sec

10 Sec 60.9 120.9

Fig. 10. The refrigerant temperature distribution in the evaporator.

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interface is clearly defined. This would allow visualisationof the modelled system, useful both in design and optimisa-

tion. As the model has been experimentally validated it

stands alongside previous lumped parameter models. For

the future, three-way comparisons between the two model-

ling methods and experiments would be interesting.

Most researchers have previously used a pre-defined

function of the void fraction in their spatially distributed

model, based on experimental results. This approach results

in the separate solution of the mass balance and energy eq-

uations, and less calculations are required. However, it is

recognized that the mass and energy equations should be

coupled during solving for more accurate solution. The spa-tial distribution model constructed here is based on the prin-

ciple of energy and mass balance to solve the velocity,

pressure, temperature and wall temperature functions in

heat exchangers simultaneously. Although more calcula-

tions are required, this method presents a clearer understand-

ing of two-phase flow region. If the initial and boundary

conditions are accurate, better precision should be achieved

in the simulation.

The lumped parameter model gives an energy balance for

each section namely superheat region, two-phase region and

sub-cooling region. This means there are only a maximum

of three mesh points in each heat exchanger. The spatial

distribution model gives more mesh points in heat

exchangers and therefore the solution should be more

accurate. The results show that mesh size is one factor inthe models performance. By examining the results from the

two models, it is believed that the results depend on the

following factors:

(1) The accuracy of heat transfer coefficient used;

(2) The boundary conditions;

(3) The initial conditions for dynamic model; and

(4) The physical model assumptions.

Although a number of heat transfer coefficients are avail-

able, there are some discrepancies and more work is needed

in this area. The models used here are not reliable enough topredict variable superheat. This may be due to theeffect of oil

on the superheat exchanger surface or the influence of droplets

in the compressor suction.

Iteration is the only procedure for numeric solution in

system modelling; a skilled iteration method will reduce

computation time. A novel iteration method for reducing

computation time has been proposed here.

Acknowledgements

The financial support of the EPSRC is gratefully

acknowledged.

References

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2.5

3.5

4.5

5.5

0 200 400 600 800 1000

Time (sec)

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22

23

24

25

26

27

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[8] F.Q. Wang, The passive use of phase change materials (PCMs)

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