dynamic simulation of natural convection bypass

8/12/2019 Dynamic Simulation of Natural Convection Bypass

1/12

Dynamic simulation of natural convection bypass

two-circuit cycle refrigeratorfreezer and its application

Part I: Component models

Guoliang Ding *, Chunlu Zhang, Zhili Lu

Department of Power and Energy Engineering, Institute of Refrigeration and Cryogenics Engineering,Shanghai Jiaotong University, No. 1954 Huashan Road, Shanghai 200030, China

Received 21 May 2003; accepted 14 December 2003

Available online 23 January 2004

Abstract

In order to reduce the greenhouse gas emissions, efficient household refrigerator/freezers (RFs) are

required. Bypass two-circuit cycle RFs with one compressor are proved to be more efficient than two-

evaporator in series cycle RFs. In order to study the characteristics and improve the design of bypass two-

circuit cycle RFs, a dynamic model is developed in this paper. In part I, the mathematic models of allcomponents are presented, considering not only the accuracy of the models but also the computation

stability and speed to solve the models. An efficiency model that requires a single calorimeter data point at

the standard test condition is employed for compressor. A multi-zone model is employed for condenser and

for evaporator, with its wall thermal capacity considered by effective metal method. The approximate

integral analytic model is employed for adiabatic capillary tube, and the effective inlet enthalpy method is

used to transfer the non-adiabatic capillary tube to adiabatic capillary tube. The z-transfer function model

is employed for cabinet load calculation.

2004 Elsevier Ltd. All rights reserved.

Keywords: Dynamic simulation; Refrigerator/freezers; Two-circuit; Natural convection; Component model

1. Introduction

The two-evaporator in series natural convection refrigeratorfreezers (RFs) charged with purerefrigerants, such as R-600a and R134a, are widely used because of the simple design and low

* Corresponding author. Tel.: +86-21-62932110; fax: +86-21-62932601.

E-mail address: [email protected](G. Ding).

1359-4311/$ - see front matter

2004 Elsevier Ltd. All rights reserved.doi:10.1016/j.applthermaleng.2003.12.009

Applied Thermal Engineering 24 (2004) 15131524

www.elsevier.com/locate/apthermeng
http://mail%20to:%[email protected]/http://mail%20to:%[email protected]/


2/12

Nomenclature

C specific heat (J kg1

K1

)d, D diameter (m)F area (m2)h enthalpy (J kg1)K heat transfer coefficient (W m2 K1)l, L length or thickness (m)m mass flow rate (kg s1)_m average mass flow rate (kg s1)M refrigerant mass (kg)n compressor speed (r.p.s)P pressure (Pa)

Q heat transfer rate (W)t time (s)T temperature (C or K)T average temperature (C or K)v specific volume (m3 kg1)W work (W or J s1)x vapor quality

Greek symbols

a heat transfer coefficient (W m2 K1)k coefficient of compressor capacity

g electrical efficiency of compressorh excess temperature (C or K)q refrigerant density (kg m3)q average density of refrigerant (kg m3)p circumference ratiow polytropic exponentd thickness of control volume (m)l dynamic viscosity (Pa s)s time variable (s)Ds time step (s)

Subscriptsamb ambient

c, con condensercap capillarycom compressor

cmpt compartmente, eva evaporator

g vapor phase

1514 G. Ding et al. / Applied Thermal Engineering 24 (2004) 15131524


3/12

costs. For this kind of products, it is difficult to control the temperatures of both compartments in

the required values in the same time. The inherent problem is that temperature can be specified inonly one compartment. The temperature in either the freezer compartment (FZC) or the fresh

food compartment (FFC) controls the operation of the compressor and, ultimately, the amount ofheat extracted from both evaporators. Thus, the evaporator that does not control the compressoroften receives more or less cooling than it requires. Also, since the evaporating temperature in

FFC is the same as the evaporating temperature in FZC, the thermodynamic irreversible loss islarge in FFC. For this cycle, extra electric heating is often necessary when the food compartment

temperature is specified and ambient temperature is very low.One method termed dual-loop system has been suggested to overcome challenge of the

unique two-temperature application of RFs [14]. The dual-loop cycle RFs had independentcycles for FFC and FZC. Every cycle has itself compressor, condenser, capillary tube, and

evaporator. One shortcoming for dual-loop cycle is that the initial cost is high since two com-pressors are used.

Another method termed bypass two-circuit cycle has been used to overcome the short-comings of two-evaporator in series RFs. Bypass two-circuit cycle is achieved by adding an

additional path in the two-evaporator in series cycle, which can bypass the former evaporator ofin series cycle. The schematic diagrams of bypass two-circuit cycle are shown in Fig. 1(a) and (b).Obviously, the initial cost for this cycle is low since only single compressor is employed. For this

cycle, since an additional path is added, the design and the characteristics are more complicatedthan two-evaporator in series cycle and dual-loop system, researches on this two-circuit cycle arenecessary and urgent.

Computer simulation is widely used in refrigerators and other refrigeration system [512],which is a cheap and time-saved means to study characteristics of refrigeration system and im-prove or optimize refrigeration units. For refrigerators simulation, it includes steady-state sim-

ulation [57] and dynamic simulation [810]. For steady-state simulation, the thermal capacity offoam insulation is neglected. For dynamic simulation, not only the refrigeration system, but also

the cabinet is considered to be dynamic, so the simulation is complicated. Furthermore, since

f liquid phasefg liquidvapor phase

sat saturation, saturatedsc subcoolingsh superheat

suc suctiontp two phasew wall

unit unit length

Superscripts

i inleto outlet

G. Ding et al. / Applied Thermal Engineering 24 (2004) 15131524 1515


4/12

bypass two-circuit cycle RF is complicated, computation stability of dynamic simulation is dif-

ficult to keep, thus reliable and stable models and its solutions are necessary.Many works on refrigeration simulation were done in our research team, such as RFs, air-

conditioners, and liquid chillers [912]. Compared with air-conditioners and liquid chillers, the

dynamic cabinet load model is necessary in dynamic simulation of RFs, it is more difficult to buildand solve the models for dynamic simulation of refrigerator. In this paper, dynamic simulation ofbypass two-circuit cycle RFs is presented as a further research on the early simulation of two-

evaporator in series cycle RFs [10]. In Part I, only the component models, including the com-pressor model, condenser and evaporator model, capillary tube model, and cabinet load model,are presented in details.

2. Bypass two-circuit cycle RF

As shown in Fig. 1(a) and (b), there are two types of bypass two-circuit cycle RFs. The cyclesystem of Fig. 1(a) can bypass the freezer evaporator, and the cycle system of Fig. 1(b) can bypassthe food evaporator. In bypass two-circuit cycle RFs, the three-way solenoid valves only control

the refrigerant flow direction. The operation of compressor and three-way solenoid valve iscontrolled not only by FFC temperature, but also by FZC temperature. In the example shown inFig. 1(a), FZC controls loop A, and FFC controls loop B. So independent compartment tem-

perature control can be maintained. The detailed control ways for Fig. 1(a) are as follows:Initially, FZC and FFC temperatures are equal to ambient temperature that is larger than the

up-limit values of FZC and FFC. The compressor starts up and the solenoid valve switches for the

refrigerant flowing through loop A. Then the RF works according to the following modes.

(a) Refrigerant is flowing through the loop A. The compressor runs at first until the FZC temper-ature reaches its low-limit value, then the FFC temperature is compared with its middle-limit

Condenser

Compressor

Capillary Tube-Suction

Line Heat Exchanger

Suction Line

Capillary

Tube A

Freezer

Evaporator

FoodEvaporator

Three-Way

Solenoid Valve

A B

Capillary

Tube B

FoodEvaporator

Freezer

Evaporator

Capillary Tube-Suction

Line Heat Exchanger

Capillary

Tube B

Condenser

Suction Line

Three-Way

Solenoid Valve

Capillary

Tube A

A B

Compressor

(a) (b)

Fig. 1. Bypass two-circuit cycle RF.



5/12

value in order to determine the working mode of the RF. If the FFC temperature is lower

than the middle-limit value, then the compressor stops and the RF works according to themode (c); otherwise the solenoid valve switches for the refrigerant flowing through loop B

and the RF works according to the mode (b).(b) Refrigerant is flowing through the loop B. In this case, only the FFC is cooled, and the FZC

temperature goes up. If FZC temperature gets to its up-limit value before the FFC gets its low-limit value, the solenoid valve switches for the refrigerant flowing through loop A, and the RF

changes its working mode to mode (a); otherwise, the compressor runs until the FFC temper-ature is below its low-limit value, and the RF changes its working mode to mode (c).

(c) Compressor stops. In order to protect the motor, off time must be longer than 3 min. After 3

min, if FZC temperature gets to its up-limit value before FFC temperature gets to its up-limitvalue, the compressor starts up, the solenoid valve switches for the refrigerant flowing through

loop A, the RF works according to mode (a); if FZC temperature gets to its up-limit value afterFFC temperature does, the RF works according to mode (b); if FZC temperature and FFC

temperature gets to their up-limit values at the same time, the RF works according to mode (a).

3. Mathematical models

3.1. Compressor

The compressor used in the RF is hermetic reciprocating one. It is composed of two parts:cylinder and shell.

The mass flow rate and input work are calculated by

m0com kVth=vsuc 1

W0com kVthpevaw

w1

pcon

peva

w1w1

" #,g 2

VthpD2Ln i=4 3

where,m0com is the mass flow rate of compressor; k is the coefficient of compressor capacity; Vth isthe theoretical piston displacement of compressor; vsuc is the specific volume of suction vapor;W0com is the input work of compressor; w is the polytropic exponent; g is electrical efficiency of

compressor; D is the diameter of the cylinder; L is the length of stroke; n is revolutions of thecrankshaft per second; i is the number of cylinders.

In order to improve the prediction accuracy by Eqs. (1) and (2), the correction coefficients ofmass flow rate and input work cm; cw are introduced here. They are given by

cm mcom=m

0com 4

cWWcom=W

0com 5

wheremcomandWcomare measured mass flow rate and input work respectively at a single point of

ASHRAE standard test condition. m0com and W0com are computed by Eqs. (1) and (2).



6/12

The final mass flow rate and work input to compressor are computed by

mcomm0com cm 6

WcomW0com cW 7

The mass conservation equation for compressor is given by

dMcom=ds moevamcom 8

The energy conservation equation for compressor shell is given by

cqVcomdTcom

ds Wcomm

icomh

icomm

ocomh

ocomacomFcomTcomTamb 9

whereacomis the convection heat transfer coefficient of outside surface of compressor shell; Fcomis

the outside heat transfer area of compressor shell. moeva is the mass flow rate of evaporator outlet.

3.2. Heat exchangers

Heat exchangers in RFs include condenser and evaporator. Heat exchanger model can be di-vided into three types: lumped-parameter model, distributed-parameter model and multi-zone

model. Lumped-parameter model cannot reflect the different heat transfer at different phase state,so its error is great. Theoretically, the precision of distributed-parameter model is best, but theprecision of the heat exchanger model is related with the heat transfer coefficients, and at present,

the local heat transfer coefficients of heat exchangers are severely limited to use, therefore, as thematter of fact, the precision of distributed-parameter model cannot be assured. At the mean time,

the convergence of distributed-parameter model is poor and it would cost much time to solve sucha model. As the precision, the convergence, and computation speed are all taken into consider-

ation, multi-zone model is reasonable in this work.As condensation and evaporation are very complicated, the following assumptions are pro-

posed to simplify the mathematical models: (1) refrigerant flowing in the tube is one-dimensional;(2) in any cross-section of the evaporator and the condenser, the refrigerant is homogeneous and,

hence, values of any parameter are identical in any cross-section of the ducts; and (3) heat transferalong the tube axis, pressure drop along the flowing direction, flowing force, and gravity force are

all neglected.Thermal capacity of wall metals has effects on the dynamic characteristics of heat exchanger. In

order to take the thermal capacity of the wall metals into account and simplify the heat exchangermodel, the effective metal method is used. Because the convective heat transfer on the refrigerant

side is greater than that on the airside, the wall temperature is very close to the refrigeranttemperature. Assume that the wall temperature is equal to the refrigerant temperature. Actually,not all of the heat exchanger metal would be the same temperature with the refrigerant, and the

metal at the same temperature with the refrigerant is termed the effective metal. The effective metalcoefficient, kw is evaluated by

kwmeff=mw 10

where meff is mass of effective metal of heat exchanger; mw is total mass of heat exchanger.



7/12

For whole heat exchanger, mass conservation equation is given by

dMshMtpMsc

ds mo mi 0 11

For each zone of heat exchanger, mass and energy conservation equations are given by

Mshlsh Aunitqsh 12

MtpltpAunitqtp 13

Msc lsc Aunitqsc 14

lcon lshltp lsc 15

dMshhsh

ds kwqwCwAwdlshTsh

ds moshh

oshm

i

hi

FunitlshKshTairTsh 16

dMtphtp

ds kwqwCwAwdltpTtp

ds motph

otpm

itph

itp FunitltpKtpTairTtp 17

dMschsc

ds kwqwCwAwdlscTsc

ds moho misch

iscFunitlscKscTairTsc 18

In Eqs. (10)(18), the refrigerant mass flow rate (mi) and enthalpy (hi) at the inlet of heat ex-

changer are known, which are equal to ones at the outlet of compressor for condenser or ofcapillary tube for evaporator, and the refrigerant mass flow rate (mo) at outlet of heat exchanger is

known too, which is equal to the mass flow rate of capillary tube for condenser or of compressorfor evaporator. The qualitative temperature and average mass flow rate of refrigerant at any zoneare given by arithmetic mean temperature of its inlet and outlet values respectively.

Eqs. (10)(18) constitute a set of eight equations with eight unknownsMsh;Mtp;Msc; lsh; ltp; lsc;Ttp; Tsc, hence the above equations of heat exchanger can be solved to get a unique group ofsolution.

The mass in any single-phase zone is calculated by

Ms-pqs-pFunitls-p 19

where Ms-p and qs-p are the mass and the average density of refrigerant in single-phase zone

respectively;ls-p is the length of single-phase zone.The mass in two-phase zone is calculated by [13]

Mtp baqg 1aqfcFunitltp 20

a 1 1

1

x

1

qg

qf

21

where Mtp is the refrigerant mass in two-phase zone; a is the void fraction in two-phase zone; qgand qfare the saturated density of vapor and liquid in two-phase zone respectively;x is averagemass quality of refrigerant in two-phase zone.



8/12

3.3. Capillary tube

The capillary tube used in RF has following characteristics:

(1) The capillary tube used in household RF is always non-adiabatic, which has a heat exchangewith the suction line. (2) During the startup and shutdown processes, the inlet conditions ofcapillary tube change greatly. Not only the subcooling state, but also the superheat state and two-phase state occur.

Since the outlet states of capillary tube responds very fast to its inlet states because of the highflow velocity in the small diameter capillary tube [14], steady-state model is employed for capillarytube.

Since the throttling process of non-adiabatic capillary tube is very complicated, generally thenon-adiabatic capillary tube is transferred to adiabatic capillary tube by certain equivalent

method, such as effective subcooling level method. In this paper, the steady-state model of adi-abatic capillary tube is established first, then, the effective inlet enthalpy method is presented to

transfer non-adiabatic capillary tube to adiabatic capillary tube.The flow in adiabatic capillary tube is generally considered to be one-dimensional and

homogeneous and thermal equilibrium. Consequently, the governing equations are as follows:

dG 0 22

d h 12G2v2

dL

0 23

dp G2 dv1

2

f

DvG2dL 24

where p, h, v, G are refrigerant pressure, specific enthalpy, specific volume, and mass fluxrespectively;fis the friction factor, DandLare the inner diameter and length of the capillary tuberespectively.

From the viewpoint of system simulation, the stable and fast approach is better. Theapproximate integral analytic method to solve Eqs. (22)(24) is presented in [15] when the inletcondition of the capillary tube is subcooled, and in [16] when the inlet condition of the capillary

tube is superheated or two-phase.The solution to Eqs. (22)(24) cannot be used directly for non-adiabatic capillary tube. In order

to predict the mass flow rate in non-adiabatic capillary tube, the effective subcooling level method

was presented at 1988 ASHRAE Equipment Handbook [17]. Such method was evaluated in paper[18]. The accuracy of this method is very well when the inlet condition is subcooled. The effective

subcooling level method is suitable only when the inlet condition of capillary tube is subcooled.When the inlet condition is superheated or two-phase, the effective subcooling level method is not

suitable. Herein, effective inlet enthalpy method is presented.Heat exchange between capillary tube and suction line can be equivalent to the effective en-

thalpy drop by

Dheffmsuch

osuch

isuc

mcap25



9/12

where Dheffis the effective enthalpy drop; msucis mass flow rate in the suction line;hisucand h

osucare

the inlet and outlet enthalpy of suction line heat exchanger, respectively. mcapis the mass flow rate

in the capillary tube.

The effective inlet enthalpy can be equivalent by

hicap;effhicap Dheff 26

where hicap;effis the effective inlet enthalpy; hicap is the inlet enthalpy.

The effective inlet enthalpy method of modeling the non-adiabatic capillary tube first requiresadding the calculated effective enthalpy drop to the actual inlet enthalpy. This results in an adi-abatic equivalent inlet enthalpy. Then the problem to predict the mass flow rate of a non-adiabatic

capillary tube is changed to do the mass flow rate of a adiabatic capillary tube at the equivalentinlet enthalpy.

When the inlet condition of capillary tube is subcooled, the effective inlet enthalpy method is

actually the same as the effective subcooling level method, so the effective inlet enthalpy method isthe extension of the effective subcooling level method.

3.4. Cabinet

The performance of RFs is not only concerned with the refrigeration cycle, but also with thecabinet load. Because the thermal properties of foam insulation and convective heat transfer

coefficients of inner and outer surfaces of insulation walls vary very small under the workingcondition of RFs, they are assumed to be constants. Therefore, the thermal insulation wall is a

linear system, and the thermal response factor method or conduction transfer function method

[1921] can be used to calculate the heat transfer through the wall.For RFs, the output parameters are the compartment temperatures. The inputs are heat

transfer in the surface of inner side and outside of the thermal insulation wall. The output andinputs can be formulated by

hcmptz1 Wz1Iz1 27

where hcmptz1is compartment temperature. All temperatures mentioned in this section are the

excess temperatures to the ambient temperature.Wz1is transfer function of cabinet load;Iz1is total cabinet load; zdenotes the zoperator.

The method to get the transfer function Wz1 is shown as follows:Take all air in the compartment as the control volume. According to energy conservation law

and superposition principle of linear system, the governing equation on the control volume isgiven by

Ccmptdhcmpts

ds Qambs Qcmpts Qcoms Qothercmpts Qhots Qcolds Qleaks

28

where Ccmpt is the thermal capacity of the control volume; Qambs is the heat transfer into thecontrol volume through the foam insulation by the ambient temperature pulse; Qcmptsis the heattransfer into the control volume through the foam insulation by the compartment temperature



10/12

pulse; Qcoms is the heat transfer into the control volume through the foam insulation by thecompressor chamber temperature pulse; Qhots is the heat transfer into the control volumethrough the foam insulation by the hot-wall condenser heat flux pulse; Qothercmpts is the heat

transfer into the control volume through the foam insulation by the other compartment tem-perature pulse;Qcoldsis the heat transfer into the control volume through the foam insulation bythe cold-wall evaporator heat flux pulse; Qleaks is the heat transfer into the control volumethrough the door and other gaskets.

The heat transfer through the gaskets could be calculated by

Qleaks bLTambTcmpts bLhcmpts 29

Backward finite-difference and z-transformation is used for Eq. (28), we can get

Ccmpt

Ds 1z1hcmptz Qambz

1 Qcmptz1 Qcomz

1 Qothercmptz1

Qhotz1

Qcoldz1

Qleakz1

30where

Qambz1 Wambz

1hcmptz1 31a

Qcmptz1 Wcmptz

1hcmptz1 31b

Qcomz1 Wcomz

1hcomz1 31c

Qothercmptz1 Wothercmptz

1hothercmptz1 31d

Qhotz1

Whotz1

qconz1

31e

Qcoldz1 Wcoldz

1qevaz1 31f

Qleakz1 bLhcmptz

1 31g

Qambz1, Qcmptz

1, Qcomz1, Qothercmptz

1, Qhotz1, Qcoldz

1 are heat flux at the inner sur-face of cabinet wall.Wambz

1,Wcmptz1,Wcomz

1,Wothercmptz1,Whotz

1,Wcoldz1are theirz-

transfer functions. These z-transfer functions can be obtained by the method described in [21].Substituting Eqs. (31ag) into Eq. (30) and rearranging the equation, we can get

hcmptCcmpt

Ds

1 z1 WambWcomprWothercmpt bLWcomprhcomprz Wothercmpthother cmptWhotqconWevaqeva 32

Compare Eqs. (32) and (27), we have

Wz 1 Ccmpt

Ds 1

z1 WambWcomprWother cmpt bL

33a

Iz Wcomprhcomprz Wothercmpthother cmptWhotqconWevaqeva 33b

So far, the cabinet model has been solved.



11/12

4. Conclusions

The component models for natural convection bypass two-circuit cycle RFs are presented here.

In order to make the simulation program run fast and its accuracy acceptable, the efficiency modelthat required a single calorimeter data point at the standard test condition is adopted for thecompressor; the multi-zone models are employed for condenser and evaporator, with its wallthermal capacity considered by effective metal method; the approximate integral analytic model is

employed for adiabatic capillary tube, and the effective inlet enthalpy method is used to transferthe non-adiabatic capillary tube to adiabatic capillary tube; the z-transfer function model is usedfor the cabinet load calculation.

Acknowledgements

The research is supported by the State Key Fundamental Research Program of China under thecontract No. 2000026309. Part of the research was financed by Refrigerator/Freezer Ltd Com-pany (R/FLC), Haier Group, China. Helps of Mr. Dongning Wang and Mr. Linfei Xu in R/FLC,

Haier Group are greatly appreciated.

References

[1] S. Won, D. Jung, R. Radermacher, An experimental study of the performance of a dual-loop refrigerator/freezer

system, Int. J. Refrig. 17 (6) (1994) 411416.

[2] E. Baskin, F.R. Delafield, Performance of a two-cycle refrigerator/freezer using HFC refrigerants, ASHRAE

Trans. 105 (Part 1) (1999) 310318.

[3] A.I. Gan, S.A. Klein, D.T. Reind, Analysis of refrigerator/freezer appliances having dual refrigeration cycles,

ASHRAE Trans. 106 (Part 1) (2000) 185191.

[4] K.E. Simmons, I. Haider, R. Radermacher, Independent compartment temperature control of LorenzMeuitzner

and modified LorenzMeuitzner cycle refrigerators, ASHRAE Trans. 102 (Part 1) (1996) 10851096.

[5] S. Rogers, D.R. Tree, Algebraic modeling of components and computer simulation of refrigeration steady-state

operation, in: Proceedings of the XVIII International Congress of Refrigeration, Montreal, Canada, 1991, pp.

12251230.

[6] D.S. Jung, R. Radermacher, Performance simulation of single-evaporator domestic refrigerators charged with pure

and mixed refrigerants, Int. J. Refrig. 14 (4) (1991) 223232.

[7] D.S. Jung, R. Radermacher, Performance simulation of a two-evaporator refrigeratorfreezer charged with pure

and mixed refrigerants, Int. J. Refrig. 14 (5) (1991) 254263.[8] V. Vidmar, B. Gaspersic, Dynamic simulation of domestic refrigerators with refrigerants R12 and R134a, in:

Proceedings of the XVIII International Congress of Refrigeration, 1991, pp. 12501254.

[9] Z.J. Chen, W.H. Lin, Dynamic simulation and optimal matching of a small-scale refrigeration system, Int. J.

Refrig. 14 (1991) 329335.

[10] Z.J. Chen, G.L. Ding, X.F. Wang, X.C. Que, Dynamic simulation and optimal matching of a small-scale

refrigeration system with multi-refrigerant, in: Proceedings of 19th International Congress of Refrigeration, The

Netherlands, August 1995, pp. 262269.

[11] G.L. Ding, C.L. Zhang, H. Li, Z.J. Chen, Fast dynamic simulation of a split unit domestic air conditioner, in:

Proceedings of International Conference of Cryogenics and Refrigeration, Hangzhou, China, April 1998, pp. 282

285.



12/12

[12] L. Fu, G.L. Ding, Z.J. Su, G.Q. Zhao, Steady-state simulation of screw liquid chillers, Appl. Thermal Eng. 22 (15)

(2002) 17311748.

[13] A. Premoli, D.D. Francesco, A. Prina, A dimensional correlation for evaluating two-phase mixture density, La

Termotecnica 25 (1) (1971).

[14] F. Escanes, C.D. Perez-Segarra, A. Oliva, Numerical simulation of capillary-tube expansion devices, Int. J. Refrig.

18 (1995) 113122.

[15] C.L. Zhang, G.L. Ding, Modified general equation for the design of capillary tubes, ASME J. Fluids Eng. 123 (4)

(2001) 914919.

[16] G.L. Ding, C.L. Zhang, H. Li, Z.J. Chen, An approximate analytic model for flow through capillary tubes, Chin.

Sci. Bull. 44 (7) (1999) (in English).

[17] ASHRAE, ASHRAE Handbook-1998 Equipment, American Society of Heating, Refrigerating and Air

Conditioning Engineers, Atlanta, 1988.

[18] R. Robert, W. Bittle, Stephenson Robert, B. Pate Michael, An evaluation of the ASHRAE method for predicting

capillary tube-suction line heat exchanger performance, ASHRAE Trans. 101 (Part 2) (1995) 434442.

[19] D.G. Stephenson, G.P. Mitalas, Calculation of heat conduction transfer functions for multi-layer slabs, ASHRAE

Trans. 77 (Part 2) (1971) 117126.

[20] C.L. Zhang, G.L. Ding, A stable series expansion method for calculating thermal response factors of multi-layerwalls, Building and Environment 38 (2003) 699705.

[21] G.L. Ding, C.L. Zhang, X.F. Wang, Z.J. Chen, Calculating the z-transfer function with state-space method for

unsteady heat transfer through thermal insulation layer, in: Proceedings of 19th International Congress of

Refrigeration, The Netherlands, August 1995, pp. 6572.


dynamic simulation of natural convection bypass

Documents