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Applied Thermal Engineering 27 (2007) 1062–1071

Numerical simulation and experimental validation of coiledadiabatic capillary tubes

O. Garcıa-Valladares *

Centro de Investigacion en Energıa, Universidad Nacional Autonoma de Mexico (UNAM), Apdo. Postal 34, 62580 Temixco, Morelos, Mexico

Received 7 February 2006; accepted 12 July 2006Available online 11 October 2006

Abstract

The objective of this study is to extend and validate the model developed and presented in previous works [O. Garcıa-Valladares, C.D.Perez-Segarra, A. Oliva, Numerical simulation of capillary tube expansion devices behaviour with pure and mixed refrigerants consid-ering metastable region. Part I: mathematical formulation and numerical model, Applied Thermal Engineering 22 (2) (2002) 173–182;O. Garcıa-Valladares, C.D. Perez-Segarra, A. Oliva, Numerical simulation of capillary tube expansion devices behaviour with pureand mixed refrigerants considering metastable region. Part II: experimental validation and parametric studies, Applied Thermal Engi-neering 22 (4) (2002) 379–391] to coiled adiabatic capillary tube expansion devices working with pure and mixed refrigerants. The dis-cretized governing equations are coupled using an implicit step by step method. A special treatment has been implemented in order toconsider transitions (subcooled liquid region, metastable liquid region, metastable two-phase region and equilibrium two-phase region).All the flow variables (enthalpies, temperatures, pressures, vapor qualities, velocities, heat fluxes, etc.) together with the thermophysicalproperties are evaluated at each point of the grid in which the domain is discretized. The numerical model allows analysis of aspects suchas geometry, type of fluid (pure substances and mixtures), critical or non-critical flow conditions, metastable regions, and transientaspects. Comparison of the numerical simulation with a wide range of experimental data presented in the technical literature will beshown in the present article in order to validate the model developed.� 2006 Elsevier Ltd. All rights reserved.

Keywords: Metastable flow; Refrigeration; Coiled tube; Two-phase flow; Expansion devices; Separated flow model

1. Introduction

A capillary tube is a fixed length of small diameter tub-ing lying between the outlet of the condenser and the inletof the evaporator. It reduces the pressure of the refrigerantfrom the high-pressure side to the low-pressure side of thesystem and controls the flow of refrigerant to the evapora-tor. Capillary tube expansion devices are widely used insmall refrigeration equipment such as household refrigera-tors, freezers and air conditioners because its simplicity,reliability and low cost. Despite its simplicity, the flowinside a capillary tube is very complex. In some practical

1359-4311/$ - see front matter � 2006 Elsevier Ltd. All rights reserved.

doi:10.1016/j.applthermaleng.2006.07.034

* Tel.: +52 55 562 29746; fax: +52 55 562 29791.E-mail address: [email protected]

applications, capillary tubes are coiled to save space andthe effect of the coiling affects the fluid flow inside them.

The capillary tube dimensions play an important role indeciding its contribution to the refrigeration system perfor-mance. For these reasons and in order to optimize the effi-ciency of the coiled adiabatic capillary tubes, and thereforethe energy consumption, more general and accurate meth-ods for predicting their thermal and fluid-dynamic behav-ior are required.

Fig. 1 shows a typical pressure and saturated pressuredistributions of a refrigerant flowing through a capillarytube, where the flow is divided into four regions: subcooledregion (zone I: when p P psat,l, xg = 0), metastable liquidregion (zone II: when psat,l > p P pv, xg = 0), metastabletwo-phase region (zone III: when pv > p P psat,g, 0 < xg 6

xgequil, 0 6 y 6 1), and thermodynamic equilibrium

mailto:[email protected]

Nomenclature

A cross-section area (m2)cp specific heat at constant pressure (J kg�1 K�1)CV control volumeD diameter (m)Dc coiled diameter (m)D 0 reference lengthf friction factorg acceleration due to gravity (m s�2)G mass velocity (kg m�2 s�1)h enthalpy (J kg�1)k mesh concentration factorK Boltzmann’s constant 1.380662 · 10�23

(J K�1 mol�1)L length (m)m mass (kg)_m mass flow rate (kg s�1)n number of control volumesp pressure (Pa)pv pressure of vaporization (Pa)Re Reynolds number GD

l

� �t time (s)T temperature (K)v velocity (m s�1)xg vapor mass fractiony mass ratio of total saturated phase to total

phasez axial coordinate

Greek letters

d rate of convergenceDt temporal discretization step (s)DTsc degree of subcooling (K)

Dz axial discretization step (m)e roughness (m)eg void fraction/ generic dependent variableU two-phase frictional multiplierl dynamic viscosity (kg m�1 s�1)h tube inclination angle (rad)l density (kg m�3)r surface tension (N m�1)s shear stress (N m�2)

Subscripts

c criticalcon condenserequil thermodynamic equilibriumev evaporatorexp experimentalg gas or vaporl liquidm superheated liquidnum numericalsat saturationtp two-phasez axial direction

Superscriptso previous instant– arithmetical average over a CV: �/ ¼ ð/iþ

/iþ1Þ=2� integral average over a CV: ~/ ¼ ð1=DzÞR zþDz

z /dz

O. Garcıa-Valladares / Applied Thermal Engineering 27 (2007) 1062–1071 1063

two-phase region (zone IV: when pv > p P psat,g, xgequil <xg 6 1). At point a, the pressure is equal to the saturatedpressure but the vaporization does not take place. A meta-stable (superheat) liquid flow occurs for a short distanceuntil the onset of vaporization. In this region the pressuredrop is almost linear. At point b, the vapor bubbles appearand pressure suddenly drops due to the beginning of thetwo-phase region. However, this is a metastable regionbecause of the existence of superheat liquid together withsaturated liquid and vapor fluid [1]. After point c, the localthermodynamic equilibrium state is reached.

Kuehl and Goldschmidt [2] presented an experimentalstudy to determine the effects of coiling on capillary tubes;the mass flow rate obtained for these authors is approxi-mately 0–5% lower than that of a straight tube.

Wei et al. [3] examined the performance of capillarytubes having R-407C and R22 as the working fluid. Theeffect of coiling was quantitatively investigates and a corre-lation was proposed to correlate their experimental data

and to describe the relation between straight and coiledcapillary tubes.

Wei et al. [4] studied the performance difference betweenstraight and coiled capillary tubes working with R22. Theresults indicate that helical effect increased with decreaseof the diameter of coiling, and the effect of inlet conditionsis relatively small. A simple relation between the straightand coiled capillary tubes based on their experimental datais obtained.

Kim et al. [5] developed a dimensionless correlation onthe basis of experimental data of adiabatic capillary tubesfor R22, R407C and R410A. The results for straight capil-lary tubes were compared with capillary tubes with coileddiameters of 40, 120 and 200 mm.

Gorasia et al. [6] developed a numerical simulation oncoiled capillary tubes, they used Mori and Nakayama [7]equation and Giri correlation for calculating liquid andtwo-phase flow friction factor; metastable regions areignored.

Position [m]

Pre

ssu

re [

bar

]

0

0

0.5

0.5

1

1

1.5

1.5

3 3

4 4

5 5

6 6

7 7

8 8

ppsat

I II III IV

I. Subcooled, single liquid phaseII. Metastable, single liquid phaseIII. Metastable, liquid-vapor two-phaseIV. Thermodynamic equilibrium, liquid-vapor two-phase

a b

cb’

Fig. 1. Typical pressure distribution along a capillary tube (dotted line:saturated pressure).

Fig. 2. Schematic diagram of characteristic control volumes, inlet andoutlet sections are represented as ‘i’ and ‘i + 1’.

1064 O. Garcıa-Valladares / Applied Thermal Engineering 27 (2007) 1062–1071

Zhou and Zhang [8] carried out a numerical modelincluding metastable regions, considering homogeneoustwo-phase flow and three methods are discussed to calcu-late friction factors for coiled adiabatic capillary tubes.The method chooses presents an average deviation of±5% with their experimental data and Kim et al. [5] datafor R22.

The objective of this study is to extend and validate themodel by Garcıa-Valladares et al. [9,10] and Garcıa-Valladares [11] to coiled adiabatic capillary tubes consider-ing separated flow model and metastable regions. Thenumerical solution has been performed by discretizationof the one-dimensional governing equations based on afinite volume formulation. The entropy equation enablesdetection of the limitation of the physical process producedunder critical flow conditions.

2. Mathematical formulation of two-phase flow inside tubes

The mathematical formulation of two-phase flow insidea characteristic CV of a tube was developed in [9], in thefollowing sections only the more relevant aspects of thisformulation or those aspects not presented in previouswork are shown.

A characteristic CV is shown schematically in Fig. 2,where ‘i’ and ‘i + 1’ represent the inlet and outlet sectionsrespectively. Taking into account the characteristic geome-try of tubes (diameter, length, roughness and coiled diam-eter), the governing equations have been integratedassuming the following assumptions:

• One-dimensional flow: p(z, t), h(z, t), T(z, t), etc.• Adiabatic flow.

• Fluid: pure and mixed refrigerant (no oil is entrained).• Axial heat conduction inside the fluid is neglected.• Constant internal and coiled diameters and uniform

roughness surface.

The semi-integrated governing equations (continuity,momentum, energy and entropy) over the above men-tioned finite CV were presented in [9]. The one-dimen-sional model requires the knowledge of the two-phaseflow structure, which is evaluated by means of the voidfraction eg. The evaluation of the shear stress is performedby means of a friction factor f. This factor is defined fromthe expression: s = U(f/4)(G2/2q), where U is the two-phase factor multiplier. The coiled effects of a tube areembodied in the calculation of friction factor. So, astraight capillary tube model can also be applied to coiledcapillary tubes by changing the corresponding friction fac-tor equations.

3. Evaluation of empirical coefficients

The mathematical model requires some additional localinformation generally obtained from empirical correla-tions. After comparing different empirical correlations pre-sented in the technical literature, the following ones havebeen selected:

3.1. Single-phase region (zone I)

The friction factor is evaluated from the followingexpressions proposed by Zhou and Zhang [8] which is theMori and Nakayama [7] equation with different coefficientsconsidering the tube wall roughness

f ¼ C1ðD=DcÞ0:5

½ReðD=DcÞ2:5�1=61þ C2

½ReðD=DcÞ2:5�1=6

( )ð1Þ


where

C1 ¼ 1:88411177� 10�1 þ 8:52472168� 101ðe=DÞ� 4:63030629� 104ðe=DÞ2 þ 1:31570014� 107ðe=DÞ3

C2 ¼ 6:79778633� 10�2 þ 2:53880380� 101ðe=DÞ� 1:06133140� 104ðe=DÞ2 þ 2:54555343� 106ðe=DÞ3

The values obtained by Zhou and Zhang [8] for coefficientsC1 and C2 for different relative roughness have been in thisarticle integrated from a polynomial regression in order toobtain continuous values between intervals. For e/D >0.0014, C1 = 0.253 and C2 = 0.0897 are considered.

3.2. Metastable liquid region (zone II)

Metastable flow model presented by Chen et al. [12] hasbeen employed for predicting the pressure of vaporization(pv). The friction factor is evaluated in the same way asfor the single-phase region.

3.3. Metastable two-phase region (zone III)

In this region, a model suggested by Feburie et al. [1] forthe evaluation of the mass ratio of total saturated phase tototal phase is used (Eq. (7)). Due to the lack of specificinformation, the friction factor is evaluated in the sameway as for equilibrium two-phase region.

3.4. Equilibrium two-phase region (zone IV)

Considering a separated two-phase flow model (vg 5 vl),the void fraction is estimated from the semi-empirical equa-tion of Premoli [13]. The friction factor is calculated fromthe same equation as in the case of the single-phase flowconsidering the two-phase dynamic viscosity correlationproposed by Dukler et al. [14]

ltp ¼xgqllg þ ð1� xgÞqgll

xgql þ ð1� xgÞqg

ð2Þ

4. Numerical resolution

The main objective is the determination of the mass flowrate through the coiled adiabatic capillary tube. Owing tothe high gradients produced at the end of the capillarytube, a non-uniform grid concentrated at the outlet sectionis generated according to the expression given by Garcıa-Valladares et al. [9].

The discretized equations have been coupled using afully implicit step by step method in the flow direction.From the known values at the inlet section, the variablevalues at the outlet of each CV are iteratively obtainedfrom the discretized governing equations (see next section).This solution (outlet values) is the inlet values for the nextCV. The procedure is carried out until the end of the coiledtube is reached.

For each CV, a set of algebraic equations is obtained bya discretization of the governing equations. The transientterms of the governing equations are discretized using thefollowing approximation: o//oT ffi (/ � /o)/Dt, where /represents a generic dependent variable (/ = h, p, T, q,etc.); superscript ‘‘o’’ indicates the value of the previousinstant. The averages of the different variables have beenestimated by the arithmetic mean between their values atthe inlet and outlet sections, that is: ~/i ffi �/i �ð/i þ /iþ1Þ=2.

Based on the numerical approaches indicated above, thegoverning equations can be discretized to obtain the valueof the dependent variables (mass flow rate, pressure andenthalpy) at the outlet section of each CV.

The mass flow rate is obtained from the discretized con-

tinuity equation,

_miþ1 ¼ _mi �ADzDtð�qtp � �qo

tpÞ ð3Þ

where the two-phase density is obtained from: qtp =egqg + (1 � eg)ql.

In terms of the mass flow rate, gas and liquid velocitiesare calculated as,

vg ¼_mxg

qgegA

" #; vl ¼

_mð1� xgÞqlð1� egÞA

� �

The discretized momentum equation is solved for the outletpressure,

piþ1 ¼ pi �DzA

Uf4

�_m2

2�qtpA2pDþ �qtpAg sin h

þ ½ _mðxgvg þ ð1� xgÞvlÞ�iþ1i

Dzþ

�_m� �_mo

Dt

!ð4Þ

where the subscript and superscript in the brackets indicate½X �iþ1

i ¼ X iþ1 � X i, i.e., the difference between the quantityX at the outlet section and the inlet section.

From the energy equation and the continuity equation,the following equation is obtained for the outlet enthalpyconsidering adiabatic flow:

hiþ1 ¼�a _miþ1 þ b _mi þ c ADz

Dt

_miþ1 þ _mi þ�qo

tpADz

Dt

ð5Þ

where

a ¼ ðxgvg þ ð1� xgÞvlÞ2iþ1 þ g sin hDz� hi

b ¼ ðxgvg þ ð1� xgÞvlÞ2i � g sin hDzþ hi

c ¼ 2ð�pi � �poi Þ � �qo

tpðhi � 2�hoi Þ � ð�q�v2

i � �qo�vo2

i Þ

In the zones with thermodynamic equilibrium: subcooled

liquid region and equilibrium two-phase region; temperature,mass fraction and all the thermophysical properties are cal-culated using matrices function of the pressure and enthal-py obtained with REFPROP v7.0 [15].

The above mentioned conservation equations of mass,momentum and energy together with the thermophysical


properties, are applicable to transient two-phase flow. Sit-uations of steady flow and/or single-phase flow (liquid orgas) are particular cases of this formulation. Moreover,the mathematical formulation in terms of enthalpy givesgenerality of the analysis (only one equation is needed forall the regions) and allows dealing in easy form with casesof refrigerant mixtures.

The metastable liquid region (zone II) begins when thepressure drops down to the saturated condition and fin-ishes at the onset of vaporization. The pressure of vapori-zation (pv) at the point of flashing (point ‘‘b’’ in Fig. 1) isevaluated by correlation proposed by Chen et al. [12]

ðpsat;l � pvÞffiffiffiffiffiffiffiffiffiffiffiffiffiKT sat;l

pr3=2

¼ 0:679ql

ql � qg

" #Re0:914 DT sc

T c

� ��0:208 DD0

� ��3:18

ð6Þ

where D 0 is a reference length given by D0 ¼ffiffiffiffiffiffiffiffiffiKT sat;l

r

q� 104.

In this region properties are estimated using the valuescorresponding to liquid saturated conditions at the fluidpressure. Temperature is estimated from the thermody-namic relation dh = cp dT + (oh/op)T dp using the enthal-pies obtained from the energy equation.

In the metastable two-phase region (zone III), Feburieet al. [1] suggest that the flow can be separated into threestates: superheated liquid (subscript m), saturated liquid(subscript l) and saturated vapor (subscript g). In this zone,the governing equations are used to evaluate the mass flowrate, the pressure and the mean enthalpy respectively.

A variable y, defined as mass ratio of total saturatedphase to total phase, y = (ml + mg)/(ml + mg + mm), is usedto evaluate the superheat liquid mass flow. This variable isevaluated by correlation proposed by Feburie et al. [1]

dydz¼ 0:02

pDA

� �ð1� yÞ

psat;l � p

pc � psat;l

" #0:25

ð7Þ

The mean enthalpy (obtained from the energy equation) isrelated to the individual enthalpies through their mass frac-tions, that is

�h ¼ xmhm þ xlhl þ xghg

¼ ð1� yÞhm þ y � xg

hl þ xghg ð8Þ

This expression allows the evaluation of the vapor massfraction (xg). Following Feburie et al. [1], the temperatureof the superheat liquid has been assumed constant in thisregion.

When y approaches unity, the superheated liquid van-ishes and the flow process enters to equilibrium two-phaseflow.

In metastable two-phase region, two temperaturesshould be distinguished: the superheated liquid tempera-ture (Tm) and the saturation liquid or gas temperature(Tequil). For this reason, an averaged temperature has beendefined as

T ¼ T equil �xg � xgequil

xgequil

� �ðT m � T equilÞ ð9Þ

where xg is the actual vapor mass fraction which resultsfrom the mean enthalpy equation and Tequil and xgequil

are the values obtained for the temperature and vapor massfraction if the fluid were in thermodynamic equilibrium.

Discharge shock wave. In the case of critical (choked)flow, a CV of larger outlet diameter than inlet diameter isdefined at the discharge. Assuming null heat transfer andaccumulative terms, the outlet vapor quality is calculatedby means of the energy equation, the outlet pressure isequal to the discharge pressure (pev), and the mass flow rateremains constant.

Using the conditions of differentiation between regionsmentioned in introduction, the CV where transition occursis detected. The CV is split into two control volumes.The length of the first CV is calculated from the momen-tum equation, imposing the pressure condition to differen-tiate between one region and the other at the outlet section.The length of the second one is calculated by simple differ-ence. The transition criterion is very important because ofthe significant changes in the friction factor betweenregions. Depending on the empirical correlations used,these coefficients can increase or decrease by factors often or more. The importance of this criterion in obtainingaccurate results with fewer control volumes represents sig-nificant important gains in computation cost [16].

In each CV, the values of the flow variables at the outletsection of each CV are obtained by solving iteratively theresulting set of algebraic equations (continuity, momen-tum, energy, entropy and state equations mentioned above)from the known values at the inlet section and the bound-ary conditions. The solution procedure is carried out in thismanner, moving forward step by step in the flow direction.At each cross-section, the shear stresses, and the void frac-tions are evaluated using the empirical correlationsobtained from the available literature (see Section 3). Thetransitory solution is iteratively performed at each timestep.

Convergence is verified at each CV using the followingcondition:

1� /�iþ1 � /i

D/

��

� �< d ð10Þ

where / refers to the dependent variables of mass flow rate,pressure and enthalpy; and /* represents their values at theprevious iteration. The reference value D/ is locally evalu-ated as /i+1 � /i. When this value tends to be zero, D/ issubstituted by /i.

The numerical global algorithm is as follows: the inletmass flow rate is iteratively estimated by means of a numer-ical Newton–Raphson method to obtain critical flux condi-tions. Critical or choke flux conditions are reached whenentropy equation is not verified in the last CV. To checkcritical conditions the criterion dp/dz!1 is sometimesreported in the literature. For the cases analyzed both


criteria are equivalent. A change of variable has been intro-duced [17] to ensure the convergence, using _m�fNR insteadof _m as independent variable. A value greater than 3.5 ofthe factor fNR fits most applications well.

After critical conditions are evaluated, the critical pres-sure and discharge (or evaporator) pressure are compared.If the critical pressure is greater than or equal to the dis-charge pressure, the flow is critical and the discharge shockwave is then solved. Otherwise, the flow is non-critical.Then, the mass flow rate that offers an outlet pressure equalto the discharge pressure is evaluated by means of anotherNewton–Raphson method, where the outlet pressure istaken as dependent variable. The transitory solution is iter-atively performed at each time step.

Input data. Internal diameter (D), coiled diameter (Dc),inclination angle (h, taken as zero for the analyzed cases),roughness (e) and length (L) of the coiled adiabatic capillary

Degree of subcooling [oC]

Ma

ssflo

wra

te[k

gh-1

]

0 5 10 15 2010

20

30

40

50

60

70

80

90

100

110

120

130

D=2.2 mm exp.D=1.6 mm exp.D=1.4 mm exp.D=1.2 mm exp.D=1.0 mm exp.D=2.2 mm num.D=1.6 mm num.D=1.4 mm num.D=1.2 mm num.D=1.0 mm num.

R22Dc=80 mmTcon=54 oCL=2 m

Coiled di

Mas

sflo

wra

te[k

gh-1

]

0 100 200 30028

30

32

34

36

38

40

42

44

46ΔTsc=15 oC exp.ΔTsc=10 oC exp.ΔTsc= 5 oC exp.ΔTsc=15 oC num.ΔTsc=10 oC num.ΔTsc= 5 oC num.

Fig. 3. Comparison of numerical simulation versus mass flow rates of R22 obtadegree of subcooling; (b) different tube length and condensing temperature; an

tube; Other data must be the following: kind of fluid (ther-mophysical and transport properties) and the boundaryconditions inlet (or condenser) pressure (pcon) and tempera-ture (Tcon) or degree of subcooling (DTsc) in the cases ofinlet single phase flow or mass fraction (xgcon) in the casesof inlet two-phase flow. From any of these values (temper-ature or mass fraction) and the pressure, enthalpy (the inde-pendent variable) is obtained; discharge (or evaporator)pressure (pev) or temperature (Tev) must be also data.

Numerical parameters: number of control volumes (n),mesh concentration factor (k) and rate of convergence (d)must be given.

5. Numerical results

A straight capillary tube model can also be appliedto coiled capillary tubes by changing the corresponding

Capillary tube length [m]

Ma

ssflo

wra

te[k

gh-1

]

0.0 0.5 1.0 1.5 2.0 2.535

40

45

50

55

60

65

70

75

80

85

90

95

Tcon=54 oC exp.

Tcon=48 oC exp.

Tcon=42 oC exp.

Tcon=54 oC num.

Tcon=48 oC num.

Tcon=42 oC num.

R22

Dc=40 mm

D=1.6 mm

Tev=7 oC

ΔTsc=10 oC

ε/D=0.0014

ameter [mm]400 500 600 700

R22D=1.4 mmL=2 mTcon=54 oC

ined by Zhou and Zhang [8] with: (a) different capillary tube diameters andd (c) different coiled diameters and degree of subcooling.


friction factor equations as has been shown in previoussection.

In this section comparison of numerical results withexperimental data available in the technical literature illus-trate the capabilities of the present model developed. Allsituations are analyzed using the following numericalparameters: d = 1 · 10�7, k = 3.5 and fNR = 3.5. All thenumerical results presented are grid independent solutions.

5.1. Comparison of the numerical results with experimental

data by Zhou and Zhang [8] for R22

Fig. 3a shows the comparison of the Zhou and Zhang [8]experimental mass flow rates with the results obtained bythe present model for different degree of subcooling andcapillary tubes diameters (D = 1.0, 1.2, 1.4, 1.6 and2.2 mm with a relative roughness (e/D) of 2.5 · 10�4,4 · 10�4, 12.2 · 10�4, 14 · 10�4 and 11.6 · 10�4, respec-tively). The evaporation temperature is 7 �C for all theZhou and Zhang [8] analyzed cases. Similar to a straighttube, the mass flow rate of the coiled adiabatic capillarytube increases as the tube diameter increases (due to thedecrease in the pressure drop through it). The numericalmass flow rate calculated for these 15 cases offer an averagepredicting deviation of 3.97%.

Fig. 3b presents the numerical and experimental massflow rates varying the capillary tube length and condensing

Table 1Comparison of the mass flow rate between simulation results and experimenta

Fluid pcona (bar) DTsc (�C) Dc (mm)

R22 17.2923 1.5 2005.0 200

10.0 2001.5 1205.0 120

10.0 1201.5 405.0 40

10.0 40

R407C 19.7234 1.5 2005.0 200

10.0 2001.5 1205.0 120

10.0 1201.5 405.0 40

10.0 40

R410A 27.3354 1.5 2005.0 200

10.0 2001.5 1205.0 120

10.0 1201.5 405.0 40

10.0 40

Mean deviation (%) =Pjerrorj/cases

a Bubble saturation pressure to 45 �C according to REFPROP v.7.0 [15].

temperature. Mass flow rate decreases rapidly as the tubelength increases (due to the increase in the pressure dropthrough it), and it increases as condensing temperatureincreases. The numerical mass flow rate calculated for these12 cases overestimate the experimental results with an aver-age discrepancy of 4.15%.

Fig. 3c presents the mass flow rate variations versus thecoiled diameters and degree of subcooling with both exper-imental and modeling results. The experimental resultsshown and the numerical results confirms that beyond300 mm the mass flow rate changes little and with thecoiled diameter increasing from 40 to 120 mm, the massflow rate increases about 3–4%. The numerical mass flowrate calculated for these 24 cases overestimate the experi-mental results with an average discrepancy of 2.30%.

The numerical mass flow rate obtained for these 51 casesof coiled adiabatic capillary tubes offer a relative discrep-ancy of 3.22% against the experimental data and it is about1–10% lower than the mass flow rate obtained for straightcapillary tubes with Garcıa-Valladares et al. [9] model.


data by Kim et al. [5] for R22, R407C and R410A

The experimental mass flow rate results presented byKim et al. [5] for a coiled adiabatic capillary tubes usingthree different fluids (R22, R407C [R32/125/134a, 23/25/

l results given by Kim et al. [5]

_mexp (kg h�1) _mnum (kg h�1) Error (%)

50.652 48.708 �3.83855.044 53.542 �2.72961.848 60.082 �2.85549.104 48.144 �1.95653.676 52.911 �1.42560.264 59.343 �1.52947.556 46.799 �1.59252.200 51.403 �1.52658.356 57.578 �1.333

50.904 51.472 1.11656.268 56.586 0.56563.252 63.476 0.35448.852 50.890 4.17253.496 55.914 4.52061.056 62.687 2.67147.844 49.459 3.37653.100 54.312 2.28359.400 60.814 2.380

62.352 63.354 1.60768.328 69.051 1.05876.680 76.652 �0.03759.580 62.588 5.04966.780 68.202 2.12975.420 75.676 0.33957.456 60.785 5.79561.992 66.203 6.79272.000 73.380 1.917

2.405


52% mass fraction] and R410A [R32/125, 50/50% massfraction]) have been used. The coiled adiabatic capillarytube has the following geometry and boundary conditionsfor all the cases: L = 1.0 m, D = 1.5 · 10�3 m, e/D = 6 ·10�5 and Tcon = 45 �C. Table 1 shows experimental resultsand the corresponding numerical ones obtained, togetherwith the coiled diameter geometry and working conditions.According to Kim et al. [5] outlet pressure of all their casesreported are far lower than choking pressure due to thisreason the outlet pressure is not considered. In the numer-ical model n = 300 for all these cases.

Twenty-seven different cases have been tested. FromTable 1, it is observed that the numerical results give rea-sonable and satisfactory deviation with respect to theexperimental results in all the cases; the mass flow ratemean deviation for the 27 cases analyzed is lower than2.41%. The numerical mass flow rate of coiled capillary



Mas

sflo

wra

te[k

gh-1

]

0 2 4 6 8 10 12 14 1614

16

18

20

22

24

26

28

30

32

34

36D =130 mm, p =15.7 bar exp.D =130 mm, p =15.7 bar num.D =130 mm, p =18.65 bar exp.D =130 mm, p =18.65 bar num.D =130 mm, p =21.6 bar exp.D =130 mm, p =21.6 bar num.D =52 mm, p =15.7 bar exp.D =52 mm, p =15.7 bar num.D =52 mm, p =18.65 bar exp.D =52 mm, p =18.65 bar num.D =52 mm, p =21.6 bar exp.D =52 mm, p =21.6 bar num.

R22D=1 mmε/D=0.00015L=1 m

Mas

sflo

wra

te[k

gh-1

]

0 2 4 6 8 10 12 14 1650

60

70

80

90


R22D=1.6 mmε/D=0.000056L=1 m

Fig. 4. Comparison of numerical simulation versus mass flow rates obtained b1.2 mm diameter; (c) R22 and 1.6 mm diameter; and (d) R407C and 1 mm dia

tubes for these cases is about 1–8% lower than thoseobtained for straight capillary tubes calculated withGarcıa-Valladares et al. [9] model.


data by Wang [18] for R22 and R407C

The experimental mass flow rate results by Wang [18]for coiled adiabatic capillary tubes using two different flu-ids (R22 and R407C) have been used. Due to roughnessmeasurement was not taken by the author; values consid-ered are similar to those obtained by Kim et al. [5]. Allthe numerical cases evaluated in this section present chokeexit conditions and n = 300.

Fig. 4a shows the comparison of experimental mass flowrates with the results obtained by the present model for dif-ferent degree of subcooling and condensing pressures, with



Mas

sflo

wra

te[k

gh-1

]

0 2 4 6 8 10 12 14 1624

26

28

30

32

34

36

38

40

42

44

46D =52 mm, p =15.7 bar exp.D =52 mm, p =15.7 bar num.D =52 mm, p =18.65 bar exp.D =52 mm, p =18.65 bar num.D =52 mm, p =21.6 bar exp.D =52 mm, p =21.6 bar num.

R22D=1.2 mmε/D=0.0001L=1 m

Mas

sflo

wra

te[k

gh-1

]

0 2 4 6 8 10 12 14 1612

14

16

18

20

22

24

26

28

30

32


R407CD=1 mmε/D=0.00015L=1 m

y Wang [18] with: (a) R22 and 1 mm capillary tube diameter; (b) R22 andmeter.


1 m length and 1 mm coiled adiabatic capillary tube diam-eter. As expected, the mass flow rate of the coiled adiabaticcapillary tube increases as the condensing pressure and thedegree of subcooling increases. The numerical mass flowrate calculated for these 40 cases offer an average predictingdeviation of 3.78%.

Fig. 4b presents the numerical and experimental massflow rates with a 1.2 mm capillary tube diameter. Thenumerical mass flow rate calculated for these 26 cases offeran average discrepancy of 2.6%.

Fig. 4c presents the mass flow rate variations versus thedegree of subcooling for a 1.6 mm capillary tube diameterwith both experimental and modeling results. The massflow rate calculated numerically for these 47 cases havean average deviation 1.78% against experimental data.

Fig. 4d presents the numerical and experimental massflow rates given by Wang [3,18] with a 1 mm coiled adia-batic capillary tube diameter working with R407C. Thenumerical mass flow rate calculated for these 30 cases offeran average discrepancy of 3.6%.

The numerical simulations results compared with the143 experimental data by Wang [18] offers a relative dis-crepancy of ±2.68%. Moreover, the numerical results arean average of 3% lower than the mass flow rate obtainedfor straight capillary tubes with Garcıa-Valladares et al.[9] model.

5.4. Global results

Fig. 5 shows an excellent degree of correlation betweenall the numerical and experimental results compared.97.7% of the 211 data points evaluated are within an errorof ±10%, 86% are within ±5% with a mean deviation forall the cases of ±2.89%. Experimental data was obtained

++++++++++++

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+ ++++

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+

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Experimental mass flow rate [kgh-1]

Nu

mer

ical

mas

s fl

ow

rat

e [k

gh

-1]

0 10 20 30 40 50 60 70 80 90 100 110 120 1300

10

20

30

40

50

60

70

80

90

100

110

120

130

Zhou and Zhang [8]: R22Wang [18]: R22 and R407CKim et al. [5]: R22, R407C and R410A

0% error±10% error

++10%

-10%

Fig. 5. Comparison of numerical results vs. 221 experimental data pointgiven by Zhou and Zhang [8], Kim et al. [5] and Wang [18].

from different sources and with different fluids (R22,R407C and R410A), geometries and working conditions.Moreover, because the numerical model is based on theapplications of physical laws, it is possible to extrapolatewith greater confidence to other fluids, mixtures and oper-ating conditions; it allows using the model developed as animportant tool to design and optimize these kinds ofsystems.

6. Concluding remarks

The objective of this study was to extend and validatethe model developed in previous works [9,10] for analyzingthe behavior of coiled adiabatic capillary tube expansiondevices working with pure and refrigerants mixtures.The numerical model has been developed by means of aone-dimensional analysis of the governing equations (con-tinuity, momentum, energy and entropy). Metastable con-ditions both liquid and two-phase regions have been takeninto account.

The simulation has been implemented on the basis of afinite volume formulation of the governing equations. Thesolution has been carried out using an implicit step-by-stepnumerical scheme. The calculus of the mass flow rate hasbeen made iteratively by means of a Newton–Raphsonmethod. In order to minimize computational cost, a specialtreatment has been implemented to solve the CV that con-tains transition between regions and a special mesh distri-bution has been used.

The accuracy of the detailed simulation model is demon-strated in this paper due to numerical results are comparedwith a wide range of experimental data from the technicalliterature showing a good degree of agreement in the massflow rate obtained. 97.7% of the 211 data points evaluatedare within an error of ±10%, 86% are within ±5% with amean deviation for all the cases of ±2.89%. Moreover,because the numerical model is based on the applicationsof physical laws, it is possible to extrapolate with greaterconfidence to other fluids, mixtures and operating condi-tions; it allows using the model developed as an importanttool to design and optimize these kinds of systems.

Acknowledgement

The author thanks Professor C.C. Wang for the dataand technical support provided. This study has been par-tially supported by CONACYT project U44764-Y.

References

[1] V. Feburie, M. Goit, S. Granger, J.M. Seynhaeve, A model forchoked flow through cracks with inlet subcooling, InternationalJournal of Multiphase Flow 19 (4) (1993) 541–562.

[2] S.J. Kuehl, V.W. Goldschmidt, Steady flows of R-22 through capillarytubes: test data, ASHRAE Transactions 96 (1) (1990) 719–728.

[3] C.Z. Wei, Y.T. Lin, C.C. Wang, J.S. Leu, An experimental study ofthe performance of capillary tubes for R-407C refrigerant, ASHRAETransactions 105 (2) (1999) 634–638.


[4] C.Z. Wei, Y.T. Lin, C.C. Wang, A performance comparison betweencoiled and straight capillary tubes, Heat Transfer Engineering 21 (2)(2000) 62–66.

[5] S.G. Kim, S.T. Ro, M.S. Kim, Experimental investigation of theperformance of R22, R407C and R410A in several capillary tubes forair-conditioners, International Journal of Refrigeration 25 (5) (2002)521–531.

[6] J.N. Gorasia, N. Dubey, K.K. Jain, Computer-aided design ofcapillaries of different configurations, ASHRAE Transactions 97 (1)(1991) 132–138.

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[9] O. Garcıa-Valladares, C.D. Perez-Segarra, A. Oliva, Numericalsimulation of capillary tube expansion devices behaviour with pureand mixed refrigerants considering metastable region. Part I: math-ematical formulation and numerical model, Applied Thermal Engi-neering 22 (2) (2002) 173–182.

[10] O. Garcıa-Valladares, C.D. Perez-Segarra, A. Oliva, Numericalsimulation of capillary tube expansion devices behaviour with pureand mixed refrigerants considering metastable region. Part II:

experimental validation and parametric studies, Applied ThermalEngineering 22 (4) (2002) 379–391.

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[13] A. Premoli, D.D. Francesco, A.A. Prina, A dimensional correlationfor evaluating two-phase mixture density, La Termotecnia 25 (1)(1971) 17–26.

[14] A.E. Dukler, M. Wicks, R.G. Cleveland, Frictional pressure drop intwo-phase flow Parts A and B, AICHE Journal 10 (1) (1964)38–51.

[15] REFPROP version 7.0, Reference fluid thermodynamics and trans-port properties, NIST Standard Reference Database 23, Gaithers-burg, MD 20899, USA, 2002.

[16] O. Garcıa-Valladares, C.D. Perez-Segarra, J. Rigola, Numericalsimulation of double-pipe condensers and evaporators, InternationalJournal of Refrigeration 27 (6) (2004) 979–991.

[17] F. Escanes, C.D. Perez-Segarra, A. Oliva, Numerical simulation ofcapillary tube expansion devices, International Journal of Refriger-ation 18 (2) (1995) 113–122.

[18] C.C. Wang, private communication.

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