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A generalized dimensionless local power-lawcorrelation for refrigerant flow through adiabaticcapillary tubes and short tube orifices

Liang Yang, Chun-Lu Zhang*

School of Mechanical Engineering, Tongji University, Shanghai 201804, China

a r t i c l e i n f o

Article history:

Received 3 March 2014

Received in revised form

25 April 2014

Accepted 17 June 2014

Available online 25 June 2014

Keywords:

Refrigerant

Flow rate

Capillary

Orifice

Correlation

* Corresponding author. Tel.: þ86 136 7182E-mail address: chunlu.zhang@gmail.com

http://dx.doi.org/10.1016/j.ijrefrig.2014.06.0110140-7007/© 2014 Elsevier Ltd and IIR. All rig

a b s t r a c t

This paper presents a new method to obtain generalized dimensionless correlation of

refrigerant mass flow rates through adiabatic capillary tubes and short tube orifices. The

dimensionless Pi groups were derived from the homogeneous equilibrium model, which is

available for different refrigerants entering adiabatic capillary tubes or short tube orifices

as the subcooled liquid, two-phase mixture, or supercritical fluid. To mitigate the potential

over-fitting risk in neural network, a new “local” power-law correlation reformed from the

homogeneous equilibrium model was proposed and compared with the conventional

“global” power-law correlation and recently developed neural network model. About 2000

sets of experimental mass flow rate data of R12, R22, R134a, R404A, R407C, R410A, R600a

and CO2 (R744) in the open literature covering capillary and short tube geometries,

subcritical and supercritical inlet conditions were collected for the model development.

The comparison between the recommended six-coefficient correlation and experimental

data reports 0.80% average and 8.98% standard deviations, which is comparable with the

previously developed neural network and much better than the “global” power-law

correlation.

© 2014 Elsevier Ltd and IIR. All rights reserved.

Une corr�elation locale adimensionnelle g�en�eralis�ee de loi depuissance pour l'�ecoulement du frigorig�ene dans des tubescapillaires et des orifices de tubes courts

Mots cl�es : Frigorig�ene ; Vitesse d'�ecoulement ; Capillaire ; Orifice ; Corr�elation

5 133.(C.-L. Zhang).

hts reserved.

Nomenclature

ai empirical coefficients in correlations

cp specific heat at constant pressure (J kg-1 K-1)

D tube diameter (m)

f friction factor

G mass flux (kg m-2 s-1)

L tube length (m)

N number of data

m mass flow rate (kg s-1)

p pressure (Pa)

T temperature (�C)v specific volume (m3 kg-1)

Greek symbols

m viscosity (N s m-2)

pi dimensionless Pi groups

r density (kg m-3)

Superscripts and Subscripts

f saturated liquid at the inlet saturation pressure

g saturated vapor at the inlet saturation pressure

in inlet

m mean value

max maximum

sat saturation

Fig. 1 e Definition of psat under different inlet conditions

i n t e r n a t i o n a l j o u r n a l o f r e f r i g e r a t i o n 4 6 ( 2 0 1 4 ) 6 9e7 670

1. Introduction

Capillary tubes and short tube orifices are widely used as the

throttling and flow-rate controlling device on the residential

and light commercial air conditioning, refrigeration and heat

pump units because of the simple configuration, high reli-

ability, and low cost. On the other hand, the complexity of

critical two-phase flow inside the tube and the substitution of

conventional CFC and HCFC refrigerants brought plenty of

theoretical and experimental investigations on capillary tubes

and short tube orifices since 1940’s (ASHRAE, 2010; Khan et al.,

2009; Nilpueng and Wongwises, 2012a, b).

For ease-of-use in engineering applications and for fast

and robust computation in system modeling, empirical or

semi-empirical models or correlations of refrigerant mass

flow rate through capillary tubes or short tube orifices have

been developed. Particularly, since Bittle et al. (1998) proposed

a generalized power-law correlation for adiabatic capillary

tubes based on the dimensional analysis, this type of corre-

lations has been applied to different refrigerants or data banks

by many researchers (ASHRAE, 2010; Choi et al., 2004; Choi

et al., 2003; Yang and Wang, 2008). On the other hand, to

overcome the numerical issues in the existing power-law

correlations and improve the model accuracy, Zhang (2005)

first developed a new type of capillary tube model using arti-

ficial neural network (ANN). Then the neural network model

was modified with new dimensionless Pi groups based on the

homogeneous equilibrium model (HEM) (Zhang and Zhao,

2007) and was extended to short tube orifices (Zhao et al.,

2007) and transcritical CO2 throttling device (Yang and

Zhang, 2009). In addition, Vin�s and Vacek (2009) applied the

neural network to a new refrigerant R218 and Heimel et al.

(2014) extended it to non-adiabatic capillary tubes. Shao

et al. (2013) did a comprehensive assessment on the existing

dimensionless correlations and neural networks of adiabatic

capillary tubes and found there is still room for improvement

in model generality and over-fitting risk mitigation.

In this work, we propose a generalized “local” power-law

correlation model based on the HEM for supercritical, sub-

cooled or two-phase refrigerant flowing through adiabatic

capillary tube or short tube orifice. About 2000 sets of experi-

mental data in the open literature including refrigerants R12,

R22, R134a, R404A, R407C, R410A, R600a and CO2 (R744) are

collected for data regression and comparison with the con-

ventional power-law correlation and neural networkmodel. It

turns out the new local power-law correlation is much more

accurate than the conventional power-law correlation model

and no over-fitting risk in comparison with the neural

network model.

2. Dimensionless parameter groups

In terms of the recent assessment (Shao et al., 2013), the

dimensionless parameter groups derived from the HEM (Yang

and Zhang, 2009) are more effective than those from the

conventional dimensional analysis method (Bittle et al., 1998;

Choi et al., 2003). Therefore, we keep using the dimensionless

Pi groups developed by Yang and Zhang (2009). To give a

Table 1 e Experimental data source of capillary tubes andshort tube orifices

Fluid Tube type Data Source No. ofData

R12 Capillary Melo et al. (1999) 19

Li et al. (1990) 5

Wijaya (1991) 90

Orifice Kim et al. (1994) 50

R22 Capillary Kuehl and Goldschmidt (1991) 55

Wei et al. (1999) 31

Kim et al. (2002) 11

Orifice Kim and O'neal (1994) 84

R134a Capillary Melo et al. (1999) 19

Wijaya (1991) 90

Wijaya (1992) 9

Orifice Singh et al. (2001) 422

Kim et al. (1994) 62

R410A Capillary Kim et al. (2002) 10

Augusto Sanzovo Fiorelli et al.

(2002)

24

Orifice Payne and O'Neal (1999) 68

Kim et al. (2005) 52

R407C Capillary Kim et al. (2002) 11

Wei et al. (1999) 20

Augusto Sanzovo Fiorelli et al.

(2002)

36

Motta et al. (2000) 331

Orifice Payne and O'Neal (1998) 106

R600a Capillary Melo et al. (1999) 19

R404A Capillary Motta et al. (2000) 113

CO2 Capillary Cecchinato et al. (2009) 148

da Silva et al. (2009) 66

Orifice Liu et al. (2004) 49

Table 3 e Deviations and coefficients of different correlations

Models ANN (Yang and Zhang, 2009) Eq. (1

A.D. (%) 0.65 6.37

S.D. (%) 8.24 17.7

No. of coef. 19 5

a1 – 0.14621

a2 – -0.8463

a3 – -0.0639

a4 – -0.2528

a5 – 0.11844

a6 – –

a7 – –

Table 2 e Range of Pi groups

Subcritical inlet

Capillary Short tube

P1 0.0583 ~ 0.2522 0.3202 ~ 0.8462

P2 0.4385 ~ 1.005 0.4814 ~ 1.005

P3 0.0235 ~ 0.1197 0.0205 ~ 0.3108

P4 384.6 ~ 4947.2 5.5 ~ 31.48

P5 1.031e5 ~ 1.038e6 1.499e5 ~ 1.802e6

i n t e rn a t i o n a l j o u r n a l o f r e f r i g e r a t i o n 4 6 ( 2 0 1 4 ) 6 9e7 6 71

complete description of the newmethod, we briefly introduce

how to get the dimensionless Pi groups as below.

In the HEM, refrigerant flow through an adiabatic tube can

be regarded as an isenthalpic progress. The momentum

equation for the one-dimensional steady-state homogeneous

flow (gravity effect ignored) is as below (Gu et al., 2014; Yilmaz

and Unal, 1996).

�dp ¼ G2dvþ f2D

G2vdL (1)

Equation (1) can be written in an integral form.

pin � pout ¼ G2ðvout � vinÞ þ fm2D

G2vmL (2)

Moreover, equation (2) can be rearranged in a dimension-

less form.

G2vin

pin¼

1� poutpin

voutvin

� 1þ fmL2D

vmvin

(3)

In terms of equation (3), we finally defined the following

dimensionless Pi groups (Yang and Zhang, 2009).

p1 ¼ffiffiffiffiffiffiffiffiffiffiffiffiG2vin

pin

s¼ 1:273m

D2 ffiffiffiffiffiffiffiffiffiffiffiffipinrin

p (4)

p2 ¼ psat

pin(5)

p3 ¼ vf

vg¼ rg

rf(6)

p4 ¼ L=D (7)

p5 ¼ Dffiffiffiffiffiffiffiffiffiffiffiffipinrin

p �min (8)

where, for the two-phase inlet condition,min¼xinmgþ(1�xin)mfand1/rin¼xin/rgþ(1�xin)/rf. The densities in p3 are calculated at

1) Eq. (12) Eq. (13) Eq. (14)

0.80 0.80 0.30

8.98 8.98 9.17

7 6 5

3 -3.650509 -3.6622463 -3.4817787

195 -1.4249184 -1.4311445 -1.4049856

05 -11.481192 -12.377615 -11.539162

265 0.00596366 -4.4141877 -13.338855

03 -4.5299793 -0.2874146 -0.3643278

-0.2902006 -0.0662179

-0.06782366

Supercritical inlet (CO2)

Capillary Short tube

0.0637 ~ 0.2473 0.4042 ~ 0.6973

0.5352 ~ 1.062 0.5586 ~ 1.322

0.1869 ~ 1.0 0.1869 ~ 1.0

594.1 ~ 7500 5.941 ~ 18.829

5.536e5 ~ 1.699e6 7.858e5 ~ 3.16e6

Table 4 e Deviations and statistical percentages of newcorrelations for capillary and short tubes

Data Equation A.D.(%)

S.D.(%)

Err.< ±10%

Err.< ±15%

Err.< ±20%

Capillary Eq. (12) 0.75 8.11 80.6 93.6 98.0

Eq. (13) 0.75 8.11 80.7 93.7 98.2

Eq. (14) -0.18 8.49 76.1 94.1 98.2

Short

tube

Eq. (12) 0.86 9.94 71.1 89.8 94.6

Eq. (13) 0.85 9.93 71.0 89.9 94.6

Eq. (14) 0.86 9.90 71.0 89.8 94.7

All Eq. (12) 0.80 8.98 76.3 91.9 96.4

Eq. (13) 0.80 8.98 76.3 92.0 96.5

Eq. (14) 0.30 9.17 73.7 92.2 96.6

Fig. 2 e Comparison of predicted and measured mass flow

rates through capillary tubes

i n t e r n a t i o n a l j o u r n a l o f r e f r i g e r a t i o n 4 6 ( 2 0 1 4 ) 6 9e7 672

the inlet saturation pressure psat. p3 becomes unity in the

supercritical region.

For subcooled inlet, the inlet saturation pressure psat is the

saturation pressure at the inlet temperature, as shown in

Fig. 1(a). For two-phase inlet, psat is naturally pin, as shown in

Fig. 1(b). For the supercritical inlet, we apply the pseudo-

critical line (Liao and Zhao, 2002), which is formed by the

points of the maximum specific heat at every pressure above

the critical pressure, as shown in Fig. 1(c). The pseudo-critical

line equation of CO2 is as below (Liao and Zhao, 2002).

Tpseudo ¼ �122:6þ 6:124p� 0:1657p2 þ 0:01773p2:5

� 0:0005608p3ð75 � p � 140Þ (9)

where, the temperature (Tpseudo) and the pressure (p) are in �C

and bar, respectively. Therefore, we substitute the inlet tem-

perature for Tpseudo in equation (9) to get psat in the super-

critical region.

Accordingly, refrigerant mass flow rate through adiabatic

capillary and short tubes can be expressed as follows.

p1 ¼ Fðp2;p3;p4;p5Þ (10)

where, F is the functional equation. It can be approximated by

the conventional power-law correlation, neural networks, or

the local power-law correlation to be developed hereinafter.

3. Data bank

Experimental data in the open literature are collected for

model development, as listed in Table 1. There are about 2000

Table 5 e Deviations and statistical percentages of new correla

Inlet state Equation A.D. (%) S.D. (%)

Two-phase Eq. (12) -1.23 10.07

Eq. (13) -1.30 10.07

Eq. (14) -1.29 10.06

Subcooled Eq. (12) -0.20 8.87

Eq. (13) -0.19 8.86

Eq. (14) -0.51 8.96

Supercritical Eq. (12) 3.33 8.73

Eq. (13) 3.35 8.72

Eq. (14) 2.18 9.44

sets of experimental mass flow rate data of R12, R22, R134a,

R404A, R407C, R410A, R600a and CO2 (R744) covering capillary

and short tube geometries, subcritical and supercritical inlet

conditions in the data bank.

Accordingly, the ranges of previously defined Pi groups are

given in Table 2. All the thermal properties of refrigerants are

calculated using REFPROP 9.0 (Lemmon et al., 2010).

4. Local power-law correlation

First of all, the conventional power-law correlation, equation

(11), is curve-fitted using the data bank. Since all Pi groups in

this type of power-law correlation are at the equal position,

we call it the “global” power-law correlation.

p1 ¼ a1pa22 p

a33 p

a44 p

a55 (11)

The results of data regression are given in Table 3. It turns

out that the average and standard deviations of equation (11)

is 6.37% and 17.7%, respectively. As expected, it is muchworse

than the accuracy of previously developed neural network. In

other words, the global power-law correlation is not a good

candidate for the function of equation (10).

tions under different inlet conditions

Err. < ±10% Err. < ±15% Err. < ±20%

68.23 87.96 95.65

67.89 87.96 95.65

67.89 87.63 95.32

76.06 93.86 96.84

76.15 93.77 96.75

74.25 93.77 96.93

76.6 88.72 95.45

76.6 89.23 95.96

71.21 90.07 95.96

Fig. 3 e Comparison of predicted and measured mass flow

rates through short tube orifices

Fig. 4 e Mass flow rate change with inlet conditions

i n t e rn a t i o n a l j o u r n a l o f r e f r i g e r a t i o n 4 6 ( 2 0 1 4 ) 6 9e7 6 73

To improve the accuracy, we used to bring into force the

neural network because in theory it’s a generic function

approximant. However, the over-fitting risk is still a concern

because a relatively large number of empirical coefficients (or

connecting weights and biases) were employed. Therefore we

return to the original physical-based dimensionless equation

(3) and construct a “local” power-law correlation. Namely,

p1 ¼�

1þ a1pa22

a3pa43 � 1þ a5p4p

a65 p

a73

�0:5

(12)

where Pi groups p1 ~ p5 are placed at the corresponding po-

sitions of equation (3) in a local power-law expression. Note

that the power exponent of p4 is unity because it is original

and by comparison the rest Pi groups are derived parameters.

Equation (12) is found much accurate than the global

power-law correlation (11), as shown in Table 3. The average

and standard deviations of equation (12) is 0.80% and 8.98%,

respectively, which is very close to the previously developed

neural network (A.D.¼ 0.65% and S.D.¼ 8.24%). The seven co-

efficients (a1~a7) in equation (12) are listed in Table 3, which is

much less than the number of coefficients used in neural

network.

The values of empirical coefficients (a1~a7) in equation (12)

indicate that further reduction of coefficients is possible. Two

power exponents (a4 and a7) for the density ratio p3 are small.

Given the range of p3, we have 0:977 � pa43 � 1 and

1 � pa73 � 1:30.

Sincetherangeofbothterms is limited,wecantrytopartially

or totally remove p3 from equation (12). Namely, we have

p1 ¼�

1þ a1pa22

a3 þ a4p4pa55 p

a63

�0:5

(13)

p1 ¼�

1þ a1pa22

a3 þ a4p4pa55

�0:5

(14)

The data regression results of equations (13) and (14) are

given in Table 3 as well. A more detailed statistical results of

Fig. 5 e Mass flow rate change with length-to-diameter ratio

i n t e r n a t i o n a l j o u r n a l o f r e f r i g e r a t i o n 4 6 ( 2 0 1 4 ) 6 9e7 674

the three local power-law equations are given in Table 4 and

Table 5. Table 4 breaks down the statistical results by capillary

tube and short tube orifice, while Table 5 breaks down the

statistical results by different inlet conditions. Table 4 tells

that equation (13) is almost equal to equation (12), while

equation (14) loses some accuracy in capillary tube data. In

physics, it makes sense that the impact of density ratio is

marginal because the mass flow rate largely depends on the

inlet density which has been actually taken into account in p1.

Table 5 tells that all three local power-law correlations can

well fit different inlet conditions. Only equation (14) loses

some accuracy under supercritical inlet conditions. In addi-

tion, the overall accuracy for the two-phase inlet conditions is

worse than that of other inlet conditions because the mea-

surement uncertainty is usually higher in two-phase

experiments.

In all tables, the average deviation (A.D.) and the standard

deviation (S.D.) are defined as follows.

A:D: ¼ 1N

XN

mprediction �mexperiment

mexperiment� 100% (15)

S:D:¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

N�1

XN

�mprediction�mexperiment

mexperiment�A:D:

�2

vuut �100% (16)

To balance accuracy and simplicity, the six-coefficient cor-

relation (13) is recommendedforuse.Thecomparisonsbetween

equation (13) and the experimental data are illustrated in Fig. 2

and Fig. 3 for adiabatic capillary tubes and short tube orifices,

respectively. Fig. 2 indicates 93.7% capillary tube data fall into

±15% error band. By comparison, the results of short tube ori-

fices are a little bit worse than that of capillary tubes. Fig. 3

shows 94.6% short tube orifice data fall into ±20% error band.

Since the experimental data were from different sources

with different uncertainties, the above deviation levels are

reasonable. Other research works including theoretical

models and empirical correlations got the similar error bands

and standard deviations as well when dealing withmany data

from different sources. On the other hand, we know thatmore

accurate predictions are desired in applications. Usually, two

ways could help get better accuracy. One is to particularly redo

the local power-law correlation using data of one source or

one refrigerant. It can much improve the accuracy but loses

generality. Another is to use an adjusting factor for tuning the

correlation accuracy based on some actual testing data.

5. Parametric analysis

Except the neural networks, there was no correlation covering

different inlet conditionsand ranging fromshort tubeorifices to

capillary tubes. The proposed correlation (13) is of wide range

andgoodaccuracyasneuralnetworksandsimpleexpressionas

the conventional power-law correlations. Therefore, two

typical cases are studied in the parametric analysis.

Fig. 4 shows the new correlation can well predict the mass

flow rate change with different inlet conditions. Fig. 4(a), (b)

and (c) illustrate the case study of capillary tube, short tube

orifice, and supercritical CO2, respectively. Themarkers on the

plots represent experimental data from literature. In Fig. 4(a)

and 4(b), the horizontal axis uses the mathematical exten-

sion of vapor quality definition so that the transition of inlet

conditions can be smoothly plotted on the same chart.

Fig. 5 demonstrates the prediction of mass flow rate transi-

tionfromshort tubeorificetocapillary tube.Asshowninnormal

and logarithmic scales of L/D, respectively, the mass flow rate

flattens out at L/D less than 20 or greater than 2000. This is

consistent with the common sense that the mass flow rate is

predominated by the entrance effect when L/D is small and

varies slowly when the tube is long enough. Fig. 5 also shows

that the mass flow rate increases with the inlet pressure or

condensing temperature. Moreover, we tested many re-

frigerants includingCO2andgot similar trendsasR410A inFig. 5.

6. Conclusions

This paper proposed a new generalized local power-law cor-

relation of refrigerant mass flow rate through capillary tubes

i n t e rn a t i o n a l j o u r n a l o f r e f r i g e r a t i o n 4 6 ( 2 0 1 4 ) 6 9e7 6 75

or short tube orifices covering both subcritical and supercrit-

ical inlet conditions. Five dimensionless Pi groups grounding

on the HEM were adopted to reform the conventional global

power-law correlation. Compared with 2000 sets of experi-

mental data of multiple refrigerants in the open literature, the

average and standard deviations of the recommended six-

coefficient correlation are 0.80% and 8.98%, respectively. On

one hand, the proposed local power-law correlation is of wide

range and good accuracy as neural networks, but without

over-fitting risk in neural networks. On the other hand, the

proposed local power-law correlation has simple expression

(only six coefficients needed) as the conventional global

power-law correlations, but much better accuracy and works

well for different inlet conditions. The method of local power-

law correlation could be a generic data-based modeling

approach for more applications.

Acknowledgments

This work is supported by the National Natural Science

Foundation of China (Grant No. 51206123), the Innovation

Program of Shanghai Municipal Education Commission (Grant

No. 11ZZ30), and the China Postdoctoral Science Foundation

(Grant No. 2013M541539).

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