A generalized correlation for two-phase flow of alternativerefrigerants through short tube orifices

Jongmin Choia, Jin Taek Chungb,*, Yongchan Kimb

aDepartment of Mechanical Engineering, Hanbat National University, Duckmyung-Dong, Yusung-Gu, Daejeon, 305-719, South KoreabDepartment of Mechanical Engineering, Korea University, Anam-Dong, Sungbuk-Gu, Seoul, 136-701, South Korea

Received 29 September 2003; received in revised form 28 November 2003; accepted 28 November 2003

Abstract

The objective of this study is to develop a generalized correlation for the prediction of refrigerant flow rate throughshort tube orifices using alternative refrigerants. The generalized correlations for two-phase and subcooled inlet con-ditions are separately derived from a power law form of dimensionless parameters generated by the Buckingham Pi

theorem. The database for the present correlation includes extensive experimental data for R12, R22, R134a, R407C,R410A, and R502 obtained from the open literature. For subcooled inlet conditions at the short tube entrance, thecorrelation yields an average deviation of 0.3% and a standard deviation of 6.1% based on the present database, while

for two-phase inlet conditons it predicts the database with an average deviation of 0.2% and a standard deviation of5.0%. The relative deviations of the predictions using the present correlations for two-phase and subcooled inlet con-ditions range from �21% to 18%.# 2003 Elsevier Ltd and IIR. All rights reserved.

Keywords: Orifices; Tubes; Two-phase flow; Refrigerant; Correlation

Ecoulement diphasique de frigorigenes de remplacement atravers des orifices de tubes courts : correlation generale

Mots cles : Orifices ; Tubes ; Ecoulement diphasique ; Frigorigene ; Correlation

1. Introduction

The chlorofluorocarbon (CFC) and hydrochlorofluoro-carbon (HCFC) are being replaced by hydrofluorocarbon(HFC) and HFC mixtures due to environmental concerns

on the depletion of the ozone layer and global warming[1–4]. Designing of a high efficiency air conditioner orheat pump with the new refrigerants requires more

accurate data and models for the performance of eachcomponent in the system. A short tube orifice has beenwidely used as an expansion device in air-conditioners

and heat pumps due to the advantages of simplicity, lowcost, high reliability, and the elimination of additional

check valves used for flow direction change in heatpump applications [3–7]. The manufacturers need adesign tool that predicts the mass flow rate through

short tube orifices to satisfy the redesign requirementswith alternative refrigerants [3,4]. Therefore, a generalizedcorrelation to predict the refrigerant flow rate through

short tube orifices with new alternative refrigerants mustbe developed to afford a convenient design tool.During the past few decades, many researchers have

extensively investigated the performance of short tube

orifices. Early studies on two-phase flow through shorttube orifices focused exclusively on R12 and R22[5–12]. Aaron and Domanski [5] developed an empirical

0140-7007/$35.00 # 2003 Elsevier Ltd and IIR. All rights reserved.

doi:10.1016/j.ijrefrig.2003.11.008

International Journal of Refrigeration 27 (2004) 393–400

www.elsevier.com/locate/ijrefrig

* Corresponding author. Tel.: +82-2-3290-3364; fax +82-2-

928-9766.

E-mail addresses: jchung@korea.ac.kr (J.T. Chung).

correlation for predicting R22mass flow rate through shorttube orifices at subcooled inlet conditions by modifying

the single-phase orifice equation based on their test datafor R22. Kim and O’Neal [6] also presented an empiricalflow correlation for short tube orifices using R22 by

extending the Aaron and Domanski correlation [5].Recently, several empirical correlations to predictrefrigerant mass flow rate through short tube orificesusing alternative refrigerants have been developed

[3,4,13,14]. Kim and O’Neal [3] analyzed the perfor-mance of short tube orifices and developed a two-phaseflow model with R134a by modifying the single-phase

orifice equation. Payne and O’Neal [13] also developed aflow model for HFC mixtures in a similar format to theKim and O’Neal correlation [3]. Payne [4] developed a uni-

versal correlation for both pure and oil-mixed refriger-ants flowing through short tube orifices at subcooledand two-phase inlet conditions by correlating dimen-

sionless parameters based on his own experimental data.Most empirical correlations for predicting refrigerant

mass flow rate through short tube orifices have beendeveloped by applying the modified single-phase orifice

equation. A generalized correlation for two-phase flowof alternative refrigerants through short tube orifices isvery limited in the open literature. Most correlations

with alternative refrigerants cover subcooled inlet con-ditions only and have a complicated form to apply.Furthermore, most correlations were developed andvalidated based on limited database from their own

experiments. Therefore, the objective of this study is todevelop a generalized two-phase flow correlation in asimple form for predicting refrigerant mass flow rate

through short tube orifices with alternative refrigerants.The Buckingham Pi theorem [15] is applied to generatedimensionless parameters for operating conditions,

short tube orifice geometries, and refrigerant properties.The generalized correlation is developed from a powerlaw form of dimensionless parameters based on extensive

experimental data for CFC, HCFC, and HFC refriger-ants in the open literature. Comparing the predictionswith the database and the existing correlations in theliterature validates the present correlation.

2. Correlation development

2.1. Database of the correlation

A database of the present correlation is obtained fromthe open literature [3–6,13,14] to cover wide range of shorttube geometries and operating conditions using promis-

ing alternative refrigerants of R12 and R22. Table 1shows the detailed information on the present databaseincluding data source and number of data points. Thedatabase contains a total of 2126 data points for six

refrigerants: R12, R22, R134a, R407C, R410A, R502 atsubcooled and two-phase inlet conditions. In this study,two generalized correlations for subcooled and two-

phase inlet conditions, respectively, are developed sepa-rately due to a difference in major operating parameters.The database covers wide ranges of short tube dia-

meters from 1.0 to 2.0 mm, short tube lengths from 9.5to 25.4 mm, condensing temperatures from 35 to 54 �C,evaporating temperatures from �1.1 to 16.6 �C (belowsaturation pressure according to inlet temperature), inlet

subcoolings from 0.1 to 20 �C, and inlet qualities from 1to 30%. All short tube orifices included in the databasehave a sharp-edged entrance shape. The ranges of the

database will impose limitations on the application ofthe correlation.

2.2. Correlation for subcooled inlet conditions

In order to generate appropriate dimensionless para-

meters for a generalized correlation, the variables influ-encing mass flow rate through short tube orifices areselected based on the data analysis [3,5–14]. Forsubcooled inlet conditions, operating parameters con-

sidered in this study are inlet pressure Pin, degree ofsubcooling DTsc, saturation pressure Psat correspondingto inlet temperature, and downstream pressure Pdown.

Nomenclature

D inner diameter (m)L length (m)

m:

mass flow rate (kgs�1)Pc critical pressure (Pa)Pdown exit pressure (Pa)Pin inlet pressure (Pa)

Psat saturation pressure (Pa)Tc critical temperature (�C)DTsc degree of subcooling (�C)

xin inlet quality

Greek letters� density (kgm�3)� dimensionless parameter groups surface tension (Nm�1)

� viscosity (Pa.s)

Subscriptsf saturated liquidg saturated vapor

mean averagedmeas measuredpred predicted

394 J. Choi et al. / International Journal of Refrigeration 27 (2004) 393–400

Saturation pressure Psat is included because the sub-

cooled liquid region is one of the dominant factorsaffecting mass flow rate [3,5,6]. The pressure differences,Pc � Pin, Pc � Psat, and Pc � Pdown, are used to considerrelative pressure to the reference pressure Pc for each

alternative refrigerant.The existence of the metastable region increases mass

flow rate through a short tube orifice due to a relatively

higher fluid density [8,16,17]. The influence of meta-stable flow can be represented as a function of surfacetension and pressure drop [18,19]. Therefore, the selected

properties associated with metastable flow and frictionare viscosity �, density � of both the liquid and vaporphases, and surface tension �. In addition to these

variables, critical temperature Tc is also included to

non-dimensionalize inlet subcooling DTsc. Short tube

orifice diameter D and length L are included to considergeometric effects on mass flow rate. The resulting rela-tionship between the mass flow rate and the selectedvariables at subcooled inlet conditions is represented in

the given functional form

m:¼ f

�Pc-Pinð Þ; Pc-Psatð Þ; Pc � Pdownð Þ;DTsc;

L;D; �f; �g; �f; �g; �; Tc

�ð1Þ

Nine dimensionless Pi-groups are derived fromapplying the Buckingham Pi theorem [15] to the selected

variables in Eq. (1) with the four repeating variables ofD; �f; �f; and Tc. The definitions of each Pi-group(�1��9) at subcooled inlet conditions are given in

Table 2. To generate a simple form of dimensionless

Table 1

Database for the present correlation

Data source

Refrigerant Inlet conditions No. of data point

Kim and O’Neal [3]

R-134a Subcooled/Two-phase 404

Payne [4]

R-12, R-502 Subcooled/Two-phase 138

Aaron and Domanski [5]

R-22 Subcooled 427

Kim and O’Neal [6]

R-22 Subcooled/Two-phase 417

Payne and O’Neal [13]

R-407C, R-410A Subcooled/Two-phase 335

Singh et al. [14]

R-134a Subcooled/Two-phase 405

Table 2

Dimensionless � groups of the present correlation

� group

Original parameter Modified parameter Inlet conditions

p1

m:

D�f

m:

D 2ffiffiffiffiffiffiffiffiffiffiffi�fPin

p

Subcooled/Two-phase

p2

Pc � Pinð Þ�fD

2

�2f

Pc � Pin

Pc

Subcooled/Two-phase

p3

Pc � Pdownð Þ�fD

2

�2f

Pc � Pdown

Pc

Subcooled/Two-phase

p4

Pc � Psatð Þ�fD

2

�2f

Pc � Psat

Pc

Subcooled

p5

DTsc

Tc

DTsc

Tc

Subcooled

p6

L

D

L

D

Subcooled/Two-phase

p7

�f�g

�f�g

Subcooled

p8

�g

�f

�f � �g

�g

Subcooled/Two-phase

p9

��fD

�2f

�

DPin

Subcooled

p10

xin xin

1� xin

Two-phase

p11

�mean

�f

�mean

�f

Two-phase

J. Choi et al. / International Journal of Refrigeration 27 (2004) 393–400 395

Pi-groups, the complicated repeating variable �2f =ð�fD2Þ

included in the original parameters is replaced by Pin orPc in the modified parameters. In addition, the original�5 group �g=�f is modified as �f � �g

� �=�g since the

modified parameter shows better dependence on massflow rate.The generalized correlation for the dimensionless mass

flow rate �1 at subcooled inlet conditions is generatedfrom a power law form of the remaining Pi-groups. Thecoefficient and exponents of the eight independent Pi-

groups are determined using a non-linear regressiontechnique along with the present database. All propertiesof the refrigerants were calculated using REFPROP

[20]. Finally, the generalized correlation for refrigerantmass flow rates through short tube orifices at subcooledinlet conditions is given as

�1 ¼ 0:1378� ��0:9502 �0:0333 �0:7694 �0:0825 ��0:099

6

��0:1047 �0:5548 ��0:034

9 ð2Þ

2.3. Correlation for two-phase inlet conditions

A generalized correlation for two-phase inlet condi-

tions is also developed by applying the same procedureused in the correlation for subcooled inlet conditionswith some modifications in the dimensionless para-

meters to satisfy operating conditions. Basically, forpure refrigerants at two-phase inlet conditions, the inletpressure to the test section coincides with the saturationpressure. In addition, the effects of the metastable

region can be disregarded at two-phase inlet conditions.Therefore, saturation pressure and surface tension areexcluded from the variables influencing mass flow rate

at two-phase inlet conditions, and inlet quality is intro-duced by replacing subcooling. Inlet average density,which is given by Eq. (3) [21], replaces saturated liquid

density to consider more accurate fluid density enteringinto a short tube orifice due to significant influence ofinlet density on mass flow rate [3–6].

�mean ¼ 1� xinð Þ=�f þ xin=�g� ��1

ð3Þ

By modifying Eq. (1) with a consideration of the dif-

ferences between subcooled and two-phase inlet condi-tions, the relation between mass flow rate and thevariables at two-phase inlet conditions can be given as

m:¼ f

�Pc-Pinð Þ; Pc-Pdownð Þ;xin; L;D; �f; �g;

�mean; �f; Tc

�ð4Þ

Applying the Buckingham Pi theorem [15] to theselected variables in Eq. (4) with the four repeatingvariables of D; �f; �f; and Tc derives seven dimen-

sionless Pi-groups. Quality xin is modified asxin1�xin

[3,4,6].The definitions of each Pi-group (�1,�2,�3,�6,�8,�10,�11)at two-phase inlet conditions are also given in Table 2.

The generalized correlation for two-phase inlet condi-tions is derived from a power law form of the remainingPi-groups, using the same method in the correlation forsubcooled inlet conditions. As a result, the generalized

correlation for two-phase inlet conditions is given as

�1 ¼ 0:0720� ��0:0902 �0:4063 ��0:149

6 ��0:0998

��0:01310 �0:28311 ð5Þ

3. Verification of the correlation

In order to verify the present correlation, the pre-

dicted mass flow rates are compared with the presentdatabase and the Payne correlation [4]. Relativedeviation, average deviation, and standard deviation,which are given in Eqs. (6)–(8), respectively, are used to

determine the goodness of fit.

Relative deviation %ð Þ ¼m:pred�:

m:meas

� �m:meas

� 100 ð6Þ

Average deviation %ð Þ ¼1

n

Xn1

m:pred �m

:meas

� �m:meas

� 100

�

ð7Þ

Standard deviation %ð Þ ¼1

n

Xn1

m:pred �m

:meas

� �m:meas

� 100

�2

� Average deviation½ 2

ð8Þ

Figs. 1 and 2 show the comparison of the predictedmass flow rates using the present correlations for sub-

cooled and two-phase inlet conditions, respectively, withthe present database for six refrigerants: R12, R22,R134a, R407C, R410A, and R502. As shown in Fig. 1,relative deviations of the present correlation with pure

and mixed refrigerants are from �16.5% to 15.3%, andfrom �20.7% to 17.9%, respectively. The correlationfor subcooled inlet conditions yields an average

deviation of 0.3% and a standard deviation of 6.1%based on the present database. As shown in Fig. 2, thepresent correlation for two-phase inlet conditions shows

a better agreement with the database having relativedeviations within 17%, and the predictions yield anaverage deviation of 0.2% and a standard deviation of

5.0%. For all database including subcooled and two-phase inlet conditions, relative deviations of the predic-tions are from �21% to +18%. Approximately 91% ofthe present database are correlated within a relative

deviation of 10%.In order to check the applicability of the correlation

to each group of refrigerant, the correlation is compared

396 J. Choi et al. / International Journal of Refrigeration 27 (2004) 393–400

with that of each group of CFC, HCFC, and HFC.Fig. 3 shows relative deviations of the predictions using

the present correlation based on the data for CFC andHCFC [4–6]. The predictions with CFC and HCFCyield relative deviations from �17.9% to +13.5%, anaverage deviation of �0.1% and a standard deviation of

4.8%. Fig. 4 shows the comparison of the predictedmass flow rate with the data for HFCs. Relative devi-ations of the predictions with HFCs are from �20.7% to

+17.9%, and average deviation and standard deviationare 0.7% and 6.6%, respectively. As shown in Figs. 1–4,

the deviations of the predictions are quite reasonable forall refrigerants included in this study due to a properconsideration of dimensionless parameters affecting therefrigerant mass flow rate through short tube orifices.

In order to verify the present correlation against theexisting correlation, the Payne correlation [4] is com-pared with the present database. As shown in Table 3,

Fig. 1. Comparison of the present correlation with the database at subcooled inlet conditions.

Fig. 2. Comparison of the present correlation with the database at two-phase inlet conditions.

J. Choi et al. / International Journal of Refrigeration 27 (2004) 393–400 397

the Payne correlation [4] shows good predictions for

R12, R22, R407C, R410A, and R502, but it yields rela-tively large deviations for the R134a data. Fig. 5 showsrelative deviations of the predictions with R134a as afunction of inlet subcooling and quality. The Payne

correlation shows relatively large deviations at highqualities and high subcoolings due to a limitation on thedatabase of his correlation. The Payne correlation [4]

was developed using the database covering qualities

from 0 to 10% and subcoolings from 0 to 13.9 �C.However, the present correlation yields fairly reasonableconsistency with the experimental data for all refrigerants.Based on the present database, average and standard

deviations of the present correlation are 0.1% and5.6%, respectively, while the Payne correlation [4]represents an average deviation of 0.0%, and a standard

Fig. 3. Relative deviations of the present correlation for CFC and HCFC data.

Fig. 4. Relative deviations of the present correlation for HFC data.

398 J. Choi et al. / International Journal of Refrigeration 27 (2004) 393–400

deviation of 6.2%. It should be noted that the present

correlation for two-phase inlet conditions is much easierto use than the Payne correlation [4] due to a smallernumber of dimensionless parameters and coefficients.

For two-phase inlet conditions, the Payne correlationneeds 11 dimensionless parameters and 16 coefficients,while the present correlation requires 7 dimensionless

parameters and 7 coefficients.Basically, the generalized form of the correlation can

be applied to predict refrigerant mass flow rate throughshort tube orifices with extensive ranges of operating

conditions and refrigerants. However, the accuracyof the correlation cannot be guaranteed when thecorrelation is applied beyond the range of the data-

base. Table 4 provides the limitations of the present

correlation on short tube geometry and operatingconditions.

Table 3

Comparison of the correlations with experimental data

Data source

Range of relative deviation (%)

Payne correlation [4]

Present correlation

Inlet condition

Inlet condition

Subcooled

Two-phase Subcooled Two-phase

CFC

R12 Payne [4] �13.9–19.5 �14.1–13.5

R502

Payne [4] �14.4–8.2 �8.6–7.3 �17.9–9.9 10.7–5.3

HCFC

R22 Aaron and Domanski [5] �8.9–10.4 �11.0–9.8

Kim and O’Neal [6]

�10.1–12.4 �8.0–8.8 �15.7–13.3 �7.6–6.7

HFC

R134a Kim and O’Neal [3] �14.4–12.9 �14.8–13.9 �13.6–15.3 �13.9–13.5

Singh et al. [14]

�22.4–38.3 �21.7–36.3 �16.5–15.2 �16.7–14.9

R407C

Payne and O’Neal [13] �13.3–11.4 �5.2–2.3 �13.9–14.7 �0.5–4.7

R410A

Payne and O’Neal [13] �5.5–16.6 �11.9–10.9 �20.7–17.9 �8.0–12.9

Fig. 5. Comparison of the present correlation with the Payne correlation for R134a data.

Table 4

Limitations on the applications of the present correlation

Parameter

Minimum Maximum

Short tube length (mm)

9.5 25.4

Short tube diameter (mm)

1.0 2.0

Condensing temperature (�C)

34 54

Subcooling (�C)

0.1 20

Quality (%)

1 30

J. Choi et al. / International Journal of Refrigeration 27 (2004) 393–400 399

4. Conclusions

A generalized correlation to predict the refrigerantflow rate through short tube orifices at subcooled and

two-phase inlet conditions is developed from a powerlaw form of dimensionless Pi-groups. The dimensionlessparameters are generated by applying the Buckingham

Pi theorem to the variables for operating conditions,fluid properties, and short tube geometry. The databaseconsists of extensive experimental data for R12, R22,

R134a, R407C, R410A, and R502 obtained from theopen literature. The present correlation yields a goodagreement with the database with an average deviation

of 0.3% and a standard deviation of 6.1% at subcooledinlet conditions, and an average deviation of 0.2% and astandard deviation of 5.0% at two-phase inlet condi-tions. The relative deviations of the present correlation

are from �21% to 18% for subcooled and two-phaseinlet conditions. The present correlation shows a muchsimpler form to use at two-phase inlet conditions than

the Payne correlation, while it yields comparatively bet-ter predictions at higher quality and higher subcooling.

Acknowledgements

This work was jointly supported by the CarbonDioxide Reduction & Sequestration Center, one of 21stCentury Frontier R&D Programs funded by the Minis-try of Science and Technology of Korea.

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