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 3 2 ( 2 0 0 9 ) 1 7 7 6 – 1 7 8 3

www. i ifi i r .org

ava i lab le at www.sc iencedi rec t .com

journa l homepage : www. e lsev ier . com/ loca te / i j re f r ig

Prediction and simulation of two-phase pressure drop inreturn bends

Miguel Padillaa,b,c, Remi Revellina,b,c, Jocelyn Bonjoura,b,c,*aUniversite de Lyon, CNRS, FrancebINSA-Lyon, CETHIL, UMR5008, F-69621, Villeurbanne, FrancecUniversite Lyon 1, F-69622, France

a r t i c l e i n f o

Article history:

Received 11 March 2009

Received in revised form

13 May 2009

Accepted 5 June 2009

Published online 17 June 2009

Keywords:

Refrigeration system

Piping

Geometry

Modelling

Simulation

Pressure drop

Two-phase flow

* Corresponding author. INSA-Lyon, CETHIL,E-mail address: jocelyn.bonjour@insa-lyo

0140-7007/$ – see front matter ª 2009 Elsevidoi:10.1016/j.ijrefrig.2009.06.006

a b s t r a c t

In this paper, 325 pressure drop data points measured in return bends have been collected

from the literature. The database includes 3 different fluids (R-12, R-134a and R-410A) from

two laboratories. Based on this database, a new method is proposed for predicting the

pressure gradient in return bends, which is the sum of the frictional pressure gradient that

would be obtained in straight tubes (predicted by Muller-Steinhagen and Heck, 1986

correlation) and the singular pressure gradient (proposed equation). The proposed

correlation includes only two empirical constants and exhibits the correct physical limits.

Using the proposed equation, simulations have been performed to predict the effect of

fluid, mass velocity and saturation temperature on the singular pressure gradient.

ª 2009 Elsevier Ltd and IIR. All rights reserved.

Chute de pression diphasique dans les coudes de retour :prevision et simulation

Mots cles : Systeme frigorifique ; Tuyauterie ; Geometrie ; Modelisation ; Simulation ; Chute de pression ; Ecoulement diphasique

1. Introduction

The purpose of this study is to characterize the flow distur-

bances caused by return bends and their effects on the

UMR5008, F-69621, Villeun.fr (J. Bonjour).er Ltd and IIR. All rights

hydrodynamic (i.e. pressure drop) performance of refrigerants

in refrigeration systems such as air conditioners or heat

pumps. This problem is of great interest in HVAC&R industry

especially in the design of evaporator and condenser coils. As

rbanne, France. Tel.: þ33 4 72 43 64 27; fax: þ33 4 72 43 88 10.

reserved.

Nomenclature

a empirical constant (s2/3/m1/3)

b empirical constant

D tube diameter (m)

f friction factor

G mass velocity (kg/m2 s)

J superficial velocity (m/s)

K pressure drop coefficient of Chisholm (1983)

L length (m)

MAE mean absolute error

MAE¼ 1N

PN1

���predicted value�experimental valueexperimental value

���� 100�%�

MRE mean relative error

MAE¼ 1N

PN1

���predicted value�experimental valueexperimental value

���� 100�%�

p pressure (Pa)

R curvature radius (m)

Re Reynolds number

We Weber number

x vapor quality

Greeks

L curvature multiplier of Domanski and Hermes

(2008)

m dynamic viscosity (Pa s)

F two-phase multiplier

J curvature multiplier

r density (kg/m3)

s surface tension (N/m)

Sub and superscripts

f frictional

l liquid

o turning of the flow

rb return bend

sing singular

sp single-phase

st straight tube

tp two-phase

v vapor

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 3 2 ( 2 0 0 9 ) 1 7 7 6 – 1 7 8 3 1777

a matter of fact, it is necessary to know not only the pressure

drop in straight tubes, but also that in singularities such as in

return bends. These parameters will be useful for designing

the evaporators and condensers. As a matter of fact, the effect

of sigularities on the hydrodynamic performance increases

when the evaporators and condensers become more and more

compact.

1.1. Existing experimental studies

Two-phase flow pressure drops in return bends in refrigera-

tion systems have been experimentally investigated by

several authors in the open literature. Pierre (1964) studied the

pressure drop of R-12 in return bends with two-phase flow for

the oil-free medium and for oil-refrigerant mixtures. He used

a test section which consisted of a straight tube and an

evaporator, made with six 180� return bends in copper tubing.

The evaporation temperature was 0 �C and �10 �C, and the

vapor quality in the bend was varying between 0.30 and 0.98.

The refrigerant mass flux was comprised between 134 and

208 kg/m2 s. The author made a calculation diagram for

pressure drop in evaporators with an experimental resistance

factor for different types of return bends. Using the same fluid

(R-12), Traviss and Rohsenow (1973) measured two-phase

pressure drops in a 8 mm tube in order to determine whether

the disturbance caused by a return bend was only a localized

effect or extended over a significant length of the condenser

tube. They found that the effect of a return bend on the

downstream pressure drop was negligible when averaged

over a length of 90 tube diameters or more. Nevertheless,

Hoang and Davis (1984) suggested that the length of nine tube

diameters is required to complete the phases remixing

process downstream of the return bend.

Geary (1975) investigated the two-phase adiabatic flow

pressure drop in return bends based on his R-22 data with tube

diameters from 11.05 mm to 11.63 mm with curvature ratios

(2R/D) from 2.317 to 6.54. He proposed a correlation for pre-

dicting the two-phase pressure drop for design purposes in

typical air-conditioning applications. In this paper, the author

also proposes two-phase frictional pressure drop data for

straight tubes.

Later, Chen et al. (2004) presented single-phase and two-

phase frictional data of R-410A in four type of return bends

with tube diameters ranging from 3.3 and 5.07 mm and

curvature ratios varying from 3.91 to 8.15. They proposed

a modified two-phase friction factor based on Geary’s corre-

lation. Then, Chen et al. (2007) presented a study with single-

phase and two-phase pressure drop data for R-134a/oil

mixture with oil concentration of 0%, 1%, 3% and 5%, flowing

in a wavy tube with an inner diameter of 5.07 mm and

a curvature ratio of 5.18. Very recently, Chen et al. (2008)

presented measurements of R-134a two-phase frictional

pressure gradients for vertical and horizontal arrangements of

a U-type copper wavy tube which contained nine consecutive

return bends with an inner diameter of 5.07 mm and a curva-

ture ratio of 5.18. The working temperature was near 25 �C, the

mass flux ranged from 200 to 700 kg/m2 s and the vapor

quality varied from 0.1 to 0.9. They conducted their tests for

vertical arrangement by selecting the inlet flow at the upper

tube or at the lower tube in the return bend. They found that

the pressure gradients in the return bend of vertical arrange-

ment were always higher than those of horizontal arrange-

ment regardless the flow entry was at the upper or at the lower

tube. Furthermore, the pressure gradient for a flow entering at

the upper tube was higher than that for a flow entering at the

lower tube because of the influence of buoyancy.

1.2. Prediction methods

In the open literature, among the articles related to two-phase

pressure drop correlations in return bends, only six of them

provided specific information (fluid, saturation temperature,

mass velocity, diameter, etc.) using refrigerants only. Pierre

(1964) proposed a correlation, only valid for horizontal tubes,

based on his experiments carried out with R-12 by assuming

that the total flow resistance is divided into two part: the first

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 3 2 ( 2 0 0 9 ) 1 7 7 6 – 1 7 8 31778

one corresponds to the turning of the flow, and the second one

to the friction only. This is expressed as follows:

Dprb ¼ Dpo þ Dpf (1)

where Dprb is the total pressure drop in the return bend, Dpo is

the pressure drop due to the turning of the flow and Dpf is the

pressure drop due to the friction. For the calculation of Dpo

and Dpf, one needs to refer to two different graphs so as to

determine the corresponding resistance factor. Those values

of the resistance factors obtained by Pierre for the center-to-

center distance (2R) considered in his study, showed little

influence of the center-to-center distance of the bend in the

resulting pressure drop. In that sense, Pierre concluded that

this parameter was not of major importance and did not

include it in his pressure drop correlation.

Geary (1975) presented a correlation (using 4 empirical

constants) by using the friction factor approach of the straight

tubes. The correlation of the two-phase pressure drop in

return bends consisted in a single-phase pressure drop

equation for vapor flow only. He included a dimensionless

friction factor depending on the effects of the vapor quality

and the center-to-center distance in return bends. The pres-

sure drop induced by the return bends was expressed as:

Dprb ¼ fL

D

G2x2

2rv

(2)

where L is the bend length, D is the tube diameter, rv is the

density of the vapor phase and f is a dimensionless friction

factor given by:

f ¼ 8:03� 10�4Re0:5v

expð0:215ð2R=DÞÞx1:25(3)

In Eq. (3), Rev is the Reynolds number (GDx/mv) of the vapor

phase and the term 2R/D represents the curvature ratio of the

return bend.

Chisholm (1983) proposed a correlation (using 5 empirical

constants) to calculate the pressure drop in return bends Dprb

by using a two-phase multiplier F and the single-phase flow

liquid pressure drop in the return bend Dpsp:

Dprb ¼ FDpsp (4)

where Dpsp is expressed as follows:

Dpsp ¼ KspG2

2rl

(5)

In Eq. (5), rl is the density of the liquid phase and Ksp is the all-

liquid local pressure drop coefficient obtained for single-phase

flows. To estimate the value of Ksp, Idelshik (1986) suggested

the following expression:

Ksp ¼ flLDþ 0:294

�RD

�0:5

(6)

where R is the curvature radius and fl is the single-phase

friction factor calculated from the Blasius equations.

The two-phase multiplier F in Eq. (4) is given by:

F ¼ 1þ�

rl

rv

� 1

�x½bð1� xÞ þ x� (7)

where b is expressed as:

b ¼ 1þ 2:2Kspð2þ R=DÞ (8)

The Chisholm (1983) and Idelshik (1986) method did not

provide validation against experimental measurements.

Based on Geary’s correlation and using Geary’s R-22 database

and their own R-410A database, Chen et al. (2004) presented

a new correlation for the friction factor from the empirical fit,

including new parameters such as the Weber number and

a combined vapor and liquid Reynolds number as well as the

vapor quality and the curvature ratio 2R/D. The expression of

the friction fraction is written as:

f ¼ 10�2Re0:35m

We0:12v expð0:194ð2R=DÞÞx1:26

(9)

where Rem ¼ Rev þ Rel ¼ GDðx=mv þ ð1� xÞ=mlÞ is a combined

vapor and liquid Reynolds number and Wev¼G2D/rvs is the

Weber number which takes into account the influence of the

surface tension in the two-phase frictional pressure drop. The

authors reported a good agreement with their own R-410A

database and with Geary’s R-22 database, with a mean devi-

ation of 19.1%.

Recently, Domanski and Hermes (2008) proposed a corre-

lation for the calculation of the two-phase pressure gradient

in return bends based on 241 experimental data points from

Geary’s R-22 database and Chen et al. (2004) R-410A database.

The correlation allowed predicting the two-phase pressure

gradient in a straight tube using the Muller-Steinhagen and

Heck (1986) correlation and using a multiplier that accounted

for the bend curvature:

�dpdz

�rb

¼ L

�dpdz

�st

(10)

The multiplier L includes five empirical constants and takes

into account the influence of the vapor velocity and the mass

distribution in each phase. The multiplier L is given by:

L ¼ 6:5� 10�3

�GxDmv

�0:54 �1x� 1

�0:21 �rl

rv

�0:34 �2RD

��0:67

(11)

The Domanski and Hermes (2008) correlation predicted 75% of

the 241 experimental data points within a �25% error band

and exhibited a RMS deviation of 25%.

The purpose of this study is to develop a prediction method

that includes as few empirical parameters as possible and

based, as much as possible, on the phenomena encountered

during two-phase flow in return bends.

2. Presentation of the experimental databasefrom the literature

A total of 690 experimental data points have been collected

from the literature: Pierre (1964), Traviss and Rohsenow (1973),

Geary (1975), Wang et al. (2003), Chen et al. (2004, 2007, 2008).

These data come from four different laboratories and the

database includes four different refrigerants. However, some

of these data have been removed from the database for the

following reasons:

0 2 4 6 8 10 120

2

4

6

8

10

12

Experimental frictional pressure gradient in

straight tube [kPa/m]

Pred

icted

frictio

nal p

ressu

re g

rad

ien

t in

straig

ht tu

be [kP

a/m

]

Müller−Steinhagen & Heck correlation 37.2 % of the data within ± 30% MAE = 97.0% MRE = 43.9%

R−22 (Geary’s data)

Fig. 1 – Comparison between the experimental frictional

pressure gradient in straight tubes from Geary (1975) and

the predicted one by Muller-Steinhagen and Heck (1986).

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 3 2 ( 2 0 0 9 ) 1 7 7 6 – 1 7 8 3 1779

- The pressure drop measurements in the return bend were

lower than those in the straight tube, which is obviously

not realistic and is probably a clue for too low accuracy.

- The length between two consecutive return bends was

less than 4.8D. In the literature, a length between 9D

(Hoang and Davis, 1984) and 90D (Traviss and Rohsenow,

1973) was proposed to let the flow regimes recover after

the return bend. Nevertheless, it can be observed that the

proposed correlation allows predicting the data down to

4.8D.

- The R-22 data from Geary have been removed. The author

indeed presents pressure drop data for straight tube as

well as for return bends. Regarding the data for straight

tube (Fig. 1), it is not possible to predict them using the

Muller-Steinhagen and Heck (1986) correlation whereas

this correlation is now often considered as very efficient

for such conditions. Especially it has been shown to work

satisfactorily with R-22 (Revellin and Haberschill, 2009).

Table 1 – Experimental conditions of Traviss and Rohsenow (1

Authors Refrigerant D (mm) 2R/D G (kg/m

Traviss and

Rohsenow (1973)

R-12 8 3.175 155

300

500

6.35 150

300

Chen et al. (2007) R-134a 5.07 5.18 200–700

Chen et al. (2008) R-134a 5.07 5.18 200–700

5.07a 200–700

5.07b 200–700

Chen et al. (2004) R-410A 3.25 3.91 300–900

3.3 8.15 300–900

Total 3.25–8.00 3.175–8.15 150–900

a Vertical flow - Inlet is at the upper tube.

b Vertical flow - Inlet is at the lower tube.

The difference is significant. It can be remarked that

Geary’s data are not reliable for straight tubes, and are not

reliable either for return bends. Geary’s data have been

therefore removed from the database.

Finally, the database used for developing the new predic-

tion method includes 325 data points from Traviss and Roh-

senow (1973) for R-12, Chen et al. (2004) for R-410A and Chen

et al. (2007, 2008) for R-134a. The ranges of the experimental

data, with the uncertainties reported by the authors, are pre-

sented in Table 1. As can be observed, the tube diameter varies

from 3.25 to 8 mm, the curvature ratios from 3.175 to 8.15, the

mass velocity ranges from 150 to 900 kg/m2 s and the vapor

quality from 0.0095 to 0.9367. The saturation temperature

varies from 10 to 39 �C. Each parameter exhibits a large vari-

ation and, as a consequence, the database is statistically

representative. It is worth mentioning that the data corre-

spond to the total pressure gradient in return bends, i.e. the

sum of the frictional pressure gradient in straight tubes plus

the singular pressure gradient.

3. Comparison to existing methods

Existing prediction methods have been used to check their

efficiency for predicting the experimental results of the

experimental database from two laboratories. Fig. 2 presents

the comparison between the data and the method proposed

by Chisholm (1983) and Idelshik (1986). As can be observed,

only 31.4% of the data are predicted within a �30% error band.

The MAE is around 107%. It is clear that this method does not

work at all for predicting the pressure drops in return bends.

Fig. 3 shows the results of the comparison between the

experimental data and the correlation developed by Chen

et al. (2004). 39.7% of the data are predicted within a �30%

error band. The prediction is however better for R-134a and R-

410A than for R-12. It is noteworthy that the R-134a and R-

410A data come from the same author and that the R-12 data

include a diameter (8 mm), which is different from that used

973), Chen et al. (2007, 2008, 2004).2 s) Tsat (�C) x Number of

data pointsDPrb,

Uncertainty

34 0.218–0.77 21 Not reported

38 0.0095–0.9367 35

39 0.147–0.928 24

34 0.154–0.935 24

34 0.046–0.932 35

20 0.1–0.9 27 �0.3%

25 0.1–0.9 25 �0.5%

25 0.1–0.9 25 �0.5%

25 0.1–0.9 26 �0.5%

25 0.1–0.9 33 �0.5%

10–25 0.1–0.9 50 �0.5%

10–39 0.0095–0.9367 325

0.1 1 10 100 10000.1

1

10

100

1000

Experimental pressure gradient in

return bends [kPa/m]

Pred

icted

p

ressu

re g

rad

ien

t in

retu

rn

b

en

ds [kP

a/m

]

Chisholm & Idelshik correlation 31.4 % of the data within ± 30 %MAE = 107.0 %MRE = 102.4 %

R−12R−134aR−410A

Fig. 2 – Experimental pressure gradient data compared to

the Chisholm and Idelshik method.

0.1 1 10 100 10000.1

1

10

100

1000

Experimental pressure gradient in

return bends [kPa/m]

Pred

icted

p

ressu

re g

rad

ien

t in

retu

rn

b

en

ds [kP

a/m

]

Domanski & Hermes correlation 44.0 % of the data within ± 30 %MAE = 62.9 %MRE = 54.7 %

R−12R−134aR−410A

Fig. 4 – Experimental pressure gradient data compared to

the Domanski and Hermes (2008) prediction method.

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 3 2 ( 2 0 0 9 ) 1 7 7 6 – 1 7 8 31780

by Chen et al. (2004) for developing their correlation. It was

therefore expected these data to be well predicted by Chen

et al. (2004). Fig. 4 shows the comparison between the

Domanski and Hermes (2008) correlation and the database

(Table 1). This correlation predicts 44% of the data within

a �30% error band with a MAE around 62.9%. The correlation

predicts relatively well the R-410A data, which is not

surprising since Domanski and Hermes (2008) uses the same

data for developing their correlation.

Accurately predicting the data is important but the

prediction methods must also presents the correct physical

limits. As an example, Fig. 5 shows the curvature multiplier L

proposed by Domanski and Hermes (2008) as a function of the

curvature ratio 2R/D. The ratio between the return bend

pressure gradient determined by the Chen et al. (2004) and the

Chisholm (1983) correlations and the straight tube pressure

gradient determined by the Muller-Steinhagen and Heck

(1986) correlation was also calculated. As can be observed,

when 2R/D / 0, L /þN which is correct. Nevertheless, when

0.1 1 10 100 10000.1

1

10

100

1000

Experimental pressure gradient in

return bends [kPa/m]

Pred

icted

p

ressu

re g

rad

ien

t in

retu

rn

b

en

ds [kP

a/m

]

Chen et al correlation 39.7 % of the data within ± 30 % MAE = 84.7 %MRE = 77.0 %

R−12R−134aR−410A

Fig. 3 – Experimental pressure gradient data compared to

the Chen et al. (2004) prediction method.

2R/D /þN, L< 1 which is not physical. In that case, L must

tend toward 1.

A new method must be developed not only to correctly

predict the data but also to be in agreement with the physical

limits. This new prediction method is presented hereafter.

4. New prediction method for return bends

Now, the intention is to develop a new method for predicting

the pressure gradient in return bends using a different

approach from that used in other studies. The ratio between

the pressure gradient in return bends and that in straight

tubes will not be calculated. On the contrary, by analogy with

single-phase flow, a force balance has been carried out on

both phases. As a result, the centrifugal force acting on both

phases due to the return bend has been considered. This

analysis has thus led to the use of the superficial velocities, as

encountered in a two-fluid model (separated flow model). As

0 10 20 30 40 500

1

2

3

4

5

6

7

8

Cu

rv

atu

re

m

ultip

lie

r [−

]

Curvature ratio [−]

R−134aD = 5 mmTsat = 20 °CG = 200 kg/m2sx= 0.2

Chisholm & IdelshikChen et al.Domanski & Hermes

Fig. 5 – Curvature multiplier (dp/dz)rb/(dp/dz)st as a function

of the curvature ratio (2R/D) for different prediction

methods.

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 3 2 ( 2 0 0 9 ) 1 7 7 6 – 1 7 8 3 1781

a result, the total measured pressure gradient in return bends

is given by the following relation:

��dp

dz

�rb

¼��dp

dz

�st

þ��dp

dz

�sing

(12)

where (�dp/dz)st is the frictional pressure gradient that would

be reached in straight tubes calculated by the Muller-Stein-

hagen and Heck (1986) correlation (plus eventually the static

pressure gradient for vertical orientation (Thome, 2004)). The

second term (�dp/dz)sing is the singular pressure gradient.

From the whole database (Table 1), it is given by the following

relation:

�dpdz

�sing

¼ a

�rvJ2

v

R

� �J2l

R

�b

(13)

where a¼ 0.047 s2/3/m1/3 and b¼ 1/3. This constant and this

exponent were obtained from the least square method based

on the experimental data shown in Table 1. R is the radius of

the curvature, Jv is the superficial velocity of the vapor written

as:

Jv ¼Gxrv

(14)

and Jl is the superficial velocity of the liquid expressed as

Jl ¼Gð1� xÞ

rl

(15)

G represents the mass velocity, x the vapor quality and rv and

rl are the densities of the vapor and the liquid, respectively. All

the parameters (mass velocity, densities, radius.) should be

taken in the S.I. units (see Nomenclature). The comparison

between the proposed prediction method and the database

from the literature is shown in Fig. 6. Almost 67% of the data

are predicted within a �30% error band. The mean absolute

error is less than 23.5% and the mean relative error is around

�8%. The prediction is satisfactory. Using the same singular

pressure gradient correlation but the Friedel (1979) correlation

instead of the Muller-Steinhagen and Heck (1986) relation for

the straight tube, this yields a MAE of 26% which is still

correct. In addition, if the Revellin and Haberschill (2009)

0.1 1 10 100 10000.1

1

10

100

1000

Experimental pressure gradient in

return bends [kPa/m]

Pre

dic

te

d p

re

ss

ure

g

ra

die

nt in

re

tu

rn

be

nd

s [k

Pa

/m

]

Proposed correlation 66.5 % of the data within ± 30 %MAE = 23.5 %MRE = −7.7 %

R−12R−134aR−410A

Fig. 6 – Experimental pressure gradient data for return

bends (Table 1) compared to the present prediction method.

model and the Gronnerud (1972) correlation were used, this

would have yielded MAE¼ 32.7% and MAE¼ 35.7%, respec-

tively. Comparing the proposed prediction method (Eq. (13))

and the Geary (1975) database, only 3.5% of the data are pre-

dicted within a �30% error band, with a MAE of 79.9% and

a MRE of 79.7%. These results show that it is not possible to

predict the Geary (1975) data using the proposed correlation,

probably for the same reason as it was not possible to predict

the straight tube values with a very conventional method such

as Muller-Steinhagen and Heck (1986) (cf. Section 2 and Fig. 1).

Eq. (13) presents many advantages:

- Only two empirical constants have been used for devel-

oping this equation (instead of five for Domanski and

Hermes (2008) and Chen et al. (2004)).

- The relation has been developed based on a large data-

base: 325 data points for three different fluids obtained in

two different laboratories.

- The equation works for horizontal and vertical return

bends (51 data points in vertical orientation from Chen

et al. (2004) database).

- The relation has been developed over a wide range of tube

diameters (3.25–8.00 mm), mass velocities (150–900 kg/

m2 s), saturation temperatures (10–39 �C) and over the all

range of vapor quality (0.0095–0.9367).

- The term ½rvJ2v=R� represents the centrifugal force acting

on the vapor phase due to the return bend.

- The term ½J21=R� takes into account the centrifugal force

acting on the liquid phase due to the return bend.

- The relation is independent of the diameter, which is logical

in view of the following reasoning: the curvature effect on

the pressure gradient in a straight tube is predicted. The

diameter effect is thus directly taken into account by the

Muller-Steinhagen and Heck (1986) correlation.

- There is no term including the properties such as mv,ml or

s. This seems logical since the singularity should only be

affected by the curvature effect and not the transport

properties.

- Rearranging Eq. (12), it comes:

0 10 20 30 40 500

1

2

3

4

5

6

7

8

Cu

rvatu

re m

ultip

lier [−

]

Curvature ratio [−]

R−410AD = 3.3 mmTsat = 25 °CG = 300 kg/m2sx= 0.2

Chisholm & IdelshikChen et al.Domanski & HermesProposed

Fig. 7 – Curvature multiplier (dp/dz)rb/(dp/dz)st as a function

of the curvature ratio (2R/D) for the present prediction

method and different correlations from the literature.

200

]

R−134a

a

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 3 2 ( 2 0 0 9 ) 1 7 7 6 – 1 7 8 31782

�dpdz

�¼�

dpdz

� "1þðdp

dz

sing

dp#¼�

dpdz

�J (16)

0 500 1000 1500 20000

20

40

60

80

100

120

140

160

180

S

in

gu

la

r p

re

ss

ure

g

ra

die

nt [k

Pa

/m

Mass velocity [kg/m2s]

Proposed correlationD = 8 mm2R/D =5 x = 0.3 Tsat= 10 °C

R−410A

8

/m

] Proposed correlationD = 8 mm

R−134aR−410A

b

rb stdz st

st

It can be observed that the physical limits are correct. In

Eq. (16), the curvature multiplier J represents the ratio

between the pressure gradient in return bends and that in

straight tubes calculated by the Muller-Steinhagen and Heck

(1986) correlation. When the curvature radius R /þN, the

curvature multiplier J / 1 and the resulting pressure gradient

in return bends tends toward the pressure gradient in straight

tubes. Furthermore, when R / 0, the curvature multiplier

J /þN and the pressure gradient in return bends tends

toward infinity. In order to illustrate this, Fig. 7 shows the

curvature multiplier J as a function of the curvature ratio for

four different return bend pressure gradient correlations (the

Chisholm (1983) and Idelshik (1986) method, the Chen et al.

(2004), the Domanski and Hermes (2008) and present correla-

tions). The pressure gradient in straight tubes is calculated

using the Muller-Steinhagen and Heck (1986) correlation.

0 0.2 0.4 0.6 0.8 10

5

10

15

20

25

30

35

40

45

Pressu

re g

rad

ien

t in

retu

rn

b

en

ds [kP

a/m

]

Vapor quality [−]

R−410AD = 3.25 mm2R/D = 3.91G = 400 kg/m2sTsat= 25 °C

Experimental data of Chen et al.Chisholm & IdelshikChen et al.Domanski & HermesProposedMuller−Steinhagen & Heck (Straight − tubes)

0 0.2 0.4 0.6 0.8 10

5

10

15

Pressu

re g

rad

ien

t in

retu

rn

b

en

ds [kP

a/m

]

Vapor quality [−]

R−134aD = 5.07 mm2R/D = 5.18G = 200 kg/m2sTsat= 25 °C

a

b

Chisholm & IdelshikChen et al.Domanski & HermesProposedMuller−Steinhagen & Heck (Straight − tubes)Experimental data of Chen et al.

Data by Chen et al. (2008).

Data by Chen et al. (2004).

Fig. 8 – Pressure gradient in return bends as a function of the

vapor quality using the experimental data by Chen et al.

(2004, 2008) compared to different prediction methods.

−10 0 10 20 30 400

1

2

3

4

5

6

7

Sin

gu

la

r p

re

ss

ure

g

ra

die

nt [k

Pa

Saturation temperature [°C]

2R/D = 5G = 500 kg/m2sx= 0.3

Fig. 9 – Singular pressure gradient as a function of the mass

velocity and the saturation temperature for R-134a and

R-410A. (a) Singular pressure gradient as a function of the

mass velocity. (b) Singular pressure gradient as a function

of the saturation temperature.

Fig. 8(a) and (b) shows the comparison between the Chis-

holm (1983) and Idelshik (1986) method, the Chen et al. (2004)

and the Domanski and Hermes (2008) correlations, the

proposed equation and some data available in the literature.

The simulations have been performed using R-410A and

R-134a for different conditions. It is shown that the proposed

correlation is the best method for predicting the data and for

determining the vapor quality corresponding to the maximum

value of the pressure gradient (so does the method by

Domanski and Hermes (2008)).

In addition to the previous comparisons, simulations have

been performed using the proposed equation. Fig. 9(a) shows the

effect of the mass velocity on the singular pressure gradient

(�dp/dz)sing. Note that the range of the mass velocity has been

voluntarily extrapolated in order to see any particular behavior.

As can be observed, the higher the mass velocity, the larger the

singular pressure gradient. There is indeed an increase of the

centrifugal force when G increases. The singular pressure

gradient increases as G8/3. Fig. 9(b) presents the effect of the

saturation temperature on the pressure gradient. Saturation

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 3 2 ( 2 0 0 9 ) 1 7 7 6 – 1 7 8 3 1783

conditionsact onthedensities.Whenthetemperature increases,

the vapor density increases, the vapor velocity decreases and as

a consequence the pressure gradient decreases.

5. Conclusions

In this paper, 325 pressure drop data points measured in

return bends have been collected from the literature. The

database includes 3 different fluids (R-12, R-134a and R-410A)

from two laboratories and provides a large variation of

geometries and conditions: the tube diameter varies from 3.25

to 8 mm for horizontal and vertical arrangements, the curva-

ture ratios from 3.175 to 8.15, the mass velocity ranges from

150 to 900 kg/m2 s and the vapor quality from 0.0095 to 0.9367.

The saturation temperature varies from 10 to 39 �C. Based on

this database, a new method has been proposed for predicting

the pressure gradient in return bends. The idea was to sum the

frictional pressure gradient that would be obtained in straight

tubes (predicted by Muller-Steinhagen and Heck (1986) corre-

lation) and the singular pressure gradient (present equation).

The present correlation is based on the centrifugal force

acting on the vapor and the liquid phases. This equation is

independent of the diameter and the transport properties

since only the curvature effect is taken into account. In

addition, the proposed correlation includes only two empirical

constants and exhibits the correct physical limits, i.e. when

the curvature ratio tends toward infinity, the pressure

gradient due to the return bends tends toward that of the

straight tubes. Furthermore, when the curvature ratio tends

toward zero, the pressure gradient tends toward infinity.

r e f e r e n c e s

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