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
Chen, I.Y., Wang, C.-C., Lin, S.Y., 2004. Measurements andcorrelations of frictional single-phase and two-phase pressuredrops of R-410A flow in small U-type return bends. Exp.Thermal Fluid Sci. 47, 2241–2249.
Chen, I.Y., Wu, Y.-S., Chang, Y.-J., Wang, C.-C., 2007. Two-phasefrictional pressure drop of R-134a and R-410A refrigerant-oilmixtures in straight tubes and U-type wavy tubes. Exp.Thermal Fluid Sci. 31, 291–299.
Chen, I.Y., Wu, Y.-S., Liaw, J.-S., Wang, C.-C., 2008. Two-phasefrictional pressure drop measurements in U-type wavy tubessubject to horizontal and vertical arrangements. Appl.Thermal Eng. 28, 847–855.
Chisholm, D., 1983. Two-phase Flow in Pipelines and HeatExchangers. George Godwin, London, p. 304.
Domanski, P.A., Hermes, C.J.L., 2008. An improved correlation fortwo-phase pressure drop of R-22 and R-410A in 180� returnbends. Appl. Thermal Eng. 28, 793–800.
Friedel, L., 1979. Improved friction pressure drop correlationsfor horizontal and vertical two-phase pipe flow. In:European Two-phase Flow Group Meeting, Ispra, Italy,Paper E2.
Geary, D.F., 1975. Return bend pressure drop in refrigerationsystems. ASHRAE Trans. 81, 250–264.
Gronnerud, R., 1972. Investigation of liquid hold-up, flow-resistance and heat transfer in circular type evaporators, partiv: two-phase flow resistance in boiling refrigerants. In:Annexe 1972-1, Bull. de l’Inst. du Froid.
Hoang, K., Davis, M.R., 1984. Flow structure and pressure loss fortwo-phase flow in return bends. Trans. ASME 106, 30–37.
Idelshik, I.E., 1986. Handbook of Hydraulic Resistance, second ed.Hemisphere, New York, p. 640.
Muller-Steinhagen, H., Heck, K., 1986. A simple friction pressuredrop correlation for two-phase flow in pipes. Chem. Eng.Process 20, 297–308.
Pierre, B., 1964. Flow resistance with boiling refrigerantsdPart II:Flow resistance in return bends. ASHRAE J., 73–77.
Revellin, R., Haberschill, P., 2009. Prediction of frictional pressuredrop during flow boiling of refrigerants in horizontal tubes:comparison to an experimental database. Int. J. Refrigeration32, 487–497.
Thome, J.R., 2004. Engineering Data Book III. Wolverine Tube Inc.http://www.wlv.com/products/databook/db3/DataBookIII.pdf.
Traviss, D.P., Rohsenow, W.M., 1973. The influence of returnbends on the downstream pressure drop and condensationheat transfer in tubes. In: ASHRAE Semiannual Meeting,Chicago, Paper 2269RP-63.
Wang, C.-C., Chen, I.Y., Shyu, H.-J., 2003. Frictional performanceof R-22 and R-410A inside a 5.0 mm wavy diameter tube. Int. J.Heat Mass Transf. 46, 755–760.
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