ARTICLE IN PRESS
Experimental Thermal and Fluid Science xxx (2008) xxx–xxx
Contents lists available at ScienceDirect
Experimental Thermal and Fluid Science
journal homepage: www.elsevier.com/locate/etfs
Effective property models for homogeneous two-phase flows
M.M. Awad *, Y.S. Muzychka
Faculty of Engineering and Applied Science, Memorial University of Newfoundland, St. John’s, Newfoundland, Canada A1B 3X5
a r t i c l e
i n f o
Article history:
Received 30 January 2008
Received in revised form 20 July 2008
Accepted 20 July 2008
Available online xxxx
Keywords:
Thermal conductivity
Porous media
Viscosity
Two-phase flow
Frictional pressure gradient
Circular pipes
Minichannels
Microchannels
Homogeneous flow
a b s t r a c t
Using an analogy between thermal conductivity of porous media and viscosity in two-phase flow, new
definitions for two-phase viscosity are proposed. These new definitions satisfy the following two conditions: namely (i) the two-phase viscosity is equal to the liquid viscosity at the mass quality = 0% and (ii)
the two-phase viscosity is equal to the gas viscosity at the mass quality = 100%. These new definitions can
be used to compute the two-phase frictional pressure gradient using the homogeneous modeling
approach. These new models are assessed using published experimental data of two-phase frictional
pressure gradient in circular pipes, minichannels and microchannels in the form of Fanning friction factor
(fm) versus Reynolds number (Rem). The published data include different working fluids such as R-12, R22, argon (R740), R717, R134a, R410A and propane (R290) at different diameters and different saturation
temperatures. Models are assessed on the basis minimizing the root mean square error (eRMS). It is shown
that these new definitions of two-phase viscosity can be used to analyze the experimental data of twophase frictional pressure gradient in circular pipes, minichannels and microchannels using simple friction
models.
Ó 2008 Elsevier Inc. All rights reserved.
1. Introduction
The homogeneous flow model provides the simplest technique
for analyzing two-phase (or multiphase) flows. In the homogeneous model, both liquid and vapor phases move at the same
velocity (slip ratio = 1). Consequently, the homogeneous model
has also been called the zero slip model. The homogeneous model
considers the two-phase flow as a single-phase flow having average fluid properties, which depend upon mixture quality. Thus,
the frictional pressure drop is calculated by assuming a constant
friction coefficient between the inlet and outlet sections of the
pipe.
The void fraction based on the homogeneous model (am) can be
expressed as follows:
am ¼
1
1þ
1xqg x
ð1Þ
ql
When (ql/qg) is large, the void fraction based on the homogeneous
model (am) increases very rapidly once the mass quality (x) increases even slightly above zero, as shown in Fig. 1. The prediction
of the void fraction using the homogeneous model is reasonably
accurate only for bubble and mist flows since the entrained phase
* Corresponding author. Tel.: +1 709 737 3547; fax: +1 709 737 4042.
E-mail addresses: [email protected] (M.M. Awad), [email protected]
(Y.S. Muzychka).
travels at nearly the same velocity as the continuous phase. Also,
when (ql/qg) approaches 1 (i. e. near the critical state), the void fraction based on the homogeneous model (am) approaches the mass
quality (x) and the homogeneous model is applicable at this case.
For the homogeneous model, the density of two-phase gas–liquid flow (qm) can be expressed as follows:
qm ¼
x
qg
þ
1x
!1
ð2Þ
ql
Eq. (2) can be derived knowing that the density is equal to the reciprocal of the specific volume and using thermodynamics relationship for the specific volume
vm ¼ ð1 xÞvl þ xvg
ð3Þ
Eq. (2) can also be obtained based on the volume averaged value as
follows:
qm ¼ am qg þ ð1 am Þql ¼
x
qg
þ
1x
ql
!1
ð4Þ
Eq. (2) satisfies the following limiting conditions between (qm) and
mass quality (x):
qm ¼ ql
x ¼ 1; qm ¼ qg
x ¼ 0;
)
ð5Þ
0894-1777/$ - see front matter Ó 2008 Elsevier Inc. All rights reserved.
doi:10.1016/j.expthermflusci.2008.07.006
Please cite this article in press as: M.M. Awad, Y.S. Muzychka, Effective property models for homogeneous two-phase flows, Exp. Therm.
Fluid Sci. (2008), doi:10.1016/j.expthermflusci.2008.07.006
ARTICLE IN PRESS
2
M.M. Awad, Y.S. Muzychka / Experimental Thermal and Fluid Science xxx (2008) xxx–xxx
Nomenclature
a
b
d
e
f
G
g
K
k
N
n
dp/dz
p
Re
v
x
Churchill parameter
Churchill parameter
pipe diameter, m
error
Fanning friction factor
mass flux, kg/m2 s
acceleration due to gravity, m/s2
index for summation
thermal conductivity, W/m K
number of data points
Blasius index
pressure gradient, Pa/m
pressure drop, Pa
Reynolds number = Gd/l
specific volume, m3/kg, or volume fraction of material
component
mass quality
Greek Symbols
a
void fraction
e
pipe roughness, m
q
density, kg/m3
The Reynolds number based on the homogeneous model (Rem) can
be expressed as follows:
Gd
Rem ¼
ð6Þ
lm
In the homogeneous model, there are some common expressions
for the viscosity of two-phase gas–liquid flow (lm). The expressions
available for the two-phase liquid–gas viscosity are mostly of an
empirical nature as a function of mass quality (x). The liquid and
gas are presumed to be uniformly mixed due to the homogeneous
flow. The possible definitions for the viscosity of two-phase gas–liquid flow (lm) can be divided into two groups. In the first group, the
form of the expression between (lm) and mass quality (x) satisfies
the following important limiting conditions:
lm ¼ ll
x ¼ 1; lm ¼ lg
x ¼ 0;
)
ð7Þ
For example, McAdams et al. [1], introduced the definition of twophase viscosity (lm) based on the mass averaged value of reciprocals as follows:
l
m
dynamic viscosity, kg/m s
kinematic viscosity, m2/s
angle of inclination of pipe to horizontal
h
Subscripts
1
component 1
2
component 2
a
acceleration
cont
continuous phase
disp
dispersed (discontinuous) phase
e
effective
f
frictional
g
gas
grav
gravitational
i
inlet
l
liquid
lo
liquid only
m
homogeneous mixture
o
outlet
RMS
root mean square
s
saturation
tp
two-phase
lm ¼
x
lg
þ
1x
!1
ð8Þ
ll
They proposed their viscosity expression by analogy to the expression for the homogeneous density (qm). Eq. (8) leads to the homogeneous Reynolds number (Rem) is equal to the sum of the liquid
Reynolds number (Rel) and the gas Reynolds number (Reg).
Cicchitti et al. [2], introduced the definition of two-phase viscosity (lm) based on the mass averaged value as follows:
lm ¼ xlg þ ð1 xÞll
ð9Þ
They used the above definition of lm in place of the definition proposed by McAdams et al. [1]. The only reason for doing this, in addition to simplicity, was a reasonable agreement with experimental
data.
Dukler et al. [3], introduced the definition of two-phase viscosity (lm) based on the mass averaged value of kinematic viscosity as
follows:
"
lm ¼ qm x
!
#
lg
l
þ ð1 xÞ l
qg
ql
ð10Þ
or
0
0.2
0.4
0.6
0.8
1
1
1
0.8
0.6
0.6
α
0.8
ρl / ρg =1
0.4
0.4
ρl / ρg =10
ρl / ρg =100
0.2
0.2
ρl / ρg =1000
0
0
0
0.2
0.4
x
0.6
0.8
1
Fig. 1. Void fraction as a function of mass quality for various density ratios.
mm ¼ xmg þ ð1 xÞml
ð11Þ
Beattie and Whalley [4] presented a simple two-phase pressure
drop calculation method. They adapted a theoretically based flow
pattern dependent calculation method to yield a simple predictive
method in which flow pattern influences were in an implicit method and hence need not to be explicitly taken into account when
using the method. For both bubble flow and annular flow, they proposed that the average two-phase viscosity (lm) was replaced by a
hybrid definition:
lm ¼ ll ð1 am Þð1 þ 2:5am Þ þ lg am
¼ ll 2:5ll
xql
xql þ ð1 xÞqg
!2
xql ð1:5ll þ lg Þ
þ
xql þ ð1 xÞqg
!
ð12Þ
Lin et al. [5] introduced the definition of two-phase viscosity as
follows:
Please cite this article in press as: M.M. Awad, Y.S. Muzychka, Effective property models for homogeneous two-phase flows, Exp. Therm.
Fluid Sci. (2008), doi:10.1016/j.expthermflusci.2008.07.006
ARTICLE IN PRESS
3
M.M. Awad, Y.S. Muzychka / Experimental Thermal and Fluid Science xxx (2008) xxx–xxx
lm ¼
ll lg
lg þ x1:4 ðll lg Þ
Owens [9] introduced a definition of two-phase viscosity based
on the liquid viscosity, simply as lm = ll. The rationale being that
in most two-phase flows, the liquid is the dominant phase.
García et al. [10], defined the Reynolds number of two-phase
gas–liquid flow using the kinematic viscosity of liquid flow (ml) instead of the kinematic viscosity of two-phase gas–liquid flow (mm).
They used this definition because the frictional resistance of the
mixture was due mainly to the liquid. This was equivalent to defining lm as
ð13Þ
In their study, the range of x appearing in the capillary tubes was
0 < x < 0.25. For the best fit to their experimental data, they took
the value of the exponent in Eq. (13) as 1.4.
Fourar and Bories [6] presented an unusual expression of the
two-phase viscosity as follows:
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
lm ¼ qm xvg þ ð1 xÞvl þ 2 xð1 xÞvg vl
pffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2
xvg þ ð1 xÞvl
¼ qm
lm ¼ ll
ð14Þ
pffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2
xmg þ ð1 xÞml
ð15Þ
It should be noted that Fourar and Bories [6] definition of two-phase
viscosity is similar to the Dukler et al. [3] definition of two-phase
viscosity but with the addition of an extra term.
In the second group, the form of the expression between (lm)
and mass quality (x) does not satisfy the limiting conditions of
Eq. (7).
For example, Davidson et al. [7], defined the viscosity of twophase gas–liquid flow (lm) as follows:
"
lm ¼ ll 1 þ x
ql
1
qg
ll qg
qm
¼
ql
xql þ ð1 xÞqg
ð17Þ
The main disadvantage for the various forms of lm in the second
group is that they are not accurate as the mass quality (x) approaches 1. A robust model should capture the physics of all limiting cases.
In the realm of two-phase flow viscosity models, Collier and
Thome [11] mention that the definition of lm proposed by
McAdams et al. Eq. (8), is the most common definition of lm.
Eq. (14) can be rewritten in terms of the kinematic viscosity as
follows:
mm ¼
2. Proposed methodology
In the current work, many of the existing two-phase flow viscosity models along with several new definitions are examined.
Heat transfer studies involving porous media and two component
systems, have lead to many definitions for effective thermal conductivity of the homogeneous medium. Using a one dimensional
transport analogy between thermal conductivity in porous media
[12] and viscosity in two-phase flow, new definitions for twophase viscosity will be introduced. These new definitions are generated by analogy as follows:
!#
ð16Þ
The reason for this definition is that when Davidson et al., plotted
the experimental two-phase friction factor (ftp) of their high pressure steam-water pressure drop data against the Reynolds number
for all-liquid flow (Relo), they observed that there were large discrepancies from the single-phase friction factor at Relo < 2 105.
They found that considerably better agreement with the normal
single-phase flow relationship represented by the Blasius Equation
[8] was obtained if they plotted the experimental two-phase friction factor against the Reynolds number for all-liquid flow multiplied by the ratio of the inlet to outlet mean specific volumes. It
should be noted that the above definition of the viscosity of twophase gas–liquid flow (lm), does not extrapolate to the gas viscosity
(lg) as the mass quality (x) approaches 1.
(i)
(ii)
(iii)
(iv)
lm is analogous to ke
ll is analogous to k1
lg is analogous to k2
x is analogous to v2 (volume fraction of component 2).
These new definitions for two-phase viscosity are given in Table
1. As one can see the series and parallel combination rules are analogous to existing rules proposed by McAdams et al. [1], and
Cicchitti et al. [2]. Definition 3 for two-phase viscosity is generated
by analogy to the effective thermal conductivity using the
Maxwell-Eucken 1 model [13]. Maxwell-Eucken 1 [13] is suitable
Table 1
Analogy between thermal conductivity in porous media and viscosity in two-phase flow
System
Property
Definition 1
Porous media
Thermal conductivity (k)
ke ¼
1m2
k1
þ mk22
1
(Series model)
Definition 2
ke ¼ ð1 m2 Þk1 þ m2 k2 (Parallel
model)
Definition 3
1 þk2 2ðk1 k2 Þm2
(Maxwellke ¼ k1 2k
2k1 þk2 þðk1 k2 Þm2
Eucken 1 [13])
Definition 4
2 þk1 2ðk2 k1 Þð1m2 Þ
(Maxwellke ¼ k2 2k
2k2 þk1 þðk2 k1 Þð1m2 Þ
Two-phase flow
Viscosity (l)
1
x
lm ¼ 1x
(McAdams et al. [1])
l1 þ lg
lm ¼ ð1 xÞl1 þ xlg (Cicchitti et al. [2])
ll lg Þx
lm ¼ l1 22lll lþþllgg2ð
þðll lg Þx (generated by analogy)
lg ll Þð1xÞ
lm ¼ lg 22llggþþlll l2ð
þðlg ll Þð1xÞ (generated by analogy)
Eucken 2 [13])
Definition 5
—ke
—ke
þ m2 kk22þ2k
¼ 0 Effective
ð1 m2 Þ kk11þ2k
e
e
Medium Theory (EMT [14,15])
l
m
g
m
lm ¼ 1=4 ð3x 1Þlg þ ½3ð1 xÞ 1ll þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
½ð3x 1Þlg þ ð3f1 xg 1Þll 2 þ 8ll lg
(generated by analogy)
h
i
ll lg Þx
2lg þl1 2ðlg ll Þð1xÞ
lm ¼ 12 ll 22lll lþþllgg2ð
(arithmetic mean of definition 3 and 4)
þðll lg Þx þ lg 2lg þll þðlg ll Þð1xÞ
Definition 6
Comment on
all
definitions
l l
l l
ð1 xÞ l lþ2lm þ x l gþ2lm ¼ 0
They satisfy the following
conditions:
i. m2 = 0, ke = k1
ii. m2 = 1, ke = k2
They satisfy the following conditions:
i. x = 0, lm = ll
ii. x = 1, lm = lg
Please cite this article in press as: M.M. Awad, Y.S. Muzychka, Effective property models for homogeneous two-phase flows, Exp. Therm.
Fluid Sci. (2008), doi:10.1016/j.expthermflusci.2008.07.006
ARTICLE IN PRESS
M.M. Awad, Y.S. Muzychka / Experimental Thermal and Fluid Science xxx (2008) xxx–xxx
for materials in which the thermal conductivity of the continuous
phase is higher than the thermal conductivity of the dispersed
phase (kcont > kdisp) like foam or sponge. In this case, the heat flow
essentially avoids the dispersed phase. In the case of momentum
transport, this is akin to a bubbly flow, where the dominant phase
is the liquid. Definition 4 for two-phase viscosity is generated by
analogy to the effective thermal conductivity using Maxwell-Eucken 2 model [13]. Maxwell-Eucken 2 [13] is suitable for materials
in which the thermal conductivity of the continuous phase is lower
than the thermal conductivity of the dispersed phase (kcont < kdisp)
like particulate materials surrounded by a lower conductivity
phase. In this case, the heat flow involves the dispersed phase as
much as possible. In the case of momentum transport, this is akin
to droplet flow, where the dominant phase is the gas. Definition 5
for two-phase viscosity is generated by analogy to the effective
thermal conductivity using the Effective Medium Theory (EMT
[14,15]). The Effective Medium Theory (EMT [14,15]) is suitable
for the structure that represents a heterogeneous material in which
the two components are distributed randomly, with neither phase
being necessarily continuous or dispersed. In the case of momentum transport, this averaging scheme seems reasonable given the
unstable and random distribution of phases in a liquid/gas flow. Finally, Definition 6 for two-phase viscosity is based on the arithmetic mean of Maxwell-Eucken 1 [13] and Maxwell-Eucken 2 [13]
models. This is proposed here as a simple alternative to the Effective Medium Theory.
Figs. 2 and 3 show the analogy between thermal conductivity in
porous media [12] and viscosity in two-phase flow where Fig. 2
shows ke/k1 versus v2 [12] while Fig. 3 shows lm/ll versus x for
air–water system at atmospheric conditions.
These new definitions overcome the disadvantages of some definitions of two-phase viscosity such as Davidson et al.’s definition
[7], Owens’ definition [9] and García et al.’s definition [10] that do
not satisfy the condition at x = 1, lm = lg. These new definitions of
two-phase viscosity can be used to compute the two-phase frictional pressure gradient using the homogeneous modelling
approach.
Often, it is desirable to express the two-phase frictional pressure gradient, (dp/dz)f, versus the total mass flux (G) in a dimensionless form like the Fanning friction factor (fm) versus the
Reynolds number (Rem) as follows:
0
0.2
0.4
0.6
0.8
1
1
1
air-water
McAdams et al. [1]
Cicchitti et al. [2]
Maxwell–Eucken 1 [13]
Maxwell–Eucken 2 [13]
Effective Medium Theory
(EMT) equation [14, 15]
Definition 6
0.8
0.6
μ / μl
4
0.8
0.6
0.4
0.4
0.2
0.2
0
0
0
0.2
0.4
0.6
x
0.8
1
Fig. 3. lm/ll versus x for air–water.
fm ¼
qm ðdp=dzÞf d
ð18Þ
2G2
Eqs. (2) and (6) represent the two-phase density based on the
homogeneous model (qm) and Reynolds number based on the homogeneous model (Rem).
To satisfy a good agreement between the experimental data and
well-known friction factor models, assessment of the best definition of two-phase viscosity among the different definitions (old
and new) is based on the definition that corresponds to the minimum the root mean square (RMS) error.
The fractional error (e) in applying the model to each available
data point is defined as
Predicted Available
e¼
Available
ð19Þ
For groups of data, the root mean square error, eRMS, is defined as
"
N
1X
e2
N K¼1 K
eRMS ¼
#1=2
ð20Þ
The Fanning friction factor (fm) can be predicted using the Hagen–
Poiseuille flow [16] for laminar–laminar flow and the Blasius equation [8] for turbulent–turbulent flow as follows:
fm ¼ Re16m
Rem < 2300
0:079
fm ¼ Re
0:25
Rem > 4000
m
)
ð21Þ
For the case of minichannels and microchannels, the friction
factor is calculated using the Churchill model [17] that allows for
prediction over the full range of laminar-transition-turbulent regions. The Fanning friction factor (fm) can be predicted using the
Churchill model [17] as follows:
"
fm ¼ 2
8
Rem
12
þ
"
am ¼ 2:457 ln
bm ¼
Fig. 2. ke/kl versus v2 [12].
1
#1=12
ð22Þ
ðam þ bm Þ3=2
1
#16
ð7=Rem Þ0:9 þ ð0:27e=dÞ
16
37; 530
Rem
ð23Þ
ð24Þ
Please cite this article in press as: M.M. Awad, Y.S. Muzychka, Effective property models for homogeneous two-phase flows, Exp. Therm.
Fluid Sci. (2008), doi:10.1016/j.expthermflusci.2008.07.006
ARTICLE IN PRESS
5
M.M. Awad, Y.S. Muzychka / Experimental Thermal and Fluid Science xxx (2008) xxx–xxx
The Churchill model [17] is preferable since it encompasses all
Reynolds numbers and includes roughness effects in the turbulent
regime.
The total pressure drop is the sum of frictional, acceleration and
gravitational pressure drops. The proposed correlation gives us the
frictional term using the homogeneous model. The acceleration
term can be calculated as follows:
# "
#)
x2
ð1 xo Þ2
x2
ð1 xi Þ2
þ o þ i
ql ð1 ao Þ qg ao
ql ð1 ai Þ qg ai
100000
0.01
fm
("
Dpa ¼ G2
10000
0.01
Maxwell-Eucken 1 [13]
eRMS = 21.15%
ð25Þ
Blasius [8]
Bandel [18]
Hashizume (R12) [19]
Hashizume (R22) [19]
Müller-Steinhagen [20]
The acceleration term is negligible in adiabatic channels. The
gravitational term can be calculated using the homogeneous model
as follows:
dp
g sin h
¼
dz grav;m qx þ 1x
q
g
0.001
0.001
10000
ð26Þ
Rem
100000
l
Fig. 6. fm versus Rem in circular pipes using definition 3 of two-phase viscosity.
From Eq. (26), it is clear that the gravitational term equals zero in
horizontal flow (h = 0).
10000
100000
0.01
3. Results and discussion
0.01
fm
Comparisons of the two-phase frictional pressure gradient versus mass flux from published experimental studies in circular
pipes, minichannels and microchannels are undertaken, after
expressing the data in dimensionless form as Fanning friction facMaxwell-Eucken 2 [13]
eRMS = 24.13%
10000
Blasius [8]
Bandel [18]
Hashizume (R12) [19]
Hashizume (R22) [19]
Müller-Steinhagen [20]
100000
0.01
0.01
0.001
0.001
10000
Rem
100000
fm
Fig. 7. fm versus Rem in circular pipes using definition 4 of two-phase viscosity.
McAdams et al. [1]
eRMS = 31.91%
Blasius [8]
Bandel [18]
Hashizume (R12) [19]
Hashizume (R22) [19]
Müller-Steinhagen [20]
0.001
10000
100000
0.01
0.01
0.001
Rem
100000
Fig. 4. fm versus Rem in circular pipes using definition 1 of two-phase viscosity.
fm
10000
Effective Medium Theory (EMT [14,15])
eRMS = 19.99%
10000
Blasius [8]
Bandel [18]
Hashizume (R12) [19]
Hashizume (R22) [19]
Müller-Steinhagen [20]
100000
0.01
0.01
0.001
0.001
10000
Rem
100000
fm
Fig. 8. fm versus Rem in circular pipes using definition 5 of two-phase viscosity.
Cicchitti et al. [2]
eRMS = 22.92%
tor versus Reynolds number. The published data include different
working fluids such as R-12, R-22, Argon (R740), R717, R134a,
R410A and Propane (R290) at different diameters and different saturation temperatures.
Blasius [8]
Bandel [18]
Hashizume (R12) [19]
Hashizume (R22) [19]
Müller-Steinhagen [20]
0.001
0.001
10000
Rem
3.1. Data for circular pipes
100000
Fig. 5. fm versus Rem in circular pipes using definition 2 of two-phase viscosity.
Figs. 4–9 show the Fanning friction factor (fm) versus Reynolds
number (Rem) for data obtained on circular pipes, using the six
Please cite this article in press as: M.M. Awad, Y.S. Muzychka, Effective property models for homogeneous two-phase flows, Exp. Therm.
Fluid Sci. (2008), doi:10.1016/j.expthermflusci.2008.07.006
ARTICLE IN PRESS
6
M.M. Awad, Y.S. Muzychka / Experimental Thermal and Fluid Science xxx (2008) xxx–xxx
10000
1000
100000
0.01
10000
100000
0.1
0.1
0.01
McAdams et al. [1]
eRMS = 20.76%
fm
fm
Churchill [17]
Ungar and Cornwell [21]
Tran et al.[ 22]
Cavallini et al. [23]
Field and Hrnjak [24]
0.01
0.01
Average of Maxwell-Eucken 1 and 2 [13]
eRMS = 20.06%
Blasius [8]
Bandel [18]
Hashizume (R12) [19]
Hashizume (R22) [19]
Müller-Steinhagen [20]
0.001
0.001
0.001
10000
Rem
eRMS (%)
McAdams et al. [1]
Cicchitti et al. [2]
Maxwell-Eucken 1 [13]
Maxwell-Eucken 2 [13]
Effective medium theory (EMT [14,15])
Arithmetic mean of Maxwell-Eucken 1 and 2 [13]
31.91
22.92
21.15
24.13
19.99
20.06
6000
8000
16000
McAdams et al. [1]
Cicchitti et al. [2]
Maxwell-Eucken 1 [13]
Maxwell-Eucken 2 [13]
Effective Medium Theory (EMT [14,15])
Arithmetic Mean of Maxwell-Eucken 1 and 2 [13]
0%
+30%
-30%
(dp/dz)f, predicted
12000
8000
4000
1000
10000
100000
0.1
0.1
Cicchitti eta l. [2]
eRMS = 31.06%
Churchill[ 17]
Ungar and Cornwell [21]
Tran et al.[ 22]
Cavallini et al. [23]
Field and Hrnjak [24]
fm
Definition
4000
100000
Fig. 11. fm versus Rem in minichannels and microchannels using definition 1 of twophase viscosity.
Table 2
eRMS% Values Based on Measured Fanning Friction Factor and Predicted Fanning
Friction Factor in Circular Pipes Using Different Definitions of Two-Phase Viscosity
2000
10000
Rem
Fig. 9. fm versus Rem in circular pipes using definition 6 of two-phase viscosity.
0
0.001
1000
100000
0.01
0.01
10000
16000
0.001
1000
12000
8000
0.001
100000
10000
Rem
Fig. 12. fm versus Rem in minichannels and microchannels using definition 2 of twophase viscosity.
1000
4000
10000
100000
0.1
0.1
Maxwell-Eucken 1 [13]
eRMS = 24.78%
0
0
2000
4000
6000
8000
Churchill [17]
Ungar and Cornwell [21]
Tran et al.[ 22]
Cavallini et al. [23]
Field and Hrnjak [24]
0
10000
Fig. 10. Predicted frictional pressure gradient versus measured frictional pressure
gradient in circular pipes.
different definitions of two-phase viscosity shown in Table 1 on
log–log scale. The sample of the published data includes Bandel’s
data [18] for R12 flow at x = 0.3 and Ts = 0 °C in a smooth horizontal
pipe at d = 14 mm, Hashizume’s data [19] in a smooth horizontal
pipe at d = 10 mm for R12 flow at x = 0.5 and Ts = 39 °C and R 22
flow at x = 0.5 and Ts = 20 °C, and Müller-Steinhagen’s data [20]
for argon (R740) flow at x = 0.3 and reduced pressure of 0.188 in
a smooth horizontal pipe at d = 14 mm. Eq. (18) represents the
measured Fanning friction factor while Eq. (21) represents the predicted Fanning friction factor. From Table 2, eRMS% values based on
measured Fanning friction factor and predicted Fanning friction
factor using the six different definitions of two-phase viscosity
for this sample of the published data, it can be seen that two-phase
fm
(dp/dz)f,measured
0.01
0.01
0.001
1000
10000
0.001
100000
Rem
Fig. 13. fm versus Rem in minichannels and microchannels using definition 3 of twophase viscosity.
viscosity based on Effective Medium Theory (EMT [14,15]) gives
the best agreement between the published data and the Blasius
equation [8] with the root mean square error (eRMS) of 19.99%.
Please cite this article in press as: M.M. Awad, Y.S. Muzychka, Effective property models for homogeneous two-phase flows, Exp. Therm.
Fluid Sci. (2008), doi:10.1016/j.expthermflusci.2008.07.006
ARTICLE IN PRESS
7
M.M. Awad, Y.S. Muzychka / Experimental Thermal and Fluid Science xxx (2008) xxx–xxx
1000
10000
100000
0.1
0.1
1000
10000
Maxwell-Eucken 2 [13]
eRMS = 16.47%
Average of Maxwell-Eucken 1 and 2 [13]
eRMS = 17.98%
0.01
Churchill [17]
Ungar and Cornwell [21]
Tran et al.[ 22]
Cavallini et al. [23]
Field and Hrnjak [24]
fm
fm
Churchill[ 17]
Ungar and Cornwell [21]
Tran et al.[ 22]
Cavallini et al. [23]
Field and Hrnjak [24]
0.01
0.001
1000
0.01
0.01
0.001
0.001
100000
10000
1000
10000
Fig. 14. fm versus Rem in minichannels and microchannels using definition 4 of twophase viscosity.
10000
Effective Medium Theory (EMT[ 14,15])
eRMS = 23.60%
Churchill [17]
Ungar and Cornwell [21]
Tran et al.[ 22]
Cavallini eta l. [23]
Field and Hrnjak [24]
0.01
Fig. 16. fm versus Rem in minichannels and microchannels using definition 6 of twophase viscosity.
Table 3
eRMS% values based on measured fanning friction factor and predicted fanning friction
factor in minichannels and microchannels using different definitions of two-phase
viscosity
100000
0.1
0.1
fm
0.001
100000
Rem
Rem
1000
100000
0.1
0.1
0.01
Definition
eRMS (%)
McAdams et al. [1]
Cicchitti et al. [2]
Maxwell-Eucken 1 [13]
Maxwell-Eucken 2 [13]
Effective medium theory (EMT [14,15])
Arithmetic mean of Maxwell-Eucken 1 and 2 [13]
20.76
31.06
24.78
16.47
23.60
17.98
3.2. Data for minichannels and microchannels
0.001
1000
0.001
100000
10000
Re m
Fig. 15. fm versus Rem in minichannels and microchannels using definition 5 of twophase viscosity.
The data in the current dimensionless form as fm versus Rem
(Figs. 4–9) can be presented as predicted frictional pressure gradient versus measured frictional pressure gradient as shown in Fig.
10.
0
200000
Figs. 11–16 show the Fanning friction factor (fm) versus
Reynolds number (Rem) in minichannels and microchannels using
the six different definitions of two-phase viscosity shown in Table
1 on log–log scale. The sample of the published data includes Ungar and Cornwell’s data [21] for R717 flow at Ts 74 °F (165.2 °C)
in a smooth horizontal tube at d = 0.1017 in. (2.583 mm), Tran et
al.’s data [22] for R134a flow at ps = 365 kPa and x 0.73 in a
smooth horizontal pipe at d = 2.46 mm, Cavallini et al.’s data [23]
for R410A flow at Ts = 40 °C and x = 0.74 in smooth multi-port
minichannels at hydraulic diameter of 1.4 mm, and Field and
400000
600000
(dp/dz)f, predicted
1200000
800000
McAdams et al. [1]
Cicchitti et al. [2]
Maxwell-Eucken 1 [13]
Maxwell-Eucken 2 [13]
Effective Medium Theory (EMT[ 14,15])
Arithmetic Mean of Maxwell-Eucken 1 and 2 [13]
0%
+30%
-30%
800000
400000
400000
0
0
800000
1200000
200000
400000
600000
0
800000
(dp/dz)f, measured
Fig. 17. Predicted frictional pressure gradient versus measured frictional pressure gradient in minichannels and microchannels.
Please cite this article in press as: M.M. Awad, Y.S. Muzychka, Effective property models for homogeneous two-phase flows, Exp. Therm.
Fluid Sci. (2008), doi:10.1016/j.expthermflusci.2008.07.006
ARTICLE IN PRESS
8
M.M. Awad, Y.S. Muzychka / Experimental Thermal and Fluid Science xxx (2008) xxx–xxx
Hrnjak data [24] for propane (R290) flow at reduced pressure of
0.23 and G 330 kg/m2 s in a smooth horizontal pipe at hydraulic
diameter of 0.148 mm. Eq. (18) defines the measured Fanning friction factor while Eqs. (22)–(24) represent the predicted Fanning
friction factor. Table 3 presents eRMS% values based on measured
Fanning friction factor and predicted Fanning friction factor using
the six different definitions of two-phase viscosity for this sample
of the published data. It can be seen that two-phase viscosity based
on Maxwell-Eucken 2 model [13] gives the best agreement between the published data and the Churchill model [17] with the
root mean square error (eRMS) of 16.47%. It can be seen from Fig.
14 and Table 3, that the definition of effective viscosity based on
the Maxwell Eucken 2 model [13] appears to be more appropriate
for defining two-phase flow viscosity in minichannels, and microchannels. On the basis of the data considered, a nominal 5–6% gain
in accuracy can be achieved using the homogeneous flow modeling
approach. When one considers the nature of the Maxwell-Eucken 2
definition, whereby the dominant phase is the lower viscosity
phase, i.e. the gas, and considering the context of Fig. 1, it is clear
that this definition is most appropriate for liquid/gas mixtures
which have very high density ratios. Thus, even for small mixture
qualities, a significant portion of the flow volume is occupied by
gas, making the Maxwell-Eucken 2 definition most appropriate.
The data in the current dimensionless form as fm versus Rem
(Figs. 11–16) can be presented as predicted frictional pressure gradient versus measured frictional pressure gradient as shown in Fig.
17.
4. Summary and conclusions
Using the analogy between thermal conductivity in porous
media and viscosity in two-phase flow, new definitions for twophase viscosity are given in Table 1 (Definitions 3, 4, 5 and 6,
respectively). These new definitions for two-phase viscosity satisfy
the following two conditions: namely (i) lm = ll at x = 0 and (ii)
lm = lg at x = 1. These new definitions of two-phase viscosity can
be used to compute the two-phase frictional pressure gradient
using a homogeneous modelling approach. Expressing two-phase
frictional pressure gradient in dimensionless form as Fanning friction factor versus Reynolds number is also desirable in many applications. The models are verified using published experimental data
of two-phase frictional pressure gradient in circular pipes, minichannels and microchannels after expressing it in a dimensionless
form as Fanning friction factor versus Reynolds number. The published data include different working fluids such as R-12, R-22, Argon (R740), R717, R134a, R410A and Propane (R290) at different
diameters and different saturation temperatures. To satisfy a good
agreement between the experimental data and well-known friction factor models such as the Blasius equation and the Churchill
model, selection of the best definition of two-phase viscosity is
based on the definition that corresponds to the minimization of
the root mean square error (eRMS). From eRMS% values based on
measured Fanning friction factor and predicted Fanning friction
factor using the six different definitions of two-phase viscosity, it
is shown that the new definitions of two-phase viscosity gives
the best agreement between the experimental data and wellknown friction factor models in circular pipes, minichannels and
microchannels. As a result, the new definition of two-phase viscosity can be used to analyze the experimental data of two-phase frictional pressure gradient in circular pipes, minichannels and
microchannels using the homogeneous model.
Acknowledgments
The authors acknowledge the financial support of Natural Sciences and Engineering Research Council of Canada (NSERC). Also,
the authors gratefully acknowledge ASME International Petroleum Technology Institute (IPTI) scholarship awarded to M.M.
Awad.
References
[1] W.H. McAdams, W.K. Woods, L.C. Heroman, Vaporization inside horizontal
tubes II-benzene-oil mixtures, Trans. ASME 64 (3) (1942) 193–200.
[2] A. Cicchitti, C. Lombaradi, M. Silversti, G. Soldaini, R. Zavattarlli, Two-phase
cooling experiments – pressure drop heat transfer burnout measurements,
Energia Nucleare 7 (6) (1960) 407–425.
[3] A.E. Dukler, Wicks Moye, R.G. Cleveland, Frictional pressure drop in two-phase
flow. Part A: a comparison of existing correlations for pressure loss and holdup,
and Part B: an approach through similarity analysis, AIChE J. 10 (1) (1964) 38–
51.
[4] D.R.H. Beattie, P.B. Whalley, Simple two-phase frictional pressure drop
calculation method, Int. J. Multiphase Flow 8 (1) (1982) 83–87.
[5] S. Lin, C.C.K. Kwok, R.Y. Li, Z.H. Chen, Z.Y. Chen, Local frictional pressure drop
during vaporization for R-12 through capillary tubes, Int. J. Multiphase Flow 17
(1) (1991) 95–102.
[6] M. Fourar, S. Bories, Experimental study of air–water two-phase flow
through a fracture (narrow channel), Int. J. Multiphase Flow 21 (4) (1995)
621–637.
[7] W.F. Davidson, P.H. Hardie, C.G.R. Humphreys, A.A. Markson, A.R. Mumford, T.
Ravese, Studies of heat transmission through boiler tubing at pressures from
500–3300 Lbs, Trans. ASME 65 (6) (1943) 553–591.
[8] H. Blasius, Das Ähnlichkeitsgesetz bei Reibungsvorgängen in Flüssikeiten,
Forsch. Gebiete Ingenieurw 131 (1913).
[9] W.L. Owens, Two-phase pressure gradient, ASME Int. Develop. Heat Transf.
Part II (1961) 363–368.
[10] F. García, R. García, J.C. Padrino, C. Mata, J.L. Trallero, D.D. Joseph, Power law
composite power law friction factor correlations for laminar turbulent gas–
liquid flow in horizontal pipelines, Int. J. Multiphase Flow 29 (10) (2003)
1605–1624.
[11] J.G. Collier, J.R. Thome, Convective Boiling and Condensation, third ed.,
Claredon Press, Oxford, 1994.
[12] J.K. Carson, S.J. Lovatt, D.J. Tanner, A.C. Cleland, Thermal conductivity bounds
for isotropic, porous materials, Int. J. Heat Mass Transfer 48 (11) (2005) 2150–
2158.
[13] Z. Hashin, S. Shtrikman, A variational approach to the theory of the effective
magnetic permeability of multiphase materials, J. Appl. Phys. 33 (1962) 3125–
3131.
[14] R. Landauer, The electrical resistance of binary metallic mixtures, J. Appl. Phys.
23 (1952) 779–784.
[15] S. Kirkpatrick, Percolation and conduction, Rev. Mod. Phys. 45 (1973) 574–588.
[16] F.M. White, Viscous Fluid Flow, third ed., McGraw-Hill Book Co., 2005 (Chapter
3).
[17] S.W. Churchill, Friction factor equation spans all fluid flow regimes, Chem. Eng.
84 (24) (1977) 91–92.
[18] J. Bandel, Druckverlust und Wärmeübergang bei der Verdampfung Siedender
Kältemittel im Durchströmten Waagerechten Rohr, Doctoral Dissertation,
Universität Karlsruhe, 1973.
[19] K. Hashizume, Flow pattern, void fraction and pressure drop of refrigerant
two-phase flow in a horizontal pipe. Part I: experimental data, Int. J.
Multiphase Flow 9 (4) (1983) 399–410.
[20] H. Müller-Steinhagen, Wärmeübergang ünd fouling beim strömungssieden
von argon ünd stickstoff im horizontalen rohr, Fortschr. Ber. VDI Z. 6 (1984).
[21] E.K. Ungar, J.D. Cornwell, Two-phase pressure drop of ammonia in small
diameter horizontal tubes, in: Seventeenth AIAA Aerospace Ground Testing
Conference, AIAA 92-3891, Nashville, TN, 1992.
[22] T.N. Tran, M.-C. Chyu, M.W. Wambsganss, D.M. France, Two-phase pressure
drop of refrigerants during flow boiling in small channels: an experimental
investigation and correlation development, Int. J. Multiphase Flow 26 (11)
(2000) 1739–1754.
[23] A. Cavallini, D. Del Col, L. Doretti, M. Matkovic, L. Rossetto, C. Zilio, Two-phase
frictional pressure gradient of R236ea, R134a and R410A inside multi-port
mini-channels, Exp. Therm. Fluid Sci. 29 (7) (2005) 861–870.
[24] B.S. Field, P. Hrnjak, 2007, Adiabatic two-phase pressure drop of refrigerants in
small channels, Heat Transf. Eng. 28 (8–9) (2007) 704–712. (Also presented at
The Fourth International Conference on Nanochannels, Microchannels and
Minichannels (ICNMM 2006), Session: Two-Phase Flow, Experiments in
Minichannels, ICNMM2006-96200, Stokes Research Institute, University of
Limerick, Ireland.)
Please cite this article in press as: M.M. Awad, Y.S. Muzychka, Effective property models for homogeneous two-phase flows, Exp. Therm.
Fluid Sci. (2008), doi:10.1016/j.expthermflusci.2008.07.006