7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
Phenomenological Heat Transfer Model of Micro TwoPhase Slug Flow without
Phase Change
Y. Hasegawa, N. Kasagi and A. Yamamoto
Department of Mechanical Engineering, The University of Tokyo, Hongo 731, Bunkyoku, Tokyo 1138656, Japan
Keywords: micro twophase flow, heat transfer, modelling
Abstract
We investigate the heat transfer and pressure loss characteristics of gasliquid slug flow in a micro tube. Experimental
results suggest that injection of slight amount of gas into liquid flow enhances heat transfer with favorable pressure
penalty. As a result, the heat transfer per unit pumping power becomes larger than that in the singlephase laminar
flow. Numerical results clearly show that a circulation generated in the liquid slug plays an important role in enhancing
the heat transfer. In order to predict the effect of different flow patterns on the heat transfer, we model the overall heat
transfer as onedimensional unsteady heat conduction inside the liquid film with a timedependent heat transfer rate
between the film and the slug regions. It is found that the present model predicts the overall heat transfer rate fairly
well under a wide rage of flow patterns
Nomenclature
Roman symbols
Cp thermal capacity, [J/kg K]
D tube diameter, [m]
h heat transfer coefficient, [W/K m2]
j superficial velocity, [m/s]
L longitudinal length, [m]
Nu Nusselt number
p pressure, [Pa]
Pe thermal Peclet number,pLUCpLD/AL
Pr Prandtl number
q heat flux, [W/m2]
r radial direction, [m]
R tube radius, [m]
Re Reynolds number, PLUD/hIL
t time, [s]
T temperature, [K]
u velocity
We Weber number, PLU2D/ca
z longitudinal direction, [m]
Greek symbols
a thermal diffusion coefficient [m2/s]
3 volumetric flow ratio
6 residual film thickness, [m]
C void fraction
a surface tension coefficient, [N/m]
p viscosity, [Pa s]
v kinematic viscosity, [m2/s]
A thermal conductivity, [W/m K]
7 dimensionless stream function
p density,[kg/m3]
Subscripts
g gas phase
1 liquid phase
TP twophase
Introduction
Gasliquid twophase flow without phase change is a
possible way of heat transfer enhancement for compact
heat exchangers. The presence of gas bubbles separat
ing discrete liquid slugs causes a circulation inside the
liquid phase so that the overall heat transfer is enhanced.
Moreover, such gasliquid flows are rather stable due to
absence of explosive boiling. These facts open up a pos
sibility of achieving better heat transfer with moderate
pressure penalty. The structure and behavior of such
towphase flow, however, are inherently complex, so that
basic understanding the flow and heat transfer mecha
nisms is essential.
A number of experimental visualizations have been
carried out in order to clarify the flow characteristics
of adiabatic gasliquid twophase flow in micro tubes
(Tripletter et al. 1999; Kawahara et al., 2002; Serizawa
et al., 2002.). Recently, it has been shown that the flow
pattern in a micro tube is not uniquely determined with
the flow rates of gas and liquid, but strongly depends on
the inlet condition. Once a flow pattern is reached down
stream of the inlet, it remains unchanged even far down
stream (Hayashi et al., 2007; Kawaji et al., 2009). These
facts suggest a possibility to control the flow pattern so
as to achieve favorable heat transfer characteristics by
modifying the inlet condition. However, there exist few
studies conduct simultaneous measurement of the heat
transfer and pressure loss characteristics in these flows.
One of the most fundamental problems is to predict
the heat transfer rate based on the flow parameters, such
as the slug lengths and flow rates of gas and liquid. A
typical phenomenological model is the threezone flow
boiling model proposed by Thome et al. (2003). Al
though this model has been successfully applied to boil
ing and condensation in a micro tube, there exists no
model applicable to gasliquid slug flow without phase
change.
In the present work, the pressure loss and heat trans
fer characteristics of gasliquid two phase flow without
phase change in a micro tube are measured. In order
to study the detailed mechanisms of heat transfer, we
also carry out a series of numerical simulations. Base
on the numerical results, we decompose the whole flow
field into an adherent liquid film near the wall and the
gasliquid slug regions. It is shown that the transient
temperature fluctuation inside the liquid film is a key
to predict the overall heat transfer rate. We analyze a
onedimensional transient heat conduction problem in
side the liquid film, and then develop a heat transfer
model as a function of parameters representing the flow
pattern. Finally, we verify the present model through
comparison with the numerical results.
Experimental Measurement
Experimental Setup. The schematic figure of the
present experimental setup is shown in Fig. 1. Water
is used as a continuous phase, while gaseous nitrogen
as a dispersed phase. The liquid flow rate and the gas
flow rate are controlled by a twin plunge pump (GL sci
ences, PU714) and a mass flow controller (Yamatake
Co., CMQV 5 ml/min), respectively. In order to visual
ize flow patterns, a micro glass tube with the inner and
outer diameter of 500 and 800 pm is used in the test sec
tion. The gas and liquid phases are mixed through coax
ial tubes at the inlet as shown in Fig. 2. A fine stainless
steel tube is inserted into the micro glass tube. The in
ner and outer diameter of the inner tube are 300 and 400
pm, respectively. Nitrogen is introduced from the inner
pipe, while water from the outer annular section. Table 1
shows the experimental conditions. In the present study,
the gas superficial velocity j, is systematically changed
from 0.02 to 0.3 m/s while the liquid superficial veloc
ity ji is kept constant. Throughout this manuscript, the
subscripts of I and g represent values in the liquid and
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
Manometer Cold Light Voltage supplier
V
Water / I
Nitrogen ,
Thermocouples
High speed camera
(Phantom V5)
Figure 1: Experimental Setup.
Nitroaen
Pressure port
n'?a
Figure 2: Inlet Condition.
gas phases, respectively. Two different ji, i.e., ji 0.06
and 0.2 m/s are considered. It was confirmed that the
flow patterns in the above conditions are periodic gas
liquid slug flows, and the gas and liquid slug lengths are
kept constant in each case. Under these conditions, we
measure the gas and liquid slug lengths, heat transfer
rate and pressure loss.
A mixture of ITO and silver is uniformly sputtered on
the outer surface of the glass tube for the heating ele
ment. This enable us to visualize the flow pattern with
a highspeed camera (Vision Research, Phantom v5).
Each image has 1024 x 128 pixels and the frame rate
was 1000~4000 frames/sec. From visualization images,
we estimate the gas and liquid slug lengths. The outer
wall temperature Twloa t is measured with Ktype ther
mocouples, calibrated with the accuracy of 0.1 K. By
solving onedimensional steady heat conduction prob
lem inside the tube wall, we estimate the inner wall tem
perature Twall, From the heat flux q and Twai, the
twophase heat transfer rate is obtained as follows:
hTP Ta Tbulk
Twall, Tb~u~l
We also measure the pressure at the inlet and the out
let in order to obtain the pressure loss within the test
section. The inlet pressure loss is estimated in a single
phase flow experiment and this value is subtracted from
the obtained pressure loss in the twophase flow experi
ments.
Results. In Fig. 3, the heat transfer rate, the pressure
loss and the heat transfer per unit pumping power are
shown when j, is systematically increased while jl is
kept constant. The horizontal axis represents the void
I L~
1
I I
I
Table 1: Experimental Conditions.
Inner Outer Liquid Gaseous Heat
Path Path Superficial Superficial Flux
Velocity Velocity
[m/s] [m/s] [kW/m2]
Water Nitrogen 0.06, 0.2 0.020.3 2762
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
flow with heat transfer in a cylindrical pipe. It is
assumed that each phase is incompressible and phase
change does not take place. The temperature is consid
ered as a passive scalar. The gravity is neglected due to
dominance of the surface tension. The interface is cap
tured by using the PhaseField method (Anderson et al.,
1998). The dimensionless governing equations are given
as follows:
V = 0,
J V(pJ)= Vp
+ V [j(ViJ + VJT)]
ReCT
9(pCpT) Vp
8 + & V(pC T) =
')CVf
(3)
Cn We'
V* AVT, (4)
PeC
where, p, p, C, and A denote the density, viscosity, ther
mal capacity and conductivity, respectively, and all these
quantities are normalized by the values of liquid.
The dimensionless parameters appear in Eqs. (34) are
defined as:
P*U*2pD*
0.0 0.2 0.4 0.6 0.8 1.0
Figure 3: Performances of gasliquid two phase flow
fraction c jg/(j, + ji), while the heat transfer rate,
the pressure loss and the heat transfer per unit pumping
power are all normalized by the quantities of the single
phase flow, i.e., c 0.
It is found that with increasing c, the liquid slug length
monotonically decreases, while the gas slug length in
creases. In addition, the effect of ji is minor so that
the flow pattern is mainly determined by c. In accor
dance with the change of the flow pattern, the heat trans
fer rate also monotonically decreases with increasing
c, while the pressure loss increases drastically. Conse
quently, the heat transfer per unit pumping power mono
tonically decreases with c. We note that the heat transfer
per unit pumping power becomes larger than that in a
single phase flow when the void fraction is small, i.e.,
0 < c < 0.3. This suggests that the heat transfer char
acteristics is improved from those in a single phase flow
by introducing small amount of gas in the liquid flow.
Numerical Simulation
Numerical Method. In order to study the effects of
a flow pattern on the heat transfer characteristics, we
conduct numerical simulations of a gasliquid twophase
pR U pD*
PeTp R=TpPr
Pe, = ReTpPr, = '
P* CP
*U}pD* ,
A Pr t =
1 1a1
where a value with an asterisk denotes a dimensional
quantity, while D* is the tube diameter. The thermal
diffusion coefficient is denoted by a* A*/p* C. The
characteristic velocity U>p is defined as the sum of su
perficial liquid and gas velocities, i.e., Up, = j* + j7.
Equations of (2) (4) are satisfied both in gas and liq
uid phases, and the local fluid properties of p, p, A and
C, are interpolated between those of gas and liquid ac
cording to the position of interface.
The flow is assumed to be periodic with constant gas
and liquid slug lengths. Therefore, only one period of
the flow is simulated with a pair of gas and liquid slugs.
In addition, the flow is assumed axisymmetric, so that a
twodimensional (r z) computational domain is em
ployed. The periodic length L = L* /R* defines the
computation domain, where R* is the tube radius. In
the present study, Lz is changed as 3 < Lz < 15 with
different gas and liquid slug lengths. A periodic bound
ary condition is applied at the two ends of the computa
tional domain, while the noslip and fully wetted bound
ary conditions are used on the tube wall. For the tem
perature field, a uniform heat flux q is assumed along
the wall. Because only the temperature difference is of
interest, a quasiperiodic boundary condition, namely,
aTz a z =L, (6)
az z az L0 (6)
1   
ti~ 'Ifrn'f.ldh l"Cohf er
iCMF 201:,. Tampa, FL
0.5 ::::::::;^::^:  .: .^ ^
 t i
  A
1
7I
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0
z /R
Figure 4: Bubble shape and relative streamlines at c =
0.29 and ReTp = 288. thick solid line: gasliquid inter
face, thick broken line: dividing streamline between an
adherent liquid film and gasliquid slugs
is applied on the both ends of the computation domain.
In accordance with the present experiment conditions,
water and Nitrogen at 293 K and 1 atm are assumed
as working fluids, and the surface tension o* is 0.0728
N/m. The tube diameter D* is fixed at 600pm, and
the characteristic velocity U*p is given as 0.03~ 1.5m/s.
This range covers both bubbly and slug flows according
to experimental observation (Hayashi et al., 2007). The
Prandtl number in liquid is Prl = 6.96.
Results. The contours of the dimensionless stream
function 7 relative to the bubble motion in a typical
case, where Lz = 6.0 and the void fraction c = 0.29
are shown in Fig. 4. Here, 7 is defined as:
= Uz Ububb
r Or r 9z
u, (7)
where Ububb is the velocity of the moving bubble. As
illustrated in Fig. 4, a large anticlockwise circulation is
found inside the gas phase between smaller clockwise
circulations in the front and rear of a gas bubble. The
circulation can also be found in the liquid region. The
present result agrees with the sketch by Taylor (1961)
and the visualization by Thulasidas et al. (1997).
Comparison between the Nusselt numbers obtained
by simulation and experiment under the same flow rates
and slug lengths of gas and liquid is shown in Fig. 5. Al
though qualitative agreement is confirmed, quantitative
agreement between the simulation and the experiment is
not obtained in the present study. This can be mainly at
tributed to a measurement error in the experiment. Espe
cially, when the heat transfer rate is high, the difference
between the innerwall temperature and the bulk tem
perature becomes quite small so that high accuracy in
the temperature measurement is required. This remains
as future work.
Heat Transfer Modelling
Basic Concept. The present heat transfer model is
schematically shown in Figs. 6 a, b). The whole domain
is modelled as an adherent liquid film with alternatively
an fr rate is h the ffren
1 tempeatue and the bulk tem, UcIM
Heat Transfer Modelling
1.0. I I3 ... ....... 
J9nhttrnfrPe h
L
L L L
Lbubz l h uhh a lbib
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
heat transfer rate h*p from the bottom wall to the bulk
fluid should be correlated with h*1Ug, t,, t* and 6*.
Dimensionless parameters The energy equation (9)
are normalized as follows:
OT 2T
=Fo
at dy2'
(13)
q = h(T.,, T),,)
Constant heat flux q
wall
Figure 6: Conceptual figures of the present heat transfer
model a) Decomposition of the entire flow field; b) Heat
transfer modelling inside the adherent liquid film.
stant heat flux condition is applied:
dT*
A* =q* = const. at y* = 0. (10)
09y*
At the interface between the slug and film regions, a
thermal boundary condition of the third kind is assumed:
dT*
09y*
h" (t)T* at y* = .
Note that we shift the temperature so that the bulk tem
perature is zero so that T* represents the temperature
deviation form the bulk temperature. Here, hi is the in
terfacial heat transfer rate from the liquid film to the slug
region.
Considering that the heat capacity and conductivity
of the gas phase are quite small, hi should be negligible
when a gas bubbule passes above the liquid film. In ad
dition, assuming that the heat transfer rate hsg of the
liquid slug is uniform along the streamwise direction,
hi (t) can be modelled as:
h (t)
h (t)
h*,, 0 < t* < t*
0, t < t*
Here, t* is the entire period, while t* is the duration
in which the liquid slug passes above the liquid film.
Hence, t* t* + t*, where t* is the gas slug passage
duration. In the slug region, we assume that the gas and
liquid slugs travel at the same velocity. Therefore, t*
and t* are given by L*lug/UIt and L;ubble/UI P, re
spectively. Now, the major task is to find how the total
where the temperature T, time t and distance from
the bottom wall y are normalized by AT* = q*6*/A*,
t* and 6*, respectively. The Fourier number is defined
as Fo = a*t*/6*2.
Similarly, the boundary conditions of Eqs. (10) and
(11) are respectively normalized as:
= 1 at y 0,
Bihi(t)T at y 1.
Here, Bi is the Biot number defined as Bi
and hi is normalized by its mean value as hi
so that:
hi(t) = 3, 0 < t < .1,
hi(t)= O, 31 < t < 1.
(14)
(15)
hi6*/A*
= hl/h*
(16)
Here, t* /t* is replaced with 31, since we assume that the
gas and liquid flow in the slug region is homogeneous.
From the above, the heat transfer rate is governed by
the three dimensionless parameters, i.e., Bi, Fo and 31.
TwoLayer Heat Conduction Model. It was observed
that the temperature fluctuation mainly occurs in the
nearinterface region, i.e., 1 It < y < 1, while
the temperature is almost steady in the outer region,
i.e., 0 < y < 1 It. Here, the thermal penetration
thickness rt can be estimated by the liquidslug pas
sage time 31 and the dimensionless conductivity Fo as
r]t = a/Fo1, where a ~ 0.55.
In view of the above result, we further divide the liq
uid film into two layers. In the first layer adjacent to
the interface, the temperature temporally fluctuates in
response to hi. In the second layer, the temperature is
assumed to be steady. Eventually, the Nusselt number
Nu model predicted in the present model is given by:
Nufmodel 
T(O)
132 Fop3
Bi 1t \
1
exp(_ BFo)
77t
+1}l
For detailed derivation of the above equation, please re
fer to He et al. (2010). The comparison between the
1 10 ,
S* / "25%
I 8
e 6
5 /
2/
0 2 4 6 8 10 12 14
NuT, (Predition)
Figure 7: Comparison between the numerical simula
tion and the model prediction
model predictions and the numerical results is shown in
Fig. 7. The Nusselt number obtained in the present sim
ulation varies from 2.85 to 10.25 depending on the flow
pattern. It is confirmed that the present model predict
the numerical data fairly well under a wide rage of flow
patterns.
Conclusions
We investigate heat transfer and pressure loss character
istics in micro gasliquid two phase flow. It was found
that injection of small amount of gas into liquid flow
enhances heat transfer with less increase of pressure
penalty. As a result, the heat transfer per unit pumping
power is enhanced in the two phase flow compared to
the single phase flow. In order to study the mechanisms
of heat transfer, we also conducted a series of numeri
cal simulations. It was found that the Nusselt number
strongly depends on the flow pattern, and is upt to 2.4
times higher than that of the singlephase laminar flow.
The streamline function in the entire flow domain
shows that the gasliquid slug flow is generally charac
terized by an adherent liquid film with alternate passage
of gas and liquid slugs above the film. In order to pre
dict the effects of the flow pattern on the heat transfer,
we model the overall heat transfer as onedimensional
unsteady heat conduction inside the liquid film with a
timedependent heat transfer rate between the film and
slug regions.
The present model is useful in optimizing the flow
pattern so as to achieve the highest heat transfer perfor
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
mance with the minimum pressure penalty. The opti
mization of the flow pattern remains as future work.
Acknowledgements
The authors are grateful to Dr. Q. He for providing his
valuable numerical data.
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