Microchannels and Minichannels - 2004
June 17-19, 2004, Rochester, New York, USA
Copyright © 2004 by ASME
ICMM2004 – xxxx
NUMERICAL SIMULATION OF GROWTH OF A VAPOR BUBBLE DURING FLOW
BOILING OF WATER IN A MICROCHANNEL
Abhijit Mukherjee1 and Satish G. Kandlikar2
Rochester Institute of Technology, Rochester, NY, USA
1
Email: [email protected], 2Email: [email protected]
ABSTRACT
The present study is performed to numerically analyze
growth of a vapor bubble during flow of water in a
microchannel. The complete Navier-Stokes equations along
with continuity and energy equations are solved using the
SIMPLER method. The liquid vapor interface is captured
using the level set technique. The microchannel is 200
microns in square cross-section and the bubble is placed at
the center of the channel with superheated liquid around it.
The results show steady initial bubble growth followed by a
rapid axial expansion after the bubble fills the channel with
a thin liquid film around it. The bubble then rapidly turns
into a plug and fills up the entire channel. A trapped liquid
layer is observed between the bubble and the channel as the
plug elongates. The bubble growth rate increased with the
incoming liquid superheat and formation of vapor patch at
the walls is found to be dependent on the bubble growth
rate. The upstream interface of the bubble is found to
exhibit both forward and reverse movement during bubble
growth. Results show little effect of gravity on the bubble
growth under the specified conditions. The bubble growth
features obtained from numerical results are found to be
qualitatively similar to experimental observations.
miniature heat exchangers. The studies, however, were
primarily confined to experiments. But, experiments at
microscale have their own limitations, and a small
inaccuracy in measurement can lead to a large error. Hence
there is a need to numerically model two phase flow through
microchannels that would help us better interpret the
available experimental data and also explain the underlying
physics. When boiling takes place in a microchannel,
bubbles nucleate at the wall, but soon grow large enough to
fill the entire channel. Thus the behavior of individual
bubbles determines the flow field in the microchannel. The
present work is undertaken to numerically simulate the
growth of a vapor bubble in a microchannel and analyze its
different characteristics.
LITERATURE REVIEW
Steinke and Kandlikar[2003] experimentally studied
flow boiling and pressure drop characteristics in parallel
microchannels. They used six parallel microchannels of 207
micrometers hydraulic diameter and observed conventional
flow boiling patterns. The flow patterns observed were
bubbly flow, slug flow and annular flow similar to
conventional flow boiling. They could maintain heat flux of
up to 930 kW/m2 in the microchannel. One of the major
differences noted by them was the reversed flow during
rapid growth of a bubble.
INTRODUCTION
Two phase flow through microchannels has been
studied extensively in the last decade since it has the
potential of providing very high heat transfer rates in
1
•
Hetsroni et al. [2003] studied convective boiling in
parallel microchannels using pure water and surfactants.
They observed two flow regimes in steam-water flow. The
low heat flux regime was characterized by the presence of
liquid phase in part of the parallel microchannels. The high
heat flux regime was characterized by convective boiling,
accompanied by quasi-periodical rewetting and refilling of
the microchannels. They recommended that boiling of
surfactant solutions in microchannels may be used to
provide a nearly isothermal heat sink.
Peles [2003] studied two-phase boiling in
microchannels and obtained flow regime maps. He used 16
mm long parallel triangular microchannels with hydraulic
diameters ranging from 50 to 200 micrometers. He also
observed rapid bubble growth around a nucleating bubble.
He concluded that two-phase flow instabilities were of
primary importance in micro scale and should be
comprehensively addressed.
Fogg et al. [2003] numerically studied transient boiling
in microchannels. They used a homogeneous model using
mass weighted averages of the local properties of the liquid
and the vapor. The two phases were considered to be
uniformly distributed within each grid element precluding
any models formulated for the bubble or slug structures. In
conjunction, they also solved one-dimensional transient heat
conduction equation at the wall. They concluded that future
work needs to model bubbles growing into slugs and
annular flow.
Ajaev and Homsey [2003] developed a mathematical
model of constrained vapor bubbles. They assumed that the
shape of the bubble is dominated by capillary forces away
from the wall. A lubrication type analysis was used to find
the local vapor-liquid interface shapes and mass fluxes near
the wall. The microscopic adsorbed film on the constraining
walls was assumed to be in thermodynamic equilibrium with
the vapor phase due to the action of London-van-der-Waals
forces. Solutions indicate large values of mass flux near the
contact line. Longer bubbles were obtained for higher
heater temperatures.
Son and Dhir [1999] developed a two-dimensional
numerical model of growth and departure of single vapor
bubbles during nucleate pool boiling. They used the levelset technique to implicitly capture the liquid vapor interface.
Mukherjee and Dhir [2003] extended the model to threedimensional cases and studied merger and departure of
multiple bubbles during nucleate pool boiling. The present
analysis is done using a similar model to study the growth of
vapor bubble inside a microchannel.
To determine the effect of liquid superheat and
gravity on the bubble growth rate.
NUMERICAL MODEL
Method
The complete incompressible Navier-Stokes equations
are solved using the SIMPLER method [Patankar, 1980],
which stands for Semi-Implicit Method for Pressure-Linked
Equations Revised. The continuity equation is turned into
an equation for the pressure correction. A pressure field is
extracted from the given velocity field. At each iteration the
velocities are corrected using velocity-correction formulas.
The computations proceed to convergence via a series of
continuity satisfying velocity fields.
The algebraic
equations are solved using the line-by-line technique, which
uses TDMA (tri-diagonal matrix algorithm) as the basic
unit.
The speed of convergence of the line-by-line
technique is further increased by supplementing it with the
block-correction procedure [Patankar, 1981]. The multigrid technique is employed to solve the pressure equations.
Sussman et al. [1994] developed a level set approach
where the interface was captured implicitly as the zero level
set of a smooth function. The level set function was
typically a smooth function, denoted as φ . This formulation
eliminated the problems of adding/subtracting points to a
moving grid and automatically took care of merging and
breaking of the interface. Furthermore, the level set
formulation generalized easily to three dimensions. The
present analysis is done using this level set technique.
The liquid vapor interface is identified as the zero level
set of a smooth distance function φ . The level set
function φ is negative inside the bubble and positive outside
the bubble. The interface is located by solving the level set
equation. A fifth order WENO (weighted, essentially nonoscillatory) scheme is used for left sided and right sided
discretization of φ [Fedkiw et al., 1998]. While φ is
initially a distance function, it will not remain so after
solving the level set equation. Maintaining φ as a distance
function is essential for providing the interface with a width
fixed in time. This is achieved by reinitialization of φ . A
modification of Godunov's method is used to determine the
upwind directions. The reinitialization equation is solved in
fictitious time after each fully complete time step. With
h
∆τ =
, ten τ steps are taken with a third order TVD
2u0
(total variation diminishing) Runge Kutta method.
OBJECTIVE
Governing Equations
Momentum equation r
r
r
∂u r r
ρ ( + u .∇u ) = −∇p + ρg − ρβ T (T − Tsat ) g
∂t
r
r
− σκ∇H + ∇.µ∇u + ∇.µ∇u T
The objectives of the present work are as follows –
• To obtain the flow and thermal fields around a
growing vapor bubble inside a microchannel
through numerical simulation.
• To study the behavior of the liquid vapor interface
and its interaction with the constraining walls.
2
(1)
Energy equation ∂T r
+ u.∇T ) = ∇.k∇T for φ > 0
∂t
T = Tsat for φ ≤ 0
ρC p (
Continuity equation r
r m
∇.u = 2 .∇ρ
ρ
Scaling Factors
The governing equations are made non-dimensional
using a length scale and a time scale. The length scale l0
given by the channel width is equal to 200 microns. The
Reynolds number is specified as 100 for all our cases. Thus
for our case of water at 100o C, the velocity scale u0 is
calculated as 0.146 m/s. The corresponding time scale t0 is
1.373 ms. All non-dimensional quantities hereafter are
indicated with * superscript.
The non-dimensional
temperature is defined as
T − Tsat
(14)
T* =
Tw − Tsat
(2)
(3)
The curvature of the interface is defined as -
κ (φ ) = ∇.(
∇φ
)
| ∇φ |
(4)
Computational Domain
Figure 1 shows the typical computational domain. The
total domain is 4.95x0.99x0.495 non-dimensional units in
size. Cartesian coordinates are used with uniform grid.
The mass flux of liquid evaporating at the interface r k ∇T
m= l
h fg
(5)
Y
The vapor velocity at the interface due to evaporation –
r
r
k ∇T
m
(6)
uevp =
= l
ρ v ρ v h fg
X
Z
To prevent instabilities at the interface, the density and
viscosity are defined as -
ρ = ρv + (ρl − ρv )H
µ = µ v + ( µl − µ v ) H
(7)
5
1
4
(8)
3
0.5
H is the Heaviside function given by H = 1 if φ ≥ + 1.5h
H = 0 if φ ≤ −1.5h
2
0
0.5
(9)
H = 0.5 + φ /(3h) + sin[2πφ /(3h)] /( 2π ) if | φ | ≤ 1.5h
(10)
The level set equation is solved as (11)
After every time step, the level-set function φ , is
reinitialized as –
∂φ
= S (φ0 )(1− | ∇φ |)u0
∂t
(12)
φ ( x,0) = φ0 ( x)
S is the sign function which is calculated as S (φ0 ) =
φ0
φ0 2 + h 2
0
Flow takes place in the x-direction with liquid entering
the domain at x* = 0. To take advantage of symmetry and
reduce computation time, the bubble is placed at the center
of the microchannel cross section equidistant from the walls
in the y and the z directions. The gravity vector acts in the
negative y-direction. It is shown later in this paper that
gravity has little effect on the bubble growth under the given
conditions in this study. Thus, computations are done using
the domain shown in Fig. 1 only when gravity is considered.
Otherwise, when neglecting gravity calculations are done
for only half the domain (in y-direction) of what is shown in
Fig. 1.
The number of computational cells in the domain are
400x80x40, i.e. 80 grids are used per 0.99l0. This grid size
is chosen from prior experience to ensure grid independence
and negligible volume loss. Variable time step is used
which varied typically between 1e-4 to 1e-5.
Since the vapor is assumed to remain at saturation
temperature, the thermal conductivity is given by –
r r
∂φ
+ (u + uevp ).∇φ = 0
∂t
0
Fig. 1 – Computational Domain
where h is the grid spacing
k = k l H −1
1
Initial Conditions
The bubble is placed at x* = 0.99, y* = 0.495 and z* = 0,
with 0.1l0 radius in the domain shown in Fig. 1. All
velocities in the internal grid points are set to zero. The wall
(13)
3
temperatures are set to 107o C (T* = 1). The vapor inside the
bubble is set to saturation temperature of 100o C (T* = 0).
The liquid temperature inside the domain is set equal to the
inlet liquid temperature. All physical properties are taken at
100o C. The inlet velocity is specified as u0 and the inlet
liquid temperatures Tin are varied as 102o C, 104o C and
107o C. The contact angle at the walls is specified as 50o.
Y
X
Z
0.5
0
-0.5
-0.5
Boundary Conditions
The boundary conditions are as following –
•
1
2
0.000 ms
X
uz = vz = w = Tz = 0; φ z = 0
•
0
Z
(15)
0.5
0
At the plane of symmetry (z* = 0) :-
*
0.5
5
4
Y
At the inlet (x* = 0) :u = u0; v = w = 0; T = Tin; φ x = 0
•
0
3
-0.5
-0.5
(16)
0
0.5
0
1
2
3
5
4
0.082 ms
Y
*
At the walls (y = 0, y = 0.99) :-
X
u = v = w = 0; T = Tw; φ y = − cos ϕ
where
•
ϕ
(17)
0.5
is the contact angle
0
*
At the wall (z = 0.495):u = v = w = 0; T = Tw; φ z = − cos ϕ
•
Z
-0.5
-0.5
(18)
0.5
0
1
2
5
4
0.165 ms
Y
X
At the outlet (x* = 4.95) :ux = vx = wx = Tx = 0; φ x = 0
0
3
Z
(19)
0.5
0
-0.5
-0.5
RESULTS
First the growth of a vapor bubble is considered at a
liquid inlet temperature of 102o C, with no gravity. The
flow and thermal fields around the bubble is analyzed and
its interaction with the wall is studied. Next, the effects of
the inlet liquid superheat and gravity on the bubble growth
rates are determined. Finally, the numerical results are
compared with experimental observations.
0
0.5
0
1
2
3
5
4
0.247 ms
Y
X
Z
0.5
0
-0.5
-0.5
Bubble growth with Tin = 102o C, g = 0
Figure 2 shows the growth of the vapor bubble inside
the microchannel. The wall superheat is 7oC and the liquid
inlet temperature is 102o C. Gravity is neglected in this case
specifying g = 0 in Eqn. 1. The time corresponding to each
frame is shown at the lower right corner. The frame at 0.00
ms shows the initial condition with a bubble of 0.1l0 radius
placed at the center of the channel equidistant from the
walls.
At 0.082 ms we see that the bubble has grown bigger
due to evaporation and it still retains its spherical shape.
The bubble continues to grow and at the same time moves
downstream in the direction of flow as seen at 0.165ms.
0
0.5
0
1
2
3
5
4
0.325 ms
Y
X
Z
0.5
0
-0.5
-0.5
0
0.5
0
1
2
3
5
4
0.354 ms
Fig. 2 – Growth of vapor bubble in microchannel,
Tin = 102o C, g = 0
4
towards downstream of the bubble, which can be seen at
0.082 ms. The velocity vectors are much larger downstream
of the bubble compared to the upstream. The reference
vector shown in the frame corresponds to 10u0.
At 0.165 ms, as the bubble interface approaches the
walls, vapor jets are seen normal to the walls due to
evaporation at the interface near the walls. At 0.247 ms the
bubble has started to elongate along the channel axis and
more evaporation takes place from its interface near the
walls causing the bubble to expand rapidly.
The bubble touches the wall as it expands and forms
vapor patches as seen at 0.325 ms. A thin layer of liquid is
trapped between the bubble and the wall in the downstream
of the vapor patch. Due to the presence of saturated vapor
on one side, and superheated wall on the other side, very
high rate of evaporation takes place in this liquid layer. The
vapor patch expands as this thin layer of liquid is pulled
downstream and also due to evaporation. Large velocity
vectors can be seen across the downstream interface at 0.354
ms which is pushing the liquid out of the channel.
Figure 4 shows the temperature field around the bubble
at 0.325 ms. Isotherms are plotted in ten intervals between
0 and 1. The wall is at T* = 1 and the vapor inside the
bubble is at T* = 0. Formation of thermal boundary layer
can be seen in the liquid near the wall. Crowding of
isotherms between the bubble and the wall at the
downstream of the vapor patch indicates very high heat
transfer rate in that region.
At 0.247 ms the bubble starts to elongate in the
direction of flow. Here the bubble has almost filled the
channel cross-section and cannot expand any further in the
y-z plane due to resistance from the wall. At 0.325 ms the
bubble is seen to form vapor patches at the walls. The vapor
patches keep expanding both axially and along the channel
cross section, due to both evaporation and surface tension
forces. A liquid layer can be seen trapped at the corner of
the channel as the bubble gradually turns into a plug. Our
calculations were stopped when the downstream interface
crossed 4l0 length of the channel and that corresponds to the
frame at 0.354 ms.
Y
0.5
X
Z
0
0
1
2
3
4
5
-0.5
0.000 ms
Y
(10 units)
0.5
X
Z
0
0
1
2
3
4
5
-0.5
0.082 ms
Y
0.5
X
Z
0
0
1
2
3
4
5
-0.5
0.165 ms
Y
0.3
0.3
T*
1
0.8
0.6
0.4
0.2
0
Y
0.5
Z
X
X
Z
0
-0.5
0
1
2
3
4
0.3
5
0.3
0.247 ms
1
Y
2
3
0.325 ms
0.5
X
Z
Fig. 4 – Temperature field at the central vertical plane,
Tin = 102o C, g = 0
0
0
1
2
3
4
5
-0.5
Figure 5 shows the plot of bubble diameter and the
bubble length in the axial direction against time. It also
shows the locations of the upstream and downstream
interfaces of the bubble in the channel as a function of time.
The bubble equivalent diameter is calculated assuming a
sphere of equal volume. The bubble length is the length of
the bubble at the x-axis through the center of the domain. It
is calculated as the difference between the upstream and
downstream locations.
The bubble growth rate, indicated by the equivalent
diameter is constant initially but increases as the bubble
diameter approaches the channel width at around 0.2 ms.
The initial bubble growth is linear as it is inertia controlled.
The constant bubble growth rate (dr/dt) before 0.1 ms is
found to be around 0.3 m/s. The bubble growth rate is
0.325 ms
Y
0.5
X
Z
0
0
1
2
3
4
5
-0.5
0.354 ms
Fig. 3 – Velocity vectors at the central vertical plane,
Tin = 102o C, g = 0
Figure 3 shows the velocity vectors at a vertical x-y
plane through the center of the domain. The first frame at
0.000 ms show no velocity vectors, since before the start of
the calculations all internal velocities set are to zero.
Thereafter, as the bubble grows, the liquid is pushed
5
found to decrease slightly after 0.3 ms due to formation of
the vapor patches.
The upstream interface is found to move back initially
in all cases causing reversed flow, indicating dominance of
the bubble growth over the incoming liquid flow. The
amount by which the upstream interface receded increased
with liquid inlet temperature due to higher growth rates.
This reversed flow is however, different than that observed
by investigators in parallel microchannels. In the present
case of single channel the flow at the inlet is constant, and
hence the backward movement of the upstream interface
increases the downstream liquid flow between the bubble
and the walls.
After some time, the location of the upstream interface
becomes constant in all the three cases. In the case of 102oC
inlet temperature, which has the lowest bubble growth rate,
the upstream interface is found to advance considerably in
the direction of flow after 0.2 ms. The downstream
interface moves forward with bubble growth in all the cases
and displays similar growth patterns.
Equivalent Diameter
Bubble Length
Upstream End Location
Downstream End Location
1
0.9
Dimensions (mm)
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
0.1
0.2
Time(ms)
0.3
0.4
UEL - Upstream End Location
DEL - Downstream End Location
Fig. 5 – Bubble growth rate, Tin = 102o C, g = 0
UEL, Tin =
DEL, Tin =
UEL, Tin =
DEL, Tin =
UEL, Tin =
DEL, Tin =
1
The bubble length is equal to the equivalent diameter
initially as the tiny bubble grows with superheated liquid
around it. However, as the bubble equivalent diameter
approaches the channel diameter, it faces resistance from the
wall and grows rapidly in the axial direction. This is evident
as the bubble length becomes greater than its equivalent
diameter after 0.14 ms. As the bubble elongates, more
surface area becomes available for evaporation, and the
bubble length increases exponentially between 0.14 and 0.3
ms.
The bubble upstream interface location is seen to
decrease slightly till 0.1 ms, as the bubble grows uniformly
at the initial stages. Thereafter, as the bubble starts to
elongate after 0.14 ms, the upstream interface location
increases indicating that the entire bubble is moving
downstream with the flow.
The bubble downstream interface location increases
constantly throughout the entire growth period. It initially
increases linearly similar to the bubble diameter till 0.1 ms
and afterwards exponentially as the bubble elongates into a
plug. The velocity of the bubble downstream interface is
around 2.5 m/s when the bubble length is 0.5 mm. This is
much higher than the inlet velocity of liquid which is around
0.146 m/s. Thus the liquid is pushed out of the channel at a
very high rate due to evaporation and the bubble growth.
Interface Location (mm)
0.9
102
102
104
104
107
107
deg. C
deg. C
deg. C
deg. C
deg. C
deg. C
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
0.1
0.2
Time(ms)
0.3
0.4
Fig. 6 – Effect of inlet liquid temperatures, g =0
Formation of vapor patch at the walls was observed for
inlet superheat of 2o C (Fig. 2). Interestingly, it was not
observed for the cases of 4o C and 7o C inlet superheats.
This could be due to the fact that the bubbles grew too
quickly in the latter two cases. Thus, there was insufficient
time for the thin layer of liquid between the bubbles and the
walls to evaporate, thereby preventing the formation of
vapor patches.
Effect of gravity
All the results so far were obtained neglecting gravity.
Figure 7 shows the effect of gravity on the bubble growth
rate for the case with Tin = 107o C. In the case with gravity,
calculations are carried out for the entire domain shown in
Fig. 1. The gravity acts in the negative y-direction.
The plot shows the bubble equivalent diameter against
time, till the bubble diameter reaches 0.2 mm, which is the
channel width. It indicates insignificant difference in the
bubble growth for the two cases. This justifies our
assumption of neglecting gravity for the present channel
Effect of liquid inlet temperature
Figure 6 compares the effect of inlet liquid temperature
on the bubble growth. The upstream and downstream
interface locations are plotted against time for inlet
superheats of 2o C, 4o C and 7o C. Gravity is neglected in all
the cases. The results indicate that the bubble grows
considerably faster with increase in liquid inlet temperature,
thereby decreasing the time taken by the bubble to fill the
channel.
6
Figure 8 compares the bubble shapes between our
numerical calculations and experimental observations. This
comparison is only qualitative. The times indicated on the
frames for numerical results do not correspond to the
experimental observations.
Both experiments and numerical calculations show
formation of vapor patches at the wall and thin layers of
liquid between the bubble and channel corners. The bubbles
are seen to move downstream with flow as it turns into a
plug and fills up the entire channel cross section. The
upstream and the downstream interfaces of the plug have
similar shapes in both the cases.
Future work is directed towards quantitative
comparison of the results using similar test conditions.
geometry and Reynolds number. It helps us to take
advantage of symmetry in the y-direction and reduce our
computation time.
With Gravity
Without Gravity
Equivalent Diameter (mm)
0.5
0.4
0.3
0.2
0.1
0
CONCLUSIONS
0
0.02
0.04
0.06
Time(ms)
0.08
1.
0.1
2.
Fig. 7 – Effect of gravity, Tin = 107oC
The above finding is in agreement with the
experimental observations made by Kandlikar and
Balasubramanian [2004]. They found similar flow patterns
for horizontal flows, vertical up flows and vertical down
flows during flow boiling of water in parallel minichannels.
3.
Comparison with experimental observations
The numerical results qualitatively agree with the
experimental data obtained by Balasubramanian and
Kandlikar [2004]. The experiments show increase in
bubble/plug growth rate with time, similar to our
calculations. The plug growth rate in the experiments with
subcooled liquid at the inlet was found to be 0.3 m/s at the
initial stages of bubble growth to around 3.5 m/s at the later
stages. This is comparable to the value of 2.5 m/s obtained
from our numerical calculations at Tin = 102o C. However,
the above experimental study was undertaken in 1 mm wide
parallel channels with 63.5 mm in length.
4.
5.
6.
A vapor bubble growing in superheated liquid
inside a microchannel is modeled using the levelset technique.
The bubble initially grows at a constant rate but its
length increases rapidly when it fills the channel
cross section and expands in longitudinal direction.
This increase in growth rate is due the thin layer of
liquid between the walls and its interface, where
high rate of evaporation takes place.
The upstream interface of the bubble is found to
exhibit both forward and reverse movement during
bubble growth.
The bubble growth rate is found to increase with
the incoming liquid superheat thereby decreasing
the time taken by the bubble to fill the channel.
Vapor patch formation at the walls is observed only
for the lowest liquid inlet temperature.
The effect of gravity is found to be negligible on
the bubble growth under the specified conditions.
The bubble growth features obtained from
numerical calculations are found to be qualitatively
similar to the experimental observations.
NOMENCLATURE
00 u ts
0.082 ms
Cp
g
H
h
hfg
k
l0
m
ms
p
Re
r
T
∆T
t
00 u ts
0.165 ms
00 u ts
0.247 ms
00 u ts
0.325 ms
00 u ts
0.354 ms
Fig. 8 – Comparison of bubble shapes
7
specific heat at constant pressure
gravity vector
Heaviside function
grid spacing
latent heat of evaporation
thermal conductivity
length scale
mass transfer rate at interface
milliseconds
pressure
Reynolds number
radius
temperature
temperature difference, Tw-Tsat
time
t0
u
u0
v
w
x
y
z
βT
κ
µ
ν
ρ
σ
τ
φ
ϕ
time scale
x direction velocity
velocity scale
y direction velocity
z direction velocity
distance in x direction
distance in y direction
distance in z direction
coefficient of thermal expansion
interfacial curvature
dynamic viscosity
kinematic viscosity
density
surface tension
time period
level set function
contact angle
REFERENCES
Ajaev, V. S., and Homsy, G. M., 2003, Mathematical
Modeling of Constrained Vapor Bubbles, Proc. of 1st
International Conference on Microchannels and
Minichannels 2003, Rochester, NY, pp. 589-594.
Balasubramanian, P., and Kandlikar, S. G., 2004,
Experimental study of flow patterns, pressure drop and flow
instabilities in parallel rectangular minichannels, To be
presented at the 2nd International Conference on
Microchannels and Minichannels 2004, Rochester, NY.
Fedkiw, R. P., Aslam, T., Merriman, B., and Osher, S.,
1998, A Non-Oscillatory Eulerian Approach to Interfaces in
Multimaterial Flows (The Ghost Fluid Method), Department
of Mathematics, UCLA, CAM Report 98-17, Los Angeles.
Fogg, D. W., Koo, J. M., Jiang, L., Goodson, K. E.,
2003, Numerical Simulation of Transient Boiling
Convection in Microchannels, Proc. of ASME Summer Heat
Transfer Conference, 2003, Las Vegas, Nevada, HT200347300.
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Subscripts
evp
evaporation
in
inlet
l
liquid
sat
saturation
v
vapor
w
wall
∂
x
∂x
∂
y
∂y
z
∂
∂z
Superscripts
*
non dimensional quantity
→
vector quantity
ACKNOWLEDGEMENTS
The work was conducted in the Thermal Analysis and
Microfluidics Laboratory at RIT.
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