International Journal of Heat and Mass Transfer 54 (2011) 3702–3718
Contents lists available at ScienceDirect
International Journal of Heat and Mass Transfer
journal homepage: www.elsevier.com/locate/ijhmt
Numerical study of bubble growth and wall heat transfer during flow boiling
in a microchannel
A. Mukherjee a,⇑, S.G. Kandlikar b, Z.J. Edel a
a
b
Department of Mechanical Engineering and Engineering Mechanics, Michigan Technological University, 1400 Townsend Drive, Houghton, MI 49931, USA
Department of Mechanical Engineering, Rochester Institute of Technology, 76, Lomb Memorial Drive, Rochester, NY 14623, USA
a r t i c l e
i n f o
a b s t r a c t
Article history:
Received 5 February 2010
Received in revised form 27 January 2011
Accepted 27 January 2011
Available online 12 April 2011
A numerical study has been performed to analyze the wall heat transfer mechanisms during growth of a
vapor bubble inside a microchannel. The microchannel is of 200 lm square cross section and a vapor
bubble begins to grow at one of the walls, with liquid coming in through the channel inlet. 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. Experiments have been conducted to validate the numerical model. The bubble growth rate and shapes show good agreement
between numerical and experimental results. The numerical results show that the wall heat transfer
increases with wall superheat but stays almost unaffected by the liquid flow rate. The liquid vapor surface tension value has little influence on bubble growth and wall heat transfer. However, the bubble with
the lowest contact angle resulted in the highest wall heat transfer.
Ó 2011 Elsevier Ltd. All rights reserved.
Keywords:
Flow boiling
Microchannels
Bubbles
1. Introduction
Flow through microchannels is a subject of extensive study due
to wide ranging applications in engineering and biological sciences. Microchannel heat sinks with liquid cooling are extensively
used in various applications such as electronic chip cooling. Bubble
formation inside microchannels can take place if the fluid is a mixture of a gas and a liquid or the temperature of the wall reaches
above the local liquid saturation temperature. During flow boiling,
bubbles nucleate on the microchannel walls and may grow big enough to fill up the entire channel cross-section. When the bubbles
are of the same size as the microchannel, they regulate the flow
characteristics and if applicable the wall heat transfer. The wall
heat transfer from the channel wall to the liquid is affected by
the bubble nucleation and growth inside the channels.
At microscale the surface tension forces are expected to dominate the gravitational forces and control the bubble dynamics.
Water has been the preferred coolant for flow boiling for its excellent thermal properties. Direct application of boiling water on a
chip surface may not be desirable due to its poor dielectric properties and high boiling temperature. Fluorochemicals with excellent
dielectric properties are also often used for electronics cooling. The
heat transfer mechanism in microchannels is affected not only by
the thermal properties but also by the contact angle between the
bubbles and the microchannel walls. The bubble nucleation tem⇑ Corresponding author. Tel.: +1 906 487 1174; fax: +1 906 487 2822.
E-mail addresses: [email protected]
(S.G. Kandlikar), [email protected] (Z.J. Edel).
(A.
Mukherjee),
[email protected]
0017-9310/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved.
doi:10.1016/j.ijheatmasstransfer.2011.01.030
perature is also known to be dependent on the contact angle.
Dielectric liquids typically are highly wetting in nature with much
lower contact angle and surface tension as compared to water.
2. Literature review
Excellent reviews are available on flow boiling in microchannels, e.g. Bergles et al. [1], Garimella and Sobhan [2] and Thome [3].
Kandlikar [4] carried out a critical literature review on flow
boiling through minichannels and microchannels. He identified
the effect of surface tension to be significant in microchannels
causing the liquid to form small uniformly spaced slugs that fill
the channel. He pointed out that one of the main reasons for the
boiling instability in microchannels is the explosive growth of a vapor bubble after it nucleates. It is therefore important to understand the dynamic growth characteristics of a bubble upon its
nucleation during flow boiling in microchannels.
Yen et al. [5] studied convective flow boiling in a circular pyrex
glass microtube and a square pyrex glass microchannel. Higher
heat transfer coefficient was observed in the square microchannel
as compared to the circular crossectional microtube because of
square corners acting as active nucleation sites.
Agostini et al. [6–8] studied high heat flux flow boiling of refrigerants in microchannel heat sinks in a three-part paper. The
authors compared the heat transfer results with their own theoretical models. Lee and Pan [9] studied eruptive boiling in silicon
based microchannels and argued that it may be caused by the
nano-sized cavities present at the channel walls.
A. Mukherjee et al. / International Journal of Heat and Mass Transfer 54 (2011) 3702–3718
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Nomenclature
A
Cp
D
D
G
G
H
h
hfg
k
K1
K2
L
L1
L2
l0
m
ms
Nu
p
q
Re
T
DT
t
t0
u
u0
v
w
wall area
specific heat at constant pressure
characteristic dimension
grid spacing
mass flux
gravity vector
heaviside function
heat transfer coefficient
latent heat of evaporation
thermal conductivity 2
qL
Kandlikar number 1 Ghq
qG
fg
2
Kandlikar number 2 hq rDq
fg
G
length of bubble
upstream bubble cap location
downstream bubble cap location
length scale
mass transfer rate at interface
milliseconds
Nusselt number
pressure
heat flux
Reynolds number
temperature
superheat, Tw ÿ Tsat
time
time scale
x direction velocity
velocity scale
y direction velocity
z direction velocity
Bertsch et al. [10] measured local flow boiling heat transfer
coefficient in a microchannel-based cold plate evaporator using
HFC-134a. The heat transfer coefficient showed a peak value at
0.2 vapor quality in all of the experiments. Lee and Garimella
[11] investigated flow boiling of water in a microchannel array
and presented new correlations for predicting two-phase pressure
drop and local saturated boiling heat transfer coefficient.
Wang and Cheng [12] investigated subcooled flow boiling and
microbubble emission boiling (MEB) of water in a microchannel.
The occurrence of MEB resulted in removal of high heat flux at
moderate rise in wall temperatures. Geisler and Bar-Cohen [13]
studied saturated flow boiling CHF (Critical Heat Flux) in narrow
vertical microchannels and observed that bubble confinement led
to heat transfer enhancement in the low heat flux region of the
nucleate boiling curve.
Harirchian and Garimella [14] defined a new transition criterion
to qualify microscale two-phase flow. They termed this number as
the ‘convective confinement number’ that incorporated mass flux,
fluid properties and channel cross-sectional area.
Yang et al. [15] simulated bubbly two phase flow in a narrow
channel using a numerical code FlowLab based on the LatticeBoltzmann method. Single or multiple two-dimensional Taylor
bubbles were placed in a vertical channel and their behavior was
studied for different values of surface tension and body forces.
No heat transfer or phase change was considered between the
two phases. The authors found little effect of surface tension on
the movement of the bubbles or the flow regime transition.
Jacobi and Thome [16] developed an analytical model of elongated bubble flows in microchannels and compared the results
with experimental data. The central idea to this model was the thin
film evaporation around the elongated bubbles. The model correctly predicted the heat transfer coefficient to be dependent on
x
y
z
distance in x direction
distance in y direction
distance in z direction
Greek symbols
bT
coefficient of thermal expansion
j
interfacial curvature
l
dynamic viscosity
m
kinematic viscosity
q
density
r
surface tension
s
time period
/
level set function
u
contact angle
Subscripts
evp
evaporation
G
gas or vapor
in
inlet
L, l
liquid
sat
saturation
v
vapor
w
wall
x
o/ox
y
o/oy
z
o/oz
Superscripts
⁄
non-dimensional quantity
?
vector quantity
heat flux but insensitive to mass flux. The model did not include
the possibility of formation of vapor patches at the walls and thus
completely excluded the effect of contact angle. The success of this
model depended exclusively on initial guesses of the critical nucleation radius and initial liquid film thickness. The authors argued
that the experimental studies that show heat-flux dependence of
the convection coefficient along with relative independence from
quality and mass flux cannot be ascribed only to the nucleate boiling mechanism. They developed a hypothesis that microchannel
evaporation is thin-film dominated.
There are a few studies that have appeared in the literature on
the effect of contact angle on vapor bubble growth inside microchannels. Mukherjee and Mudawar [17] studied microchannel
electronics cooling using dielectric coolant FC-72 and water which
has large differences in surface tension and contact angle. The results showed opposite trends in critical heat flux (CHF) with decrease in channel size for the two liquids. Water produced large
diameter bubbles that blocked liquid flow through the channel
cross-section and hence CHF decreased with decrease in channel
gap below the departure diameter of the bubbles. In the case of
the dielectric liquid, the departure bubble diameters were much
smaller as compared to water and these bubbles blocked the liquid
flow only below a channel gap of 0.13 mm. When the channel gap
varied between 3.56 mm and 0.13 mm the CHF increased with a
decrease in channel diameter because partial blockage caused by
the tiny bubbles increased the two-phase mixture velocity and improved heat transfer.
Thome et al. [18] and Dupont et al. [19] developed a three-zone
flow boiling heat transfer model to describe evaporation of elongated bubbles in a microchannel and compared the time-averaged
local heat transfer coefficients from several independent experimental studies. Their numerical model consisted of sequential
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A. Mukherjee et al. / International Journal of Heat and Mass Transfer 54 (2011) 3702–3718
and cyclic passage of (a) a liquid slug, (b) an evaporating elongated
bubble with a thin liquid film around it and (c) a vapor slug,
through a microchannel. The model had three adjustable parameters, the initial thickness of the liquid film, the minimum thickness
of the liquid film at dry-out, and the bubble departure frequency.
The comparison of the results with experimental data indicated
limited success of the model.
Kandlikar [20] observed that at low Reynolds number, the convective boiling is diminished in microchannels and nucleate boiling plays a major role with periodic flow of liquid and vapor
slugs in rapid succession. He compared the transient conduction
under the approaching rewetting liquid slug and the heat transfer
in the evaporating meniscus region of the liquid–vapor interfaces
in the contact line region to the nucleate boiling phenomenon.
The author also listed several non-dimensional groups relevant
to study of two-phase flow in microchannels. He developed a
mechanistic model of the flow boiling phenomena based on the
different forces acting on a growing vapor bubble. He introduced
two new non-dimensional groups K1 and K2 relevant to flow boil 2
qL
ing in microchannels. K 1 ¼ Ghq
q represents the ratio between
fg
G
the evaporation momentum force and the inertia force whereas
2
K 2 ¼ hq rDq represents the ratio between evaporation momenfg
G
tum force and the surface tension force. K1 is important for overall
heat transfer considerations since the boiling number embedded in
it is used extensively in flow boiling correlations. K2 is relevant to
the interface movement during bubble growth phenomena and is
more pertinent to the present study. When K2 is divided by the cosine of the contact angle it takes into account the actual surface
tension force acting at the bubble base. The author identified transient heat conduction under the approaching rewetting film and
thin film evaporation around the bubbles as the main mechanisms
of heat transfer during flow boiling inside microchannels. He concluded that the wetting characteristic of the liquid which is indicated by the contact angle plays a significant role during dry out
and rewetting and also during the events leading to CHF.
Lee et al. [21] experimentally studied bubble dynamics in trapezoidal microchannels with hydraulic diameter of 41.3 lm and recorded bubble departure size and frequency using high speed
digital camera. The bubble departure radius was found to decrease
with heat flux whereas there was a mixed effect of mass flux on the
bubble departure radius. The authors concluded that the bubble
departure radius was primarily influenced by surface tension
forces and drag due to bulk flow.
Steinke and Kandlikar [22] experimentally studied flow boiling
of water in microchannels. The most common flow boiling regime
observed was annular-slug flow where a vapor bubble filled the
entire channel length with a thin liquid film around it. Flow reversal in parallel microchannels occurred during explosive growth of
nucleating bubbles. They recorded visual observations of the liquid
vapor interface near the wall during dry-out conditions. They observed that the liquid vapor surface contact line shifted from an
advancing contact position to receding contact orientation at the
onset of dry out due to rapid evaporation of the liquid at the contact line region.
Kandlikar et al. [23] used inlet restrictor and fabricated nucleation sites to to improve flow boiling instability in microchannels. The fabricated nucleation sites in absence of pressure
drop elements increased flow boiling instability. Kosar et al.
[24] used inlet orifices to suppress flow boiling instability in parallel microchannels. They defined a non-dimensional pressure
drop multiplier and correlated it with the suppression of flow
boiling oscillations. Kuo and Peles [25] studied the effect of reentrant cavities on suppression of flow boiling instabilities inside
microchannels. Kuo and Peles [26] reported that an increase in
system pressure prevented rapid bubble growth thereby delaying
flow boiling instability.
Wang et al. [27] tested the effect of different inlet/outlet configurations on flow boiling instability in parallel microchannels. They
found that inlet throttling led to stable flow boiling with no oscillations of temperature and pressure.
Mukherjee and Kandlikar [28] numerically computed bubble
growth in a microchannel with inlet constriction. The results
showed presence of higher inlet velocities due to the restriction
which suppressed the bubble growth and flow reversal. The
authors suggested parallel microchannel designs with increasing
cross-sectional area to prevent flow instability.
Recently Zhang et al. [29] derived a lumped oscillator model to
predict and control dynamic pressure drop instability in flow boiling microchannel systems.
Several analytical and numerical models of bubble growth in
microchannels have appeared in the literature. However, most of
them include many simplifying assumptions that do not represent
the complete physics of dynamic bubble growth. Ajaev and Homsey [30] mathematically modeled a three dimensional steady vapor bubble inside a microchannel with a square cross-section.
The bottom wall of the channel was heated where evaporation
took place while condensation occurred near the top wall. The
inertia terms in the momentum equation and the convective heat
transfer terms in the energy equation were neglected. Stress contributions from vapor recoil and thermocapillarity at the vaporliquid interface were assumed to be insignificant. The problem
was solved in the limits of small capillary numbers and temperature gradients. A local solution at the contact line was obtained
based on the lubrication theory. Results indicated that highest
evaporative mass flux occurred in the macroscopic regions of the
thin liquid film at the contact line where the disjoining pressure
was not important. The overall bubble length was found to
increase with the heater temperature.
Mukherjee and Dhir [31] developed a three dimensional numerical model using the level-set method to study lateral merger of vapor bubbles during nucleate pool boiling. Mukherjee and Kandlikar
[32] developed a complete numerical model of a growing vapor
bubble during flow boiling inside a microchannel and studied the
bubble growth rate for different liquid superheats and flow velocities. However, in that model the contact angle at the contact region of the liquid–vapor interface with the wall as well as the
liquid and vapor thermal properties were maintained constant.
The results indicated that bubble growth rate increased with increase in liquid superheat but decreased with incoming liquid flow
rate. Effect of gravity was found to be negligible on bubble growth.
Lee and Son [33] simulated bubble dynamics and heat transfer
during nucleate boiling in a microchannel using level-set method.
Later Suh et al. [34] used a similar model to simulate flow boiling
in parallel microchannels.
3. Objective
The various numerical and experimental studies indicate that
wall heat transfer during flow boiling inside microchannels depends strongly on the wall heat flux but weakly on the mass flux.
However, there is no general agreement on whether the dominant
wall heat transfer mechanism is nucleate boiling or thin film evaporation. The wide difference in thermal properties and contact angle of water and dielectric liquids which are often employed for
electronics cooling makes it difficult to properly ascertain their
influence on the wall heat transfer. It is important to distinguish
between the effects of surface wettability and thermal properties
on the bubble growth and wall heat transfer in order to understand
the underlying physics that governs the wall heat transfer
A. Mukherjee et al. / International Journal of Heat and Mass Transfer 54 (2011) 3702–3718
mechanisms. As a first step to that effect a vapor bubble growing
inside a microchannel during flow boiling of water is simulated
in the present study. All liquid and vapor properties are kept constant except the value of surface tension and contact angle which
are systematically varied. The objective is to analyze and explain
the effect of wall superheat, liquid mass flux, surface tension and
contact angle on bubble growth and the corresponding wall heat
transfer.
4. Numerical model
4.1. Computational domain
Fig. 1 shows the typical computational domain. The domain is
3.96 0.99 0.99 non-dimensional units in size. Cartesian coordinates are used with uniform grid.
The liquid enters the domain at x⁄ = 0 and leaves the domain at
⁄
x = 3.96. To take advantage of symmetry and reduce computation
time, a nucleating cavity is placed equidistant from the walls in the
x–y plane. The two horizontal walls in the x–z planes are named as
South Wall (y⁄ = 0) and North Wall (y⁄ = 0.99). The vertical wall in
the x–y plane is named the Top Wall (z⁄ = 0.495).
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function. The level set function was typically a smooth function,
denoted as /. 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 /. The level set function / 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 Non-Oscillatory) scheme is used for left
sided and right sided discretization of / [38]. While / is initially
a distance function, it will not remain so after solving the level
set equation. Maintaining / as a distance function is essential for
providing the interface with a width fixed in time. This is achieved
by reinitialization of /. 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 Ds ¼ 2ud0 ; ten s steps are taken with a third order total variation diminishing (TVD) Runge Kutta method.
4.4. Governing equations
Momentum equation:
4.2. Grid independence check
The number of computational cells in the domain is
320 80 40, i.e. 80 grids are used per 0.99l0. This grid size is chosen from previous work of Mukherjee and Kandlikar [32] to minimize numerical error and optimize computation time. Variable
time step is used which varied typically between 1eÿ4 and
1eÿ5. Negligible change in the results is observed when calculations are repeated with half the time step, which ensured that calculations are time step independent.
4.3. Method
o~
u
þ~
u r~
g ÿ rjrH þ r
u ¼ ÿrp þ q~
g ÿ qbT ðT ÿ T sat Þ~
ot
q
u þ r lr~
uT :
lr~
ð1Þ
Energy equation:
qC p
oT
þ~
u rT ¼ r krT
ot
for u > 0;
ð2Þ
for u 6 0:
T ¼ T sat
Continuity equation:
The complete incompressible Navier–Stokes equations are
solved using the SIMPLER method [35], 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 tri-diagonal
matrix algorithm (TDMA) 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 [36]. The multigrid technique is employed to solve the pressure equations.
Sussman et al. [37] developed a level set approach where the
interface was captured implicitly as the zero level set of a smooth
r ~
u¼
~
m
q2
rq:
ð3Þ
The curvature of the interface:
jðuÞ ¼ r ru
:
jruj
ð4Þ
The mass flux of liquid evaporating at the interface:
~¼ÿ
m
kl rT
:
hfg
ð5Þ
The vapor velocity at the interface due to evaporation:
~
uevp ¼
~
m
qv
¼ÿ
Fig. 1. Computational domain.
kl rT
qv hfg
:
ð6Þ
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A. Mukherjee et al. / International Journal of Heat and Mass Transfer 54 (2011) 3702–3718
To prevent instabilities at the interface, the density and viscosity are defined as:
q ¼ qv þ ql ÿ qv H;
ÿ
l ¼ lv þ ll ÿ lv H:
ÿ
ð7Þ
ð8Þ
100 °C (T⁄ = 0). The initial liquid temperature inside the domain
is set equal to the inlet liquid temperature of 102 °C. All physical
properties are taken at 100 °C. The contact angle at the walls is
set as 40° unless otherwise specified, which is obtained from the
experimental data of Balasubramanian and Kandlikar [39].
H is the Heaviside function given by [37]:
H ¼ 1 if u P þ1:5d;
H ¼ 0 if u 6 ÿ1:5d;
4.7. Boundary conditions
ð9Þ
H ¼ 0:5 þ u=ð3dÞ þ sin ½2pu=ð3dފ=ð2pÞ if juj 6 1:5d:
At the inlet (x⁄ = 0):
where d is the grid spacing.
Since the vapor is assumed to remain at saturation temperature,
the thermal conductivity is given by
k ¼ kl Hÿ1 :
ð10Þ
The level set equation is solved as:
ou ÿ
þ ~
u þ~
uevp ru ¼ 0:
ot
ð11Þ
After every time step the level-set function /, is reinitialized as:
ou
¼ Sðu0 Þð1 ÿ jrujÞu0 ;
ot
uðx; 0Þ ¼ u0 ðxÞ:
u
ð13Þ
1
A
Z
ð14Þ
A
hdA;
ð15Þ
0
where A is the wall area and h is obtained from:
ÿkl oT
jwall
oy
for horizontal walls;
T w ÿ T sat
ÿkl oT
jwall
oz
and h ¼
for vertical walls:
T w ÿ T sat
h¼
ux ¼ 0:
At the outlet (x⁄ = 3.96):
ux ¼ 0:
0
hl
:
kl
ð19Þ
At the plane of symmetry (z = 0):
⁄
uz ¼ 0:
ð20Þ
At the walls (y = 0, y = 0.99):
u ¼ v ¼ w ¼ 0;
⁄
T ¼ Tw;
uy ¼ ÿ cos /;
ð21Þ
u ¼ v ¼ w ¼ 0;
T ¼ Tw;
uz ¼ ÿ cos /:
ð22Þ
4.8. Experimental validation
Experimental studies were conducted to obtain single bubble
growth data during flow boiling inside a microchannel. The data
is used to validate the numerical model.
The experimental flow loop consists of a syringe pump, a constant temperature bath, an in-line heater, a particle filter, and a
microchannel test section (see Fig. 2). The syringe pump provides
a constant flow rate of de-gassed, de-ionized water. After the syringe pump, the water is passed through a constant temperature
bath consisting of a metal tube running through a beaker of continuously boiling water. The flow is then passed through an insulated
in-line heater to adjust the inlet temperature of the water. A 7-lm
ð16aÞ
ð16bÞ
The wall Nusselt number is defined as:
Nu ¼
ð18Þ
where u is the contact angle.
At the wall (z⁄ = 0.495):
The Nusselt number (Nu) is calculated based on the area-averaged
heat transfer coefficient (h) at the wall given by
h¼
T ¼ T in ;
Constant inlet flow velocity has been specified in the numerical
calculations. In parallel microchannel heat exchangers constant inlet flow velocity is necessary to maintain stable operating conditions, which can be achieved using flow restrictions at the inlet,
[23]:
⁄
The governing equations are made non-dimensional using a
length scale and a time scale. The length scale l0 given by the channel width/height and is equal to 200 lm. Thus for water at 100 °C,
and Re = 100, the velocity scale u0 is calculated as 0.146 m/s. The
corresponding time scale t0 is 1.373 ms.
The non-dimensional temperature is defined as:
T ÿ T sat
:
T w ÿ T sat
v ¼ w ¼ 0;
uz ¼ v z ¼ w ¼ T z ¼ 0;
4.5. Scaling factors
T ¼
u ¼ u0 ;
ux ¼ v x ¼ wx ¼ T x ¼ 0;
ð12Þ
S is the sign function which is calculated as:
0
:
Sðu0 Þ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
u20 þ d2
The boundary conditions are as following:
ð17Þ
4.6. Initial conditions
The bubble is placed at x⁄ = 0.99, y⁄ = 0 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 liquid inlet temperature is set to 102 °C
and the wall temperature is set to the specified superheat (T⁄ = 1).
The vapor inside the bubble is set to saturation temperature of
Fig. 2. Schematic of experimental microchannel flow loop.
A. Mukherjee et al. / International Journal of Heat and Mass Transfer 54 (2011) 3702–3718
Fig. 3. Comparison of bubble growth rates between experimental data and
numerical simulation.
particle filter is located after the in-line heater, followed by a tee
with a 0.25 mm type T thermocouple submersed in the flow to
measure the fluid inlet temperature. After the microchannel test
section, the water is discharged into a beaker.
The microchannel test section consists of a single microchannel cut into a 400 micron-thick piece of brass using a micro-mill-
3707
ing machine. The cross-section is slightly trapezoidal, with the
bottom of the trench about 40 lm smaller than the top of the
trench. The nominal rectangular cross-section dimensions are
266 lm deep by 201 lm wide, giving a hydraulic diameter of
229 lm. The channel is 25.4 mm long with 7 mm of uncut brass
surrounding it on all sides. A microheater is placed behind the
brass with three 0.25 mm type T thermocouples located directly
underneath the channel on the brass surface, between the microheater and the brass. Water is introduced into the microchannel
through a polycarbonate face plate which is bolted in-place between a steel cover and a block of Teflon with a pocket milled
in it. The steel cover has a through-slot cut in the center for flow
visualization.
The syringe pump was operated at a constant flow rate of
0.41 ml/min, which corresponds to a liquid Reynolds number of
100 using the saturation temperature of water. A fiber-optic cold
light source illuminated the microchannel and high speed video
was recorded at 10,000 fps over the downstream end of the channel. Temperature readings were taken every 100 ms using a laboratory data acquisition system. Bubble growth rates were measured
from the high speed video images using the Phototron PFV video
analysis software package.
Fig. 3 shows the comparison of bubble growth rates between
the simulation results and the experimental data. Numerical simulation was conducted in a square channel with 229 lm hydraulic
diameter assuming saturated flow at the inlet. The wall superheat
was set to the experimental value of 102.1 °C with an initial linear
temperature profile in the liquid between North and South walls.
The North wall boundary condition was defined as adiabatic
assuming insignificant heat loss from the Plexi-glass cover used
Fig. 4. Comparison of bubble shapes: (a) numerical simulation, (b) experimental data.
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A. Mukherjee et al. / International Journal of Heat and Mass Transfer 54 (2011) 3702–3718
in the experiments. The wall contact angle was set to 30° as obtained from the experiments.
The bubble growth rates show good agreement between the
numerical simulation and experimental data. The simulation results, however, overpredicts the data with increase in time, the
largest difference being around 16%. This is probably due to the
assumption of constant wall temperature in the numerical simulation, whereas, local variation in wall temperatures around the bubble occurs in the experiments. There is a slight decrease in the
bubble growth rate observed around 1.0 ms in both results. This
is due to the fact that as the bubble expands, it touches the side
walls of the microchannel leading to the formation of vapor
patches that decreases heat transfer into the bubble.
Fig. 4 compares the bubble shapes between the numerical
simulation and experimental data. Fig. 4(a) shows the numerically obtained bubble shapes. Fig. 4(b) shows the experimental
data at similar times with the bubble outline indicated in each
frame. In both figures, the bubbles are seen to move downstream along with the flow. The bubbles are seen to touch the
side walls around 1.0 ms. Vapor patches can be observed on
the plexi-glass surface between the bubble end caps in the
experimental data at 1.4 and 1.8 ms. Similar vapor patch formation on the North wall is observed in the numerical simulation
at 1.39 and 1.8 ms.
Comparison of bubble growth rate and shapes indicate good
agreement between the numerical results and experimental data
Fig. 5. Bubble shapes (DT = 8 K, Re = 100, r = 0.589 N/m, / ¼ 40 ).
A. Mukherjee et al. / International Journal of Heat and Mass Transfer 54 (2011) 3702–3718
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5. Results
Fig. 6. Bubble cap locations inside the computational domain (DT = 8 K, Re = 100,
r = 0.589 N/m, / ¼ 40 ).
and provides validation for the numerical model used in the following parametric study.
A parametric study has been carried out to analyze the effect of
wall superheat, liquid flow rate, surface tension and contact angle
on bubble growth inside a microchannel during flow boiling of
water with constant incoming liquid superheat of 2 K. The wall
superheat is varied as 5, 8 and 10 K. The Reynolds no. is varied as
50, 100 and 200, which correspond to inlet liquid velocities of
0.073, 0.146 and 0.29 m/s, respectively. The values of contact angle
used are 20°, 40°, 60° and 80°. The surface tension values studied
are 0.03, 0.04, 0.05 and 0.0589 N/m.
Fig. 5 shows the growth of a vapor bubble inside the microchannel with Re = 100, wall superheat of 8 K, surface tension value of
0.0589 N/m and contact angle of 40°. The bubble grows predominantly in the direction of flow, due to heat transfer from the wall
and the surrounding superheated liquid. The time in each frame
is indicated at the upper right corner. The bubble is also seen to
form vapor patches on the vertical walls at 0.311 ms.
Fig. 6 plots the bubble cap locations from the microchannel inlet
as a function of time, corresponding to Fig. 5. The upstream and
downstream bubble cap locations, L1 and L2 respectively, are measured from the channel inlet as indicated in the third frame in
Fig. 5. The upstream cap distance L1 stays almost constant initially
and increases slightly as the bubble fills up the channel length. The
Fig. 7. Bubble growth rates: (a) effect of wall superheat, (b) effect of inlet Reynolds number, (c) effect of surface tension, (d) effect of contact angle.
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downstream cap distance L2 increases linearly at the beginning of
the bubble growth but afterwards increases exponentially as the
bubble turns into a vapor plug. The results indicate that the bubble
grows faster with time as the spherical bubble turns into an elongated bubble.
Fig. 7 plots the bubble equivalent diameters as a function of
time for the different values of wall superheat, Reynolds number,
surface tension and contact angle. The bubble equivalent diameter
increases as the bubble grows due to heat transfer. The bubble
equivalent diameter is calculated assuming a sphere of equal volume. The bubble equivalent diameter is 0.04 mm at 0 ms as the initial radius was set to 0.1l0.
Fig. 7(a) shows that the bubble growth rate increases with increase in wall superheat. However, in Fig. 7(b) it can be seen that
bubble growth rate decreases slightly with increase in the Reynolds number. Comparing Fig. 7(a) and (b), it can also be inferred
that the effect of Reynolds number on the bubble growth is less
as compared to the effect of wall superheat for the range of parameters used in the calculations.
Fig. 7(c) compares the bubble equivalent diameters against time
for different values of surface tension. There is little difference between the equivalent diameters for different values of surface tension indicating that the bubble growth rate is unaffected by the
changes in the values of surface tension. Fig. 7(d) plots the bubble
equivalent diameters against time for the cases with different con-
tact angle. The case with 20° contact angle clearly shows the highest growth rate compared to the other three values of contact
angle. In this case, the presence of liquid film around and below
the downstream cap of the bubble contributed to the increased
wall heat transfer which will be explained further in the subsequent sections of this paper.
Fig. 8 shows the effect of the wall superheat on the bubble size
and shape after 0.3 ms of growth. The bubble is seen to grow comparatively slowly at 5 K wall superheat as it elongates along the
channel in the direction of flow. The bubble at 10 K wall superheat
is the biggest in size and has filled the entire computational domain along the length of the channel. Both the bubbles at 8 and
10 K wall superheat are seen to have developed vapor patches on
the vertical walls along the x–y planes.
Fig. 9 compares the bubble shapes for the two limiting values of
surface tension used in this study, i.e. 0.0589 and 0.03 N/m. The
frames show the bubbles when it has grown large enough to almost fill the entire channel length. The time taken for the bubbles
to grow is 0.3 ms which is shown in the lower right corner of each
frame. The two bubbles show very similar shapes and sizes in spite
of widely varying surface tension values. One noted difference is
that the bubble with higher surface tension has formed vapor
patches at the vertical walls. This is because, higher surface tension
causes the bubble to try to maintain its spherical shape and hence
it grows comparatively more in the lateral direction. The bubble
Fig. 8. Effect of wall superheat on bubble shapes at 0.3 ms.
A. Mukherjee et al. / International Journal of Heat and Mass Transfer 54 (2011) 3702–3718
3711
Fig. 9. Effect of surface tension on bubble shapes.
with lower value of surface tension does not show vapor patches at
the walls in the x–y plane indicating higher wall heat transfer locally around the bubble.
Fig. 10 shows the effect of contact angle on bubble shapes inside
the microchannel after they grew and elongated along the channel
length. Calculations were stopped just before the bubbles reached
the channel outlet. The contact angle and the time corresponding
to each case have been indicated at the lower right corner of each
frame. The figure shows that for the case with 20° contact angle,
the bubble has taken the least time to fill the channel length. Vapor
patch formation on the vertical walls in the x–y plane can be seen
for all the cases except for the 20° case. The extent of the vapor
patch formation is found to increase with the contact angle indicating that the surface wettability decreases with increase in contact
angle. The bubble base and the vapor patches at the vertical walls
have merged for the case with 80° contact angle exposing large
area of the walls to the vapor thereby decreasing wall heat transfer.
The case with 20° contact angle shows a thin layer of liquid
trapped between the wall and the bubble below the downstream
end (starting from around x⁄ = 2.6). The extent of this liquid layer
between the bubble base and the downstream bubble cap location
can be seen to decrease with increase in contact angle.
Figs. 11–13 compare the area averaged heat transfer at the
North, South and Top Wall (refer Fig. 1) respectively for all the
cases as a function of time. The Nu is calculated using Eqs. (16)
and (17) as explained earlier. The wall heat transfer in each case
is very high initially (t 0) as the incoming liquid contacts the
heated wall. The wall heat transfer decreases with time as the thermal boundary begins to develop. At all the microchannel walls, it
can be seen that the heat transfer improves with increase in the
wall superheat. This is because the bubble grows faster with increase in the wall superheat, pushing the liquid towards the opposite wall, thereby inhibiting the thermal boundary layer
development.
Fig. 11 plots the effect of the different parameters on wall heat
transfer at the North Wall. It is seen in Fig. 11(a) that the wall heat
transfer decreases initially but can be seen to increase at the later
stages of bubble growth, as the bubble changes to vapor plug and
elongates. The vapor plug elongates and in a sweeping action
presses the already developed thermal boundary layer at its downstream end towards the wall, thereby causing increased wall heat
transfer.
Fig. 11(b) shows the effect of Reynolds number on the heat
transfer at the North Wall. Compared to Fig. 11(a) it can be seen
that Reynolds number, or the mass flux has little effect on the wall
heat transfer at the North Wall. This is because the velocities associated with bubble growth are much higher compared to the
incoming liquid velocity. From Fig. 5 it can be seen that the rate
of change of L2 is initially about 1 m/s which increases to 2 m/s
at later stages of bubble growth, whereas the specified incoming liquid velocity is only around 0.1 m/s. Thus, the boundary layer
development and the wall heat transfer are primarily influenced
by the bubble growth and not by the incoming liquid mass flux.
Fig. 11(b) also shows that the heat transfer at the North Wall increases slightly after 0.3 ms with a decrease in Re. This is because
the bubble growth rate increases with a decrease in Re, which opposes the expansion of the thermal boundary layer.
Fig. 11(c) compares the wall averaged Nusselt numbers for all
the different values of surface tension as function of time at the
North wall. The heat transfer decreases initially as the flow develops and the thermal boundary layer thickens, but as the bubble
grows and approaches the wall it pushes the thermal boundary
layer against the wall causing the heat transfer to increase after
0.25 ms. There is little difference noted between the heat transfers
at the North wall for different values of surface tension.
Fig. 11(d) shows the effect of contact angle on the wall averaged
Nusselt number at the North Wall as a function of time. As the
thermal boundary layer thickens with time, the Nusselt number
decreases until the bubble growth on the opposite South wall prevents it from thickening any further. As the bubble growing on the
South wall elongates along the channel it eventually starts to push
back the thermal boundary layer at the North Wall increasing the
heat transfer as seen around 0.2 ms for the 20° contact angle case.
This case has the highest bubble growth rate as was explained
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A. Mukherjee et al. / International Journal of Heat and Mass Transfer 54 (2011) 3702–3718
Fig. 10. Effect of contact angle on bubble shapes.
earlier with reference to Fig. 7. The larger bubble influences the
thermal boundary layer development over a wider area and hence
it results in the highest wall heat transfer compared to the other
three cases.
Fig. 12 shows the area averaged heat transfer at the South Wall
for all the cases. The heat transfer decreases continuously from
time 0 ms as the bubble base expands. At the bubble base, vapor
is in contact with the wall and since the vapor has much lower
thermal conductivity as compared to liquid, the wall heat transfer
decreases with increase in the bubble base diameter.
Fig. 12(a) shows the effect of wall superheat. At a time of 0.1 ms,
the wall heat transfer is seen to increase with the increase in the
wall superheat. This is because the bubble grows faster at a higher
wall superheat, which causes the liquid to move faster downstream in the channel resulting in a thinner thermal boundary
layer at the South Wall. However, as the bubble base area increases
with the bubble growth, the wall heat transfer decreases with time
for all the cases due to the vapor contact. At around 0.3 ms, the
wall heat transfer at the South Wall for the bubble at 10 K superheat has decreased considerably and is close to that of the bubble
at 5 K superheat, due to its larger base area.
Fig. 12(b) shows the effect of the inlet liquid flow rates. The
bubble growth rate is suppressed by higher flow rates and hence
the bubble base also expands at a slower rate. Thus, the higher flow
A. Mukherjee et al. / International Journal of Heat and Mass Transfer 54 (2011) 3702–3718
3713
Fig. 11. Heat transfer at the north wall: (a) effect of wall superheat, (b) effect of inlet Reynolds number, (c) effect of surface tension, (d) effect of contact angle.
rate not only improves the convective heat transfer at the walls but
also ensures that a larger area at the wall is in contact with the
liquid.
Fig. 12(c) shows the effect of surface tension on the heat transfer at the South Wall. The bubble with least surface tension 0.03 N/
m has expanded most along the channel length and has a larger
base area and hence it shows least wall heat transfer at around
0.3 ms.
Fig. 12(d) shows the effect of contact angle on the wall heat
transfer at the South wall as a function of time. The increase in
bubble base area depends on surface tension forces as well as
evaporation momentum forces. Decrease in contact angle increases
the surface tension forces at the bubble base resulting in decreased
bubble base diameter. At 0.2 ms, the Nu at the South wall increases
with decrease in contact angle due to smaller bubble base diameters except for the case with 20° contact angle. The evaporation
momentum force in this case is high (due to the thin liquid layer
around the bubble base) increasing the bubble base diameter and
decreasing wall heat transfer at the South wall as compared to
the 40° contact angle case.
Fig. 13 shows the effect of the different parameters at the Top
Wall (refer Fig. 1). The heat transfer is high initially (t 0) which
decreases due to the thermal boundary layer development. The
wall heat transfer decreases more rapidly at the later stages of bubble growth due to formation of vapor patches at the Top Wall as
was seen earlier in Fig. 5.
Fig. 13(a) shows that heat transfer improves with increase in
wall superheat due to increased bubble growth rates. Fig. 13(b)
indicates little effect of the liquid flow rate on the wall heat transfer until the vapor patch forms at the Top Wall which is around
0.25 ms. After initiation of the vapor patch formation, heat transfer
improves with Re. This is because higher liquid flow rates inhibit
further expansion of the vapor patch. Fig. 13(c) shows that surface
tension values too have little effect on the wall heat transfer at the
Top Wall until the initiation of a vapor patch at around 0.27 ms.
However, the vapor patch formation is slightly delayed by the lower surface tension values causing a small improvement in the wall
heat transfer. Fig. 13(d) shows the effect of the contact angle values. The bubble with the 20° contact angle shows the highest wall
heat transfer at around 0.2 ms due to its higher growth rate. However, after the initiation of the vapor patch the wall heat transfer
decreases rapidly for all the cases.
Fig. 14 compares the temperature field around the vapor bubbles at the central x–y plane inside the microchannel, for wall
superheats of 5 and 10 K, after 0.3 ms of bubble growth. Nondimensional temperature contours are plotted between 0 and 1
with intervals of 0.1. The plot shows the thermal boundary layers
at the North and South Walls. The wall heat transfer increases with
wall superheat at the North Wall due to two reasons. The faster
growing bubble pushes the thermal boundary layer thinner and
at the same time the bigger bubble affects a larger area along the
bubble length. At the South wall, larger temperature gradient is
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A. Mukherjee et al. / International Journal of Heat and Mass Transfer 54 (2011) 3702–3718
Fig. 12. Heat transfer at the south wall: (a) effect of wall superheat, (b) effect of inlet Reynolds number, (c) effect of surface tension, (d) effect of contact angle.
observed in the liquid at the upstream end of the bubble at 10 K
superheat. At the downstream end of the bubble, higher temperature gradient is also present in the liquid below the bubble nose
(x⁄ 3.5) for 10 K wall superheat. However, at 10 K wall superheat,
a significant portion of the South Wall is in contact with the vapor
which has much lower thermal conductivity as compared to the liquid. Hence, the net heat transfer at the South Wall is found to be
almost equal at 0.3 ms for the two wall superheats as was seen earlier in Fig. 12(a).
Fig. 15 compares the effect of the wall superheat on the velocity
field at the center of the computational domain as in Fig. 14. The
reference vector has been indicated in each frame and is equal to
10 units. A significantly higher rate of evaporation is observed near
the upstream and downstream interfaces of the bubble at the wall
for the case with 10 K superheat. The rate of evaporation is also
found to be significantly higher along the bubble interface close
to the North Wall at 10 K superheat. The bubble at 5 K wall superheat is growing at a significantly slower rate which is indicated by
the smaller velocity vectors at the downstream of the microchannel as compared to the case with 10 K wall superheat.
Fig. 16 compares the temperature field in the central vertical x–
y plane in the computational domain for the cases with 20° and 80°
contact angle at the same time of 0.232 ms. Isotherms are plotted
of non-dimensional temperature T⁄ from 0 to 1 with intervals of
0.1. T⁄ = 0 inside the bubble in the vapor and T⁄ = 1 at the wall.
The bubble with 20° contact angle has grown much bigger compared to the bubble with 80° contact angle. Thus, the former bubble has pushed the thermal boundary layer at the North Wall more
effectively increasing the local wall heat transfer as mentioned earlier with respect to Fig. 11(d). A thin layer of liquid is clearly seen
below the bubble downstream cap for the case with 20° contact
angle. The crowding of the isotherms between the bubble interface
and the wall indicates an area of increased wall heat transfer.
Fig. 17 compares the velocity field in the central vertical x–y
plane in the computational domain for the cases with 20° and
80° contact angle at the same time of 0.232 ms. The reference
velocity vector equal to 20 units is indicated in each frame. High
rate of evaporation near the North wall can be seen in the upper
frame indicated by larger velocity vectors in the vapor near the
interface inside the bubble. Large velocity vectors can also be observed near the thin liquid film at the downstream end below
the bubble cap indicating intense evaporation. This explains the
higher rate of bubble growth for the case with 20° contact angle.
The velocity vectors near the channel inlet are much smaller compared to the channel outlet in both the frames indicating that liquid is being pushed out at a faster rate due to bubble growth.
The liquid flow rate indicated by the velocity vectors at the channel
outlet for the case with 20° contact angle is much greater as compared to the case with 80° contact angle, indicating faster bubble
growth rate in the former case.
A. Mukherjee et al. / International Journal of Heat and Mass Transfer 54 (2011) 3702–3718
3715
Fig. 13. Heat transfer at the top wall: (a) effect of wall superheat, (b) effect of inlet Reynolds number, (c) effect of surface tension, (d) effect of contact angle.
6. Discussion
The above results indicate that the wall heat transfer inside the
microchannel is primarily influenced by the wall superheat and the
contact angle made by the moving liquid–vapor interfaces at the
walls. With increase in wall superheat, bubble growth rate increases pushing the liquid more rapidly against the wall and thereby increasing the wall heat transfer. This explains the experimental
findings that the heat transfer coefficient during flow boiling inside
microchannels is directly dependent on the wall heat flux. However, increase in bubble growth rate at higher heat fluxes also
accelerates vapor patch formation at the walls which is detrimental to wall heat transfer and may lead to early CHF.
Comparison of heat transfer at the three different wall shows
that the wall heat transfer decreases rapidly due to growth of bubble base and vapor patch at the South and Top walls respectively.
However, at the North wall, heat transfer improvement is observed
as the bubble growth prevents thickening of the thermal boundary
layer at that wall.
The incoming liquid mass flux suppresses bubble growth inside
the microchannel thereby limiting the enhancement in the wall
heat transfer due to the liquid motion generated by the bubble
growth. On the other hand an increase in incoming liquid mass flux
delays vapor patch formation thereby increasing the wall heat
transfer. The liquid velocities generated from the bubble growth
are found to be much higher in this study as compared to the
incoming liquid velocities in the microchannel. Thus as a net effect,
the incoming liquid mass flux does not significantly affect the wall
heat transfer, as seen in Figs. 11–13.
The contact angle also has significant influence on the bubble
growth and wall heat transfer inside microchannels. Contact angle
not only affects the surface tension forces acting at the bubble base
but also influences formation of liquid layer between the bubble
and the walls. Small values of contact angle increases surface tension force parallel to the heater surface that opposes bubble base
expansion. At the same time smaller contact angle leads to formation of thin liquid layer between the bubble and the walls where
high rate of evaporation leads to large evaporation momentum
force that assists expansion of bubble base. These opposing forces
affect the bubble growth and movement of the upstream and
downstream interfaces which in turn affects the wall heat transfer.
The bubble with 20° contact angle is seen to have the highest
growth rate and wall heat transfer as compared to all other cases
presented in this study.
The effect of surface tension on bubble growth and corresponding wall heat transfer is found to be insignificant for the cases studied. The net surface tension force acting on a bubble at the walls
depends on the surface tension values as well as the contact angle.
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A. Mukherjee et al. / International Journal of Heat and Mass Transfer 54 (2011) 3702–3718
Fig. 14. Effect of wall superheat on the thermal field at 0.3 ms.
Fig. 15. Effect of wall superheat on the velocity field at 0.3 ms.
However, we have observed earlier that there is a significant influence of the contact angle on bubble growth and wall heat transfer
for similar values of surface tension. This indicates that surface
wettability rather than the surface tension plays a more important
role in determining the wall heat transfer inside microchannels.
7. Conclusions
Numerical simulation of a vapor bubble growing on a heated
wall inside a microchannel during flow boiling of water has been
performed. The inlet liquid flow rates, the wall superheat, the
A. Mukherjee et al. / International Journal of Heat and Mass Transfer 54 (2011) 3702–3718
3717
Fig. 16. Effect of contact angle on the thermal field at 0.232 ms.
Fig. 17. Effect of contact angle on the velocity Field at 0.232 ms.
surface tension values and the contact angle at the walls have been
systematically varied keeping all other properties constant. Experiments were conducted to validate the numerical model. The bubble growth rate and shapes show good agreement between
numerical and experimental results.
The following observations are made from the parametric
numerical study:
(1) The wall heat transfer is found to improve with increase in
wall superheat and the bubble growth rates.
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(2) The wall heat transfer is found to be unaffected by the
changes in the incoming liquid flow rates.
(3) There is little effect of the liquid surface tension on bubble
growth. The bubble shapes are affected by the liquid surface
tension values with lower surface tension producing longer
and thinner bubbles. The effect of surface tension on wall
heat transfer is found to be negligible.
(4) The movements of the upstream and downstream interfaces
of the bubble as well as formation of vapor patches at the
walls are found to be dependent on the contact angle at
the walls. A decrease in contact angle causes formation of
a liquid layer between the bubble downstream interface
and the wall that has significant influence on bubble growth
and wall heat transfer. The bubble with the lowest contact
angle exhibited the highest growth rate and also the highest
wall heat transfer.
(5) The wall heat transfer increases primarily due to the motion
of the evaporating liquid–vapor interface. The bubble
growth is found to push the liquid against the microchannel
walls, thus preventing the growth of the thermal boundary
layers.
Acknowledgments
The work was conducted in the Thermal Analysis and Microfluidics Laboratory at RIT and in the Advanced Energy Systems and
Microfluidics Laboratory at MTU.
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