HT TOC
Proceedings of IMECE04
2004 ASME International Mechanical Engineering Congress and Exposition
November 13-20, 2004, Anaheim, California USA
Proceedings of IMECE04
2004 ASME International Mechanical Engineering Congress and RD&D Expo
November 13-19, 2004, Anaheim, California USA
IMECE2004-59976
IMECE2004-59976
EFFECT OF DYNAMIC CONTACT ANGLE ON SINGLE BUBBLES DURING NUCLEATE
POOL BOILING
Abhijit Mukherjee,1
Satish G. Kandlikar2
Mechanical Engineering Department
Rochester Institute of Technology
Rochester, NY 14623
1
Email: [email protected]
2
Email: [email protected]
ABSTRACT
Nucleate pool boiling at low heat flux is typically
characterized by cyclic growth and departure of single vapor
bubbles from the heated wall. It has been experimentally
observed that the contact angle at the bubble base varies
during the ebullition cycle. In the present numerical study,
dynamic advancing and receding contact angles obtained
from experimental observations are specified at the base of a
vapor bubble growing on a wall. The complete NavierStokes equations are solved and the liquid-vapor interface is
captured using the level-set technique. The effect of
dynamic contact angle on the bubble dynamics and vapor
removal rate are compared to results obtained with static
contact angle. The results show that bubble base exhibits a
slip/stick behavior with dynamic contact angle though the
overall effect on the vapor removal rate is small. Higher
advancing contact angle is found to increase the vapor
removal rate.
departure stages. The dynamic contact angle is different
from static or equilibrium contact angle, which depends on
the liquid, vapor and the material of the solid surface. Use
of a single contact angle may not be justified, as even under
equilibrium conditions, the static advancing contact angle is
different (larger) from the static receding contact angle.
Figure 1 shows a nucleating bubble at the wall during
pool boiling [1]. The frame on the left shows a bubble just
after nucleation. The bubble base is expanding in this case,
and the contact angle at the wall is receding. The frame on
the right shows the same bubble just prior to departure. The
bubble base is contracting in this case and the contact angle
is advancing. It can be seen from the figures that the
advancing contact angle is larger than the receding one.
INTRODUCTION
Bubbles nucleate from the cavities at the wall during
nucleate pool boiling. During its growth period, the bubbles
stay attached to the wall at the base. The bubble base
diameter increases initially, then stays constant for a period
of time and finally decreases as the bubble departs. Intense
evaporation is believed to take place near the bubble base
that results in very high wall heat flux.
The liquid-vapor interface at the bubble base
experiences dynamic advancing and dynamic receding
contact angles at the wall during bubble growth and
Receding
contact
angle
Advancing
contact
angle
Fig. 1 – Advancing and receding contact angle
1
Copyright © 2004 by ASME
Son et al. [7] carried out complete numerical simulation
of a single bubble on a horizontal surface during nucleate
pool boiling. They assumed a static contact angle at the
bubble base and accounted for the microlayer evaporation
by including the disjoining pressure effect. The results
showed that the departing bubble became larger with
increase in contact angle.
Sobolev et al. [8] measured dynamic contact angle of
water in thin quartz capillaries with radii varying from 200
to 40 nm. The value of dynamic contact angle was found to
depend on the degree of surface coverage by the absorbed
water molecules. At velocities lower than 5 microns/s, the
dynamic contact angle was found to be linearly dependent
on the velocity whereas, at higher velocities, it was found to
be rate independent.
Kandlikar [9] developed a theoretical model of CHF
with dynamic receding contact angle. The model was based
on the assumption that CHF occurs when the force due to
the momentum change pulling the bubble interface into the
liquid along the heated surface exceeds the sum of the forces
from surface tension and gravity holding the bubble. He
assumed the dynamic receding contact angle for various
liquid-solid systems. The model indicated decrease in CHF
with increase in contact angle.
Kandlikar and Steinke [10] made photographic
observations of liquid droplets impinging on a heated
surface. They studied the effects of surface roughness and
surface temperatures on the dynamic advancing and
receding contact angles. They found that the equilibrium
contact angle first decreased and then increased with surface
roughness. The dynamic advancing and receding contact
angles were found to be equal for high wall superheats at
critical heat flux conditions.
Barazza et al. [11] determined advancing contact angle
during spontaneous capillary penetration of liquid between
two parallel glass plates by measuring the transient height
attained by it. The data was fitted into an averaged NavierStokes model integrated over a cross section away from the
liquid front. The calculated contact angle values failed to
predict the dependence of the dynamic contact angle on
velocity as suggested by classical hydrodynamics and
molecular theories in the nonwetting case.
Lam et al. [12] carried out dynamic one-cycle and
cyclic contact angle measurements for different solids and
liquids. Four different patterns of receding contact angle
were obtained: (a) time dependent receding contact angle;
(b) constant receding contact angle; (c) stick/slip pattern and
(d) no receding contact angle. The authors identified liquid
sorption and retention as the primary cause of contact angle
hysteresis.
Abarajith and Dhir [13] studied the effect of various
contact angles on single bubbles during nucleate pool
boiling. The contact angle was kept fixed throughout the
bubble growth and departure process. The effect of
microlayer evaporation was included in the study. The
contact angle was related to the magnitude of the Hamaker
constant, which was found to change with surface
wettability.
The surface tension force acting at the bubble base
depends on the dynamic contact angle. This affects the
overall bubble dynamics and the wall heat transfer. The
present numerical calculations are performed to study the
effect of the dynamic contact angle at the bubble base as
compared to a static contact angle.
LITERATURE REVIEW
A brief literature survey is presented to review some of
the papers dealing with static and dynamic contact angles
associated with evaporating liquid-vapor interface on a
heated surface.
Schulze et al. [2] empirically determined the
equilibrium contact angle for certain low energy solids and
pure liquids. They varied the roughness of the polymer
solid surfaces and used a sessile drop experiment to measure
the advancing and receding contact angles. The contact
angle hysteresis was assumed to be dependent on the surface
roughness and the equilibrium contact angle was linearly
approximated setting the hysteresis to be zero.
Shoji and Zhang [3] experimentally measured contact
angle of water on copper, glass, aluminum and Teflon
surfaces. The receding contact angle was found to decrease
with surface roughness while the advancing contact angle
remained almost constant. They also developed a model for
evaluating surface wettability by introducing a surface
roughness parameter and a surface energy parameter. They
concluded that the advancing and receding contact angles
are unique for any liquid-solid combination and the surface
condition.
Kandlikar and Stumm [4] developed a model to analyze
the forces acting on a vapor bubble during subcooled flow
boiling. They also measured experimentally the upstream
and downstream contact angles as a function of flow
velocity. They found that the upstream and downstream
contact angles went through a maxima and minima
respectively with increase in flow velocity.
Brandon and Marmur [5] simulated contact angle
hysteresis for a two-dimensional drop on a chemically
heterogeneous surface. The intrinsic contact angle was
assumed to vary periodically with distance from the center
of the drop. The changes in free energy of the system, the
contact angle and the size of the base of the drop were
calculated with increase and decrease in volume of the drop.
The authors concluded that the quasi-static analysis of the
dependence of the free energy of the system on the drop
volume could explain the contact angle hysteresis
measurements.
Ramanujapu and Dhir [6] studied dynamic contact
angle at the base of a vapor bubble during nucleate pool
boiling. A silicon wafer was used as the test surface with
micromachined cavities for nucleation. The bubble base
diameter was measured as a function of time and the
interface velocity was calculated. The results show that
though the contact angle varied during different stages of
bubble growth, it was weakly dependent on the interface
velocity. They concluded that contact angle could be
determined primarily based on the sign of the interface
velocity.
2
Copyright © 2004 by ASME
Kandlikar and Kuan [14] experimentally studied an
evaporating meniscus on a smooth heated rotating copper
surface. They studied the size and shape of the meniscus
and its receding and advancing contact angles. The
meniscus is comparable to a nucleating bubble base near its
contact line region. The results show that for a stationary
meniscus, the contact angle was almost independent of the
flow rate and heat flux. In the case of a moving meniscus,
large difference was observed between the advancing and
receding contact angles.
βT
κ
µ
ν
ρ
σ
τ
φ
ϕ
∂
z
∂
∂y
∂z
Superscripts
*
non dimensional quantity
→
vector quantity
NUMERICAL MODEL
Method
The numerical analysis is done by solving the complete
incompressible Navier-Stokes equations using the SIMPLE
method [15], which stands for Semi-Implicit Method for
Pressure-Linked Equations. A pressure field is extracted
from the given velocity field. The continuity equation is
turned into an equation for the pressure correction. During
each iteration, the velocities are corrected using velocitycorrection formulas. The “consistent” approximation [16] is
used for the velocity correction. The resulting velocity field
exactly satisfies the discretized continuity equation,
irrespective of the fact that the underlying pressure
corrections are only approximate.
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 [17]. Multi-grid technique is used to solve the
pressure fields.
NOMENCLATURE
Cp
d
g
H
hfg
k
l0
m
ms
p
Re
r
T
∆T
t
u
u0
V
v
w
x
y
z
y
specific heat at constant pressure
grid spacing
gravity vector
Heaviside function
latent heat of evaporation
thermal conductivity
length scale
mass transfer rate at interface
milliseconds
pressure
Reynolds number
bubble base radius
temperature
temperature difference, Tw-Tsat
time
x direction velocity
velocity scale
interface velocity at bubble base
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
Sussman et al. [18] developed a level set approach
where the interface was captured implicitly as the zero level
set of a smooth distance 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. 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 5th order WENO (weighted, essentially nonoscillatory) scheme is used for left sided and right sided
discretization of φ [19]. 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
d
∆τ =
, ten τ steps are taken with a 3rd order TVD
2u0
(total variation diminishing) Runge Kutta method.
Subscripts
a
advancing
evp
evaporation
l
liquid
lim
limiting
r
receding
sat
saturation
v
vapor
w
wall
∂
x
∂x
3
Copyright © 2004 by ASME
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
After every time step, the level-set function
reinitialized as –
Continuity equation r
r m
∇.u = 2 .∇ρ
ρ
∇φ
)
| ∇φ |
S is the sign function which is calculated as (2)
S (φ0 ) =
φ0
(3)
(4)
at the interface (5)
φ0 + d 2
The vapor velocity at the interface due to evaporation –
r
r
m k l ∇T
uevp =
(6)
=
ρ v ρ v h fg
Y
(7)
µ = µ v + ( µl − µ v ) H
(8)
H = 0.5 + φ /(3d )
1
0.5
0
1
(9)
+ sin[ 2πφ /(3d )] /( 2π ) if | φ | ≤ 1.5d
Since the vapor is assumed to remain at saturation
temperature, the thermal conductivity is given by –
0 0
0.5
1
Scaling Factors
The governing equations are made non-dimensional using a
length scale (l0) and a time scale (t0) defined by,
(10)
l0 =
Level set equation is solved as r r
∂φ
+ (u + uevp ).∇φ = 0
∂t
0.5
Fig. 2 – Computational domain
where d is the grid spacing
k = k l H −1
Z
1.5
H is the Heaviside function H = 1 if φ ≥ + 1.5d
H = 0 if φ ≤ −1.5d
X
2
To prevent instabilities at the interface, the density and
viscosity are defined as -
ρ = ρv + (ρl − ρv )H
(13)
2
Computational Domain
Figure 2 below shows the computational domain. The
total domain is 0.99x1.98x0.99 non-dimensional units in
size. Cartesian coordinates are used with uniform grid. The
bottom of the domain is defined as the wall. The bubble is
placed at the wall. Taking advantage of symmetry,
calculations are done for one quarter of the bubble.
The number of computational cells in the domain are
72x144x72 i.e. 72 grids are used per 0.99l0. It was selected
based on experience [20] to optimize numerical accuracy
and computation time.
No separate effect of microlayer evaporation at the
bubble base [7, 13 and 20] is incorporated in the present
calculations.
The mass transfer rate of liquid evaporating
r k ∇T
m= l
h fg
(12)
φ ( x,0) = φ0 ( x)
The curvature of the interface is defined as -
κ (φ ) = ∇.(
is
∂φ
= S (φ0 )(1− | ∇φ |)u0
∂t
(1)
Energy equation ∂T r
ρC p (
+ u .∇T ) = ∇.k∇T for φ > 0
∂t
T = Tsat for φ ≤ 0
φ
(11)
t0 =
4
σ
g (ρl − ρv )
l0
g
(14)
(15)
Copyright © 2004 by ASME
The characteristic velocity is thus given by,
u0 =
l0
t0
(16)
The non-dimensional temperature is defined as,
T* =
T − Tsat
Tw − Tsat
(17)
with T* = 0 when T = Tsat and T* = 1 at the wall when T =
Tw.
Initial Conditions
A spherical bubble of radius 0.1 non-dimensional
length is placed in the domain on the wall, with the given
contact angle. The initial coordinates for the center of a
single bubble would be (0, r cos ϕ , 0). All initial velocities
are set to zero. The initial thermal boundary layer thickness
is calculated from the correlation for the turbulent natural
convection heat transfer. The initial thickness is given by δ t = 7.14(ν lα l / gβT ∆T )1 / 3
(18)
Fig. 3 – Experimental observation of dynamic contact
angle [6]
The liquid and vapor properties are taken for water at
100oC. The vapor temperature is set to the saturation
temperature i.e. 100oC. The wall temperature is set to
110oC for all cases.
Case I – Static Contact Angle
Boundary Conditions
The boundary conditions are as follows At the wall (y =0) :
u*=v*=w*=0; T*=1,
where
ϕ
dφ
= − cos ϕ
dy
(19)
3
Diameter (mm)
•
Equivalent diameter
Base diameter
4
*
is the contact angle
*
*
• At the planes of symmetry (x =0, x =1) :
u*=v*x=w*x=T*x=0
2
(20)
1
(21)
0
(22)
Fig. 4 – Bubble growth, static contact angle 54o
• At the planes of symmetry (z*=0, z*=1) :
u*z=v*z=w*=T*z=0
•
*
0
10
20
30
40
Time(ms)
50
60
70
At the top of the domain (y =2) :
u*y=v*y=w*y=0, T*=0
Numerical simulation for a single bubble is carried out
with a static contact angle of 54o. Figure 4 plots the
equivalent bubble diameter and the bubble base diameter as
a function of time. The bubble equivalent diameter is
calculated assuming a sphere of equal volume. The bubble
base diameter is found to increase initially and stay constant
at 1.85 mm at around 30 ms. The base diameter decreases
thereafter and becomes zero at 54 ms. This indicates bubble
departure with an equivalent diameter of 3.5 mm.
RESULTS
Figure 3 shows the experimental data obtained by
Ramanujapu and Dhir [6] for variation of contact angle at
the base of a single vapor bubble. They have plotted the
contact angle as a function of interface velocity. The plot
shows an approximate maximum advancing contact angle of
61o and an approximate minimum receding contact angle of
48o. The fitted curve shows a linear variation of contact
angle between limiting interface velocities. The static
contact angle was 54o for the test surface. The data from
Fig. 3 will be used in the following numerical simulations.
5
Copyright © 2004 by ASME
Case 2 - Constant Advancing and Receding
Contact Angles
dr
< −Vlim
dt
dr
ϕ = ϕ r for > Vlim
dt
ϕ r − ϕ a dr ϕ r + ϕ a
dr
≤ Vlim
( )+
for
ϕ=
2Vl
2
dt
dt
ϕ = ϕ a for
Equivalent diameter
Base diameter
4
dr
is the rate of change of bubble base radius,
dt
ϕ r = 48o and ϕ a = 61o . It may be noted here that in Fig. 3,
the interface velocity is defined as the rate of change of
bubble base diameter (and not radius) with time.
where
3
Diameter (mm)
(23)
2
1
Equivalent diameter: V lim = 0.04
Base diameter: V lim = 0.04
Equivalent diameter: V lim = 0.02
Base diameter: V lim = 0.02
4
0
0
10
20
30
40
Time(ms)
50
60
70
3
Diameter (mm)
Fig. 5 – Bubble growth, receding contact angle 48o,
advancing contact angle 61o
Figure 5 shows the results of numerical calculations
with different advancing and receding contact angles. In
this case, the contact angle is specified depending on the
sign of the interface velocity. Thus when the bubble base
diameter is increasing, the contact angle is specified as 48o
whereas when the bubble base diameter is decreasing, the
contact angle is specified as 61o. The bubble base diameter
is seen to increase initially till 28 ms. Thereafter, as the
base diameter starts decreasing, the contact angle
immediately changes to advancing contact angle and
becomes higher. This affects the surface tension force at the
bubble base. The increase in contact angle decreases the
surface tension force that was causing the bubble base to
contract. As a result, the bubble base starts to expand again.
The contact angle at the bubble base keeps changing
depending on the increase or decrease of base diameter.
However, as a net result, the bubble base starts to recede at a
higher rate after 28 ms. The overall base diameter increases
and exhibits a stick/slip pattern till 44 ms, after which it
decreases till bubble departure. The bubble departs at 66 ms
with a higher equivalent bubble diameter compared to the
previous case where a static contact angle was assumed.
Comparing Figs. 4 and 5, it can be seen that
incorporating different (but constant) advancing and
receding contact angles results in (1) a stick/slip interface
movement during the bubble growth and (2) a larger bubble
departure diameter.
2
1
0
0
10
20
30
40
Time(ms)
50
60
70
Fig. 6 – Bubble growth, contact angle between 48o and
61o, as a function of interface velocity
Figure 6 shows plot of the bubble and base diameters
against time for this case. Results of two calculations are
presented with Vlim=0.02m/s and 0.04 m/s. In these cases,
the base contact angle changes dynamically during the
bubble growth between the specified limiting advancing and
receding contact angles. For the case with Vlim = 0.04 m/s,
the base diameter does not show any overall fluctuations. In
fact, the variation in the bubble base diameter is smooth and
comparable to the results in Case 1 with a static contact
angle of 54o. For the case with Vlim = 0.02 m/s, the bubble
base diameter does show some fluctuations similar to Case
2. Thus as Vlim is decreased, the behavior of the bubble base
approaches the stick/slip pattern observed in Case 2 and as
Vlim is increased, the behavior of the bubble base approaches
static constant angle pattern of Case 1. In both cases of
dynamic contact angle, the bubble departure time is around
60 ms which is comparatively longer than that in Case 1 but
less than Case 2.
Case 3 - Dynamic Contact Angle as a function of
Interface Velocity
Numerical calculations are carried out with dynamic
contact angle at the bubble base as obtained from Fig. 3.
The contact angle is specified as follows –
6
Copyright © 2004 by ASME
Case 4 - Effect of Increase in Hysteresis with
Constant Advancing Contact Angle of 900
5
Diameter (mm)
Calculations were performed for a case with static
advancing contact angle of 90o and static receding contact
angle of 54o. It has been previously demonstrated [7] that
the bubble departure diameter increases with increase in
contact angle. Thus a bigger computation domain of size
1.43x2.86x1.43 non-dimensional units is used for this case.
00.0 ms
16.0 ms
4
3
2
1
0
47.8 ms
63.8 ms
79.8 ms
85.5 ms
87.4 ms
0
10
20
30
40
50
Time(ms)
60
70
80
90
100
Fig. 8 – Bubble growth, receding contact angle 54o,
advancing contact angle 90o
It is seen from this analysis that the advancing contact
angle influences the departure diameter considerably. It is
also seen that using different but constant advancing and
receding contact angles results in a stick/slip behavior of the
bubble base.
Figure 9 compares the vapor removal rate for the four
cases. The vapor removal rate is calculated by dividing the
departure volume with departure time. Case 3 represents the
simulation of the actual experimental conditions shown in
Fig. 3 with Vlim = 0.04 m/s.
1
Vapor Removal Rate (mm3/ms)
31.8 ms
Equivalent diameter
Base diameter
6
Fig. 7 – Bubble shapes, receding contact angle 54o,
advancing contact angle 90o
0.8
0.6
0.4
0.2
0
Figure 7 shows the bubble shapes obtained for Case 4.
The time corresponding to the shapes is indicated on each
frame. The bubble initially grows with a spherical shape but
gradually turns into a hemispherical shape due to the effect
of high advancing contact angle. When the bubble departs,
the bubble base forms a neck which is seen in the frame
corresponding to 85.5 ms.
Figure 8 shows the bubble growth rate corresponding to
Case 4. The bubble initially grows similar to case 1 with
constant receding angle of 54o. After 32 ms, as the bubble
base starts to contract the advancing contact angle becomes
90o. The effect of the hysteresis is found to be similar on
bubble growth as in Case 2. The bubble base starts to
recede again and around 50 ms, the bubble diameter and
base diameter becomes almost equal. The bubble departs at
around 88 ms.
1
2
3
4
Case
Fig. 9 – Comparison of vapor removal rate (Case 3
with Vlim = 0.04 m/s)
It can be seen that for the first three cases, there is little
difference between the vapor removal rates. Therefore the
effect of contact angle hysteresis on the vapor removal rate
is negligible for the cases studied. In Case 4 with 90o
advancing contact angle, the vapor generation rate is
considerably higher compared to the previous cases.
Thus in partial nucleate pool boiling, the bubble
dynamics is affected due to variation of contact angle at the
base during the bubble growth period. The bubble base
exhibits a stick/slip behavior with different advancing and
receding contact angles. The vapor removal rate, however,
7
Copyright © 2004 by ASME
during nucleate boiling on a horizontal surface,” Journal of
Heat Transfer, 121, pp. 623-631.
[8] Sobolev, V. D., Churaev, N. V., Velarde, M. G. and
Zorin, Z. M., 2000, “Surface tension and dynamic contact
angle of water in thin quartz capillaries,” Journal of Colloid
and Interface Science, 222, pp. 51-54.
[9] Kandlikar, S. G., 2001, “A theoretical model to predict
pool boiling CHF incorporating effects of contact angle and
orientation,” Journal of Heat Transfer, 123, pp. 1071-1079.
[10] Kandlikar, S. G. and Steinke, M. E., 2002, “Contact
angles and interface behavior during rapid evaporation of
liquid on a heated surface,” International Journal of Heat
and Mass Transfer, 45, pp. 3771-3780.
[11] Barraza, H. J., Kunapuli, S. and O’Rear, E. A., 2002,
“Advancing contact angles of Newtonian fluids during
“high” velocity, transient, capillary-driven flow in a parallel
plate geometry,” 2002, J. Phys. Chem. B, 106, pp. 49794987.
[12] Lam, C. N. C., Wu, R., Li, D., Hair, M. L. and
Neumann, A. W., 2002, “Study of the advancing and
receding contact angles: liquid sorption as a cause of contact
angle hysteresis,” Advances in Colloid and Interface
Science, 96, pp. 169-191.
[13] Abarajith, H. S. and Dhir, V. K., 2002, “A numerical
study of the effect of contact angle on the dynamics of a
single bubble during pool boiling,” Proceedings of the
ASME IMECE, New Orleans, LA, IMECE2002-33876.
[14] Kandlikar, S. G. and Kuan, W. K., 2003, “Heat transfer
from a moving and evaporating meniscus on a heated
surface,” Proceedings of the ASME Summer Heat Transfer
Conference, Las Vegas, NV, HT2003-47449.
[15] Patankar, S. V., 1980, Numerical Heat Transfer and
Fluid Flow, Hemisphere Publishing Company, Washington
D.C.
[16] Van Doormaal, J. P. and Raithby, G. D., 1984,
“Enhancements of the SIMPLE method for predicting
incompressible fluid flows,” Numerical Heat Transfer, 7,
pp. 147-163.
[17] Patankar, S. V., 1981, “A calculation procedure for
two-dimensional elliptic situations,” Numerical Heat
Transfer, 4, pp. 409-425.
[18] Sussman, M., Smereka, P. and Osher S., 1994, “A
Level Set Approach for Computing Solutions to
Incompressible
Two-Phase
Flow,”
Journal
of
Computational Physics, 114, pp. 146-159.
[19] 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,
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[20] Mukherjee, A. and Dhir, V. K., 2003, “Numerical
Study of Lateral Merger of Vapor Bubbles during Nucleate
Pool Boiling,” Proc. of ASME Summer Heat Transfer
Conference, 2003, Las Vegas, Nevada, HT2003-47203.
is found to be little affected by the dynamic contact angle.
The advancing contact angle is found to play a major role on
both bubble dynamics and the vapor removal rate.
CONCLUSIONS
Numerical simulation is carried out for single
nucleating vapor bubble on a wall with dynamic contact
angle at the bubble base. The contact angle at the wall is
specified from the data obtained from experimental
observations.
When different but constant advancing and receding
contact angles are specified, the bubble base diameter goes
through a series of fluctuations and exhibits a stick/slip
pattern during bubble growth period.
When dynamic contact angle is specified at the bubble
base based on the interface velocity, the bubble growth
behavior is found to be dependent on the limiting velocity
for the advancing and receding contact angles.
The results indicate that the overall effect of dynamic
contact angle on the vapor removal rate is small for the
same limiting advancing and receding contact angles. The
vapor removal rate is found to increase with increase in the
advancing contact angle.
ACKNOWLEDGMENTS
The work was conducted in the Thermal Analysis and
Microfluidics Laboratory at RIT. The support extended by
the Mechanical Engineering Department and Gleason Chair
Endowment is gratefully acknowledged.
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