Proceedings of ASME ICNMM2006
Proceedings
of ICNMM2006
and Minichannels
4th International Conference on Nanochannels, Microchannels
Fourth International Conference on Nanochannels,
Microchannels
and Minichannels
June
19-21, 2006, Limerick,
Ireland
June 19-21, 2006, Limerick, Ireland
ICNMM2006-96050
Paper No. ICNMM2006-96050
NUMERICAL STUDY OF THE EFFECT OF SURFACE TENSION ON VAPOR BUBBLE GROWTH
DURING FLOW BOILING IN MICROCHANNELS
Abhijit Mukherjee Satish G Kandlikar
[email protected] [email protected]
Rochester Institute of Technology, Rochester, NY, USA
ABSTRACT
Microchannel heat sinks typically consist of parallel
channels connected through a common header. During flow
boiling random temporal and spatial formation of vapor
bubbles may lead to reversed flow in certain channels which
causing an early CHF condition. Inside the microchannels the
liquid surface tension forces is expected to play an important
role and impact the vapor bubble growth and corresponding
wall heat transfer. In the present study growth of a vapor
bubble inside a microchannel during flow boiling is
numerically studied by varying the surface tension but keeping
the value of contact angle constant. The complete NavierStokes 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 fluid
properties used are of water but the surface tension value is
varied systematically. The effect of surface tension on bubble
growth rate and wall heat transfer is quantified. The results
indicate that for the range of parameters investigated surface
tension has little influence on bubble growth and wall heat
transfer.
INTRODUCTION
Flow through microchannels is a matter of extensive study
due to wide ranging applications in engineering and biological
sciences. Bubble formation inside microchannels can take
place if the fluid is a mixture of gas and liquid or the
temperature of the liquid is above its saturation temperature
corresponding to the pressure. When the bubbles are of the
same size as the microchannel hydraulic diameters, they
regulate the flow characteristics and if applicable the wall heat
transfer. In the microscale the surface tension forces are
expected to dominate relative to the gravitational forces and
control the bubble dynamics.
Kandlikar (2004) listed several non-dimensional groups
relevant to study of two-phase 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 a new non-dimensional group K2 which is the ratio
of the evaporation momentum force and the surface tension
force but it did not include the contact angle. He also provided
a plot of K2 employed in different experimental investigations
in minichannels.
Mukherjee and Mudawar (2003) developed a smart
pumpless loop for microchannel electronic cooling and tested it
with both water and dielectric FC-72. The dielectric has much
lower surface tension and contact angle values as compared to
water and produced much smaller vapor bubbles. The authors
concluded that bubbles formed in FC-72 provided less
obstruction to the liquid flow as compared to water and hence
dielectrics are more appropriate for microchannel heat
exchangers.
Lee et al. (2004) experimentally studied bubble dynamics
in trapezoidal microchannels with hydraulic diameter of 41.3
microns 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.
Mukherjee and Dhir (2005) 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 (2005) extended the model to study
vapor bubble growth inside a microchannel during flow boiling.
The bubble growth was studied for various values of incoming
liquid flow rate and temperature. The effect of gravity was
found to be negligible on the bubble dynamics. The model
however, used a fixed value of surface tension and the contact
angle.
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Copyright © 2006 by ASME
inlet
liquid
saturation
vapor
wall
∂
∂x
∂
∂y
x
y
∂
z
∂z
Superscripts
*
non-dimensional quantity
vector quantity
→
NUMERICAL MODEL
Computational Domain
Figure 1 shows the typical computational domain. The
domain is 3.96x0.99x0.99 non-dimensional units in size.
Cartesian coordinates are used with uniform grid.
North Wall
---->
1
--------------> Outlet
0.75
0.5
<--
0.25
Inlet -------------->
<--- Bubble at wall
0
-0.5
-0.25
Z*
2
0
0.25
1
0.5
0
4
X*
--->
NOMENCLATURE
A
wall area
specific heat at constant pressure
Cp
d
grid spacing
g
gravity vector
H
Heaviside function
h
heat transfer coefficient
latent heat of evaporation
hfg
k
thermal conductivity
L
length of bubble
L1
upstream bubble cap location
L2
downstream bubble cap location
length scale
l0
m
mass transfer rate at interface
ms
milliseconds
Nu
Nusselt number
p
pressure
Re
Reynolds number
ST
surface tension
T
temperature
∆T
temperature difference, Tw-Tsat
t
time
time scale
t0
u
x direction velocity
velocity scale
u0
v
y direction velocity
w
z direction velocity
x
distance in x direction
y
distance in y direction
z
distance in z direction
βT
coefficient of thermal expansion
κ
interfacial curvature
µ
dynamic viscosity
kinematic viscosity
ν
ρ
density
σ
surface tension
τ
time period
φ
level set function
ϕ
contact angle
Subscripts
evp
evaporation
in
l
sat
v
w
Y*
Yang et al. (2002) simulated bubbly two phase flow in a
narrow channel using a numerical code FlowLab based on the
Lattice-Boltzmann method.
Single or multiple twodimensional 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.
In the present study a vapor bubble growing on a heated
wall inside a microchannel during flow boiling is numerically
studied. All liquid and vapor properties are kept constant
except the value of surface tension which is systematically
varied. The contact angle of the liquid vapor interface at the
contact line region with the solid wall is also kept constant.
The objective is to study the effect of surface tension without
any effect of contact angle on the bubble dynamics and
corresponding wall heat transfer.
-- Top Wall
3
South Wall
Figure 1 – Computational domain
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).
The number of computational cells in the domain is
320x80x40, i.e. 80 grids are used per 0.99l0. This grid size is
chosen from previous work of Mukherjee and Kandlikar (2005)
to minimize numerical error and optimize computation time.
The paper also provides qualitative comparison of the
numerical results with experimental data. Variable time step is
used which varied typically between 1e-4 to 1e-5. Negligible
change in the results is observed when calculations are repeated
with half the time step, which ensured that calculations are time
step independent.
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
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Copyright © 2006 by ASME
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 (TriDiagonal 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 multi-grid 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 Non-Oscillatory) 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
d
∆τ =
, ten τ steps are taken with a third order TVD (Total
2u0
Variation Diminishing) Runge Kutta method.
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
Energy equation ∂T r
ρC p (
+ u.∇T ) = ∇.k∇T for φ > 0
∂t
T = Tsat for φ ≤ 0
Continuity equation r
r m
∇.u = 2 .∇ρ
ρ
The curvature of the interface ∇φ
κ (φ ) = ∇.(
)
| ∇φ |
r k ∇T
m= l
h fg
(5)
The vapor velocity at the interface due to evaporation –
r
r
k ∇T
m
uevp =
= l
(6)
ρ v ρ v h fg
To prevent instabilities at the interface, the density and
viscosity are defined as ρ = ρv + (ρl − ρv )H
(7)
µ = µv + (µl − µ v ) H
(8)
H is the Heaviside function given by H = 1 if φ ≥ + 1.5d
H = 0 if φ ≤ −1.5d
(9)
H = 0.5 + φ /(3d ) + sin[ 2πφ /(3d )] /(2π ) if | φ | ≤ 1.5d
where d is the grid spacing
Since the vapor is assumed to remain at saturation
temperature, the thermal conductivity is given by
−1
k = kl H
(10)
The level set equation is solved as r r
∂φ
+ (u + uevp ).∇φ = 0
∂t
(11)
After every time step the level-set function φ , is
reinitialized as∂φ
= S (φ0 )(1− | ∇φ |)u0
(12)
∂t
φ ( x,0) = φ0 ( x)
S is the sign function which is calculated as S (φ0 ) =
φ0
φ0 2 + d 2
(13)
(1)
(2)
(3)
(4)
The mass flux of liquid evaporating at the interface -
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/height and is equal to 200 microns. Thus for
water at 100o 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 − Tsat
(14)
T* =
Tw − Tsat
The Nusselt number (Nu) is calculated based on the areaaveraged heat transfer coefficient ( h ) at the wall given by,
1A
h = ∫ hdA
(15)
A0
where A is the wall area and h is obtained from
∂T
− kl
| wall
∂y
for horizontal walls
(16a)
h=
Tw − Tsat
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Copyright © 2006 by ASME
(16b)
(17)
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 102o C and the wall temperature is set to 108o C (T* =
1). The vapor inside the bubble is set to saturation temperature
of 100o C (T* = 0). The initial liquid temperature inside the
domain is set equal to the inlet liquid temperature of 102oC.
All physical properties are taken at 100o C (the surface tension
value is systematically varied). The contact angle at the walls
is specified as 40o, which is obtained from the experimental
data of Balasubramanian and Kandlikar (2004).
Boundary Conditions
The boundary conditions are as following –
At the inlet (x* = 0) :u = u0; v = w = 0; T = Tin; φ x = 0
(18)
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, (Kandlikar et al. 2005.)
At the outlet (x* = 3.96) :ux = vx = wx = Tx = 0; φ x = 0
At the plane of symmetry (z* = 0) :uz = vz = w = Tz = 0; φ z = 0
At the walls (y* = 0, y* = 0.99) :u = v = w = 0; T = Tw; φ y = − cos ϕ
where ϕ is the contact angle
At the wall (z* = 0.495):u = v = w = 0; T = Tw; φ z = − cos ϕ
Figure 2 – Comparison of bubble shapes
Figure 2 compares the bubble shapes for the two limiting
values of surface tension used in this study, i.e. 0.0589 N/m 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 the higher surface tension causes the
bubble to try to maintain its spherical shape and hence it grows
comparatively more in the lateral direction.
ST - 0.03 N/m
ST - 0.04 N/m
ST - 0.05 N/m
ST - 0.0589 N/m
1
0.9
0.8
Dimensions (mm)
∂T
| wall
∂z
and h =
for vertical walls
Tw − Tsat
The wall Nusselt number is defined as,
hl
Nu = 0
kl
− kl
(19)
(20)
(21)
0.7
0.6
<----- L2
0.5
0.4
0.3
<-----L1
0.2
0.1
0
(22)
RESULTS AND DISCUSSION
Calculations are carried out with Re = 100 and Twall =
108oC for different values of surface tension. The surface
tension of water at 100oC is 0.0589 N/m. The surface tension
value has been changed systematically to 0.05 N/m, 0.04 N/m
and 0.03 N/m with all other properties remaining constant.
0
0.1
0.2
Time(ms)
0.3
0.4
Figure 3 – Comparison of bubble end locations
Figure 3 compares the upstream and downstream locations
of the bubble interface as a function of time for all the four
cases. The upstream interface location is denoted as L1 and the
downstream interface location as L2 measured from the
channel inlet as indicated in the top frame of Fig. 2. At time 0
ms the bubble center is located at 0.2 mm from the channel
inlet as shown in the figure. Thereafter the downstream
interface location L2 increases continuously as the bubble
elongates in the direction of the flow. The upstream interface
location L1 initially decreases as it opposes the incoming flow.
Eventually after 0.2 ms L2 increases as the entire bubble starts
to move downwards with the flow.
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Copyright © 2006 by ASME
ST - 0.03 N/m
ST - 0.04 N/m
ST - 0.05 N/m
ST - 0.0589 N/m
0.4
50
0.3
0.2
0.1
0
ST
ST
ST
ST
40
Nu North
Equivalent Diameter (mm)
0.5
- 0.03 N/m
- 0.04 N/m
- 0.05 N/m
- 0.0589 N/m
30
20
10
0
0.1
0.2
Time(ms)
0.3
0
0.4
0.1
0.2
Time(ms)
0.3
0.4
Figure 5 – Comparison of wall heat transfer at the
North wall
Figure 4 – Comparison of bubble equivalent
diameters
50
ST
ST
ST
ST
40
Nu South
Figure 3 shows small difference on the bubble interface
locations even though the surface tension value has been
reduced almost by a factor of two. The bubble length is seen to
increase slightly with decrease in surface tension. This result is
in agreement with the bubble shapes seen in Fig. 2 since the
bubble with the higher surface tension has expanded
comparatively more in the lateral direction. Thus a decrease is
surface tension results in long and narrow bubbles.
Figure 4 compares the bubble equivalent diameters against
time for all the four cases. 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
0.1l0. The bubble equivalent diameter increases as the bubble
grows due to heat transfer. There is only small difference
between the equivalent diameters for different values of surface
tension indicating that the bubble growth rate is almost
unaffected from the changes in the values of surface tension.
Figure 5 compares the wall averaged Nusselt numbers for
all the four cases as function of time. The Nu is calculated
using Eq. 17 as explained earlier. The plot shows the wall
averaged Nu 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.
0
- 0.03 N/m
- 0.04 N/m
- 0.05 N/m
- 0.0589 N/m
30
20
10
0
0
0.1
0.2
Time(ms)
0.3
0.4
Figure 6 – Comparison of wall heat transfer at the
South wall
Figure 6 plots the heat transfer at the South wall, where the
bubble is attached to the wall. 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 which has very
low thermal conductivity. 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.
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Copyright © 2006 by ASME
Figure 7 – Comparison of temperature fields
Figure 7 shows the temperature field around the bubbles
for the cases with surface tension of 0.0589 N/m and 0.03 N/m.
Isotherms are plotted for T* from 0 to 1 with intervals of 0.1.
The plot shows formation of thermal boundary layer at the
North and South walls as the liquid enters the microchannel.
The isotherms are affected due to the bubble growth with
increased heat transfer at the North wall as the bubble pushes
the liquid. Around the bubble base the isotherms are pushed
back at the upstream end of the base. However, at the
downstream end there is crowding of isotherms near the bubble
base which indicates region of high heat transfer. The effect of
surface tension on the temperature field is found to be small
since the bubble shapes and sizes are almost similar.
Thus the effect of surface tension on bubble growth and
corresponding wall heat transfer is found to be insignificant for
the cases studied. The surface tension force acting on the
bubble at the walls is also dependent on the contact angle. In a
separate study the authors have noted significant effect of
contact angle on bubble growth and wall heat transfer for the
same value of surface tension which indicates surface
wettability rather than surface tension plays a more important
role in wall heat transfer in microchannels.
CONCLUSIONS
Numerical simulation has been carried out for vapor
bubble growing on a heated wall inside a microchannel during
flow boiling of water. The surface tension values have been
varied keeping all other properties constant including the
contact angle. The wall superheat and the incoming liquid flow
rate are also kept fixed. The results show little effect of surface
tension on bubble growth. The bubble shapes are affected by
the surface tension values with low surface tension producing
longer and thinner bubbles. The effect of surface tension on
wall heat transfer is also found to be negligible.
REFERENCES
Balasubramanian, P., and Kandlikar, S. G., 2004,
Experimental Study of Flow Patterns, Pressure Drop and Flow
Instabilities in Parallel Rectangular Minichannels, Proc. of 2nd
International Conference on Microchannels and Minichannels
2004, Rochester, pp. 475-481, ICMM2004-2371.
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.
Kandlikar, S. G., 2004, Heat Transfer Mechanisms during
Flow Boiling in Microchannels, Journal of Heat Transfer, 126,
pp. 8-16.
Kandlikar, S. G., Willistein, D. A., and Borrelli, J., 2005,
Experimental Evaluation of Pressure Drop Elements and
Fabricated Nucleation Sites for Stabilizing Flow Boiling in
Minichannels and Microchannels, Proc. of 3rd International
Conference on Microchannels and Minichannels 2005,
Toronto, Canada, ICMM2005-75197.
Lee, P. C., Tseng, F. G., and Pan, C., 2004, Bubble
Dynamics in Microchannels. Part I: Single Microchannel,
International Journal of Heat and Mass Transfer, 47, no. 25,
pp 5575-5589.
Mukherjee, A., and Dhir, V. K., 2004, “Study of Lateral
Merger of Vapor Bubbles during Nucleate Pool Boiling,”
Journal of Heat Transfer, 126, pp. 1023-1039.
Mukherjee, A., and Kandlikar, S. G., 2005, Numerical
Study of Growth of a Vapor Bubble during Flow Boiling of
Water in a Microchannel, Journal of Microfluidics and
Nanofluidics, 1, no. 2, pp. 137-145.
Mukherjee, S., and Mudawar, I., 2003, Smart Pumpless
Loop for Micro-Channel Electronic Cooling Using Flat and
Enhanced Surfaces, IEEE Transactions on Components and
Packaging Technologies, 26, no. 1, pp 99-109.
Patankar, S. V., 1980, Numerical Heat Transfer and Fluid
Flow, Hemisphere Publishing Company, Washington D.C.
Patankar, S. V., 1981, A Calculation Procedure for TwoDimensional Elliptic Situations, Numerical Heat Transfer, 4,
pp. 409-425.
Sussman, M., Smereka, P., and Osher S., 1994, A Level
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Yang, Z. L., Palm, B., and Sehgal, B. R., 2002, Numerical
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International Journal of Heat and Mass Transfer, 45, no. 3, pp
631-639.
ACKNOWLEDGMENTS
The work was conducted in the Thermal Analysis and
Microfluidics Laboratory at RIT.
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