Effect of Dynamic Contact Angle on Single

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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


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


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)


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


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


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


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


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


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, Los Angeles, CA. [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. REFERENCES [1] Mukherjee, A., 2003, “Numerical and experimental study of lateral merger of vapor bubbles formed on a horizontal surface during nucleate pool boiling,” Ph.D. Thesis, University of California, Los Angeles, CA, pp. 206. [2] Schulze, R. -D., Possart, W., Kamusewitz, H. and Bischof, C., 1989, “Young’s equilibrium contact angle on rough solid surfaces. Part I. An empirical determination,” J. Adhesion Sci. Technol., 3, no. 1, pp. 39-48. [3] Shoji, M., and Zhang, X. Y., 1994, “Study of contact angle hysteresis (In relation to boiling surface wettability),” JSME International Journal, Series B, 37, no. 3, pp. 560567. [4] Kandlikar, S. G., and Stumm, B. J., 1995, "A control volume approach for investigating forces on a departing bubbles under subcooled flow boiling,” Journal of Heat Transfer, 117, pp. 990-997. [5] Brandon, S. and Marmur, A., 1996, “Simulation of contact angle hysteresis on chemically heterogeneous surfaces,” 1996, Journal of Colloid and Interface Science, 183, pp. 351-355. [6] Ramanujapu, N. and Dhir, V. K., 1999, “Dynamics of contact angle during growth and detachment of a vapor bubble at a single nucleation site”, Proceedings of the 5th ASME/JSME Joint Thermal Engineering Conference, San Diego, CA, AJTE99/6277. [7] Son, G., Dhir, V. K. and Ramanujapu, N., 1999, “Dynamics and heat transfer associated with a single bubble 8
