HT Proceedings of HT2003 ASME Summer Heat Transfer Conference July 21-23, 2003, Las Vegas, Nevada, USA

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1 Proceedings of HT3 ASME Summer Heat Transfer Conference July 1-3, 3, Las Vegas, Nevada, USA HT Numerical Simulation and Experimental Validation of the Dynamics of a Single Bubble during Pool Boiling under and Time-Varying Reduced Gravity Conditions * H.S. Abarajith, D.M. Qiu and V.K. Dhir! University of California, Los Angeles, Mechanical and Aerospace Engineering Department, Los Angeles, CA 995, U.S.A, vdhir@seas.ucla.edu ABSTRACT The numerical simulation and experimental validations of the growth and departure of a single bubble on a horizontal heated surface during pool boiling under reduced gravity conditions have been performed here. A finite difference scheme is used to solve the equations governing mass, momentum and energy in the vapor liquid phases. The vaporliquid interface is captured by level set method, which is modified to include the influence of phase change at the liquidvapor interface. The effects of reduced gravity conditions, wall superheat and liquid subcooling and system pressure on the bubble diameter and growth period have been studied. The simulations are also carried out under both constant and timevarying gravity conditions to benchmark the solution with the actual experimental conditions that existed during the parabolic flights of KC-135 aircraft. In the experiments, a single vapor bubble was produced on an artificial cavity, 1 µm in diameter microfabricated on the polished silicon wafer, the wafer was heated electrically from the back with miniature strain gage type heating elements in order to control the nucleation superheat. The bubble growth period and the bubble diameter predicted from the numerical simulations have been found to compare well with the data from experiments. INTRODUCTION Boiling, being the most efficient mode of heat transfer is employed in various energy conversion systems and component cooling devices. Applications of boiling heat transfer in space applications include thermal management, fluid handling, control and power systems. The key factors that are to be addressed for the space systems based on Rankine cycle are the boiling heat transfer coefficients and the critical heat flux under depleted gravity conditions. Keshock and Siegel (196) showed that the bubbles grow larger and show higher growth periods before getting detached from the heater surface under reduced gravity conditions owing to the reduced buoyancy force acting on the bubbles. Merte (199) and Lee and Merte (1997) have reported the results of pool boiling experiments conducted in the space shuttle for a surface similar to that used in drop tower tests. The subcooled boiling was found to be unstable during long periods of micro gravity conditions. It was concluded that the subcooling has negligible influence on the steady state heat transfer coefficient. Ma and Chung (1) experimentally studied single bubble dynamics in flow boiling of FC7 at terrestrial gravity and reduced gravity conditions in a 1 second drop tower. It was observed that the bubble departure diameter at reduced gravity conditions is larger than that of terrestrial gravity case. However no lift- off of the bubble from the heater surface was observed during the short micro gravity conditions. Straub, Zell and Vogel (199, 199) conducted series of nucleate boiling experiments using thin platinum wires and gold coated flat plate as heaters at low gravity conditions in the flights of ballistic rockets and in KC-135 aircraft. For a flat plate heater with R1 as the test liquid, boiling curves similar to those of normal gravity cases were obtained. Using R113 as the test liquid, rapid bubble growth and large bubbles were observed. However, neither the bubble growth rate nor the bubble diameter at the departure was given. For subcooled micro gravity cases, they observed a reduction of up to 5% in * This work received support from NASA under the Microgravity Fluid Physics Program.! author for correspondance 1 Copyright 3 by ASME

2 heat transfer coefficient in comparison with the normal gravity cases. Qiu et al. () conducted experimental studies on the growth and detachment mechanisms of a single bubble during the parabolic flights of the KC-135 aircraft. Experiments were carried out under normal and reduced gravity conditions for various wall superheats and liquid subcoolings at system pressure varying from 1.7 to 16.6 Psia. Artificial cavities were made on the polished silicon wafer, which was heated on the backside in order to control the nucleation superheat. A single bubble was produced on an artificially etched cavity at the center of the wafer in degassed and distilled water. The gravity in the experiments was not constant but varied with time. These experimental results are used to compare and validate the present simulations. Several attempts have been made in the past to model bubble growth on a heated wall and bubble departure processes. Lee and Nydahl (1989) calculated the bubble growth rate by solving the flow and temperature fields numerically from the momentum and energy equations. They used the formulation of Cooper and Llyod (1969) for micro layer thickness. However they assumed a hemispherical bubble and wedge shaped microlayer and thus they could not account for the shape change of the bubble during growth. Zeng et al. (1993) used a force balance approach to predict the bubble diameter at departure. They included the surface tension, inertial force, buoyancy and the lift force created by the wake of the previous departed bubble. But there was empiricism involved in computing the inertial and drag forces. The study assumed a power law profile for growth rate and the coefficients were determined from the experiments. Mei et al. (1995) studied the bubble growth and departure time assuming a wedge shaped microlayer. They also assumed that the heat transfer to the bubble was only through the microlayer, which is not totally correct for both subcooled and the saturated boiling. The study did not consider the hydrodynamics of the liquid motion induced by the growing bubble and introduced empiricism through the shape of the growing bubble. Welch (1998) has published a scheme using a finite volume method and an interface tracking method. The conduction in the solid wall was also taken into account. However, the microlayer was not modeled explicitly. It has been noted from the above discussion that there is no mechanistic model that describes the observed physical behavior of a vapor bubble and its dependence on various parameters such as wall superheat, liquid subcooling etc. under reduced gravity conditions. Son et al. (1999) numerically simulated the bubble growth during the nucleate boiling by using the Level Set method. This method has been applied to adiabatic incompressible two-phase flow by Sussman et al. (199) and to film boiling near critical pressures by Son and Dhir (1998). The computational domain was divided into two regions viz. micro and macro regions. The interface shape, position and velocity and temperature field in the liquid were obtained from the macro region by solving the conservation equations. The micro region equations, which include the disjoining pressure in the thin liquid film, were solved using the lubrication theory. The solutions of the micro region and macro region were matched at the outer edge of the micro layer. Singh and Dhir (1999) have obtained numerical results for low gravity conditions by exercising the numerical simulation model of Son et al. (1999). However the gravity level was taken to be time invariant, which is not always the case in the experiments. For experiments in which gravity level varied with time, the predicted growth behavior during the life of the bubble differed somewhat from the numerical results. However the bubble lift off diameters matched with the experimental data where the gravity level at the time of the bubble departure was used in the numerical simulations. In the present study, the above-cited work is extended to obtain the bubble growth during pool boiling under constant and time varying reduced gravity conditions. This study focuses on growth of a single bubble in a pool of liquid and on the effect of system variables on growth characteristics. This work neglects the effects of adjoining bubbles on the bubble of interest. The complete understanding of the effect of subcooling, level of gravity and wall superheat on a single bubble is essential for development of a predictive model for nucleate boiling. NOMENCLATURE A = dispersion constant, J c p = specific heat at constant pressure, kj/(kg K) D = diameter of the bubble, m g e = gravitational acceleration at earth level, m/s g = gravitational acceleration at any level, m/s H = step function h = grid spacing for the macro region h ev = evaporative heat transfer coefficient, W/(m K) h = latent heat of evaporation, J/Kg fg Ja sub = Jakob No. based on liquid subcooling ( ρ * Ja = Jakob No. based on wall super heat ( ρ * T w l l C pl* T sub ) ρ * h v fg C pl* T w ) ρ * h K = interfacial curvature, 1/m l = characteristic length, m M = molecular weight m = evaporative mass rate vector at interface, kg/(m s) p = pressure, Pa q = heat flux, W/m R = radius of computational domain, m v fg Copyright 3 by ASME

3 R = universal gas constant, - R = radius of dry region beneath a bubble, m R 1 = radial location of the interface at y=h/, m r = radial coordinate, m T = temperature, K = time, s t t = characteristic time, l / u, s u = velocity in r direction, m/s u = interfacial velocity vector, m/s int u = characteristic velocity, m micro = evaporative mass rate from micro layer, kg/s V c = volume of a control volume in the micro region, m 3 v = velocity in y direction, m/s Z = height of computational domain, m z = vertical coordinate normal to the heating wall, m α = thermal diffusivity, m /s β = coefficient of thermal expansion, 1/K δ t = liquid thin film thickness, m δ T = thermal layer thickness, m ε ( ) = apparent contact angle, deg δ φ = smoothed delta function ϕ φ = level set function θ = dimensionless temperature, (T-T s )/(T wall - T s ) = thermal conductivity, W/mK κ µ = viscosity, Pa s ν = kinematic viscosity, m /s ρ = density, kg/m 3 σ = surface tension, N/m Γ = mass flow rate in the micro layer, kg/s = heating wall superheat, K T s Subscripts lv, = liquid and vapor phase rzt,, = / r, / y, / t swall, = saturation, wall int = interface = infinite MATHEMATICAL DEVELOPMENT OF THE MODEL ASSUMPTIONS The assumptions made in the model are 1) The process is two dimensional and axisymmetric ) The flows are laminar. 3) The wall temperature remains constant throughout the process. ) Pure water at atmospheric pressure is used as the test fluid. 5) The thermodynamic properties of the individual phases are assumed to be insensitive to the small changes in temperature and pressure. The assumption of constant property is reasonable as the computations are performed for low wall superheat range. 6) Static contact angle is assumed to be known. Variations of contact angle during advancing and receding phases of the interface are not included ANALYSIS The model of Son et al. (1999) is extended to study single bubble growth in nucleate boiling under constant and timevarying micro gravity conditions. The computational domain is divided into two regions viz. micro region and macro region as shown in Fig.1. The micro region is a thin film that lies underneath the bubble whereas the macro region consists of the bubble and it s surrounding. Both the regions are coupled and modeled and solved simultaneously. The calculated shapes of the interface in the micro region and macro region are matched at the outer edge of the micro layer. MICRO REGION The equation of mass conservation in micro region is written as, q 1 δ = ρ l. rudz h r r () fg where q is the conductive heat flux from the interface, defined kl( Twall Tint ) as with δ as the thickness of the thin film. δ Lubrication theory and one dimensional heat transfer in the thin film have been assumed in a manner similar to that in the earlier works by Stephan and Hammer (199), Lay and Dhir (1995) and Wayner (1999). According to the lubrication theory, the momentum equation in the micro region is written as, pl r u = µ z () where pl is the pressure in the liquid. Heat conducted through the thin film must match that due to evaporation from the vapor-liquid interface. By using modified Clausius-Clapeyron equation, the energy conservation equation for the micro region yields, 3 Copyright 3 by ASME

4 kl( Twall Tint ) ( pl pv) T v = hev Tint Tv + δ ρlhfg (3) [ ] Γ rq = r h fg (9) The evaporative heat transfer coefficient is obtained from kinetic theory as, h 1/ M ρvhfg ev = and Tv Ts pv π RTv Tv = ( ) () The pressure of the vapor and liquid phases at the interface are related by, A q p = l p v σ K 3 δ + ρ h (5) where A v fg is the dispersion constant. The second term on the right-hand side of equation (5) accounts for the capillary pressure caused by the curvature of the interface, the third term is for the disjoining pressure, and the last term originates from the recoil pressure. The curvature of the interface is defined as, 1 δ δ K = r / 1+ r r r r (6) The combination of the mass conservation, equation (), momentum conservation, equation (), mass balance and energy conservation, equation (3) and pressure balance equation, (5) along with equation (6) for the curvature for the micro-region yields a set of three nonlinear first order ordinary differential equations (7),(8) and (9) 3/ δr δr(1 + δr ) (1 + δr ) = + r r σ ρlhfg q A T + Tv int Tv 3 hev δ ρvhfg Tint qδ 3T h µ Γ = + r κ + h δ κ + h δ ρ h rδ r v ev l ev ( l ev ) l fg q (7) (8) The above three differential equations (7)-(9) can be simultaneously integrated by using a Runge-Kutta method, when initial conditions at r = R are given. In present case, the radial location R1 the interface shape obtained from micro and macro solutions are matched. As such this is the end point for the integration of the above set of equations. The radius of dry region beneath a bubble, R, is related to R 1 from the definition of the apparent contact angle, The boundary conditions for film thickness at the end points are: δ = δ, δ =, Γ= at r = R δ = h/, δ = at r = R r rr For numerically analyzing the macro region, the level set formulation developed by Son et al. (1999) for nucleate boiling of pure liquid is used. The interface separating the two phases is captured by φ which is defined as a signed distance from the interface. The negative sign is chosen for the vapor phase and the positive sign for the liquid phase. The discontinuous pressure drop across vapor and liquid caused by surface tension force is smoothed into a numerically continuous function with a δ - function formulation (refer to Sussman et al., 199, for detail). The continuity, momentum and energy, conservation equations for the vapor and liquid in the macro region are written as, 1 (1) where, δ is the interline film thickness at the tip of microlayer, which is calculated by combining Eqs. (3) and () and requiring that T int = T wall at r = R and h is the spacing of the two dimensional grid for the macro-region. For a given Tint, at r = R, a unique vapor-liquid interface is obtained. The static contact angle, φ, for water-silicon system based on measurements was taken to be 5 and the corresponding value of A o chosen was 1-9. MACRO REGION ρ + = t ( ρu ) T ρ ( ut + u u) = p+ µ u+ µ u + ρg ρβ ( T T) g( t) σk H T s (11) (1) Copyright 3 by ASME

5 ρc T u T T H p ( + ) = κ for > t (13) T = T ( p ) for H = (1) s v 1 1 κ κ l H v l v H = (17) where, H is the Heaviside function, which is smoothed over three grid spaces as described below, m u = ρ ρ (3) The vapor produced as a result of evaporation from the micro region is added to the vapor space through the cells adjacent to the heated wall, and is expressed as, The fluid density, viscosity and thermal conductivity of water are defined in terms of the step function H as, ρ = ρv + ( ρl ρv) H (15) 1 dv m mic = δε ( φ) Vc dt V mic cρv µ = µ + ( µ µ ) (16) () where, Vc is the volume of a control volume, m mic is the evaporation rate from the micro-layer and is expressed as, κ ( T T ) R1 l w int mic = rdr (5) R hfgδ The volume expansion contributed by micro layer is 1 if φ 1.5h smoothed at the vapor-liquid interface by the smoothed delta H = if φ 1.5h (18) function φ πφ sin /( π) if φ 1.5h δε ( φ) = H / φ (6) 3h 3h In level set formulation, the level set function φ is used to The mass conservation equation (11) can be rewritten as, keep track of the vapor-liquid interface location as the set of points where φ =, and it is advanced by the interfacial u = ( ρt + u ρ) / ρ (19) velocity while solving the following equation, The term on right hand side of equation (19) is the volume φt = u int φ (7) expansion due to liquid-vapor phase change. From the conditions of the mass continuity and energy balance at the vapor-liquid interface, the following equations are obtained, To keep the values of φ close to that of a signed distance m = ρ( u int u ) = ρl( u int u l) = ρv( u int u function, φ = 1, φ is reinitialized after every time step, v ) () m= κ T / hfg (1) φ φ = (1 φ ) (8) t where m is the water evaporation rate vector, and u φ + h int is interface velocity. If the interface is assumed to advect in the where, φ is a solution of equation (7). same way as the level set function, the advection equation for The boundary conditions for velocity, temperature, density at the interface can be written as, concentration, and level set function for the governing equations, (11)-(1) are: ρt + u int ρ = () u = v=, T = Twall, φz = cosφ at z = Using equations (18), () and (1), the continuity uz = vz =, T = Ts, φz = at z = Z (9) equation (19) for macro region is rewritten as, u = v = T = φ = at r =, R m r r r 5 Copyright 3 by ASME

6 For the numerical calculations, the governing equations for micro and macro regions are non-dimensionalized by defining the characteristic length, l, the characteristic velocity,, and the characteristic time, t as, l v l = σ /[ g( ρ ρ )]; u = gl ; t = l / u (3) SOLUTION The governing equations are numerically integrated by following the procedure of Son et al. (1999). The computational domain is chosen to be ( R/ l, Z / l ) = (1,), so that the bubble growth process is not affected by the boundaries of the computational domain. The initial velocity is assumed to be zero everywhere in the domain. The initial fluid temperature profile is taken to be linear in the natural convection thermal boundary layer and the thermal boundary layer thickness, δ T, is evaluated using the correlation for the turbulent natural convection on a horizontal plate as, 1/3 δt = 7.1( να l l / gβt T) (31) The gravity level at the lift-off point in the experiments is used to calculate the normalized quantities in the numerical simulations. The grid independence study is shown in Fig.. The variation of bubble diameter with time is plotted for various mesh sizes in Fig.. It can be seen that the change in lift-off diameter and the growth period between 98 x 98 and 196 x 596 are very small. The mesh size for all calculations is chosen as It represents the best trade-off in calculation accuracy and computing time, also as shown by Son et al. (1999). RESULTS AND DISCUSSION BUBBLE DIAMETER Figure 3 (a) shows the variation of equivalent bubble diameter with time for a time-invariant gravity level of.g e and a wall superheat of 5.5 K in saturated water. Fig 3 (b) shows the variation of base diameter of the bubble on the wall with time for the same case. The bubble lifts of at 3.8 sec with a final diameter of 1.1mm. The bubble base initially expands and then shrinks as the bubble starts to detach, as seen from the variation of bubble base in Fig.3 (b). In Fig. 3 (a) and 3 (b), the experimental results of Qiu et al. () are also plotted and the numerical results are found to match well with the experimental results. Figure (a) shows the variation of equivalent bubble diameter with time for a gravity of.5g e and a wall superheat of.5 K in water with a subcooling of. K. Fig. (b) shows the variation of base diameter of the bubble on the wall with time for the same case. u EFFECT OF WALL SUPERHEAT AND LIQUID SUBCOOLING Figure 5 (a) shows the variation of bubble diameter with time for various wall superheats for a gravity of.g e. The lift off diameter increases with the increase in the wall superheat. The growth period decreases with the increase in the wall superheat. The effect of superheat on bubble lift-off diameter is not very significant in the range of our consideration. Figure 5 (b) shows the variation of bubble diameter with time for various liquid subcoolings under a gravity of.g e. The bubble lift off diameter decreases with the increase in the liquid subcooling. The growth period increases with the increase in the liquid subcooling. The effect of liquid subcooling on the bubble growth and lift off diameter seems to be more significant than the effect of wall superheat in the experimental ranges of our consideration. EFFECT OF TIME-VARYING GRAVITY The magnitude of gravity normal to heater varies constantly with time in the KC-135 experiments as shown in Fig 6. The effect of time-varying gravity is introduced in the model by introducing piecewise linear functions of gravity obtained from the experiments and the results obtained are compared with the experimental results as well as with the constant gravity case. The initial boundary layer thickness δ T is calculated using a gravity level that corresponds to the bubble lift-off in the experiments. The shapes of the bubble at various instants obtained from the experiments and from the numerical simulation for the gravity level corresponding to Fig. 6 are given in Fig. 7. As seen from the figure, the bubble becomesflattened or streched in the vertical axis as the gravity level goes from positive to negative or from negative to positive direction respectively in the time-varying gravity case. Figure 8 shows the comparison of numerically calculated bubble shapes between constant gravity case and time-varying gravity case corresponding to Fig. 6 at various instants. As it can be seen from Fig. 8, the base area of the bubble in the negative gravity levels of the time-varying gravity case is larger than that of the constant gravity case. This increase in the base area causes an increase in the heat transfer and thereby increase in the bubble growth rate. Figure 9 shows the comparison of velocity vectors at various instants between constant gravity case and time-varying gravity case corresponding to Fig. 6. As seen from the figure, the velocity vectors in the time-varying gravity case are smaller at the instants where the gravity becomes negative than for the constant gravity case. The evaporation rate from the microlayer into the bubble is determined by the heat transfer rate into the bubble, which in turn is determined by the temperature difference between interface and wall and the thermal layer thickness. During the negative gravity case, the vapor velocity is reduced because of the positive pressure gradient developed in the vapor space. The vapor velocities in the time-varying 6 Copyright 3 by ASME

7 gravity case become similar to that of constant gravity case when the gravity level becomes positive because new pressure gradient is negative for both cases. Figure 1 shows the comparison of temperature contours around the bubble between constant gravity case and timevarying gravity case corresponding to Fig. 6 at time, t= sec. The thermal boundary layer thickness in the time-varying gravity case during the period of negative gravity is higher than the constant gravity case as seen from the figure. This is due to the fact that the negative gravity level in the time-varying gravity case causes the thermal boundary layer to expand. This increase in thermal boundary layer thickness augments the heat transfer rate into the bubble. Figure 11 shows the variation of bubble diameter with time for a liquid subcooling of.9 K and wall superheat of.6 K for the gravity level corresponding to Fig. 6. The bold line shows the variation of bubble diameter with time under timevarying gravity condition whereas dotted lines correspond to constant gravity case. The numerical results of time-varying gravity case match well with the experimental results as seen from the figure. The bubble lift off diameter is higher in the case of time-varying gravity case than the constant gravity case. After 5.5 seconds, the magnitude of the gravity reaches.1g e and the bubble has grown to a size of 5mm, which corresponds to the lift-off diameter at that gravity level. The bubble base has already started shrinking as shown in Fig. 7 at that instant. So the bubble will depart soon irrespective of the gravity level thereafter. HEAT TRANSFER INTO VAPOR BUBBLE Figure 1 (a) and Fig. 1 (b) show the variation of heat transfer rates from microlayer, from surroundings and the total heat transfer rate to the bubble in the constant gravity and timevarying gravity cases corresponding to Fig 6. The heat transfer rate in the time-varying gravity case is higher than the constant gravity case since the bubble is bigger and has a higher base area in the former case. The microlayer heat transfer rate is also higher in the time-varying gravity case because of greater base area as seen from the comparison of bubble shapes in Fig. 9 and higher thermal boundary layer thickness during instants where the gravity level is lower than that for constant gravity case as seen from Fig. 1. WALL HEAT FLUX Figure 13 (a) shows the variation of heat flux with radial distance, r for various instants for a super heat of.6 K and liquid subcooling of.9 K under a constant gravity of.1g e. As seen from the Fig. 13 (a), the local heat flux increases with radial distance because of the increase in temperature difference between wall and the bubble interface, but at the same time the thermal boundary layer thickness also increases with radial distance, which reduces the local heat flux. Therefore the local heat flux attains a peak and then starts decreasing because of the increasing thermal boundary layer thickness and eventually attains a value corresponding to the natural convection heat flux. The peak in the heat flux occurs at the contact line between the bubble and the wall, which is caused by the high evaporation rate of liquid into the bubble. During the bubble growth, the peak moves outwards first, reaches a position of maximum radius and then moves back towards the cavity. The farthest position of the peak from the cavity occurs when the base diameter reaches its maximum value. It is also noticed that the local heat flux attains a peak at the bubble lift off time. The cold liquid occupies the free space left behind by the bubble. This sets up a transient conduction for a small period of time in that region. The heat flux in the transient state goes as the inverse square root of the time. This causes a sharp increase in the heat flux in that region for that small period of time and thereafter heat flux decreases until natural convection is established. Fig 13 (b) shows the variation of heat flux with radial distance, r for various instants for a superheat of.6 K and liquid subcooling of.9 K under time-varying gravity conditions shown in Fig. 6. Another similar time-varying gravity case shown in Fig. 1 (a) has also been numerically calculated. Only the effect of variation of gravity in the direction normal to the heater is considered. Fig 1 (b) shows the variation of bubble diameter with time for a liquid subcooling of.9 K and wall super heat of 9 K under constant gravity of.35g e, marked by the dashed line. The solid line shows the variation of bubble diameter with time under time-varying gravity condition. The numerical results of time-varying gravity case match well with the experimental results as seen from Fig. 1 (b). CONCLUSIONS The numerical simulation and experimental analysis of a single vapor bubble in pool boiling under micro gravity conditions have been carried out. Bubble shape and the growth process are numerically computed. The following conclusions have been drawn : 1) Larger bubble diameters and longer bubble growth periods are the characteristic features in the reduced gravity cases due to the decrease in the buoyancy force. ) The effect of liquid subcooling is to increase the bubble growth period significantly and to decrease the bubble lift off diameter. 3) The effect of wall superheat is to decrease the bubble growth period and to increase the bubble lift off diameter. But this effect is not as significant as the effect of liquid subcooling over the range of consideration. ) The effect of the time-varying gravity is to increase the heat transfer rate and the bubble growth rate. The bubble gets flattened in the vertical direction when gravity is negative. Negative gravity results in the increase in base diameter and thereby an increase in heat transfer rate and bubble growth rate. 5) In the time-varying gravity case, the bubble departs soon after reaching the bubble lift-off diameter corresponding to the gravity level at that instant. The gravity level succeeding that point has little effect on the bubble departure diameter. 7 Copyright 3 by ASME

8 buoyancy and the surface tension play major role in the bubble growth process. ACKNOWLEDGMENTS This work received support from NASA under the Microgravity Fluid Physics program. REFERENCES 1. Siegel, R. and Keshock, E.G., "Effect of Reduced Gravity on Nucleate Bubble Dynamics in Water", Journal of AIChE, Vol. 1., 196, pp Merte, H., "Pool and Flow Boiling in Variable and Microgravity", nd Microgravity Fluid Physics Conference, Paper No.33, Cleveland, OH, June 1-3, Lee, H.S., and Merte, H., "Pool Boiling Curve in Microgravity", Journal of Thermophysics and Heat Transfer, Vol. 11, No., 1997, pp Qiu, D.M., Dhir, V.K., Hasan, M.M. and Chao, D., et al., Single Bubble Dynamics during Nucleate Boiling under Low Gravity Conditions, Microgravity Fluid Physics and Heat Transfer, Editor: V.K. Dhir, Begell House, New York,, pp Lee, R.C and Nyadhl, J.E., Numerical Calculation of Bubble Growth in Nucleate Boiling from inception to departure, Journal of Heat Transfer, Vol. 111, pp Zeng, L.Z., Klausner, J.F. and Mei, R., 1993, A unified Model for the prediction of Bubble Detachment Diameters in Boiling Systems-1.Pool Boiling, International Journal of Heat and Mass Transfer, Vol. 36, pp Mei, R., Chen, W. and Klausner, J. F., 1995, Vapor Bubble Growth in Heterogeneus Boiling-1.Growth Rate and Thermal Fileds, International Journal of Heat and Mass Transfer, Vol. 38 pp Straub, J., Zell, M. and Vogel, B., "Boiling under Microgravity Conditions", Proceedings of 1 st European Symposium on FLUIDS IN SPACE, Ajaccio, France, Nov. 18-, Straub, J., "The Role of Surface Tension for Two-Phase Heat and Mass Transfer in the Absence of Gravity", Experimental Thermal and Fluid Science, Vol. 9, 199, pp Welch, S. W. J., 1998, Direct Simulation of Vapor Bubble Growth, International Journal of Heat and Mass Transfer, Vol. 1,pp Son, G., Dhir, V.K., and Ramanujapu, N. "Dynamics and Heat Transfer Associated With a Single Bubble During Nucleate Boiling on a Horizontal Surface", Journal of Heat Transfer, Vol.11, Aug., 1999, pp Sussman, M., Smereka, P and Osher, S., 199, A Level Set Approach for Computings Slutions to Incompressible Twophase flow, Journal of Computational Physics, Vol. 11, pp Singh, S. and Dhir, V.K., "Effect of Gravity, Wall Superheat and Liquid Subcooling on Bubble Dynamics during Nucleate Boiling", Microgravity Fluid Physics and Heat Bubble Diameter, mm Transfer, Editor: V.K. Dhir, Begell House, New York,, pp Fig.1 Macro and Micro Regions of the mathematical model for numerical simulation z δ o g Wall r =R o T w = 5.5 K T sub = K g =.g e z=z Vapor Wall Macro Region T sat Micro Region 196 X X 98 6 X 19 Liquid Fig.. Grid Independence study for T w = 5.5 K, T sub = K, g =.g e δ h ϕ r=r Copyright 3 by ASME

9 16 1 Bubble Diameter, mm T w = 5.5 K T sub = K g =.g e Bubble Diameter, mm T w =.5 K T sub =. K g =.5g e 1 3 Fig. 3 (a). The Variation of Equivalent Bubble Diameter with Time for T w = 5.5 K, T sub = K, g =.g e Fig. (a). The Variation of Equivalent Bubble Diameter with Time for T w =.5 K, T sub =. K, g =.5g e. 6 5 Bubble Base Diameter, mm T w = 5.5 K T sub = K g =.g e Bubble Base Diameter, mm 3 1 T w =.5 K T sub =. K g =.5g e Fig. 3 (b). The Variation of Bubble Base Diameter with Time for T w = 5.5 K, T sub = K, g =.g e. Fig. (b). The Variation of Equivalent Bubble Base Diameter with Time for T w =.5 K, T sub =. K, g =.5g e. 9 Copyright 3 by ASME

10 16 1 T w = 6.5 K T w = 5.5 K..3 Bubble Diameter, mm T w = 3.7 K T sub = K g =.g e e g/g Z-Direction T w =.6 K T sub =.9 K Y-Direction X-Direction Time, s Fig. 5 (a) The variation of Bubble Diameter with time for various wall superheats. Fig. 6 (a). The Variation of Gravity levels with Time in the experiments 18 Bubble Diameter, mm T sub = K T sub =. K Bubble Diameter, mm Varying g constant g =.1g e T w =.6 K T sub =.9 K Fig. 5 (b) The variation of Bubble Diameter with time for various liquid subcoolings Fig. 6 (b). The Variation of Equivalent Bubble Diameter with Time for T w =.6 K, T sub =.9 K. 1 Copyright 3 by ASME

11 Experimental Bubble Shapes g =.1g e g =.1g e t = sec t =.5 sec t = 5. sec Numerical Bubble Shapes t=1 sec Experimental Bubble Shapes g =.1g e g = -.1g e t = 5.7 t = 5.78 sec t = 5.8 sec Numerical Bubble Shapes t=.5 sec Fig. 7 Comparison of Bubble shapes between Experimental and Numerical Results for the time-varying gravity case in Fig 6(a). Fig 8. Comparison of Bubble shapes between constant gravity case and time-varying gravity case in Fig. 6(a) for various time instants. 11 Copyright 3 by ASME

12 g =.1g e g = -.g e g =.1g e g =.1g e t= sec t=1 sec g =.1g e g =.g e g =.1g e g = -.1g e t=5.5 sec 5 1 Fig 8. (contd) Comparison of Bubble shapes between constant gravity case and time-varying gravity case in Fig 6(a) for various time instants. 5 1 t=.5 sec 5 1 Fig 9. Comparison of velocity vectors between constant gravity case and time-varying gravity case in Fig 6(a) for various time instants. 1 Copyright 3 by ASME

13 g =.1g e g = -.g e g =.1g e T = 37 K T = 376 K T = 378 K T = 38 K T = 38 K T = 383 K t= sec t= sec g =.1g e g =.g e g = -.g e T = 37 K T = 376 K T = 378 K T = 38 K T = 38 K T = 383 K t=5.5 sec Fig 9 (contd.). Comparison of velocity vectors between constant gravity case and time-varying gravity case for various time instants. Fig 1. Comparison of Temperature contours between constant gravity case and time-varying gravity case for time, t= sec. 13 Copyright 3 by ASME

14 1. 1 Total Microlayer Surroundings 3 5 t= sec t=.5 sec t=3. sec t=3.5 sec Heat Transfer Rate, W.8.6. Heat Flux, W/cm 15 1 Lift off t=6 sec T w =.6 K T sub =.9 K g =.1g e. 5 t=. sec r, mm Fig. 11 (a) The variation of heat transfer rates from microlayer, from surroundings and total heat transfer rate with Time for T w =.6 K, T sub =.9 K. for contant gravity case. Fig. 1 (a). The Variation of Heat flux with Time for constant gravity case, g=.1 g e. Heat Transfer Rate, W Time- Heat Flux, W/cm t=.5 sec t= sec t=3. sec Lift off t=6 sec t=3.5 sec t=. sec 6 r, mm Fig. 11(b). The variation of heat transfer rates from microlayer, from surroundings and total heat transfer rate with Time for T w =.6 K, T sub =.9 K. for time-varying gravity case. Fig. 1 (b). The Variation of Heat flux with Time for T w =.6 K, T sub =.9 K. for time-varying gravity case. 1 Copyright 3 by ASME

15 Z-Direction Y-Direction X-Direction T w = 9 K T sub =.9 K g/g e Time, s Fig. 13 (a) The Variation of Gravity in the experiments. Bubble Diameter, mm T w = 9 K T sub =.9 K Varying g 1 5 constant g =.5g e Fig.13 (b). The Variation of Equivalent Bubble Diameter with Time for T w = 9 K, T sub =.9 K. 15 Copyright 3 by ASME