A New Method for Estimating Bubble Diameter at Different Gravity Levels for Nucleate Pool Boiling

Nucleate pool boiling is an efficient mode of heat transfer, wherein high heat transfer rates can be accomplished at low temperature differentials between the wall and the liquid bubble point, alternatively known as wall superheat. This characteristic of nucleate pool boiling can therefore provide an efficient heat transfer mode that can be used to make components more compact for space-based applications. Nucleate pool boiling is influenced significantly by gravity, and it is therefore vital to understand its impact. We briefly summarize the behavior of pool boiling at earth gravity next and outline some of the areas where it differs at microgravity conditions. Typically, the life cycle of single vapor bubble in earth gravity consists of four stages as shown in Fig. 1:

Departure Phase: In the departure phase, bubble base radius starts to shrink, as the buoyancy force increases due to larger volume of the vapor bubble.

Lift Off: Bubble lifts off or pinches off once it has reached the peak size or departure size depending on the balance between the buoyancy, surface tension, and inertia.

Once a bubble lifts off, a new bubble generally forms or nucleates at the same location. There is a general consensus on this process in the research community. However, regarding the effect of reduced and microlevels of gravity, there is no general consensus yet. There have been multiple experimental, as well as numerical studies performed. A brief review is presented in Secs. 1.1 and 1.2.

It is well known that under reduced gravity, nucleate pool boiling behavior is different. Siegel and Usiskin [1] performed one of the first studies to investigate nucleate boiling under reduced gravity conditions using water as the test liquid. They performed nucleate boiling experiments under free fall conditions and observed that vapor bubbles remained attached to the heater surface instead of departing as under earth gravity. This could be attributed to the reduced buoyancy force with reduction in gravity. The reduction in buoyancy does not allow the buoyancy force to overcome the surface tension force, which holds the vapor bubble at the surface. Siegel and Keshock [2] observed that under reduced gravity the bubble growth period was longer and the bubble departure diameter was larger. Their observation is consistent with that of Qiu et al. [3], who conducted experiments on parabolic flights using the same test liquid (water). Lee et al. [4] performed nucleate boiling experiments in reduced gravity (⁠$g/ge=10−4$⁠) environment on a space shuttle. Kannengieser et al. [5] conducted boiling experiments on board the Sounding Rocket Maser 11. They found a large primary bubble stayed attached to the heater surface during the experiment. Using the Microheater Array Boiling Experiment (MABE) on the International Space Station (ISS), Raj et al. [6] performed nucleate boiling experiment using perfluoro-n-hexane (PFnh), which is a linear isomer of $C6F14$⁠. This isomer is the principal constituent of FC-72, which is widely used as an electronic cooling fluid. They reported a large vapor bubble covering the entire heater surface and did not observe bubble lift off. Warrier et al. [7] also performed experiments under microgravity conditions using PFnh as the test liquid and reported a large bubble being formed near the heater in their experiment on the ISS. The large bubble formation could be seen as a result of reduced buoyancy and merging of small bubbles. At low wall superheat temperature, the bubble departure was not observed; however at high wall superheats, the large bubble could depart from the surface. Even if the bubble departed, the authors observed the bubble to continue to hover near the heater wall. They concluded that under microgravity conditions surface tension and inertia forces dominate during most of the bubble growth period, until a bubble reaches a relatively large size. Once the bubble reaches a large size, it is observed to sit in the middle of the surface and pull smaller surrounding bubbles into it for merger. This large vapor bubble acts as a sink for vapor generated on the heated surface. With a large sized bubble, the growth rate is slower and surface tension balances buoyancy force. In microgravity conditions, buoyancy is weak due to reduced gravity, so bubble departure is not observed. These observations are similar to that reported by Merte et al. [8–10], who conducted pool boiling experiments in a space shuttle.

The experimental data of Usiskin and Siegel [11] and Straub et al. [12] indicated that nucleate boiling heat transfer rate was insensitive to the gravity. However, Zhao et al. [13], conducted nucleate pool boiling experiment aboard a recoverable satellite and observed that under microgravity the nucleate pool boiling heat transfer rate was lower than that under terrestrial gravity. The critical heat flux values were about three orders of magnitude lower than that obtained at earth gravity. Warrier et al. [7] performed nucleate boiling experiments at microgravity and also at earth normal gravity using identical heating surface. Their results showed that heat transfer coefficients decrease with the decrease in gravity level. Their observation is consistent with that of Zhao et al. [13]. Raj et al. [6] showed that at lower wall superheats microgravity boiling heat fluxes were larger than those obtained at normal earth gravity, but the trend reversed at higher wall superheats. The anomaly may be caused by the small heater sizes and high wall superheats which resulted in film boiling instead of nucleate boiling in the experiments of Raj et al. [6]. Kannengieser et al. [5] showed that when the wall temperature was lower than the saturation temperature the Marangoni convection was the dominant heat transfer mechanism while when it was higher than the saturation temperature, evaporation at the bubble base was the dominant heat transfer mechanism. However, they found little difference in the boiling heat transfer rates under microgravity and earth gravity conditions. Experiments conducted on the ISS also showed the dependence of the heat flux on wall superheat decreased as the pressure or liquid subcooling increased under microgravity (Warrier et al. [7]). However, the dependence of heat flux on wall superheat was weakened at earth normal gravity in comparison to that under microgravity conditions. It is believed that geometry of the heated surface is a major factor in the differences observed. In general, experiments conducted using wires showed little or no change in nucleate boiling heat transfer as gravity was changed while experiments using flat heated surfaces generally showed reduced heat transfer under microgravity. With the advent of computational resources, computational fluid dynamics (CFD) has also been used to study nucleate boiling by performing numerical simulations. A review of the significant past work of numerical simulations is provided in Sec. 1.2.

Lee and Nydahl [14] modeled bubble dynamics in saturated nucleate pool boiling on a heated horizontal surface by solving the axisymmetric Navier–Stokes and energy equations. They calculated the bubble growth rate, but they did not consider the change in bubble shape as their model assumed that the bubble maintained a hemispherical shape during its growth. Their model included a wedge-shaped microlayer whose thickness was adjusted in an ad hoc way to match the experimentally measured bubble growth. Mei [15] investigated bubble growth and departure. However, their study ignored the hydrodynamics of the liquid motion by the growing bubble. Furthermore, their model assumed that heat transfer to the bubble was only through the microlayer, which is not an accurate assumption for saturated liquid. Welch [16] used an interface tracking method to study the boiling problem. However, he only considered small distortions due to the limited capability of his method in handling topology changes. Kunkelmann and Stephan [17] simulated the contact line evaporation for nucleate boiling using volume of fluid method. Son et al. [18] simulated nucleate boiling by assuming axisymmetric and laminar flow. They used the level set method to capture the vapor–liquid interface. In their model, the computational domain was divided into micro- and macroregions. The macroregion included the bubble and the liquid surrounding the bubble. The microregion consists of the thin liquid microlayer that forms underneath the bubble. In the microregion, the lubrication theory was used to model the microlayer. The two regions were matched near the outer layer of the microlayer. The model predicted the bubble growth but not the relation between the heat flux and wall superheat. Abarajith et al. [19] used this model to investigate the effect of contact angle. Later, Abarajith et al. [20] used that model to study bubble merger in reduced gravity. They found that bubble merger in a plane lead to early bubble liftoff under low gravity conditions due to the additional liftoff force developed from the merger process. They also found that in most cases the bubble departure diameter for merged bubbles was much smaller than that for a single bubble case. The bubble merger also increased time averaged wall heat flux. However, as pointed out by Tanguy et al. [21], the use of smoothing of the velocity jump condition at interface can lead to misleading mass prediction. Shin et al. [22] simulated three-dimensional nucleate boiling using the level contour reconstruction method. Their model included the effect of nucleation site density and was capable of predicting the relationship between the heat flux and the wall superheat in a realistic surface. But they did not include contact line dynamics in their calculations. Wu and Dhir [23] conducted numerical simulations to investigate the effect of noncondensable gases. They found that accumulation of noncondensable gas caused a drop in the local vapor pressure and reduced the effect of subcooling.

Direct numerical simulation of nucleate boiling imposes significant challenges due to the issues of wall contact. Further, the coupling of unsteady mass, momentum, and energy transport with the complicated liquid–vapor interface dynamics and interfacial physics such as surface tension and discontinuity in material properties pose numerical challenges. Tanguy et al. [21] conducted extensive simulation to compare different numerical methods for the simulation of boiling flows. They found that solving the thermal boundary layer accurately around the bubble and computation of the boiling mass flow rate are critical for the boiling flow simulation. One of the major challenges of numerical simulations of multiphase problems is to capture the interface between two phases. In the moment-of-fluid (MoF) method used in this work, comparison between reference centroid information corresponding to the interface is made with actual centroid information. The difference between the actual centroid and reference centroid is minimized while keeping the actual volume fraction and reference volume fraction same. This ensures a sharp interface. Other numerical methods such as Front Tracking method (Juric and Tryggvason [24]), Level-Set method (Son et al. [18]), and Volume of Fluid method (Sato and Ničeno [25]) have been used for simulations of nucleate boiling; however, they exhibit certain limitations such as conservation of mass or volume is not maintained in some or the interface is not sharp or less accurate. Moment of fluid method used in this work guarantees conservation of mass and volume, and interface is both sharp and accurate. More detailed discussion about interface representation is presented in Sec. 2.3.

The remainder of the paper is divided into the following sections: A brief description of the numerical method and its implementation is provided in Sec. 2. Results and Discussion are reported in Sec. 3, beginning with the validation of the code, followed by novel results at earth gravity, reduced gravity, and microgravity simulations. Next, a new method is proposed for estimating the bubble diameter at different gravity levels. This method is based on an analysis of empirical data at different gravity values and using power-series curve fitting to obtain a generalized bubble growth curve irrespective of the gravity value. Conclusions are provided in Sec. 4.

where for material m:

where $nlv$ is the normal vector at the interface Γ, which points from liquid to vapor phase, h_fg is the latent heat of vaporization. This equation denotes the energy flux jump condition across the interface. It is based on the Rankine–Hugoniot condition.

Temperature at the phase change front (vapor–liquid interface) is always assumed to be the saturation temperature T_sat.

The numerical method is described on a rectangular Cartesian grid where the velocities (u and v) are discretized on the cell face centers and other variables such as the pressure (p), cell centroids (x), level-set functions (⁠$Φm$⁠), volume fractions (F_m), and temperature are discretized at the cell centers. The location of the definitions of the variables is shown in Fig. 2, where Γ is the interface.

At the vapor–liquid interface, both liquid and vapor phase meet, so a jump condition needs to be applied at the boiling front Γ to maintain the conservation of mass, momentum, and energy.

where $[f]=fvap−fliq$ is the jump operator; and $m˙=ρvap(uvap−VI)·nlv$ is the change of local mass flow rate.

where σ is the surface tension, κ is the local interface curvature and $∂un∂n$⁠, is the normal derivative of the normal component of velocity. The third term on the right-hand side is the recoiling pressure occurring with phase change, the recoil pressure has been ignored in the numerical implementation as it has been found to be of small value. $tl(i=1,2)$ represents the two tangential normals in three-dimensional space.

The multiphase flow solver solves the three-dimensional Navier–Stokes equations using the variable density pressure projection algorithm [27] on block structured adaptive mesh refinement grid (AMR) [28]. It can handle both compressible [29] and incompressible flows [30,31]. The solver employs the state of the art MoF method to represent multiphase interfaces [29,31–34]. It employs dynamic contact models for droplet impact problems [35]. Tests have showed that the code has a high parallel efficiency of more than 96% on a 48-core workstation.

During the MoF interface reconstruction process, a reference volume fraction function F_ref and a reference centroid, $Xrefc$ both corresponding to the real interface, are given in Fig. 3 (left) and the actual volume fraction function, F_A and the actual centroid, $Xactc$ corresponding to the reconstructed interface, are then computed as shown in Fig. 3 (right).

As discussed in Jemison et al. [31], the use of centroid information ensures the MoF method maintains a sharp interface.

The contribution to the interface velocity due to phase change is divided into two parts:

Next, the level-set functions and volume fractions are updated using the calculated phase change rates. The interface is perfectly sharp. This property is an extension to the work of Weymouth and Yue [36] to the phase problem. This property in the method addresses the over shoot and undershoot problem for material advection and volume conservation is maintained.

Validation of the code is done by comparison with the theoretical case of Scriven [37] which deals with the case of vapor bubble growth in superheated liquid under absence of wall-heat transfer in Sec. 3.1. Next, simulation results at earth gravity and at reduced gravity level are compared with experimental results in Secs. 3.2 and 3.3, respectively, followed by comparison of results with experiments on the ISS at microgravity level using Perfluoro-n-hexane as the test liquid in Sec. 3.4. A new method is proposed in Sec. 3.5 for estimating the bubble diameter at different gravity levels. This method is based on an analysis of empirical data at different gravity values and using power-series curve fitting to obtain a generalized bubble growth curve irrespective of the gravity value.

For the validation case, earth gravity, and reduced gravity cases, water is used as the test liquid and perfluoro-n-hexane is the test liquid for the case of microgravity conditions. The values used for the physical properties of both liquids are provided in Table 1 for reference. The values are at the saturation temperature of the liquid and atmospheric pressure.

Simulation of a growing vapor bubble in a superheated liquid has been performed in r-z coordinate system. There is an analytical solution available for this problem in the literature; hence, this provides a good basis for the code validation. The analytical solution was provided by Scriven [37]. In that paper, the author provided an equation which relates the bubble radius as a function of time for a specific superheat.

where ρ_v and c_pv are the density and specific heat capacity of the vapor phase and h_fg is the latent heat of vaporization. $ΔT$ is the superheat (i.e., the difference between the saturation temperature of the liquid and the actual temperature of the liquid).

where D_x and D_y are the bubble diameters in x and y directions, respectively.

A grid convergence study is presented in Fig. 4 for the case of $ΔT=5K$⁠. Here, the domain is divided into 32 by 32, 64 by 64, and 96 by 96 points, respectively, in each case. However, since the adaptive mesh refinement feature in the code has been used AMR = 1, this corresponds to a double value of the number of grid points in the multiphase regions. For the finest case of 96 by 96, the initial number of data points inside the bubble in each direction is about 80. From Fig. 4, it can be concluded that the 96 by 96 with AMR = 1 resolution has converged. Hence, this is the chosen resolution for the validation study.

The rate of increase of the bubble radius provides good agreement with the analytical solution. It should be noted that the numerical simulation data points start from a radius of 0.05 mm, as that is the initial radius value used for the simulations in Fig. 4. The comparison of the growth rates for different superheats (5 K, 2.5 K, and 1.25 K) with the respective analytical solutions is shown in Fig. 5. The results are similar to that of Sato and Ničeno [25]. The mean percentage error for each superheat for the MoF predictions from the Scriven equation values are presented in Table 2. The results show that error for all three superheats are lower than 4%.

Vapor bubble growth in presence of a wall heater is being considered under earth gravity. The wall is maintained at a constant temperature T_w. In this case, as the heat transfer from the wall is considered and due to the presence of gravity, bubble departure is expected. Urbano et al. [38] and Guion et al. [39] showed the significance of a microlayer and provided a figure to determine the cases for which the microlayer contribution would be significant. They showed in Fig. 9(a) of their paper that whether the case falls in the microlayer regime (i.e., a microlayer is formed and hence contributes to bubble growth) or Contact Line regime depends on a combination of Jakob number $(Ja=ρlcpl(Tw−Ts)ρvLvap)$ (where ρ corresponds to the density, c_p corresponds to the specific heat, T corresponds to temperature, and L_vap is the latent heat of vaporization, subscripts $l,v,w,s$ correspond to liquid, vapor, wall, and saturation), and Contact Angle. Figure 9(a) in Urbano et al. shows that our simulation case (with Ja = 21.03 and contact angle = 50 deg) falls under the Contact Line regime; hence, no microlayer consideration is required. This is further verified with good agreement between the MoF simulations with the experiments. Similar results were also reported by Tryggvason and Lu [40], who did not consider the microlayer contribution as well. For cases with higher Ja and Contact Angle, some form of microlayer modeling approach can be implemented as an extension to the MoF method. The results reported are a comparison with experimental results reported by Dhir [41]. In that paper, the authors provide numerical simulation as well as experimental results. The test liquid used is saturated water, with a wall superheat of $ΔTw=7K$ and static contact angle of $θ=50 deg$⁠. Domain size of 64 by 128 was used, with AMR = 1, which makes the effective grid size to be 128 by 256 points. We simulate half of the bubble and use symmetric boundary conditions. Thus, for the initial bubble size of diameter = 0.00075 m, the number of grid points inside the bubble along the diameter is ∼26. Figure 6 shows the snapshot of the bubble growth life cycle at different time instants.

Dhir [41] did not provide bubble shape for the case of $ΔT=7 K$⁠, so we compare the bubble shape at departure with Dhir [41] results for the case of $ΔT=8.5 K$ in Fig. 7.

Results in Fig. 7 show that the MoF method provides reasonably good prediction for bubble shape at departure in comparison to experiments.

A comparison plot of the MoF method is presented with both the numerical result and experimental result of Dhir [41] for the bubble growth rate indicating the departure diameter and the departure time in Fig. 8. It should be noted that in this case, as per Dhir [41], the diameter is calculated as equivalent bubble diameter for an equal volume sphere. Figure 8 shows the comparison of departure diameter and departure time. In their paper, Dhir [41] reported that their numerical simulations overpredicted the departure time and the departure diameter. For departure diameter, the percent difference between MoF simulation and experimental case is 1.12%, Dhir [41] is at 0.54%. But, the percentage difference of departure time between MoF simulation with experimental result is 16.72%, which is more accurate than that of Dhir [41] at 21.43%. Since the bubble in the MoF results has an initial radius, the data have been shifted such that in all the cases in Fig. 8, the vapor bubbles are at the same initial radius at the initial time to maintain consistency and accuracy between the reported results and the comparison results.

Dynamic contact angle has been used in the studies by Ajaev et al. [42], Mukherjee et al. [43], Jo et al. [44] among others. Ajaev et al. [42] explore dynamic contact angle for evaporating liquids on inclined surfaces, not nucleate boiling. However, the study by Mukherjee et al. [43] explores the effect of static contact angle, and different cases of dynamic contact angles on single bubble nucleate boiling. They conclude that the vapor-removal rate shows very little change for three following cases: static contact angle (case 1), dynamic contact angle with constant value advancing and receding angle (case 2), and dynamic contact angle as a function of interfacial velocity (case 3).

where θ_s is the static contact angle, Ca (Capillary number) is given as: $Ca=μVclσ$⁠, where μ is the dynamic viscosity of the liquid, V_cl is the contact line velocity, and σ is the surface tension. One drawback of the Jiang model is that it is not as accurate for large values of Ca [47]. However, since Ca values are significantly small in our simulations, that drawback is not applicable. Using the Jiang model, we found the maximum range of the dynamic contact angle to be about 2 deg, which shows that the effect of the dynamic contact angle in this case is negligible. The results for growth rate for nucleate boiling in earth gravity using these two models in comparison to the static contact angle model, and experimental results from Dhir [41] are reported in Fig. 9.

The results in Fig. 9 show that the departure time and diameter for the Kistler model is under-predicted, whereas for the Jiang model, it is similar to the static contact angle model for the earth gravity case. The growth rate, however, is almost similar for both Jiang and Kistler models in comparison to the static contact angle model. These results are in agreement with the conclusions of Mukherjee et al., that the vapor removal rate sees very little change. Hence, for the results reported in this paper, a static contact angle model has been used.

Next, a plot of the heat transfer rate as a function of time is reported in Fig. 10. Here, the heat transfer is calculated as the heat energy per unit time required to vaporize the volume of liquid which the bubble grows in that time. Since it considers the total heat supplied to the bubble, it takes into account both the heat transfer from the wall and the heat transfer from the surrounding liquid. The bubble growth rate is also plotted on the right Y-axis of Fig. 10. A peak could be observed in the heat transfer rate at a later stage of the bubble life ycle. During the departure phase, the bubble base shrinks, which means increased quantity of colder water surrounds the bubble, this makes the vapor bubble to lose some of its heat by conduction to the colder water, i.e., conduction of heat from the vapor bubble is increased; hence, more heat is transferred to the bubble. The peak signifies this additional heat transfer. Finally, there is a drop in the heat transfer rate, as the bubble enters departure phase; this is evident from the shrinking size of the bubble base diameter from about 0.03 s mark to the ultimate liftoff. Heat transfer rate continues to drop till the bubble lifts off and departs indicated by the vertical line, which is in agreement to the reasoning provided above.

In the case of reduced gravity, a comparison of the MoF simulations is presented in comparison with results from experiments by Siegel and Keshock [2]. In this paper, the authors provide experimental results for reduced gravity cases which were performed using a drop tower. The results are for a case with wall superheat of $ΔTw=11.1 K$⁠, and $gz=1.4%ge$⁠, where g_z corresponds to the effective gravity value and g_e corresponds to the value of gravity at earth. The simulations were carried out using a grid resolution of 128 by 256 and AMR = 1, which makes the effective grid size to be 256 by 512 in the X-direction and Y-directions, respectively. The initial number of grid points inside the bubble along the diameter is 26, as only half the bubble is simulated due to symmetric boundary conditions. The bubble shape at different time instants is shown in Fig. 11. With reduced gravity environment, the bubble did not observe departure similar to that in the experiments.

Figure 12 shows the bubble growth rate comparison of numerical results of MoF method with experimental results of Siegel and Keshock [2]. The growth rate results provided by the MoF simulation are in good agreement with the experimental results. Next, the heat transfer as a function of time is provided for the reduced gravity case in Fig. 13.

Heat transfer in the reduced gravity case does not observe the drop in comparison to the earth gravity case, as the bubble does not enter the departure phase in this case. The growth rate is also plotted along the right y-axis for reference.

MoF simulation of boiling in microgravity conditions is compared with experiments on the ISS by Dhir et al. [48]. For the experiments, the wall superheat was in a range of 4 °C–7 °C and the liquid subcooling varied from 5 °C to 1 °C due to variance in pressure. For the comparison of the numerical simulation using MoF method with the microgravity simulations, we use a saturated liquid (i.e., no subcooling). Dhir et al. [48] reported multiple challenges with their experiments on ISS, one of the key challenges being a heater malfunction, which resulted in the system pressure to vary and this resulted in an increase in the liquid saturation temperature. As a consequence of this increase in the liquid saturation temperature, the liquid subcooling got introduced. Dhir et al. then increased the wall temperature to compensate for the increase in liquid subcooling. However, in our simulations, the system pressure, liquid saturation temperature, and wall temperature were maintained to be constant through-out the simulation. Hence, there was no need for subcooling or increasing wall temperature in the MoF simulations. The wall superheat for our simulations is set to be constant at 16 °C. The MoF simulation is run for a domain of 32 mm × 32 mm with a base resolution of 144 × 144 and a refinement setting of 2. As mentioned earlier, a refinement value of 2 corresponds to a resolution of 576 × 576. This results in about 110 initial grid points inside the bubble in along the diameter. For our results, we performed the simulations for about 5 s of total run time, where in the bubble diameter reached a size of about 20 mm. For the microgravity case, since the bubble size is larger, the domain is also larger, making the required resolution to be very fine. This makes the simulation computationally very expensive and time-consuming. The reason for that is the time-step size is very small due to the stability time constraint for surface tension. The stability time constraint due to surface tension $tsurf∝(Δx)3/2$⁠. Since $Δx$ is very small, t_surf is also very small, which requires a large number of time-steps to reach a significant total simulation time value. Figure 14 shows a comparison of the experimental results with the MoF simulations for the same test liquid (perfluoro-n-hexane) in microgravity conditions.

The results from the MoF simulation are in good overall agreement with the experimental ones. As in the experiments, the simulations also did not observe bubble departure in the microgravity environment. A zoomed in plot for the portion of the run time of the simulation is added on the right side of Fig. 14. In order to provide a qualitative understanding of the above simulation under microgravity, the heat transfer rate is also plotted in Fig. 15 as a function of time. In Fig. 15, the growth rate is also plotted on right y-axis for reference. The slight difference in the growth rate of the simulations in comparison to the experiments could be attributed to the uncertainties in the experiment itself which are mentioned in details in their paper. Moreover, only one experiment for single bubble case was completed in the experiments, which makes the statistical uncertainty considerably high.

Based on the heat transfer plots provided for each gravity case, it can be seen that for the same amount of heat transfer of 0.5 W, the bubble diameter is different. For earth gravity with water, the diameter at 0.5 W of heat transfer is about 2.5 mm, the same for water in reduced gravity is 6 mm, and the same for Pfnh in microgravity in 8.3 mm. From the results, it can be concluded that as gravity is reduced, less heat transfer is required to reach the same bubble diameter.

So far, bubble growth and departure results for cases at different gravity levels have been reported. A closer analysis of the results shows that there is a similar pattern to the growth rate curves of the bubbles at each gravity level. The growth rate changes with gravity. For this case, we use the experimental results at reduced gravity by Siegel and Keshock [2]. In their paper, they used a drop tower to control the value of the gravity. We use their results for three different gravity values of 1.4%, 3.0%, and 6.1% of earth gravity. This has been shown in Fig. S1 (available in the Supplemental Materials on the ASME Digital Collection).

where $d∗$ is the nondimensional diameter, D is the diameter of the bubble at any time instant, $g∗$ is the nondimensional gravity, g is the gravity value in question and g_e is the earth gravity value. C is the coefficient for the specific case and D_e is the departure diameter at earth gravity. $t∗$ is the nondimensional time, where T is the time instant of the simulation or the experiment and T_e is the departure time at earth gravity. It should be noted that T_e and De could be any value for time and length, respectively. Both T_e and De-are used as constant value and their significance lies only in nondimensionalization of the time and diameter.

We start by choosing one time instant from Fig. S1 (available in the Supplemental Materials) for all three gravity values. In this case, we chose $t=0.3003 s$⁠. Next, we plot the diameters at this time-instant for all three gravity values and perform a curve fitting, as shown in Fig. S2 (available in the Supplemental Materials).

Based on the curve fitting equation, we can generate the model equations, which is shown above. Using the model equations, we plot the growth rates at three different gravity level in Fig. S3 (available in the Supplemental Materials). It should be noted that the values of the coefficients of the equation are specifically for the case of Siegel and Keshock [2]. Only data from Seigel and Keshock are used since this is the only paper, which provides experimental results at different gravity levels, by keeping all other parameters constant. Since the proposed equation is mainly dependent on gravity, it is imperative to keep other variables such as wall superheat and system pressure as constant. The test liquid (water in this case) parameters, the superheat value, and all other parameters that contribute to the growth of a bubble also contribute toward the calculation of the coefficients. The coefficient C for this case was calculated to be $C=−0.1587$⁠.

This single curve in Fig. S3 (available in the Supplemental Materials) can then predict the effective diameter of a bubble if the time instant and the gravity value are known. We validate this claim by calculating the $d∗$ from the 3.0% gravity value, and then plug in the $d∗$ to predict the diameters at 1.4% and 6.1% gravity values at those time instants. This is compared with the experimental results in Fig. S4 (available in the Supplemental Materials).

It shows that the model is accurate at gravity levels for both higher and lower values than the one, which was used to calculate $d∗$⁠. Additionally, although the diameter from only one time instant was used for performing the curve fit to generate the coefficient, the model predicts accurate results for all time instants.

An adaptive Moment of Fluid method has been used to perform numerical simulations of nucleate boiling of water vapor bubble in a pool of test liquid (water) under different heat transfer conditions. Comparison of results of single bubble dynamics especially bubble growth rate, departure radius, and departure time has been made with different results. Validation with analytical solution of Scriven for bubble growth in superheated liquid under absence of wall heat transfer is reported as the first case. Simulations in earth gravity, reduced gravity, and microgravity are presented in comparison with experimental results from Dhir [41], Siegel and Keshock [2], and Dhir et al. [48], respectively. The specific cases that were simulated did not have high wall superheat values for reduced and microgravity cases. Some prior works cited have shown that for high wall, superheat values may observe bubble departure and after departure, the bubble continued to hover around the heater in microgravity conditions. MoF simulations with higher values of wall superheat should be performed in a future work to compare with these experimental observations. A new method is proposed for estimating the bubble diameter at different gravity levels for a specific liquid. This method is based on an analysis of empirical data at different gravity values and using power-series curve fitting to obtain a generalized bubble growth curve irrespective of the gravity value. The significant conclusions are listed below:

1. The new proposed method of diameter prediction is shown to provide a good estimate of the bubble diameter for any gravity value at any time.

2. From the reduced gravity and micro gravity results, it can be observed that as gravity is reduced, less heat transfer is required to reach the same bubble diameter.

3. MoF simulation results for earth gravity, reduced gravity, and micro gravity are in good agreement with experiments in pertinence to bubble growth rate and bubble shape.

4. Bubble departure is not observed in reduced gravity and micro gravity cases, which is in agreement with experiments due to lower buoyancy.