1. Introduction

2. Materials and Methods

2.3. Computational modeling

2.3.1. Description of the case and geometry

An axis-symmetric domain is constructed with the cylinder specimen placed in the middle of the domain height, as shown in Figure 1. The computational domain spans over the area x ∈ [-1.025e-1, 1.025e-1] m × y ∈ [0, 1,025e-1] m, and is composed of fluid and solid part, separated via the fluid-solid interface. At the domain top, a 15 mm high outlet region is prescribed, containing the vapor phase. In the vicinity of the solid specimen, a one-cell vapor layer is initially placed, with a thickness of 0.61 mm, required for the mass transfer model in order to induce the mass transfer. In Figure 1, the vapor blanked is, for the sake of simplicity, enlarged.

Figure 1. A schematic representation of the validation model with the boundary conditions in two dimensional axisymmetric coordinate system. The dimensions are in mm.

Figure 1. A schematic representation of the validation model with the boundary conditions in two dimensional axisymmetric coordinate system. The dimensions are in mm.

The domain is composed of 7938 quadrilateral cells (Figure 2a), with the refined cell zone adjacent to the specimen surface (Figure 2b). The specimen is made of silver and has equal diameter and height, 45 mm.

Figure 2. (a) The entire computational mesh; (b) A magnified view in the vicinity of the fluid-solid boundary.

Figure 2. (a) The entire computational mesh; (b) A magnified view in the vicinity of the fluid-solid boundary.

2.3.2. Boundary conditions

The boundary conditions are assigned to the domain’s top, bottom, and left side surface for both the phases, whilst the domain symmetry axis is defined as “axis” in the applied software. Thus, the top of the domain is set to pressure outlet condition, with prescribed outlet temperature 100 ^oC for both phases and the unit backflow vapor volume fraction. The bottom and left surface are defined as solid walls. The specimen surface is composed of three walls that interact with the fluid region via “coupled” condition.

2.3.3. Initial conditions

The initial conditions involve the initial distributions of phase volume fractions, temperatures and velocities, supplemented with the initial profile of turbulent kinetic energy that is, in contrast to other aforementioned dependent variables, maintained constant through the simulation, since the turbulence equations are not being solved - so-called „frozen turbulence” approach. The velocity fields of both phases were initially set to zero elsewhere in the domain, although the turbulent kinetic energy is initially set to a non-zero value everywhere except the initial vapor layer.

The initial temperature of a solid specimen is set to 600 ^oC, as per experiment in Momoki et al. [9]. The liquid and outlet zone were set to the initial temperature of 100 ^oC. Inside the vapor layer, an initial temperature field is prescribed as follows (see Figure 3 and Figure 4).

Figure 3. A schematic representation of initial temperature field distribution: (a) In radial direction, in vicinity of the vertical surface; (b) In the immediate neighborhood of top and bottom horizontal surface.

Figure 3. A schematic representation of initial temperature field distribution: (a) In radial direction, in vicinity of the vertical surface; (b) In the immediate neighborhood of top and bottom horizontal surface.

Figure 4. A conceptual sketch on determination of initial temperature in the edge control volume: (a) The position of the control volume; (b) The neighboring cell center values used in computation.

Figure 4. A conceptual sketch on determination of initial temperature in the edge control volume: (a) The position of the control volume; (b) The neighboring cell center values used in computation.

In radial direction along the cylinder height, due to the analytical solution of the heat conduction equation in the radial direction of a cylindrical coordinate system (see Figure 3a for a qualitative description), a logarithmic temperature profile is imposed as follows:

$$T_{cell,i}=\frac{T\prime -T_w}{ln\frac{H+\Delta x_c}{H}}ln\frac{y_{c,i}}{H}+T_w,$$

(13)

whilst in the axial direction, at top and bottom surfaces, respectively, it follows the linear law due to the same reason as in the former case (the analytical solution to heat conduction PDE in the axial directions of the cylindrical coordinate system; see Figure 3b for a qualitative description):

$$T_{c,i}=\frac{T\prime -T_w}{\Delta x_c}\left(x_{c,i}-H\right)+T_w;$$

(14)

$$T_{c,i}=\frac{T\prime -T_w}{-\Delta x_c}\left(x_{c,i}+H\right)+T_w$$

(15)

The edge element, however, is filled with the initial temperature computed as (Figure 4):

$$T_{c,i}=\frac{T_{ln}+T_{lin}+2T\prime }{4},$$

(16)

that stems from an energy balance on a knot in the cell center assuming thereby that the face temperatures of neighboring cells are those of their respective cell centers. This yields, however, the initial temperature in the edge control volume (CV) that is lower than these other two temperatures, due to cooling contributions from two sides that remain at, significantly lower, saturation temperature.

2.3.4. Time stepping procedure applied in the simulation

The time step applied in simulation is divided into two parts. Firstly, ten time steps with a size of 1e-6 s were carried out. Then, a 2e-4 s time step size has been applied in the remainder of the simulation. Extensive study on the applied time-step size, and a brief comment on the applied equation- discretization techniques can be found in [10].

3. Results

3.1. Temperature distribution

The temperature distribution in a solid specimen agrees well with the experimental data by Momoki et al. [9] in approximately ten percent of total film boiling period2, as shown in Figure 5.

Figure 5. The temperature evolution in a solid specimen during approximately the first 15 s of the simulation; the data is compared to the experimental results from Momoki et al. [9], extracted using [26]. The dashed line denotes ± 2 % error band.

Figure 5. The temperature evolution in a solid specimen during approximately the first 15 s of the simulation; the data is compared to the experimental results from Momoki et al. [9], extracted using [26]. The dashed line denotes ± 2 % error band.

The time shift is accomplished using initial vapor thickness as shown in [10], since there is no known analytical solution to Stefan problem in cylindrical coordinates, as outlined by Galović [27]; yet there is a quasi-steady approximate solution that is valid for low Stefan number, Ste = c_p(T_w – T’)/r₀, that is, in the case of the low wall superheat.

3.2. Comparison of calculated heat transfer coefficients

The area weighted average heat transfer coefficients were extracted using the graphical-user interface; the data is stored approximately every 0.2 s. The calculated heat transfer coefficient, h, in the performed numerical simulation stems from the Newton’s cooling law:

$$h=\frac{q_w}{T_w-T_{\infty }},$$

(17)

where q_w is the heat flux at the specimen wall, T_w and T_∞ are the wall and free stream temperature, respectively. The area-weighted heat transfer coefficient is calculated as a post-process, considering thereby the heat transfer rates at bottom, top and vertical (circumferential) surface of the cylinder; that is, the overall heat transfer coefficient is taken into account; instead of separate, per-surface heat transfer. The analytical solution to average overall heat transfer coefficient is presented in Momoki et al. [9] and reads:

$$\underset{\_}{h}=\left[\frac{{\underset{\_}{h}}_A+4\frac{\left\{{\underset{\_}{h}}_{B1}{L}_{B1}+{{\underset{\_}{h}}_{B2}}_{}\left(L-L_{B1}\right)\right\}}{D}+{{\underset{\_}{h}}_C}_{}}{2+4\left(\frac{L}{D}\right)}\right],$$

(18)

where ${{\underset{\_}{h}}_A}_{}$ and ${\underset{\_}{h}}_C$ are, respectively, the average heat transfer coefficient at the bottom and top horizontal surface of the cylinder; ${{\underset{\_}{h}}_{B1}}_{}$ and ${{\underset{\_}{h}}_{B2}}_{}$ are the heat transfer coefficients at the vertical surface in the region of smooth and wavy vapor-liquid interfaces, respectively, whilst L and L_B1 denote the total and smooth interface length, respectively; D is the diameter of the cylinder. In their study, the authors impose the ± 15 % error bandwidth on the estimated average heat transfer coefficient results. Hence, we will stick here to the value obtained using the Equation (18), and the result is shown in Figure 6.

Figure 6. The calculated heat transfer coefficients using numerical simulation in comparison to ones obtained using Equation (18).

Figure 6. The calculated heat transfer coefficients using numerical simulation in comparison to ones obtained using Equation (18).

In the formulae for average heat transfer coefficient during film boiling around a finite length cylinder specimen, Equation (18), the contributions from each heat transfer surface via its own heat transfer coefficient noted above, the vapor phase is considered and superheated and thus the thermal properties of the vapor phase were taken for mean temperature, ϑ_m = (ϑ_w + ϑ’)/2, from [28] using the linear interpolation technique, and are listed in Table 1.

Table 1. The thermal properties of superheated vapor are used in estimation of film boiling heat transfer coefficient at the temperature 350 ^oC.

Table 1. The thermal properties of superheated vapor are used in estimation of film boiling heat transfer coefficient at the temperature 350 ^oC.

Phase	$\rho$, kg/m3	cp, kJ/(kg K)	λ, W/(m K)	Pr, -
Vapor	0.35	2.04	0.041	0.91

The liquid (water) is considered as saturated, hence among the thermal properties of the water, only the density is used, ${\rho }_l$ = 958.41 kg/m³, also taken from [28].

The discrepancy is found to be slightly more than 30 % in the present numerical simulation. The average heat transfer coefficient from the numerical simulation is calculated as:

$${{\underset{\_}{h}}_{simul}}_{}=\frac{{\sum }_{}^{}{h}_i\Delta t_i}{t_{total}},$$

(19)

where, ${{\underset{\_}{h}}_{simul}}_{}$ refers to a calculated heat transfer coefficient using numerical simulation, $\Delta t_i$ is the time segment used for the averaging of heat transfer coefficient, and $t_{total}$is the total simulation time, ~15 s.

In Figure 6, furthermore, one can observe the cyclic behavior of the heat transfer coefficient. This is due to cyclic behavior of the vapor-liquid interface during the film boiling process, that has been also noted in study by Tsui et al. [29]. The heat transfer coefficient, obtained via the numerical simulation, enters the upper limit of the error band, that is 15 %, prescribed by the authors of the correlation.

The vapor evolves from the top horizontal surface and is being advected due to buoyancy to the free surface of water, as shown in Figure 7; forming thereby the vapor jet, as observed also in other studies. Similar flow patterns were observed with photographic study in Tsui et al. [29] and Jurić and Tryggvason [30]; both being calculated for film boiling on a horizontal surface.

Figure 7. Transient behavior of: (a) vapor film, (b) temperature in the vapor phase and (c) temperature in the liquid phase.

Figure 7. Transient behavior of: (a) vapor film, (b) temperature in the vapor phase and (c) temperature in the liquid phase.

The temperature field, on the other hand, exhibits slightly overheating above the top of the cylinder, while being uniform elsewhere. The superheats may be found comparable, at least quantitatively, to those in the film boiling study by Sato and Ničeno [7]. The uniform temperature distribution in a solid is due to its small Biot number and, therefore, the inexistence of spatial temperature gradients in a solid. Since the vapor temperature doesn’t differ significantly from the liquid temperature, the latter would be studied any further in this text.

An analysis of a single bubble using Newton’s second law, that is, the momentum conservation equation, shows how the bubble shape is influenced by inertia, buoyancy, viscous and surface tension forces. The study by Yamada [31] reveals the true behavior during film boiling of a cylinder specimen studied here, but with different dimensions. It is rather difficult to make a direct comparison with the experimental observation, but at least qualitatively one may obey the similarity between the performed simulation and the experiment that is conducted using silver specimens of similar dimensions and wall superheat.

In Table 2, a comparison of the flow fields obtained in the numerical simulation and the one in experiment is made. The influence of surface tension is in the tendency to minimize the contact area between the bulk fluid and a vapor phase, i. e., the surface tension force acts on the vapor, and tends to minimize the shape of the vapor bubble.

Table 2. Joint comparison of experimental and the flow fields (volume fraction) obtained with usage of numerical simulation.

Table 2. Joint comparison of experimental and the flow fields (volume fraction) obtained with usage of numerical simulation.

Quantity	Simulation; d × L = 45 × 45 mm	Experiment [31]; d × L = 32 × 32 mm
Flow field
t, s	10.22	15
ΔTw, K	444.9	~345

3.3. The applied turbulent kinetic energy value

The effective thermal conductivity in a liquid phase, as outlined before, has been found to be crucial in appropriate modeling of heat extraction from a solid specimen. Hence, since the molecular thermal conductivity of the liquid phase remains constant thorough the computation, one should affect the turbulent part of the effective thermal conductivity; which, on the other hand, is a function of a turbulent viscosity since the turbulent Prandtl number is assumed as constant. The turbulent viscosity, from the viewpoint of truly relevant dependent variables, is a function of turbulent kinetic energy, although there is also a factor C_μ, that in the case of the selected two-equation turbulence model is a variable, that is, it has not its default value 0.09 as in standard and RNG instances of the model. However, we will now focus on turbulent kinetic energy as the key parameter in turbulence modeling; similar has been also pointed out in the thesis by Subhash [32]. Within the framework of the present work, the TKE value has been obtained using a parametric investigation. A parametric study has been carried out in order to determine the TKE value for which the temperature evolution would agree with the experimental result. In this case it was 0.25 m²/s². This value, however, may be obtained from Equation (6) if the velocity input is taken from numerical simulation; the volume-weighted average velocity magnitude field of the vapor phase in the boundary layer in vicinity of the vertical surface of the cylinder. In doing so, the one-cell boundary layer thickness may be found as appropriate since the two-fluid model is used, and the mesh resolution does not play an important role in obtaining the accurate solution, as shown in study by Gauss et al. [33]. In addition, the usage of fine mesh resolution is regarded as inappropriate in the framework of Euler-Eulerian modeling, as noted in the study by Liu and Pointer [34]. Figure 8 compares the calculated TKE values using Equation (6) and the one obtained with parametric study and used in this validation case.

Figure 8. The comparison of the applied TKE value, obtained by use of a parametric analysis, and the ones obtained using Equation (6).

Figure 8. The comparison of the applied TKE value, obtained by use of a parametric analysis, and the ones obtained using Equation (6).

It is, however, obvious that the applied TKE value obtained using the parametric investigation is slightly overestimated in the very beginning of the film boiling process (first 5 s of the process), whilst being completely overestimated in the later section of the process; when compared to calculated values using Equation (6).

This range of TKE values, say, ~ 0.15 m²/s² to ~ 0.25 m²/s², could be related to the studies that follow. The study by Kashima et al. [18] brings us the study from Svendsen [35], in whose model, if the TKE values in the range from 0.05 m²/s² to 0.15 m²/s² are used together with a certain turbulent length scale, one may obey the energy dissipation rate, in the context of breaking surf zone wave analysis. It is interesting to note that these values coincide with the TKE values obtained in instability of a stable vapor flow study presented in Zimmer and Bolotnov [36]. In their study, the authors found 0.15 m²/s² value of TKE as the threshold value in causing the instability of the stable slug flow. Also, the authors have proposed the correlation for which average value of TKE the instability may occur. Furthermore, these findings, obtained in the present research, are in close correspondence with the TKE value obtained in an analysis of a single jet flow in Kimber [16], wherein a 0.2 m²/s² is obtained slightly beneath the jet core.

On the other hand, the TKE value of 0.25 m²/s² has been chosen as the highest TKE input value in the simulation of dispersed two-phase flow using LES method in the context of atmospheric turbulent boundary layer modeling in Vinković et al. [37]. Contrasting the temperature field in a solid specimen (Figure 5) with the calculated TKE values (Figure 8) may reveal that the constant TKE value resulted in too strong cooling after initial say 5 s of the process; yet the variable TKE value should be considered in further studies.

The visual observation of turbulent viscosity and corresponding vapor volume fraction fields shown in Figure 9., reveals how the former closely follow the pattern of the latter. This may be addressed to the selected mode of the applied turbulence model, i. e., the dispersed mode, wherein the turbulence is being modeled only in the continuous phase, whilst Tchen correlation has been used in the dispersed phase. In addition to this, there is also a changeable turbulent viscosity coefficient, Cμ, that is present in the selected, although not solved, turbulence model. This may be found relevant in obtaining the turbulent viscosity adjacent to the interface, in the liquid phase; the transition values from zero to a constant value that is in the bulk, since the k is assumed as a constant. Hence, it is obvious that a laminar flow is concerned within the vapor phase, implying the existence of the BIT in the performed numerical simulation. Speaking in terms of analogy with a turbulent jet flow, the realizable k-epsilon turbulence model is recommended for such flows.

Figure 9. The distributions of (a) turbulent viscosity and (b) volume fraction of vapor in two different time instances.

Figure 9. The distributions of (a) turbulent viscosity and (b) volume fraction of vapor in two different time instances.

In the present case, the Cμ is a function of strain rate, that is, the non-diagonal elements of a strain rate tensor (momentum equation) that can be found in standard continuum mechanics or fluid mechanics textbooks and reads [38]:

$${\epsilon }_{ij}=\frac{1}{2}\left(\frac{\partial u_i}{\partial x_i}+\frac{\partial u_j}{\partial x_j}\right),$$

(20)

where i and j are indexes of spatial coordinates in index notation (i, j = 1, 2, 3; meaning: x-, y- and z-coordinate), with u as the fluid velocity in the i-th direction and correspond to tangential stresses (friction). Hence, the distribution of the turbulent viscosity exhibits the pattern shown before in Figure 9; it experiences smooth transition in the areas in the liquid that are occupied with a vapor-liquid interface. This friction is further transferred into the liquid in the nearby zone to the interface between the phases. Once more, we prove, at least quantitatively, the necessity of the application of such an instance of the selected turbulence model (realizable) in modeling such a phenomenon (film boiling).

A joint view of velocity magnitude of the vapor phase and its strain rate shows how the friction is about to occur at the interface nearby zone, say, the “shear layer” in the context of the turbulent jet flow, Figure 10, where velocity gradients are the highest. A discussion on the application of this model in the context of jet flow in a confined space is available in Heschl [39].

Figure 10. An observation in vapor phase distributions of (a) velocity magnitude and (b) strain rate. The strain rate follows closely the interface shape, indicating the presence of friction at the interface between the phases.

Figure 10. An observation in vapor phase distributions of (a) velocity magnitude and (b) strain rate. The strain rate follows closely the interface shape, indicating the presence of friction at the interface between the phases.

4. Conclusions

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