### 1. Introduction

### 2. Subcooled Boiling Model

### 2.1 Rohsenow Boiling Model

*μ*is the dynamic viscosity of the liquid phase;

_{l}*h*is the latent heat of vaporization;

_{lat}*g*is the gravitational acceleration;

*ρ*is the density of the liquid;

_{l}*ρ*is the density of the vapor;

_{v}*σ*is the surface tension coefficient at the liquid-vapor interface;

*C*is the specific heat of the liquid phase;

_{Pl}*T*is the temperature of the wall;

_{w}*T*is the saturation temperature;

_{sat}*C*is an empirical coefficient varying with the liquid-surface combination; Pr

_{qw}*is the Prandtl number of the liquid;*

_{l}*n*is the Prandtl number exponent that depends on the surface-liquid combination.

_{p}### 2.2 Wall Boiling Model

*q*represents convective heat flux, which refers to the contribution of the heat flux transferred from the wall to the convection of the liquid, and is defined by Eq. (4). where

_{C}*h*indicates the single-phase heat transfer coefficient;

_{C}*T*is the average liquid temperature;

_{l}*A*is a coefficient that depends on the bubble departure diameter and the nucleation site density;

_{b}*K*is an empirical constant;

*N*is the nucleation site density;

_{W}*d*is the bubble departure diameter;

_{W}*Ja*is the subcooled Jacob number; ∆

_{sub}*T*=

_{sub}*T*−

_{sat}*T*represents the subcooling temperature of the liquid;

_{l}*h*is the latent heat of evaporation.

_{lat}*q*indicates the evaporation heat flux. It refers to the contribution of the heat flux transferred from the wall to the boiling of the liquid, and is expressed using Eq. (8). where

_{E}*V*indicates the volume of vapor bubbles for the bubble departure diameter, and

_{d}*f*represents the bubble departure frequency defined by Cole (1960).

*q*is the quenching heat flux, which refers to the average heat flux transferred to the liquid when the liquid occupies an empty space immediately after bubble departure, and it is defined using Eq. (11). where

_{Q}*k*indicates thermal conductivity;

_{l}*λ*indicates the diffusivity of the liquid;

_{l}*τ*is the periodic time of bubble departure.

*N*,

_{W}*d*and

_{W}*f*. As these parameters vary according to diverse factors, they are generally derived empirically through experiments, and the correlations between these parameters have been studied widely (Kurul, 1990; Alglart, 1993; Alglart and Nylund, 1996; Tu and Yeoh, 2002; Lee et al., 2002; Koncar et al., 2004; Krepper et al., 2007; Chen and Cheng, 2009; Krepper and Rzehak, 2011; Nemitallah et al., 2015; Gu et al., 2017). Previous studies have been confined mostly to selecting the optimal combination of submodels (nucleation site density and bubble departure diameter models), which is suitable only for a specific pressure section of a pipe. Gu et al. (2017) compared five combinations of submodels using ANSYS Fluent, a commercial analysis program. Compared to ANSYS Fluent, STAR-CCM+ used in this study has limitations in that it does not include the model proposed by Unal (1976) among the bubble departure diameter models. On the other hand, as some combinations were omitted from the comparison in Gu et al. (2017), it was considered necessary to conduct a comparative study using the combinations provided by STAR-CCM+. Therefore, in the present study, an overview of the and models provided by STAR-CCM+ were presented, and a parametric study on the combination of

*N*and

_{W}*d*models and the minimum bubble departure diameter was investigated.

_{W}####
2.2.1 Nucleate site density, *N*_{W}

_{W}

*C*and

*n*are coefficients determined based on experimental data; they were estimated to be 210 and 1.805, respectively, in Kurul and Podowski (1991).

*N*proposed by Kocamustafaogullari and Ishii (1983) is as follows (KI model): where ∆

_{W}*T*represents the effective superheat, and

_{e}*S*is the suppression factor.

####
2.2.2 Bubble departure diameter, *d*_{W}

_{W}

*d*proposed by Tolubinsky-Kostanchuk (1970) is as follows (TK model):

_{W}*d*as follows (KI model): where

_{W}*ϕ*represents the contact angle of bubbles, and

*ϕ*was set to be 80° in Rogers and Li (1994).

### 3. Numerical Simulation

### 3.1 Setting Up the Problem of Flow Boiling in a Tube

^{2}·s into a vertical tube made of stainless 1Cr18Ni9Ti steel with a height (

*H*) of 2.0m and a diameter (

*d*) of 15.4mm. A constant heat flux of 570 kW/m

^{2}was applied to the surrounding wall to induce a phase change of the water into vapor, and the temperature and vapor content according to the flow enthalpy were measured. At this time, the initial pressure inside the tube was set to 45 bar (= 45 × 10

^{5}Pa).

*L*) was set to 0.5m to allow subcooled water to achieve fully developed flow. At the outlet, the vertical length of the outlet region (

_{inlet}*L*

_{outle}_{t}) was also set to 0.1m to allow the flow to exit smoothly. Furthermore, by applying axial symmetry, only one wall in Fig. 3 was heated with the heat flux level observed in the experiment. Adiabatic conditions were set for other walls except for the heated section. The simulations were conducted by assuming a steady state based on Sontireddy and Hari’s (2016) results, who reported no significant differences in the simulation results between the steady and unsteady states.

### 3.2 Simulation Results

#### 3.2.1 Comparison between different boiling models

#### 3.2.2 Comparison of the simulation results by different combinations of nucleation site density and bubble departure diameter models (Cases 1-3 & 9)

#### 3.2.3 Grid convergence test (Cases 4–6, 9, 12 & 13)

*H*= 1.1–1.7 m (the ONB-SNB section), where the initial bubbles started to be generated on the wall, and the results closely approximated the experimental results. At the outlet region, however, there was no significant difference in the void fraction according to the grid system.

*H*≈ 1.3–1.7 m), where the ratio of vapor in the internal liquid becomes relatively higher, large differences according to the grid size were observed. This is believed to be related to the grid resolution, which indicates the degree of ability to represent the mixture of the generated vapor and liquid numerically. On the other hand, in the SNB section (

*H*≈ 1.7–2.0 m), where saturation boiling is dominant, there was a slight difference between the simulated and experimental values regardless of the grid resolution. This disparity was attributed to the increased outlet region for a stable numerical calculation and the constant temperature for the added outlet region. Another cause is believed to be differences due to the combination of the nucleate site density model and bubble departure diameter model. Gu et al. (2017) applied the LC model to the nucleate site density and the KI model to the bubble departure diameter. They reported that the numerical results for the liquid temperature distribution in the same section (

*H*≈ 1.7–2.0 m) showed good agreement with the experimental results. On the other hand, a comparison of the void fraction distribution showed large differences. These results suggest that when multiphase-thermal flow in a pipe is simulated, the model combination selected may vary depending on which physical quantities are used as the basis for analysis. Therefore, it is important to apply the optimal boiling model through the combination of submodels and parametric studies.