Evaluation on Coupling of Wall Boiling and Population Balance Models for Vertical Gas-Liquid Subcooled Boiling Flow of First Loop of Nuclear Power Plant

An accurate prediction of the interphase behaviors of the vertical gas-liquid subcooled boiling flow is meaningful for the first loop of a nuclear power plant (NPP). Therefore, the interphase behaviors including the bubble size distribution in the first loop of the NPP are analyzed, evaluated, and validated using various wall boiling models coupled with the population balance model (PBM) kernels in this paper. Firstly, nondimensional numbers of the first loop of the NPP and DEBORA (Development of Borehole Seals for High-Level Radioactive Waste) experiment test cases are analyzed with approximation. Secondly, five active nucleation site density models Nn coupled with the PBM kernel combination, four kernel combinations (C1~C4) with the Nn models are calculated and analyzed. Lastly, various behaviors including the bubble size distribution Sauter mean diameter (SMD) dp, void fraction α, gas superficial velocity jg, and liquid superficial velocity jl are compared and validated with the experimental data of the DEBORA-1 (P = 2.62 MPa). The results indicate that the two Nn models are suitable for the calculations of thefirst loop of the nuclear power plant. For instance, for the bubble size distribution SMD dp, the specified Nn model with C1 (maximum relative error 9.63%) has relatively better behaviors for the first loop of the NPP. Especially, the combination C1 is applicable for the calculation of the bubble size distribution dp, void fraction α and liquid superficial velocity jl while C4 is suitable for the calculation of the gas superficial velocity jg. These results can provide guidance for the numerical computation of the subcooled boiling flow in the first loop of the NPP.


Gas-Liquid Subcooled Boiling Flow
Gas-liquid subcooled boiling flow is of enormous interest in many industrial applications like the thermal engineering systems, electronic systems, chemical reactors, and nuclear reactors [1]. On the one hand, it includes the interactive and dynamic phenomena such as the wall nucleation boiling, bubble departure, condensation, coalescence, and breakup [2]. On the other hand, it does have advantages in the heat transfer processes compared with that of the single phase forced convection flow [3]. In particular, an accurate prediction on the gas-liquid subcooled boiling flow is crucial for the operation safety and efficiency of the specific high-pressure application [4].
For instance, the gas-liquid subcooled boiling could occur in the first loop of the nuclear power plant (NPP) which highly affects the reactor safety and efficiency [5]. Especially, the phenomenon of the departure from nucleate boiling (DNB) relates with the critical heat flux (CHF), and the validations of the improvement on the CHF in the design of the nuclear

Coupling of Wall Boiling and PBM Model
In the past decades, many wall heat flux partitioning models were proposed for the wall-to-flow heat transfer in the subcooled boiling flow [17]. The most widely adopted wall boiling model, namely the Rensselaer Polytechnic Institute (RPI) wall boiling model, is developed [18]. In which, the wall heat flux is split into three parts: the evaporation heat flux q e , the liquid-phase convection heat flux q c and quenching heat flux q q as shown in Equation (1). Q = q e + q c + q q , where Q is the wall heat flux, W/m 2 . Then, these heat flux q e , q c and q q can be expressed as presented in Equation (2). From Equation (2), the three components of the wall heat flux have close relationships with the following three closure parameters: the active nucleation site density N n , bubble departure frequency f and bubble departure diameter D d .
where ρ l , ρ g are density of the liquid phase and gas phase respectively, kg/m 3 ; h lg is the latent heat, J/(kg); h s is the single-phase heat transfer coefficient, W/(m 2 ·K); T w is the temperature of the heated wall, • C or K; T l is the temperature of the liquid phase, • C or K; A b is the proportional of the heated wall coved by the nucleation bubbles, and ), K is the empirical constant; k l is the liquid thermal conductivity, W/(m·K); C Pl is the specific heat capacity of the liquid phase at a certain pressure, J/(kg·K).
Recently, many authors derived refined versions of the RPI model to predict the wall boiling phenomenon [7,8,19,20]. Especially, there are some researches analyze various wall boiling models in the subcooled boiling flow [2,21]. Moreover, few evaluations on the PBM kernels in the subcooled boiling flow [10,22]. Furthermore, very few studies relate the wall boiling models with PBM kernels in the subcooled boiling flow [8,20]. For both the wall boiling models and PBM kernels, each model or kernel has a limited-range application due to the complexity and dynamics of the interphase behaviors in the gas-liquid subcooled boiling flow [23,24].
However, existing research lacks descriptions the performance of the various wall boiling mechanisms coupled with PBM kernels for the high-pressure application like the first loop of the NPP [11]. In which, Hu [11] evaluated various PBM kernels for the first loop of the NPP. Nevertheless, it still needs the evaluation on the various combinations of the wall boiling mechanisms and PBM kernels. Furthermore, the accuracy of the CFD calculation relies on the validation with the experiment data [20]. Therefore, it is necessary to analyze, evaluate and validate the wall boiling models coupled with PBM kernels in the vertical gas-liquid subcooled boiling flow for the first loop of the NPP.

Experiment Setup and First Loop of NPP
Confidence in the CFD analysis relies on the validation of the numerical calculations against with the experiment data. For this purpose, parameters in the typical experiment facilities especially for the subcooled boiling flow are illustrated in Table 1. It summarizes the physical properties and nondimensional parameters from various facilities in the SUBO (Subcooled boiling facility) [25,26], DEBORA [6,27,28], Arizona State University (ASU) [20,29], Seoul National University (SNU) [30,31], Purdue University [1,32], respectively. In particular, the first loop of NPP is taken into consideration in which the pressure P = 15.70 MPa. The parameters of the first loop of the NPP are based on the previous reference [28].  In Table 1, typical experimental parameters like the working fluid, channel, orientation, equivalent hydraulic diameter D, pressure P, wall heat flux Q, mass flow rate G, and inlet liquid subcooling temperature ∆T sub greatly differ in various experimental facilities. Hence, the relevant nondimensional numbers need to be introduced. Additionally, the nondimensional numbers for the subcooled boiling flow of the typical experiment facilities including the SUBO, DEBORA, ASU, SNU, and Purdue facility are presented in Table 1. For the wall boiling phenomenon in the vertical subcooled boiling flow, the liquid Reynolds number (Re l ), ratio of two phases (ρ l /ρ g ), effective Jacob number (Ja e ) and Boiling number (Bo) are most important [8]. Especially, the experiment test case analysis based on the typical experimental facilities are presented in Section 3.1. For the wall boiling coupled with the PBM model, the bubble size distribution is described with the population balance equation (PBE) as shown in Equation (3) [16]. For the wall boiling coupled with the PBM model, the bubble size distribution is described with the population balance equation (PBE) as shown in Equation (3) [16].
where → x is the spatial vector; → m is the mass of bubbles, kg; t is the time, s; . → x is the time change rate of the bubble physical position; n is the number density of bubbles, 1/m 3 ; . → m is the time change rate of the bubble property (mass or volume) state. Furthermore, B C , D C are the source and sink term respectively due to the bubble coalescence, and B B , D B are also the source and sink term from the bubble breakup behaviors. Particularly, R i is the sink term for the bubble condensation.
In addition, the representations of the source and sink term for the bubble coalescence, breakup and condensation with the integrated form are given in Equation (4) [11]. where n(m − m ), n(m ) mean the bubbles number density of mass (m − m ) and m respectively; h(m − m ; m ) indicates the collision frequency between the bubbles of mass (m − m ) and m ; Ω(m ) represents the overall breakup frequency of the parent bubble with a mass m ; β(m ; m) is the daughter bubble size distribution with the parent bubble of a mass m .

Two-Fluid Model
With the treatment of the continua of both the gas and liquid phases, the two-fluid model includes the mass, momentum, and energy equations of each phase in Figure 1. For the closure of the two-fluid model, the interfacial transfer terms are introduced [33,34]. Both the wall boiling model, PBM kernel, and interfacial force model contribute to the interfacial transfer terms, and the two-fluid model is a typical model that was illustrated in many previous research [7,9,14,20]. Consequently, this paper skips the presentations of the equations of the two-fluid model and focuses on the wall boiling and PBM models.

Wall Boiling Model
For the wall boiling mechanism, the bubble nucleation occurs on the heated wall. Unlike the sink term related with the phase change due to the bubble condensation, the wall boiling model is handled as the boundary condition in Equation (3) [7], and the expression of the wall nucleation interfacial area rate Φ of the wall boiling model is presented in Equation (5) [2].

Typical Kernels of Bubble Coalescence
Typical kernels of the a bubble coalescence rate a(di, dj) are illustrated in Table 5 [53][54][55][56][57]. As the bubble coalescence at a collision with a probability, a(di, dj) is represented as the product of the collision frequency h(di, dj) and coalescence efficiency λ(di, dj) in Equation (10) [24]. At the same time, λ(di, dj) is described as the ratio of the time required for coalescence of bubbles tij and contact time for two bubbles τij in Equation (10) with various kernels from Table 5. Considering the coalescence rate kernel of Guo [57] includes four collision frequency mechanisms, it is most comprehensive and will be focused on in this paper. a(di, dj) = h(di, dj)λ(di, dj), (10) where di, dj is diameter of two colliding bubbles, m. ] oefficients in coalescence rate kernels, c1 = 0.28~1.11, c2 = 0.25ln(hi/hf) = 2.3, c3 = 1.12, c4 = 0.0157, c5 = 1.29, c6 ic viscosity of liquid phase, Pa·s; dij is equivalent diameter of two colliding bubbles with unequal size in els, dij = (1/ri + 1/rj) −1 , m; ucrit is critical velocity for coalescence, m/s; u' is turbulent fluctuating velocity, m/s; void faction; Γij is ratio of distance between bubbles and bubble turbulent path length; is ratio of ble i and j; Weij is equivalent Weber number of two colliding bubbles in coalescence rate model; γ is tual mass, γ = 0.5; Weij is equivalent Weber number of two colliding bubbles; h(di, dj) T , h(di, dj) B , h(di, dj) W collision frequency due to turbulent fluctuation, buoyancy driven, wake entrainment, and viscous shear,

Typical Kernels of Bubble Breakup
For decades, typical kernels of the bubble breakup presented in Table 6 [55,[57][58][59][60]. In the bubble breakup process, the breakup frequency and daughter size distribution matter [23]. In Table 6, the kernels of the breakup frequency Ω(Vp) and its daughter size distribution β(Vd, Vp) or the dimensionless form of the daughter size distribution β(fbv, 1) where ow in this paper. . .
bble Coalescence a bubble coalescence rate a(di, dj) are illustrated in Table 5 [53ence at a collision with a probability, a(di, dj) is represented as on frequency h(di, dj) and coalescence efficiency λ(di, dj) in ame time, λ(di, dj) is described as the ratio of the time required tij and contact time for two bubbles τij in Equation (10) with 5. Considering the coalescence rate kernel of Guo [57] includes echanisms, it is most comprehensive and will be focused on in  Table 6 [55,[57][58][59][60]. In s, the breakup frequency and daughter size distribution matter els of the breakup frequency Ω(Vp) and its daughter size e dimensionless form of the daughter size distribution β(fbv, 1) H is the heated perimeter of the boiling channel, m; A C is the cross-sectional area of the boiling channel, m 2 .
As reasonably concluded from Equations (2) and (5), the wall boiling mechanisms include the model of the active nucleation site density N n , bubble departure diameter D d and bubble departure frequency f. These three models are necessary to compute the wall nucleation interfacial area rate of the wall boiling model. To this end, typical models of the active nucleation site density N n , bubble departure diameter D d and bubble departure frequency f are summarized in Tables 2-4. 2.3.1. Typical Models of Active Nucleation Site Density N n Typical models of the active nucleation site density N n are illustrated in Table 2. In  Table 2, firstly, correlation in the form of the power laws depending on the wall superheat ∆T sup (∆T sup = T w − T sat ) was proposed by Lemmert and Chawla [35]. Then, Kocamustafaogullari and Ishii [36] derived the model of N n which is highly dependent on the nondimensional cavity radius R * c and ratio ρ * as shown in Equation (6). As the N n model of Kocamustafaogullari and Ishii [36], it is derived from the pooling boiling condition while it was widely adopted for the subcooled flow boiling condition [37][38][39].
where R c is the critical cavity radius based on the wall superheat, m; ∆ρ is the density difference of the two phases, ∆ρ = (ρ l − ρ g ), kg/m 3 ; f (ρ * ) is the function; ρ * is the ratio between the density difference and gas density, ρ * = ∆ρ/ρ g ; σ is the surface tension, N/m; T sat is the saturation temperature, • C or K; ∆T e is the effective wall superheat in the gasliquid flow boiling, ∆T e = S∆T sup = S(T w − T sat ), • C or K; S is the suppression factor in the subcooled convective boiling, 0 < S < 1; T g is the temperature of the gas phase, • C or K; R is the ideal gas constant, R = 8.3144 J/(K·mol). Additionally, Basu [40] put forward the model of N n for the subcooled convective boiling. It is a function of both wall superheat ∆T sup and contact angle θ. Furthermore, Hibiki and Ishii [21] obtained the model of N n for both the pool boiling and subcooled convective boiling from the experiment data. It shows that the N n is a function of the critical cavity radius R c , contact angle θ and model parameter ρ + , ρ + = log(ρ * ). Similarly in Equation (6), R c = 2σT sat /(ρ g h lg ∆T e ) for ρ g ρ l and h lg (T g − T sat )/(RT g T sat ) 1 in this model. Lastly, a new model with a parametric analysis based on the existing experiment data for the vertical subcooled boiling flow was developed [5]. It is a function of the wall superheat ∆T sup , pressure P and contact angle θ. And in this model, the additional calculation formulas for the contact angel θ and pressure P are shown in Equation (7).
where θ 0 is the contact angle at the room temperature, • ; T 0 is the room temperature, K; T c is critical temperature at which contact angle becomes 0, • C or K; γ is the model parameter.
For subcooled flow boiling Basu et al., 2002 [40] N For subcooled flow boiling Hibiki and Ishii, 2003 [21]  where N nc is active nucleation site density in subcooled convective boiling, m −2 ; N * nc is nondimensional active nucleation site density in subcooled convective boiling; ∆T sub is liquid subcooling temperature, ∆T sub = T sat − T l , • C or K; N n , µ, λ , A and B are model parameters.

Typical Models of Bubble Departure Diameter D d
In the past decades, typical models of the bubble departure diameter D d are elaborated in Table 3. In Table 3, firstly, correlation in the function of the liquid subcooling temperature ∆T sub was described by Tolubinsky and Kostanchuk [41]. In 1983, Kocamustafaogullari and Ishii [36] predicted the model of D d with a wide range of the pressure (P = 0.1 − 19.8 MPa). Then, Situ [4] proposed the nondimensional bubble departure diameter D * d with a function of the Jacob (Ja) number and Prandtl number (Pr) at the ambient pressure. At last, Krepper [8] derived the model of D d with the liquid subcooling temperature ∆T sub in the high-pressure condition. Hence, the model of the bubble departure diameter of Krepper [8] is adopted in this paper.

Typical Models of Bubble Departure Frequency f
As shown in Table 4, the expression of the bubble departure frequency f was proposed with multiple parameters [42]. A formula for f by considering the effects of buoyancy, drag and thermodynamics was given [43]. Then, f was derived when the heat flux is low and the bubbles do not noticeably influence with each other [44] in the subcooled boiling. Furthermore, the expression of f was presented in combination with the study of bubble number density in the subcooled boiling [45]. In addition, f was obtained based on the interphase heat flux q and the latent heat of the phase change h lg [46]. It was already validated with many experiment data sets. In this paper, the model of the bubble departure frequency of Brooks and Hibiki [46] is carried out for the calculation.
Ja 2 e Pr −1 l P = 0.1 MPa; High pressure; For subcooled flow boing where C l is shear lift coefficient; D dF is the departure diameter with Fritz equation, m; g is gravity acceleration, g = 9.80 m/s 2 ; C Pl is specific heat capacity of liquid phase at a certain pressure, J/(kg·K); j l is superficial liquid velocity, m/s; j r is relative velocity between two phases, m/s; j s is nondimensional fluid velocity gradient; Re g , bubble Reynolds number; Ja e , effective Jakob number, Ja e = SJa = ρ l C Pl ∆ T e /(ρ g h ig ); Pr l is Prandtl number of liquid phase, Pr l = C Pl µ l /k l ; b is model parameter. where η l is liquid thermal diffusivity, m 2 /s; D is hydraulic equivalent diameter, m.

Population Balance Model
Different mechanisms including the bubble condensation, bubble coalescence, and breakup are considered in the PBM model when the bubbles enter and leave a control volume in the gas-liquid subcooled flow [23]. The details mechanisms are illustrated below.

Bubble Condensation Model
The bubble condensation model concerns with the sink term of the PBM as shown in Equation (8), and the interphase heat transfer coefficient h is proportional to the Nusselt number (Nu) in Equation (8). Hence, the sink term R i due to the bubble condensation is mainly determined by the interface area concentration a i and Nu number. In addition, there are many models of the interface area concentration a i [47][48][49] and Nu number [50][51][52]. where α is the void fraction; n i is the number density of bubbles in the discrete bubble class i, m −3 ; d sm is the Saturn mean diameter, m. Among them, the models of a i derived by Kocamustafaogullari [47] and Nu number proposed with Tomiyama [52] in Equation (9) is most applicable for the high-pressure condition [11]. Hence, the two models are adopted for the bubble condensation in the gas-liquid subcooled boiling flow in this paper.
where ε is the turbulent dissipation rate, m 2 /s 3 .

Typical Kernels of Bubble Coalescence
Typical kernels of the a bubble coalescence rate a(d i , d j ) are illustrated in Table 5 [53][54][55][56][57]. As the bubble coalescence at a collision with a probability, a(d i , d j ) is represented as the product of the collision frequency h(d i , d j ) and coalescence efficiency λ(d i , d j ) in Equation (10) [24]. At the same time, λ(d i , d j ) is described as the ratio of the time required for coalescence of bubbles t ij and contact time for two bubbles τ ij in Equation (10) with various kernels from Table 5. Considering the coalescence rate kernel of Guo [57] includes four collision frequency mechanisms, it is most comprehensive and will be focused on in this paper.
where d i , d j is diameter of two colliding bubbles, m. Prince and Blanch, 1990 [53] h condition [11]. Hence, the two models are adopted for the bubble condensation in the gasliquid subcooled boiling flow in this paper.
with high pressure, where ε is the turbulent dissipation rate, m 2 /s 3 .

Typical Kernels of Bubble Coalescence
Typical kernels of the a bubble coalescence rate a(di, dj) are illustrated in Table 5 [53][54][55][56][57]. As the bubble coalescence at a collision with a probability, a(di, dj) is represented as the product of the collision frequency h(di, dj) and coalescence efficiency λ(di, dj) in Equation (10) [24]. At the same time, λ(di, dj) is described as the ratio of the time required for coalescence of bubbles tij and contact time for two bubbles τij in Equation (10) with various kernels from Table 5. Considering the coalescence rate kernel of Guo [57] includes four collision frequency mechanisms, it is most comprehensive and will be focused on in this paper.
where di, dj is diameter of two colliding bubbles, m.

Typical Kernels of Bubble Breakup
For decades, typical kernels of the bubble breakup presented in Table 6 [55,[57][58][59][60]. In the bubble breakup process, the breakup frequency and daughter size distribution matter [23]. In Table 6, the kernels of the breakup frequency Ω(Vp) and its daughter size distribution β(Vd, Vp) or the dimensionless form of the daughter size distribution β(fbv, 1) condition [11]. Hence, the two models are adopted for the bubble condensation in the gasliquid subcooled boiling flow in this paper. = .
with high pressure, where ε is the turbulent dissipation rate, m 2 /s 3 .

Typical Kernels of Bubble Coalescence
Typical kernels of the a bubble coalescence rate a(di, dj) are illustrated in Table 5 [53][54][55][56][57]. As the bubble coalescence at a collision with a probability, a(di, dj) is represented as the product of the collision frequency h(di, dj) and coalescence efficiency λ(di, dj) in Equation (10) [24]. At the same time, λ(di, dj) is described as the ratio of the time required for coalescence of bubbles tij and contact time for two bubbles τij in Equation (10) with various kernels from Table 5. Considering the coalescence rate kernel of Guo [57] includes four collision frequency mechanisms, it is most comprehensive and will be focused on in this paper.
di, dj is diameter of two colliding bubbles, m.

Typical Kernels of Bubble Breakup
For decades, typical kernels of the bubble breakup presented in Table 6 [55,[57][58][59][60]. In the bubble breakup process, the breakup frequency and daughter size distribution matter [23]. In Table 6, the kernels of the breakup frequency Ω(Vp) and its daughter size distribution β(Vd, Vp) or the dimensionless form of the daughter size distribution β(fbv, 1) condition [11]. Hence, the two models are adopted for the bubble condensation in the gasliquid subcooled boiling flow in this paper. = .
with high pressure, where ε is the turbulent dissipation rate, m 2 /s 3 .

Typical Kernels of Bubble Coalescence
Typical kernels of the a bubble coalescence rate a(di, dj) are illustrated in Table 5 [53][54][55][56][57]. As the bubble coalescence at a collision with a probability, a(di, dj) is represented as the product of the collision frequency h(di, dj) and coalescence efficiency λ(di, dj) in Equation (10) [24]. At the same time, λ(di, dj) is described as the ratio of the time required for coalescence of bubbles tij and contact time for two bubbles τij in Equation (10) with various kernels from Table 5. Considering the coalescence rate kernel of Guo [57] includes four collision frequency mechanisms, it is most comprehensive and will be focused on in this paper.
di, dj is diameter of two colliding bubbles, m.

Typical Kernels of Bubble Breakup
For decades, typical kernels of the bubble breakup presented in Table 6 [55,57-60]. In the bubble breakup process, the breakup frequency and daughter size distribution matter [23]. In Table 6, the kernels of the breakup frequency Ω(Vp) and its daughter size distribution β(Vd, Vp) or the dimensionless form of the daughter size distribution β(fbv, 1)   condition [11]. Hence, the two models are adopted for the bubble condensation in the gasliquid subcooled boiling flow in this paper. with high pressure, where ε is the turbulent dissipation rate, m 2 /s 3 .

Typical Kernels of Bubble Coalescence
Typical kernels of the a bubble coalescence rate a(di, dj) are illustrated in Table 5 [53][54][55][56][57]. As the bubble coalescence at a collision with a probability, a(di, dj) is represented as the product of the collision frequency h(di, dj) and coalescence efficiency λ(di, dj) in Equation (10) [24]. At the same time, λ(di, dj) is described as the ratio of the time required for coalescence of bubbles tij and contact time for two bubbles τij in Equation (10) with various kernels from Table 5. Considering the coalescence rate kernel of Guo [57] includes four collision frequency mechanisms, it is most comprehensive and will be focused on in this paper.
di, dj is diameter of two colliding bubbles, m. condition [11]. Hence, the two models are adopted for the bubble condensation in the gasliquid subcooled boiling flow in this paper. with high pressure, where ε is the turbulent dissipation rate, m 2 /s 3 .

Typical Kernels of Bubble Coalescence
Typical kernels of the a bubble coalescence rate a(di, dj) are illustrated in Table 5 [53][54][55][56][57]. As the bubble coalescence at a collision with a probability, a(di, dj) is represented as the product of the collision frequency h(di, dj) and coalescence efficiency λ(di, dj) in Equation (10) [24]. At the same time, λ(di, dj) is described as the ratio of the time required for coalescence of bubbles tij and contact time for two bubbles τij in Equation (10) with various kernels from Table 5. Considering the coalescence rate kernel of Guo [57] includes four collision frequency mechanisms, it is most comprehensive and will be focused on in this paper.
di, dj is diameter of two colliding bubbles, m. condition [11]. Hence, the two models are adopted for the bubble condensation in the gasliquid subcooled boiling flow in this paper.
where ε is the turbulent dissipation rate, m 2 /s 3 .

Typical Kernels of Bubble Coalescence
Typical kernels of the a bubble coalescence rate a(di, dj) are illustrated in Table 5 [53][54][55][56][57]. As the bubble coalescence at a collision with a probability, a(di, dj) is represented as the product of the collision frequency h(di, dj) and coalescence efficiency λ(di, dj) in Equation (10) [24]. At the same time, λ(di, dj) is described as the ratio of the time required for coalescence of bubbles tij and contact time for two bubbles τij in Equation (10) with various kernels from Table 5. Considering the coalescence rate kernel of Guo [57] includes four collision frequency mechanisms, it is most comprehensive and will be focused on in this paper.
di, dj is diameter of two colliding bubbles, m. Table 5. Typical kernels of bubbles coalescence rate.

Typical Kernels of Bubble Breakup
where c 1~c6 are coefficients in coalescence rate kernels, c 1 = 0.28~1.11, c 2 = 0.25ln(h i /h f ) = 2.3, c 3 = 1.12, c 4 = 0.0157, c 5 = 1.29, c 6 = 1.11; µ l is dynamic viscosity of liquid phase, Pa·s; d ij is equivalent diameter of two colliding bubbles with unequal size in coalescence kernels, d ij = (1/r i + 1/r j ) −1 , m; u crit is critical velocity for coalescence, m/s; u' is turbulent fluctuating velocity, m/s; α max is maximum void faction; Γ ij is ratio of distance between bubbles and bubble turbulent path length; VIEW 9 of 33 condition [11]. Hence, the two models are adopted for the bubble condensation in the gasliquid subcooled boiling flow in this paper.
where ε is the turbulent dissipation rate, m 2 /s 3 .

Typical Kernels of Bubble Coalescence
Typical kernels of the a bubble coalescence rate a(di, dj) are illustrated in Table 5 [53][54][55][56][57]. As the bubble coalescence at a collision with a probability, a(di, dj) is represented as the product of the collision frequency h(di, dj) and coalescence efficiency λ(di, dj) in Equation (10) [24]. At the same time, λ(di, dj) is described as the ratio of the time required for coalescence of bubbles tij and contact time for two bubbles τij in Equation (10) with various kernels from Table 5. Considering the coalescence rate kernel of Guo [57] includes four collision frequency mechanisms, it is most comprehensive and will be focused on in this paper.
di, dj is diameter of two colliding bubbles, m. Table 5. Typical kernels of bubbles coalescence rate.
we ] osity of liquid phase, Pa·s; dij is equivalent diameter of two colliding bubbles with unequal size in (1/ri + 1/rj) −1 , m; ucrit is critical velocity for coalescence, m/s; u' is turbulent fluctuating velocity, m/s; ij is ratio of diameter of bubble i and j; We ij is equivalent Weber number of two colliding bubbles in coalescence rate model; γ is coefficient of virtual mass, γ = 0.5; We ij is equivalent Weber number of two colliding bubbles; are collision frequency due to turbulent fluctuation, buoyancy driven, wake entrainment, and viscous shear, m 3 /s.

Typical Kernels of Bubble Breakup
For decades, typical kernels of the bubble breakup presented in Table 6 [55,[57][58][59][60]. In the bubble breakup process, the breakup frequency and daughter size distribution matter [23]. In Table 6, the kernels of the breakup frequency Ω(V p ) and its daughter size distribution β(V d , V p ) or the dimensionless form of the daughter size distribution β(f bv , 1) are elaborated. In the previous research [11], it suggests that the breakup kernel of Luo and Svendsen [60] may be suitable to the high-pressure condition. It will be highlighted in the calculation.

Typical Kernels of Bubble Coalescence
Typical kernels of the a bubble coalescence rate a(di, dj) are illustrated in Table 5 [53][54][55][56][57]. As the bubble coalescence at a collision with a probability, a(di, dj) is represented as the product of the collision frequency h(di, dj) and coalescence efficiency λ(di, dj) in Equation (10) [24]. At the same time, λ(di, dj) is described as the ratio of the time required for coalescence of bubbles tij and contact time for two bubbles τij in Equation (10) with various kernels from Table 5. Considering the coalescence rate kernel of Guo [57] includes four collision frequency mechanisms, it is most comprehensive and will be focused on in this paper.
where di, dj is diameter of two colliding bubbles, m. Table 5. Typical kernels of bubbles coalescence rate.

Typical Kernels of Bubble Breakup
For decades, typical kernels of the bubble breakup presented in Table 6 [55,[57][58][59][60]. In the bubble breakup process, the breakup frequency and daughter size distribution matter [23]. In Table 6, the kernels of the breakup frequency Ω(Vp) and its daughter size distribution β(Vd, Vp) or the dimensionless form of the daughter size distribution β(fbv, 1) min (1+ where ε is the turbulent dissipation rate, m 2 /s 3 .

Typical Kernels of Bubble Coalescence
Typical kernels of the a bubble coalescence rate a(di, dj) are illustrated in Table 5 [53][54][55][56][57]. As the bubble coalescence at a collision with a probability, a(di, dj) is represented as the product of the collision frequency h(di, dj) and coalescence efficiency λ(di, dj) in Equation (10) [24]. At the same time, λ(di, dj) is described as the ratio of the time required for coalescence of bubbles tij and contact time for two bubbles τij in Equation (10) with various kernels from Table 5. Considering the coalescence rate kernel of Guo [57] includes four collision frequency mechanisms, it is most comprehensive and will be focused on in this paper.
where di, dj is diameter of two colliding bubbles, m. Table 5. Typical kernels of bubbles coalescence rate.

Typical Kernels of Bubble Breakup
For decades, typical kernels of the bubble breakup presented in Table 6 [55,57-60]. In the bubble breakup process, the breakup frequency and daughter size distribution matter [23]. In Table 6, the kernels of the breakup frequency Ω(Vp) and its daughter size distribution β(Vd, Vp) or the dimensionless form of the daughter size distribution β(fbv, 1) where ε is the turbulent dissipation rate, m 2 /s 3 .

Typical Kernels of Bubble Coalescence
Typical kernels of the a bubble coalescence rate a(di, dj) are illustrated in Table 5 [53][54][55][56][57]. As the bubble coalescence at a collision with a probability, a(di, dj) is represented as the product of the collision frequency h(di, dj) and coalescence efficiency λ(di, dj) in Equation (10) [24]. At the same time, λ(di, dj) is described as the ratio of the time required for coalescence of bubbles tij and contact time for two bubbles τij in Equation (10) with various kernels from Table 5. Considering the coalescence rate kernel of Guo [57] includes four collision frequency mechanisms, it is most comprehensive and will be focused on in this paper.
di, dj is diameter of two colliding bubbles, m. Table 5. Typical kernels of bubbles coalescence rate.

Typical Kernels of Bubble Breakup
For decades, typical kernels of the bubble breakup presented in Table 6 [55,57-60]. In the bubble breakup process, the breakup frequency and daughter size distribution matter [23]. In Table 6, the kernels of the breakup frequency Ω(Vp) and its daughter size distribution β(Vd, Vp) or the dimensionless form of the daughter size distribution β(fbv, 1) 11 3 exp − 12C f σ where ε is the turbulent dissipation rate, m 2 /s 3 .

Typical Kernels of Bubble Coalescence
Typical kernels of the a bubble coalescence rate a(di, dj) are illustrated in Table 5 [53][54][55][56][57]. As the bubble coalescence at a collision with a probability, a(di, dj) is represented as the product of the collision frequency h(di, dj) and coalescence efficiency λ(di, dj) in Equation (10) [24]. At the same time, λ(di, dj) is described as the ratio of the time required for coalescence of bubbles tij and contact time for two bubbles τij in Equation (10) with various kernels from Table 5. Considering the coalescence rate kernel of Guo [57] includes four collision frequency mechanisms, it is most comprehensive and will be focused on in this paper.
di, dj is diameter of two colliding bubbles, m. Table 5. Typical kernels of bubbles coalescence rate.

Typical Kernels of Bubble Breakup
For decades, typical kernels of the bubble breakup presented in Table 6 [55,57-60]. In the bubble breakup process, the breakup frequency and daughter size distribution matter [23]. In Table 6, the kernels of the breakup frequency Ω(Vp) and its daughter size distribution β(Vd, Vp) or the dimensionless form of the daughter size distribution β(fbv, 1) where ε is the turbulent dissipation rate, m 2 /s 3 .

Typical Kernels of Bubble Coalescence
Typical kernels of the a bubble coalescence rate a(di, dj) are illustrated in Table 5 [53][54][55][56][57]. As the bubble coalescence at a collision with a probability, a(di, dj) is represented as the product of the collision frequency h(di, dj) and coalescence efficiency λ(di, dj) in Equation (10) [24]. At the same time, λ(di, dj) is described as the ratio of the time required for coalescence of bubbles tij and contact time for two bubbles τij in Equation (10) with various kernels from Table 5. Considering the coalescence rate kernel of Guo [57] includes four collision frequency mechanisms, it is most comprehensive and will be focused on in this paper.
di, dj is diameter of two colliding bubbles, m. Table 5. Typical kernels of bubbles coalescence rate.

Typical Kernels of Bubble Breakup
For decades, typical kernels of the bubble breakup presented in Table 6 [55,57-60]. In the bubble breakup process, the breakup frequency and daughter size distribution matter [23]. In Table 6, the kernels of the breakup frequency Ω(Vp) and its daughter size distribution β(Vd, Vp) or the dimensionless form of the daughter size distribution β(fbv, 1) , and . .
with high pressure, where ε is the turbulent dissipation rate, m 2 /s 3 .

Typical Kernels of Bubble Coalescence
Typical kernels of the a bubble coalescence rate a(di, dj) are illustrated in Table 5 [53][54][55][56][57]. As the bubble coalescence at a collision with a probability, a(di, dj) is represented as the product of the collision frequency h(di, dj) and coalescence efficiency λ(di, dj) in Equation (10) [24]. At the same time, λ(di, dj) is described as the ratio of the time required for coalescence of bubbles tij and contact time for two bubbles τij in Equation (10) with various kernels from Table 5. Considering the coalescence rate kernel of Guo [57] includes four collision frequency mechanisms, it is most comprehensive and will be focused on in this paper.
di, dj is diameter of two colliding bubbles, m. Table 5. Typical kernels of bubbles coalescence rate.

Typical Kernels of Bubble Breakup
For decades, typical kernels of the bubble breakup presented in Table 6 [55,[57][58][59][60]. In the bubble breakup process, the breakup frequency and daughter size distribution matter [23]. In Table 6, the kernels of the breakup frequency Ω(Vp) and its daughter size distribution β(Vd, Vp) or the dimensionless form of the daughter size distribution β(fbv, 1) . .
with high pressure, where ε is the turbulent dissipation rate, m 2 /s 3 .

Typical Kernels of Bubble Coalescence
Typical kernels of the a bubble coalescence rate a(di, dj) are illustrated in Table 5 [53][54][55][56][57]. As the bubble coalescence at a collision with a probability, a(di, dj) is represented as the product of the collision frequency h(di, dj) and coalescence efficiency λ(di, dj) in Equation (10) [24]. At the same time, λ(di, dj) is described as the ratio of the time required for coalescence of bubbles tij and contact time for two bubbles τij in Equation (10) with various kernels from Table 5. Considering the coalescence rate kernel of Guo [57] includes four collision frequency mechanisms, it is most comprehensive and will be focused on in this paper.
di, dj is diameter of two colliding bubbles, m. Table 5. Typical kernels of bubbles coalescence rate.

Typical Kernels of Bubble Breakup
For decades, typical kernels of the bubble breakup presented in Table 6 [55,57-60]. In the bubble breakup process, the breakup frequency and daughter size distribution matter [23]. In Table 6, the kernels of the breakup frequency Ω(Vp) and its daughter size distribution β(Vd, Vp) or the dimensionless form of the daughter size distribution β(fbv, 1) . .
with high pressure, where ε is the turbulent dissipation rate, m 2 /s 3 .

Typical Kernels of Bubble Coalescence
Typical kernels of the a bubble coalescence rate a(di, dj) are illustrated in Table 5 [53][54][55][56][57]. As the bubble coalescence at a collision with a probability, a(di, dj) is represented as the product of the collision frequency h(di, dj) and coalescence efficiency λ(di, dj) in Equation (10) [24]. At the same time, λ(di, dj) is described as the ratio of the time required for coalescence of bubbles tij and contact time for two bubbles τij in Equation (10) with various kernels from Table 5. Considering the coalescence rate kernel of Guo [57] includes four collision frequency mechanisms, it is most comprehensive and will be focused on in this paper.
di, dj is diameter of two colliding bubbles, m.

Typical Kernels of Bubble Breakup
For decades, typical kernels of the bubble breakup presented in Table 6 [55,[57][58][59][60]. In the bubble breakup process, the breakup frequency and daughter size distribution matter [23]. In Table 6, the kernels of the breakup frequency Ω(Vp) and its daughter size distribution β(Vd, Vp) or the dimensionless form of the daughter size distribution β(fbv, 1) . . with high pressure, where ε is the turbulent dissipation rate, m 2 /s 3 .

Typical Kernels of Bubble Coalescence
Typical kernels of the a bubble coalescence rate a(di, dj) are illustrated in Table 5 [53][54][55][56][57]. As the bubble coalescence at a collision with a probability, a(di, dj) is represented as the product of the collision frequency h(di, dj) and coalescence efficiency λ(di, dj) in Equation (10) [24]. At the same time, λ(di, dj) is described as the ratio of the time required for coalescence of bubbles tij and contact time for two bubbles τij in Equation (10) with various kernels from Table 5. Considering the coalescence rate kernel of Guo [57] includes four collision frequency mechanisms, it is most comprehensive and will be focused on in this paper.
di, dj is diameter of two colliding bubbles, m.

Typical Kernels of Bubble Breakup
For decades, typical kernels of the bubble breakup presented in Table 6 [55,[57][58][59][60]. In the bubble breakup process, the breakup frequency and daughter size distribution matter [23]. In Table 6, the kernels of the breakup frequency Ω(Vp) and its daughter size distribution β(Vd, Vp) or the dimensionless form of the daughter size distribution β(fbv, 1) . . with high pressure, where ε is the turbulent dissipation rate, m 2 /s 3 .

Typical Kernels of Bubble Coalescence
Typical kernels of the a bubble coalescence rate a(di, dj) are illustrated in Table 5 [53][54][55][56][57]. As the bubble coalescence at a collision with a probability, a(di, dj) is represented as the product of the collision frequency h(di, dj) and coalescence efficiency λ(di, dj) in Equation (10) [24]. At the same time, λ(di, dj) is described as the ratio of the time required for coalescence of bubbles tij and contact time for two bubbles τij in Equation (10) with various kernels from Table 5. Considering the coalescence rate kernel of Guo [57] includes four collision frequency mechanisms, it is most comprehensive and will be focused on in this paper.
di, dj is diameter of two colliding bubbles, m.

Typical Kernels of Bubble Breakup
For decades, typical kernels of the bubble breakup presented in Table 6 [55,57-60]. In the bubble breakup process, the breakup frequency and daughter size distribution matter [23]. In Table 6, the kernels of the breakup frequency Ω(Vp) and its daughter size distribution β(Vd, Vp) or the dimensionless form of the daughter size distribution β(fbv, 1) 1 0 1 condition [11]. Hence, the two models are adopted for the bubble condensation in the gasliquid subcooled boiling flow in this paper. . . with high pressure, where ε is the turbulent dissipation rate, m 2 /s 3 .

Typical Kernels of Bubble Coalescence
Typical kernels of the a bubble coalescence rate a(di, dj) are illustrated in Table 5 [53][54][55][56][57]. As the bubble coalescence at a collision with a probability, a(di, dj) is represented as the product of the collision frequency h(di, dj) and coalescence efficiency λ(di, dj) in Equation (10) [24]. At the same time, λ(di, dj) is described as the ratio of the time required for coalescence of bubbles tij and contact time for two bubbles τij in Equation (10) with various kernels from Table 5. Considering the coalescence rate kernel of Guo [57] includes four collision frequency mechanisms, it is most comprehensive and will be focused on in this paper.
di, dj is diameter of two colliding bubbles, m.

Typical Kernels of Bubble Breakup
For decades, typical kernels of the bubble breakup presented in Table 6 [55,[57][58][59][60]. In the bubble breakup process, the breakup frequency and daughter size distribution matter [23]. In Table 6, the kernels of the breakup frequency Ω(Vp) and its daughter size distribution β(Vd, Vp) or the dimensionless form of the daughter size distribution β(fbv, 1) min (1+ condition [11]. Hence, the two models are adopted for the bubble condensation in the gasliquid subcooled boiling flow in this paper. . . with high pressure, where ε is the turbulent dissipation rate, m 2 /s 3 .

Typical Kernels of Bubble Coalescence
Typical kernels of the a bubble coalescence rate a(di, dj) are illustrated in Table 5 [53][54][55][56][57]. As the bubble coalescence at a collision with a probability, a(di, dj) is represented as the product of the collision frequency h(di, dj) and coalescence efficiency λ(di, dj) in Equation (10) [24]. At the same time, λ(di, dj) is described as the ratio of the time required for coalescence of bubbles tij and contact time for two bubbles τij in Equation (10) with various kernels from Table 5. Considering the coalescence rate kernel of Guo [57] includes four collision frequency mechanisms, it is most comprehensive and will be focused on in this paper.
di, dj is diameter of two colliding bubbles, m.

Typical Kernels of Bubble Breakup
For decades, typical kernels of the bubble breakup presented in Table 6 [55,57-60]. In the bubble breakup process, the breakup frequency and daughter size distribution matter [23]. In Table 6, the kernels of the breakup frequency Ω(Vp) and its daughter size distribution β(Vd, Vp) or the dimensionless form of the daughter size distribution β(fbv, 1) ) 2 condition [11]. Hence, the two models are adopted for the bubble condensation in the gasliquid subcooled boiling flow in this paper. . . with high pressure, where ε is the turbulent dissipation rate, m 2 /s 3 .

Typical Kernels of Bubble Coalescence
Typical kernels of the a bubble coalescence rate a(di, dj) are illustrated in Table 5 [53][54][55][56][57]. As the bubble coalescence at a collision with a probability, a(di, dj) is represented as the product of the collision frequency h(di, dj) and coalescence efficiency λ(di, dj) in Equation (10) [24]. At the same time, λ(di, dj) is described as the ratio of the time required for coalescence of bubbles tij and contact time for two bubbles τij in Equation (10) with various kernels from Table 5. Considering the coalescence rate kernel of Guo [57] includes four collision frequency mechanisms, it is most comprehensive and will be focused on in this paper.
di, dj is diameter of two colliding bubbles, m.

Typical Kernels of Bubble Breakup
For decades, typical kernels of the bubble breakup presented in Table 6 [55,57-60]. In the bubble breakup process, the breakup frequency and daughter size distribution matter [23]. In Table 6, the kernels of the breakup frequency Ω(Vp) and its daughter size distribution β(Vd, Vp) or the dimensionless form of the daughter size distribution β(fbv, 1) [11]. Hence, the two models are adopted for the bubble condensation in the gasliquid subcooled boiling flow in this paper. . . with high pressure, where ε is the turbulent dissipation rate, m 2 /s 3 .

Typical Kernels of Bubble Coalescence
Typical kernels of the a bubble coalescence rate a(di, dj) are illustrated in Table 5 [53][54][55][56][57]. As the bubble coalescence at a collision with a probability, a(di, dj) is represented as the product of the collision frequency h(di, dj) and coalescence efficiency λ(di, dj) in Equation (10) [24]. At the same time, λ(di, dj) is described as the ratio of the time required for coalescence of bubbles tij and contact time for two bubbles τij in Equation (10) with various kernels from Table 5. Considering the coalescence rate kernel of Guo [57] includes four collision frequency mechanisms, it is most comprehensive and will be focused on in this paper.
di, dj is diameter of two colliding bubbles, m.

Typical Kernels of Bubble Breakup
For decades, typical kernels of the bubble breakup presented in Table 6 [55,57-60]. In the bubble breakup process, the breakup frequency and daughter size distribution matter [23]. In Table 6, the kernels of the breakup frequency Ω(Vp) and its daughter size distribution β(Vd, Vp) or the dimensionless form of the daughter size distribution β(fbv, 1) condition [11]. Hence, the two models are adopted for the bubble condensation in the gasliquid subcooled boiling flow in this paper. . . with high pressure, where ε is the turbulent dissipation rate, m 2 /s 3 .

Typical Kernels of Bubble Coalescence
Typical kernels of the a bubble coalescence rate a(di, dj) are illustrated in Table 5 [53][54][55][56][57]. As the bubble coalescence at a collision with a probability, a(di, dj) is represented as the product of the collision frequency h(di, dj) and coalescence efficiency λ(di, dj) in Equation (10) [24]. At the same time, λ(di, dj) is described as the ratio of the time required for coalescence of bubbles tij and contact time for two bubbles τij in Equation (10) with various kernels from Table 5. Considering the coalescence rate kernel of Guo [57] includes four collision frequency mechanisms, it is most comprehensive and will be focused on in this paper.
di, dj is diameter of two colliding bubbles, m.

Typical Kernels of Bubble Breakup
For decades, typical kernels of the bubble breakup presented in Table 6 [55,57-60]. In the bubble breakup process, the breakup frequency and daughter size distribution matter [23]. In Table 6, the kernels of the breakup frequency Ω(Vp) and its daughter size distribution β(Vd, Vp) or the dimensionless form of the daughter size distribution β(fbv, 1) condition [11]. Hence, the two models are adopted for the bubble condensation in the gasliquid subcooled boiling flow in this paper. with high pressure, where ε is the turbulent dissipation rate, m 2 /s 3 .

Typical Kernels of Bubble Coalescence
Typical kernels of the a bubble coalescence rate a(di, dj) are illustrated in Table 5 [53][54][55][56][57]. As the bubble coalescence at a collision with a probability, a(di, dj) is represented as the product of the collision frequency h(di, dj) and coalescence efficiency λ(di, dj) in Equation (10) [24]. At the same time, λ(di, dj) is described as the ratio of the time required for coalescence of bubbles tij and contact time for two bubbles τij in Equation (10) with various kernels from Table 5. Considering the coalescence rate kernel of Guo [57] includes four collision frequency mechanisms, it is most comprehensive and will be focused on in this paper.
di, dj is diameter of two colliding bubbles, m. where c1~c6 are coefficients in coalescence rate kernels, c1 = 0.28~1.11, c2 = 0.25ln(hi/hf) = 2.3, c3 = 1.12, c4 = 0.0157, c5 = 1.29, c6 = 1.11; µl is dynamic viscosity of liquid phase, Pa·s; dij is equivalent diameter of two colliding bubbles with unequal size in coalescence kernels, dij = (1/ri + 1/rj) −1 , m; ucrit is critical velocity for coalescence, m/s; u' is turbulent fluctuating velocity, m/s; αmax is maximum void faction; Γij is ratio of distance between bubbles and bubble turbulent path length; is ratio of diameter of bubble i and j; Weij is equivalent Weber number of two colliding bubbles in coalescence rate model; γ is coefficient of virtual mass, γ = 0.5; Weij is equivalent Weber number of two colliding bubbles; h(di, dj) T , h(di, dj) B , h(di, dj) W and h(di, dj) V are collision frequency due to turbulent fluctuation, buoyancy driven, wake entrainment, and viscous shear, m 3 /s.

Typical Kernels of Bubble Breakup
For decades, typical kernels of the bubble breakup presented in Table 6 [55,57-60]. In the bubble breakup process, the breakup frequency and daughter size distribution matter [23]. In Table 6, the kernels of the breakup frequency Ω(Vp) and its daughter size distribution β(Vd, Vp) or the dimensionless form of the daughter size distribution β(fbv, 1) Energies 2021, 14, x FOR PEER REVIEW 9 of 33 condition [11]. Hence, the two models are adopted for the bubble condensation in the gasliquid subcooled boiling flow in this paper.
. . with high pressure, where ε is the turbulent dissipation rate, m 2 /s 3 .

Typical Kernels of Bubble Coalescence
Typical kernels of the a bubble coalescence rate a(di, dj) are illustrated in Table 5 [53][54][55][56][57]. As the bubble coalescence at a collision with a probability, a(di, dj) is represented as the product of the collision frequency h(di, dj) and coalescence efficiency λ(di, dj) in Equation (10) [24]. At the same time, λ(di, dj) is described as the ratio of the time required for coalescence of bubbles tij and contact time for two bubbles τij in Equation (10) with various kernels from Table 5. Considering the coalescence rate kernel of Guo [57] includes four collision frequency mechanisms, it is most comprehensive and will be focused on in this paper.
di, dj is diameter of two colliding bubbles, m.

Typical Kernels of Bubble Breakup
For decades, typical kernels of the bubble breakup presented in Table 6 [55,[57][58][59][60]. In the bubble breakup process, the breakup frequency and daughter size distribution matter [23]. In Table 6, the kernels of the breakup frequency Ω(Vp) and its daughter size distribution β(Vd, Vp) or the dimensionless form of the daughter size distribution β(fbv, 1) condition [11]. Hence, the two models are adopted for the bubble condensation in the gasliquid subcooled boiling flow in this paper.

Typical Kernels of Bubble Coalescence
Typical kernels of the a bubble coalescence rate a(di, dj) are illustrated in Table 5 [53][54][55][56][57]. As the bubble coalescence at a collision with a probability, a(di, dj) is represented as the product of the collision frequency h(di, dj) and coalescence efficiency λ(di, dj) in Equation (10) [24]. At the same time, λ(di, dj) is described as the ratio of the time required for coalescence of bubbles tij and contact time for two bubbles τij in Equation (10) with various kernels from Table 5. Considering the coalescence rate kernel of Guo [57] includes four collision frequency mechanisms, it is most comprehensive and will be focused on in this paper.
di, dj is diameter of two colliding bubbles, m.

Typical Kernels of Bubble Breakup
For decades, typical kernels of the bubble breakup presented in Table 6 [55,[57][58][59][60]. In the bubble breakup process, the breakup frequency and daughter size distribution matter [23]. In Table 6, the kernels of the breakup frequency Ω(Vp) and its daughter size distribution β(Vd, Vp) or the dimensionless form of the daughter size distribution β(fbv, 1) Energies 2021, 14, x FOR PEER REVIEW 9 of 33 condition [11]. Hence, the two models are adopted for the bubble condensation in the gasliquid subcooled boiling flow in this paper.

Typical Kernels of Bubble Coalescence
Typical kernels of the a bubble coalescence rate a(di, dj) are illustrated in Table 5 [53][54][55][56][57]. As the bubble coalescence at a collision with a probability, a(di, dj) is represented as the product of the collision frequency h(di, dj) and coalescence efficiency λ(di, dj) in Equation (10) [24]. At the same time, λ(di, dj) is described as the ratio of the time required for coalescence of bubbles tij and contact time for two bubbles τij in Equation (10) with various kernels from Table 5. Considering the coalescence rate kernel of Guo [57] includes four collision frequency mechanisms, it is most comprehensive and will be focused on in this paper.
di, dj is diameter of two colliding bubbles, m.

Typical Kernels of Bubble Breakup
For decades, typical kernels of the bubble breakup presented in Table 6 [55,[57][58][59][60]. In the bubble breakup process, the breakup frequency and daughter size distribution matter [23]. In Table 6, the kernels of the breakup frequency Ω(Vp) and its daughter size distribution β(Vd, Vp) or the dimensionless form of the daughter size distribution β(fbv, 1) condition [11]. Hence, the two models are adopted for the bubble condensation in the gasliquid subcooled boiling flow in this paper.

Typical Kernels of Bubble Coalescence
Typical kernels of the a bubble coalescence rate a(di, dj) are illustrated in Table 5 [53][54][55][56][57]. As the bubble coalescence at a collision with a probability, a(di, dj) is represented as the product of the collision frequency h(di, dj) and coalescence efficiency λ(di, dj) in Equation (10) [24]. At the same time, λ(di, dj) is described as the ratio of the time required for coalescence of bubbles tij and contact time for two bubbles τij in Equation (10) with various kernels from Table 5. Considering the coalescence rate kernel of Guo [57] includes four collision frequency mechanisms, it is most comprehensive and will be focused on in this paper.
di, dj is diameter of two colliding bubbles, m.

Typical Kernels of Bubble Breakup
For decades, typical kernels of the bubble breakup presented in Table 6 [55,[57][58][59][60]. In the bubble breakup process, the breakup frequency and daughter size distribution matter [23]. In Table 6, the kernels of the breakup frequency Ω(Vp) and its daughter size distribution β(Vd, Vp) or the dimensionless form of the daughter size distribution β(fbv, 1) dp ) 2/3 d condition [11]. Hence, the two models are adopted for the bubble condensation in the gasliquid subcooled boiling flow in this paper.

Typical Kernels of Bubble Coalescence
Typical kernels of the a bubble coalescence rate a(di, dj) are illustrated in Table 5 [53][54][55][56][57]. As the bubble coalescence at a collision with a probability, a(di, dj) is represented as the product of the collision frequency h(di, dj) and coalescence efficiency λ(di, dj) in Equation (10) [24]. At the same time, λ(di, dj) is described as the ratio of the time required for coalescence of bubbles tij and contact time for two bubbles τij in Equation (10) with various kernels from Table 5. Considering the coalescence rate kernel of Guo [57] includes four collision frequency mechanisms, it is most comprehensive and will be focused on in this paper.
di, dj is diameter of two colliding bubbles, m.

Typical Kernels of Bubble Breakup
For decades, typical kernels of the bubble breakup presented in Table 6 [55,[57][58][59][60]. In the bubble breakup process, the breakup frequency and daughter size distribution matter [23]. In Table 6, the kernels of the breakup frequency Ω(Vp) and its daughter size distribution β(Vd, Vp) or the dimensionless form of the daughter size distribution β(fbv, 1)

Binary breakup
where c 1~c 7 are coefficients in breakup models, c 1 = 0.00481, c 2 = 0.08, c 3 = 1, c 4 = α 2 (2π) 5/3 , c 5 = 2, c 6 = 1.59, c 7 = 2; d p is diameter of parent bubble in breakup model, m; α 1 is damping constant, 0.05 ≤ α 1 ≤ 0.3; α 2 is ratio of minimum energy required for bubble breakup to bubble surface energy, 0.1 ≤ α 2 ≤ 0.5; m is number of daughter bubbles in breakup model; f bv is bubble breakup volume fraction; f bv,min is minimum bubble breakup volume fraction; d ed is eddy size in breakup model, m; Γ is gamma function; Energies 2021, 14, x FOR PEER REVIEW 9 of 33 condition [11]. Hence, the two models are adopted for the bubble condensation in the gasliquid subcooled boiling flow in this paper.
. . with high pressure, where ε is the turbulent dissipation rate, m 2 /s 3 .

Typical Kernels of Bubble Coalescence
Typical kernels of the a bubble coalescence rate a(di, dj) are illustrated in Table 5 [53][54][55][56][57]. As the bubble coalescence at a collision with a probability, a(di, dj) is represented as the product of the collision frequency h(di, dj) and coalescence efficiency λ(di, dj) in Equation (10) [24]. At the same time, λ(di, dj) is described as the ratio of the time required for coalescence of bubbles tij and contact time for two bubbles τij in Equation (10) with various kernels from Table 5. Considering the coalescence rate kernel of Guo [57] includes four collision frequency mechanisms, it is most comprehensive and will be focused on in this paper.
di, dj is diameter of two colliding bubbles, m. Table 5. Typical kernels of bubbles coalescence rate.

Typical Kernels of Bubble Breakup
For decades, typical kernels of the bubble breakup presented in Table 6 [55,[57][58][59][60]. In the bubble breakup process, the breakup frequency and daughter size distribution matter [23]. In Table 6, the kernels of the breakup frequency Ω(Vp) and its daughter size distribution β(Vd, Vp) or the dimensionless form of the daughter size distribution β(fbv, 1) is ratio of diameter of eddy size and diameter of parent bubble; 9 of 33 e, the two models are adopted for the bubble condensation in the gasiling flow in this paper.  Table 5 [53oalescence at a collision with a probability, a(di, dj) is represented as collision frequency h(di, dj) and coalescence efficiency λ(di, dj) in t the same time, λ(di, dj) is described as the ratio of the time required ubbles tij and contact time for two bubbles τij in Equation (10) with Table 5. Considering the coalescence rate kernel of Guo [57] includes ncy mechanisms, it is most comprehensive and will be focused on in

Frequency h(di, dj)
Coalescence Efficiency λ(di, dj) . we ] ; dij is equivalent diameter of two colliding bubbles with unequal size in cal velocity for coalescence, m/s; u' is turbulent fluctuating velocity, m/s; nce between bubbles and bubble turbulent path length; is ratio of eber number of two colliding bubbles in coalescence rate model; γ is ent Weber number of two colliding bubbles; h(di, dj) T , h(di, dj) B , h(di, dj) W ent fluctuation, buoyancy driven, wake entrainment, and viscous shear, ls of Bubble Breakup pical kernels of the bubble breakup presented in Table 6 [55,[57][58][59][60]. In process, the breakup frequency and daughter size distribution matter e kernels of the breakup frequency Ω(Vp) and its daughter size p) or the dimensionless form of the daughter size distribution β(fbv, 1) min is minimum value of 9 of 33 n [11]. Hence, the two models are adopted for the bubble condensation in the gasbcooled boiling flow in this paper. pical Kernels of Bubble Coalescence ical kernels of the a bubble coalescence rate a(di, dj) are illustrated in Table 5 [53he bubble coalescence at a collision with a probability, a(di, dj) is represented as uct of the collision frequency h(di, dj) and coalescence efficiency λ(di, dj) in (10) [24]. At the same time, λ(di, dj) is described as the ratio of the time required scence of bubbles tij and contact time for two bubbles τij in Equation (10) with ernels from Table 5. Considering the coalescence rate kernel of Guo [57] includes ision frequency mechanisms, it is most comprehensive and will be focused on in er.
, dj is diameter of two colliding bubbles, m. ypical kernels of bubbles coalescence rate.

Collision Frequency h(di, dj)
Coalescence Efficiency λ(di, dj) . we ] ence rate kernels, c1 = 0.28~1.11, c2 = 0.25ln(hi/hf) = 2.3, c3 = 1.12, c4 = 0.0157, c5 = 1.29, c6 d phase, Pa·s; dij is equivalent diameter of two colliding bubbles with unequal size in m; ucrit is critical velocity for coalescence, m/s; u' is turbulent fluctuating velocity, m/s; ratio of distance between bubbles and bubble turbulent path length; is ratio of quivalent Weber number of two colliding bubbles in coalescence rate model; γ is eij is equivalent Weber number of two colliding bubbles; h(di, dj) T , h(di, dj) B , h(di, dj) W ue to turbulent fluctuation, buoyancy driven, wake entrainment, and viscous shear, pical Kernels of Bubble Breakup decades, typical kernels of the bubble breakup presented in Table 6 [55,[57][58][59][60]. In le breakup process, the breakup frequency and daughter size distribution matter Table 6, the kernels of the breakup frequency Ω(Vp) and its daughter size ion β(Vd, Vp) or the dimensionless form of the daughter size distribution β(fbv, 1) ; C f is coefficient of surface area; τ critical is critical stress force, kg/m 3 ; Binary breakup means it assuming binary breakup in kernel.

Numerical Solutions of PBM
As the analytical solutions of the PBM are only valid in quite few special cases, the numerical methods are mostly preferred [61]. Firstly, the average bubble number density transportation (ABND) approach is introduced for its relatively less calculation while it has low accuracy [19,34,62]. Then, the method of moments (MOM) is conducted. It includes the quadrature method of moments (QMOM) approach [63] and direct quadrature method of moments (DQMOM) model [13,64], etc. However, it may lead to the monovariate issues.
In addition, the multiple size group (MUSIG) method [16,31] and hence advanced inhomogeneous MUSIG model [8,65,66] are adopted. Although it has similar principles, the inhomogeneous MUSIG approach demands more computation resources than that of the MUSIG method. With the development of computation ability and trade-off between them, the superior of the MUSIG approach dominates [16]. Hence, the MUSIG method will be carried out in this paper.

Interfacial Force Model
For the modeling of the interfacial momentum forces, the drag force, lift force, wall lubrication force, and turbulent dispersion force are considered [67]. It integrates with the wall boiling model and PCM kernels as illustrated in Figure 1. These models are presented as follows.

Drag Force
Based on the similarity criterion and the mixed viscosity model, Ishii and Zuber [68] established the relationship of the drag coefficient of the two-phase flow. This relationship was widely used in both fluid particle system and solid particle system. The expression is given as follows: where C D is the drag coefficient; u g and u l are the average velocity of the gas phase and liquid phase, respectively, m/s.

Lift Force
In the shear flow or turbulence liquid flow, the lift force functions on bubbles by the velocity gradient and negligible viscosity. On this basis, the lift force model proposed by Tomiyama [69] is adopted in this paper. In addition, the lateral net shear lift force F L induced by the wake and external shear flow field is expressed as shown in Equation (12). The lift coefficient C L is mainly related to the Re g number, Eo number, and its empirical relationship is given in Equation (12).

Wall Lubrication Force
The wall lubrication force primarily acts in radial direction away from the wall. Compared with that of the liquid phase velocity between the bubble and bulk flow field, the liquid phase velocity within the bubble and wall is smaller. This kind of hydrodynamic pressure difference drives bubble away from the wall. From of the wall lubrication force F W is shown in Equation (13) [70].
where C w1 , C w2 are the wall lubrication coefficients; y w is the distance between the bubble and wall, m; n w is the unit outward normal vector on the wall surface.

Turbulent Dispersion Force
In this paper, the turbulent dissipative force model of Burns [71] is adopted to calculate the turbulent dissipative force. It describes the turbulent dissipation force F TD using Farveaveraged variables in Equation (14).
where C TD is the turbulent diffusion coefficient, C TD = 1; K TD is the empirical constant related with the Schmidt number, K TD = 90; β g , β l are the gas volume fraction and liquid volume fraction correspondingly.

Results and Discussion
In this section, firstly, nondimensional numbers of the first loop of the NPP (P = 15.70 MPa) and DEBORA experiment test cases are analyzed with approximation. Then, various combinations of wall boiling models and PBM kernels are calculated with ANSYS fluent software using User-Defined Functions (UDF). In addition, the bubble size distribution d p , void fraction α, gas superficial velocity j g and liquid superficial velocity j l are evaluated with the experiment test case (DEBORA-1). The model of the bubble departure diameter D d [8] and bubble departure frequency f [46] are specified for efficiency. It is selected from the explanations in Sections 2.3.2 and 2.3.3.

Experiment Test Case for First Loop of NPP
Under high-pressure conditions, the availability of the experimental data are highly limited and experimental work are quite difficult. Furthermore, none of the previous experiments provides a complete description [20]. Then, the subcooled boiling experiment with the refrigerant have great advantages. It can obtain the similar nondimensional number with a relative low system pressure [28]. To compare the first loop of the NPP to typical refrigerant experiments with relatively low pressures, the relevant nondimensional numbers should be similar.

Nondimensional Number Approximation Analysis
As illustrated in Section 1.4, the Re l number, ratio of two phases ρ l /ρ g , Ja e number and Bo number are most considered for the wall boiling phenomena in the vertical subcooled boiling flow. Commonly, the relevant nondimensional numbers should be better within 1 order of the magnitude (<10 times) [8].
In Table 7, typical parameters including the Re l number, ratio ρ l /ρ g , Ja e number, Bo number of the DEBORA facility are quite close to that in the first loop of the NPP. For the first loop of the NPP (P = 15.70 MPa), the above 4 nondimensional numbers of the DEBORA facility with P = 2.62 MPa condition are similar with each other (all within 3 times). Other nondimensional numbers like the bubble Reynolds number (Re g ) and Eötvös number (Eo) are also less than 1 order of the magnitude with that in the first loop of the NPP (see Section 1.4). In contrast, the ratio ρ l /ρ g , Re g number and Ja e number of the SUBO, SNU and Purdue facilities are over 1 order of the magnitude than that in the first loop of the NPP as shown in Table 1 in Section 1.4. Therefore, the numerical calculations are compared and analyzed with the experiment data of the DEBORA facility (P = 2.62 MPa) for the first loop of the NPP (P = 15.70 MPa).

Experimental Test Case
The illustration of the DEBORA facility is presented in Figure 2. In Figure 2, the length of the inlet section, heated section and outlet section are 1 m, 3.5 m, and 0.5 m, respectively [28]. Meanwhile, the radial profiles of volume fraction and gas velocities at the end of the head section are measured by the optical probe with the coordinate z = 3.5 m. Moreover, a r-z coordinate system is established in Figure 2. Additionally, the detail parameters are given in Table 1. As explained in Section 3.1.1, the experiment test case of DEBORA-1 (P = 2.62 MPa) is approximate with the first loop of the NPP. Hence, experimental data of the DEBORA-1 of the DEBORA facility are selected for the CFD validation.

Analysis of Bubble Size Distribution d p
In this section, the wall boiling model coupled with the coalescence and breakup kernels are considered for the analysis of the bubble size distribution. Additionally, the calculations of wall boiling models and combinations of coalescence and breakup kernels are analyzed and compared with DEBORA-1 (P = 2.62 MPa) for the first loop of the NPP (P = 15.70 MPa). Before the calculation, the mesh size independencies are carried out.
Three mesh sizes of 250,000, 500,000, and 1,000,000 are conducted. For the Sauter mean diameter (SMD) d p , the former two calculation results are compared with the finest one. Its relative errors of the d p at the radius r = 0.007 m with z = 3.5 m are 2.9%, 0.2%, respectively. To ensure the accuracy and time, an appropriate mesh size of 500,000 are selected.
3.2.1. Influence of Wall Boiling Model with DEBORA-1 for First Loop of NPP For the first loop of NPP, the SMD d p with various typical models of the active nucleation site density N n are shown in Figure 3. For the evaluation of the wall boiling models, the coalescence rate kernel of Guo [57] and breakup kernel of Luo and Svendsen [60] are adopted for the PBM model.

Analysis of Bubble Size Distribution dp
In this section, the wall boiling model coupled with the coalescence and breakup kernels are considered for the analysis of the bubble size distribution. Additionally, the calculations of wall boiling models and combinations of coalescence and breakup kernels are analyzed and compared with DEBORA-1 (P = 2.62 MPa) for the first loop of the NPP (P = 15.70 MPa). Before the calculation, the mesh size independencies are carried out. Three mesh sizes of 250,000, 500,000, and 1,000,000 are conducted. For the Sauter mean diameter (SMD) dp, the former two calculation results are compared with the finest one. Its relative errors of the dp at the radius r = 0.007 m with z = 3.5 m are 2.9%, 0.2%, respectively. To ensure the accuracy and time, an appropriate mesh size of 500,000 are selected.

Influence of Wall Boiling Model with DEBORA-1 for First Loop of NPP
For the first loop of NPP, the SMD dp with various typical models of the active nucleation site density Nn are shown in Figure 3. For the evaluation of the wall boiling models, the coalescence rate kernel of Guo [57] and breakup kernel of Luo and Svendsen [60] are adopted for the PBM model. . Bubble size distribution of SMD dp of various typical models of Nn with model of Dd [8], f [46], a(di, dj) [57], Ω(Vp), β(fbv, 1) [54] and DEBORA-1 experiment (P = 2.62 MPa) [28] at z = 3.5 m.

Combinations of Coalescence and Breakup Kernel with DEBORA-1 for First Loop of NPP
With the active nucleation site density models of Kocamustafaogullari and Ishii [36] and Basu [40], the SMD d p with four combinations of coalescence and breakup kernels are conducted in the calculations as shown in Figure 4. In Figure 4a, numerical results of four combinations of C1, C2, C3, and C4 with the N n model of Kocamustafaogullari and Ishii [36] are obtained. The maximum relative errors between numerical results of C1, C2, C3, and C4 with the DEBORA-1 experiment data are 9.63%, 37.54%, 12.97%, and 29.88%, correspondingly. Consequently, the N n model of Kocamustafaogullari and Ishii [36] coupled with C1 have better results.  . Bubble size distribution of SMD dp of various combinations of coalescence and breakup kernel with model of Dd [8], f [46] and DEBORA-1 experiment (P = 2.62 MPa) [28] at z = 3.5 m. (a) with Nn of Kocamustafaogullari and Ishii [36], and (b) with Nn of Basu [40].
Similarly in Figure 4b, the maximum relative errors between numerical results of C1, C2, C3, C4 under the Nn model of Basu [40] with the DEBORA-1 experiment data are 11.18%, 34.86%, 25.72%, and 21.99%, respectively. Hence, the Nn model of Basu [40] coupled with C1 performs well. Furthermore, the Nn model of Kocamustafaogullari and  [8], f [46] and DEBORA-1 experiment (P = 2.62 MPa) [28] at z = 3.5 m. (a) with N n of Kocamustafaogullari and Ishii [36], and (b) with N n of Basu [40]. Figure 4b, the maximum relative errors between numerical results of C1, C2, C3, C4 under the N n model of Basu [40] with the DEBORA-1 experiment data are 11.18%, 34.86%, 25.72%, and 21.99%, respectively. Hence, the N n model of Basu [40] coupled with C1 performs well. Furthermore, the N n model of Kocamustafaogullari and Ishii [36] with C1 or Basu [40] with C1 have better performance for the bubble size distribution. Moreover, the N n model of Kocamustafaogullari and Ishii [36] with C1 performs best in the DEBORA-1 experiment for the first loop of the NPP.

Error Analysis on Bubble Size Distribution
The maximum relative errors of the SMD d p between numerical results with DEBORA-1 experiment data (P = 2.62 MPa) for the first loop of the NPP (P = 15.70 MPa) is shown in Table 8. For the first loop of the NPP, the N n model of Kocamustafaogullari and Ishii [36] with C1 (maximum relative error 9.63%) performs better than others. n the one hand, the N n model of Kocamustafaogullari and Ishii [36] relates with the nondimensional cavity radius R * c , wall superheat ∆T sup and density ratio ρ * . On the other hand, the N n model of Basu [40] relies on the wall superheat ∆T sup and contact angle θ. These two models have more insights into the wall boiling mechanisms for the subcooled boiling flow in the first loop of the NPP.
For the CO kernel of Guo [57], it contains four collision frequency mechanisms and more complete than others. At the same time, the BR kernel of Luo and Svendsen [54] which is based on the BR kernel of Lee [59] are more sophisticated. Therefore, the combinations of C1 (CO-Guo [57] &BR-Luo and Svendsen [54]) with the N n model of Kocamustafaogullari and Ishii [36] or Basu [40] lead to better results.

Calculation of Void Fraction α
Then, the wall boiling model coupled with the coalescence and breakup kernels are adopted for the calculation of the void fraction α in this section. Meanwhile, the influence of wall boiling models and combinations of coalescence and breakup kernels are considered with DEBORA-1 (P = 2.62 MPa) for the first loop of the NPP.

Influence of Wall Boiling Model with DEBORA-1 for First Loop of NPP
For the first loop of the NPP, the void fraction α with various typical models of the active nucleation site density N n [5,21,35,36,40] are given in Figure 5.

Combinations of Coalescence and Breakup Kernel with DEBORA-1 for First Loop of NPP
With the active nucleation site density models of Kocamustafaogullari and Ishii [36] and Basu [40], void fraction α with four combinations of coalescence and breakup kernels are conducted in the calculations in Figure 6.
In Figure 6a, the maximum relative errors between four combinations of C1, C2, C3 and C4 under the N n model of Kocamustafaogullari and Ishii [36] with the DEBORA-1 experiment data are 29.64%, 64.06%, 100.00% and 95.07% correspondingly. Consequently, the N n model of Kocamustafaogullari and Ishii [36] coupled with C1 has relatively better results. Similarly in Figure 6b, the maximum relative errors between four combinations of C1, C2, C3, C4 under the N n model of Basu [40] with the experiment data are 100.00%, 72.16%, 68.57%, and 69.82% respectively. Finally, the N n model of Kocamustafaogullari and Ishii [36] coupled with C1 has nice behaviors.

Error Analysis on Void Fraction α
The maximum relative errors of the void fraction α between numerical results with DEBORA-1 experiment data (P = 2.62 MPa) for the first loop of the NPP (P = 15.70 MPa) is shown in Table 9.
For the first loop of the NPP, the N n model of Kocamustafaogullari and Ishii [36] with C1 (maximum relative error 29.64%) performs best than others. Models of Kocamustafaogullari and Ishii [36] have more insights into the wall boiling mechanisms for the subcooled boiling flow in the first loop of the NPP. Lastly, the combinations of C1 (CO-Guo [57] &BR-Luo and Svendsen [54]) are more complete and fully described than other combinations. This is consistent with that of the bubble size distribution in Section 3.2.3. In Figure 6a, the maximum relative errors between four combinations of C1, C2, C3 and C4 under the Nn model of Kocamustafaogullari and Ishii [36] with the DEBORA-1 Figure 6. Void fraction α distribution of various combinations of coalescence and breakup kernel with model of D d [8], f [46] and DEBORA-1 experiment (P = 2.62 MPa) [28] at z = 3.5 m. (a) with Nn of Kocamustafaogullari and Ishii [36], (b) with N n of Basu [40]. Furthermore, the wall boiling model coupled with the coalescence and breakup kernels are adopted for the calculation of the gas superficial velocity j g in this section. At the same time, the influence of wall boiling models and combinations of coalescence and breakup kernels are taken into account with DEBORA-1 (P = 2.62 MPa) for the first loop of the NPP (P = 15.70 MPa).

Influence of Wall Boiling Model with DEBORA-1 for First Loop of NPP
For the first loop of the NPP, the gas superficial velocity j g with various typical models of the active nucleation site density N n [5,21,35,36,40] are presented in Figure 7.  . Superficial gas velocity jg distribution of various typical models of Nn with model of Dd [8], f [46], a(di, dj) [57], Ω(Vp), β(fbv, 1) [54] and DEBORA-1 experiment (P = 2.62 MPa) [28] at z = 3.5 m.
From Figure 7, except the near wall region, the maximum relative errors between various models of Lemmert and Chawla [35], Kocamustafaogullari and Ishii [36], Basu [40], Hibiki and Ishii [23] and Li [5] with experiment data from DEBORA-1 are 19.84%, 18.56%, 93.50%, 60.42%, and 13.65%, respectively. Therefore, the models of Kocamustafaogullari and Ishii [36] and Li [5] perform better than other models. To ensure the consistency with the previous analysis, the model of Kocamustafaogullari and Ishii [36] will be focused on as well as the model of Basu [40].

Combinations of Coalescence and Breakup Kernel with DEBORA-1 for First Loop of NPP
The gas superficial velocity j g with four combinations of coalescence and breakup kernels are conducted in the calculations as given in Figure 8. It is coupled with the N n models of Kocamustafaogullari and Ishii [36] and Basu [40]. In Figure 8a, except the near wall region (r > 8.5 mm), the relative errors between four combinations of C1, C2, C3, and C4 under the Nn model of Kocamustafaogullari and Ishii  [36], and (b) with N n of Basu [40]. In Figure 8a, except the near wall region (r > 8.5 mm), the relative errors between four combinations of C1, C2, C3, and C4 under the N n model of Kocamustafaogullari and Ishii [36] with the experiment data are 18.56%, 8.06%, 16.65%, and 12.77% correspondingly. Thus, the N n model of Kocamustafaogullari and Ishii [36] coupled with C2 has relatively better results.
As shown in Figure 8b, except the near wall region (r > 8.5 mm), the relative errors between four combinations of C1, C2, C3, and C4 under the N n model of Basu [40] with the experiment data are 100.00%, 14.74%, 16.65%, and 4.67%, respectively. Hence, the N n model of Basu [40] coupled with C4 behaves well. In summary, the N n model of Basu [40] coupled with C4 performs best.

Error Analysis on Gas Superficial Velocity j g
The maximum relative errors of the gas superficial velocity j g between numerical results with DEBORA-1 experiment data (P = 2.62 MPa) for the first loop of the NPP (P = 15.70 MPa) is shown in Table 10. onsidering the calculation of the gas superficial velocity j g near the wall region is influenced by the wall effect and measurement discrepancy, the error analysis on the gas superficial velocity j g are mainly except the near wall region. For the first loop of the NPP, the model of Basu [40] coupled with C4 (maximum relative error 4.67%) has nice behaviors.
For the first loop of the NPP, the combination C4 is more suitable than C1 for the calculation of the gas superficial velocity j g . Compared C4 (CO-Prince and Blanch [53]&BR-Luo and Svendsen [54]) with C1 (CO-Guo [57]&BR-Luo and Svendsen [54]), the breakup kernel is the same while the coalescence kernel differs. For the collision frequency h(d i , d j ), it is mainly due to four mechanisms: turbulence-induced collision; buoyancy-induced collision; wake-entrainment collision; viscous shear-induced collision. In addition, the coalescence kernel of Prince and Blanch [53] considers the turbulence-induced collision while the coalescence kernel of Guo [57] takes the four mechanisms together. Hence, the results show that the gas superficial velocity j g could mainly dependent on the turbulenceinduced collision which leads to better results.

Tendency of Liquid Superficial Velocity j l
Lastly, the wall boiling model coupled with the coalescence and breakup kernels are adopted for the calculation of the liquid superficial velocity j l in this section. In the meantime, the results of wall boiling models and combinations of coalescence and breakup kernels are compared with DEBORA-1 (P = 2.62 MPa) for the first loop of the NPP (P = 15.70 MPa).

Influence of Wall Boiling Model with DEBORA-1 for First Loop of NPP
For the first loop of the NPP, the liquid superficial velocity j l with various typical models of the active nucleation site density N n [5,21,35,36,40] are presented in Figure 9. For the first loop of the NPP, the liquid superficial velocity jl with various typical models of the active nucleation site density Nn [5,21,35,36,40] are presented in Figure 9. Figure 9. Superficial liquid velocity jl distribution of various typical models of Nn with model of Dd [8], f [46], a(di, dj) [57], Ω(Vp), β(fbv, 1) [54], and DEBORA-1 experiment (P = 2.62 MPa) [28] at z = 3.5 m.

Combinations of Coalescence and Breakup Kernel with DEBORA-1 for First Loop of NPP
The liquid superficial velocity jl with four combinations of coalescence and breakup kernels are conducted in the calculations as given in Figure 10. In Figure 10a, except the near wall region (r > 8.5 mm), the maximum relative errors between four combinations of C1, C2, C3, and C4 under the Nn model of Kocamustafaogullari and Ishii [36] with the experiment data are 6.97%, 6.97%, 7.19%, and 6.97%, correspondingly. Thus, the Nn model of Kocamustafaogullari and Ishii [36] coupled with all combinations separately has relatively better results. Figure 9. Superficial liquid velocity j l distribution of various typical models of N n with model of D d [8], f [46], a(d i , d j ) [57], Ω(V p ), β(f bv , 1) [54], and DEBORA-1 experiment (P = 2.62 MPa) [28] at z = 3.5 m.

Combinations of Coalescence and Breakup Kernel with DEBORA-1 for First Loop of NPP
The liquid superficial velocity j l with four combinations of coalescence and breakup kernels are conducted in the calculations as given in Figure 10. In Figure 10a, except the near wall region (r > 8.5 mm), the maximum relative errors between four combinations of C1, C2, C3, and C4 under the N n model of Kocamustafaogullari and Ishii [36] with the experiment data are 6.97%, 6.97%, 7.19%, and 6.97%, correspondingly. Thus, the N n model of Kocamustafaogullari and Ishii [36] coupled with all combinations separately has relatively better results.
As shown in Figure 10b, except the near wall region, the maximum relative errors between four combinations of C1, C2, C3, and C4 under the N n model of Basu et al. (2002) with the experiment data are 6.50%, 6.63%, 7.68%, and 7.21%, respectively. Hence, the N n model of Basu [40] coupled with all combinations alone behaves well.  [36], and (b) with N n of Basu [40]. The maximum relative errors of the liquid superficial velocity j l between numerical results with DEBORA-1 experiment data (P = 2.62 MPa) for the first loop of the NPP (P = 15.70 MPa) is shown in Table 11. imilarly, the liquid superficial velocity j l near the wall region has large deviations due to the wall effect and measurement discrepancy. Hence, the error analysis in this section excludes the near wall region. For the first loop of the NPP, the N n model of Basu [40] coupled with C1 (maximum relative error 6.50%) also has nice behaviors.

Conclusions
In this work, the wall boiling coupled with the PBM models including the wall nucleation, bubble departure, bubble condensation, coalescence, and breakup are analyzed for the gas-liquid subcooled boiling flow in the first loop of the NPP. Firstly, nondimensional numbers of the first loop of the NPP (P = 15.70 MPa) and experiment test cases DEBORA-1 (P = 2.62 MPa) are analyzed with approximation. Then, five active nucleation site density models coupled with the kernel combination C1 are calculated and analyzed with ANSYS fluent software using UDF. In addition, four combinations of the PBM kernels with the active nucleation site density models of Kocamustafaogullari and Ishii [36] and Basu [40] are evaluated for the first loop of the NPP. Lastly, various behaviors including the bubble size distribution d p , void fraction α, gas superficial velocity j g and liquid superficial velocity j l are compared and validated with the DEBORA-1 (P = 2.62 MPa) of the DEBORA facility. Main conclusions are as below:

7.
Especially, the combination C4 is more suitable than C1 for the calculation of the gas superficial velocity j g for the first loop of the NPP. The gas superficial velocity j g could mainly dependent on the turbulence-induced collision which the C4 is mainly related with. 8.
At last, the CO kernel of Guo [57] contains four collision frequency mechanisms and more complete than others. At the same time, the BR kernel of Luo and Svendsen [54] which is based on the BR kernel of Lee [59] are more sophisticated. Therefore, the combinations of C1 are applicable for the calculation of the bubble size distribution d p , void fraction α, and liquid superficial velocity j l .
The results indicate that the detailed evaluation and comparison of the wall boiling coupled with PBM models are particularly crucial for the practical engineering issues. Evaluations on the wall boiling models coupled with PBM kernels are elaborated for the first loop of the NPP (P = 15.70 MPa). In the end, this paper can provide guidance for the numerical computation of the subcooled boiling flow of the first loop of the NPP. The combination of the coalescence rate kernel of Hibiki and Ishii (2002b) and breakup kernel of Luo and Svendsen (1996) C4 The combination of the coalescence rate kernel of Prince and Blanch (1990)  The proportional of the heated wall coved by the nucleation bubbles, The cross-sectional area of the boiling channel, m 2 B C , D C The source term and sink term related to the coalescence of bubbles B B , D B The source term and sink term related to the breakup of bubbles c 1~c6 The coefficients in the coalescence rate models c 1~c 7 The coefficients in the breakup models C D , C D,i , C D,j The drag coefficient C f The coefficient of the surface area C L The lift coefficient C Pl The specific heat capacity of the liquid phase at a certain pressure, J/(kg·K) C R The circulation ratio of the steam generator C TD The turbulent diffusion coefficient, C TD = 1 C w1, C w2 The wall lubrication coefficients d i , d j The diameter of two colliding bubbles in the coalescence rate model, m F u The area of the ascending channel area of the secondary side in the steam generator, m 2 F W The wall lubrication force, N g The gravity acceleration, g = 9.80 m/s 2 G The mass flux, kg/(m 2 ·s) h i , h f The initial and critical film thickness of bubbles in the coalescence model, m H The height of the unaerated liquid, m h The inter-phase heat transfer coefficient, W/(m 2 ·K) h lg The latent heat, J/(kg) h s The single-phase heat transfer coefficient, W/(m 2 ·K) h(d i , d j ) The collision frequency, m 3 /s h(d i , d j ) T The collision frequency due to turbulent fluctuation, m 3 /s h(d i , d j ) B The collision frequency due to buoyancy driven, m 3 /s h(d i , d j ) W The collision frequency due to wake entrainment, m 3 /s h(d i , d j ) V The collision efficiency due to viscous shear, m 3 /s h fe The feed water enthalpy, J/kg h gs The saturated gas enthalpy, J/kg h ls The saturated liquid enthalpy, J/kg j g The superficial gas velocity, m/s j go The outlet superficial gas velocity, m/s j l The superficial liquid velocity, m/s j lo The outlet superficial liquid velocity, m/s j r The relative velocity between the two phases, m/s j s The nondimensional fluid velocity gradient k l The liquid thermal conductivity, W/(m·K) The number of U-shaped tubes in the steam generator n 2 The number of support bars in the steam generator n w The unit outward normal vector on the wall surface P The pressure, MPa q The interphase heat flux, W/m 2 q c The liquid-phase convection heat flux, W/m 2 q e The evaporation heat flux, W/m 2 q q The quenching heat flux, W/m 2 Q The wall heat flux, W/m 2 Q total The total heat exchange, W or kW r The radius of the pipe, m R The ideal gas constant, R = 8.3144 J/(K·mol) R c The critical cavity radius based on the wall superheat, m R * c The nondimensional cavity radius, R * c = (R c /(D d /2)) S The suppression factor in the forced convective boiling, 0 < S < 1 t The time, s t ij The time required for coalescence of bubbles in the coalescence model, s T The temperature, • C or K T 0 The room temperature, • C or K T c The critical temperature at which contact angle becomes 0, • C or K T g The temperature of the gas phase, • C or K T l The temperature of the liquid phase, • C or K T w The temperature of the heated wall, • C or K T sat The saturation temperature, • C or K u crit The critical velocity for coalescence, m/s u bi , u bj The rising velocity of bubbles of size d i and d j , m/s u g The average velocity of gas phase, m/s u l The average velocity of liquid phase, m/s u wi , u wj The wake entrainment velocity of bubbles of size d i and d j , m/s u' The The ratio between the density difference and gas density, ρ * = ∆ρ/ρ g ρ + The model parameter, ρ + = log(ρ * ) ∆t ln The logarithmic average temperature difference between the primary side of the steam generator, ∆t ln = (∆t max − ∆t min )/ln(∆t max /∆t min ), • C or K ∆T ln The logarithmic average temperature difference of the liquid subcooling temperature, ∆T ln = (∆T max − ∆T min )/ln(∆T max /∆T min ), • C or K ∆T e The effective wall superheat in the forced convective boiling, ∆T e = S∆T sup = S(T w − T sat ), • C or K ∆T sub The inlet liquid subcooling temperature, ∆T sub = T sat − T l , • C or K ∆T sup The wall superheat above saturation, ∆T sup = T w − T sat , • C or K ∆T ONB The wall superheat above saturation at the onset of nucleate boiling, • C or K The coalescence efficiency