development of cfd code for subcooled boiling two-phase flow...
TRANSCRIPT
KAERI/TR-3679/2008
Development of CFD Code for Subcooled Boiling
Two-Phase Flow with Modeling of the Interfacial Area
Transport Equation
계면면적 수송방정식 모델을 통한
미포화 비등 이상유동 해석코드 개발
한 국 원 자 력 연 구 원
기술보고서
제 출 문
한국원자력연구원장 귀하
본 보고서를 20O8 년도 “핵심 열수력 개별현상 고정밀 실험” 과제의
기술보고서로 제출합니다.
2008. 12.
주 저 자 : 배 병 언
공 저 자 : 윤 병 조
윤 한 영
어 동 진
송 철 화
i
Summary
I. Title
Development of CFD Code for Subcooled Boiling Two-Phase Flow with Modeling of
the Interfacial Area Transport Equation
II. Background and objective
The subcooled boiling is a crucial phenomenon for a nuclear reactor safety. In order to
analyze the subcooled boiling two-phase flow, the two-fluid model is the most
appropriate model which treats the behavior of each phase separately. To precisely close
the two-fluid model, an accurate prediction for the interfacial area concentration is
required. Thus a formulation of an interfacial area transport equation has been suggested
as an improved approach, which estimates the dynamic and multi-dimensional behavior
of the interfacial area concentration. In this study, the interfacial area transport equation
available for the subcooled boiling flow was developed with a mechanistic model for
the wall boiling source term. To evaluate the model, the computational analysis has
been conducted.
III. Contents
In the analysis, the one-group interfacial area transport equation was developed with
modeling the source terms for the wall nucleation. It included the bubble lift-off
diameter model and lift-off frequency reduction factor model, which took into account
the bubble sliding on the heated wall followed by the departure from a nucleate site.
With such models, EAGLE (Elaborated Analysis of Gas-Liquid Evolution) code was
developed for a multi-dimensional analysis of two-phase flow with the interfacial area
transport equation. The code structure was based on the two-fluid model and the
Simplified Marker And Cell (SMAC) algorithm was modified to be available for a two-
phase flow simulation with a phase change.
ii
IV. Results
The results of the computational analysis revealed that the interfacial area transport
equation with the bubble lift-off diameter model agreed well with the experimental
results of SUBO facility and the experiment in Seoul National University. It
demonstrates that the source term for the wall nucleation by considering a bubble
sliding and lift-off mechanism improved the prediction capability for a multi-
dimensional behavior of void fraction or interfacial area concentration in the subcooled
boiling flow. Also, the modeling of the turbulence induced by boiling bubbles
contributed a better prediction of the bubble velocity near the heated wall.
V. Application
In the future, EAGLE code with the mechanistic model of the bubble lift-off is expected
to contribute an accurate multi-dimensional analysis of the complex two-phase flow
behavior in the subcooled boiling. Also, EAGLE code will make it possible to validate
the two-group interfacial area transport equation in the future.
iii
요 약 문
I. 연구제목
계면면적 수송방정식 모델을 통한 미포화 비등 이상유동 해석코드 개발
II. 연구 배경 및 목적
원자로 안전에서 중요한 현상 중의 하나인 미포화 비등 현상을 해석하기 위한 방법
중 이유체 모델(Two-fluid model)은 각 상의 거동을 개별적으로 고려함으로써
이상유동 해석에 가장 적절한 모델로 알려져 있다. 이유체 모델을 통해 현상을
정확히 해석하기 위해서는 계면면적밀도에 대한 정확한 예측이 요구되며, 계면면적
수송방정식을 통해 계면면적밀도의 동적 및 다차원적 거동에 대한 해석 능력을
향상시킬 수 있는 방법이 기존의 연구들을 통해 제안되어 왔다. 본 연구에서는
미포화 비등 현상에 적용할 수 있는 현상학적 생성항 모델 개발을 통해 계면면적
수송방정식을 개선하고 이를 적용한 전산유체해석 코드를 개발 및 검증하고자 한다.
III. 연구 내용 및 범위
벽면 비등에 대한 계면면적 수송방정식 생성항 모델 개발을 위해 기포의 벽면 이탈
현상만을 고려한 기존 모델과는 달리 기포의 Sliding, Coalescence 및 Lift-off되는
현상을 모델링하였다. 이를 적용한 계면면적 수송방정식을 삽입하기 위해, 이유체
모델을 기반으로 이상유동의 다차원 거동을 해석하기 위한 EAGLE (Elaborated
Analysis of Gas-Liquid Evolution) 코드를 개발하였다. 상변화가 존재하는 유동에 대해
연속방정식 및 운동량 보존방정식 간의 수렴을 위해 SMAC (Simplified Marker And
Cell) 알고리즘을 확장시켜 적용하였다.
IV. 연구 결과
개발된 계면면적 수송방정식 및 EAGLE 코드의 검증을 위해 SUBO (Subcooled
Boiling) 실험에 대한 검증 계산을 수행하였다. 또한 열유속과 질량유속이 낮은
조건의 실험결과에 대한 벤치마크를 위해 서울대학교 (SNU) 실험결과를
활용하였다. SUBO 및 SNU 실험의 국소기포인자 데이터와 EAGLE 해석 결과의
iv
비교를 통해, 기포의 벽면 Lift-off 모델을 적용한 계면면적 수송방정식이 미포화
비등 채널 내의 이상유동 거동을 적절하게 예측하는 것으로 나타났다. 즉 비등
벽면에서 기포가 생성될 때 발생하는 Sliding 및 Lift-off 현상에 대한 생성항 모델을
통해 국소 기포인자들의 다차원적인 해석 능력이 향상되었다고 할 수 있다. 또한
벽면에서 생성되는 기포에 의해 액상의 난류성분 및 벽면마찰이 증대되는 효과를
모델링함으로써, 가열벽면 근처에서 기포속도 분포를 더 적절히 예측할 수 있었다.
V. 연구 결과의 활용
본 연구에서 개발된 계면면적 수송방정식 모델 및 이상유동 해석코드 EAGLE은
향후 미포화 비등 유동에 대한 다차원적 해석 능력을 향상시킬 것으로 기대된다.
또한 확장된 유동 영역에 대한 해석 및 모델 개발 시 EAGLE 코드를 활용할 수
있다.
v
List of Contents
1. Introduction ................................................................................................ 1
1.1. Background ........................................................................................ 1
1.2. Literature survey ................................................................................ 2
1.3. Objectives and scope of this study......................................................... 3
2. Code Structure ............................................................................................ 4
2.1. Governing equations............................................................................ 4
2.2. Numerical scheme : Extended SMAC algorithm .................................... 6
2.3. Constitutive relations........................................................................... 9
(1) Subcooled boiling and condensation model .......................................... 9
(2) Closure relation for the interfacial momentum transfer......................... 10
(3) Turbulence model in the two-phase flow............................................ 11
2.4 Numerical formulation ....................................................................... 12
(1) Phase change at interface................................................................. 12
(2) Implicit formulation of momentum and energy ................................... 13
(3) Standard k-ε model......................................................................... 15
(4) Heat partition model ....................................................................... 16
(5) Discretization of governing equation ................................................. 18
3. Mechanistic Modeling of Interfacial Area Transport Equation...................... 28
3.1. One-group interfacial area transport equation .................................... 28
3.2. Modeling of bubble lift-off diameter ................................................... 30
3.3. Modeling of lift-off frequency reduction factor .................................... 33
4. Analysis Results and Model Evaluation ....................................................... 38
4.1. Benchmark analysis for a single-phase flow......................................... 38
(1) Benchmark problem ....................................................................... 38
(2) Analysis results.............................................................................. 38
4.2. Analysis of SUBO experiment ............................................................ 39
(1) Description of the calculation........................................................... 39
(2) Result and discussion...................................................................... 41
4.3. Benchmark analysis for SNU experiments........................................... 43
(1) Description of the experiment .......................................................... 43
(2) Result and discussion...................................................................... 44
5. Conclusion ................................................................................................ 78
vi
Nomenclatures .............................................................................................. 80
References .................................................................................................... 82
vii
List of Tables
Table 2.1 Models for active nucleation site density "( )N in literature ................... 21
Table 2.2 Models for bubble departure diameter ( )dD in literature ....................... 22
Table 2.3 Models for bubble departure frequency ( )f in literature ....................... 23
Table 2.4 Wall heat flux partitioning model in CFX-4........................................... 24
Table 2.5 Major subroutines in EAGLE code ...................................................... 25
Table 4.1 Test geometry for the natural convection............................................... 46
Table 4.2 Comparison of average the Nusselt number........................................... 46
Table 4.3 Test matrix of SUBO experiment......................................................... 46
Table 4.4 Geometry of the SNU experiment........................................................ 46
Table 4.5 SNU Test condition for the subcooled boiling........................................ 47
viii
List of Figures
Figure 2.1 Velocity around a bubble................................................................... 26
Figure 2.2 Flow chart of EAGLE code ............................................................... 27
Figure 3.1 Mechanism of sliding and lift-off of a bubble....................................... 36
Figure 3.2 Force balance on a bubble at the wall (Yeoh and Tu, 2005) .................... 36
Figure 3.3 Fitted relation of Eq. (3.19) ............................................................... 37
Figure 4.1 Benchmark problem for single-phase natural convection ....................... 48
Figure 4.2(a) Temperature distribution, Ra=103 ................................................... 49
Figure 4.2(b) Temperature distribution, Ra=104 ................................................... 49
Figure 4.2(c) Temperature distribution, Ra=105 ................................................... 50
Figure 4.2(d) Temperature distribution, Ra=106 ................................................... 50
Figure 4.3(a) Velocity field, Ra=103 .................................................................. 51
Figure 4.3(b) Velocity field, Ra=104 .................................................................. 51
Figure 4.3(c) Velocity field, Ra=105 .................................................................. 52
Figure 4.3(d) Velocity field, Ra=106 .................................................................. 52
Figure 4.4 Geometry and measuring position of SUBO facility.............................. 53
Figure 4.5(a) Comparison of the void fraction in Base case ................................... 54
Figure 4.5(b) Comparison of the void fraction in Q1 case ..................................... 54
Figure 4.5(c) Comparison of the void fraction in Q2 case...................................... 55
Figure 4.5(d) Comparison of the void fraction in V1 case ..................................... 55
Figure 4.5(e) Comparison of the void fraction in V2 case...................................... 56
Figure 4.5(f) Comparison of the void fraction in T1 case ...................................... 56
Figure 4.6(a) Comparison of IAC in Base case .................................................... 57
Figure 4.6(b) Comparison of IAC in Q1case ....................................................... 57
Figure 4.6(c) Comparison of IAC in Q2 case ...................................................... 58
Figure 4.6(d) Comparison of IAC in V1 case ...................................................... 58
Figure 4.6(e) Comparison of IAC in V2 case ...................................................... 59
Figure 4.6(f) Comparison of IAC in T1 case ....................................................... 59
Figure 4.7 Sensitivity on boiling source term in Base case .................................... 60
Figure 4.8 Sensitivity on boiling source term in Q2 case ....................................... 62
Figure 4.9(a) Comparison of the bubble velocity in Base case ............................... 64
Figure 4.9(b) Comparison of the bubble velocity in Q1 case.................................. 64
Figure 4.9(c) Comparison of the bubble velocity in Q2 case .................................. 65
ix
Figure 4.9(d) Comparison of the bubble velocity in V1 case.................................. 65
Figure 4.9(e) Comparison of the bubble velocity in V2 case .................................. 66
Figure 4.9(f) Comparison of the bubble velocity in T1 case................................... 66
Figure 4.10 Comparison of the bubble velocity in the Base case ............................ 67
Figure 4.11 Comparison of the bubble velocity in the V1 case ............................... 68
Figure 4.12 Effect of grid refinement in Base case ............................................... 69
Figure 4.13 Annulus channel in SNU experiment................................................. 70
Figure 4.14 Void fraction in SNU test cases ........................................................ 71
Figure 4.15 Interfacial area concentration in SNU test cases.................................. 73
Figure 4.16 Bubble velocity in SNU test cases .................................................... 75
Figure 4.17 Comparison of bubble velocity with CFX calculation .......................... 77
1
1. Introduction
1.1. Background
Two-phase flow phenomena with a boiling or condensation are known to be crucial for
a nuclear reactor design and safety analysis. Especially, the subcooled boiling
phenomena have become one of important issues in a design, operation and safety
analysis of a nuclear power plant. For an example of the phenomena, it can be observed
in the downcomer boiling during a Large-Break Loss-of-Coolant Accident (LBLOCA)
reflood phase (Song et al., 2007). For the analysis of the subcooled boiling two-phase
flow, the two-fluid model is considered as a state-of-the-art model which deals with the
mass, momentum and energy of each phase separately (Ishii and Mishima, 1984). As
given in the formulation of the two-fluid model, the interaction between two phases
such as the interfacial momentum or heat transfer plays an important role in the
dynamics of each phase and is proportional to the interfacial area. So the interfacial area
concentration (IAC), which is defined as the area of interface per unit mixture volume,
is one of the most significant parameters governing the behavior of each phase. In
conventional approaches as implemented in the nuclear system analysis codes, IAC
models have been developed for a fully developed flow based on a flow regime map.
However, due to the static characteristics of those models, it has been reported that they
had a weakness in predicting a gradual transition of interfacial structure and induced
artificial discontinuities or instabilities during the estimation of interfacial interaction
terms (Kelly, 1997). Consequently, the two-phase flow analysis with a dynamic
approach for IAC has been required.
2
1.2. Literature survey
In order to resolve the problems of the conventional models for IAC, an interfacial area
transport equation (IATE) has been developed for an adiabatic bubbly flow or nucleate
boiling flow. As derived in the research of Kocamustafaogullari and Ishii (1995), it
describes the transport phenomena of the IAC with the source term for adiabatic
interactions and a phase change. Wu et al. (1998) and Hibiki and Ishii (2002)
investigated the source terms for a bubble coalescence and breakup in the adiabatic air-
water bubbly flow. To extend the applicability of the interfacial area transport equation
to a bubbly-slug transition flow, Fu and Ishii (2002) and Sun et al. (2004) developed a
two-group interfacial area transport equation for a vertical air-water flow. From these
studies, the interfacial area transport equation has been utilized to dynamically estimate
the multi-dimensional distribution of IAC and enhance the capability to predict the
behavior of the adiabatic air-water flow, rather than the static approach for modeling the
IAC.
In the view-point of the boiling source terms, Kocamustafaogullari and Ishii (1983)
suggested a basic formulation of the source term in a bubble number density transport
equation, which is composed of the bubble departure diameter, the departure frequency,
and the nucleate site density. Recently, Yao and Morel (2004) selected some models in
the literature for those three parameters and applied them to the analysis of a boiling
flow with a one-group interfacial area transport equation. On the other hand, Yeoh and
Tu (2005) pointed out a complex mechanism for the bubble on the heated wall, which
includes a sliding on the wall after departure and a lift-off toward the bulk fluid. Situ et
al. (2005) set up a force balance for a bubble, so that a bubble lift-off diameter model
was developed without considering the bubble departure mechanism. Hence, in order to
perform a more realistic two-phase flow analysis for the subcooled boiling, it is required
to develop a mechanistic model for the bubble lift-off diameter and to apply the model
into the source term of the interfacial area transport equation.
3
1.3. Objectives and scope of this study
For the investigation of the subcooled boiling flow with a dynamic modeling of the
interfacial structure, this study aims for the development of an interfacial area transport
equation with a mechanistic model for the bubble lift-off mechanism. To implement the
model, a computational fluid dynamics (CFD) code for a two-phase flow analysis is
developed, which adopts the two-fluid model and various constitutive models
Consequently, the experimental databases are utilized to validate the interfacial area
transport equation developed in this study.
In this paper, Chapter 2 represents the procedure of the CFD code development. Chapter
3 deals with the mechanistic modeling for source terms of the interfacial area transport
equation in the code. The numerical analysis results are discussed in Chapter 4
including the evaluation of the developed model.
4
2. Code Structure
2.1. Governing equations
To implement the interfacial area transport equation, the computational fluid dynamics
(CFD) code is developed in this study, which is named as EAGLE (Elaborated Analysis
of Gas-Liquid Evolution). The code aims for the multi-dimensional analysis of
subcooled boiling two-phase flow, with the dynamic approach of the interfacial area
transport equation.
EAGLE code uses the two-fluid model, which is beneficial to treat the behavior of each
phase separately and to consider a phase interaction term properly. As derived in the
two-fluid model by Ishii and Mishima (1984), the mass balance equation for a phase is
given as,
( ) ( )k kk k k kt
α ρα ρ
∂+ ∇ ⋅ = Γ
∂u (2.1)
where kΓ is the rate of a phase change for the k phase.
The momentum equations are given as follows.
( ) ( ) ( ) ( )k k k Tkk k k k k k k k k
ki k ik k ki k
uu p
tu F p
α ρα ρ α α τ τ α ρ
α τ α
∂ ⎡ ⎤+ ∇ ⋅ = −∇ + ∇ ⋅ + +⎢ ⎥⎣ ⎦∂+ Γ + − ∇ ⋅ + ∇
u g (2.2)
where kτ and Tkτ are the molecular stress tensor and the turbulent stress tensor,
respectively. ikF denotes the term of an interfacial momentum transfer including the
interfacial drag force, the wall lubrication force, the lift force, the turbulent dispersion
force and the virtual mass force (Ishii and Hibiki, 2006).
Energy equations are expressed with a form of the enthalpy ( )kH transport of each
phase, which is given as,
5
( ) ( )
( ) "
k k kk k k k
T kk k k ki k ki i kk
HH
tD p H q aDt
α ρα ρ
α α
∂+ ∇ ⋅
∂⎡ ⎤= −∇ ⋅ + + + Γ + + Φ⎢ ⎥⎣ ⎦
u
q q (2.3)
where kq is a diffusive flux by a conduction and the superscript ‘T’ means the
enhanced flux by a turbulence. "kiq is the interfacial heat flux between two phases,
defined as ( )i s kh T T− , and ia is the interfacial area concentration. kΦ is an external
source term for a phase. From the energy equation for a boiling flow, the phase changes
due to a wall boiling ( wΓ ) and a bulk condensation ( igΓ ) can be estimated as follows,
respectively,
ew
sg f
qH H
Γ =−
(2.4)
( ) ( )i i s f i i s gig
g sf
h a T T h a T TH H
− + −Γ = −
− (2.5)
where eq is the amount of evaporative heat transfer from the heated and it is estimated
from the heat partitioning model. From Eqs. (2.4) and (2.5), the rate of a phase change
can be found as follows.
g f w igΓ = −Γ = Γ + Γ (2.6)
Boussinesq approximation was used in the momentum equation, so that the
gravitational acceleration was modified as given in Eq. (2.7). This approximation makes
it possible to utilize a constant density of each phase.
( )0
1 1 1g gp p Tβρ ρ
− ∇ + ≅ − ∇ + + ∆ (2.7)
Liquid turbulence is estimated by the standard k-ε model (Ferziger and Peric, 2002),
which is a kind of Reynolds-Averaged Navier-Stokes (RANS) equations. In a two phase
flow with a phase change, transport equations for the turbulent kinetic energy, k, and the
dissipation, ε, are formulated as follows.
( ) ( )f f tf f f f f f f
k
kk k P
tα ρ µα ρ α µ α α ρ ε
σ∂ ⎡ ⎤⎛ ⎞
+ ∇ ⋅ = ∇ ⋅ + ∇ + −⎢ ⎥⎜ ⎟∂ ⎝ ⎠⎣ ⎦u (2.8)
6
( ) ( ) ( )1 2f f ft
f f f f fC P Ct k ε ε
ε
α ρ ε α εµα ρ ε α µ ε ρ εσ
∂ ⎡ ⎤⎛ ⎞+ ∇ ⋅ = ∇ ⋅ + ∇ + −⎢ ⎥⎜ ⎟∂ ⎝ ⎠⎣ ⎦
u (2.9)
where P is the production rate of a turbulent kinetic energy. From the turbulent kinetic
energy and dissipation, the turbulent diffusive flux of momentum or energy can be
calculated with a turbulent viscosity. The coefficients used in this study are 1.0kσ = ,
1.4εσ = , 1 1.44Cε = , and 2 1.92Cε =
2.2. Numerical scheme : Extended SMAC algorithm
A semi-implicit method is usually used for transient problems of the fluid dynamics,
while an implicit method struggles with a convergence of the solution. Since two-phase
flow phenomena have transient characteristics in most of cases, the semi-implicit
method can be applied effectively. Among the various semi-implicit methods, the
Simplified Marker And Cell (SMAC) algorithm (Amsden et al., 1971), which was
originally developed for a single-phase flow, is known to be advantageous for avoiding
repeated iterations. For an application of the algorithm to the subcooled boiling flow in
this study, the original SMAC algorithm has been extended to a two phase flow as
follows.
For the stability of a numerical calculation, the momentum transport equations for two
phases were coupled by an interfacial drag and virtual mass term as defined in Eqs.
(2.10) and (2.11), so that they could be solved simultaneously with an implicit
procedure.
( )18
drag dragg f f i D g f g fF F a Cρ= − = − − −u u u u (2.10)
vm vmg fF F= − =
* *g g f f
f vm g vm
D DDt Dt
φ φα ρ α ρ− (2.11)
where kφ means the superficial velocity k kuα , vmρ is the characteristic density for
the virtual mass force as derived by Drew et al. (1979), and Dk/Dt is the convective
derivative. With the assumption of an incompressible flow, Eq. (2.2) can be rewritten
for each phase as follows.
7
( )* * *
* *18
g g g g f fn ng g g f i D R g f f vm g vm
D D DF a C
Dt Dt Dtφ φ
α ρ ρ α ρ α ρ⎛ ⎞
= − − − −⎜ ⎟⎜ ⎟⎝ ⎠
uu u u (2.12)
( )* * *
* *18
g g g g f fn nf f f f i D R g f f vm g vm
D D DF a C
Dt Dt Dtφ φ
α ρ ρ α ρ α ρ⎛ ⎞
= + − + −⎜ ⎟⎜ ⎟⎝ ⎠
uu u u (2.13)
( )T NDkk k k k k k kF p Fα α τ τ α ρ⎡ ⎤≡ − ∇ + ∇ ⋅ + + +⎢ ⎥⎣ ⎦
g (2.14)
where NDkF is a term for non-drag interfacial momentum transfer of each phase, which
will be discussed in Section 2.3. In these equations, the superscript ‘n’ means the value
at a previous time. And the superscript ‘*’ stands for the value estimated at an advanced
time. However, the velocity at the advanced time step from Eqs. (2.12) and (2.13), *gu
or *fu , does not satisfy the mass conservation required in Eq. (2.1).
In order to calculate an accurate velocity at the next time step, it should be corrected by
considering a mass conservation. In the SMAC algorithm, the velocity corrections for
two phases are found by considering a transport equation with respect to the velocity at
an advanced time step as follows.
( ) ( )1 1
1 1 1int
n ng f n n n n
g g f vm f g vm g gc g fp F Ct t
α ρ α ρ α α ρ α+ +
+ + +∂ ∂+ − = − ∇ + − −
∂ ∂
u uu u (2.15)
( ) ( )1 1
1 1 1int
n nf g n n n n
f f g vm g f vm f fc g fp F Ct t
α ρ α ρ α α ρ α+ +
+ + +∂ ∂+ − = − ∇ + + −
∂ ∂
u uu u (2.16)
( )gc g g g f vm g gF F α ρ α ρ≡ − + ⋅∇u u (2.17)
( )fc f f f g vm f fF F α ρ α ρ≡ − + ⋅∇u u (2.18)
where int18 f i D RC a Cρ= u . When subtracting the Eq. (2.12) and (2.13) from Eq. (2.15)
and (2.16) respectively, a velocity correction, ' 1 *nk k k
+≡ −u u u , can be formulated with
respect to a pressure correction, ' 1n np p p+= − , and time step, t∆ . ' '
int ''int
g f vm
f g m vm m
C tp
t C tρ ρ
ρ ρ ρ ρ ρ+ + ∆
= − ∇∆ + + ∆
u (2.19)
' 'int '
'int
f g vm
f g m vm m
C tp
t C tρ ρ
ρ ρ ρ ρ ρ+ + ∆
= − ∇∆ + + ∆
u (2.20)
8
where mρ is a mixture density defined as m f f g gρ α ρ α ρ= + and 'int int / f gC C α α= .
To compute the pressure correction, Eq. (2.19) and (2.20) are multiplied by a phase
fraction and taken with a divergence. Then, adding the two equations yields a Poisson
equation as,
( ) ( )1 1 * *
, ,
'g fn ng f g f
g eff f eff
t pα α
φ φ φ φρ ρ
+ +⎛ ⎞⎛ ⎞
∇ ⋅ + − ∇ ⋅ + = −∆ ⋅∇ ⋅ + ∇⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠ (2.21)
'int
, 'int
f g m vm mg eff
f vm
C tC t
ρ ρ ρ ρ ρρ
ρ ρ+ + ∆
=+ + ∆
(2.22)
'int
, 'int
f g m vm mf eff
g vm
C tC t
ρ ρ ρ ρ ρρ
ρ ρ+ + ∆
=+ + ∆
(2.23)
The term ( )1 1n ng fφ φ+ +∇ ⋅ + in Eq. (2.21) can be derived from the mass conservation of
each phase as,
( )1 1 f g f f g gn nf g
f g f g
D DDt Dt
α ρ α ρφ φ
ρ ρ ρ ρ+ +
⎛ ⎞ ⎛ ⎞Γ Γ∇⋅ + = + − +⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟
⎝ ⎠ ⎝ ⎠ (2.24)
For an incompressible flow without a phase change, the right-hand side of Eq. (2.24) is
zero, and is identical to that of the SMAC algorithm for a single phase flow. However,
when the phase change such as a boiling or a condensation exists in the flow, this term
should be additionally considered as a source in the Poisson equation. With the
assumption of an incompressible flow for each phase, Eq. (2.24) is reduced to Eq. (2.25).
( )1 1 f gn nf g
f g
φ φρ ρ
+ +⎛ ⎞Γ Γ
∇⋅ + = +⎜ ⎟⎜ ⎟⎝ ⎠
(2.25)
Therefore, by substituting Eq. (2.25) and solving the matrix given in Eq. (2.21), the
pressure and velocity corrections can be estimated for the next time step. To solve the
matrix, ICCG solver has been adopted in EAGLE code.
For a numerical discretization of the governing equations, the finite volume method was
adopted in this study, where a complex geometry can be easily analyzed by using an
unstructured grid. In calculating a convection term, the upwind scheme has been applied
for a numerical stability of the solution.
9
2.3. Constitutive relations
(1) Subcooled boiling and condensation model
In the subcooled boiling flow, the amount of vapor generation can be computed by a
wall heat flux partitioning model. The mechanisms of a heat transfer from the wall
consist of the surface quenching (qq), evaporative heat transfer (qe), and single phase
convection (qc), which are modeled in CFX-4 code as follows (AEA, 1997).
( )22
q w f f pf f w fq t k C f A T Tρπ
⎛ ⎞= −⎜ ⎟⎝ ⎠
(2.26)
" 3
6e d g fgq N f D hπ ρ⎛ ⎞= ⎜ ⎟⎝ ⎠
(2.27)
( )1c c f w fq h A T T= − (2.28)
where 1 fA and 2 fA are the fraction of the effective area subjected to quenching and
evaporation, defined in the following equations, respectively.
( )1 2max 1 ,0f fA A= − ,
2"
2 4d
fDA N Kπ
= ⋅ (2.29)
K is a bubble influence factor, which means the ratio of the area influenced by a
nucleate boiling heat transfer to the projected area at a bubble departure. Various
models for the active nucleate site density (N”), the bubble departure diameter (Dd) and
the bubble departure frequency (f) are summarized in Tables 2.1 to 2.3. In EAGLE code,
the mechanistic models in the heat partition model of CFX-4 was adopted as
summarized in Table 2.4, where St is a Stanton number and Wu is a friction velocity.
Eq. (2.27) estimates the evaporative heat flux based only on the bubble departure
mechanism. To mechanistically consider the motion of bubble during the evaporation, a
modeling of the bubble lift-off phenomena is required. In this study, instead of Eq.
(2.27), the term of the evaporative heat flux is replaced by the lift-off diameter model
and the lift-off frequency reduction factor model, which will be discussed in Chapter 3.
Interfacial heat transfer coefficient ( ih ) for a condensation is given as a function of the
Nusselt number. As a default correlation, the Ranz’s model (1952) was used, which is
10
based on the boundary layer theory around a sphere.
0.5 0.32 0.6 Re Pri smb
f
h DNuk
= = + (2.30)
where Reb is a bubble Reynolds number, defined as Re /b f R b mdρ µ= u .
(2) Closure relation for the interfacial momentum transfer
Interfacial drag coefficient in Eq. (2.10), DC , is taken from Ishii’s model (2006) which
is based on Ishii and Zuber’s model (1979) and modified to take into account the effect
of a multi-particle system. It is dependent on the flow regime as follows.
- Viscous regime : ( )0.751
24 1 0.1ReReD b
b
C = + (2.31)
- Distorted regime : ( )
( )
21.3
2 1.5
1 17.67 12 Re3 18.67 1
gD
g
C Nµ
α
α∞
⎡ ⎤+ −⎢ ⎥=⎢ ⎥−⎣ ⎦
(2.32)
- Spherical cap regime : ( )2
38 13D gC α= − (2.33)
- 1 2D DC C> : 1D DC C=
1 2D DC C< : ( )2 3min ,D D DC C C=
11
m
f g
µµ α
=−
, 1/ 2f
f
N
g
µ
µ
σρ σρ
=⎛ ⎞⎜ ⎟∆⎝ ⎠
(2.34)
As one of the non-drag forces, wall lubrication force is acting on a bubble motion in a
lateral direction near the wall, so it dominantly governs the radial distribution of a void
fraction. The formulation of the wall lubrication force is shown in Eq. (2.35), where
coefficients of Krepper’s model (2007) was used in this study. 2
lub lub1 2max , 0g f R d
g f w w wd bw
rF F C Cr y
α ρ ⎡ ⎤⎛ ⎞= − = +⎢ ⎥⎜ ⎟
⎢ ⎥⎝ ⎠⎣ ⎦
un
1 0.0064wC = − , 2 0.016wC = (2.35)
where ybw is the distance from a wall to the center of a control volume. Lift force on a
11
bubble is induced by a rotational motion of the liquid phase as follows (Ishii and Hibiki,
2006).
( ) ( )lift liftg f f L g f fF F Cαρ= − = − − × ∇×u u u (2.36)
where the coefficient LC is set to 0.01 for a viscous flow. The turbulent dispersion
force is defined as Eq. (2.37) (Ishii and Hibiki, 2006), with the coefficient TDC equal to
0.1. TD TD
g f TD fF F C kρ α= − = − ∇ (2.37)
(3) Turbulence model in the two-phase flow
In the standard k-ε model as mentioned in Eqs. (2.8) and (2.9), turbulent viscosity can
be estimated from the turbulent kinetic energy (k) and the energy dissipation (ε), with
the coefficient of 0.09Cµ = . For the case of bubbly two-phase flow, the effect of
bubble’s motion on the turbulence has been modeled as a term of the Sauter-mean
bubble diameter and relative velocity and considered additionally (Lahey, 2005). Then,
total turbulent viscosity of the liquid phase is defined as, 2
0.6ft sm g r
f
kC D uµµ α
ε= + (2.38)
In case of boiling flow, boiling bubbles at the heated surface was known to affect the
turbulence structure near the wall by altering the velocity field in a laminar sublayer
considerably. Kataoka and Serizawa (1997) modeled an enhanced turbulence by boiling
bubbles and modified the turbulence mixing length in the boiling region.
'61boil e
TP TPg fg p l
ql lh uρ α
⎛ ⎞= +⎜ ⎟⎜ ⎟
⎝ ⎠ (2.39)
pα in Eq. (2.39) is the void fraction at / 2dy D= and 'lu means the fluctuation of the
velocity, that is, k is substituted in this study. In the definition of turbulence mixing
length model, turbulent viscosity is proportional to the turbulence mixing length, that is, '
t l TP ll uµ ρ= . Therefore, the increased turbulence near the heated surface and wall shear
stress were modeled with the same multiplication factor of Eq. (2.39).
12
'61boil e
t tg fg p l
qh u
µ µρ α
⎛ ⎞= +⎜ ⎟⎜ ⎟
⎝ ⎠, '
61boil ew w
g fg p l
qh u
τ τρ α
⎛ ⎞= +⎜ ⎟⎜ ⎟
⎝ ⎠ (2.40)
In above equations, the term for evaporative heat flux from wall heat partitioning model
was utilized.
Moreover, Kataoka and Serizawa (1997) derived a source term for the turbulent kinetic
energy induced by the boiling bubbles, as shown in Eq. (2.41)
( )'2 3" 1
3/ 2 2 6KE B bwB bw
N u dt d
πΦ = ⋅ ⋅ (2.41)
where 'Bu is the fluctuating velocity and ' 2 /B B Bu d t≈ was assumed in Kataoka’s
research. From the definition of the evaporative heat flux and the bubble departure
frequency ( 1/ Bf t= ), a reduced form of Eq. (2.41) was utilized in this study as follows.
89
evKE
f g fg
g qh
ρρ ρ∆
Φ = ⋅ ⋅ (2.42)
To mechanistically model the turbulence structure, the source term defined in Eq. (2.42)
was additionally considered in the right-hand side of k-transport equation, Eq. (2.8).
2.4 Numerical formulation
(1) Phase change at interface
The left-hand side of momentum equation, Eq. (2.2), can be rearranged with considering
the continuity equation.
( )k kk k k k k k k
kk k k k k k
uLHS u ut t
uu ut
αρ α α
ρ α
⎡ ⎤∂ ∂⎛ ⎞ ⎛ ⎞= + ∇ ⋅ + + ⋅∇⎜ ⎟ ⎜ ⎟⎢ ⎥∂ ∂⎝ ⎠ ⎝ ⎠⎣ ⎦∂⎛ ⎞= Γ + + ⋅∇⎜ ⎟∂⎝ ⎠
u u
u
(2.43)
When the term of k ku Γ is integrated over a control volume of a cell according to the
finite volume method, the volume integral is transformed as,
, ,k k ki j k jCVj
u dV uΓ = Γ∑∫ (2.44)
where j is an index for each bubble in the cell and the subscript i indicates the interface.
13
In most of bubbly flows, an increasing or decreasing rate of the bubble size is negligible
when compared to the bubble velocity. Consequently, to get the value of an
instantaneous interfacial velocity, the bubble can be assumed to be a rigid body within
the continuous liquid. Therefore, as shown in Figure 2.1, the velocity distribution
around a bubble represents that the interfacial velocity for both phases is equivalent to
that of the bubble. That is,
gi fi gu u u= ≈ (2.45)
From Eqs. (2.44) and (2.45), k ku Γ term in the left-hand side of momentum equation is
eliminated with the ki ku Γ term in the right-hand side as revealed in Eq. (2.2). Then the
left-hand side is reduced to Eq. (2.46) and the momentum equation has a form of
describing a transport of the velocity, ku , not the superficial velocity, k kuα .
kk k k k
uLHS ut
ρ α ∂⎛ ⎞= + ⋅∇⎜ ⎟∂⎝ ⎠u (2.46)
Similarly, the energy equation given in Eq. (2.3) can be reduced to the transport
equation of an enthalpy, kH , as shown in Eq. (2.47), where the ki kH Γ terms in left-
hand side and right-hand side of Eq. (2.3) are eliminated.
kk k k k
HLHS Ht
ρ α ∂⎛ ⎞= + ⋅∇⎜ ⎟∂⎝ ⎠u (2.47)
(2) Implicit formulation of momentum and energy
In the momentum equation, the wall lubrication force and the lift force are estimated
implicitly, since it plays an important role on a radial distribution of the void in a two-
phase flow. Lift force in Eq. (2.36) can be derived in an axisymmetric channel as
follows.
( )fz frlift liftg f f L Rz Rr
u uF F C u u
r zαρ
∂ ∂⎛ ⎞= − = − − ⋅ −⎜ ⎟∂ ∂⎝ ⎠
r z (2.48)
Usually, in a convective two-phase flow in a vertical channel, the relative velocity in a
radial direction ( Rru ) is negligible when compared to that in an axial direction ( Rzu ). So
the lift force in Eq. (2.48) can be assumed to have a component only in a radial direction.
14
Then the momentum equation in radial (x) and axial (y) direction of each phase in Eqs.
(2.12) and (2.13) can be replaced as follows.
( )( ) ( ) ( )
* *
* * * * * *
n ngx gx fx fx
g g f vm f g vm
n n n n ngx g f gx fx Lift gy fy Lub gy fy gy fy
u u u ut t
F C u u C u u C u u u u
α ρ α ρ α α ρ− −
+ −∆ ∆
= − − − − − + − −u u (2.49a)
( )( ) ( ) ( )
* *
* * * * * *
n ngx gx fx fx
f g vm f f g vm
n n n n nfx g f gx fx Lift gy fy Lub gy fy gy fy
u u u ut t
F C u u C u u C u u u u
α α ρ α ρ α ρ− −
− + +∆ ∆
= + − − + − − − −u u (2.49b)
( ) ( )* *
* *n n
gy gy fy fy n n ng g f vm f g vm gy g f gy fy
u u u uF C u u
t tα ρ α ρ α α ρ
− −+ − = − − −
∆ ∆u u (2.50a)
( ) ( )* *
* *n n
gy gy fy fy n n ng g f vm f g vm gy g f gy fy
u u u uF C u u
t tα ρ α ρ α α ρ
− −+ − = − − −
∆ ∆u u (2.50b)
where the coefficients for drag force, wall lubrication force, and lift force are defined
from Eqs. (2.10), (2.35), and (2.48), respectively.
18 f i DC a Cρ= ⋅ (2.51a)
n nfz fr
Lift f L
u uC C
r zαρ
⎛ ⎞∂ ∂= −⎜ ⎟⎜ ⎟∂ ∂⎝ ⎠
(2.51b)
1 2
ng f R d
Lub w w wd bw
u rC C C nr y
α ρ ⎡ ⎤⎛ ⎞= +⎢ ⎥⎜ ⎟
⎢ ⎥⎝ ⎠⎣ ⎦ (2.51c)
As indicated in Eqs. (2.49) and (2.50), a solution of the momentum equation in y-
direction is independent on the velocity in x-direction, so that it is not complex to solve
those equation when *kyu is solved at first and substituted in Eq. (2.49) to get the
solution of *kxu .
In case of the energy equation, the term of the interfacial heat transfer between two
phases is implicitly considered. Rearranging Eq. (2.3) with an implicit interfacial heat
transfer yields Eq. (2.52).
( )k kk k k i i sk k
D H SRC h a T TDt
α ρ = + − (2.52)
where kSRC is including other source terms on the right-hand side of Eq. (2.3) which
15
are calculated by the value at a previous time step. As a form of a difference equation,
Eq. (2.52) is converted with specific heat of the phase.
( )*
*n
k k i ik k k sk k
pk
H H h aSRC H Ht C
α ρ −= + −
∆ (2.53)
where *kH is the phase enthalpy including a convective term.
(3) Standard k-ε model
Similarly to Eqs. (2.46) and (2.47), disassembling the left-hand side of Eq. (2.8) yields a
form of a transport equation of the kinetic energy.
( ) ( )f ff f f f f f f
k kk k kt t
α ρα ρ α ρ
∂ ∂⎛ ⎞+ ∇ ⋅ = Γ + + ⋅∇⎜ ⎟∂ ∂⎝ ⎠u u (2.54)
Substituting Eq. (2.54) into Eq. (2.8), k-transport equation is formulated as follows.
1f tf f
f f f k
k k k k Pt
να ν εα ρ α σ
⎡ ⎤Γ ⎛ ⎞∂+ ⋅∇ = − + ∇ ⋅ + ∇ + −⎢ ⎥⎜ ⎟∂ ⎝ ⎠⎣ ⎦
u (2.55)
where production term is defined as ( ) :Tt KEP u u uν= ∇ + ∇ ∇ + Φ . The same approach
can be applied to the transport equation of the dissipation, Eq. (2.9).
( )1 21f t
f ff f f
C P Ct k ε ε
ε
ε ν εε ε α ν ε εα ρ α σ
⎡ ⎤Γ ⎛ ⎞∂+ ⋅∇ = − + ∇ ⋅ + ∇ + −⎢ ⎥⎜ ⎟∂ ⎝ ⎠⎣ ⎦
u (2.56)
To implicitly estimate the transport of the kinetic energy and dissipation, Eqs. (2.55)
and (2.56) are rearranged in a difference equation as follows. 1
1 1n n
f n nk k
f f
k k Conv k Diff Pt
εα ρ
++ +Γ−
+ = − + + −∆
(2.57)
11 11 2
nn nf n n
nf f
C P CConv Difft k
ε εε ε
εε ε ε εα ρ
++ +Γ −−
+ = − + +∆
(2.58)
where Conv and Diff are the convection and diffusion term, respectively. As revealed in
the equations, Eq. (2.58) does not include an implicit value of the turbulent kinetic
energy, 1nk + . Therefore two equations can be solved sequentially as shown in Eq.
(2.59).
16
( )11 21n
f n nn
f f
C P Ct t Conv Diff tk
ε εε ε
ε ε εα ρ
+⎛ ⎞Γ −
+ ∆ − ∆ = + − + ∆⎜ ⎟⎜ ⎟⎝ ⎠
(2.59a)
( )1 11 f n n nk k
f f
t k k Conv Diff P tεα ρ
+ +⎛ ⎞Γ
+ ∆ = + − + + − ∆⎜ ⎟⎜ ⎟⎝ ⎠
(2.59b)
This formulation is available for a cell not adjacent to the wall. For a cell adjacent to the
wall, a proper wall function should be applied to estimate the kinetic energy or the
dissipation. The dissipation for those cells is approximated as follows. (Ferziger and
Peric, 2002) 3/ 4 3/ 2
PP
C kn
µεκ
= (2.60)
where the subscript P means the cell next to the wall. 0.41κ = is utilized in EAGLE
code as the von Karman constant. Therefore, in order to satisfy Eqs. (2.57) and (2.60)
simultaneously for the cell adjacent to the wall, following procedure is adopted.
( )3/ 4 1/ 2
11n
f n nk k
f f
C kt t k k Conv Diff P t
nµ
α ρ κ+
⎛ ⎞Γ+ ∆ + ∆ = + − + + ∆⎜ ⎟⎜ ⎟
⎝ ⎠ (2.61a)
3/ 4 1 3/ 21
nn C k
nµε
κ
++ = (2.61b)
Moreover, to compute the wall shear stress according to the wall function (Ferziger and
Peric, 2002), Eq. (2.62) is applied at the surface adjacent to the wall.
( )1/ 4
lnt
w f P B
uC kn eµ κ
τ ρ κ+
= ⋅ (2.62)
where ( )1/ 4 /fn C k nµρ µ+ = and n is a normal distance from the wall. B is an
empirical constant related to the thickness of viscous sublayer and EAGLE code
adopted B=5.5. (Ferziger and Peric, 2002)
(4) Heat partition model
To compute each heat flux term in Eqs. (2.26), (2.27), (2.28), wall temperature should
be known. It can be solved by an equation of the heat flux conservation,
q e cq q q q= + + , where q is the total heat flux at the heated wall. In those equations, the
17
active nucleate site density includes a term of ( )1.805w satT T− as shown in Table 2.1. So
Newton-Rhapson method is adopted to get a solution of the wall temperature and heat
partition. Substituting Eqs. (2.26), (2.27), and (2.28) into the heat flux conservation
equation, following relation is derived.
( ) ( ) ( ) ( )( )( )
1.805 1.805
0w q A w sat w l e w sat
c w w l
F T C C T T T T C T T
C g T T T q
= − − + −
+ − − = (2.63a)
2q w f f plC t k C fρ
π= ,
21.805185
4d
ADC Kπ
= ⋅ (2.63b)
3 1.8051856e d g fgC f D hπ ρ⎛ ⎞= ⋅⎜ ⎟
⎝ ⎠, c l pl lC St C uρ= ⋅ (2.63c)
2"1 0
4dDN Kπ
− > : ( )2 1.805"( ) 1 1
4d
w A w fDg T N K C T Tπ
= − = − − (2.63d)
2"1 0
4dDN Kπ
− ≤ : ( ) 0wg T = (2.63e)
To solve the equation ( ) 0wF T = , following iteration is utilized.
( )( )
1'
kwk k
w w kw
F TT T
F T+ = − (2.64a)
( ) ( ) ( ) ( )
( ) ( )( ) ( )
0.805 1.805'
0.805
1.805
1.805 '
w q A w sat w l w sat
e w sat c w w l w
F T C C T T T T T T
C T T C g T T T g T
⎡ ⎤= − − + −⎣ ⎦
⎡ ⎤+ − + − +⎣ ⎦
(2.64b)
2"1 0
4dDN Kπ
− > : ( )0.805'( ) 1.805w A w lg T C T T= − − (2.64c)
2"1 0
4dDN Kπ
− ≤ : '( ) 0wg T = (2.64d)
where k is an iterative index. The iteration continues until the difference between 1kwT +
and kwT becomes negligible. After the end of the iterative calculation, each heat flux of
the surface quenching, evaporation, and single-phase convection is estimated from the
solution of wT . On the other hand, when the bubble lift-off model is adopted as
discussed in Section 2.3, eC is replaced by Eq. (2.65).
18
3 1.8051856e lo e g fgC f D R hπ ρ⎛ ⎞= ⋅⎜ ⎟
⎝ ⎠ (2.65)
In above equation, loD is the bubble lift-off diameter and eR is the reduction factor of
the bubble lift-off frequency. The detailed modeling for each parameter will be
described in Chapter 3.
(5) Discretization of governing equation
According to the finite volume method, each term in the governing equations should be
integrated over a cell. For the time integration, the first-order Euler method is adopted. 1
i
n ni i
iVdV V
t tϕ ϕϕ + −∂
=∂ ∆∫ (2.66)
where iV is the volume of cell i. ϕ is a conservative parameter in each transport
equation, which becomes a phase fraction in the continuity equation, a phase velocity in
the momentum equation, and an enthalpy in the energy equation. Transport equations of
k and ε also adopt the same approach for the time integration.
For a convection term in the continuity equation, Gauss’s theorem is applied as given in
Eq. (2.67)
( ) ( )i i
k k k k kj k jV Sj
dV dα α α∇ ⋅ = ⋅ = ⋅∑∫ ∫u u S u S (2.67)
where jS is a normal vector at the j-th surface of the cell. In cases of the momentum
and energy equation, convection terms can be derived as follows.
( ) ( )
( ) ( )i i i
k k k k k kV V V
kj k k kj jj j
dV dV dVϕ ϕ ϕ
ϕ ϕ
⋅∇ = ∇ ⋅ − ∇ ⋅
= ⋅ − ⋅
∫ ∫ ∫∑ ∑
u u u
u S u S (2.68)
In Eqs. (2.67) and (2.68), volumetric flux at the surface, ( )k j⋅u S , is estimated by an
interpolation of neighboring cells. That is,
( ), ,1kj ij k nj ij k iu fac u fac u= + − where ij iij
nj i
fac−
=−
r r
r r (2.69)
In Eq. (2.69), subscript ij denotes the adjacent surface and nj means the center of the
19
neighboring cell. On the other hand, the property at the surface ( kjα in Eq. (2.67) and
kjϕ in Eq. (2.68)) is estimated according to the upwind differencing scheme for a
stability of the calculation. It depends on a direction of the volumetric flux at the surface
as indicated in Eq. (2.70).
( )( )
,
,
0
0kj k i k j
k nj k j
α α
α
= ⋅ ≥
= ⋅ <
u S
u S (2.70a)
( )( )
,
,
0
0kj k i k j
k nj k j
ϕ ϕ
ϕ
= ⋅ ≥
= ⋅ <
u S
u S (2.70b)
Conv and Diff terms in the k or ε transport equations can be integrated according to
the same procedure as given in Eqs. (2.68) to (2.70).
For a momentum diffusion term in Eq. (2.2), a stress tensor is integrated over a cell as
follows.
( )( ) ( )( ) ( )
i i
i
Tk k k eff kV V
k eff k k eff k jjSj
dV dV
d
α τ τ α µ
α µ α µ
∇ ⋅ + = ∇ ⋅ ∇
= ∇ = ∇ ⋅
∫ ∫∑∫
u
u S u S (2.71)
where the effective viscosity is defined as eff tµ µ µ= + . Similarly, the diffusion term in
the energy equation is given in follows.
( )( ) ( )( ) ( )
i i
i
Tk k k eff kV V
k eff k k eff k jjSj
dV k T dV
k T d k T
α α
α α
∇ ⋅ + = ∇ ⋅ ∇
= ∇ = ∇ ⋅
∫ ∫∑∫
q q
S S (2.72)
where the effective thermal conductivity is eff tk k k= + . tk can be estimated from tµ ,
according to the turbulent Prandtl number as defined in Eq. (2.73). Pr /t t p tC kµ=
(2.73)
In this study, Pr 0.9t = is utilized.
Other source term such as a gravity term can be discretized by integrating over a cell as
follows.
ii iV
dV VΦ = Φ∫ (2.73)
Major subroutines in above procedures in EAGLE code are listed in Table 2.5 and the
20
flow chart is depicted in Figure 2.2.
21
Table 2.1 Models for active nucleation site density "( )N in literature
Author Model
Kocamustafaogullari
and Ishii (1983) ( ) ( )
4.4
*4.4*** 2−
−
⎟⎟⎠
⎞⎜⎜⎝
⎛
∆≅=
satfgg
satcn Th
TfRfNρ
σρρ
Yang and Kim (1988) ( ) ( )
( )sat
cnn
TKC
Rdss
NN
∆−=
−⋅⎟⎟⎠
⎞⎜⎜⎝
⎛ −−= ∫
/exp
exp2
exp21
0 2
2
λβββπ
θ
Wang and Dhir
(1993) ( ) 0.629 cos11081.7 −− −×= cnp RN θ
Benjamin and
Balakrishnan (1997) 34.063.1 1Pr8.218 wnp TN ∆Θ⎟⎟
⎠
⎞⎜⎜⎝
⎛= −
γ
Basu, Warrier and
Dhir (2002)
( ) 0.24 cos11034.0 wnc TN ∆−×= θ KTT wONB 15<∆<∆
( ) 0.24 cos11034.0 wnc TN ∆−×= θ wTK ∆≤15
Hibiki and Ishii
(2003) ( )
2"
2'1 exp exp 1
8nc
N N fR
θ λρµ
+⎛ ⎞⎛ ⎞⎛ ⎞ ⎛ ⎞= − − −⎜ ⎟⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟⎜ ⎟ ⎝ ⎠⎝ ⎠ ⎝ ⎠⎝ ⎠
Lemmert and Chwala
(1977) ( ) 1.805" 210 w satN T T= −⎡ ⎤⎣ ⎦
22
Table 2.2 Models for bubble departure diameter ( )dD in literature
Author Model
Levenspiel (1959) ( )( )
( ) ( ) 2/12/13/1
3/1709.05
max 21042.2
uufgg
ulssp
bahCukYTThqp
Dφπρ
−−×=
−
Farajisarir (1993) 62.1
65.192
max 1002.10−
−⎟⎟⎠
⎞⎜⎜⎝
⎛−−
×=sw
lww
l TTTTJa
aDρ
σ
Kocamustafaogullari
and Ishii (1983)
0.952.64 10d
gD
gρ σθ
ρ ρ− ⎛ ⎞∆
= × ⎜ ⎟⎜ ⎟ ∆⎝ ⎠
Unal(1976) 5 0.7092.42 10
dp aD
b
−×=
Φ
Fritz (1935) 0.208dDgσθ
ρ=
∆
Tolubinskiy and
Kostanchuk (1970) ( )min 0.0006exp / 45 ,0.0014d subD T⎡ ⎤= −∆⎣ ⎦
Cole (1967) 24 10 f pfd
g fg
C TD
g hρσ
ρ ρ− ∆
= ×∆
Cole and Rosenhow
(1968)
5/ 441.5 10 f pf
dg fg
C TD
g hρσ
ρ ρ− ⎛ ⎞∆
= × ⎜ ⎟⎜ ⎟∆ ⎝ ⎠
23
Table 2.3 Models for bubble departure frequency ( )f in literature
Author Model
Cole (1960) ( )43
f g
d l
gf
Dρ ρ
ρ
−=
Ivey (1967) 0.9d
gfD
=
Stephan (1992) 0.5
2
1 412 d d
gfD D g
σπ ρ
⎛ ⎞= +⎜ ⎟
⎝ ⎠
Zuber (1963) 0.25
2
1.18
d f
gfD
σ ρρ
⎛ ⎞∆= ⎜ ⎟⎜ ⎟
⎝ ⎠
24
Table 2.4 Wall heat flux partitioning model in CFX-4
Parameter Model
Bubble influence factor
K 4
Active nucleate site density
"N ( ) 1.805
185 w satT T⎡ ⎤−⎣ ⎦
Bubble departure frequency
f ( )43
f g
d f
gDρ ρ
ρ
−
Bubble waiting time
wt 0.8
wtf
=
Heat transfer coefficient
ch f pf fSt C uρ⋅
25
Table 2.5 Major subroutines in EAGLE code
Subroutine Function
Calc_iat Calculate IAC according to interfacial area transport
equation model
Calc_interphase Calculate interfacial heat transfer coefficient
Calc_turb Solve the standard k-ε equation and calculate
turbulent viscosity
Calc_h Solve the energy equation
Calc_gamma Calculate the amount of phase change according to
heat partition model and condensation model
Calc_void Solve the continuity equation
Calc_vel Solve the momentum equation and calculate the
pseudo-velocity
Calc_press Calculate the pressure matrix according to the
extended SMAC algorithm
26
Figure 2.1 Velocity around a bubble
27
t t t= + ∆
Figure 2.2 Flow chart of EAGLE code
28
3. Mechanistic Modeling of Interfacial Area
Transport Equation
3.1. One-group interfacial area transport equation
For a multi-dimensional calculation of the IAC (interfacial area concentration), Yao and
Morel (2004) derived an interfacial area transport equation available for a boiling flow
as follows.
( ) 23
gi ii g ig co bk ph
g
da aaVt dt
ρα φ φ φ
αρ⎡ ⎤∂
+ ∇ ⋅ = Γ − + + +⎢ ⎥∂ ⎣ ⎦ (3.1)
where coφ , bkφ , and phφ mean the variance of IAC by a coalescence, breakup and
nucleation, respectively. The first term on the right-hand side of Eq. (3.1) is the term of
a bubble size variance due to a condensation heat transfer or a pressure drop.
Noting that the subcooled boiling flow in this study is a bubbly flow, the coalescence by
a random collision (RC) and the breakup by a turbulent impact (TI) are considered for
the second and the third terms on the right-hand side of Eq. (3.1), respectively. Yao and
Morel (2004) modeled those terms as follows.
( )
2 2
2 1/3 2
1 311/32
1 1 1 13 2 3 2
1 1 exp3 /
c cRC
i c i cf ci
c ci sm cc c
n na T a T T
WeK Ka D Weg K We We
η ηα αφψ ψ
α ε αψ α α
⎛ ⎞ ⎛ ⎞= − ⋅ = − ⋅⎜ ⎟ ⎜ ⎟ +⎝ ⎠ ⎝ ⎠
⎛ ⎞⎛ ⎞= − ⋅ ⋅ ⋅ −⎜ ⎟⎜ ⎟ ⎜ ⎟+⎝ ⎠ ⎝ ⎠
(3.2)
( )( )
2 2
2 1/3
1 11/32
1 13 3
11 1 exp3 1 1 /
b bTI
i b i bf bi
bi sm cc c
n na T a T T
WeKa D WeK We We
η ηα αφψ ψ
ε α ααψ α
⎛ ⎞ ⎛ ⎞= ⋅ = ⋅⎜ ⎟ ⎜ ⎟ +⎝ ⎠ ⎝ ⎠
−⎛ ⎞ ⎛ ⎞= ⋅ ⋅ ⋅ −⎜ ⎟ ⎜ ⎟
+ −⎝ ⎠ ⎝ ⎠
(3.3)
where ψ is a bubble shape factor, 1/36π for a spherical bubble, and η and n are the
29
interaction efficiency of neighboring bubbles and the bubble number density,
respectively. We is a Weber number and ε is the dissipation, which can be obtained
from the k-ε model. ( )g α is a modification factor defined as ( )1/3max1 /α α− , and the
coefficients in the equations are 1cK =2.86, 2cK =1.922, 3cK =1.017, cWe =1.24,
maxα =0.52, 1bK =1.6, 2bK =0.42. Rather than other models suggested by Wu et al.
(1998) and Hibiki et al. (2002), Yao’s model considers the free-traveling time ( cfT for a
coalescence and bfT for a breakup) and the interaction time ( ciT for a coalescence and
biT for a breakup) of bubbles separately. This approach has enhanced the capability for
predicting IAC by mechanistically modeling a coalescence or breakup process. Recently,
a commercial CFD-code analysis of Cheung et al. (2007) represented that the model of
Yao showed a better agreement for an air/water adiabatic flow.
The last term on the right-hand side in Eq. (3.1) denotes an increase of the IAC by a
bubble nucleation at the heated wall, that is, the boiling source term in the interfacial
area transport equation. Similarly to the evaporative heat flux as defined in Eq. (2.27), it
is composed of a product of the active nucleate site density (N”), the bubble departure
diameter (Dd) and the bubble departure frequency (f), as presented in Eq. (3.4). "
2
VolH
ph dN f ADφ π ⋅ ⋅
= ⋅ (3.4)
where HA is the area of heated surface and Vol is the volume of a unit cell.
In the previous studies, the boiling source term of an interfacial area transport equation
has been modeled according to a bubble departure mechanism as shown in Eq. (3.4).
That is, it estimates the evaporation induced by the departure of bubble at the heated
wall. In that modeling, a bubble generated at a nucleate site is assumed to depart from
the wall with the bubble departure frequency. However, it is observed that the actual
behavior of the bubbles at the wall is more complex than the departure mechanism. The
bubble departing from a nucleate site slides along the wall without directly moving to
the bulk liquid as depicted in Figure 3.1. During the sliding, the bubble size can be
varied in the superheated liquid layer on the wall, so that the bubble diameter at the lift-
off is different with the bubble departure diameter. Moreover, when a sliding bubble
encounters another departing bubble at a nucleation site, coalescence between two
30
bubbles occurs and the bubble size is enlarged. Therefore, in this study, a modified form
of the boiling source term is discussed in following sections considering the lift-off
diameter and the reduction factor for the lift-off frequency.
3.2. Modeling of bubble lift-off diameter
To determine the bubble size at a lift-off from the heated wall, a force balance on a
sliding bubble along the wall should be considered. Yeoh and Tu (2005) investigated the
force balance on the bubble from studies of Klausner et al. (1993) and Zeng et al. (1993),
which is presented in Figure 3.2. On a vertical wall, forces on the bubble in x-direction
and y-direction are as follows.
- In x-direction (Normal to a vertical wall)
cos cossx WF d β αα β
πσ θ θθ θ
⎡ ⎤= − −⎣ ⎦− (3.5a)
cosdux du iF F θ= − (3.5b)
2 212sl sl f rF C u rρ π= (3.5c)
221 9
2 4 4w
h f rdF u πρ= ⋅ (3.5d)
2 24
wcp
r
dFr
π σ= ⋅ (3.5e)
- In y-direction (Parallel to a vertical wall)
( )( )22
sin sinsy WF d α βα β
α β
π θ θσ θ θ
π θ θ
−⎡ ⎤= − +⎣ ⎦
− − (3.6a)
sinduy du iF F θ= − (3.6b)
2 212qs D f rF C u rρ π= (3.6c)
( )343b f gF r gπ ρ ρ= − (3.6d)
31
sF is the surface tension force, duF is the unsteady drag force due to a bubble growth,
which is given by Zeng et al. (1993)
2 232du f sF r C r rrρ π ⎛ ⎞= − +⎜ ⎟
⎝ ⎠& && (3.7)
slF is the shear lift force, hF is the hydrodynamic pressure force, and cpF is the
contact pressure force. qsF is the quasi-steady drag force and bF is the buoyancy
force.
At the moment of a bubble lift-off, the force balance on the bubble in x-direction is
violated and the contact diameter (dw) at wall becomes zero, that is, 0xF =∑ and
0wd = . Therefore, the force balance on the lifting bubble is given as follows.
2 2 2 23 1cos 02 2f s i sl f rr C r rr C r uρ π θ ρ π⎛ ⎞− + + =⎜ ⎟
⎝ ⎠& && (3.8)
where iθ is the bubble inclination angle and ru is the relative velocity between the
bubble and liquid. slC , the shear lift force coefficient, has been modeled by Klausner et
al. (1993).
( )1/ 41/ 2 2 23.877 Re 0.014l s b sC G G−= + (3.9)
where sG and Reb are given as,
l bs
r
du rGdx u
= , Re b rb
f
D uν
= (3.10)
The coefficient sC is given as 20/3 according to Zeng’s study (1993). Then Eq. (3.8) is
reduced to Eq. (3.11).
2 21102cos sl r
i
r rr C uθ
+ =& && (3.11)
To resolve Eq. (3.11), the function for a bubble growth with respect to time is required.
Zuber (1961) derived it considering a thermal diffusion in the superheated liquid.
( )w satg fg
k T Tdrhdt t
ρπη−
= (3.12)
where η is a thermal diffusivity of liquid. In the literature, Situ (2005) estimated this
32
term without considering the departure diameter as follows.
( ) 2br t Ja tηπ
= (3.13)
where b is an adjustable constant for the aspherical effect of the bubble and Zeng et al.
(1993) suggested a constant of 1.73. The equation means that the bubble size at t=0 is
zero. However, a sliding and lift-off of the bubble can occur after the departure from a
nucleate site on the wall. Regarding that the bubble departure occurs at t=0, the bubble
growth function can be derived with integrating Eq. (3.12) from t=0, as follows.
2( ) dbr t r Ja tηπ
= + (3.14)
where dr is the bubble radius at the departure from the wall. Ja is the Jacob number
defined as,
( )f pf w sat
g fg
C T TJa
hρ
ρ−
= (3.15)
From Eq. (3.14), the time-derivatives of bubble growth were derived as, 121
2r At
−=& ,
321
4r At
−= −&& where 2bA Ja η
π= (3.16)
On the other hand, the bubble radius at the moment of lift-off, lor , can be related with
the lift-off time, lot , where lo d lor r A t= + . Then substituting Eqs. (3.14) and (3.16)
into Eq. (3.11), a relation for the lift-off time is found as, 3
2 1 22 29coslo d lo sl r
i
A t Ar t C uθ
−− − = (3.17)
For a non-dimensional form, a ratio of the bubble growth is defined as *r with the
following relation.
* lo d
d
r rrr−
≡ (3.18)
By substituting Eq. (3.18) and the definition of bubble lift-off time into Eq. (3.17), Eq.
(3.17) can be formulated with respect to the bubble growth ratio. 2
*2 *3 2
9 1 2cos
d rsl
i
r uCr r Aθ
⎛ ⎞− = ⎜ ⎟⎝ ⎠
(3.19)
33
Eq. (3.19) is not an explicit form to estimate *r , so that the left-hand side of the
equation is fitted by an exponential function as shown in Figure 3.3.
* 1.41*2 *3
9 1 9.95rr r
−− ≈ (3.20)
Hence, the explicit formulation for the bubble lift-off diameter is derived as,
( )0.72
*21 1 8.34
cossl d r
lo d di
C D uD D r DAθ
−⎛ ⎞⎡ ⎤⎛ ⎞⎜ ⎟= + = + ⋅⎢ ⎥⎜ ⎟⎜ ⎟⎝ ⎠⎢ ⎥⎣ ⎦⎝ ⎠ (3.21)
For the bubble departure diameter, this study adopts Unal’s model (1976), which is
applicable to the flow condition in a wide range as given in the following. 5 0.7092.42 10
dp aD
b
−×=
Φ
( )1/3
1/32 /w sub f w w pw
f f pffg f f pf g
q h T k k Ca
k CC h k Cρρπ ρ ρ
− ∆= ,
( )2 1 /sub
g f
Tbρ ρ
∆=
−,
0.47
max ,1.00.61
lu⎛ ⎞⎛ ⎞Φ = ⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠
( 0.1<p<17.7MPa, 0.47<qw<10.64MW/m2, 0.08<ul<9.15m/s ) (3.22)
3.3. Modeling of lift-off frequency reduction factor
During the sliding of a departed bubble, coalescences can occur with another bubble at a
nucleate site. It reduces the number of actual bubble lift-offs from the wall, with respect
to the nucleate site density for the bubble departure, so that it affects an evaporative heat
flux and a nucleation source term in the interfacial area transport equation. Therefore
lift-off frequency reduction factors are considered for the evaporative heat flux and the
boiling source term in interfacial area transport equation as follows.
3 "
6e g fg lo eq h D N f Rπρ= ⋅ ⋅ (3.23)
2 " Hph lo a
AD N f RVol
φ π= ⋅ ⋅ (3.24)
where eq is the evaporative heat flux, eR and aR are reduction factors for the
34
evaporation term and the interfacial area concentration, respectively. The interfacial area
transport equation given in Eqs. (3.1), (3.2), (3.3) and (3.24) was integrated over the
control volume and the IAC at the next time step was computed by an explicit
calculation of the source terms.
In order to model the lift-off frequency reduction factors, the sliding length and the
spacing of a nucleate site should be considered. The sliding length, 0l , is the distance
from the departure to the lift-off of a bubble on the wall, assuming that the departed
bubble does not encounter any coalescence with another bubble during the sliding. The
spacing, s , is an averaged distance between two neighboring nucleate sites. When the
spacing is shorter than the bubble departure diameter, dD , the bubble at a nucleation
site cannot grow until the size reaches the bubble departure diameter and it lifts off the
wall with the diameter of s without a sliding. Hence, for a consistency of the
formulation in Eqs. (3.23) and (3.24), the reduction factors are determined as follows. 3
3elo
sRD
= , 2
2alo
sRD
= for ds D< (3.25)
When 0l is shorter than the spacing, there is no coalescence among departed bubbles
on the wall and all bubbles lift off the wall with a diameter of loD as defined in Eq.
(3.21). It yields that the reduction factors become unity, that is,
1e aR R= = for 0l s< (3.26)
When the spacing is longer than the departure diameter ( dD ) and shorter than 0l , the
bubble generated at a nucleation site can begin sliding and make the coalescences with
other bubbles during the sliding. In this region, the lift-off frequency reduction factor is
assumed to be a linear function between the no-sliding region (Eq. (3.25)) and no-
coalescence region (Eq. (3.26)), so that those are estimated as follows. 3 3
3 30
1 d d de
lo d lo
D s D DRD l D D
⎛ ⎞⎛ ⎞−= − +⎜ ⎟⎜ ⎟−⎝ ⎠⎝ ⎠
for 0dD s l< < (3.27a)
2 2
2 20
1 d d da
lo d lo
D s D DRD l D D
⎛ ⎞⎛ ⎞−= − +⎜ ⎟⎜ ⎟−⎝ ⎠⎝ ⎠
for 0dD s l< < (3.27b)
where the ratio between dD and loD can be estimated from Eq. (3.21).
The sliding length in Eqs. (3.25) to (3.27) can be determined from a momentum balance,
35
where the initial velocity at the moment of a departure is assumed to be zero and the
buoyancy force is considered.
3 34 43 3
gg
dur r g
dtρ π ρ π⋅ = ∆ (3.28)
Then Eq. (3.28) is integrated over the lift-off time, lot , shown in Eq. (3.17). Thus, the
sliding length of a departed bubble can be derived as, 4
2 *0
1 12 32
dlo
g g
Dl gt g rA
ρ ρρ ρ∆ ∆ ⎛ ⎞= ⋅ = ⋅ ⎜ ⎟
⎝ ⎠ (3.29)
The average spacing can be formulated as a function of the nuclear site density, "N , as
follows.
"NKs
N= (3.30)
where NK is a proportional coefficient. If it is assumed that nucleation sites are
distributed on a square grid, NK is equal to 1.
36
Figure 3.1 Mechanism of sliding and lift-off of a bubble
Figure 3.2 Force balance on a bubble at the wall (Yeoh and Tu, 2005)
37
0 1 2 3 4 5
0
20
40
60
80
100
120
LHS of Eq. (3.19) Eq. (3.20)
r*
Figure 3.3 Fitted relation of Eq. (3.19)
38
4. Analysis Results and Model Evaluation
4.1. Benchmark analysis for a single-phase flow
(1) Benchmark problem
Before an analysis of the complex phenomena in a two-phase flow using EAGLE code,
it is necessary to confirm that the mass, momentum, and energy equations in each phase
provide a reasonable prediction independently. As a benchmark problem for a single
phase flow analysis, the problem proposed by G. de Vahl Davis (1983) was selected,
where a two-dimensional buoyancy-driven cavity flow was computed with constant
wall temperatures and adiabatic boundaries, as depicted in Figure 4.1.
The test condition is given with respect to the Rayleigh number (Ra) as defined in Eq.
(4.1) and the standard solutions are known for the cases of Ra=103, 104, 105, and 106.
( )2 3
2 PrH Cg T T LRa
ρ βµ
−= (4.1)
For an application in this study, the working fluid was water at an atmospheric pressure
and the temperatures at the hot and cold wall were fixed at 80 and 30 , respectively. ℃ ℃
The size of the square (L) was determined to satisfy the given Rayleigh number. A
40 40 structured grid was ⅹ used for this analysis. Consequently, the test geometry was
determined as listed in Table 4.1.
(2) Analysis results
Figures 4.2 and 4.3 represent the temperature and velocity field analysis results. As
shown in the results, the thermal conduction between hot and cold plates is a dominant
mechanism for a heat transfer in the case of a small Rayleigh number, whereas a natural
convection driven by buoyancy force causes a vertical stratification of the temperature
39
as the Rayleigh number is larger. The overall results of the temperature and velocity
profile show that the developed code simulates the two-dimensional behavior of the
natural convection reasonably.
For the quantitative comparison, a local Nusselt number (Nuy) is defined as the non-
dimensional temperature gradient at wall, and then average Nusselt number ( Nu ) for
the whole length of wall can be obtained, as follows. *
*ywH Cw
T L TNux T T x
⎛ ⎞∂ ∂⎛ ⎞= =⎜ ⎟ ⎜ ⎟∂ − ∂⎝ ⎠⎝ ⎠ (4.2)
0
1 1L
y ii
Nu Nu dy Nu yL L
= = ∆∑∫ (4.3)
The Nusselt numbers in this study and in the literature were compared in Table 4.2.
Both results show that the larger Rayleigh number enhanced the convective heat transfer
near the wall, so that the Nusselt number is increased. And it is also confirmed that
reasonable agreement exists between the developed code and the standard solution. The
increased deviation between Nusselt numbers in EAGLE code and standard solution is
caused by the uncertainties in predicting the wall friction. Since the major focus of
EAGLE analysis is on the turbulent two-phase flow, this deficiency does not affect the
calculation significantly.
4.2. Analysis of SUBO experiment
(1) Description of the calculation
To validate the EAGLE code with the bubble lift-off model, the experimental data of
the SUBO (Subcooled Boiling) tests (Yun, 2008) were utilized. In the experiment, the
subcooled boiling phenomena in a vertical annulus channel were observed in SUBO
facility as shown in Figure 4.4. The inner diameter of the test section is 35.5mm and the
outer diameter of the heater rod is 10.02mm. The heater rod consists of three parts. The
first part is an unheated section with 222mm in length for regulating the water condition
at the inlet, the second part is a heated section with 3098mm in length for the simulation
40
of boiling, and the third part is an unheated section with 800mm in length for the bubble
condensation at the top region. Local two-phase flow parameters such as a void fraction,
interfacial area concentration, bubble velocity were measured by an optical fiber two-
sensor probe, which are traversed through 12 positions in a radial direction at 6 levels.
The test conditions of SUBO experiments are summarized in Table 4.3. Compared with
other facilities in the literature, SUBO has a capacity for simulating experimental
conditions of the higher heat flux and mass flux in a longer vertical channel. Outlet
pressure was maintained at around 155kPa in all the cases. When compared to the Base
case, the Q1 and Q2 cases are tested to investigate the effect of heat flux. Moreover, in
order to observe the subcooled boiling phenomena according to various conditions, the
V1 and V2 cases have a higher mass flux than the base case and the T1 case has a
higher inlet subcooling condition.
To validate the EAGLE code and the interfacial area transport equation model in this
study, SUBO experiment was analyzed. The analysis adopted the models for the bubble
lift-off diameter and lift-off frequency reduction factor in the evaporative heat flux and
IATE source term, which are derived in Eqs. (3.21), (3.23), and (3.24).
Analysis was conducted for modeling the heated section within a grid composed of 10
(radial)ⅹ200 (axial) axisymmetric cells in a cylindrical coordinate. The grid in the
EAGLE analysis was constructed with referring those of previous studies for similar
subcooled boiling flow channels. In the CFX analysis of Yeoh and Tu (2005), the grid
of 13 (radial) 30 (axial) x 3 (circumference) cells was used for an annulus channel ⅹ
with a 9.25mm gap size and Yao and Morel (2004) used the grid of 13 (radial) 30 ⅹ
(axial) cells for an annulus of 9.6mm gap in NEPTUNE code. In this study, the
comparison with a more refined grid was conducted additionally, so that the results
between 10 cells and 15 cells in a radial direction will be discussed. The inlet condition
and the heated wall boundary condition were given as a constant mass flux and heat flux
at the cell surface, and a zero-gradient condition was taken into account at the outlet
boundary. The inlet turbulence intensity was set to 5% of the mean velocity.
41
(2) Result and discussion
Figure 4.5 compares the radial distribution of void fraction in all test cases of the SUBO
experiment. As shown in the figure, the axial development of bubbly boundary layer in
the subcooled boiling was predicted reasonably well by the calculation of EAGLE code.
It means that the evaporative heat flux in the heat partitioning model according to the
bubble lift-off mechanism could appropriately estimate the amount of vapor generation
at a heated wall. The difference in the bubbly boundary layer thickness between the
experiment and analysis was due to the limitation of the interfacial heat transfer model,
which should be precisely improved depending on the local two-phase flow parameters.
Moreover, the radial distribution of the void in a two-phase flow is governed by the
non-drag forces such as the wall lubrication force or the lift force which have been
described in Eqs. (2.35) and (2.36). The reason for the discrepancy of void fraction
profile between the analysis and experiment can be also found by incompleteness in the
constitutive models for the non-drag interfacial momentum transfer.
From the comparison of interfacial area concentration as revealed in Figure 4.6, it was
interpreted that the modeling of interfacial area transport equation predicted the
appropriate distribution of the interfacial area in SUBO experiments. The advanced
source term of interfacial area transport equation according to the bubble lift-off
mechanism induced a peak of the interfacial area concentration near the heated wall,
while the condensation and bubble interaction mechanism such as a breakup or a
coalescence affected the radial distribution of the interfacial area concentration
effectively.
For a comparison, Situ’s model of the bubble lift-off diameter (Situ et al., 2005) was
implemented in an additional calculation for the Base case, instead of the bubble lift-off
diameter derived in this study. That model is given as, 2
* 2 14 22 / 3 Prr lolo l e f
f
u D bD C Jaν π
−⎛ ⎞
≡ =⎜ ⎟⎜ ⎟⎝ ⎠
(4.4)
Figure 4.7(b) compares the interfacial area concentration in Base case with the
calculation results adopting Situ’s bubble lift-off diameter model. As shown in the
figure, it is proved that the bubble lift-off diameter model derived in Section 3 yields a
42
more reasonable agreement in the multi-dimensional distribution of interfacial area
concentration near the heated wall, since the model considers the growth of a departed
bubble on the wall mechanistically. In the condition of a higher heat flux, the bubble
lift-off model in this study also represented a better prediction as shown in the analysis
results of Q2 case in Figure 4.8(b).
Figures 4.7(c) and 4.8(c) represent the calculation result where the model for a
frequency reduction factor was not included in Base case and Q2 case, respectively. In
both cases, the exclusion of the reduction factor made an overestimation of the
interfacial area concentration, since it did not take into account the effect of
coalescences between the departed bubbles on the heated wall. On the other hand, a
sensitivity test was tried to calculate the boiling source term of interfacial area transport
equation with the conventional bubble departure mechanism according to Eq. (3.4). As
illustrated in Figures 4.7(d) and 4.8(d) for Base case and Q2 case, considering the
phenomenon of bubble departure only showed an over-prediction of interfacial area
concentration when compared with the calculation with the bubble lift-off mechanism in
this study. From the above sensitivity tests, it is concluded that the bubble lift-off model
for the boiling source term in the interfacial area transport equation indicated an
advanced prediction of the distribution of interfacial area concentration.
Figure 4.9 represents the analysis results of bubble velocity and compares them with the
experimental ones. In the experimental result, the axial bubble velocity showed a peak
around the center of the channel due to a larger buoyant force of large bubbles. EAGLE
code analysis also showed a peak of the bubble velocity at around the center in all test
cases and indicated a reasonable agreement with the experimental results as shown in
the figure. Especially, the modeling of an increased turbulence in the laminar sublayer
as described in Section 2.3 played an important role in enhancing a prediction capability
of the bubble velocity near the heated surface. The modeling effect of boiling bubbles at
the surface is compared for the analysis results of Base case in Figure 4.10, where the
calculation by excluding the enhanced turbulence overestimated the bubble velocity
near the heated wall. The similar behavior was observed in the analysis for a higher
mass flux condition in V1 case as revealed in Figure 4.11. The wall function according
to the single phase flow turbulence was applied to calculate the wall shear stress, which
is equal to a diffusive flux term acting on the surface of a cell adjacent to the wall. The
43
shape of velocity profile as depicted in EAGLE analysis results could be parabolic by
the effect of the wall shear stress. The assumption of velocity profile according to the
wall function theory can be available only in a region which is closely adjacent to the
wall. So the parabolic shape of the velocity in Figures 4.10 and 4.11 is not directly
related with the velocity profile according to the wall function itself.
Figure 4.12 compares the analysis results of Base case with a more refined grid. (15
radial cells x 200 axial cells) As shown in the figure, the more refined grid in a radial
direction did not significantly influence the analysis results.
4.3. Benchmark analysis for SNU experiments
(1) Description of the experiment
To extend the applicability of the developed model and EAGLE code in the extended
test condition, the subcooled boiling experiment at Seoul National University (SNU)
(Kim et al., 2004) was selected for the two phase flow analysis. That experiment
focused on a boiling and condensation for a vertical upward flow in a concentric
annulus as shown in Figure 4.13, whose geometrical dimensions are listed in Table 4.4.
Major measured parameters were the local void fraction, the IAC, and the bubble
velocity, which had been measured at 13 points in the radial direction and at 3 levels,
L/Dh=58.4, 68.0, 77.5, in the axial direction. The test conditions selected for the
benchmark in the SNU experiments are listed in Table 4.5. The selected test cases have
a lower heat flux and a lower mass flux condition than the SUBO experiment, so that
the data of the SNU experiment can be good benchmark data for evaluating the
calculation capability of the EAGLE code in low heat flux and mass flux conditions.
Analysis was conducted for modeling the heated section within a grid composed of 10
(radial)ⅹ100 (axial) axisymmetric cells in a cylindrical coordinate. Similarly with the
analysis of SUBO experiments, the inlet condition and the heated wall boundary
condition were given as a constant mass flux and heat flux at the cell surface, and a
zero-gradient condition was taken into account at the outlet boundary. The inlet
turbulence intensity was set to 5% of the mean velocity.
44
(2) Result and discussion
Figure 4.14 represents the radial distribution of local void fraction in all cases of SNU
experiment and EAGLE analysis results. As explained in the experimental results of
SUBO facility, the subcooled boiling in a low heat flux and mass flux condition also
showed an existence of the bubbly boundary layer near the heated wall and the
thickness of that increased in a larger heat flux condition. The result of EAGLE code
analysis presented a reasonable behavior of the multi-dimensional distribution of the
void fraction, which are taking into account the phenomena of the bubble sliding and
lift-off at the heated surface. From the results, the mechanistic model of the bubble lift-
off diameter considering both of the bubble departure and sliding was validated to have
a capability in predicting the amount of evaporation for a low heat flux condition of the
subcooled boiling.
Figure 4.15 depicts the distribution of interfacial area concentration. In the experiment,
due to the coalescence effect in a high void fraction condition, Case 3 with a larger heat
flux showed a lower interfacial area concentration than Case 2. As shown in the figure,
the EAGLE analysis represented a similar trend of the interfacial area concentration
when compared to the experimental results in all test cases. Therefore, it confirms the
validity of the interfacial area transport equation with the bubble lift-off diameter model
and lift-off frequency reduction factor model.
The bubble velocity profiles in the test cases are shown in Figure 4.16. Similarly with
the analysis results of SUBO experiment, SNU test data with a lower mass flux reveals
a peak of the bubble velocity at around the center of flow channel, which is due to a
larger buoyancy force of the large bubbles. All cases of the computational analysis
indicated a sufficiency in predicting the velocity profile of the gas phase. Although the
bubble velocity adjacent to the heated wall was underestimated by a larger turbulent
shear stress in Case 3 with a high mass flux condition, the enhanced turbulence by
boiling bubbles at the heated surface contributed a prediction of the bubble velocity near
the heated wall. On the other hand, Figure 4.17 compares the analysis results of bubble
velocity in Cases 1 and 2 with the calculation of CFX 4-2 (Kim et al., 2001). It shows
that the modeling of turbulence enhanced a prediction capability of the velocity
distribution in EAGLE code, while CFX overestimated the bubble velocity near the
45
heated surface in both cases.
From the above results of benchmark test and analysis for the low heat flux and mass
flux conditions, it is ascertained that the EAGLE code with the interfacial area transport
equation has an improved capability for the subcooled boiling two-phase flow analysis.
46
Table 4.1 Test geometry for the natural convection
Ra L(m) Cell size (=L/40, m)
103 9.910 × 10-4 2.478 × 10-5
104 2.135 × 10-3 5.338 × 10-5
105 4.600 × 10-3 1.150 × 10-4
106 9.910 × 10-3 2.478 × 10-4
Table 4.2 Comparison of average the Nusselt number
Ra Analysis G. de Vahl Davis(1983) Error (%)
103 1.125 1.118 0.63
104 2.296 2.243 2.31
105 4.868 4.519 7.17
106 10.06 8.8 12.5
Table 4.3 Test matrix of SUBO experiment
Case Heat flux
(kW/m2)
Mass flux
(kg/m2s)
Inlet
subcooling
(K)
Inlet
pressure
(kPa)
Outlet
pressure
(kPa)
Base 470.6 1132.6 19.1 192.9 157.3
Q1 363.7 1119.6 19.0 192.7 156.7
Q2 563.0 1126.9 18.3 188.9 155.7
V1 465.7 2126.5 19.6 196.9 156.9
V2 567.9 2128.8 19.5 197.6 158.0
T1 465.5 1103.9 29.6 190.7 155.0
Table 4.4 Geometry of the SNU experiment
Flow area 9.72615cm2
Heating length 1870mm
Hydraulic diameter 21mm
Outer diameter of heater 19mm
Inner diameter of channel 40mm
47
Table 4.5 SNU Test condition for the subcooled boiling
Case 1 Case 2 Case 3
Mass flux 339.6 kg/m2s 342.2 kg/m2s 673.7 kg/m2s
Heat flux 96.7 kW/m2 212.7 kW/m2 358.8 kW/m2
Inlet pressure 1.21bar 1.21bar 1.42bar
Inlet subcooling 12.7K 21.7K 19.4K
48
TH=80'C T
C=30'C
L
L
Adiabatic
Adiabatic
Figure 4.1 Benchmark problem for single-phase natural convection
49
X
Y
0 0.0002 0.0004 0.0006 0.0008 0.0010
0.0002
0.0004
0.0006
0.0008
0.001Tf
8077.57572.57067.56562.56057.55552.55047.54542.54037.53532.530
Figure 4.2(a) Temperature distribution, Ra=103
X
Y
0 0.0005 0.001 0.0015 0.0020
0.0005
0.001
0.0015
0.002Tf
8077.57572.57067.56562.56057.55552.55047.54542.54037.53532.530
Figure 4.2(b) Temperature distribution, Ra=104
50
X
Y
0 0.001 0.002 0.003 0.004 0.0050
0.001
0.002
0.003
0.004
Tf8077.57572.57067.56562.56057.55552.55047.54542.54037.53532.530
Figure 4.2(c) Temperature distribution, Ra=105
X
Y
0 0.002 0.004 0.006 0.008 0.010
0.002
0.004
0.006
0.008
0.01Tf
8077.57572.57067.56562.56057.55552.55047.54542.54037.53532.530
Figure 4.2(d) Temperature distribution, Ra=106
51
X
Y
0 0.0002 0.0004 0.0006 0.0008 0.0010
0.0002
0.0004
0.0006
0.0008
0.001
0.001m/s
Figure 4.3(a) Velocity field, Ra=103
X
Y
0 0.0005 0.001 0.0015 0.0020
0.0005
0.001
0.0015
0.002
0.005m/s
Figure 4.3(b) Velocity field, Ra=104
52
X
Y
0 0.001 0.002 0.003 0.004 0.0050
0.001
0.002
0.003
0.004
0.005m/s
Figure 4.3(c) Velocity field, Ra=105
X
Y
0 0.002 0.004 0.006 0.008 0.010
0.002
0.004
0.006
0.008
0.01
0.005m/s
Figure 4.3(d) Velocity field, Ra=106
53
Figure 4.4 Geometry and measuring position of SUBO facility
54
0.0 0.2 0.4 0.6 0.8 1.00.0
0.2
0.4
0.6
0.8
1.0
q"=470.7kW/m2
G = 1132.6kg/m2sTsub=19.1K Exp Analysis
L/Dh=42.5 L/Dh=42.5 L/Dh=66.4 L/Dh=66.4 L/Dh=91.7 L/Dh=91.7 L/Dh=116.2 L/Dh=116.2
Voi
d fra
ctio
n
r*((r-ri)/(ro-ri))
Figure 4.5(a) Comparison of the void fraction in Base case
0.0 0.2 0.4 0.6 0.8 1.00.0
0.2
0.4
0.6
0.8
1.0
q"=363.7kW/m2
G = 1119.6kg/m2sTsub=19.0K Exp Analysis
L/Dh=42.5 L/Dh=42.5 L/Dh=66.4 L/Dh=66.4 L/Dh=91.7 L/Dh=91.7 L/Dh=116.2 L/Dh=116.2
Voi
d fra
ctio
n
r*((r-ri)/(ro-ri))
Figure 4.5(b) Comparison of the void fraction in Q1 case
55
0.0 0.2 0.4 0.6 0.8 1.00.0
0.2
0.4
0.6
0.8
1.0q"=563.0kW/m2
G = 1126.9kg/m2sTsub=18.3K Exp Analysis
L/Dh=42.5 L/Dh=42.5 L/Dh=66.4 L/Dh=66.4 L/Dh=91.7 L/Dh=91.7 L/Dh=116.2 L/Dh=116.2
Voi
d fra
ctio
n
r*((r-ri)/(ro-ri))
Figure 4.5(c) Comparison of the void fraction in Q2 case
0.0 0.2 0.4 0.6 0.8 1.00.0
0.2
0.4
0.6
0.8
1.0
q"=465.7kW/m2
G = 2126.5kg/m2sTsub=19.6K Exp Analysis
L/Dh=42.5 L/Dh=42.5 L/Dh=66.4 L/Dh=66.4 L/Dh=91.7 L/Dh=91.7 L/Dh=116.2 L/Dh=116.2
Voi
d fra
ctio
n
r*((r-ri)/(ro-ri))
Figure 4.5(d) Comparison of the void fraction in V1 case
56
0.0 0.2 0.4 0.6 0.8 1.00.0
0.2
0.4
0.6
0.8
1.0
q"=567.9kW/m2
G = 2128.8kg/m2sTsub=19.5K Exp Analysis
L/Dh=42.5 L/Dh=42.5 L/Dh=66.4 L/Dh=66.4 L/Dh=91.7 L/Dh=91.7 L/Dh=116.2 L/Dh=116.2
Voi
d fra
ctio
n
r*((r-ri)/(ro-ri))
Figure 4.5(e) Comparison of the void fraction in V2 case
0.0 0.2 0.4 0.6 0.8 1.00.0
0.2
0.4
0.6
0.8
1.0
q"=465.5kW/m2
G = 1103.9kg/m2sTsub=29.6K Exp Analysis
L/Dh=42.5 L/Dh=42.5 L/Dh=66.4 L/Dh=66.4 L/Dh=91.7 L/Dh=91.7 L/Dh=116.2 L/Dh=116.2
Voi
d fra
ctio
n
r*((r-ri)/(ro-ri))
Figure 4.5(f) Comparison of the void fraction in T1 case
57
0.0 0.2 0.4 0.6 0.8 1.00
500
1000
1500
2000
2500
3000
q"=470.7kW/m2
G = 1132.6kg/m2sTsub=19.1K Exp Analysis
L/Dh=42.5 L/Dh=42.5 L/Dh=66.4 L/Dh=66.4 L/Dh=91.7 L/Dh=91.7 L/Dh=116.2 L/Dh=116.2
IAC
(1/m
)
r*((r-ri)/(ro-ri))
Figure 4.6(a) Comparison of IAC in Base case
0.0 0.2 0.4 0.6 0.8 1.00
500
1000
1500
2000
2500
3000
q"=363.7kW/m2
G = 1119.6kg/m2sTsub=19.0K Exp Analysis
L/Dh=42.5 L/Dh=42.5 L/Dh=66.4 L/Dh=66.4 L/Dh=91.7 L/Dh=91.7 L/Dh=116.2 L/Dh=116.2
IAC
(1/m
)
r*((r-ri)/(ro-ri))
Figure 4.6(b) Comparison of IAC in Q1case
58
0.0 0.2 0.4 0.6 0.8 1.00
500
1000
1500
2000
2500
3000
q"=563.0kW/m2
G = 1126.9kg/m2sTsub=18.3K Exp Analysis
L/Dh=42.5 L/Dh=42.5 L/Dh=66.4 L/Dh=66.4 L/Dh=91.7 L/Dh=91.7 L/Dh=116.2 L/Dh=116.2
IAC
(1/m
)
r*((r-ri)/(ro-ri))
Figure 4.6(c) Comparison of IAC in Q2 case
0.0 0.2 0.4 0.6 0.8 1.00
500
1000
1500
2000
2500
3000
q"=465.7kW/m2
G = 2126.5kg/m2sTsub=19.6K Exp Analysis
L/Dh=42.5 L/Dh=42.5 L/Dh=66.4 L/Dh=66.4 L/Dh=91.7 L/Dh=91.7 L/Dh=116.2 L/Dh=116.2
IAC
(1/m
)
r*((r-ri)/(ro-ri))
Figure 4.6(d) Comparison of IAC in V1 case
59
0.0 0.2 0.4 0.6 0.8 1.00
500
1000
1500
2000
2500
3000
q"=567.9kW/m2
G = 2128.8kg/m2sTsub=19.5K Exp Analysis
L/Dh=42.5 L/Dh=42.5 L/Dh=66.4 L/Dh=66.4 L/Dh=91.7 L/Dh=91.7 L/Dh=116.2 L/Dh=116.2
IAC
(1/m
)
r*((r-ri)/(ro-ri))
Figure 4.6(e) Comparison of IAC in V2 case
0.0 0.2 0.4 0.6 0.8 1.00
500
1000
1500
2000
2500
3000
q"=465.5kW/m2
G = 1103.9kg/m2sTsub=29.6K Exp Analysis
L/Dh=42.5 L/Dh=42.5 L/Dh=66.4 L/Dh=66.4 L/Dh=91.7 L/Dh=91.7 L/Dh=116.2 L/Dh=116.2
IAC
(1/m
)
r*((r-ri)/(ro-ri))
Figure 4.6(f) Comparison of IAC in T1 case
60
0.0 0.2 0.4 0.6 0.8 1.00
500
1000
1500
2000
2500
3000
q"=470.7kW/m2
G = 1132.6kg/m2sTsub=19.1K Exp Analysis
L/Dh=42.5 L/Dh=42.5 L/Dh=66.4 L/Dh=66.4 L/Dh=91.7 L/Dh=91.7 L/Dh=116.2 L/Dh=116.2
IAC
(1/m
)
r*((r-ri)/(ro-ri))
(a) Bubble lift-off model in this study
0.0 0.2 0.4 0.6 0.8 1.00
500
1000
1500
2000
2500
3000
q"=470.7kW/m2
G = 1132.6kg/m2sTsub=19.1K Exp Analysis
L/Dh=42.5 L/Dh=42.5 L/Dh=66.4 L/Dh=66.4 L/Dh=91.7 L/Dh=91.7 L/Dh=116.2 L/Dh=116.2
IAC
(1/m
)
r*((r-ri)/(ro-ri))
(b) Situ’s lift-off diameter model (2005)
Figure 4.7 Sensitivity on boiling source term in Base case
61
0.0 0.2 0.4 0.6 0.8 1.00
500
1000
1500
2000
2500
3000
3500
q"=470.7kW/m2
G = 1132.6kg/m2sTsub=19.1K Exp Analysis
L/Dh=42.5 L/Dh=42.5 L/Dh=66.4 L/Dh=66.4 L/Dh=91.7 L/Dh=91.7 L/Dh=116.2 L/Dh=116.2
IAC
(1/m
)
r*((r-ri)/(ro-ri))
(c) Exclusion of frequency reduction factor
0.0 0.2 0.4 0.6 0.8 1.00
500
1000
1500
2000
2500
3000
q"=470.7kW/m2
G = 1132.6kg/m2sTsub=19.1K Exp Analysis
L/Dh=42.5 L/Dh=42.5 L/Dh=66.4 L/Dh=66.4 L/Dh=91.7 L/Dh=91.7 L/Dh=116.2 L/Dh=116.2
IAC
(1/m
)
r*((r-ri)/(ro-ri))
(d) Bubble departure mechanism
Figure 4.7 Sensitivity on boiling source term in Base case (Continued)
62
0.0 0.2 0.4 0.6 0.8 1.00
500
1000
1500
2000
2500
3000
q"=563.0kW/m2
G = 1126.9kg/m2sTsub=18.3K Exp Analysis
L/Dh=42.5 L/Dh=42.5 L/Dh=66.4 L/Dh=66.4 L/Dh=91.7 L/Dh=91.7 L/Dh=116.2 L/Dh=116.2
IAC
(1/m
)
r*((r-ri)/(ro-ri))
(a) Bubble lift-off model in this study
0.0 0.2 0.4 0.6 0.8 1.00
500
1000
1500
2000
2500
3000
q"=563.0kW/m2
G = 1126.9kg/m2sTsub=18.3K Exp Analysis
L/Dh=42.5 L/Dh=42.5 L/Dh=66.4 L/Dh=66.4 L/Dh=91.7 L/Dh=91.7 L/Dh=116.2 L/Dh=116.2
IAC
(1/m
)
r*((r-ri)/(ro-ri))
(b) Situ’s lift-off diameter model (2005)
Figure 4.8 Sensitivity on boiling source term in Q2 case
63
0.0 0.2 0.4 0.6 0.8 1.00
500
1000
1500
2000
2500
3000
q"=563.0kW/m2
G = 1126.9kg/m2sTsub=18.3K Exp Analysis
L/Dh=42.5 L/Dh=42.5 L/Dh=66.4 L/Dh=66.4 L/Dh=91.7 L/Dh=91.7 L/Dh=116.2 L/Dh=116.2
IAC
(1/m
)
r*((r-ri)/(ro-ri))
(c) Exclusion of frequency reduction factor
0.0 0.2 0.4 0.6 0.8 1.00
500
1000
1500
2000
2500
3000
q"=563.0kW/m2
G = 1126.9kg/m2sTsub=18.3K Exp Analysis
L/Dh=42.5 L/Dh=42.5 L/Dh=66.4 L/Dh=66.4 L/Dh=91.7 L/Dh=91.7 L/Dh=116.2 L/Dh=116.2
IAC
(1/m
)
r*((r-ri)/(ro-ri))
(d) Bubble departure mechanism
Figure 4.8 Sensitivity on boiling source term in Q2 case (Continued)
64
0.0 0.2 0.4 0.6 0.8 1.00.0
0.5
1.0
1.5
2.0
2.5
3.0
q"=470.7kW/m2
G = 1132.6kg/m2s, Tsub=19.1K Exp Analysis
L/Dh=42.5 L/Dh=42.5 L/Dh=66.4 L/Dh=66.4 L/Dh=91.7 L/Dh=91.7 L/Dh=116.2 L/Dh=116.2
Bub
ble
velo
city
(m/s
)
r*((r-ri)/(ro-ri))
Figure 4.9(a) Comparison of the bubble velocity in Base case
0.0 0.2 0.4 0.6 0.8 1.00.0
0.5
1.0
1.5
2.0
2.5
3.0q"=363.7kW/m2
G = 1119.6kg/m2sTsub=19.0K Exp Analysis
L/Dh=42.5 L/Dh=42.5 L/Dh=66.4 L/Dh=66.4 L/Dh=91.7 L/Dh=91.7 L/Dh=116.2 L/Dh=116.2
Bubb
le v
eloc
ity(m
/s)
r*((r-ri)/(ro-ri))
Figure 4.9(b) Comparison of the bubble velocity in Q1 case
65
0.0 0.2 0.4 0.6 0.8 1.00.0
0.5
1.0
1.5
2.0
2.5
3.0
q"=563.0kW/m2
G = 1126.9kg/m2sTsub=18.3K Exp Analysis
L/Dh=42.5 L/Dh=42.5 L/Dh=66.4 L/Dh=66.4 L/Dh=91.7 L/Dh=91.7 L/Dh=116.2 L/Dh=116.2
Bub
ble
velo
city
(m/s
)
r*((r-ri)/(ro-ri))
Figure 4.9(c) Comparison of the bubble velocity in Q2 case
0.0 0.2 0.4 0.6 0.8 1.00.0
0.5
1.0
1.5
2.0
2.5
3.0
q"=465.7kW/m2
G = 2126.5kg/m2sTsub=19.6K Exp Analysis
L/Dh=42.5 L/Dh=42.5 L/Dh=66.4 L/Dh=66.4 L/Dh=91.7 L/Dh=91.7 L/Dh=116.2 L/Dh=116.2
Bub
ble
velo
city
(m/s
)
r*((r-ri)/(ro-ri))
Figure 4.9(d) Comparison of the bubble velocity in V1 case
66
0.0 0.2 0.4 0.6 0.8 1.00.0
0.5
1.0
1.5
2.0
2.5
3.0
q"=567.9kW/m2
G = 2128.8kg/m2sTsub=19.5K Exp Analysis
L/Dh=42.5 L/Dh=42.5 L/Dh=66.4 L/Dh=66.4 L/Dh=91.7 L/Dh=91.7 L/Dh=116.2 L/Dh=116.2
Bub
ble
velo
city
(m/s
)
r*((r-ri)/(ro-ri))
Figure 4.9(e) Comparison of the bubble velocity in V2 case
0.0 0.2 0.4 0.6 0.8 1.00.0
0.5
1.0
1.5
2.0
2.5
3.0
q"=465.5kW/m2
G = 1103.9kg/m2sTsub=29.6K Exp Analysis
L/Dh=42.5 L/Dh=42.5 L/Dh=66.4 L/Dh=66.4 L/Dh=91.7 L/Dh=91.7 L/Dh=116.2 L/Dh=116.2
Bubb
le v
eloc
ity(m
/s)
r*((r-ri)/(ro-ri))
Figure 4.9(f) Comparison of the bubble velocity in T1 case
67
0.0 0.2 0.4 0.6 0.8 1.00.0
0.5
1.0
1.5
2.0
2.5
3.0
q"=470.7kW/m2
G = 1132.6kg/m2s, Tsub=19.1K Exp Analysis
L/Dh=42.5 L/Dh=42.5 L/Dh=66.4 L/Dh=66.4 L/Dh=91.7 L/Dh=91.7 L/Dh=116.2 L/Dh=116.2
Bubb
le v
eloc
ity(m
/s)
r*((r-ri)/(ro-ri))
(a) With the turbulence of boiling bubbles
0.0 0.2 0.4 0.6 0.8 1.00.0
0.5
1.0
1.5
2.0
2.5
3.0q"=470.7kW/m2
G = 1132.6kg/m2sTsub=19.1K Exp Analysis
L/Dh=42.5 L/Dh=42.5 L/Dh=66.4 L/Dh=66.4 L/Dh=91.7 L/Dh=91.7 L/Dh=116.2 L/Dh=116.2
Bubb
le v
eloc
ity(m
/s)
r*((r-ri)/(ro-ri))
(b) Without the turbulence of boiling bubbles
Figure 4.10 Comparison of the bubble velocity in the Base case
68
0.0 0.2 0.4 0.6 0.8 1.00.0
0.5
1.0
1.5
2.0
2.5
3.0
q"=465.7kW/m2
G = 2126.5kg/m2sTsub=19.6K Exp Analysis
L/Dh=42.5 L/Dh=42.5 L/Dh=66.4 L/Dh=66.4 L/Dh=91.7 L/Dh=91.7 L/Dh=116.2 L/Dh=116.2
Bub
ble
velo
city
(m/s
)
r*((r-ri)/(ro-ri))
(a) With the turbulence of boiling bubbles
0.0 0.2 0.4 0.6 0.8 1.00.0
0.5
1.0
1.5
2.0
2.5
3.0
q"=465.7kW/m2
G = 2126.5kg/m2sTsub=19.6K Exp Analysis
L/Dh=42.5 L/Dh=42.5 L/Dh=66.4 L/Dh=66.4 L/Dh=91.7 L/Dh=91.7 L/Dh=116.2 L/Dh=116.2
Bub
ble
velo
city
(m/s
)
r*((r-ri)/(ro-ri))
(b) Without the turbulence of boiling bubbles
Figure 4.11 Comparison of the bubble velocity in the V1 case
69
0.0 0.2 0.4 0.6 0.8 1.00.0
0.2
0.4
0.6
0.8
1.0
q"=470.7kW/m2
G = 1132.6kg/m2sTsub=19.1K Exp Analysis
L/Dh=42.5 L/Dh=42.5 L/Dh=66.4 L/Dh=66.4 L/Dh=91.7 L/Dh=91.7 L/Dh=116.2 L/Dh=116.2
( 10X200 grids) ( 15X200 grids)
Voi
d fra
ctio
n
r*((r-ri)/(ro-ri))
(a) Void fraction
0.0 0.2 0.4 0.6 0.8 1.00.0
0.5
1.0
1.5
2.0
2.5
3.0q"=470.7kW/m2
G = 1132.6kg/m2sTsub=19.1K Exp Analysis
L/Dh=42.5 L/Dh=42.5 L/Dh=66.4 L/Dh=66.4 L/Dh=91.7 L/Dh=91.7 L/Dh=116.2 L/Dh=116.2
( 10X200 grids) ( 15X200 grids)
Bub
ble
Velo
city
(m/s
)
r*((r-ri)/(ro-ri))
(b) Bubble velocity
Figure 4.12 Effect of grid refinement in Base case
70
UnheatedSection
(Condensation)
HeatedSection(Boiling)
UnheatedSection
(Developing)
Inlet
Heater19mm O.D.
Flow Channel
40mm I.D.
Figure 4.13 Annulus channel in SNU experiment
71
0.0 0.2 0.4 0.6 0.8 1.00.0
0.2
0.4
0.6
0.8
1.0
q"=96.7kW/m2
G = 339.7kg/m2sTsub=12.4K Exp Analysis
L/Dh=58.4 L/Dh=58.4 L/Dh=68.0 L/Dh=68.0 L/Dh=77.5 L/Dh=77.5
Voi
d fra
ctio
n
R*( =(R-Rin)/(Rout-Rin) )
(a) Case 1
0.0 0.2 0.4 0.6 0.8 1.00.0
0.2
0.4
0.6
0.8
1.0
q"=212.7kW/m2
G = 342.2kg/m2sTsub=19.1K Exp Analysis
L/Dh=58.4 L/Dh=58.4 L/Dh=68.0 L/Dh=68.0 L/Dh=77.5 L/Dh=77.5
Voi
d fra
ctio
n
R*( =(R-Rin)/(Rout-Rin) )
(b) Case 2
Figure 4.14 Void fraction in SNU test cases
72
0.0 0.2 0.4 0.6 0.8 1.00.0
0.2
0.4
0.6
0.8
1.0
q"=358.8kW/m2
G = 673.7kg/m2sTsub=19.4K Exp Analysis
L/Dh=58.4 L/Dh=58.4 L/Dh=68.0 L/Dh=68.0 L/Dh=77.5 L/Dh=77.5
Voi
d fra
ctio
n
R*( =(R-Rin)/(Rout-Rin) )
(c) Case 3
Figure 4.14 Void fraction in SNU test cases (Continued)
73
0.0 0.2 0.4 0.6 0.8 1.00
500
1000
1500
q"=96.7kW/m2
G = 339.7kg/m2sTsub=12.4K Exp Analysis
L/Dh=58.4 L/Dh=58.4 L/Dh=68.0 L/Dh=68.0 L/Dh=77.5 L/Dh=77.5
IAC
(1/m
)
R*( =(R-Rin)/(Rout-Rin) )
(a) Case 1
0.0 0.2 0.4 0.6 0.8 1.00
500
1000
1500
IAC
(1/m
)
q"=212.7kW/m2
G = 342.2kg/m2sTsub=19.1K Exp Analysis
L/Dh=58.4 L/Dh=58.4 L/Dh=68.0 L/Dh=68.0 L/Dh=77.5 L/Dh=77.5
R*( =(R-Rin)/(Rout-Rin) )
(b) Case 2
Figure 4.15 Interfacial area concentration in SNU test cases
74
0.0 0.2 0.4 0.6 0.8 1.00
500
1000
1500
IAC
(1/m
)
q"=358.8kW/m2
G = 673.7kg/m2sTsub=19.4K Exp Analysis
L/Dh=58.4 L/Dh=58.4 L/Dh=68.0 L/Dh=68.0 L/Dh=77.5 L/Dh=77.5
R*( =(R-Rin)/(Rout-Rin) )
(c) Case 3
Figure 4.15 Interfacial area concentration in SNU test cases (Continued)
75
0.0 0.2 0.4 0.6 0.8 1.00.0
0.4
0.8
1.2
1.6
2.0
q"=96.7kW/m2
G = 339.7kg/m2sTsub=12.4K Exp Analysis
L/Dh=58.4 L/Dh=58.4 L/Dh=68.0 L/Dh=68.0 L/Dh=77.5 L/Dh=77.5
Bub
ble
Vel
ocity
(m/s
)
R*( =(R-Rin)/(Rout-Rin) )
(a) Case 1
0.0 0.2 0.4 0.6 0.8 1.00.0
0.4
0.8
1.2
1.6
2.0
q"=212.7kW/m2
G = 342.2kg/m2sTsub=19.1K Exp Analysis
L/Dh=58.4 L/Dh=58.4 L/Dh=68.0 L/Dh=68.0 L/Dh=77.5 L/Dh=77.5
Bub
ble
Vel
ocity
(m/s
)
R*( =(R-Rin)/(Rout-Rin) )
(b) Case 2
Figure 4.16 Bubble velocity in SNU test cases
76
0.0 0.2 0.4 0.6 0.8 1.00.0
0.4
0.8
1.2
1.6
2.0
q"=358.8kW/m2
G = 673.7kg/m2sTsub=19.4K Exp Analysis
L/Dh=58.4 L/Dh=58.4 L/Dh=68.0 L/Dh=68.0 L/Dh=77.5 L/Dh=77.5
Bub
ble
Vel
ocity
(m/s
)
R*( =(R-Rin)/(Rout-Rin) )
(c) Case 3
Figure 4.16 Bubble velocity in SNU test cases (Continued)
77
0.0 0.2 0.4 0.6 0.8 1.00.0
0.4
0.8
1.2
1.6
q"=96.7kW/m2
G = 339.7kg/m2sTsub=12.4K, L/Dh=58.4
Exp EAGLE analysis CFX (Kim, 2001)
Bubb
le V
eloc
ity (m
/s)
R*( =(R-Rin)/(Rout-Rin) )
(a) Case 1
0.0 0.2 0.4 0.6 0.8 1.00.0
0.4
0.8
1.2
1.6
q"=212.7kW/m2
G = 342.2kg/m2sTsub=19.1K, L/Dh=58.4
Exp EAGLE analysis CFX (Kim, 2001)
Bub
ble
Vel
ocity
(m/s
)
R*( =(R-Rin)/(Rout-Rin) )
(b) Case 2
Figure 4.17 Comparison of bubble velocity with CFX calculation
78
5. Conclusion
This study focused on the development of the wall nucleation source term in the
interfacial area transport equation during an analysis of the subcooled boiling two-phase
flow. To evaluate the model, SUBO experiment was performed and the test results were
utilized in the validation of EAGLE code.
To mechanistically model the dynamic behavior of the interfacial area concentration, the
wall boiling source term in the interfacial area transport equation was improved with
taking into account the bubble sliding and lift-off phenomena on the wall. The
interfacial area transport equation with the developed wall nucleation model has been
implemented in the multi-dimensional two-phase flow analysis code, EAGLE. The code
adopted the two-fluid model and SMAC algorithm was extended to be applicable to the
two-phase flow with a phase change.
From a comparison with SUBO tests and the subcooled boiling experiments performed
in SNU, EAGLE analysis with the mechanistic model of bubble lift-off diameter model
and lift-off frequency reduction factor was confirmed to predict the experimental data
reasonably, rather than the conventional bubble lift-off diameter model. Since the
developed bubble lift-off model considered the actual force balance of bubble and the
effect of coalescences among the sliding bubbles, it reasonably improved the estimation
of the evaporative heat flux and the wall nucleation source term in the interfacial area
transport equation. On the other hand, the inclusion of the turbulence model with
respect to boiling bubbles at the heated wall indicated a better prediction of the bubble
velocity. The increased turbulence of liquid phase by the nucleation bubbles was
modeled in EAGLE code and it is ascertained that the modeling showed a better
prediction of the bubble velocity near the heated wall.
In conclusion, the development of the EAGLE code with the mechanistic model of
interfacial area transport equation will enhance the analysis capability of a multi-
dimensional two-phase flow in the subcooled boiling. As a further improvement
following this work, development of a two-group interfacial area transport equation for
79
a boiling flow is essential to cover the flow regime of a bubbly-to-slug transition flow or
a slug flow.
80
Nomenclatures
ai Interfacial area concentration [1/m]
Cp Heat capacity [J/kgK]
Dd Bubble departure diameter [m]
Dh Hydraulic diameter [m]
Dlo Bubble lift-off diameter [m]
Dsm Sauter-mean diameter [m]
f Bubble departure frequency [1/s]
g Gravitational acceleration [m/s2]
G Mass flux [kg/m2s]
H Enthalpy [J/kg]
Hfg Latent heat [J/kg]
k Thermal conductivity [W/mK]
L Length [m]
N” Active nucleation site density [1/m2]
n Normal vector at wall
Nu Nusselt number
Pr Prandtl number
q” Heat flux [W/m2]
R Lift-off frequency reduction factor
Re Reynolds number
rd Bubble radius [m]
St Stanton number
T Temperature [K]
u Velocity [m/s]
ybw Distance from wall [m]
Greek Letters
α Phase fraction
81
β Thermal expansion coefficient [1/K]
Γ Phase change rate [kg/m3s]
θ Bubble contact angle at wall
φ Superficial velocity [m/s]
µ Viscosity [Ns/m2]
ρ Density [kg/m3]
σ Surface tension [N/m]
τ Shear stress [N/m2]
Φ Bulk source term of energy [J/m3s]
Subscripts
D Drag force
e Evaporation
f Liquid phase
g Gas phase
i Interphase
L Lift force
lo Bubble lift-off
m Mixture
R Relative motion
s Saturated
vm Virtual mass
w Wall
82
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서 지 정 보 양 식
수행기관보고서번호 위탁기관보고서번호 표준보고서번호 INIS 주제코드
KAERI/TR-3679/2008
제 목 / 부제 계면면적 수송방정식 모델을 통한 미포화 비등 이상유동해석코드 개발
연구책임자 및 부서명 (주저자)
배병언 (열수력안전연구부)
연구자 및 부서명 윤병조, 어동진, 송철화 (열수력안전연구부)
윤한영 (중소형원자로기술개발부)
출 판 지 대전 발행기관 한국원자력연구원 발행년 2008
페 이 지 98 p. 도 표 있음( ○ ), 없음( ) 크 기 210x296cm
참고사항
비밀여부 공개( ○ ), 대외비( ),
-__ 급비밀 보고서종류 기술보고서
연구수행기관 계약 번호
초 록
본 연구에서는 계면면적 수송방정식을 이용한 이상유동 해석을 목표로, 미포화 비등 시 벽면에서 발생하는 기포에 대한 Sliding 및 Lift-off 를 현상학적으로 고려하여 계면면적 수송방정식의 생성항 모델을 개발하였다. 이를 다차원 전산유체해석 코드 EAGLE(Elaborated Analysis of Gas-Liquid Evolution)에 삽입하여 SUBO 실험 및 SNU 미포화 비등 유동에 대한 검증을 수행하였다. 환형 수직 유로 내의 미포화 비등 실험에 대한 해석을 수행한 결과, 본 연구에서 개발된 모델 및 코드가 기포가 생성 및 전파되는 거동을 적절하게 모사함을 확인하였다. 추후 난류 모델 및 계면 운동량 전달 모델 개선 등을 통해 더 정확한 다차원적 이상유동 해석 및 계면면적 수송방정식에 대한 평가가 가능할 것이다.
주제명키워드 (10단어내외)
미포화 비등, 이상유동, 계면면적 수송방정식, 전산유체해석코드, 기포 Lift-off 모델, EAGLE 코드
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Report No.
Sponsoring Org.
Report No. Standard Report No. INIS Subject Code
KAERI/TR-3679/2008
Title / Subtitle Development of CFD Code for Subcooled Boiling Two-Phase Flow with
Modeling the Interfacial Area Transport Equation
Project Manager
and Department
(Main author)
Byoung-Uhn Bae (Thermal Hydraulic Safety Research Division)
Researcher and
Department
Byong-Jo Yun, Dong-Jin Euh, Chul-Hwa Song (Thermal Hydraulic Safety
Research Division) Han-Young Yoon (Fluid System Engineering Division)
Publication
Place Daejeon Publisher KAERI Publication Date 2008
Page 98 p. Ill. & Tab. Yes ( ○ ), No ( ) Size 210x296cm
Note
Classified Open( ○ ), Restricted( ),
-___ Class Document Report Type Technical report
Performing Org. Contract No.
Abstract
The interfacial area transport equation for the subcooled boiling flow was developed with a
mechanistic model for the wall boiling source term. It included the bubble lift-off diameter model and
lift-off frequency reduction factor model. To implement the model, the two-phase flow CFD code was
developed, which was named as EAGLE (Elaborated Analysis of Gas-Liquid Evolution). The
developed model and EAGLE code was validated the experimental data of SUBO and SNU facilities.
The computational analysis revealed that the interfacial area transport equation with the bubble lift-off
diameter model agreed well with the experimental results. It presents that the source term for the wall
nucleation enhanced the prediction capability for a multi-dimensional behavior of void fraction or
interfacial area concentration.
Subject Keywords
(About 10 words)
Subcooled boiling, Two-phase flow, Interfacial area transport equation, CFD code, Bubble lift-off model, EAGLE code