A THREE-DIMENSIONAL TWO-FLUID MODELING OF STRATIFIED FLOW WITH CONDENSATION FOR PRESSURIZED THERMAL SHOCK INVESTIGATIONS
THERMAL HYDRAULICS KEYWORDS: condensation, stratified flow, 3-D two-fluid model
W. YAO, D. BESTION,* and P. COSTE CEA Grenoble DEN0DTP0SMTH0LMDS, 17 rue des Martyrs, 38054 Grenoble, Cédex 9, France M. BOUCKER EDF-DRD Département Mécanique des Fluides et Transferts Thermiques 6, Quai Watier 75400 Chatou Cedex, France Received August 3, 2004 Accepted for Publication January 13, 2005
A three-dimensional (3-D) two-fluid model for a turbulent stratified flow with and without condensation is presented, in view of investigating pressurized thermal shock (PTS) scenarios when a stratified two-phase flow takes place in the cold legs of a pressurized water reactor. A modified turbulent K-« model is proposed with turbulence production induced by interfacial friction. A model of interfacial friction based on an interfacial sublayer concept and three interfacial heat transfer models— namely, a model based on the small eddies–controlled surface renewal concept, a model based on the asymptotic behavior of the eddy viscosity, and a model based
on the interfacial sublayer concept—are implemented into a preliminary version of the NEPTUNE code based on the 3-D module of the CATHARE code. As a first step, the models are evaluated by comparison of calculated profiles of velocity, turbulent kinetic energy, and turbulent shear stress with data in a turbulent air-water stratified flow in a rectangular channel and with data for a water jet impacting the free surface of a water pool. Then, a turbulent steam-water stratified flow with condensation is calculated, and some first conclusions are drawn on the interfacial heat transfer modeling and on the applicability of the model to PTS investigations.
I. INTRODUCTION
or totally uncovered cold leg where the main heat source to the liquid is due to steam condensation in the cold leg and in the top of the downcomer. Condensation is mainly dependent on the interfacial structure and on the turbulent mixing in the liquid phase. There might be significant effects on the transient, particularly when it gives rise to condensation-driven instability.1 Such a situation is not investigated here; this paper focuses on slow transients following a small-break LOCA with the rather simple interfacial structure of a stratified flow in the cold leg. Previous analyses of PTS by Theofanous et al.2,3 do not apply to our case since they refer to other scenarios with pure single-phase liquid flows in the cold legs, where the mixing of the cold plume with hot water is the main process to model. Also, the rapid progress of computer power will allow using fine-resolution computational fluid dynamics tools for PTS investigations in the near future.
This paper presents a modeling work performed in the frame of the NEPTUNE code development. NEPTUNE is a common project of Electricité de France ~EDF! and Commissariat à l’Energie Atomique ~CEA! also supported by Framatome-ANP and the Institute for Radiological Protection and Nuclear Safety ~IRSN! for modeling two-phase flow in water-cooled nuclear reactors. Pressurized thermal shock ~PTS! may occur in a pressurized water reactor ~PWR! when overcooling of the pressure vessel wall in the case of irradiation-induced loss of ductility may lead to failure. Small-break loss-ofcoolant accident ~LOCA! scenarios exist with an emergency core cooling system ~ECCS! injection in a partially *E-mail:
[email protected] NUCLEAR TECHNOLOGY
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This paper is a first step in an attempt to model heat transfers in a cold leg with a stratified flow using a threedimensional ~3-D! two-fluid model. The main objective is to predict the liquid temperature field, which depends mainly on interfacial heat and mass transfer related to direct-contact condensation of steam on a subcooled liquid and on the turbulence diffusion within the liquid. Many research works support that turbulence behavior near the interface plays a dominant role for the interfacial transfers. For ECCS injection cases, the turbulence mainly comes from the impact effect of a water jet 1 and shear ~at the wall and at the gas-liquid interface!. Thus, as a first step to simulate such scenarios, separate effects in simple geometry are investigated, i.e., interfacial friction and turbulence production, interfacial heat transfer, and turbulence in a water pool induced by a water jet, to establish and validate the developed models in this paper. Only steady-state tests are considered here and no heat transfer to the wall is considered. For interfacial transfers, though there are extensive studies on gas absorption at a gas-liquid interface in chemical engineering and on condensation on a thin liquid film that is slightly subcooled,4,5 the modeling of interfacial transfer at a stratified interface is still open to question because of the theoretical and experimental difficulties concerning a deformable interface. Recently, special attention was paid to the safety analysis of nuclear reactors in which rapid direct-contact condensation of vapor on a thick subcooled liquid layer may occur during a sequence of safety injections of cold water.6–10 After the thin film model introduced by Lewis and Whitman,11 many theoretical models for interfacial transfers are proposed that can be mainly divided into two classes of models, one based on a surface renewal concept 12 and another based on the eddy diffusivity concept.13 The two approaches deal with the same problems from different aspects. For example, the first one is a microscale phenomenological analysis that focuses on the dynamic behavior of a typical eddy element, and the second one is a macroscale Reynolds average analysis that focuses on the average effect of all the eddies. For the first class of models, one should point out that until now there are still conflicting opinions about which kind of eddies renew the surface. Based on their experimental analysis, some researchers 14–16 argued that large energy-containing eddies generated from the bottom wall renew the gas-liquid interface and control interfacial heat0mass transfer. On the contrary, others 7,17 believed that small eddies of Kolmogorov microscale control the transport across an interface. Furthermore, when the models based on the small renewal eddies concept are attempted to apply to the interfacial heat transfer problems in a steam-water system 6,7 with Pr '1, it should be noted that this kind of model 18 lies in the fact that the penetration depth d during the contact time should be less than the eddy size l e , i.e., 130
d ⬇ 4Pr ⫺102 ⬍⬍ 1 le
or
Pr .. 16 .
~1!
Although it can be used in gas absorption problems with very large Sc numbers, the direct application of these models to steam-water flow is questionable in theory. For the second class of the models based on the eddy viscosity concept, the asymptotic behavior of eddy viscosity at an interface is considered because the transfer resistance mainly lies on a thin layer near an interface, especially for large Sc or Pr number fluids. Ueda et al.19 surveyed several published eddy viscosity profiles that have asymptotic characteristics at an interface as nt ⫽ Cysn , where C depends on the physical properties of fluid and flow condition near an interface. The power n appears to be from 1 to 4, according to different authors, and Ueda et al.19 concluded that the eddy viscosity decreases to zero in proportion to the distance from the interface with the power of more than 2 by checking several gas absorption experimental data sets. Recently, direct numerical simulation ~DNS! results 20,21 showed that there is a nonzero value of eddy viscosity nei at a free surface ~no interfacial shear!. This value is very small and comparable to that of molecular viscosity n. Unlike a solid-fluid interface, the asymptotic behavior of a gasliquid interface remains unclear because of the measurement difficulties. Thus, the interfacial transfer models based on the eddy viscosity concept inherit these uncertainties ~such as uncertainties on the coefficients C and n!. Another approach is to use the concept of “wall function.” Wall function has successfully been applied to the momentum, heat, and mass transfer problems at a fluidsolid interface. Experimental and DNS evidence have shown that near a gas-liquid interface, a similar sublayer exists and the wall function may be useful to predict the interfacial heat transfer problems for engineering application. Turbulence, especially at an interface, plays a dominant role for the interfacial transfers in a turbulent flow. Thus, the key issue of the multidimensional prediction is to produce good estimation of turbulence near an interface. For a stratified flow, three different sources, say, turbulence diffused from wall boundaries, turbulence production by the interfacial friction, and turbulence induced by interfacial waves, should be taken into account. The problem is further complicated if the interaction between turbulence and interfacial waves becomes important. In this paper, a local 3-D two-fluid model for a turbulent stratified flow with0without condensation is presented. A modified turbulent K-« model is proposed with turbulence production induced by interfacial friction. A model of interfacial friction based on an interfacial sublayer concept and three interfacial heat transfer models— namely, a model based on the small eddies–controlled surface renewal concept 7 ; a model based on the asymptotic behavior of the eddy viscosity, called the eddy NUCLEAR TECHNOLOGY
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viscosity model ~EVM!; and a model based on the interfacial sublayer concept, called the interfacial sublayer model ~ISM!—are implemented into a preliminary version of the NEPTUNE code based on the 3-D module of the CATHARE code. First, an experiment with adiabatic turbulent air-water stratified flow is calculated to evaluate the models that control the velocity and turbulence profiles. Then an experiment of turbulent steam-water stratified flow with condensation is applied to compare the three aforementioned interfacial heat transfer models. The capability of the K-« model to predict turbulence in a water pool induced by a water jet is also evaluated in comparison with experimental results.
II. A 3-D TWO-FLUID MODEL FOR A TURBULENT STRATIFIED FLOW A transient 3-D two-fluid model is used for predicting local flow parameters of a stratified flow in a horizontal channel with or without condensation at the free surface. Following the work of Ishii,22 a time averaging ~or ensemble averaging! is applied to basic local instantaneous mass momentum and energy equations multiplied by phase characteristic functions. This averaging allows us to filter all turbulent fluctuations in a steadystate or slow transient flow. Any point of a stratified flow is either single-phase gas, single-phase liquid, or two phases in the free surface region with possible interfacial waves, and our purpose is to use a single set of equations for all cases. Such equations in the two-phase region of the interface are also assumed to filter all interfacial wave phenomena, and the resulting void fraction may allow us to predict the average free surface location. The
single-phase gas region and single-phase liquid region can then be treated as single-phase flows with a moving boundary at the free surface and solid boundaries along the walls. Although the liquid phase ~respectively, gas phase! balance equations are also solved in the pure gas region ~respectively, pure liquid region! their predicted flow parameters have no physical meaning and must not be used in the liquid region ~respectively, gas region!. Only the two-phase region equations are coupled with both single-phase regions and contain interfacial transfers between phases. In practice, after identification of the free surface location, the two-phase region is assumed to be smaller in thickness than the computing cell size. This makes the physics somewhat dependent on numerical features, but it was considered as reasonably applicable when the predicted interface region is not smeared on several computational cells. For modeling the turbulent stresses in momentum equations and turbulent heat transfers in energy equations, the rather simple K-« model is selected to be first evaluated in such flow conditions. In single-phase regions, the classical formulation of single-phase K-« equations is used with constants as recommended by Schiestel.23 In the two-phase region, additional terms due to interface have to be modeled. Following Morel 24 and Yao and Morel,25 the following set of equations is obtained: II.A. Balance Equations II.A.1. Mass Balance Equations ] ~ak rk ! ⫹ ¹{~ak rk V1 k ! ⫽ Gki , ]t
~2!
where Gki is the volumetric production rate of phase k. II.A.2. Momentum Balance Equations ak rk
冋
册 再 冋
册冎
2 ]V1 k a ⫹ ~ V1 k{¹V1 k ! ⫽ ¹{ ak rk ntk ~¹V1 k ⫹ ¹ T V1 k ! ⫺ ~Kk ⫹ ntk¹{V1 k ! I ]t 3
⫺ ak¹P ⫹ tki a i ⫹ ak rk g5 ,
~3!
where tki ⫽ interfacial friction per unit area a i ⫽ volumetric interfacial area. The eddy viscosity ntk and the turbulent kinetic energy Kk are calculated by a K-« model presented later in this paper. The molecular diffusion terms are here neglected. The turbulent stress tensor of phase k is assumed to depend only on local strain of phase k even in the two-phase region and independent of phase change. II.A.3. Energy Balance Equations
冋冉
rk ntk ] ~ak rk ek ! ⫹ ¹{~ak rk ek V1 k ! ⫽ ¹{ ak l k¹Tk ⫹ ¹ek ]t Prt NUCLEAR TECHNOLOGY
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冊册 冋 ⫺P
册
]ak ⫹ ¹{~ak V1 k ! ⫹ Gki Hki ⫹ qki a i , ]t
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where qki is the interfacial heat transfer per unit area. The turbulent Prandtl number Prt is taken equal to 0.9 as recommended 23 for single-phase flows. II.B. K-« Turbulence Equations
] ~ak rk Kk ! ⫹ ¹{~ak rk Kk V1 k ! ]t
冉
⫽ ¹{ ak rk
II.C. Local Closure Models at an Interface and Interfacial Boundary Condition Realization
冊
ntk a ¹Kk ⫺ ak rk «k ⫹ ak tkRe : ¹V1 k PrtK
⫹ Kki Gki ⫹ ak PKi
II.C.1. Interfacial Friction (ISM) ~5!
and ] ~ak rk «k ! ⫹ ¹{~ak rk «k V1 k ! ]t
冉
⫽ ¹{ ak rk ⫹ C«1
冊
ntk «k2 ¹«k ⫺ C«2 ak rk Prt« Kk
«k a ak tkRe : ¹V1 k Kk
2 ⫺ ak rk «k¹{V1 k ⫹ «ki Gki ⫹ ak P«i , 3
^ tkRe ⫽
⫺rku i' u j'k
~6!
«k
u G* ⫽
1 ln yG⫹ ⫹ B , k
冪
ti , rG
y ⫹ ⱖ yv⫹ ,
~7!
^ u⫹ G ⫽
uG ⫺ ui , u G*
^ yG⫹ ⫽
u G* yG nG
and yL ⫽
~1 ⫺ a!DY , 2
yG ⫽
aDY . 2
~8!
,
where PKi and P«i are terms for the turbulence production induced by interfacial friction; they will be discussed later. In Eqs. ~5! and ~6!, the default values 23 are PrtK ⫽ 1, Prt« ⫽ 1.3, C«1 ⫽ 1.44, C«2 ⫽ 1.92, and Cm ⫽ 0.09. Classical jump conditions at the interface are used for mass momentum and energy. Single-phase wall functions 23 are used for solid boundaries in momentum, energy, and turbulence equations. The principal variables of the set of equations are the three components of the phase k velocities Vk ~k ⫽ g, l !, the phase k internal energies ek ~k ⫽ g, l !, the pressure, the void fraction ag , the phase k turbulent kinetic energies Kk ~k ⫽ g, l !, and the phase k turbulent dissipations «k ~k ⫽ g, l !. Models for interfacial transfers are still required for closing the system of equations. The inter132
y ⫹ ⱕ yv⫹
where
⫽ rk ntk ~¹V1 k ⫹ ¹ V1 k !
and ^ Cm ntk ⫽
u⫹ G ⫽
T
2 a ⫺ rk ~Kk ⫹ ntk¹{V1 k ! I 3 Kk2
In the two-phase region of the free surface, the average location of the surface is determined and this interface is treated as a moving boundary for the gas region and the liquid region. The wall function method is applied as it is for solid boundaries. The effects of the surface roughness due to interfacial waves are not taken into account in this first approach. Because of the significant difference in density of the liquid and gas phases, the gas phase “sees” the liquid phase like a moving solid wall ~see Fig. 1!. Thus, u i ⫽ u L is a good approximation. For the gas region near an interface, a two-sublayer model similar to the wall function is adopted. ⫹ u⫹ G ⫽ yG ,
where the Reynolds stress tensor and the turbulent eddy viscosity are given by a
facial friction tki must be modeled. Gki is related to interfacial heat transfers qki , which must be modeled. The interfacial production terms PKi and P«i in K and « equations have to be modeled. Boundary conditions at the interface have to be given. The interfacial area a i has to be calculated. These necessary closure relations are presented hereafter.
Fig. 1. A mesh with an interface. NUCLEAR TECHNOLOGY
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The values of k ⫽ 0.41, B ⫽ 4.9, and yv⫹ ⫽ 10.67 are adopted here. II.C.2. Interfacial Heat Transfer In a turbulent stratified flow, the interfacial heat transfer is controlled by turbulence in the vicinity of an interface. One part of the turbulence near the interface is diffused through the bulk flow from the wall region, and another part is due to the turbulence production at the interface by the interfacial friction and interfacial waves. In this paper, we propose to evaluate two models. One is based on the ISM, which is a direct extension of the wall sublayer model, and another one is based on the asymptotic behavior of the EVM. In addition, a model based on the surface renewal concept with small eddies,7 the Hughes and Duffey model ~HDM! is also evaluated. II.C.2.a. A Model Based ISM. The wall function in transfer problems at a fluid-solid interface has been successfully and widely applied to solve the engineering heat transfer problems. Experimental and DNS evidence showed that, like a fluid-solid boundary, similar sublayers exist at a gas-liquid interface. It seems possible to introduce an “interfacial function,” which is similar to a wall function, to the transfer problems at a gas-liquid interface. The temperature profile 23 at a wall boundary layer is used to calculate the interfacial heat transfer as follows: T ⫹ ⫽ Pr y ⫹ ,
y ⫹ ⱖ yT⫹ .
Jayatilleke 26 proposed the following: P ⫽ 9.24
冋冉 冊 册冋 Pr Prt
304
⫺1
~9!
冉
Pr 1 ⫹ 0.28 exp ⫺0.007 Prt
冊册
.
In Eqs. ~9! and ~10!, u ⫺ ui , u*
y⫹ ⫽
u * ⫽ Cm104 K 102 ,
yu * , n
T* ⫽
T⫹ ⫽
Ti ⫺ T , T*
qi . rCp u *
~11!
Accordingly, we have rCp u ~Ti ⫺ T ! *
qi ⫽
T⫹
.
~12!
Note that u ⫹ is calculated by Eq. ~7!, and yT⫹ is determined by Pr yT⫹ ⫽ Prt ~u ⫹ ⫹ P !. One should mention that u * ⫽ !t0r is used in the original model, while u * ⫽ Cm104 K 102 is adopted here using turbulent kinetic energy for estimating the friction velocity. This modification, together with a turbulence NUCLEAR TECHNOLOGY
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冉 冊册
冋
nt ⫽ nt` 1 ⫺ exp ⫺C
ys2 L2t
,
where ys ⫽ distance to the interface L t ⫽ length scale of the energy-containing eddies. If it is further assumed that when ys ⫽ L t , nt ⫽ 0.99nt` , we have, C ⫽ 4.6. Furthermore, it shows that near a free surface there is no significant variation of turbulent kinetic energy and turbulence dissipation rate. Thus, K` ⬇ Ki ,
«` ⬇ «i ,
nt` ⫽ Cm
冒冕
`
qi ⫽ ~Ti ⫺ T !
0
~10!
^ u⫹ ⫽
II.C.2.b. A Model Based on the Asymptotic Behavior of EVM. Shen 20 investigated asymptotic behavior of the turbulence viscosity by DNS method and found that the turbulent eddy viscosity in a boundary layer fits well with the Gaussian function. In fact, because of the boundary constraint, when the energy-containing eddies move from bulk flow to the free surface, the fluctuation component of vertical velocity ~and accordingly eddy viscosity! starts to decrease. It is assumed that
K`2 K2 ⬇ Cm i . «` «i
If Cm ⫽ 0.09 and Prt ⫽ 0.9, we have
y ⫹ ⱕ yT⫹
T ⫹ ⫽ Prt ~u ⫹ ⫹ P ! ,
production term as is proposed later in this paper, can take into account two effects: turbulence from the wall and turbulence production by interfacial shear.
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dYs l ⫹ rCp nt 0Prt
冉 冊
« 102 ~Ti ⫺ T ! . ~13! K II.C.2.c. A Model Based on Surface Renewal Concept with Small Eddies (HDM). The simplest surface renewal model, which studies the transient transport on stagnant eddies without internal velocity gradient, is the penetration theory proposed by Higbie.12 Based on the same idea, Banerjee 27 gave the following interfacial heat transfer relation: ⬇ 0.794rCpL!a
qi ⫽
冪
2 a rCp ~Ti ⫺ T ! . !p t
~14!
Hughes and Duffey 7 adopted the above correlation and determined the renewal period in Eq. ~14! by the Kolmogorov time scale of small eddies. Accordingly, qi ⫽
冉冊
2 « rC !a !p p n
104
~Ti ⫺ T ! .
~15!
We notice that Bankoff ’s 6 model for small eddies, which gives 102 , Nut ⫽ 0.25Re 304 t Pr
~16! 133
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can be written as qi ⫽ 0.25rCp!a
冉冊 « n
104
~Ti ⫺ T ! .
~17!
It has the same form but with different coefficients compared with the HDM ~Ref. 7!. The coefficient of Bankoff’s model 6 is from the experimental data of gas absorption,18 while the HDM ~Ref. 7! is a theoretical one. As we have previously mentioned, Eq. ~1! should be satisfied if the HDM is adopted. In a steam-water system with Pr ' 1, the application of Eq. ~15! is questionable. In addition, the thermal property of subcooled liquid strongly depends on the temperature ~the Pr number may decrease eight times from 208C to saturation temperature under 5 MPa conditions!. Considering that the interfacial transfers are determined by the local fluid properties at the interface, it is suggested that the saturated properties instead of bulk properties should be used.28
Several attempts have been made to predict the turbulence distribution in a stratified two-phase flow by K-« equations. Akai et al.29 used low Reynolds K-« equations to predict the turbulence distribution in a stratified gasliquid flow in a rectangular channel, and Issa 30 adopted the same method for a circular channel. At a smooth interface, the following boundary conditions are adopted: KLi ⫽ KGi ⫽ 0
«Li ⫽ 2
1 Re 2
~18!
冉! 冊 冉! 冊 ] KL ]y
nG 1 «Gi ⫽ 2 nL Re 2
«i ⫽
] ]z
,
134
yv
nt sk
ruv
du dy , dy
~22!
We assume that the velocity profile near an interface obeys the same law as a wall function with respect to relative velocity. In the viscous sublayer there should be no turbulence production, and in the inertial sublayer the Reynolds stress is assumed to be the same as interfacial friction, ti ⬇ ⫺ruv .
~23!
Accordingly, turbulence production by interfacial friction is as follows:
冉 冊
yp ti u * max 0, ln yp k yv
2
.
~19!
]K ]y
,
P«i ⫽ C«1
.
~24!
KGi ⫽
~21!
« PKi . K
~25!
As for a gas phase, the gas sees the interface almost like a moving solid wall, as previously mentioned. We set the turbulence value near an interface by the following equations, which are widely applied at a wall boundary:
~20!
i
where Ka is the turbulence kinetic energy at the centerline between two plates of a single-phase flow. In fact, many experiments showed that the turbulence kinetic energy at an interface is not zero even for a free surface ~zero-shear interface! because there is no constraint for the horizontal velocity components in this region.31 Asymptotic analysis near a free surface showed 32 that the Neuman conditions should be satisfied, ]« ]K ⫽ ⫽0 , ]y ]y
yp
The corresponding term in the turbulence dissipation balance equation is given by
冋冉 冊 册 n⫹
冕
yp ⫽ distance to the interface
2
] KL ]y
1 yp
where
PKi ⫽
On the contrary, Murata et al.8 suggested nonzero values at the interface, Ki ⫽ 0.8Ka ,
PK ⫽ ⫺
yv ⫽ viscous sublayer thickness.
II.C.3. Turbulence Production Induced by Interfacial Friction
and
which means that no turbulence flux is transferred through an interface. Generally speaking, turbulence at an interface on the liquid side is partially transported from bulk flow, and partially produced by interfacial friction and interfacial waves. It cannot be predetermined as the previously mentioned authors have suggested. In this work, zero diffusion flux conditions at an interface are adopted and the turbulence production induced by the interfacial friction is taken into account by a source term in K-« equations. The detailed derivation is as follows: For the liquid phase, near an interface, the turbulence production in K equation can be calculated by
*2 u Gi
!C
,
«Gi ⫽
m
*3 u Gi . ky
~26!
II.C.4. Interface Identification Criterion As previously mentioned, the position of the free surface has to be determined in this approach. A very simple method was used in these applications with a clear stratified geometry and a Cartesian structured meshing. Computing grids that contain a stratified interface were identified by the following criterion: In a vertical raw of meshes, the interface is in the cell with a void fraction value that is nearest to the value of 0.5, that is, NUCLEAR TECHNOLOGY
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6 a~iz interface ! ⫺ 0.56 ⫽ min~6 a~iz! ⫺ 0.56! , where iz ⫽ 1 . . . nz .
~27!
In practice, in our simple applications, the values of the void fraction in a vertical raw of meshes are zero or one, with only one cell having a value between zero and one. However, this criterion gives a position of the interface even if the two-phase region is extended over two or more meshes. The method has to be further extended when the flow is more complex or when more complex body-fitted meshing is used. II.C.5. Volumetric Interfacial Area In the two-phase region, the volumetric interfacial area has to be calculated to express the interfacial transfers of momentum and heat. At a first step, we assume a horizontal flat interface ~see Fig. 2! in the region of interest of our applications. In these applications, with a clear stratified geometry and when a Cartesian structured meshing is used, the volumetric interfacial area in a cell containing a stratified interface is simply calculated by ai ⫽
1 DX ⫽ . DZDX DZ
facial area will have to be determined in a more general way when applying the model to more complex cases for PTS investigations. In single-phase gas and single-phase liquid domains, a i is set to zero. II.C.6. Interfacial Boundary Conditions For a grid containing a stratified interface, the diffusion terms in momentum and energy are replaced by interfacial transfer terms, which may be seen as boundary conditions for each phase, using extended wall functions. In K-« equations, diffusion terms are set to zero to satisfy the nondiffusion conditions across the interface.32 II.D. Numerical Method Our numerical calculations have been performed by using the 3-D module of the CATHARE code.33 This module is based on a finite volume discretization scheme using staggered grids and the donor cell principle. A semi-implicit scheme 33 is used for time discretization, which may be seen as an extension of the Implicit Continuous fluid Eulerian method of Harlow and Amsden.34
~28!
This is not a general method at all since it only applies to a horizontal flat interface modeled with a Cartesian meshing, but it allowed us to evaluate interfacial transfers in our simple cases. The possible presence of interfacial waves is not taken into account but, according to the averaging of the equation, only the average interface position is calculated and the effects of waves could be later modeled as surface roughness effects like in wall friction ~or wall heat transfer! models. Both the interface identification criterion and the evaluation of the inter-
III. CALCULATION RESULTS AND DISCUSSION III.A. Prediction of Turbulence Distribution in Air-Water Stratified Flow In Sec. III.A, the experimental data of Fabre et al.,31 for a stratified air-water cocurrent flow in a horizontal channel of rectangular cross section ~0.1 m high, 0.2 m wide, and 12.6 m long!, are used to evaluate prediction capabilities of the local 3-D two-fluid model. In this experiment, systematic measurements of the components of the mean velocities and Reynolds stresses were performed under carefully controlled inlet conditions. The measured data at a test section that is 9.1 m from the inlet are used in this paper for comparison. The experimental conditions of the selected test cases are listed in Table I.
TABLE I Experimental Conditions of Fabre et al.* Run Number
Bulk velocity of water ~m0s! Bulk velocity of air ~m0s! Mean water depth ~cm! Fig. 2. A cell with a horizontal flat interface. NUCLEAR TECHNOLOGY
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T000
T250
T400
0.387 0.00 3.88
0.395 3.66 3.80
0.476 5.50 3.15
*Ref. 31. 135
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Fig. 3. Grid scheme for the 2-D calculation of air-water cocurrent flow tests.
In our calculation, considering that the width of the flow duct is comparatively large, a 2-D simplifying assumption is adopted here and a nodalization with ~see Fig. 3! X * Z ⫽50 * 20 meshes ~Z is the vertical direction! is adopted with 10 cells for water and steam layers, respectively. For the sake of computing convenience, the air-water is replaced by a steam-water flow with
zero interfacial heat transfer in the same pressure and flow rate conditions. The computing physical time is 195 s to reach a steady-state flow condition. The upstream boundary conditions are flat profiles of gas and liquid velocities and some values for K and «, which have no more effect at the section X ⫽ 9.1 m. The whole length of the test section is not modeled and the downstream boundary conditions are simplified by a constant pressure imposed at a downward-facing exit. The free surface level and the gas and liquid flow rates being imposed at the inlet section, which was found sufficient in these test conditions to predict reasonably well the free surface level at the section X ⫽ 9.1 m. Figures 4, 5, and 6 present the calculated profiles of mean longitudinal velocity, turbulent kinetic energy, and turbulent shear stress for three test cases ~T250, T000, and T400, respectively!. They show that for the case of zero ~T000! and medium gas velocity ~T250!, the calculations predicted reasonably well the liquid velocity, turbulent kinetic intensity, and shear profiles in the liquid layer, compared to the experimental data, though with a little underestimation of turbulent kinetic energy and
Fig. 4. Predicted and experimental profiles in an air-water stratified flow ~T250!: ~a! gas velocity profile, ~b! liquid velocity profile, ~c! turbulent kinetic energy profile in the liquid, and ~d! turbulent shear stress profile in the liquid. 136
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Fig. 5. Predicted and experimental profiles in an air-water stratified flow 31 ~T000!: ~a! liquid velocity profile, ~b! liquid turbulent kinetic energy profile, and ~c! liquid turbulent shear stress profile.
Fig. 6. Predicted and experimental profiles in an air-water stratified flow 31 ~T400!: ~a! liquid turbulent kinetic energy profile and ~b! liquid turbulent shear stress profile.
turbulent shear stress. The gas velocity in the gas layer is less accurately predicted than the liquid velocity ~T250! mainly because of a slight underestimation of the free surface level. NUCLEAR TECHNOLOGY
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On the other hand, for the case of high gas velocity ~T400! in which an important mean secondary flow is found in the experiment, and interfacial waves may also be significant, the present model in which these effects 137
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TABLE II
model has proven some capability to capture the momentum transfers and turbulence characteristics, at least with a rather smooth interface and without heat transfers.
Experimental Conditions of Lim et al.* Run Number 1
2
6
Mass flow rate of steam ~kg0s! 0.041 0.065 0.065 Steam temperature ~8C! 111 116 116 Mass flow rate of water ~kg0s! 0.657 0.657 1.44 Water temperature ~8C! 25 25 25 *Ref. 35.
are not taken into account fails to predict the vertical variation of liquid velocity, turbulence kinetic energy, and shear stress in the liquid layer along the vertical direction. The present calculations are not fully converged in space, and sensitivity tests suggest that the converged solution might have up to a few percent difference for turbulent kinetic energy and shear stress. However, the
III.B. Prediction of Condensation in Turbulent Steam-Water Stratified Flow Lim et al.35 presented their experimental results on interfacial condensation for a turbulent stratified steamwater flow in a horizontal channel of rectangular cross section ~0.0635 m high, 0.3048 m wide, and 1.601 m long!. There are five test sections that locate at x ⫽ 0.15, 0.30, 0.59, 0.87, and 1.23 m from the channel inlet. In each test section, pitot tubes are used to measure the local mean steam velocity, and the conductivity probes are used to measure the water height. The selected test cases with experimental conditions are listed in Table II. Observations found that ~a! test 1 was with a glossy-looking interface; ~b! at the entrance region of test 2, it was a glossy interface, while around x ⫽0.4 m it turns to a wavy interface; and ~c! test 6 was with a wavy interface. Experimental results showed a significant increment of condensation if the transition from a glossy to a wavy interface occurred.
Fig. 7. Axial evolution of steam mass flow rate in a steam-water stratified flow, with experimental data versus calculations: ~a! test 1, ~b! test 2, and ~c! test 6. 138
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The same type of grid scheme as in Fig. 3 is adopted here for the numerical simulation. The computing physical time is 195 s with inlet liquid temperature decreasing linearly from saturated temperature to experimental value within 60 s. In the absence of information about the inlet velocity and temperature profiles, flat profiles were used as boundary conditions, and the inlet turbulent conditions of Lim et al.,35 KL ⫽ 10⫺3 and EPSL ⫽ 10⫺3, were adopted for the calculation. Mesh sensitivities tests show that the differences of the predicted condensation by different grid schemes are not significant compared with the model errors, though refining meshes can improve the predicted condensation. Figure 7 represents the calculation results with three interfacial heat transfer models: EVM, ISM, and HDM. The comparison to the experimental data of the condensed stratified steam-water flow showed that for the glossy interface conditions ~test 1, Fig. 7a!, ISM gave the best predicted results, while EVM underestimated and HDM significantly overestimated interfacial heat transfer. For the wavy interface ~tests 2 and 6, Figs. 7b and 7c!, ISM and EVM underestimated the condensation effects, probably because the enhancement effects of interfacial waves on the interfacial frictions and turbulence were not taken into account in these two models. In these tests, HDM highly overestimated the condensation effects. It agrees with the conclusion of a one-dimensional evaluation by the TRAC code of Zhang et al.9 It confirms that application of Eq. ~15! is questionable when Eq. ~1! is not satisfied. III.C. Turbulence Induced by Impact of a Water Jet in a Water Pool One possible reason for the strong enhancement of condensation near a water jet is the turbulence induced by a water jet when it impacts a water pool. In Sec. III.C, the experimental data of Iguchi et al.36 are used for evaluating our model. In their experiment, the water was injected through a straight circular pipe with an inner diameter dinj of 5 mm and a length L of 0.6 m vertically into a cylindrical water bath with an inner diameter R of 0.2 m and a height H of 0.39 m ~see Fig. 8!. In our calculation, the cylindrical coordinates and the cell number of 40 * 40 ~z * r! are adopted. Two turbulence inlet conditions, i.e., Kin ⫽ 0.001, «in ⫽ 0.001 and Kin ⫽ 0.046, «in ⫽ 36.7, of the water jet are used in the calculations. It appears that the predicted results of axial mean velocity on the centerline along the vertical direction ~r! are quite good compared to the experimental data ~see Fig. 9a!. The root-mean-square value of axial velocity fluctuation on the centerline along the vertical direction ~r! is either well predicted ~see Fig. 9b! or underestimated by up to 30%. The predicted radial distribution of normalized axial mean velocity at different vertical positions can be approximated by a Gaussian distribution ~see Fig. 10a!, which is suggested by NUCLEAR TECHNOLOGY
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Fig. 8. Experiment apparatus of Iguchi et al.36
McKeogh and Ervine 37 and also verified by the experimental results of Iguchi et al.36 Furthermore, the predicted value of root-mean-square values of axial velocity fluctuation at different vertical positions are either well predicted ~see Fig. 10b! or underestimated by up to 45%. In addition, the turbulence level at the inlet of the jet may slightly influence the velocity and turbulence near the surface ~and thus having different interfacial transfers! although it has nearly no influence in the deeper water region ~see Fig. 9!. Gas bubbles are entrained below the free surface by the jet, whereas this phenomenon is not predicted by the model. The nonpredicted presence of gas bubbles below the free surface, which may significantly increase the turbulence level, may be a reason for the underestimation of the turbulence intensity. In subcooled water such bubbles should rapidly collapse because of the condensation, and this phenomenon might not be so important. However, further analysis of this effect is required. Though there is some confidence in the predicting capability of the standard K-« model on the water jet impact effect, it still remains uncertain if the standard K-« can predict the turbulent parameters near the free surface ~which is very important to precisely predict the 139
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Fig. 9. Comparison of the predicted profiles on the centerline along the vertical direction with experimental data: ~a! axial mean velocity ~along Z ! and ~b! root-mean-square values of axial fluctuation velocity ~along Z !.
IV. CONCLUSIONS In this paper, a 3-D two-fluid model for turbulent stratified flows with interfacial heat and mass transfers is presented in view of modeling PTS situations when ECCS water is injected in an uncovered cold leg of a PWR. Modified turbulent K-« equations with turbulence production induced by interfacial friction are proposed. A method for subgrid interfacial friction ~for example, ISM! and three methods for subgrid interfacial heat transfer ~for example, ISM, EVM, and HDM! are evaluated by comparison of predictions with several experimental data: turbulence induced by the impact of a jet, an adiabatic turbulent air-water stratified flow, and a turbulent steamwater stratified flow with condensation. The main findings are as follows: 1. The prediction capability of the two-fluid model for stratified flow with turbulence induced by a water jet impacting on the water level is demonstrated to a certain extent by comparison with experimental data. Further analysis of interfacial wave effects and of entrained bubbles are still required.
Fig. 10. Comparison of the predicted radial distribution at different vertical positions with experimental data: ~a! radial mean velocity ~along R! and ~b! root-meansquare values of radial fluctuation velocity ~along R!.
interfacial transfers! because of the lack of experimental data near the free surface. Moreover, the effects of water jet impact on gas entrainment and waves also need to be clarified. 140
2. The three aforementioned interfacial heat transfer models are evaluated. It shows that the ISM predicted well the condensation on a smooth stratified interface. The ISM and the EVM underestimate the condensation when interfacial waves exist. The HDM highly overestimated interfacial heat transfers even on a wavy interface. An extension of the ISM to wavy interface should be considered. Further efforts are identified before concluding on the capabilities of modeling PTS scenarios with a 3-D two-fluid modeling: 1. The effect of interfacial waves on interfacial shear, interfacial heat transfer, and interfacial turbulence should NUCLEAR TECHNOLOGY
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be further investigated, especially for the cases with high gas velocity.
l ⫽ thermal conductivity
2. The effect of temperature stratification on the turbulent transfers in the liquid should be addressed.
k ⫽ Karman constant
3. The effects of bubbles entrained by the jet should be clarified. 4. The modeling of the condensation on the jet itself before entering the free surface should also be investigated. 5. The model has to be validated with data in situations with heat transfer to the walls. 6. Flow features and heat transfers in the downcomer also have to be considered for PTS applications. New experimental data with adequate local measurements of temperature and velocity fields are required for further evaluation of the modeling.
NOMENCLATURE
Cp ⫽ isobaric thermal capacity D ⫽ hydraulic diameter
⫹ ⫽ dimensionless quantity *
⫽ reference quantity
Subscripts i
⫽ interface
k ⫽ k phase L ⫽ liquid phase G ⫽ gas phase
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e ⫽ internal energy
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g5 ⫽ gravity acceleration H ⫽ enthalpy, height
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q ⫽ heat flux Re ⫽ Reynolds number
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T ⫽ temperature t
Superscripts
The authors are fully indebted to the CEA, EDF, Framatome-ANP, and the IRSN for their financial support of this work.
a i ⫽ volumetric interfacial area
t
tki ⫽ interfacial friction
ACKNOWLEDGMENTS
a ⫽ thermal diffusivity
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G ⫽ mass transfer
⫽ time ⫽ stress tensor
u ⫽ velocity
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V1 ⫽ velocity vector y ⫽ distance
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Greek Symbols
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