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Flow Dependent Turbulent Schmidt Number in Containment Atmosphere Mixing

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Flow Dependent Turbulent Schmidt Number in Containment Atmosphere Mixing Flow Dependent Turbulent Schmidt Number in Containment Atmosphere Mixing Rok Krpan, Iztok Tiselj, Ivo Kljenak Jožef Stefan Institute, Reactor Engineering Division Jamova cesta 39 SI-1000, Ljubljana, Slovenia [email protected], [email protected], [email protected] ABSTRACT In turbulence modelling, constant value of turbulent Schmidt number is usually used. However, when compared to experiments, constant value does not always provide the best results. A new model for prescribing different values of turbulent Schmidt number to different flow regions is proposed and validated against three experiments on containment mixing performed in SPARC and PANDA experimental facilities. 1 INTRODUCTION During a severe accident in a light water nuclear reactor, hydrogen combustion could threaten the integrity of the nuclear power plant containment, which could lead to release of radioactive material into the environment. The study of hydrogen distribution in the containment is thus important to predict the occurrence of regions with high local hydrogen concentrations and flammable mixture in order to effectively install hydrogen mitigation systems in containments. Various experiments are being performed to simulate atmosphere mixing occurring in containment during severe accidents and results are used to validate Computational Fluid Dynamics (CFD) codes in order to simulate phenomena in actual power plants [1]. Turbulent Schmidt number (푆푐푡) and turbulent Prandtl number (푃푟푡) are two non- dimensional numbers used in turbulence modelling to describe the turbulent transport of mass and heat, respectively. These two numbers are also connected to turbulent fluctuations and turbulent transport of momentum. In CFD calculations of the injection of fluid from a nozzle of circular cross-section into a reservoir containing stagnant fluid of similar density, constant values of 푆푐푡 and 푃푟푡 are usually used. Typically, their values are specified with comparison of calculation and experimental results, and are in the range of 0.7 to 1 [2]. However, a constant value does not always provide the best results when compared to experiments. Values of these turbulent numbers may change throughout the flow field [3] and within the boundary layer [4]. Several authors proposed different models for 푃푟푡 and 푆푐푡 values. Yimer et al. [5] proposed a parabolic curve fit for 푆푐푡. This function is based on several turbulent round free jet experiments, which showed that the 푆푐푡 increases monotonously from a value of 0.62 on the jet axis to 0.82 on the jet edge. However, a constant average value of 0.7 was recommended for use in CFD applications involving axisymmetric free-jet flows. Sturgess and McManus [6] suggested a formulation for 푆푐푡 based on the 푘 − 휀 turbulence model. One of the model constants, 퐶휇, is modelled using the ratio of turbulence kinetic energy production to its dissipation rate, instead of keeping its value constant. Keistler et al. [7] proposed a new set of transport equations for enthalpy and convection-diffusion equation for mean mass fraction including variance and dissipation rate in order to determine 푆푐푡 and 푃푟푡. However, this set of 717.1 717.2 equations involve 26 additional constants. Goldberg [8] proposed a method for calculating 푆푐푡 and 푃푟푡 values based on algebraic Reynolds stress model and Reynolds stress anisotropy. This method gives in the bulk flow, a constant 푆푐푡value of 0.7, while in the near wall region a lower 푆푐푡 mean value of 0.34 is obtained. An experiment on containment mixing performed in SPARC experimental facility is used to validate the model proposed in the present work. The interaction of vertical axisymmetric air or steam jet on a horizontal layer of helium-steam or helium-air mixture is simulated with open- source CFD code OpenFOAM. 2 LIMITED VERTICAL JET Vertical jet, injected in the opposite direction of the gravity, considered in the present work is limited in vertical flow (streamwise) direction, at first with a light gas layer and later with the ceiling of the vessel (Figure 1). Consequently, the jet at some point changes direction, flows downstream around the main jet and generates a recirculation flow. As shown in Figure 1, different flow regions can be defined using the mean flow properties: A. main rising jet, B. shear flow region, C. returning jet, D. quiescent environment. Figure 1: Absolute value of vertical velocity in a vertical jet. Let us define a Cartesian coordinate system with origin located in the axis of the injection. Figure 2 left shows vertical velocity radial (x-direction) profile. The velocity is the highest at the jet centre (in the coordinate system origin), meaning its gradient in radial direction is zero. Further from the jet origin, vertical velocity decreases towards the jet edge (gradient is negative), becomes zero on the boundary between main jet and the down flowing returning jet (in the centre of the shear flow region), gets slightly negative in the returning down flowing jet (gradient is zero again in the axis of returning jet), and then the velocity increases (gradient is positive) back to zero in the quiescent environment. Figure 2 right shows vertical velocity vertical profile in the jet’s axis. The velocity decreases (velocity gradient is negative) until the jet reaches layer consisting of light gas or ceiling, where it changes direction. Proceedings of the International Conference Nuclear Energy for New Europe, Portorož, Slovenia, September 7 ̶ 10, 2020 717.3 0.8 3.5 0.7 3 0.6 2.5 0.5 2 0.4 Vertical velocity [m/s] Vertical velocity [m/s] 1.5 0.3 1 0.2 0.1 0.5 0 0 -0.1 -0.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 Radial distance [m] Vertical distance [m] Figure 2: Vertical velocity radial profile (left) and vertical velocity vertical profile (right). In Figure 3 are shown vertical velocity gradients in radial x-direction (휕푢푧/휕푥) (Figure 3 left) and vertical (휕u푧/휕푧) direction (Figure 3 right. 휕푢푧/휕푥 is multiplied with x-coordinate sign to maintain the same sign when crossing the origin. Since the jet is axisymmetric, the gradient in y-direction 휕푢푧/휕푦 is equal to 휕푢푧/휕푥. Regarding the sign of 휕푢푧/휕푥, an area which comprises main jet, shear flow region, and part of returning jet to its axis, can be defined. With the sign of 휕u푧/휕푧 similar area can be defined with details on top of the jet. Vertical artefacts on both sides of the jet, are visual errors generated by the visualization software, while computing the gradients. 휕푢 휕푢 푧 푧 휕푥 휕푧 Figure 3: Vertical velocity radial gradient (left) and vertical velocity vertical gradient (right). 3 PHYSICAL MODEL The containment mixing experiments were simulated with the OpenFOAM CFD code. The atmosphere in the vessel was considered as a compressible mixture of ideal gases. The erosion and mixing process was modelled as a two-component single-phase flow, where single continuity, momentum and total energy equation were solved. Convection-diffusion equation was solved only for a single specie, while the other species mass fraction was calculated using ∑푖=1,2 푌푖 = 1. Proceedings of the International Conference Nuclear Energy for New Europe, Portorož, Slovenia, September 7 ̶ 10, 2020 717.4 Turbulent Schmidt (푆푐푡) and turbulent Prandtl (푃푟푡) numbers are two non-dimensional numbers defined as the ratio of momentum eddy diffusivity (휈푡) to mass eddy diffusivity (퐷푡), and as the ratio of momentum eddy diffusivity to thermal eddy diffusivity (훼푡), respectively. In other words, they describe the ratio of the rates of turbulent transport of momentum (momentum diffusion) to the turbulent transport of mass (turbulent mass diffusion) or heat (turbulent heat diffusion) [9]: 휈푡 휈푡 푆푐푡 = , 푃푟푡 = (1) 퐷푡 훼푡 푆푐푡 takes place in the convection-diffusion equation: 휕휌푌퐻푒 휕 휕 휇푡 휕푌퐻푒 + (휌푢푗푌퐻푒) = ((휌퐷 + ) ) (2) 휕푡 휕푥푗 휕푥푗 푆푐푡 휕푥푗 where 휌, 푌퐻푒, 푡, 푢푗, 퐷, 휇푡 and 푆푐푡 are density, helium mass fraction, time, velocity, diffusion coefficient and eddy viscosity, respectively. On the other hand, 푃푟푡 takes place in the total energy equation: 휕휌ℎ 휕 휕휌퐾 휕 휕푝 + (휌푢푗ℎ) + + (휌푢푗퐾) − = 휕푡 휕푥푗 휕푡 휕푥푗 휕푡 (3) 휕 휇 휇푡 휕ℎ 휕 휇푡 휕푌푖 (( + ) ) − ∑ ( (ℎ푖 ⋅ (휌퐷 + )) ) + 휌푘푢푘 휕푥푗 푃푟 푃푟푡 휕푥푗 휕푥푗 푆푐푡 휕푥푗 푖 where ℎ, 퐾, 푝, 푃푟 and ℎ푖 are enthalpy, kinetic energy, pressure, Prandtl number and gas species enthalpy, respectively. 3.1 Variable Turbulent Schmidt and Turbulent Prandtl Number Model The vertical velocity and its gradients are used to determine the shear flow in the vertical limited jet (i.e. the jet boundary) and to specify the regions with different 푆푐푡 value. The proposed model to prescribe different 푆푐푡 value to different flow regime region is: 1 Sc = Sc + (Sc − Sc ) t t,min 2 t,max t,min (4) 푟푣 휕푢푧 휕푢푧 휕푢푧 |푢| 1 − tanh c푘,1 ( + + ) − − 푐푘,2 max(|푢|) ⏟휕푥 휕푦 휕푧 ⏟max (| 푢 |) velocity gradients normalized velocity { [ ( ) ]} where 푆푐푡,푚푖푛, 푆푐푡,푚푎푥, 푐푘,1, 푐푘,2 and 푟푣 are minimal and maximal 푆푐푡 values specified by the user, two model constants and inlet radius, respectively. Term with velocity gradients prevails in the main and in the returning jet, while term with normalized velocity prevails in the axis of the main jet. The constants 푐푘,1 defines the behaviour of the function regarding the mean flow properties. The constant 푐푘,2 lowers the function argument, where vertical velocity and its gradients are small, but their values are still noticeable. It sets the relative velocity value cut-off and selects the regions to which the function is applied. The values of the constants were set in such manner, that the selected area contains main jet, shear flow region, and part of the returning jet.
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