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

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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] velocity Vertical 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.

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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 + )) ) + 휌𝑔푘푢푘 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 c ( + + ) − − 푐 푘,1 max(|푢|) 휕푥 휕푦 휕푧 ⏟max (| 푢 |) 푘,2 ⏟ 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. Their values are listed in Table 1.

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Table 1: Turbulent Schmidt number model constants.

Constant 푆푐푡,푚푖푛 푆푐푡,푚푎푥 푐푘,1 푐푘,2 Value 0.4 0.9 50 0.05

To maintain 푆푐푡 number value in the order of unity and to limit the upper and lower value of 푆푐푡, regardless the extreme velocity gradient values during simulations, hyperbolic tangent function is used. The function written in such form, acts as an “if” function. Namely, where function arguments are positive, it prescribes lower 푆푐푡 value and vice versa. Furthermore, in the interval of small arguments around zero the result of such function is continuously differentiable, which avoids numerical instabilities.

As is common in turbulence modelling, 푃푟푡 value is adjusted according to 푆푐푡.

4 EXPERIMENT USED FOR VALIDATION

The model was validated on a single experiment performed in SPARC experimental facility. The main purpose of this experiment was to observe the interaction of a vertical steam or air jet with a previously established horizontal layer of helium-steam or helium-air mixture in the upper part in the vessels. The SPARC test facility (Figure 4 left) consists of a single cylindrical vessel with a volume of 80 m3 [10]. During the experiment, the helium-air layer at the top of the vessel was eroded with an axisymmetric vertical air jet with mass flow of 100.8 kg/h. The jet with a diameter of 0.1 m was injected at the axis of the vessel at an elevation of 5.15 m and had the same temperature as the previously established atmosphere. The constant temperature and pressure in the vessel were maintained using an open nozzle at the bottom of the vessel [10].

Figure 4: Schematics of SPARC experimental facility (left) with sampling positions [10].

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5 RESULTS

Figure 5 shows 푆푐푡 values given by Eq. (4). The model prescribes higher 푆푐푡 in the jet region, where turbulence kinetic energy (푘) and turbulence kinetic energy dissipation rate (휀) are high (Figure 6), and lower in the environment, where velocity and its gradients are low and flow is almost laminar.

Figure 5: Turbulent Schmidt number given by the Eq. (4).

Figure 6: Turbulence kinetic energy (left) and turbulence kinetic energy dissipation rate (right) in a limited vertical jet.

5.1 SPARC

On Figure 7 are compared time-dependent helium volume fractions at several locations in the SPARC facility. The helium concentrations obtained in calculations with constant 푆푐푡 and variable 푆푐푡 value in lower measuring positions H2_4, H2_6 (Figure 7 bottom) and H2_8 (Figure 7 middle right), match the experimental results. It must be noted, that despite the discrepancies in H2_4 and H2_6 seem big, the difference is only 1 vol. %. At the measuring positions directly above the injection H2_10 (Figure 7 middle left), H2_12 and H2_14 (Figure 7 top) the experimental concentrations and concentration obtained with constant 푆푐푡 value at first coincide. Later, the simulated erosion rate, i.e., the rate of concentration decrease, is reduced and final concentration values are reached later than observed in the experiment. On the other hand, the results given by the 푆푐푡 model are significantly improved, when compared

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to experimental results. Where lower 푆푐푡 is prescribed, turbulent transport of mass (turbulent mass diffusion) is increased and the mixing process is intensified. Mixing is especially increased in regions with low fluid velocity, far from the jet. This results could be interpreted as that the underestimation of turbulent diffusion of momentum (휇푡) given by the turbulence model and numerical instabilities, are compensated by a smaller value of 푆푐푡. Namely, smaller 푆푐푡 increases the turbulent mass diffusion and enhances mixing. Similar behaviour were observed in RANS computations also by other authors (Tominaga et al., [11]).

Experiment Constant Variable H2_14 H2_12 35 35 30 30 25 25 20 20 15 15 10 10 5 5 0 0 0 2000 4000 6000 8000 10000 0 2000 4000 6000 8000 10000 H2_10 H2_8 35 18 30 16 14 25 12 20 10 Helium volume fraction [vol. %] 15 8 10 6 4 5 2 0 0 0 2000 4000 6000 8000 10000 0 2000 4000 6000 8000 10000 H2_6 H2_4 5 4.5 4.5 4 4 3.5 3.5 3 3 2.5 2.5 2 2 1.5 1.5 1 1 0.5 0.5 0 0 2000 4000 6000 8000 10000 0 2000 4000 6000 8000 10000 Time [s] Time [s]

Figure 7: SPARC: helium volume fractions at different measuring locations (see Figure 4).

6 CONCLUSIONS

According to experiments, the changes in turbulent Schmidt number usually occurs within the shear flow layer, which is in our case located at the boundary of the jet. Mean flow properties are used to define different flow regions and to prescribe different values of turbulent Schmidt number. With higher turbulent Schmidt number value in the jet region and lower in the quiescent environment the underestimation of turbulent diffusion of momentum in the regions with low velocity and almost laminar flow is compensated and the discrepancies between simulation and experimental results are minimized. The proposed function greatly affects the results only when the discrepancies are mostly generated by the numerical instabilities. In other cases, the differences are minimal.

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ACKNOWLEDGMENTS

The authors acknowledge the financial support from the Slovenian Research Agency (research core funding No. P2-0026 “Reactor engineering” and project No. PR-08992).

REFERENCES

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[11] Y. Tominaga, A. Mochida, S. Murakami, S. Sawaki, “Comparison of various revised k– ε models and LES applied to flow around a high-rise building model with 1:1:2 shape placed within the surface boundary layer”, J. Wind. Eng. Ind. Aerodyn., 96, 2008, pp. 389-411.

Proceedings of the International Conference Nuclear Energy for New Europe, Portorož, Slovenia, September 7 ̶ 10, 2020