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Advection Diffusion Model for Particles Deposition in Rayleigh-Bénard Turbulent Flows

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Advection Diffusion Model for Particles Deposition in Rayleigh-Bénard Turbulent Flows International Conference Nuclear Energy for New Europe 2005 Bled, Slovenia, September 5-8, 2005 Advection Diffusion Model for Particles Deposition in Rayleigh-Bénard Turbulent Flows Paolo Oresta, Antonio Lippolis DIASS – Politecnico di Bari Viale del Turismo 8, 74100 Taranto, Italia [email protected], [email protected] Roberto Verzicco DIMeG and CEMeC – Politecnico di Bari Via Re David, 200, 70125 Bari, Italia [email protected] Alfredo Soldati DEM and CIFI – Università degli Studi di Udine Via delle Scienze, 208, 33100 Udine, Italia [email protected] ABSTRACT In this paper, Direct Numerical Simulation (DNS) and Lagrangian Particle Tracking are used to precisely investigate the turbulent thermally driven flow and particles dispersion in a closed, slender cylindrical domain. The numerical simulations are carried out for Rayleigh (Ra) and Prandtl numbers (Pr) equal to Ra = 2·108 and Pr = 0.7, considering three sets of particles with Stokes numbers, based on Kolmogorov scale, equal to Stk = 1.3, Stk = 0.65 and Stk = 0.13. This data are used to calculate a priori the drift velocity and the turbulent diffusion coefficient for the Advection Diffusion model. These quantities are function of the Stokes, Froude, Rayleigh and Prandtl numbers only. One dimensional, time dependent, Advection- Diffusion Equation (ADE) is presented to predict particles deposition in Rayleigh-Bénard flow in the cylindrical domain. This archetype configuration models flow and aerosol dynamics, produced in case of accident in the passive containment cooling system (PCCS) of a nuclear reactor. ADE results show a good agreement with DNS data for all the sets of particles investigated. 1 INTRODUCTION The passive containment cooling system (PCCS) consists of an inner steel shell and an outer concrete shell [1]. During a severe nuclear accident, the radioactive core meltdown and the energy produced is responsible for the thermal gradient between the lower and upper zones of PCCS. Specifically, the energy released generates a strong convective flow, which carries passively radioactive aerosols in the confined containment. The steel shell provides to remove the heat from the containment, to have the steam temperature and pressure below the max 173.1 173.2 design value. During an accident, the water, located in a tank on the top of the containment, flows down on the outside of the steel shell to improve the heat removal away from the PCCS. In this way, the consequences of a severe core meltdown accident will be restricted inside the containment. Dispersion, deposition and re-entrainment in buoyancy-driven flow are of fundamental issue to evaluate nuclear reactor safety problems. Particles deposition plays a significant role in limiting the release of radioactivity to the environment; yet, there is still a lack of knowledge about the nuclear aerosol release mechanisms in the containment. The convective flow occurring in this containment system can be modelled as a confined Rayleigh-Bénard flow. The flow field produced in a closed domain with the lower plate maintained at a temperature higher than the upper plate, is dominated by Rayleigh-Bénard convection. We use a cylindrical domain with aspect ratio (diameter over cell height) 1/2, filled with air as archetype configuration to model the PCCS device. The temperature gradient kept between the plates and the natural inhomogeneity of the temperature field on the cylindrical domain concurs to produce different dynamics in terms of vortical structures. In our case, the flow field is turbulent, three-dimensional and time dependent. In such flow, three sets of inertial particles are randomly dispersed on the cylindrical domain. Particles clustering and dispersion are due to their interactions with turbulent eddies, characterized by a specific dimension and by a specific vortical time. Particles are sensitive to small size eddies and their simulated behaviour can be significantly modified if these vortical structures are not directly computed, but described with sub-grid turbulence model as RANS and LES. For this reason, the solution of the flow field with DNS approach is very important. The advantage of the DNS is the possibility of obtaining an accurate velocity field, without using turbulence models. This means that we can calculate also the high frequency velocity fluctuations - that has to be modelled in other kinds of numerical simulations like LES and RANS - and evaluate their effect on the particle behaviour. With DNS, the turbulence properties along particle trajectories are directly available; with this technique, the local mechanism driving particles transfer and particle preferential distribution can be investigated. The main problem of Direct Numerical Simulation is the computational cost, due to the very fine spatial and temporal resolution in comparison to the turbulent modelling. In such flow, Lagrangian Particle Tracking is used to have an accurate temporal and spatial history of particles deposition in the cylindrical domain. In several engineering applications, a model simpler than Direct Numerical Simulation and Lagrangian Particle Tracking is needed to describe the time history of particles deposition. For instance, in case of severe nuclear reactor accident, we are mainly interested in evaluating the time history of particle concentration averaged on the horizontal plate. Specifically, time dependent, one dimensional, Advection Diffusion model is presented in this paper to predict, with sufficient accuracy, particle deposition in the cylindrical domain dominated by the turbulent convective flow. In this work, Direct Numerical Simulation and Lagrangian Particle Tracking are used to model the drift velocity and turbulent diffusion coefficient. These quantities are calculated a priori because they are functions of dimensionless parameters only. Advection-Diffusion model was tested against numerical simulations database. 2 METHODOLOGY 2.1 Flow Field The thermal convection was developed in a closed cylindrical domain of diameter d, distance between the heated plates h, filled with a fluid characterized by kinematics viscosity ν, thermal diffusivity k, coefficient of thermal expansion α and thermal conductivity λ. The Proceedings of the International Conference “Nuclear Energy for New Europe 2005” 173.3 aspect ratio Γ=d/h was set equal to 0.5 and the magnitude of the gravity vector is g. The lateral surface is adiabatic and the boundary conditions are no-slip. Hot Th and cold temperatures Tc are enforced respectively at the lower and upper plates as thermal boundary conditions; ∆=Th-Tc is the temperature difference kept between the horizontal plates. We can define the Rayleigh and the Prandtl numbers as follows: g∆h3 Ra = ; Pr = k k In our case, the fluid that fills the cylindrical domain is the air and, for this reason, we have set Prandtl number equal to 0.7. Different combinations between the Rayleigh and Prandtl numbers and the aspect ratio concur to produce the different dynamics in terms of vortical structures. Starting with the flow at rest and increasing Rayleigh number, there are three fundamental flow regimes and their onset is characterized by three critical Rayleigh numbers. At the onset of the convection the flow has a steady state and if the nonlinear terms exceed the viscous one the unsteady state of the flow is reached. The transition to chaotic and turbulent regimes is more gradual in comparison to the transition from the steady state to the time dependent regimes. In the turbulent regime the flow dynamics are not strongly dependent on the cell geometry. Critical Rayleigh numbers, in a cylindrical cell with Γ = 1/2 and Pr = 0.7, 4 5 5 results equal to Rac = 2.35·10 , Rau = 5.6·10 and Rat = 9·10 , respectively for the beginning of convection, unsteady and turbulent regimes [2], respectively. We carried out the numerical simulations with Rayleigh number set equal to Ra = 2·108. With Rayleigh number equal to Ra = 2·108 and Prandtl number Pr = 0.7, the flow is turbulent, three-dimensional and time dependent. For these set of values, despite of the axysimmetric domain, the flow is organized in a single roll that fills the whole domain and in two toroidal structures close to the plates [3]. The domain of Rayleigh-Bénard problem can be divided in two regions: the thermal boundary layers and the core region. Core region shows a Kolmogorov turbulence cascade mechanism. The energy is pumped into the system at large scale h and is transferred to the inertial range scales r and, finally, the energy is dissipated, because the viscosity at the Kolmogorov scales η. In the thermal boundary layer, there is the whole temperature gradient and the turbulence dynamics follow the Bolgiano theory. The Bolgiano scale represents the scale at which energy is injected in the system as “potential energy”. Buoyancy force converts this energy in kinetic energy. In the region close to the thermal boundary layer, the buoyancy force is the dominant effect in the system [3],[4]. For the present flow configuration (Pr=0.7, Γ=1/2), the thermal layer is smaller than the viscous one. The flow field is obtained solving the unsteady Navier- Stokes equations with the Boussinesq approximation, written in dimensionless form as follows: 1/ 2 Du Pr ’ → = −∇p + ∇2u + Q z , Dt Ra (1) DQ 1 = ∇2Q, Dt (Pr Ra)1/ 2 → with z the unity vector pointing in the opposite direction respect to gravity, u the velocity vector, p the pressure and Q = (T-Tc)/(Th-Tc) the dimensionless temperature in which T is the Proceedings of the International Conference “Nuclear Energy for New Europe 2005” 173.4 temperature in the cylindrical domain. Equations are written in dimensionless form using the 1/2 free-fall velocity Uf = [gαL(Th – Tc)] , the distance between hot and cold plates L = h and the time τf = L/Uf . The method used to solve the Navier-Stokes equations is Direct Numerical Simulation (DNS). The flow field is calculated with finite differences code in cylindrical coordinates [5]. Navier-Stokes equations are solved by a fractional step method.
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