ACHEMA 2006, 15-19 May 2006, Frankfurt am Main CFD modelling of mixing and separation in multiphase flows Simon Lo CD-adapco, UK ABSTRACT Mixing and separation of materials in multiphase flows are extremely common and frequent operations in chemical and process engineering. For example, stirred reactor is used to give good mixing and uniform dispersion of catalyst particles in the reactor to ensure uniform quality product is produced. At the end of the process, the catalyst particles may need to be recovered and removed from the product by separation equipment such as a settling tank. The commonly asked questions by plant operators in relation to these processes are: (a) What is the optimum rotating speed should be set for the stirrer to give uniform dispersion of the catalyst particles. (b) What is the appropriate settling time to allow for complete separation and recovery of the catalyst particles in the settling tank. Answers to these questions could have considerable financial impact in the operations of the equipment. Increasingly numerical analyses based on computational fluid dynamics (CFD) are used to help providing these answers. In this paper the multiphase flow model in the commercially available CFD software, STAR-CD, is used in simulations of (a) mixing and suspension of catalyst particles in a stirred tank and (b) separation of particles in a settling tank. In the stirred tank experiment, three different stirrer speeds were considered. At different stirrer speed the particles were lifted to a different level in the tank. The levels of particle suspension computed by STAR-CD for all three stirrer-speeds were found to be in good agreement with the measured data from the laboratory. For the settling tank, the computed settling time and the height of the settled layer were also found to be in good agreement with experimental and analytical values. Keywords: CFD, two-phase flow, mixing, suspension, separation, sedimentation INTRODUCTION Many important operations in the chemical and process industries involve suspension or separation of solid particles in a liquid. Mixing and suspension of particles in a liquid are usually achieved in stirred tanks and separation of particles in settling tanks. These operations involve two-phase flows and can be rather difficult to analyse by analytical methods. In recent years, numerical analyses of flow processes based on computational fluid dynamics (CFD) are becoming more widely used. For examples, the analyses of solid-liquid flows in stirred vessels by Gosman [2], Bakker [1], Micale [3] and Montante [5]. For CFD analyses to gain acceptance by engineers in the industry, it is important to describe clearly the equations and the solution method used and to demonstrate the 1
ACHEMA 2006, 15-19 May 2006, Frankfurt am Main accuracy of the method by validation exercises comparing numerical results against experimental data. The main objective of this paper is to make a contribution to the validation of CFD in modelling mixing and separation in two phase flows. MATHEMATICAL MODEL The Eulerian multiphase flow model in STAR-CD [8] was used to solve the two- phase flow problems presented in this paper. In the Eulerian multiphase flow model, the phases are treated as interpenetrating continua coexisting in the flow domain. Equations for conservation of mass, momentum and energy are solved for each phase. The share of the flow domain occupied by each phase is given by its volume fraction and each phase has its own velocity, temperature and physical properties. Interactions between phases due to differences in velocity and temperature are taken into account via the inter-phase transfer terms in the transport equations. The Inter-Phase Slip Algorithm (IPSA) of Spalding [7] is used to solve the system of multiphase flow equations. In this solution method, all the phases share a common pressure field. Since isothermal flows are considered in this paper, the main equations solved are the conservation of mass, and momentum for each phase, the energy equation is not considered. Continuity The conservation of mass for phase k is: ∂ ( ) ( ) α ρ + ∇ α ρ = . u 0 , (1) ∂ k k k k k t α is the volume fraction of phase ρ is the phase density, where k , u is the phase k k k velocity. The sum of the volume fractions is equal to unity, ∑ α = 1 . (2) k k Momentum The conservation of momentum for phase k is: ∂ ( ( ) ) ( ) ( ) α ρ + ∇ α ρ − ∇ α τ + τ = − α ∇ + α ρ + + t u . u u . p g F F , (3) ∂ k k k k k k k k k k k k k D T t τ and τ t where is the laminar and turbulence shear stresses, p is pressure, g is k k gravitational acceleration, F and F are the mean drag and turbulent drag forces. D T Mean Drag Force When the local particle volume fraction is less than 20%, the mean drag force can be obtained from: 2
ACHEMA 2006, 15-19 May 2006, Frankfurt am Main α ρ u 3 = α α ≤ d c r n F C u 0 . 2 , (4) D D r c d 4 d otherwise the Ergun equation is used: ⎡ ⎤ α ρ α µ 2 u = + α > 0.2 . ⎢ d c r ⎥ d c 150 1 . 75 (5) F u α D r d 2 ⎢ ⎥ d d ⎣ ⎦ c = − In the equations above, C is the drag coefficient, u ( u u ) is the relative D r c d µ is the dynamic viscosity of the continuous velocity between the two phases and c phase. Subscript stands for continuous phase and c d for dispersed phase. With higher particle concentration, inter-particle forces have an effect on the particle α n velocity, the hindered settling effect. The factor in (4) is used to model the c hindered settling effect and the exponent is: = − n 1 . 7 . (6) The correlations for drag coefficient by Schiller and Naumann [6] are used in the calculations: ( ) 24 = + 0 < ≤ 1000 , 0 . 687 C 1 0 . 15 Re Re (7) D d d Re d = > 1000 . C 0 . 44 Re (8) D d The particle Reynolds number, Re , is defined as: d ρ u d = c r Re . (9) µ d c Turbulent Drag Force The turbulent drag force accounts for the additional drag due to interaction between the dispersed phase and the surrounding turbulent eddies, ν t = − ∇ α c F A , (10) α α σ T D d α d c where α ρ C u 3 = D d d r , (11) A D 4 d 3
ACHEMA 2006, 15-19 May 2006, Frankfurt am Main ν is the continuous phase turbulent kinematic viscosity and σ is the turbulent t and α c Prandtl number (value of 1 is used). Turbulence Model To calculate the continuous and dispersed phase turbulence stresses, values for k and ε are required. These are computed using the extended k - ε equations containing extra source terms that arise from the interphase forces present in the momentum equations. The additional terms account for the effect of particles on the turbulence ε equations are: field. The k and ( ) ⎛ ⎞ ∂ α µ + µ t ( ) α ρ + ∇ α ρ = ∇ ⎜ ∇ ⎟ + α − ρ ε + c c c k . u k . k G S , (12) ⎜ ⎟ ∂ σ c c c c c c c k 2 ⎝ ⎠ t k ( ) ⎛ ⎞ ∂ α µ + µ t ( ) α ρ ε + ∇ α ρ ε = ∇ ⎜ ∇ ε ⎟ + α − ρ ε + c c c . u . C G C S , (13) ⎜ ⎟ ε ∂ c c c c c σ c 1 2 c 2 t ⎝ ⎠ ε where ν t ( ) ( ) = − − ∇ α + − c S A u u . 2 A C 1 k , (14) α α σ k 2 D d c d D t α c d ( ) = − ε S 2 A D C 1 , (15) ε 2 t ( ) = µ ∇ + ∇ ∇ T G u u : u . (16) c c c c In the equations above, C is a response coefficient defined as the ratio of the t dispersed phase velocity fluctuations to those of the continuous phase: ′ u = d . (17) C ′ t u c τ t The turbulent stress in the continuous phase momentum equation can be modelled c using the eddy-viscosity concept: ⎛ ⎞ 2 2 τ = µ ∇ + ∇ − ∇ − ρ ⎜ ⎟ t t T u u . u I kI , (18) c c c c c c ⎝ ⎠ 3 3 with the turbulent viscosity given by 2 k µ c = ρ t C . (19) µ ε c 4
ACHEMA 2006, 15-19 May 2006, Frankfurt am Main τ t The dispersed-phase turbulent stress is correlated to the continuous-phase d τ t turbulent stresses via the response coefficient C such that c t ρ τ = C τ t 2 t d . (20) ρ d t c c = C 1 is used in the model. t MIXING AND SUSPENSION OF PARTICLES IN STIRRED TANK Experimental work of Micale et al Micale et al [4] carried out a series of experiments to measure the level of particle suspension in a stirred vessel. A sketch of their stirred vessel is shown in Figure 1. Their system consists of a 0.19 m diameter (T) cylindrical tank with 4 baffles and a six bladed Rushton turbine (D=T/2) placed very close to vessel bottom. The off bottom clearance of the turbine, C= 0.018 m, was small enough to ensure a "single- loop" flow configuration in the tank. Silica particles in the size range 212-250µm and a measured density of 2580 kg/m 3 were used in the experiments. The particle loading in the vessel corresponds to a volume fraction of 9.6%. Fig 1 Stirred vessel of Micale et al Fig 2 CFD model of stirred vessel 5
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