CHAPTER THEORETICAL MODELS OF TWO-PHASE FLOW 3.1 Governing Equations In formulating the governing equations for air-water two phase flow in centrifugal pump impeller, the following assumptions are made: (1) The mixture is a homogeneous bubbly flow entraining fine bubbles. The bubble size is small compared to a characteristic length of the impeller channel. (2) In this case, air bubble is treated as the dispersed phase and water is treated as the continuous phase. (3) The bubbles maintain their spherical shape. Neither fragmentation nor coalescence of bubble occurs. The liquid phase is incompressible. (4) The drag coefficient of a bubble is the same as that of a solid particle. The influence of interactions between bubbles is negligible. (5) The mixture flow is steady in a relative frame or reference, which rotates around an axis with a constant velocity. Neither mass nor heat transfer takes place between the two phases. 3.1.1 Governing Equation of Liquid Phase Water flow (liquid phase) within the pump impellers can be described by the three-dimensional Reynolds-averaged Navier-Stokes (RANS) equations with a suitable turbulence model. Assuming a constant property flow, the RANS equations for liquid phase in a relative frame of reference are as follows: 30 Chapter Theoretical Models of Two-Phase Flow ∂ [(1 − α ) ρ l ] + ∇ • [(1 − α ) ρ lU l ] = ∂t (3.1) ∂ [(1 − α ) ρ lU l ] + ∇ • [(1 − α ) ρ lU l × U l ] + (1 − α ) ρ l ω × (2U l + ω × r ) ∂t (3.2) T = −(1 − α )∇p + ∇ • {(1 − α ) µ leff [∇U l + (∇U l ) ]} + M l where “× ” is a vector cross-product, U l is the velocity of liquid phase, ρ l the density of water, α is the void fraction, ω is angular velocity of impeller, r is radial vector, µ leff is the effective viscosity of liquid phase accounting for turbulence ( µ leff = µ l + µ lt ; µ l and µ lt are the molecular and turbulent viscosities of liquid phase respectively), p is the pressure, M l is interfacial forces acting on liquid phase due to the presence of gas phase. 3.1.2 Governing Equation of Gas Phase The governing equations for bubbly flow (gas phase) in a relative frame of reference are written as follows under the assumption ∂ (αρ g ) + ∇ • (αρ g U g ) = ∂t (3.3) ∂ (αρ g U g ) + ∇ • (αρ g U g × U g ] + αρ g ω × (2U g + ω × r ) ∂t = −α∇p + ∇ • {αµ geff [∇U g + (∇U g ) T ]} + M g (3.4) where U g is the velocity of gas phase, ρ g the density of air, µ geff is the effective viscosity of gas phase accounting for turbulence ( µ geff = µ g + µ gt ; µ g and µ gt are the molecular and turbulent viscosities of gas phase respectively), M g describes interfacial forces acting on gas phase due to the presence of liquid phase. 31 Chapter Theoretical Models of Two-Phase Flow The density of air ρ g in Eqs. (3.3) and (3.4) may be described as a function of temperature, pressure and one or more additional variables: ρ g = ρ ( p, T , AV1 , AV2 AVn ) For an ideal gas, the density is defined by the Ideal Gas Law: ρg = wp R0 T (3.5) where w is the molecular weight of gas, and R0 is the Universal Gas constant. Eqs. (3.1), (3.2), (3.3), (3.4) and (3.5) form a closed set of non-linear partial differential equations to solve the unknowns α , U l , U g , ρ g and p. M l and M g in Eqs. (3.2) and (3.4) are interfacial forces which can be expressed by using suitable models. The effective viscosities of liquid phase and gas phase µ leff and µ geff can also be solved by using suitable turbulence models. 3.2 Interfacial Forces Generally, the total interfacial forces acting between two phases may arise from several independent physical effects. If two phases are labelled using Greek indices α and β respectively and the total force on phase α due to interaction with phase β is denoted M α , then we will have the following relation for M α : D L VM TD + M αβp + M αβ + M αβ + M αβ + M α = M αβ (3.6) D where the term M αβ is the interphase drag force, M αβp the force due to the pressure L VM TD gradient, M αβ the lift force, M αβ the force due to virtual mass, and M αβ the turbulence dissipation force. 32 Chapter Theoretical Models of Two-Phase Flow In the present stage of simulation, the fluid motion in the centrifugal impeller VM is assumed as a steady turbulent flow. Therefore, the transient term M αβ makes a negligible contribution to the intergrated results, because its value changes sign frequently and its overall integrated effect becomes very small. The lift force and pressure force can also be negligible because of assumed small bubble size. Based on the above analyses and assumptions, only interphase drag force and turbulence dissipation force are considered in our current two-phase fluid simulation. Thus the interphase forces for liquid and gas phase in Eqs. (3.2) and (3.4) can be written as follows: M l = − M g = M lgD + M lgTD (3.7) 3.2.1 Interphase Drag Force The following general form can be used to model interphase drag force acting on liquid phase due to gas phase: M lgD = C lg (U g − U l ) (3.8) where C lg is the drag coefficient. For spherical particle, the coefficient C lg can be derived analytically. The area of a single particle projected in the flow direction, A p , and the volume of a single particle V p are given by Ap = πd , Vp = πd where d is the mean diameter. The number of particles per unit volume, n p is given by 33 Chapter np = Theoretical Models of Two-Phase Flow α 6α = V p πd The drag exerted by a single particle on the continuous liquid phase can be written as Dp = C D ρ l A p U g − U l (U g − U l ) Hence, the total drag force per unit volume on the continuous liquid phase is M lgD = Dlg = n p D p = CD αρ l U g − U l (U g − U l ) d (3.9) Compare Eq. (3.9) with Eq. (3.8), we get C lg = CD αρ l U g − U l d (3.10) where C D is non-dimensional drag coefficient. For a particle of simple shape, immersed in a Newtonian fluid and which is not rotating relative to the surrounding free stream, the drag coefficient, C D , depends only on the particle Reynolds number Re p . The function C D (Re p ) may be determined experimentally, and is known as the drag curve. The particle Reynolds number Re p is defined using the particle mean diameter, and the continuous liquid phase properties, as follows: Re p = ρl d p U g − U l µl where µ l is the viscosity of the continuous liquid phase, d p is the diameter of the particle. 34 Chapter Theoretical Models of Two-Phase Flow In the present simulation, the Schiller Naumann drag model is employed for solving C D because of its good agreement with experimental data for solid spherical particles, or for fluid particles that are sufficiently small that they may be considered spherical. The empirical relation for C D is written as CD = 24 (1 + 0.15 Re 0p.687 ) Re p (3.11) To ensure the correct limiting behavior in the inertial regime, the above Schiller Naumann model drag model is modified as 24 C D = (1 + 0.15 Re 0p.687 ),0.44 Re p (3.12) 3.2.2 Interphase Turbulent Dispersion Force In Eq. (3.7), M lgTD represents turbulent dispersion force acting on the continuous liquid phase due to the dispersed gas phase. In such case, Lopez de Bertodano Turbulent Dispersion Model (Lopez de Bertodano, 1991) is implemented and written as M lgTD = −CTD ρ l k l ∇rl (3.13) where k l is the turbulent kinetic energy of liquid phase; rl is the volume fraction of liquid phase which is equal to (1 − α ); CTD is the non-dimensional turbulent dispersion coefficient, its values of 0.1-0.5 have been used successfully for bubbly flow with bubble diameters of order a few millimetres. See Lopez de Bertodano (1998) for a general discussion on recommended values of CTD . 35 Chapter Theoretical Models of Two-Phase Flow 3.3 Turbulence Modelling in Multiphase Flow This section describes the extension of the single-phase turbulence models to multiphase simulation. According to our assumption (1) and (2), the water flow through the impeller is treated as continuous phase, and dilute fine bubbly flow is treated as dispersed phase. In such case, it is possible to mix algebraic and k − ε models between two phases for simplicity. A recommended model for dilute dispersed two-phase flow uses a k − ε model for the continuous phase, and an algebraic eddy viscosity model for the dispersed phase, which simply sets the dispersed phase viscosity propotional to the continuous phase eddy viscosity. 3.3.1 k − ε Turbulence Model for the Continuous Phase The eddy viscosity hypothesis is assumed to hold for the continuous turbulence phase. Diffusion of momentum is governed by an effective viscosity µ leff , which is equal to molecular viscosity coefficient of liquid phase µ l plus turbulent eddy viscosity coefficient of liquid phase µ lt : µ leff = µ l + µ lt (3.13) Similar to the derivation in Chapter for single-phase flow, the turbulence viscosity of liquid phase, µ lt , can be linked to the turbulence kinetic energy and dissipation of liquid phase via the relation: k2 µ lt = C µ ρ l l εl (3.14) where C µ is a constant, the value is 0.09, k l is the turbulent kinetic energy of liquid phase and ε l is the turbulent dissipation rate of liquid phase. 36 Chapter Theoretical Models of Two-Phase Flow The transport equations for k l and ε l are assumed to take a similar form to the single-phase transport equations: µ ∂ [(1 − α ) ρ l k l ] + ∇ • {(1 − α )[ ρ lU l k l − ( µ l + lt )∇k l ]} ∂t σk (3.15) = (1 − α )( Pk − ρ l ε l ) + T (k ) lg µ ∂ [(1 − α ) ρ l ε l ] + ∇ • {(1 − α )[ ρ lU l ε l − ( µ l + lt )∇ε l ]} ∂t σε ε = (1 − α ) l (Cε Pk − Cε ρ l ε l ) + Tlg(ε ) kl (3.16) where Pk is the turbulent kinetic energy production term, its definition is given in Eq. (2.11), Cε , Cε , σ k , σ ε are empirical constants for k − ε turbulence model, their values are also given in Chapter 2, the additional terms Tlg( k ) and Tlg(ε ) represent interphase transfer for k l and ε l respectively, but not considered in the current simulation. 3.3.2 Algebraic Turbulence Model for the Dispersed Phase The algebraic equation model is only available for the dispersed fluid when the continuous phase is set to use a turbulence model. µ gt = ρ g µ lt ρl σ (3.17) where µ gt and µ lt are turbulent viscosities of gas phase and liquid phase respectively; the parameter σ is a turbulent Prandtl number. In situations where the particle relaxation time is short compared to turbulence dissipation time scales, e.g. bubbles or very small particles, we can safely use the default value σ =1. 37 Chapter Theoretical Models of Two-Phase Flow 3.4 Boundary Conditions The implementation of boundary conditions for multiphase flow is very similar to that for single-phase flow. The main differences are: • Boundary conditions need to be specified for both fluids for all variables except the shared pressure field. • Volume fractions of both phases must be specified on inlet boundary condition. These must sum to unity. • For multifluid flow, pressure boundary conditions at inlet and outlet are always defined in terms of static pressure. Considering these differences, the boundary conditions for two-phase flow are specified as follows: Inlet boundary: The inlet velocity of both phases and particle inlet volume fraction are given; the inlet turbulent kinetic energy is given according to 3.7 percent of turbulence intensity; the inlet dissipation rate of turbulent kinetic energy is given as the same as that for single-phase flow. Outlet boundary: The outlet boundary for multiphase flow is specified as the same as that for single-phase flow. However, the static pressure is assumed zero at the outlet. Wall boundary: A no-slip condition is imposed for the continuous liquid (water) phase, and a slip condition is applied to dispersed particle phase. Wall functions are used to model near-wall flow. 38 . Law: T R wp g 0 = ρ (3. 5) where w is the molecular weight of gas, and 0 R is the Universal Gas constant. Eqs. (3. 1), (3. 2), (3. 3), (3. 4) and (3. 5) form a closed set of non-linear partial. 30 CHAPTER 3 THEORETICAL MODELS OF TWO-PHASE FLOW 3. 1 Governing Equations In formulating the governing equations for air-water two phase flow in centrifugal pump impeller,. force. Chapter 3 Theoretical Models of Two-Phase Flow 33 In the present stage of simulation, the fluid motion in the centrifugal impeller is assumed as a steady turbulent flow. Therefore,