# Development of a consistent and conservative Eulerian - Eulerian algorithm for multiphase flows

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1 Development of a consistent and conservative Eulerian - Eulerian algorithm for multiphase flows Ana Cubero Alberto Sanchez-Insa Norberto Fueyo Numerical Fluid Dynamics Group University of Zaragoza Spain 6th OpenFOAM Workshop Penn State University, June 2011 File version: 14 Jun 11

2 Introduction Motivation Simulation of fluidized beds Chemical reactor with solid particles and a fluid, which together have fluid-like properties Issues: Bed hydrodynamics Heat transfer Mass transfer Combustion (homogeneous, heterogeneous)

3 Introduction Outline Development of a consistent and conservative Eulerian - Eulerian algorithm for multiphase flows Motivation Eulerian-Eulerian multi-phase equations Formulations: OpenFOAM alpha-intensive Conservative Consistent Momentum Interpolation Results Illustration and comparison Fluidized beds

4 Equations 2. Heating Eulerian multiphase equations Multiple fluids (continua) share the space Fraction of shared space is the volume fraction The exchange mass, momentum and other properties as they move 1. Drag and turbulence dispersion 3. Drying (H2O) 5. Heterogeneous combustion (CO, CO2) Water Coal Char Ash 4. Pyrolisis (CH4)

5 Equations The Eulerian multiphase equations Phase-mass conservation r p ρ p t + (r p ρ p v p ) (Γ r p r p )= x p and q are phases r is volume fraction x is a mass-exchange process (eg, drying) q ṁ x qp mx qp is mass transfer rate (kg/s) from phase p into phase q due to process x

6 Equations The Eulerian multiphase equations Conservation of a variable ϕ p r p ρ p φ p t + (r p ρ p v p φ p )+ (r p Γ φ p φ p )= = f φ qp(φ q φ p )+ ṁ x qpφ x + Sp φ q x q f is an interphase transport coefficient ϕ x is the value of ϕ in the donor phase in massexchanging process x Examples of ϕ: velocity, enthalpy/temp, water, volatile matter...

7 Approaches Related OpenFOAM efforts Alpha-intensive formulation in OpenFOAM (twophaseeulerfoam) (Alpha is the volume fraction in OpenFOAM) Euler-Euler gas-solid flow solver for OpenFOAM, under development by Alberto Passalacqua (Politecnico di Torino/Iowa State University)

8 Approaches The alpha-intensive formulation Oliveira and Issa, Weller, Rusche Motivation: Division of (non-conservative form) of the mom eqs by alpha, for preventing numerical difficulties when phase fractions tend to zero

9 Approaches Shortcomings 1. Non-conservative 2. Division by α requires setting an arbitrary min value 3. Momentum eq s not fully solved 4. Density presumed constant 5. Two phases (As implemented in twophaseeulerfoam)

10 Approaches Proposed approach CIPSAmultiPhaseEuler; C for consistent IPSA-based, with pressure [correction] from global mass conservation (IPSA: InterPhase Slip Algorithm, DB Spalding, Imperial College, circa 1982) PISO-like solution algorithm

11 Approaches Advantages CIPSAmultiPhaseEuler; C for consistent Consistent: steady state independent from relax coeffs and time-step size IPSA-based, with pressure [correction] from global mass conservation Equations solved in conservative form PISO-like solution algorithm Fully implicit, eqs fully satisfied at each time step

12 Approaches Other features Variable density (suitable for combustion) Multiple phases, inc Kinetic Theory of Granular Flows Heat and species conservation equations

13 Algorithmics Algorithm (IPSA-PISO) 1. Calculate interfacial terms, constitutive relations, etc. 2. Build all U equations (for consistency in drag term) 3. Solve U equations 4. Calculate fluxes 5. Build p equation from overall mass continuity ( r p =1) 6. Solve p equation 7. Correct velocities 8. Build r p equations 9. Solve r p equations 10.Normalize so that r p =1; r p min=1.0e Goto 1 unless converged

14 Algorithmics Phase segregation with CIPSA Prompted Oliveira and Issa to develop alpha-intensive form, because it did not converge in conservative form: [...] the standard method fails altogether because of the effect of phase segregation Solid volume fraction Solid volumetric flux Mixture velocity

15 Algorithmics Consistent Momentum Interpolation Extension to multiphase flows of CMI by Cubero and Fueyo Enhancement to Rhie and Chow s procedure Robust, prevents wiggles and checker-board profiles, even for small time steps Results independent of time step and relaxation coeff Compact numerical interpolation, even for higher order temporal schemes Details (single-phase flows): Cubero, A, Fueyo N A compact momentum interpolation procedure for unsteady flows and relaxation. Numerical Heat Transfer, B. 52: (23).

16 Algorithmics CMI: essence (1)

17 Algorithmics CMI: essence (2)

18 Algorithmics CMI: essence (3)

19 Algorithmics Sample results Riser+elbow Solid density: 240 kg/m3; fluid density: 1.2 kg/m3 Inlet conditions: Us = 1.0 m/s; alpha_solid=0.001; Uf = 3.1m/s deltat = 1.0e-3 s, 1.0e-4 s Domain and gas velocity Mesh detail

20 Algorithmics Solid volume fraction Solid density 24 kg/m3, fluid 1.2 kg/m3 Using IPSA and both classic MI and Consistent MI IPSA + Momentum Interpolation CIPSA: IPSA + Consistent MI

21 Algorithmics Solid volume fraction IPSA + Momentum Interpolation CIPSA: IPSA + Consistent MI

22 Algorithmics Alpha-intensive solution deltat=1.0e-4s, restart from previous OpenFOAM alpha-intensive

23 Algorithmics Solid volume fraction in the bed Temporal evolution Note time-step dependence of IPSA+MI solid mass / Kg IPSA+CMI deltat=1.0e-3 IPSA+CMI deltat=1.0e-4 IPSA+MI OF-MI deltat=1.0e-3 IPSA+MI OF-MI deltat=1.0e time / s IPSA + MI CIPSA: IPSA + Consistent MI

24 Results::circulating beds Simulation of a dilute bed 5-meter riser Initial conditions Only gas Developed flow Kinetic theory of granular Text flows (KTGF) for solid viscosity and pressure Outlet Air Solid Syamlal-O Brien model for constitutive relations and drag Inlet

25 Results::circulating beds Dilute fluidized bed: results Solid volume fraction Solid vertical velocity Solid viscosity Particle cluster......going down......ktgf calculated Text

26 Results::circulating beds Animation Solid volume fraction Solid vertical velocity Solid viscosity Text

27 Results::circulating Tarea 2/9: Modelos Multifásicos/Modelado beds de configs Dense bed simulation 11-meter riser Initial conditions Only gas Developed flow Kinetic theory of granular Text flows (KTGF) for solid viscosity and pressure Air Outlet Solid Syamlal-O Brien model for constitutive relations and drag Inlet

28 Results::circulating beds Dense bed, results (t=18s) Solid volume fraction Solid vertical velocity Text Solid viscosity

29 Results::circulating Tarea 2/9: Modelos Multifásicos/Modelado beds de configs Animation Solid vertical velocity Solid volume fraction Solid viscosity Text

30 Results::circulating beds Dense bed: average flow fields Core-annulus structure Solid volume fraction Solid vertical velocity Text

31 Results::circulating beds Dense bed: average fields Radial profiles at 5.6 m Core-annulus structure Solid volume fraction Text Solid vertical velocity

32 Results::circulating beds From the start With initial r_solid=1.0e-30 (low, algorithmically demanding) Solid vertical velocity Solid volume fraction Solid viscosity Text

33 Results::bubbling beds A bubbling bed Two solid phases: alpha, alpha2 Volume fractions Velocities

34 Results::bubbling beds Fluent-OpenFOAM comparison Fluent OpenFOAM 0.5s 1.0s 1.5s 2.0s 3.0s 5.0s 10s 15s 20s

35 Further work WIP/Further work Full integration of heat, mass transfer, chemical reaction (via Cantera) Validation Vol frac solid Mass frac CH4 Mass frac ash Gas temperature (Pneumatic transport regime)

36 6th OpenFOAM Workshop Penn State University, June 2011 Thank you Ana Cubero Alberto Sanchez-Insa Norberto Fueyo Numerical Fluid Dynamics Group University of Zaragoza Spain Funding acknowledgements: Fundacion Ciudad de la Energia (Government of Spain), project OXY-CFD Spanish Ministry for Science and Innovation, CONSOLIDER INGENIO 2010, project CSD

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