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Multiphase LB Models

Multiphase LB Models. Single Component Multiphase. Single Phase (No Interaction). Attractive. Interaction Strength. Number of Components. Nature of Interaction. Multi- Component Multiphase. Repulsive. Miscible Fluids/Diffusion (No Interaction). Immiscible Fluids. Inherent Parallelism.

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Multiphase LB Models

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  1. Multiphase LB Models Single Component Multiphase Single Phase (No Interaction) Attractive Interaction Strength Number of Components Nature of Interaction Multi- Component Multiphase Repulsive Miscible Fluids/Diffusion (No Interaction) Immiscible Fluids Inherent Parallelism High Low

  2. Equations of State • Perfect (Ideal) gas law • van der Waals gas law • Molar volumes • Temperature dependence and Critical Points • Liquid-vapor coexistence and the Maxwell Construction • Water and the non-quantitative nature of the van der Waals equation • Alternative presentation: P(r) • Modern EOS for water

  3. Perfect or ideal gas law (EOS) • P is pressure (ATM) • V is volume (L) • n is number of mols • R is gas constant (0.0821 L atm mol-1 K-1) • T is temperature (K)

  4. Molar Volume Vm = V/n is the volume occupied by one mole of substance. The gas laws can be re-written to eliminate the number of moles n Perfect or ideal gas law (EOS):

  5. Single Component D2Q9 LBM (with c = 1 lu ts-1) So, if 1/Vm (mol L-1) is density, cs2 = RT

  6. van der Waals EOS Perfect or ideal gas law (EOS): van der Waals EOS: • ‘a’ term due to attractive forces between molecules [atm L2 mol-2] • ‘b’ term due to finite volume of molecules [L mol-1]

  7. Molar Volume Vm is the volume occupied by one mole of substance. The gas laws can be re-written to eliminate the number of moles n. van der Waals gas law (EOS):

  8. CO2: P-V Space

  9. Liquid-Vapor Coexistence: CO2, P-V Space a = 3.592 L6 atm mol-2 b = 0.04267 L3 mol-1 Vapor Pressure Vapor Molar Volume Liquid Molar Volume

  10. Maxwell Construction: CO2, P-V Space Maxwell Construction: Area A = Area B Gives: --vapor pressure--densities of coexisting liquid and vapor Vapor Pressure B Vapor Molar Volume A Liquid Molar Volume Flat interfaces only!

  11. Water, 298K: P-V Space

  12. van der Waals Non-Quantitative:Water, 298K: P- r Space Pvap

  13. Including Directional H-Bonds:Water, 298K: P- r Space Spinodal

  14. Recap • van der Waals equation gives a simple qualitative explanation of phase separation based on molecular attraction and finite molecular size • Maxwell construction gives the vapor pressure and the densities of coexisting liquid and gas at equilibrium, FOR FLAT INTERFACES • van der Waals equation fails to quantitatively reproduce the EOS of water

  15. FHP LBM Fluid Cohesion • An attractive force F between nearest neighbor fluid particles is induced as follows: • G is the interaction strength • Y is the interaction potential: • Y0 and r0 are arbitrary constants Other forms possible/common Shan, X. and H. Chen, 1994. PRE 49, 2941-2948.

  16. D2Q9 SCMP LBM Fluid-Fluid Interaction wa is 1/9 for a = 1, 2, 3, 4 and is 1/36 for a = 5, 6, 7, 8 G <0 for attraction between particles Force is stronger when the density is higher

  17. Incorporating Forces F = ma = m du/dt U = u + tF1/r + tF2/r + ...

  18. LBM Non-ideal Equation of State Non-ideal Component Shan and Chen, 1994. PRE 49, 2941-2948 No repulsive potential in LB model Realistic EOS for water: Follows ideal gas law at low density, compressibility of water at high density and spinodal at high tension Liquid/vapor coexistence at equilibrium (and flat interface) determined by Maxwell construction Water EOS after Truskett et al., 1999. J. Chem. Phys. 111, 2647-2656.

  19. Simplified EOS

  20. Single Component Multiphase Modifications to code

  21. Eqs. 60 and 61 // Compute psi, Eq. (61). for( j=0; j<LY; j++) for( i=0; i<LX; i++) if( !is_solid_node[j][i]) { psi[j][i] = 4.*exp( -200. / ( rho[j][i])); }

  22. Eqs. 60 and 61 // Compute interaction force, Eq. (60) assuming periodic domain. for( j=0; j<LY; j++) { jp = ( j<LY-1)?( j+1):( 0 ); jn = ( j>0 )?( j-1):( LY-1); for( i=0; i<LX; i++) { ip = ( i<LX-1)?( i+1):( 0 ); in = ( i>0 )?( i-1):( LX-1); Fx = 0.; Fy = 0.; if( !is_solid_node[j][i]) { Fx+= WM*ex[1]*psi[j ][ip]; Fy+= WM*ey[1]*psi[j ][ip]; Fx+= WM*ex[2]*psi[jp][i ]; Fy+= WM*ey[2]*psi[jp][i ]; Fx+= WM*ex[3]*psi[j ][in]; Fy+= WM*ey[3]*psi[j ][in]; Fx+= WM*ex[4]*psi[jn][i ]; Fy+= WM*ey[4]*psi[jn][i ]; Fx+= WD*ex[5]*psi[jp][ip]; Fy+= WD*ey[5]*psi[jp][ip]; Fx+= WD*ex[6]*psi[jp][in]; Fy+= WD*ey[6]*psi[jp][in]; Fx+= WD*ex[7]*psi[jn][in]; Fy+= WD*ey[7]*psi[jn][in]; Fx+= WD*ex[8]*psi[jn][ip]; Fy+= WD*ey[8]*psi[jn][ip]; Fx = -G * psi[j][i] * Fx; Fy = -G * psi[j][i] * Fy; } } }

  23. Phase Separation

  24. Young-Laplace Eqn (1805) ∞ r

  25. Young-Laplace Eqn (1805)

  26. Measuring Surface Tension

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