Multiphase LB Models

Multi- Component Multiphase

Miscible Fluids/Diffusion (No Interaction)

Immiscible Fluids

Single Component Multiphase

Single Phase

(No Interaction)

Num

ber

of

Com

pone

ntsInteraction Strength

Nat

ure

of

Inte

ract

ion

Attractive

Repulsive

LowHigh

Inherent Parallelism

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()• Modern EOS for water

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)

nRTPV

V

nRTP

Molar Volume

mV

RTP Perfect or ideal gas law

(EOS):

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

V

nRTP

Single Component D2Q9 LBM (with c = 1 lu ts-1)

mV

RTP

32 scP

So, if 1/Vm (mol L-1) is density, cs2 = RT

van der Waals EOS

2

V

na

nbV

nRTP

•‘a’ term due to attractive forces between molecules [atm L2 mol-2]

•‘b’ term due to finite volume of molecules [L mol-1]

V

nRTP Perfect or ideal gas law (EOS):

van der Waals EOS:

Molar Volume

21

mm Va

bV

RTP

van der Waals gas law (EOS):

Vm is the volume occupied by one mole of substance. The gas laws can be re-written to eliminate the number of moles n.

2

V

na

nbV

nRTP

CO2: P-V Space

0

50

100

150

200

250

300

0 0.1 0.2 0.3 0.4 0.5Vm (L)

P (

atm

)

Ideal, 298K

Supercritical, 373K

Critical, 304K

Subcritical, 200K

Liquid-Vapor Coexistence: CO2, P-V Space

6061626364656667686970

0 0.1 0.2 0.3Vm (L)

P (

atm

)

Subcritical, 293KVapor Pressure

Liquid Molar Volume

Vapor Molar Volume

Vapor Pressure

a = 3.592 L6 atm mol-2

b = 0.04267 L3 mol-1

Maxwell Construction: CO2, P-V Space

6061626364656667686970

0 0.1 0.2 0.3Vm (L)

P (

atm

)

Subcritical, 293KVapor Pressure

A

B

Maxwell Construction: Area A = Area B

Liquid Molar Volume

Vapor Molar Volume

)( ,,

,

,lmgm

V

V m VVPPdVgm

lm

Vapor Pressure

Flat interfaces only!

Gives: --vapor pressure--densities of coexisting liquid and vapor

Water, 298K: P-V Space

-1000

-800

-600

-400

-200

0

200

400

600

800

1000

0 0.5 1 1.5 2Vm (L)

P (a

tm)

van der Waals

Ideal

van der Waals Non-Quantitative:Water, 298K: P- Space

-2000

-1500

-1000

-500

0

500

1000

1500

2000

0 200 400 600 800 1000 1200 1400 (kg/m3)

P (a

tm)

Ideal Gas

van der Waals

IsothermalCompressibilityof Water

Pvap

Including Directional H-Bonds:Water, 298K: P- Space

-2,000

-1,500

-1,000

-500

0

500

1,000

1,500

2,000

0 200 400 600 800 1000 1200 1400 (kg/m3)

P (a

tm)

Ideal Gas

Truskett et al(1999)

IsothermalCompressibilityof Water

Spinodal

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

FHP LBM Fluid Cohesion• An attractive force F between nearest neighbor fluid

particles is induced as follows:

6

1

)(),(),(a

aa ttGt eexxxF

00 exp

Shan, X. and H. Chen, 1994. PRE 49, 2941-2948.

•G is the interaction strength

• is the interaction potential:

• 0 and 0 are arbitrary constants

0

0.5

1

1.5

2

2.5

3

3.5

0 100 200 300 400 500 600 700 800 900 1000

Density (mu lu-2)

Other forms

possible/common

D2Q9 SCMP LBM Fluid-Fluid Interaction

0

0

e

8

1

),(),(),(a

a ttwtGt aa eexxxF

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 particlesForce is stronger when the density is higher

Incorporating Forces

F = ma = m du/dt

F

u

U = u + F1/ + F2/ + ...

LBM Non-ideal Equation of State

Water EOS after Truskett et al., 1999. J. Chem. Phys. 111, 2647-2656.

Non-ideal Component

Liquid/vapor coexistence at equilibrium (and flat interface) determined by Maxwell construction

Realistic E

OS

for w

ater: Fo

llow

s ideal g

as law

at low

den

sity, com

pressib

ility of w

ater at h

igh

den

sity and

spin

od

al at hig

h ten

sion

No repulsive potential in LB model

Shan and Chen, 1994. PRE 49, 2941-2948

0

0

e 2

2 GRT

RTP

Simplified EOS

2

63 G

P

22

GRTRTP

3

1RT

0

0

e

Single Component Multiphase

Modifications to code

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]));

}

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;

}

}

}

Phase Separation

Young-Laplace Eqn (1805)

21

11

rrP

rP

2 0P

∞

r

Young-Laplace Eqn (1805)

21

11

rrP

rP

Measuring Surface Tension

y = 14.332x - 0.0016

R2 = 0.995

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07

1/r (lu-1)

P

(m

u t

s-2)