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)
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