Multiphase LB Models
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![Page 1: Multiphase LB Models](https://reader035.fdocuments.in/reader035/viewer/2022081801/56812be6550346895d90645c/html5/thumbnails/1.jpg)
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
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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
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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
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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
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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
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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:
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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Incorporating Forces
F = ma = m du/dt
F
u
U = u + F1/ + F2/ + ...
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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
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Simplified EOS
2
63 G
P
22
GRTRTP
3
1RT
0
0
e
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Single Component Multiphase
Modifications to code
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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]));
}
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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;
}
}
}
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Phase Separation
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Young-Laplace Eqn (1805)
21
11
rrP
rP
2 0P
∞
r
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Young-Laplace Eqn (1805)
21
11
rrP
rP
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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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