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nadya
New member
Username: nadya
Post Number: 1
Registered: 04-2010
| | Posted on Monday, April 12, 2010 - 08:57 am: | |
I am trying to model a two-phase gas-liquid flow in a bubble column (bottom gas injection into liquid) using standard Eulerian approach.
Within this approach both phases are described as continuous fluids which interpenetrates each other.
The pressure is considered to be common to the two phases, and both gas and liquid are assumed to be incompressible.
Mass conservation equation for each phase:
dt(dens_l*r_l)+div(dens_l*r_l*u_l)=0
dt(dens_g*r_g)+div(dens_g*r_g*u_g)=0
where
r_l- the volume fraction of liquid,
r_g- the volume fraction of gas,
r_l+r_g=1
In analogy to the mass conversation the momentum conversation for multiphase flows is described by The Navier-Stokes equation extended by the phase volume fraction and interphasial exchange terms:
dt(dens_l*r_l*U_l)+U_l*dx(dens_l*r_l*U_l)+V_l*dy(dens_l*r_l*U_l)=-r_l*dx(P)+r_l* visc_l*del2(U_l)+Fd_l_x
dt(dens_l*r_l*V_l)+U_l*dx(dens_l*r_l*V_l)+V_l*dy(dens_l*r_l*V_l)=-r_l*dy(P)+r_l* visc_l*del2(V_l)-dens_l*r_l*g+Fd_l_y
where
U_l and V_l are X- and Y- components of the liquid velocity
dens_l - liquid density
P - pressure
visc_l - liquid viscosity
Fd_l_x, Fd_l_y are X- and Y- components of the drag force (other forces (lift force, virtual mass, etc) are neglected)
Fd_l_x =50*r_g*dens_l*(U_g-U_l)
Fd_l_y =50*r_g*dens_l*(V_g-V_l)
For gas phase
dt(dens_g*r_g*U_g)+U_g*dx(dens_g*r_g*U_g)+V_g*dy(dens_g*r_g*U_g)=-r_g*dx(P)+r_g* visc_g*del2(U_g)-Fd_l_x
dt(dens_g*r_g*V_g)+U_g*dx(dens_g*r_g*V_g)+V_g*dy(dens_g*r_g*U_g)=-r_g*dy(P)+r_g* visc_g*del2(V_g)-dens_g*r_g*g-Fd_l_y
I've got 7 equations (2 N-S for each phase and 2 continuity + equation for volume fraction)
The system is underdetermined in pressure and overdetermend in velocities.
I don't know how to set a proper equation for pressure in such system.
You've used a "Penalty Pressure" approach to N-S in the example problem "Samples | Steady_State | Fluids | Viscous.pde".
I'm not sure about using this approach in two-phase flow case.
Is it possible to solve such systems in FlexPde?
If yes, then how can i find the equation for pressure?
Thank you in advance
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nadya
New member
Username: nadya
Post Number: 2
Registered: 04-2010
| | Posted on Monday, April 12, 2010 - 10:49 am: | |
Sorry, I've made a mistake in the last equation for gas phase
The right equation is
dt(dens_g*r_g*V_g)+U_g*dx(dens_g*r_g*V_g)+V_g*dy(dens_g*r_g*V_g)=-r_g*dy(P)+r_g* visc_g*del2(V_g)-dens_g*r_g*g-Fd_l_y |
rgnelson
Moderator
Username: rgnelson
Post Number: 1349
Registered: 06-2003
| | Posted on Friday, April 16, 2010 - 02:03 pm: | |
I do not know what the proper equation for pressure is in this problem. You should consult the literature.
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