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Old   June 2, 2000, 08:30
Default Flow through filter media
Franco Belosi
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I'd like to simulate the flow through a membrane filter. Really speaking I 'd just be interested in simulating the homogeneous streamline dispersion due to the high filter resistance (membrane of 0.8 um porosity). The patch "porous" with the porosity value seems to me not be enough to simulate the filter resistance to the flow. Anyone knows ho to do that?
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Old   June 6, 2000, 08:14
Default Re: Flow through filter media
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Hi Franco!

When simulating flow through a porous medium you must also define the resistance tensor in USRBF.

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Old   June 8, 2000, 03:30
Default Re: Flow through filter media
Duane Baker
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Hi Franco,

If you think about the process of volume averaging over the details of the interconnected passages in the porous media one finds that a number of terms have to be modelled. Essentially the volume averaging process results in a loss of detailed information at length and time scales that are not resolved. The processes that are not resolved through the volume and the surfaces of the porous media must be modelled.

For a more thourough understanding see Momentum, energy, and mass transfer in continua by John Charles Slattery he did a bunch of fundamental work on porous media in the 70's.

We can show that this volume average of a divergence of a flux is equal to 2 terms:

1. the divergence of the volume average of the flux

2. the surface integral of the flux over the inner neglected) porous surfaces

This is then applied to the conservation equations.

For continuity (steady) <diff_j(u_j)>=diff_j<u_j> where <> is this local space averaging operator. And we basically have the usual continuity equation except that we use the volume averaged velocity (sometimes called superficial velocity) and not the local velocity. Average local velocity is <u_j>/porosity. This all assumes no mass flux across the surfaces and hence term 2 is zero.

OK this seems a bit of a complex process and many texts jump to this conclusion for continuity right away see for example Bejan's Convection Heat Transfer. This result was easy for 2 reasons firstly, the mass flux is linear and secondly, there is no flux across the inner surfaces. But what about for the other conservation equations. Well this is where it gets complicated and depending on the terms necessary has led to a whole host of different models. Darcy, Brinkman, Forscheimer, Morgan's law bring a few to mind.

For the steady momentum equations then we have to model the advection terms over the local volumes <diff_j(rho*u_j*u_i)>, the flux of momentum (shear stress) on the inner surfaces, and any source terms ie bouyancy, centrifugal forces, etc. Well, these need some empirical info and some scale analysis but basically we get 2 momentum sink terms: one proportional to viscosity*velocity and the second proportional to density*velocity^2. The first is then due to laminar viscous shear at the surfaces and the second is turbulent. This is sometimes called Morgan's law and the coeficients are determined empirically...a good starting point is the Ergun equation.

Also of note for any non-linear transport equation, including momentum there is an effective diffusion which results from the averaging over the nonlinear advection terms. This is somewhat analogous to the modelling of turbulent fluxes. For example, these nonlinear averaged advection terms in the termperature equation is modelled as an added or effective porous diffusion coefficient * the average thermal gradient ie. k_eff = k+(rho*Cp*length_scale) where the length_scale is of the order of the pore size. See Burnmeister's text and a bunch of work by Zehnder and Schlunder.

Anyhow, I hope this helps.................Duane
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Old   June 8, 2000, 03:36
Default Correction: Flow through filter media
Duane Baker
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Hi, in the first message, something did not print correctly:

For steady incompressible continuity: <diff_j(u_j)>=diff_j<u_j> and we simply have the continuity equation for the volume averaged velocities.

I hope this clarifies.....Duane
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Old   June 8, 2000, 03:41
Default Re: Correction: Flow through filter media
Duane Baker
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Again the terms in <> did not print corretly:

For steady incompressible continuity: Ave(diff_j(u_j))= diff_j(U_j) where u is local velocity and U is volume averaged. Therefore we simply have the continuity equation for the volume averaged velocities.

I hope this clarifies.....Duane
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