Fluid Flow Through Filters

Darcy's Law

\[Q = \frac{\kappa }{\eta }\frac{{\Delta P}}{{\Delta
x}}\]

Darcy's law states that the flux of fluid through a
filter is proportional to the pressure gradient across it. It is also
dependent on the fluid viscosity and filter permeability. Initially there
is a transient state where the fluid travles at a slower speed in the
filter than either side of it. Eventually the fluid flows faster through
the filter as there is less free volume in the filter and the flux is
constant everywhere (in and outside the filter).

Carman-Kozeny Equation

\[\kappa = \frac{{{P^3}}}{{\lambda \left( {1 - P} \right){}^2{S^2}}}\]

Porosity is the volume fraction of free space in the
filter. Specific surface area is the surface area of particles or fibers
the filter is made of per unit volume. Specific surface area can be expressed
as the total surface area of the solid constituents - for example, 2πRL
for fibres - divided by the total volume, which is the volume of solid
over the solid fraction - for example, LπR^{2}/(1–P)
for fibres. Particles can be modelled as randomly packed spheres, fibers
are modelled as cylinders; touching at single points so their full surface
area is assumed.

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