With deep-bed filtration, illustrated by Fig. 3-2, the particles to be captured are entrapped in the interstices among particles held in a deep bed. The sizes of the interstices are roughly the same order as the size of the particles in the bed, so, in order to maximize the capacity of the bed, the particles are laid down in such a way that a gradient of particle sizes exists in the bed. Larger bed particles at the top, or beginning, of the bed will tend to trap the larger particles and smaller bed particles at the bottom of the bed will trap the smaller ones. The example on the sketch shows a gradation in sand particle sizes produced by taking the sand from different locations along the eastern seaboard. In practice, although sand is frequently used in such deep-bed filters, it usually is river bed sand and not salt-water sand, because the former tends to be more free of extraneous materials.
The pressure drop for laminar flow (which will certainly hold for molten
polymer melts and polymer spin dopes) can be calculated from the mechanical
energy balance equation, MEB, and the Kozeny-Carman equation. The MEB (from
Bennett & Myers) is written as:
ub is the bulk, or average velocity, z
is the change in elevation, p is pressure, is
density, lwf the lost work due to friction, Ws is the shaft work introduced
to the system. The Kozeny-Carman equation is:
f is the friction factor, D the particle diameter, ubs is the superficial liquid velocity, L is the length of the bed, is the liquid density, and is the void fraction of the bed.
On account of the high liquid viscosities, pressure drop through the
bed can be enormous. This can lead to significant temperature increase,
since one can demonstrate from thermodynamics:
H = P/+ CpT = 0 (3-7)
The enthalpy change is zero since the operation is adiabatic and there is no shaft work in the process. Temperature increases of 20 F are readily possible.