Theory and Technical Background
Uni-Minn Development Corporation was asked to design a filtration system
capable of filtering 500 gallons per hour of a slurry solution containing four percent
Celite-500 (17.3 μm particle diameter) by mass. The scaled up filtration system is to be
of rotary drum type, and capable of continuous operation. Two days of bench-scale
testing were performed in the Uni-Minn Development Corporation’s Unit Operations
Laboratory to determine the necessary parameters for scale up.
Basic Filtration Theory
Filtration is a mechanical separation process that separates the two phases of a
suspension from each other. Filtration plays a critical role throughout industry, and is
particularly important in water treatment, paper making, and mineral processing.
Filtration is used to recover solids, to clarify liquids, or to recover both phases.
The filtration apparatus in the Uni-Minn Development Cooperation’s Unit
Operations Laboratory is a vertical pressure leaf filter. Specific details of the apparatus
are contained in the apparatus subsection, but a basic schematic is shown in Figure 1.1
below. A vacuum is utilized to force flow of a slurry solution through a filter medium of
cross-sectional area A. Out the top of the filter, flows a volume, V, of filtrate. A filter
cake, of length L, forms on the bottom of the filter medium. The figure is drawn at time t
in the filtration cycle.
Figure 1.1 – Vertical Pressure Leaf Apparatus Setup.
The filter cake that forms on the bottom of the filter medium can be modeled as a
packed bed of particles. As such, the Carmen-Kozeny relation for laminar flow in a
packed bed of particles holds. That is,
pc k1 v 1 S0
where k1 is a constant (4.17 for random particles of definite size and shape), μ is the
viscosity of the filtrate, ε is the void fraction of the cake, Δpc is the pressure drop across
the cake, S0 is the specific surface area of the particle per volume, and v is the linear
velocity is given by
Additionally, a material balance gives
LA 1 p cS V LA
where ρp is the density of the solid particles in the cake and cs is the mass of solids per
volume of filtrate. The volume of liquid entrained in the solid particles is usually small
compared to the amount filtered. As such, an approximation that the εLA term is
negligible is justified and thus this approximation will be made in the subsequent
Combining equations 1, 2 and 3, yields
k1 1 S0 csV
where α is the specific cake resistance, which is defined as
k1 1 S02
An analogous analysis for the resistance of the filter medium itself can be made.
This results in the equation
1 dV p f
where Rm is the resistance or the filter medium to filtrate flow. Since the resistances to