In all actual filters the resistance to the flow
of filtrate varies with time as the precipitate
deposit on the filtering sand in sand bed
filters, or as the filter cake building up on
the cloth, screen or other filter medium. The
filter medium holds back the solids as the
filtrate passes through, and the filter cake
continues to increase in thickness, adding
its resistance to the flow of filtrate. This
action continues during filtration. At the end
of the filtration, the products are filtrate
porous filtrate cake, and fluid in the porous
of the cake.
During filtration the operation is laminar
flow and the linear velocity V are
v = (1/A) (d V/d t) = (K ρ Lw)/ L μ
= K (-Δ P) / L μ [1]
where
v linear velocity
V volume f iltrate
A area of filter media
(gc Dp
2 FRe)
K permeability of cake K = -----------
32 ff
L thickness of cake
t time of filtration
ΔP pressure drop through the cake
Relation between Volume of filtrate
and Time of filtration
In order to obtain an expression relating
filtration capacity (expressed as either the
quantity of filtrate V or the cake thickness L
) with the time of filtering t, it is necessary
to obtain the relation between the variable, L
and V . This can be done by making a
material balance between solid in slurry
filtered and the solid in cake.
Mass of solid in cake = mass of solids in
slurry.
(V+ε L A) ρ x
( 1- ε ) L A ρ s = [2]
(1-x)
where
L A volume of cake
( 1- ε ) L A volume of solid in cake
ρs density of solid in cake
ρ density of filtrate
x mass fraction of solid in slurry
ε porosity of cake
note: [(mass of filtrate/total mass) /(mass of solid
/total mass)] = (1-x)/x
Then,
mass of solid = mass of filtrate ( x / 1-x )
Then
ρ s (1-x) ( 1- ε ) - ρ x ε
V = [ ] L A [3]
ρ x
V ρ x
L = [ ] [4]
A [ρ s (1-x) ( 1- ε ) - ρ x ε]
According to equation [1]
(d V/d t) = K A (-Δ P) / L μ
K A2 [ρ s (1-x) (1- ε ) - ρ x ε ] (-ΔP)
∴(dV/dt)= [5]
μ V ρ x
This equation is an expression for the
instantaneous rate of filtration in term of
properties of the slurry, cake, quantity of
filtrate and pressure drop through the cake.
For a given slurry, the only variables subject
to
the control of the operator are pressure drop
(-ΔP), filtrate volume V, an time t. If we but :
μ ρ x
Cv = [62 K [ρ s (1-x) (1- ε ) - ρ x ε]
A2 (-ΔP)
∴ (d V/d t) = [7]
2 Cv V
If the cake porosity remains essentially
constant during filtration (as is true with a
so-called non-compressible cake and may
also occur for constant pressure drop
filtration in general).
Cv may be considered as a constant, and
equation [7] is easily integrated. For
constant pressure drop and constant
porosity, this integrates to:
Cv V2
t = [8]
A2 (-Δ P)]. absolute viscosity of filtrate
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