Page 317 - Biosystems Engineering

294 Cha pte r Ni ne

where A = filtration area, ft 2
2
G = conversion factor, 32.17 (lb mass) (ft)/(lb force) (s )
c
P = filtration pressure (i.e., pressure drop of filtrate through
the filter), lb force/ft 2
R = filter cake resistance, L/ft
c
R = initial filter stance (resistance of filter medium and filter
f
channels), L/ft
U = the filtration rate, ft/s
3
dV/dθ = filtration rate, ft /s filtrate flow
U = viscosity of filtrate (lb mass)/(ft) (s)
e = filtration time, s
The filtrate cake resistance can be written as
α wv
R = (9.9)
c
A
where w is pounds of solids deposited per cubic foot of filtrate and
A is proportionality constant, also known as specific resistance [ft/
(lb mass)].
Substituting the value of R in Eq. (9.8), we can write
c
2
dV = APg c
+
dθ μ α wv AR ) (9.10)
(
f
Considering R as being equivalent to the resistance of a fictitious
f
layer of filter cake of equal resistance, Eq. (9.10) can be written as
ρ
2
dV = Ag c
dθ αμ ( + (9.11)
wV V )
0
where V is the volume of filtrate required to form a filter cake of
0
resistance equal to the initial filter resistance R . The time required to
f
3
filter V ft of filtrate will be θ .
0 0
Filtration at Constant Pressure
During constant pressure filtration, the filtrate flow rate goes on
decreasing with time. Integrating Eq. (9.10), we can write
θ = αμw + μR f = +
1
V 2 APG APG KV C 1 (9.12)
2
c c
Similarly, by integrating Eq. (9.11), we get
2
2 APG
+
( +
=
θθ
(vV ) 2 c ( + ) K θθ ) (9.13)
0 αμ w 0 0