CHEN 4820 Fall 2010 Homework 5
(filtration equipment)
(1) Harrison prob 4.4.
(scale-up of rotary vacuum filter) Note: interpret “How significant is the medium
resistance” to mean how resistance from the filter medium alone compares to the resistance from the filter
cake.
Ans:
see Harrison key.
(2) Harrison prob 4.6 (feed and bleed microfiltration).
Note – in this problem, the variables x
f
, x
r
, and Q
f
are “known”.
Also, how Q
p
depends on x, i.e. Q
p
(x),
is “known” (so how x depends on Q
p
is also known).
Use material balances for the flows and for the cells to find the unknowns, x and R, in terms of these
knowns.
Hint: use material balances drawn around the entire system and use material balances around
the point where the recycle stream and feed stream join together.
Ans:
First get an expression for Q
p
in terms of knowns, by using the whole system balance for streams
and for cells.
Whole system balance for streams:
Q
p
= Q
f
– Q
r
Whole system balance for cells:
x
f
Q
f
= x
r
Q
r
(there are no cells in the permeate stream)
Combine them and solve for Q
p
:
g
G
=
g
±
(
1
−
²
³
²
´
)
.
Everything on the right side is known, so this eqn
yields a value for Q
p
.
However, Q
p
is a known function of x,
therefore it can be inverted, i.e. we can get
a value for x given a known value for Q
p
.
With the known relationship, x = x(Q
p
), we get a value for x.
With x now known, develop an expression for R, the recycle ratio,
in terms of other known variables,
including x, from material balances applied to the point where the recycle stream and the feed stream
merge:
Balance for streams at merger point:
Q
f
+ RQ
r
= Q
.
Balance for cells at merger point: x
f
Q
f
+ x
r
RQ
r
=
xQ
.
Multiply the first balance by x on both sides, then combine this equation with the second balance to
eliminate the xQ term:
xQ
f
+
xRQ
r
=
x
f
Q
f
+
x
r
RQ
r
.
Rearrange this eqn to solve for R:
µ
=
(
²
¶
²
³
)
(
²
´
¶
²
)
·
³
·
´
.
Use the whole system balance for streams again to eliminate Q
r
, leaving:
µ
=
(
¸
−
¸
±
)
(
¸
¹
−
¸
)
g
±
(
g
±
−
g
G
)
Everything on the right side is known, so now R is known.