
344 Green Chemical Engineering
dC
dr
e
rrr
A
mr
2
3
2
=−+
mm
2
2
(4.161)
Substituting Equations 4.159, 4.160 and 4.161 for C
A
, dC
A
/dr and d
2
C
A
/dr
2
, respectively,
into Equation 4.157, we get
e
rrrr
e
rr
e
r
mr mr
mr
m
m
m
2
2
−+
+−
23 2
22 1
0
α
which on cancellation of terms reduces to
e
mr
m
2
α 0
Thus, Equation 4.159 is a solution to the second-order differential Equation 4.157 for
=± α. Hence, the general solution to Equation 4.157 can be written as
C
A
e
A
e
A
kD Rr
DR
pA pA
=+
()
−
(/)/ (/
(4.162)
Dening the Thiele modulus
RkD
ρ
Equation 4.162 can be written in terms of
Φ as
CA
e
A
e
A
rR rR
=+
−
(/
(4.163)
The denition of the Thiele mod ...