
225Homogeneous Reactors
The solution to the second-order differential Equation 3.340 is
CA
e
A
()B
(3.344)
where
m
Pe
m
Pe
1
2
1
2
1
=
+
=
−
α
α
(3.345)
are the roots of the characteristic equation (1/Pe)m
2
− m − kτ = 0 and
α
τ
=+1
4k
Pe
(3.346)
The integration constant A
1
and A
2
are calculated using the boundary conditions (3.342)
and (3.343). Evaluating the derivative
dC
d
A
ζ
at ʓ = 0 and ʓ = 1, we have
dC
d
APeAPe
A
B
B=
=
+
+
−
0
1
2
1
2
()
(3.347)
dC
d
APe
e
APe
e
A
Pe Pe
ζ
B=
+
()
−
=
+
+
−
1
1
2
1
2
2
1
1
2
1
2
() ()
(3.348)
Substituting Equations 3.347 and 3.348 in the boundary conditions (3.342) and (3.343)
and solving A
1
and A
2
, we set
A
e
Pe
Pe Pe
1
2
22
21
=
−
−− ...