
155Homogeneous Reactors
τ
2
02 1
2
=
−
−
Cx x
rx
AA A
AA
()
(3.40)
For a xed value of X
A2
= X
Af
(nal conversion specied), X
A1
is calculated such that the
total space time τ = τ
1
+ τ
2
is minimum. Applying the necessary condition for minimum,
we have
d
dx
d
dx
Cx
rx
Cx x
rx
AA
AO A
AA
AO Af A
AAf
τ
11
1
1
1
=
−
()
+
−
−
(
)
()
()
= 0
(3.41)
⇒
−
()
+
−
()
−
−
(
Cx
d
dx rx rx
rx
AA
AAAAA
AAf
01
11 1
111
() ()
()
))
= 0
(3.42)
q
q
C
A0
C
A2
q
C
A1
X
A1
V
1
V
1
V
2
C
A2
X
Af
= X
A
C
A1
τ
1
=
q
V
2
τ
2
=
FIGURE 3.10
A system of two CSTRs connected in series.
0 0.1 0.2 0.3 0.4 0.5 0.6
–0.6
–0.4
–0.2
0
0.2
0.4
0.6
X
Ai
Ai–1
X
A1
X
A2
X
A3
X
A4
X
A5
FIGURE P3.7
Plot of X
Ai − 1
versus X
Ai
.