Page 308 - Modeling of Chemical Kinetics and Reactor Design

278 Modeling of Chemical Kinetics and Reactor Design

Integrating Equation 5-52 between the limits of fractional con-
version t = 0, X = 0 and t = t, X = X gives
A
A
A
X A 1+ ( X ) N t
∫ A 2 dX A = k AO ∫ dt (5-53)

0 1− ( X A ) V O 0

Expressing the left side of Equation 5-53 into partial fraction gives

Ι= 1 + X A 2 ≡ A + B 2
− (1 X A ) 1 − X A − (1 X A ) (5-54)

1+ X = ( 1− X )+ B (5-55)
A
A A
Putting X = 1 into Equation 5-55 gives 2 = B. Putting X = 0
A
A
into Equation 5-55 yields
1 = A + B (5-56)

and A = –1. Substituting the values of A and B into the right side of
Equation 5-54 and integrating between the limits results in

X A X A
Ι= ∫ − 1 dX A ) + ∫ 2 dX A 2 (5-57)
0 − (1 X A 0 − (1 X A )

−
Ι= ( [ ln 1 X )] X A +   2   X A = kC t
A AO
0 1 − X
 A  0
 2X 
(
 ln 1− X A ) + A  = kC AO t
 1− X A 

or

t = 1  (1 − X )+ 2 X A  (5-58)

 ln
A
kC AO  1 − X A 