Batch Reactor - Bioprocess Engineering: Kinetics, Biosystems, Sustainability and Reactor Design - page 151

or

(E4-3.9)

C A ¼ C A0 exp ð k 1 tÞ

Substituting Eqns (E4-3.9) and (E4-3.4) into Eqn (E4-3.2) , we obtain
d C B
d

¼ r B ¼ k 1 C A k 2 C B ¼ k 1 C A0 e k 1 t k 2 C B

(E4-3.10)

t

which can be rearranged to yield
e k 2 t d C B þ k 2 C B d e k 2 t k 1 C A0 e k 1 t d t ¼ t (E4-3.11)

or

d C B e k 2 t

e k 2 t d C B þ C B de k 2 t ¼ ¼ k 1 C A0 e ðk 2 k 1 Þt d t (E4-3.12)

Integration of Eqn (E4-3.12) yields

8
<

k 1
k 2 k 1 C A0 ½

e ðk 2 k 1 Þt 1; k 2 s k 1

C B e k 2 t ¼ (E4-3.13)

:

k 2 ¼ k 1

k 1 C A0 t;

Thus, the concentration of B in the isothermal constant volume reactor is given by

(E4-3.14)

8
<

k 1

e k 1 t e k 2 t Þ;

k 2 k 1 C A0 ð

k 2 s k 1

C B ¼

: k 2 ¼ k 1

e k 1 t ;

k 1 C A0 t

can be obtained either by substituting Eqns (E4-3.14) and (E4-3.5)
into Eqn (E4-3.2) or via stoichiometry. Since there is only A in the reactor initially, the total
concentration of A, B, and C is not going to change with time based on the stoichiometry
as given by the series reaction (all the stoichiometry coefficients are unity). Thus,

The concentration of C

C C ¼ C A 0 C A C B (E4-3.15)

which gives

C C ¼ C A0 k 1 e k 2 t k 2 e k 1 t
k 1 k 2

C A0 (E4-3.16)

Figure E4-3.2 shows the change of concentrations with time based on Eqns (E4-3.9) ,
(E4-3.14) , and (E4-3.16) . One can observe that there is a maximum for concentration C B
that changes with k 1 and k 2 . This maximum can be obtained by setting

t m ¼ r B ¼ k 1 C A k 2 C B

d C B
d

0 ¼

t

(E4-3.17)

k 1
k 2 k 1 C A0 ð

¼ k 1 C A0 e k 1 t m k 2 e k 1 t m e k 2 t m Þ