Batch Reactor - Bioprocess Engineering: Kinetics, Biosystems, Sustainability and Reactor Design - page 150

t = 0, C A =C A0 ,

C B = 0,
C C = 0

A B C

A,
B,
C

t = t ,C A ,

C B ,
C C

FIGURE E4-3.1 A schematic diagram of the batch reactor.

Solution. This is a set of two reactions occurring in series. Figure E4-3.1 shows a sketch of the
reactor system. Mole balance for species j leads to

d n j

d t ¼ r j V

(E4-3.1)

Since the volume is constant, we have

d C j
d

¼ r j (E4-3.2)

t

We next find the rates of reaction. Applying Eqn (3.66), we obtain

r A ¼ n 1A r 1 þ n 2A r 2
¼r 1 ¼ k 1 C A

(E4-3.3)

r B ¼ n 1B r 1 þ n 2B r 2
¼ r 1 r 2 ¼ k 1 C A k 2 C B

(E4-3.4)

r C ¼ n 1C r 1 þ n 2C r 2
¼ r 2 ¼ k 2 C B

(E4-3.5)

Substituting Eqn (E4-3.3) into Eqn (E4-3.2) , we obtain

d C A
d

¼ r A ¼k 1 C A (E4-3.6)

t

Separation of variables and integration,

Z C A Z t

d C A
C A

¼ k 1 d t (E4-3.7)

C A0 0

We obtain

ln C A

C A0 ¼k 1 t

(E4-3.8)