Enzymes - Bioprocess Engineering: Kinetics, Biosystems, Sustainability and Reactor Design - page 364

Equations (E8-4.5) and ( E8-4.6 ) lead to

k 1 þ k 2 ½

k 1 ½ S

½ E $ S ¼ E (E8-4.7)

k 3
k 3 ½

k 3
k 3 k 1 þ k 2 ½

k 1 ½ S

½ S $ E $ S ¼ E $ S ½ S ¼ E ½ S (E8-4.8)

The total amount of enzyme in the reactor is the sum of free enzyme [E] and those asso-
ciated with the substrate, [E $ S] and [S $ E $ S]. Thus,

½ E 0 ¼½ E þ½ S $ E $ S þ½ E $ S

(E8-4.9)

k 3
k 3 k 1 þk 2 ½

k 1 ½ S

k 1 þ k 2 ½

k 1 ½ S

¼½ E þ E ½ S þ E

which leads to

½ E 0

½ E ¼ (E8-4.10)

k 3
k 3 k 1 þ k 2 ½

k 1 ½ S

k 1 þk 2

k 1 ½ S

1þ S þ

and

k 1 ½ 0

k 1 þ k 2

½

S E

½ E $ S ¼ (E8-4.11)

k 3
k 3 k 1 þ k 2 ½

k 1 ½ S

k 1 þ k 2

k 1 ½ S

1þ S þ

Substitute Eqns (E8-4.10) and (E8-4.11) into Eqn (E8-4.4) , we obtain

k 2 ½ E 0 ½ S

r P ¼ (E8-4.12)

k 1 þ k 2
k 1

k 3
k 3 ½

2

þ½ S þ S

(a) Equation (E8-4.12) leads to
(1) When [S] 0;

(2) When [S] is low, r P increases with S;
(3) When [S] is very high, r P decreases with increasing [S] because of the [S] 2 term in the
denominator;
(4) There is only one maximum (peak) value for r P as only [S] in the numerator and [S]
and [S] 2 in the denominator.
Therefore, the data or curve shown in Fig. E8-4.1 is consistent with Eqn (E8-4.12) .

¼ 0, r P ¼

(b) Mole balance on the substrate around the CSTR ( Fig. 8-4.2 ) leads to:

QS

0

QS þ r s V ¼ 0