Batch Reactor - Bioprocess Engineering: Kinetics, Biosystems, Sustainability and Reactor Design - page 143

which can be rearranged to yield

t ¼ C A0 V 0
V

f A
r A

d

d (4.5)

Since the total amount of mass is not changing with reaction or temperature and mass ¼ r V ,

we have

t ¼ C A0 r
r 0

f A
r A

d

d (4.6)

The density is a function temperature and pressure. For condensed phase reactions, the
density can be considered as constant. As we encounter liquid or liquid e gas phase reactions
quite often, constant density or constant volume batch reactors have been studied
extensively.
For a single-phase reaction mixture or for a condensed reaction mixture, if the effective
reactor volume or density is constant, Eqn (4.3) can be reduced to

d C A
d

r A ¼ (4.7)

t

which is a commonly used equation. In fact, Eqn (4.7) has been abused in many situations by
many people unintentionally. Now we know that this equation is only valid for constant
density or constant volume batch reactors.
In order to solve for a batch reactor problem, one needs to solve for a differential equation
like Eqns (4.3), (4.6) ,or (4.7) . For example, Eqn (4.6) can be integrated to yield

Z f Af

r
r 0

f A
r A

d

t f ¼ C A0 (4.8)

0

In general, the rate of reaction is a function of concentration and temperature. From
Chapter 3, we learned that the stoichiometry can be applied to relate the amount of every
species in the reaction mixture. The amount change of a component participating in the reac-
tion divided by its stoichiometric coefficient is the universal extent of reaction for a single
reaction. The stoichiometry can be written in a batch reactor as

d n j

n j

d n A

n A

¼ ¼ rV (4.9)

Equation (4.9) can be integrated to give

n j n j0
n j

¼ n A n A0
n A

(4.10)