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Encyclopedia of Physical Science and Technology EN014A-654 July 28, 2001 16:35

Reactors in Process Engineering 33

region, partially reacts, and then flows out of the region. i. Example. A → B + C (irreversible, aqueous reac-
Doing a material balance, we find rate in − rate out + rate tion). The rate expression can be written as
of generation = rate of accumulation.
r A =−kC A .
In equation form
dN i Using this rate expression and the constant density ideal
˙ n i0 − ˙ n i + G i = ,
dt batch reactor equation gives
where ˙ n i0 is the molar flow rate of i in, ˙ n i the molar flow
dC A
rate of i out, G i the rate of generation of i by chemical =−kC A .
dt
reaction, and dN i /dt the rate of accumulation of i in the
region. The rate of generation of i by chemical reaction is Integrating with an initial concentration C A0 at t = 0gives
directly related to the rate of reaction by C A
t
ln =−kt [constant volume, V ],
V t

dN i C A0
G i = r i dV = .
0 dt where t is the time for the batch reaction.
It is often convenient to work with fractional conversion
2. Ideal Batch Reactor Equation of a reactant species. Let i = A, a reactant, then
t
A batch reactor has no inlet or outlet flows, so ˙ n i0 = ˙ n i = 0. N A0 − N A N A0 /V − N A /V t
Perfect mixing is assumed for this ideal reactor, and the X A = = t
N A0 N A0 /V
rate r i is independent of position. This changes our gener- t
ation term in the general reactor design equation to and if V is constant,
V t C A0 − C A
t
X A = [constant V ]
t
r i dV = r i V . C A0
0
Substituting into the ideal batch reactor equation gives
Then, by the general design equation, our ideal batch
reactor equation becomes dX A t
−C A0 = r i [constant V ]
1 dN i dt
= r i .
t
V dt ii. Example. A → B + C (elementary, constant vol-
This equation does not define the rate r i , which is an al- ume reaction). The rate expression can then be written as
gebraic expression independent of reactor type such as
2
r i = kC . r A =−kC A =−kC A0 (1 − X A ),
i
where C A = C A0 (1 − X A ). Therefore,
a. Constant volume batch reactors. For the
special case of constant volume or constant density (usu- dX A =−kC A0 (1 − X A ).
−C A0
ally values for the mixture, not the reactor), we can dt
simplify the ideal batch reactor equations. Starting with Integrating with the boundary condition X A = 0at t = 0,
the ideal batch reactor equation gives
t
1 dN i −ln(1 − X A ) = kt [constant V ]
= r i ,
V dt
t
Given a rate constant k and a desired conversion, the time
the volume is placed inside the differential and changed
for the batch reaction can be calculated.
to concentration:
t
d(N i /V ) dC i b. Variable volume batch reactors. In general, the
= = r i .
dt dt equations developed previously assumed constant volume
t
[constant V , ideal batch reactor]. or constant density. For gas-phase reaction such as
A + B = C, the total number of moles decrease, and the
This equation is usually valid for liquid-phase reactions
volume (or density) changes.
and for gas reactions where the sum of the stoichiometric
Our ideal batch reactor equation, written in terms of any
numbers equals zero, but it is invalid for constant pressure
reactant A, can be changed to reflect a change in volume.
gas-phase reactions with mole changes.
For example,
When the rate expression is known, this equation yields
t
the major design variable, time, for a batch reaction of dN A d(C A V )
−r A =− =− ,
t
t
given concentration or conversion. V dt V dt