Batch Reactor - Bioprocess Engineering: Kinetics, Biosystems, Sustainability and Reactor Design - page 155

which can be rendered to

k 1 k 1

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

k 1 (E4-3.18)

or

(E4-3.19)

lnðk 2 =k 1 Þ
k 2 k 1

t m ¼

and the corresponding concentration of B is given by

k 2
k 1

k 2
k 2 k 1

(E4-3.20)

C Bmax ¼ C A0

One can observe that at short times, the formation of intermediate product is noticeable. The
time at whichmaximal concentration of B occurs decreases with increasing k 2 . However, at long
timescale, only the formation of the final product C is important and the series reaction may be
simplified to a first-order reaction from reactant A to product C if timescale is long. The reaction
rate constant can be approximated by the smaller of the two for the approximate reaction.
When simple kinetics is combined with constant volume or constant density for single
reactions in a batch reactor, closed form analytic solutions are usually easy to achieve. A
list of some of these cases is shown in Table 4.2 .
While simple problems can be solved by hand, this last example showed how tedious for
even a simple problem could be. At this point, one can appreciate that an automatic inte-
grator is helpful in solving batch reactor problems. There are many software programs one
can use for integration: Maple, Mathcad, Matlab, to name a few. We have also supplied
a visual basic code that can run in Microsoft Excel. This is a good option as Microsoft Excel
can be applied to solve most of the programs a process engineer encounter when an auto-
matic integrator is added. The excel visual basic program provided is ODExLims. We shall
discuss the solution in Numerical Solutions of Batch Reactor Problems.

4.2. BATCH REACTOR SIZING

We have learned now how to compute the required reaction time for a given reaction
kinetics to proceed onto a desired reaction extent (or conversion). How big a reactor do we
need to carry out the bioprocess? To do this, let us look at Fig. 4.2 to find the answer.
Mass balance of the raw materials leads to
Q ¼ V

V
t P þ t f þ t U t B ¼

(4.21)

where Q is the rate of raw materials coming into the facility, V is the volume of the raw mate-
rials can be loaded for each batch, which is the effective reactor volume, and t B is the batch
time or the total time required to complete one batch operation. The batch time required
includes the preparation time, t P , for loading the reactor with reaction mixture and raising
the temperature to reaction temperature; the reaction time, t f , to achieve the desired