Batch Reactor - Bioprocess Engineering: Kinetics, Biosystems, Sustainability and Reactor Design - page 166

by default, you only need to change the lines between
“Sub Func(f, ByVal x As Double, ByVal y, Optional c)” and “End Sub.”
In between these lines, you are to input the differential equations. You will need to write
each equation separately on a different line. In the example shown in the code, there are two
differential equations:

d y 1

d

ð Þ¼ ð1Þ ð Þ

x ¼ f 1 ¼ c 1 y 1

f 1 c y 1 0

f 2 ¼ c 2 þ c 3 y 2 ^ c 4 0:1 y 1 0

d y 2

d

x ¼ f 2 ¼ c 2 x þ c 3 y c 4

2 0:1y 1

where c j 's are reserved as parameters (i.e. true constants) that one can pass from excel work-
sheet to the visual basic program at run time. This gives us freedom to make each code to be
more general and thus capable of solving a class of problems. When you change the values of
c j 's, the solution to the set of equations changes. You will not need to change the visual basic
code just for the different values of c j 's.
Example 4-8. Concentration profile for a series reaction with feedback regulation. A liquid
reaction system is carried out in a batch reactor:
A ! B ! C

(E4-8.1)

with

k 1 C A
K 1 þ C A þ K C C C

r 1 ¼ (E4-8.2)

k 2 C B
K 2 þ C B

r 2 ¼ (E4-8.3)

1 mol $ L 1 $ h 1 , k 2 ¼ 0.5 mol $ L 1 $ h 1 , K 1 ¼ 0.1 mol $ L 1 , K 2 ¼ 0.2 mol $ L 1 , and where k 1 ¼

K C ¼ 100.

Find the change of concentrations of A, B, and C as a function of time between 0 and 100 h
in a constant volume isothermal batch reactor starting with
a. pure A initially at C A0 ¼ 10 mol $ L 1 ;C B0 ¼ 0 mol $ L 1

C C0 ¼

10 mol $ L 1 ,C B0 ¼ 0 mol $ L 1 ,C C0 ¼ 10 mol $ L 1

b. C A0 ¼

Solution. This is a set of two reactions occurring in series. Mole balance for species j leads to
d n j

d

t ¼ r j V

(E4-8.4)

Since the volume is constant, we have

d C j
d

¼ r j (E4-8.5)

t

We next find the rates of reaction ( Fig. E4.8-1 ). 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
K 1 þ C A þ K C C C

(E4-8.6)