How Cells Grow - Bioprocess Engineering: Kinetics, Biosystems, Sustainability and Reactor Design - page 598

Equation (E11-3.2) can be reduced to

t B min ¼ t B min

dP
dt B

P P 0
t B

(E11-3.3)

Equation (E11-3.3) implies that the production rate (based on overall time required by the
process) at the time of discharge (each batch) and the average production rate (based on total
batch time required by the process) are identical when the production rate is maximized (or
process time at minimum). In other words, the line connects (t P 0 ) and the

optimum point (t, P) on the product generation curve, and the tangential line (in terms of t B
and P) passing the optimum point (t, P) on the product generation curve have the same slope.
Therefore, one can determine the optimum point by plot the pinch line to the product gener-
ation curve passing the “feed” or initial point (t

¼ t B t P ¼ t P , P ¼

P 0 ).
The initial point I has a total batch time of zero and production concentration of zero. The
total batch time zero corresponding to the growth time of e t p , here t p is the total preparation
time. Therefore, in Fig. E11-3.1 provided, the initial point is (t

¼ t P , P ¼

¼ 6h,P ¼ 0). We plot the pinch

line passing point I (t 0), the pinch point is then the optimumpoint for the fermen-

tation discharge. The graphical solution is illustrated in Fig. E11-3.2 . Therefore, we can read
out the fermentation time and the concentrations obtained from a batch operation:

¼ 6h,P ¼

P max ¼ 0 : 562 g = L ; t f ¼ 16 : 66 h ðt ¼ 22 : 66 h Þ; and X max ¼ 8 g = L :

The production rate is given by

PV
t

F P ¼ (E11-3.4)

8 0.8

7 0.7

6 0.6

Optimum

Pinch p oint

0.5

5

0.4

4

Pinch line

X

P

3 0.3

0.2

2

0.1

1

I

0 0

0 2 4 6 8 10 12 14 16 18

(6)

t , h

FIGURE E11-3.2