Continuous Cultivation - Bioprocess Engineering: Kinetics, Biosystems, Sustainability and Reactor Design - page 613

-r Smax

Growth rate
(from batch data)

Substrate mass balance
Slope = - D
-r S = D ( S 0 -S )

-r S

0

0

S S 0

FIGURE 12.6 A sketch of substrate mass balance on the substrate concentration vs substrate consumption rate
plane.

which is the general mass balance for substrate over an ideal chemostat. Figure 12.6 shows
a schematic of substrate mass balance on the substrate concentration vs substrate disappear-
ance rate plane. Clearly, substrate mass balance is a straight line passing through ( S

¼ S 0 ,

D. The nontrivial solution is the intercept of the mass
balance line and the substrate consumption curve (by the cells). The substrate consumption
rate can be obtained through stoichiometry (or yield factor) by

r S ¼ 0) with a negative slope of

m G X
YF X = S

r S ¼ (12.15)

where YF X/S is the yield factors (g-cell/g-S). If no cell growth and an extracellular product is
generated due to the consumption of substrate, stoichiometry leads to

m P X
YF P = S

r S ¼ (12.16)

m P is the specific rate of extracellular product formation, g-P/(L $ g-cells h), and YF P/S
is the yield factor (g-P/g-S).
In most cases, cell growth is apparent, Eqn (12.14) is reduced to

where

DðS 0 SÞ¼ m G X
YF X = S

(12.17)

Since D ¼ m net ¼ m G k d at steady state,

D
D þ k d YF X = S ðS 0 SÞ

X ¼ (12.18)

Using Eqn (12.11) , the steady-state cell concentration can be expressed as

S 0
D þ k d

K S
m max D k d

X ¼ YF X = S D (12.19)