How Cells Grow - Bioprocess Engineering: Kinetics, Biosystems, Sustainability and Reactor Design - page 589

where k L is the mass transfer coefficient in the liquid medium, a c is the cell surface
area per unit volume of the cell, and r cell is the density of the cells. At pseudosteady
state, the substrate uptake rate is identical to the substrate transfer rate to the cells.
Therefore,

m max S c X

r cell ðS S c Þ

Xm G ¼

K S þ S c X ¼

YF X=S N S c ¼ YF X=S k L a c (11.58)

which yields,

K S m G r cell m G

YF X=S k L a c

S ¼

m max m G þ

(11.59)

Rearranging Eqn (11.59) in the explicit form of growth rate m G , we obtain

2
4

3
5

t

r cell m max
YF X=S k L a c

4S r cell m max
YF X=S k L a c

S þ K S þ

m G ¼ 1 1 (11.60)

! 2

r cell
YF X=S k L a c

2

S þ K S þ r cell m max
YF X=S k L a c

which is the exact specific rate of growth when external diffusion to the cells are considered.
However, this equation may be approximated by noting
" # 2 " # 2 " # 2

r cell m max
YF X=S k L a c

r cell m max
YF X=S k L a c

r cell m max
YF X=S k L a c

4S r cell m max
YF X=S k L a c

S þ K S þ > S þ ¼ S þ (11.61)

since all the parameters are positive. Equation (11.61) leads to

4S r cell m max
YF X=S k L a c

! 2 <

1 (11.62)

S þ K S þ r cell m max
YF X=S k L a c

Thus, the first order approximation to Eqn (11.60) gives

r cell m max
YF X=S k L a c

4S r cell m max
YF X=S k L a c

S þ K S þ

m G ¼ (11.63)

2

! 2

r cell
YF X=S k L a c

2

r cell m max
YF X=S k L a c

S þ K S þ

That is,

m max S
S þ K S þ

¼ m max S
S þ K 0 S

m G ¼ (11.64)

r cell m max
YF X=S k L a c