Bioreactor Design and Operation - Bioprocess Engineering: Kinetics, Biosystems, Sustainability and Reactor Design - page 934

Since Eqn (18.30) is an asymptotic series, it diverges if too many terms are taken. At small
value of n, increasing the number of terms (n) leading to convergent value for ET(x). The
value of n is taken such that

m¼0 ð1Þ m x nþm1 E ad

m1

m¼0 ð1Þ m x nþm2 E ad

m1

X n nþ1

n !

RT a x

ðnþ1Þ !

RT a x

(18.32)

i.e. maintaining a converging trend of the series. We stop the series before it starts to diverge.

T ¼ T 0 1þ b 1 t
1þb 0 t

, then the integral

If the temperature is a hyperbolic function of time,

is given by

" X n
i¼0 ð1Þ i ði þ1Þ !

#

i2

ETðxÞ¼ ð 2 b 0 b 1 ÞE ad
ðb 1 b 0 Þ

x E ad
RT 0

b 0
b 1 b 0

expðxÞ þ / (18.33)

2

RT 0

which is also an asymptotic series. Restriction on the number of terms to monotonously
decreasing value of successive terms applies.
If the temperature changes with time linearly, T ¼ T 0 (1 þ bt), the integral can be estimated

by

RbT 0 expðxÞx 2 h 12x 1

þ . þð1Þ n ðnþ1Þ ! x n i ETðxÞ z E ad

n < x (18.34)

Thus, the dimensionless time can be computed if the temperature profile is known.
The higher t S is, the more likelihood of being successful in a fermentation. For example,
a dimensionless time t S of 4.605 yields one unsuccessful fermentation in 100 fermentations,
whereas doubling the dimensionless time t S to 9.21 yields one unsuccessful fermentation
in 10,000 fermentations. Therefore, a dimensionless time of 10 should be sufficient for
most applications.

18.7.2. Continuous Thermal Sterilization
Like reactor operations, the sterilization can be operated resembling those of PFR and
CSTR operations.

18.7.2.1. Thermal Sterilization in a CSTR
Fig. 18.10 shows a sketch of a well-mixed sterilizer. Viable cell balance in the CSTR leads to

d ðC X VÞ

d

QC X 0 QC X k d C X V ¼ ¼ 0 (18.35)

t

Therefore, the probability of a microorganism surviving the heat treatment in

aCSTRis

pð s Þ¼ C X

1
1þ k d s

C X 0 ¼

(18.36)