Bioreactor Design and Operation - Bioprocess Engineering: Kinetics, Biosystems, Sustainability and Reactor Design - page 933

TABLE 18.7 General Equations for Temperature Change with Time During Heating and
Cooling Periods in Batch Sterilizations. T 0 Is the Initial Temperature of the
Medium; T S Is the Temperature of the Isothermal (Saturated) Steam; T ci Is the
Temperature of Water Entering the Jacket; b's Are Constants Changing Only
with the Properties of Medium and Heat Transfer Fluid; as well as Heat Transfer
Coefficient or Heating rate (for Electrical Heating)

Heat transfer method Temperature e time profile Integral function: E ( x )

T ¼ T 0 1þ b 1 t
1þb 0 t

Direct sparging with steam Eqn (18.33)

Electrical heating T ¼ T 0 (1 þ bt) Eqn (18.34)

T ¼ T S 1þ T 0 T S
T S

e bt

Heating from isothermal steam
in jacket

Eqn (18.31)

T ¼ T ci 1þ T 0 T ci
T ci

e bt Cooling with cold water in jacket Eqn (18.31)

as temperature is a function of time. The total number of cells can be computed by

N 0 ¼ VC X 0 (18.28)

with V being the reactor volume and C X0 being the number concentration of cells initially
present in the reactor.
The temperature change in the vessel is a function of the heating/cooling methods.
Table 18.7 shows some of the functional forms of temperature change during the heating
up and cooling down periods of the sterilization process. In Chapter 4, we have learned how
to integrate Eqn (18.27) , i.e. H-Factor. Commonly, there are three types of time e temperature
profiles: linear (electrical heating), exponential(noncontactheatexchanger),andhyper-
bolic (direct mixing/contact with another fluid stream). The integral in Eqn (18.27) can
be represented by

E ad
RT

E ad
RT 0

Z t

e E ad

RT d

t ¼ ET ET (18.29)

0

where ET(x) is the integral function with a form dependent on the temperature profile, T is
the absolute temperature (K) at time t, and T 0 is the absolute temperature at time t ¼ 0 or the

starting point of the integration. Here

x ¼ E ad
RT

(18.30)

e bt , the corre- If the temperature changes with time exponentially T ¼ T a þðT 0 T a Þ

sponding integral can be estimated by

" X n
i¼0 ð1Þ i i !

#

m¼0 ð1Þ m x iþm1 E ad

m1

X i

ETðxÞ¼ E ad
bRT a

expðxÞ

RT a x

þ / (18.31)