Bioreactor Design and Operation - Bioprocess Engineering: Kinetics, Biosystems, Sustainability and Reactor Design - page 938

which corresponding to the dimensionless sterilization time of

1
2 k d s ln

4ðk d s þ2Þ

t S ¼

k d s ðk d s þ8Þ lnðC X 0 VÞ

(18.49)

Therefore, the effectiveness of a laminar flow sterilizer is about half of that of a PFR.

18.7.2.4. Thermal Sterilization in a Turbulent Flow Tubular Reactor
Let us now visit the tubular flow reactor analysis by incorporating the backmixing into the
cell balance. Along the axis (x) of the tube, mass balance in a differential volume ( Fig. 18.11 )at
steady state yields

! x !

d C X

d

d C X

d

QC X D eff A cr QC X D eff A cr k d C X A cr d x ¼ 0 (18.50)

x x

xþ d x

where D eff is the dispersion coefficient (or flow-induced diffusion coefficient), and A cr is
the cross-sectional area of the tube. Divide Eqn (18.50) by dx and letting dx / 0, we

obtain

"
QC X D eff A cr

#

d
d

d C X

d

k d A cr C X þ ¼ 0 (18.51)

x x

There are two Eigen values for this differential equation:

p

Q

2 cr

l 1;2 ¼ Q

þ4D eff A k d

(18.52)

2D eff A cr

We therefore need two boundary conditions to fully specify the system (well-posedness).
Physically, we know that the exit condition does not affect the progress of sterilization inside
the tubular reactor. Backward dispersion of contaminant cells is not important for evaluating
the sterilization operations. When D eff /

0,

p

Q

! 0 Q

2

þ4D eff A

cr

k d

lim

! 0 l 1;2 ¼

lim

2D eff A cr

D D

eff eff

(18.53)

¼þN; k d A c
Q

With the negative sign in Eqn (18.53) , the solution for PFR is recovered. Therefore, only one
mode is physical and that mode should be the monotonous decay mode. We neglect the posi-
tive Eigenvalue. The solution is thus given by

!

p

Q

pð s Þ¼ C X

Q

2

þ4D eff A

cr k d

C X0 ¼ exp

x (18.54)

2D eff A cr