Population Balance Equations
335
Figure 8.8 Effect of energy input (W L'') on the total hyphal in
of
P. chrysogenum
in a submerged
culture, The two data sets (Mertz, 1976) are from a series of batch fermentations (batch) and from a series
of chemostat experiments (continuous,
D
= 0.05 h ‘). The lines are model simulations.__________________
PROBLEM S
Problem 8.1 Derivation of single-cell mass distribution functions
Consider an organism for which division occurs at the cell mass
M,
birth at the cell mass
Mi
2, and the
single-cell mass growth rate follows first-order kinetics, i.e.,
f(m) = km
,
a.
Find the normalized, steady-state cell mass distribution
for this organism in a chemostat with
dilution rate
D.
b.
Find the relation between
D
and
k,
and use it to eliminate
k
from the expression for ^(w).
c.
Find the distribution function for zeroth-order kinetics (i.e.,
r(m) = fc), and compare the
distribution functions for, respectively, zeroth- and first-order kinetics.
d.
Assuming a constant yield
of mass per individual cell from the substrate, write a steady-state
substrate balance and simplify this to obtain an equation among D,
k,
and
n
(the total cell number).
e.
Assume that the substrate dependence of
k
follows a Monod-type expression. Find the substrate and
cell number concentrations as functions of the dilution rate.
Problem 8.2 Linear single-cell kinetics
Consider an organism for which division occurs at cell mass
M,,
birth at cell mass
M!
2, and the single-cell
mass growth rate follows
r{m)
=
kYm + k2.
a.
Find the normalized, steady-state cell mass distribution
\$(m)
for this organism in a chemostat with