Population Balance Equations
337
distribution,
<f>(a),
in a chemostat with dilution rate
D.
b.
Show that the doubling time, defined as the duration of the cell cycle
does not equal the
doubling time defined on the basis of the specified growth rate of the population.
Problem 8.5 Derivation of conversion rates in a yeast model
Cazzador (1991) evaluated the parameter A„ and
K
in his morphologically structured model (see Section 5.2
and Problem 5.3). In this evaluation he assumed
The critical mass for budding is assumed to be constant and equal to 1, i.e.,
m
= 1.
The critical mass for cell division is assumed to be a known function of the substrate concentration
mdiv
(\$) such that 1 <
mdiv <2.
The size of the mother cells after division is assumed to be
m
= 1 and that of newborn daughters
m = mdiv-
1.
The specific growth rate of the single cell is assumed to be
in the first (unbudded) phase and
jja
in the second (budded) phase of the cell cycle.
a.
Write the steady-state balances for the distribution function of cell mass for cells in the two phases.
Specify the renewal equations.
b.
Solve the balances for the distribution functions.
c.
The transfer rates between the two morphological forms can be expressed as
K xZu= mJ uQ)
(1)
kbxZb = (mdiv
-1
)fibmdivf h
(
mdiv
)
(2)
Use these equations to derive expressions for
kj
and At as functions of
function of the limiting substrate concentration) in the two cases
tndiv
(which again is a
ii.
mu
* Mb
Now let
mdiv
and
jj.
be given as
§
mdi -1.2 +
0.5-t------ r
div
s8+108
(3)
/i = 0.3
5
s + 4
(4)
Plot
and At as functions of the limiting substrate concentration s. Simulate using a PC the
dynamic equations for
Z^,Zb,s
(see Problem 7.4), and
x
in a chemostat operated withD = 0.2 h '1,
sf
=
400 g L'1, au = 10, and ab = 2. Show that a stable limit cycle is obtained.