Population Balance Equations
331
of different cell types may also be important (see Section 7.6.2). We now assume that each hyphal element
is characterized completely by its total length (0 and the number of actively growing tips («); i.e.,
fij.n) dl
dn
is the number of hyphal elements with length
l
and
n
actively growing tips. The hyphal diameter is
normally constant (at least for certain environmental conditions) and similarly holds for the hyphal density.
The hyphal length is therefore proportional with the hyphal mass, and the total tip extension rate for the
hyphae is therefore given by the
p(l,n,t)l. n
is in reality an integer, but it is here taken to be a real number.
The length of the hyphal element increases due to growth with the specific rate
and new actively
growing tips are formed due to branching with the frequency
(f£l,n,t).
It is now assumed that both the
specific growth rate of the hyphal element and the branching frequency are independent of the hyphal
element properties, i.e., they are not functions of / and
n.
Consequently, the rate of change of length is
^t)l
and the rate of change of actively growing tips is
\$
(t). The dynamic mass balance for the distribution
function
f {m, n)
is therefore given by
^
(l, n,,
f)]+ “
\\$(t)f{ly
«,0] =
h(l,n,t
) -
Df{l,n,t
)
(
1
)
ot
dl
on
The dynamic mass balance is illustrated in Fig. 8.7, where it is shown that for a given control volume, i.e.
the distribution function, there are inputs and outputs due to tip extension and branching. Furthermore, there
are inputs due to formation of new hyphal elements by fragmentation and spore germination. Finally there
are hyphal elements leaving the control volume due to hyphal fragmentation and washout.
We now consider growth of a mycelium in an agitated tank where shear stress causes the hyphae to break
up (hyphal fragmentation). With binary fission of hyphal elements, the net rate of formation of hyphal
elements with property
{l,n}
formed upon fragmentation is given by Eq. (8.7), i.e.,
h(l,n,t) =
2
(2)
b{l,n,t)
is the breakage function, which describes the rate of fragmentation of hyphal elements with property
{/,«}, and
p({l*,n*},{l,n})
is the partitioning function.
Figure 8.7 Illustration of the different elements in the population balance equation for individual hyphal
elements in a filamentous fungal culture.