Population Balance Equations
333
fonction at time
t
= 0:
/(0 ,n ,r) = 0 ;
f( m t
0 ,/)= 0
(6)
lim /(/,« ,£ ) = 0 ;
lim /(/,n ,f) = 0
(7)
f(m ,
n,0) = / 0
(m, n)
(8)
Solving the dynamic balance for the distribution function is complicated, since the property space is two
dimensional and since an integration step is involved in the calculation of
h(l,n,i).
For comparison of the
population model with measurements obtained in a continuous bioreactor at steady state, it is, however,
sufficient to find the steady-state solution of the population balance. For a population balance with a one-
dimensional property space Singh and Ramkrishna (1977) used the method of weighted residuals to find the
steady-state solution. This method could probably be extended to find the steady-state solution for the
population balance derived here. Alternatively one may introduce discrete variables and then solve
differential equations for each discrete variable (Krabben
et al.,
1997). However, an institutively simpler
approach is to apply Monte-Carlo simulations where a large number of individual hyphal elements are
simulated, and the elements may then together form the distribution function. Krabben
et al.
(1997) applied
this approach to simulate the distribution function for different fragmentation kinetics, i.e. different
partitioning functions and different breakage functions. It is practically impossible to obtain sufficient
experimental data - even with fully automated image analysis, on the distribution function in two
dimensions to validate different models. Krabben
et al.
(1997) therefore took a different approach. Based on
the simulated distribution functions they generated contour plots for the function. These contour plots were
then statistically used to evaluate experimentally obtained data for the hyphal element properties. Hereby it
was possible to evaluate the different models, and based on this it was concluded that hyphal fragmentation
only takes place for hyphal elements above a certain size, and then follows second order kinetics. It was,
however, not possible to discriminate between two models for the partitioning function, i.e. whether there is
largest probability for fragmentation at the centre of the hyphae or whether there is equal probability for
fragmentation throughout the hyphae.
Even though the Monte Carlo simulations offer a simple method for simulation of even complex models, it
is often sufficient to look at the average properties of the hyphal elements, as these may be directly
compared with measured data. Using the balance for the distribution function of above Nielsen (1993)
derived dynamic balances for these variables for the average hyphal length and the average number of tips.
These balances are given by:
at
(9)
{^-y/)(l)=<p{n)-y/{l)
(10)
{J
-*-*<■*)
(11)
where 9 is the average tip extension rate for all the hyphal tips in the population. Using these balances it is
possible to obtain information about the growth kinetics, i.e. the branching frequency and the tip extension
rate from measurements of the average total hyphal length and the average number of tips. The total
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