Applying Mechanistic Models in Bioprocess Development - Measurement, Monitoring, Modelling, and Control of Bioprocesses - page 143

E
E þ K E

K i
G þ K i

r E ¼ r E ; max ð 5 Þ

O
O þ K O

r O ¼ r O ; max ð 6 Þ

l Total ¼þ Y Oxid r Oxid

G

þ Y Red

XG r Red

þ Y XE r Oxid

ð 7 Þ

XG G E

The rate of oxidation and the rate of reduction of glucose are defined based on
the maximum oxygen uptake rate: if the oxygen demand that is stoichiometrically
required for oxidation of the total glucose flux (Y OG 9 r G Total ) exceeds the max-
imum oxygen uptake rate (r O,max ), the difference between the two fluxes corre-
sponds to the overflow reductive flux. With regard to the oxidation of ethanol, the
observed rate of ethanol oxidation depends on the ethanol availability (Eq. 5 ) and
it is further limited by the respiratory capacity: not only the maximum capacity of
the cell, but also the capacity remaining after considering metabolism of glucose
(Table 2 ).
In addition to the reactions taking place in the cells, oxygen is continuously
supplied to the bioreactor. This supply is described based on the mass transfer
coefficient (k L a) and the difference between the dissolved oxygen concentration
(O) and the saturation concentration of oxygen in water (O * ) as a driving force.
k L a is dependent on the aeration intensity and the mixing conditions in a given
fermentor. It is also dependent on the biomass concentration, although this
dependence is often disregarded. The rates for each component can be obtained
from the process model matrix (Table 2 ) by multiplying the transpose of the
stoichiometric matrix (Z') by the process rate vector (q): r m ; 1 ¼ Z 0 nxm q nx1 ,
where m corresponds to the number of components (or model variables) and n is
the number of processes. In Table 3 , a nomenclature list of vectors and matrices is
presented.
The model matrix in Table 2 provides a compact overview of the model
equations. In the example here, it contains information about the biological
reactions and the transfer of oxygen from the gas to the liquid phase. Of course,
depending on the purpose of the model, the model matrix could be extended with
additional equations, for instance, aiming at a more detailed description of the
biological reactions, e.g., by including additional state variables, or aiming at the
description of the mass transfer of additional components, e.g., CO 2 stripping from
the fermentation broth. Sin and colleagues [ 14 ] provided an example of the
extension of the model matrix with chemical processes for the kinetic description
of mixed weak acid-base systems. The latter is important in case pH prediction is
part of the purpose of the model. In the work of Sin and colleagues [ 14 ], the yield
coefficients are all part of the stoichiometric matrix. In our case here, an alternative
rate vector is presented, where all rates are normalized with regard to glucose.