the type of bioreactor in study, is obtained by applying general mass, energy and
momentum balances (Roels, 1983).
In practical applications the choice of a sub-set of state variables may be sufficient,
this depending on the objectives of the model and on the operating conditions. When, for
instance, the temperature is kept constant by means of an efficient control, there is no
need to include the variable temperature in the state space vector. This avoids the need to
employ macroscopic energy balances for describing and analysing the process state. The
energy equations are however required if the objective of the model is the design of a
temperature controller or, just to give another example, where cooling problems are likely
to be found as is the case in large scale bioreactors (Humphrey, 1998).
A problematic issue concerns the rheological effects in bioreactors. In most cases
modelling of flow patterns is too complex and much for this reason almost invariably the
assumption is taken that both liquid and gas phases are well mixed, i.e. the medium is
assumed to be homogeneous, in both industrial and research applications. From the
mathematical and numerical point of view this introduces a significant simplification
since, otherwise, a mathematically complex distributed parameter analysis would have to
be performed (e.g. Reuss, 1996). Under this assumption, and by considering that
temperature, pressure and pH are usually controlled quantities being kept at constant
values, there is solely the necessity of applying mass balance principles to the relevant
components. This leads to a set of ordinary differential equations that almost invariably
are the backbone of simplified bioreactor models,
(1)
being x a vector of concentrations (the state space vector), r a vector of reaction kinetics,
F the input feed rate into the bioreactor, V the working volume, xin a vector of
concentrations in the input feed with the feed rate F, and q a vector of flows associated
with the gas phase (such as oxygen and carbon dioxide transfer rates).
Mass transfer and hydrodynamic issues are addressed in detail in Chapter 1 of this
book. It is however worth stressing at this point that oxygen transfer limitations represent
an important constraint in bioprocess optimisation and control. With the exception of
cooling limitation problems usually found in large bioreactors (Kneinstreuer, 1987), the
most frequent limitation to growth in aerobic fermentations is dissolved oxygen
(Thornhill and Roy ce, 1991). Oxygen has a very low solubility in fermentation media in
comparison to other typical substrates, thus needing to be continuously supplied, usually
by aeration. The mass transfer capacity between gas and liquid phases, quantified by the
volumetric mass transfer coefficient kLa is a central concern in aerobic bioprocesses
design and operation. From a practical point of view, a very important issue in carbon
limited bioprocesses is that the maximum kLa imposes the most important constraint in
bioprocess optimisation. The correlation between kLa, stirrer speed and air flow (see
Humphrey, 1998) is also important for designing control systems for dissolved oxygen.
Cell ModelsThe reaction term r in eqn. (1) is the result of a complex metabolic reactions network at
both cell level and at the whole cell population level. In general cell models can be
Multiphase bioreactor design 68