There is a long history of mathematical work analysing flow reactor models with application in bioprocess engineering. We outline some of the historical developments in this area that are relevant to our work. The basic model is discussed in section 2. The basic model was found not to accurately describe the concentration of microorganisms at large residence times. This led to the introduction of expanded models including the death of microorganisms (or inactivation) and maintenance energy requirements; these are described in sections 3 and 4 respectively. In section 5 we discuss early models for the activated sludge process.

The state of a continuous flow bioreactor is described in
terms of two variables, a microorganism and a growth limiting
substrate. An equation for the product is not required unless
the concentration of the
product species appears in the specific growth rate law.
Many growth rate laws, which relate the consumption of the substrate to
the microorganism growth rate, have been determined
experimentally, the most frequently used is the Monod model
(Monod, 1948)

(1)
`μ(S) = μ _{m}S/(K_{S}+S)`.

The basic model was first derived by
Monod (1950)
and Novick and Szilard (1950).
Monod (1950)
found the non-trivial steady-state solution branch and
showed that it exists only when the residence
time is sufficiently large.
Novick and Szilard (1950)
analysed the non-trivial steady-state solutions for a general
growth law (`μ(S)`), showing that the steady-state
substrate concentration is independent of the substrate concentration
in the feed stream (`S _{0}`).
They also introduced the word `chemostat' to describe a
continuously stirred bioreactor.

Herbert `et al`
(1956) plotted steady-state diagrams,
showing the \textit{stable} substrate and microbial concentrations as
a function of the flow-rate for different values of the
feed concentration (`S _{0}`).
They identified the value of the residence time
below which washout occurs
(

Koga and Humphrey (1967) analysed the basic model using phase-plane methods. They showed that sustained oscillations are not possible. The non-trivial steady-state was found and its stability determined.

The basic model predicts that there is a critical value of the
residence time,
`τ _{cr}`, such that for residence times lower
than criticality
`washout' occurs, i.e.\ the steady-state concentration of microorganisms
(

From studies on continuous-flow culture Herbert
(1958a)
postulated that the deviation between theoretical and experimental
results at large residence times could be explained by including a term
representing a constant
`endogenous metabolism' in the microorganism equation. This means that
microorganisms convert endogenous substrates within themselves to
furnish some of the energy that is required to maintain themselves.
Herbert (1958b) reported that the inclusion of
a volumetric death term `-k _{d}XV` in the
biomass equation, representing
`endogenous metabolism', eliminated the
discrepancy between experimentally measured cellmass concentrations
and the theoretically predicated curve at large residence times,
but did not actually present his analysis.

In addition to representing endogenous metabolism a variety of
processes leading to a decrease in cell mass
can be modelled by a term
`-k _{d}XV` in the microorganism
equation. These include endogenous respiration,
predation, and cell death followed by subsequent lysis
(Pavlostathis & Giraldo-Gomez, 1991).
As it is not possible
experimentally to distinguish between these processes,
the coefficient

Model equations containing a constant endogenous metabolism were written down by McKinney (1962) and Dawes and Ribbons (1964). McKinney (1962) did not use a specific growth rate to relate the growth of microorganisms to the substrate concentration. Instead, he assumed that the interaction between substrate and microorganisms was first order with respect to the former and zeroth order with respect to the latter. Although Dawes and Ribbons (1964) wrote down an equation for the concentration of microorganisms using the Monod expression they did not further analyse their model.

The extension of the basic model to include a `death' term
seems to have been first analysed by
McCarty (1966)
and
Wase and Hough (1966).
McCarty (1966)
provided a steady-state diagram,
showing the stable solutions as a function of the residence time.
However, the parametric form of the steady-state solutions was
not stated nor was a stability analysis carried out. It seems to have been
\textit{assumed} that the no-washout solution is stable when it is
physically meaningful and that the washout solution is stable when
the no-washout solution is not physically meaningful.
Wase and Hough (1966)
obtained an approximation for the steady-state concentration of
microorganisms
at high residence times which was used to estimate the value of
the parameter `k _{d.
The non-trivial steady-state solutions for the basic model with death
were given by
Lawrence and McCarty (1970).
A stability analysis of the basic model including death of
microorganisms does
not appear to have been provided in the literature
until Nelson et al
(2008).
}`

A second extension to the basic model
is to allow for a `maintenance energy'.
This recognises that
some of the energy that is generated by consumption of the substrate
is used for functions other than cell growth, such as
maintaining cell integrity and supplying the energy for
cellular processes; only the surplus energy is
available for growth. In the substrate equation the addition
of a volumetric term
`-m _{s}X`
represents the amount of substrate diverted from growth.
By the early 1960s there was a long tradition, dating back to the
late 1920s, of
theoretical opinion supporting the concept of maintenance energy
for microorganisms
(McGrew and Mallette, 1962).
However, experimental techniques were not
sensitive enough to offer support for the concept until 1962, when
the first study to support the concept was published
(McGrew and Mallette, 1962).

This concept was investigated using experimental data obtained from
continuous bioreactors by
Marr `et al` (1963).
They analysed their data
using a steady-state value for the microorganisms that assumes
a large residence time. However, these authors did not
write down the differential equations for their model. This is an
important observation
because the expression for the steady-state concentration of
microorganisms
that they use can be derived either from
the basic model with maintenance and no death
(`m _{s}>0` {and

The concept of maintenance energy was first clearly applied to
a continuous bioreactor model by Schulze and Lipe
(1964). They
replaced the Monod growth model, equation 1,
by the Tessier growth model
(Tessier, 1936)

`μ(S) = &mu _{m}(1-`exp

The introduction of the maintenance energy led to an improved agreement between the theoretical and experimentally measured cell concentration ( Schulze and Lipe, 1964).

Maintenance energy was first clearly applied to the basic model,
using a Monod growth model, by Pirt (1965)
who gave the steady-state value for the concentration of microorganisms
for an arbitrary residence time. Although the differential equations
for the model are not provided, it is clear from the expression for the
steady-state that the basic model
with maintenance and no death
(`m _{s}>0` and

The simplest model for biological waste treatment processes is
the basic model augmented by both recycle and a term representing
death of microorganisms.
Early investigations of this extended model include those of
Pearson (1968),
Lawrence and McCarty (1970) and
Lawrence (1971).
The account by Pearson is confusing
with steady-state solutions not clearly identified as functions of
process parameters.
Lawrence and McCarty (1970) give the
steady-state microorganism concentration whilst
Lawrence (1971)
provides the steady-state microorganism and substrate concentrations.
In none of these papers is stability determined. The results of
Lawrence and McCarty (1970) have been used to
analyse subsequent experimental data, for example by
Yenkie ` et al` (1992).

- Dawes, E.A., Ribbons, D.W. (1964)
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In
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The DOI (Digital Object Identifier) link for this article is
`
http://dx.doi.org/10.1002/apj.106`.

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