Simple models in bioreactor engineering (2006present)
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A continuous stirred flow bioreactor is a wellstirred vessel
containing microorganisms (X) through which a substrate
(S) flows at a continuous rate. The microorganisms grow in the
vessel through the consumption of the substrate to produce microorganisms
and a product (P). Unused substrate, microorganisms and the
product flow out of the reactor at the same rate at which the feed
is admitted. In membranebased bioreactors a permeable membrane,
such as a microfiltration membrane, is used to physically retain
microorganisms inside the reactor whilst allowing the substrate and
product to move through the reactor. Entrapping the microorganisms
in this manner increases their concentration compared to a flow reactor.
This results in a greater conversion of the substrate, allowing for a more
rapid and efficient process.
In this work simple models for biochemical processes occurring in
wellstirred bioreactors are analysed. Typically the steadystate
behaviour is analysed, in both a bioreactor and a membrane reactor, as
a function of the residence time.
A variety of features may be included in
such models:
 the specific growth rate (Monod kinetics, Contois kinetics etc).
 Microbiological features: maintenance energy, death of
organisms.
 models for noncompetitive product inhibition.
 Reactor design: recycle.
The classic Monod growth rate expression assumes that microbial growth is
an increasing function of the substrate concentration and does not
depend upon the concentration of any other substance. However, in some
biochemical processes growth is inhibited by the growth product through
either competitive or noncompetitive means. The classic case of
noncompetitive product inhibition occurs in ethanol fermentation by
yeasts, in which ethanol is an inhibitor at concentrations above 5%.
Several forms for noncompetitive product inhibition have been suggested
in the literature. These include
(1)  &mu(S,P) = 
μ(S)*(1+P/K_{p})^{1} 
(2)  &mu(S,P) = 
μ(S)*exp(P/K_{p}) 
(3)  &mu(S,P) = 
μ(S)*(1P/P_{m})
H(P_{m}P) 
In these expressions, μ(S) is the growth rate expression in
the absence of product, K_{p} is the product inhibition
constant, P_{m} is a concentration at which growth stops
and H is the Heaviside function. The third of these expressions can be
derived as a Taylor series expansion of the second for small product
concentrations.
On the basis of kinetic studies of lactic acid fermentation
Luedeking and Piret (1959a)
proposed a kinetic model for product formation
involving both growthassociated product formation, i.e. the product
is formed as result of the primary metabolic function of the cell,
and nongrowthassociated product formation, i.e. the product
is formed from secondary metabolism of the cell.
This kinetic model was used to formulate a model
for lactic acid fermentation in
a continuously stirred flow bioreactor
(Luedeking & Piret, 1959b).
Model equations were derived for the
case when growth is limited by noncompetitive
product inhibition using an expression
similar to equation (3).
The steadystate
solutions of this model were obtained and stability was explained using
a graphical technique. However, the
growth rate expression was assumed
to be independent of the substrate concentration,
i.e. μ(S)=μ_{max}.
The classic example of noncompetitive
production inhibition occurs in the fermentation of ethanol.
The inhibitory effect of ethanol on
the kinetics of ethanol production in a continuously stirred tank reactor
was first studied by Aiba and coworkers
(Aiba et al, 1968;
Aiba & Shoda, 1969;
Nagatani et al, 1968).
In their originally analysis
(Aiba et al, 1968;
Nagatani et al, 1968)
the inhibitory effect of ethanol on cell
growth was found to fit the exponential model
for inhibition (2). A reassessment of
their data showed that it also fitted
expression (1).
Investigations into noncompetitive
product inhibition have a long history. In fact,
the basic model equations, without
a term representing microbial death, were
written down, but not analysed, by
Fredrickson et al (1970).
Product inhibition has become a standard topic
in biochemical engineering and is addressed in textbooks such as
Bailey & Ollis (1977, chapter 7),
Blanch & Clark (1997, chapter 3.3.8),
and Shuler & Kargi (2002, chapter 6).
Given the long history of this topic it is
surprising that a stability analysis
of the basic model
has not appeared in the literature.
In Nelson et al (2009) the first
of these expressions was investigated for the cases of a continuous
flow bioreactor and an idealized continuous flow membrane reactor.
The steadystate solutions were found and their stability determined
as a function of the residence time. The performance of the
reactor at large residence times was obtained. The key dimensionless
parameter that controls the degree of noncompetitive product inhibition
is identified and the effect that this has on the reactor performance is
quantified in the limits when product inhibition is `small' and
`large'.
Bibliography
 S. Alba and M. Shoda. (1969).
J. Ferment. Technol., 47, 790794.
 S. Aiba, M. Shoda and M. Nagatani. (1968)
Biotechnol. Bioeng., 10, 845864.
 J.E. Bailey and D.F. Ollis. (1977)
Biochemical Engineering Fundamentals.
McGrawHill Company, New York, first edition.
 H.W. Blanch and D.S. Clark. (1997).
Biochemical Engineering.
Marcel Dekker, Inc., New York, first edition.
 A.G. Fredrickson, R.D. Megee III and
H.M. Tsuchiya. (1970). Adv. Appl. Microbiol.,
13, 419465.
 R. Luedeking and E.L. Piret. (1959a)
J. Biochem. Microbiol., 1, 393412.
 R. Luedeking and E.L. Piret. (1959b)
J. Biochem. Microbiol., 1, 431459.
 M. Nagatani, M. Shoda, S. Aiba. (1968)
J. Ferment. Technol., 46, 241248.
 M.L. Shuler & F. Kargi. (2002).
Bioprocess Engineering.
Prentice Hall International Series in the Physical and Chemical
Engineering Sciences. Prentice Hall, New Jersey, USA, second edition,
In the following:
 a superscript ^{p} denotes an author
who was a PhD student at the time the research was carried out.
 a superscript ^{m} denotes an author
who was an undergraduate at the time the research was carried out.
 a superscript ^{u} denotes an author
who was an undergraduate at the time the research was carried out.
Book Chapters
 M.I. Nelson, X.D. Chen and H.S. Sidhu.
Reducing the emission of pollutants in industrial wastewater
through the use of membrane reactors. In
R.J. Hosking and E. Venturino (Editors),
Aspects of Mathematical Modelling, Birkhäuser, Basel,
95107,
2008.
Referred journal papers

M.I. Nelson, T. Kerr^{}u
and X.D. Chen.
A fundamental
analysis of continuous flow bioreactor and membrane reactor models
with death and maintenance included.
AsiaPacific Journal of Chemical Engineering,
3(1), 7080,
2008.
http://dx.doi.org/10.1002/apj.106.

M.I. Nelson, E. Balakrishnan, H.S. Sidhu and
X.D. Chen.
A fundamental
analysis of continuous flow bioreactor models and membrane reactor
models to process industrial wastewaters.
Chemical Engineering Journal, 140,
521528, 2008.
http://dx.doi.org/10.1016/j.cej.2007.11.035.
 M.I. Nelson,
J.L. Quigley^{}u and X.D. Chen.
A fundamental analysis of continuous flow bioreactor and membrane reactor
models with noncompetitive product inhibition
AsiaPacific Journal of Chemical Engineering,
4(1), 107117,
2009.
http://dx.doi.org/10.1002/apj.234.
 M.I. Nelson and H.S. Sidhu.
Analysis of a chemostat model with variable yield coefficient:
Tessier kinetics.
The Journal of Mathematical Chemistry,
46(2), 303321,
2009.
http://dx.doi.org/10.1007/s1091000894637.
 M.I. Nelson and A. Holder^{u}.
A fundamental analysis of continuous flow bioreactor models
governed by Contois kinetics. II. Reactor cascades.
Chemical Engineering Journal, 149 (13),
406416, 2009.
http://dx.doi.org//10.1016/j.cej.2009.01.028.
 M.I. Nelson, E. Balakrishnan and and H.S. Sidhu.
A fundamental analysis of continuous flow bioreactor and membrane reactor
models with Tessier kinetics
Chemical Engineering Communications, 199(3),
417433,
2012.
http://dx.doi.org/10.1080/00986445.2010.525155.
 R.T. Alqahtani^{p}.
M.I. Nelson and A.L. Worthy.
A fundamental analysis of continuous flow bioreactor models with
recycle around each reactor governed by Contois kinetics. III. Two
and three reactor cascades.
Chemical Engineering Journal, 183,
422432, 2012.
http://dx.doi.org/10.1016/j.cej.2011.12.061.
 Mark Ian Nelson and Wei Xian Lim
^{u}.
A fundamental analysis of
continuous flow bioreactor and membrane reactor models with
noncompetitive product inhibition. II. Exponential inhibition.
AsiaPacific Journal of Chemical Engineering,
7(1), 2432, 2012.
http://dx.doi.org/10.1002/apj.485.
Referred conference proceedings
 R.T. Alqahtani ^{p},
M.I. Nelson and A.L. Worthy.
A mathematical analysis of continuous flow bioreactor models
governed by contois kinetics: A two reactor cascade.
In Proceedings of the Australasian Chemical Engineering
Conference, CHEMECA 2011, pages 111. Engineers Australia,
2011.
On CDROM. ISBN 978 085 825 9225.
 M.I. Nelson and E. Balakrishnan.
An analysis of an activated sludge process containing a sludge
disintegration system.
In Proceedings of the Australasian Chemical Engineering
Conference, CHEMECA 2011, pages 111. Engineers Australia,
2011.
On CDROM. ISBN 978 085 825 9225.
 Providing an analysis of the simplest bioreactor model
(a twovariable model for substrate and biomass using monod kinetics)
including "bells and whistles": cell death; maintenance requirements
and reactor recycle
(Nelson et al 2008).
Steadystate solutions were found and their stability characterised
as a function of the residence time.
Showed that for simple models the behaviour of a reactor with
idealised recycle (all microorganisms are recycled)
is the same as an idealised membrane reactor
(no microorganisms leave the reactor).
At large residence times it is shown that the behaviour of
a flow reactor with/without recycle and an idealised membrane
are identical. Thus the main advantage of a membrane reactor,
or a flow reactor with recycle, for the treatment of industrial
wastewaters and slurries is to improve the performance at low
residence times.
 Providing the first comprehensive analysis of a bioreactor model
for the processing of industrial wastewaters and slurries based upon
the Contois specific growth rate
(Nelson et al 2008b).
 Provided the first comprehensive analysis of a bioreactor model
for noncompetitive product inhibition based upon
equation (1)
(Nelson et al 2009).
Two reactor models were considered: a wellstirred flow reactor
and a wellstirred membrane reactor.
 Providing the first comprehensive analysis of a bioreactor model
for a process governed by Tessier kinetics subject to a variable
yield
(Nelson & Sidhu, 2009b).
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Page Created: 11th November 2009.
Last Updated: 9th February 2012.