Simple models in bioreactor engineering (2006-present)

18.10.11 I am only updating the published papers section of this web-page. I don't have time to update the content!

• Background and motivation.
• Published papers.
• Highlights of research on simple models in bioreactor engineering.
• My collaborators in simple models in bioreactor engineering.

Background and information

A continuous stirred flow bioreactor is a well-stirred 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 membrane-based 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 well-stirred bioreactors are analysed. Typically the steady-state 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.

Noncompetitive product inhibition

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/Kp)-1
(2)	&mu(S,P) =	μ(S)*exp(-P/Kp)
(3)	&mu(S,P) =	μ(S)*(1-P/Pm) H(Pm-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 growth-associated product formation, i.e. the product is formed as result of the primary metabolic function of the cell, and nongrowth-associated 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 non-competitive product inhibition using an expression similar to equation (3). The steady-state 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 co-workers (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 non-competitive 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 steady-state 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

1. S. Alba and M. Shoda. (1969). J. Ferment. Technol., 47, 790--794.
2. S. Aiba, M. Shoda and M. Nagatani. (1968) Biotechnol. Bioeng., 10, 845--864.
3. J.E. Bailey and D.F. Ollis. (1977) Biochemical Engineering Fundamentals. McGraw-Hill Company, New York, first edition.
4. H.W. Blanch and D.S. Clark. (1997). Biochemical Engineering. Marcel Dekker, Inc., New York, first edition.
5. A.G. Fredrickson, R.D. Megee III and H.M. Tsuchiya. (1970). Adv. Appl. Microbiol., 13, 419--465.
6. R. Luedeking and E.L. Piret. (1959a) J. Biochem. Microbiol., 1, 393--412.
7. R. Luedeking and E.L. Piret. (1959b) J. Biochem. Microbiol., 1, 431--459.
8. M. Nagatani, M. Shoda, S. Aiba. (1968) J. Ferment. Technol., 46, 241--248.
9. 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,

Simple models in bioreactor engineering: Published papers

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

1. 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, 95-107, 2008.

Referred journal papers

2. M.I. Nelson, T. Kerru and X.D. Chen. A fundamental analysis of continuous flow bioreactor and membrane reactor models with death and maintenance included. Asia-Pacific Journal of Chemical Engineering, 3(1), 70-80, 2008. http://dx.doi.org/10.1002/apj.106.
3. 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, 521-528, 2008. http://dx.doi.org/10.1016/j.cej.2007.11.035.
4. M.I. Nelson, J.L. Quigleyu and X.D. Chen. A fundamental analysis of continuous flow bioreactor and membrane reactor models with non-competitive product inhibition Asia-Pacific Journal of Chemical Engineering, 4(1), 107-117, 2009. http://dx.doi.org/10.1002/apj.234.
5. 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), 303-321, 2009. http://dx.doi.org/10.1007/s10910-008-9463-7.
6. 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 (1-3), 406-416, 2009. http://dx.doi.org//10.1016/j.cej.2009.01.028.
7. 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), 417-433, 2012. http://dx.doi.org/10.1080/00986445.2010.525155.
8. 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, 422-432, 2012. http://dx.doi.org/10.1016/j.cej.2011.12.061.
9. Mark Ian Nelson and Wei Xian Lim ^u. A fundamental analysis of continuous flow bioreactor and membrane reactor models with non-competitive product inhibition. II. Exponential inhibition. Asia-Pacific Journal of Chemical Engineering, 7(1), 24-32, 2012. http://dx.doi.org/10.1002/apj.485.

Referred conference proceedings

10. 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 1--11. Engineers Australia, 2011. On CDROM. ISBN 978 085 825 9225.
11. 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 1--11. Engineers Australia, 2011. On CDROM. ISBN 978 085 825 9225.

Highlights of research on simple models in bioreactor engineering

1. Providing an analysis of the simplest bioreactor model (a two-variable model for substrate and biomass using monod kinetics) including "bells and whistles": cell death; maintenance requirements and reactor recycle (Nelson et al 2008). Steady-state 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.
2. 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).
3. 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 well-stirred flow reactor and a well-stirred membrane reactor.
4. Providing the first comprehensive analysis of a bioreactor model for a process governed by Tessier kinetics subject to a variable yield (Nelson & Sidhu, 2009b).

My collaborators in simple models in bioreactor engineering