# Chemical Reactor Engineering

## Bioreactors: Models based upon monod kinetics

### 1. Flow reactors: mathematical models

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.

### 2. The basic model

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) = μmS/(KS+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 (S0). 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 (S0). They identified the value of the residence time below which washout occurs (τ = (1/μm)(1+KS/S0)). They also found the value of the residence time which maximised the productivity of the reactor in producing microorganisms. They derived a relationship between the quantity of microorganisms produced per unit weight of substrate. Finally, they showed that the efficiency of removal of substrate (100(1-S/S0) is an increasing function of the residence time, reaching 100 at an infinite residence time.

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, τcr, such that for residence times lower than criticality washout' occurs, i.e.\ the steady-state concentration of microorganisms (X) is zero. For residence times greater than criticality the microorganism concentration is an increasing function of the residence time, reaching its maximal value at an infinite residence time. The agreement between experimental results and the predictions of the basic model are good at low and medium residence times. However, there is a noticeable discrepancy at high residence times. Experimental data (Herbert, 1958a; Wase & Hough, 1966) reveals that the concentration of microorganisms reaches a global maximum at a finite residence time (&taumax) and approaches zero as the residence times approaches infinity. To overcome this deficiency the basic model can be made to include either the death/inactivation of microorganisms, (see section 3) or/and a maintenance energy requirement (see section 4).

### 3. Death 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 -kdXV 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 -kdXV 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 kd is best thought of as an overall coefficient representing a combination of these mechanisms. The term -kdXV can also represent a first-order inactivation mechanism, which is widely used in models for sterilisation processes.

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 kd. 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).

### 4. Maintenance energy

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 -msX 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 (ms>0 {and kd=0) or from the basic model with no maintenance and death (ms=0 {and kd>0) Thus it is not possible to state for certain which model they considered.

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) = &mum(1-exp[-S/Ks]).
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 (ms>0 and kd=0) was considered. The basic model with maintenance was analysed by Koga and Humphrey (1967), who found the non-trivial steady-state and showed that it is stable when it is physically meaningful.

### 5. Recycle reactor

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).

### References

1. Dawes, E.A., Ribbons, D.W. (1964) Bacteriol Rev, 28:126.
2. Herbert, D., Elsworth, R., Telling, R.C. (1956). J. Gen. Microbiol 14:601.
3. Herbert, D. (1958a). In Continuous Cultivation of Microorganisms. A Symposium, Malek, I., (ed). Publishing House of the Czechoslovak Academy of Sciences: Prague, Czechoslovakia; pages 45-.
4. Herbert, D. (1958b) In Recent Progress in Microbiology, VII International Congress for Microbiology,, Tunevall, G., (ed), Almquist and Wiksell: Stockholm, Sweden; 381.
5. Koga, S., Humphrey, A.E. (1967). Biotechnol Bioeng, 9:375.
6. Lawrence, A.W., McCarty, P.L. (1970). J. Sanit. Eng. Div. ASCE, 96:757.
7. Lawrence, A.W. (1971). In Anaerobic Biological Treatment Processes, volume 105 of Advances in Chemistry Series, Pohland, F.G., (ed). American Chemical Society; pages 163.
8. Marr, A.G., Nilson, E.H., Clark, D.J. Ann NY Acad Sci, 102:536.
9. McCarty, P.L. (1966). In Developments in Industrial Microbiology,/VAR>, volume 7. Society of Industrial Microbiology: page 144-.
10. McGrew, S.B., Mallette, M.F. (1962) J. Bacteriol, 83:844.
11. McKinney, R.E. (1962). J. Sanit. Eng. Div. ASCE, 88:87.
12. Monod, J. (1949). Annu. Rev. of Microbiol, 3:371.
13. Monod, J. (1950). Ann. I. Pasteur Paris 79:390.
14. 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.

16. Novick, A., Szilard, L. (1950) P. Natl. Acad. Sci. USA., 36:708.
17. Pavlostathis, S.G., Giraldo-Gomez, E. (1991). Water Sci. Technol, 24:35.
18. Pearson, E.A. (1968). In Advances in water quality improvement, Gloyna, E.F., Eckenfelder, W.W., (eds). University of Texas Press: Austin; 381.
19. Pirt, S.J. (1965) P. Roy. Soc. Lond. B., 163224.
20. Schulze, K.L., Lipe, R.S. (1964). Arch. Mikrobiol, 48:1.
21. Tessier, G. (1936). Annales de Physiologie et de Physiochimie Biologique, 12:527.
22. Wase, D.A.J., Hough, J.S. (1966). J. Gen. Microbiol., 42
23. Yenkie, M.K.N., Geissen, S.U., Vogelpohl, A. (1992). Chem Eng J, 49:B1.

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