where Ks is the half-saturation coefficient. Ks determines how rapidly p. approaches |i and is defined as the substrate concentration at which p. is equal to half of [i, as shown in Figure 3.1. The smaller it is, the lower the substrate concentration at which |X approaches (1. Because of his pioneering efforts in defining the kinetics of microbial growth, Eq. 3.36 is generally referred to as the Monod equation.
Because of the similarity of Eq. 3.36 to the Michaelis-Menten equation in enzyme kinetics, many people have erroneously concluded that Monod proposed it on mechanistic grounds. While the Michaelis-Menten equation can be derived from consideration of the rates of chemical reactions catalyzed by enzymes, and has a mechanistic basis, the Monod equation is strictly empirical. In fact, Monod himself emphasized its empirical nature.X1
The Monod equation has been found to fit the data for many pure cultures growing on single substrates, both organic and inorganic, and has been used extensively in the development of models describing the continuous cultivation of microorganisms. It has not been blindly accepted, however, and other workers have proposed alternative equations that fit their data better.*417Nevertheless, it is still the most widely used equation.
Because the Monod equation was developed for pure cultures of bacteria growing on single organic substrates, two significant questions arise when its adoption is considered for modeling biochemical operations for wastewater treatment. The first concerns whether it can be used to express removal of a substrate that is really a mixture of hundreds of organic compounds measured by a nonspecific test like COD, since that is the nature of the organic matter in wastewater. Can the Monod equation adequately describe the effect of biodegradable COD on the specific growth rate of bacteria? The second question arises from consideration of the microbial communities present in wastewater treatment operations. As seen in Chapter 2, those communities are highly complex, containing not only many bacterial species, but higher life forms as well. Can the growth of such a heterogeneous assemblage be expressed simply as "biomass" by the Monod equation? Many researchers have investigated these questions, and it is generally agreed that the answer to both is yes.' '
Nevertheless, it should be recognized that the manner in which the culture is grown will have a strong impact on its community structure, and that the values of |i and Ks obtained from mixed culture systems are in reality average values resulting from many interacting species.1'1,1 Consequently, it has been recommended that |i and Ks be characterized by ranges, rather than by single values, just as was recommended for Y. It can be concluded, however, that the Monod equation is a reasonable model with which to describe the kinetics of microbial growth on complex organic substrates in wastewater treatment systems, and consequently, it is widely used. There are situations, however, in which it would be desirable to model the effects on microbial growth rates of individual organic compounds in complex mixtures. This situation is very complicated, however, and will be covered in Chapter 22.
Simplifications of the Monod Equation. Examination of Eq. 3.36 reveals that two simplifications can be made, and this is often done in the modeling of wastewater treatment systems. First, it can be seen that if Ss is much larger than Ks. the equation may be approximated as:
This is called the zero-order approximation because under that condition the specific growth rate coefficient is independent of the substrate concentration, i.e., it is zero order with respect to Ss, and equal to the maximum specific growth rate coefficient. In other words, the bacteria will be growing as rapidly as possible. Second, if Ss is much smaller than Ks, the term in the denominator may be approximated as Ks and the equation becomes:
This is called the first-order approximation because p. is first order with respect to Ss. Although Eq. 3.38 is often easier to use than the Monod equation, care should be exercised in its use because serious error can result if Ss is not small relative to Ks. When COD is used as a measure of the total quantity of biodegradable organic matter, Ks can be relatively large, with the result that Ss in activated sludge reactors is often less than Ks. Consequently, Eq. 3.38 is sometimes used to model such systems.
Garrett and Sawyer'^ were the first to propose the use of Eqs. 3.37 and 3.38 because they had observed that the specific growth rate coefficient for bacteria was directly proportional to the substrate concentration at low values and independent of p. = (X
it at high values. Although they recognized that these two conditions were special cases of the Monod equation, others who adopted their first-order equation incorrectly considered it to be an alternative expression.
Inhibitory Substrates. On occasion, particularly in the treatment of synthetic (xenobiotic) organic compounds in industrial wastewaters, situations are encountered in which the specific growth rate of the microorganisms reaches a maximum and then declines as the substrate concentration is increased, as illustrated in Figure 3.2. Obviously, the Monod equation is not adequate for depicting this situation, and consequently, considerable effort has been expended to determine an appropriate equation.,:!,,> ",J As with normal, naturally-occurring, noninhibitory (biogenic) substrate, many different models could be used to represent the observed relationship between the substrate concentration and |x, and from a statistical point of view there is little to recommend one over another.13"" Consequently, as with the Monod equation, it has been argued that model selection should be based on familiarity and ease of use, leading to a recommendation that an equation based on the enzymatic model of Haldane4" should be used. Andrews2 was the first to propose general use of such a function for depicting the effects of inhibitory organic substrates on bacterial growth rates, and thus it will be called the Andrews equation herein. Its form is:
Examination of Eq. 3.39 reveals that it is similar to the Monod equation, containing only one additional parameter, K,, the inhibition coefficient. Note that when K[ is very large the Andrews equation simplifies to the Monod equation, demonstrating that (1 and Ks have the same meaning in both equations. Unlike the situation for a noninhibitory substrate, however, |i cannot actually be observed and is a hypothetical maximum specific growth rate that would be attained if the substrate were not inhibitory. Furthermore, since (1 cannot be observed, Ks also takes on a hypothetical meaning. The most outstanding characteristic of the curve in Figure 3.2 is that p. passes through a maximum, |x*, at substrate concentration Si, where
Equation 3.40 is important because it demonstrates that the degree of inhibition is determined by Ks/K„ and not just by K, alone. The larger Ks/K,, the smaller p.* is relative to ji, and thus, the greater the degree of inhibition. Furthermore, because they are measurable, p.* and S* are important in the determination of the kinetic parameters for inhibitory substrates. Equation 3.39 has been used widely in the modeling of various wastewater treatment systems, and will be adopted herein for depicting the effect of an inhibitory substrate on the specific growth rate of bacteria degrading it.
Effects of Other Inhibitors. Sometimes one compound may act to inhibit microbial growth on another compound. For example, some organic chemicals are known to inhibit the growth of nitrifying bacteria,51"" whereas others inhibit the
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