## Substrate Cone mgL

Figure 3.2 Typical plot ol the relationship between (he specific growth rate coefficient and the concentration of an inhibitory substrate. The parameter values given were used to construct the curve with the Andrews equation (3.39). Note that the values of jj. and Ks are the same as in Figure 3.1.

growth of heterotrophic bacteria on biogenic organic matter. " In those cases it is necessary for the kinetic expression to depict the effect of the concentration of the inhibitor (SJ on the relationship between p. and Ss. If the Monod equation can be used to relate (jl to Ss in the absence of the inhibitor, then the effect of the inhibitor can be expressed as an effect on p. and/or Ks. " Several types of inhibitors have been defined by analogy to enzyme inhibition, but all can be modeled by an extension of the Monod model proposed by Han and I.evenspiel:ls where S;* is the inhibitor concentration that causes all microbial activity to cease and in and 11 are exponents that reflect the impact of increasing inhibitor concentrations on Ks and p., respectively. Equation 3.42 has been used successfully to model the effects of various xenobiotic compounds on the removal of biogenic organic matter' " Its use will be discussed in Chapter 22.

### 3.2.8 Specific Substrate Removal Rate

In earlier sections it was stated that the basis for writing stoichiometric equations was arbitrary and that the reference component was the choice of the investigator.

Thus, it is not surprising that many investigators71124 have selected substrate removal, rather than biomass growth, as their basic event and have written their rate equations accordingly. Combining Eqs. 3.34 and 3.35 yields:

The term p./Y has been called the specific substrate removal rate and given the symbol q.J' (Note that the subscript H has been dropped from Y and X„ to emphasize the general nature of Eq. 3.43.) Obviously, q will be influenced by Ss in exactly the same way as |x, and Eqs. 3.37 through 3.42 can be written in terms of it. When this is done, the maximum specific substrate removal rate, q, is used in place of (1, where:

Both first- and zero-order approximations have been used for the relationship between q and Ss, just as they have for p.. In fact, the ratio of q over Ks has been called the mean reaction rate coefficient and given the symbol

All restrictions that apply to the approximate expressions for the effect of Ss on p. also apply to q.

### 3.2.9 Multiple Limiting Nutrients

In the broad sense, nutrients can be divided into two categories: complementary and substitutable.1'' Complementary nutrients are those that meet entirely different needs by growing microorganisms. For example, ammonia provides the nitrogen needed for protein synthesis while glucose provides carbon and energy. If either was missing from the growth medium and no substitute was provided, no growth would occur. Substitutable nutrients, on the other hand, are those that meet the same need. For example, ammonia and nitrate can both provide nitrogen whereas glucose and phenol can both provide carbon and energy. Thus, ammonia and nitrate are substitutable for each other, as are glucose and phenol. In this section, we will consider simultaneous limitation of specific growth rate by two complementary nutrients. As stated previously, consideration of the effects of multiple carbon sources, i.e., multiple substitutable nutrients, is very complex,72 and will be covered in Chapter 22.

In spite of its potential importance in the environment, relatively little is known about how microorganisms respond to simultaneous limitation by two or more complementary nutrients.1' Because the uncertainty increases greatly as the number of nutrients involved increases, we will limit our considerations to only two.

Interactive and Noninteractive Relationships. Consider two complementary nutrients, Ss, and Ss:. Both are required for biomass growth and are present at low concentration in the environment in which the biomass is growing. Which will control the specific growth rate? Two different philosophies have been developed to answer this question, and the models representing them have been classified as interactive and noninteractive.1'

An interactive model is based on the assumption that two complementary nutrients can both influence the specific growth rate at the same time. If both are rss = -(jul/Y>X,

required for growth and are present at concentrations equal to their half-saturation coefficients, then each alone can reduce to one-half of (jl. However, since both effects are occurring simultaneously, the result would be to reduce pi to one-fourth of (jl. The most common type of interactive model in use is the multiple Monod equation:' '

Any time the concentrations of Ss, and Ss- are such that both Ssl/(Ks, -+- Ssi) and SV/(KV + Sv ) are less than one, they both act to reduce p. below |i. This has two impacts. First. for a given value of SM, p, will be lower when Sv is also limiting than it would be if Sv- were present in excess. Second, there is not a unique value of (jl associated with a given value of SM or Ss„ as there was with Eq. 3.36. Rather, it depends on both.

A noninteractive model is based on the assumption that the specific growth rate of a microbial culture can only be limited by one nutrient at a time. Therefore, p. will be equal to the lowest value predicted from the separate single-substrate models: r

If Ss; (KSi F Ss,) < Ss../(Ks: -r Sv-), nutrient SM is rate limiting, and vice versa. If Ss (Ksl t- Ssl) = Ss,/(Ks + Ss ). then both arc rate limiting, but that occurs only under special conditions. In the noninteractive conceptualization, the normal Monod equation (Eq. 3.36) would apply for whichever nutrient was rate limiting and the concentration of the other would have no impact on

Only limited experimental evidence is available to support one model over the other. Bae and Rittmann have shown both theoretically and experimentally that the interactive model is more appropriate when the two limiting constituents are the electron donor and acceptor. Furthermore, Bader" has compared the mathematical characteristics of the two expressions. The noninteractive model, by its very nature, causes a discontinuity at the transition from one nutrient limitation to another. It also predicts significantly higher growth rates in the region where Ssl/Ksl and Sv'Kv are small. The interactive model does not cause discontinuities, but may err on the side of predicting lower growth rates when SM/KM and Ss:/Ks_- are both small. Both functions become asymptotically the same if either nutrient is present in excess. Finally, the interactive model is mathematically preferable for modeling dynamic situations because it is continuous.

Equation 3.46. the interactive model, will be adopted for use herein. There are three reasons for this choice. First is the evidence provided by Bae and Rittmann. Second, for the type of situation likely to be encountered in biochemical operations for wastewater treatment, the interactive model is more conservative. Third, it works well when one nutrient is the electron donor (the substrate) and the other is the electron acceptor (oxygen or nitrate),1"" "' a common occurrence in wastewater treatment systems.

A special case of multiple nutrients occurs when an increase in the concentration of one nutrient acts to diminish microbial activity. For example, consider the

growth of heterotrophic bacteria under anoxic conditions. Because nitrate reduction can serve as an alternative to aerobic respiration, the enzymes involved in the transfer of electrons to nitrate and its reduced products are influenced negatively by the dissolved oxygen concentration and consideration must be given to this fact when expressing the kinetics of growth under anoxic conditions. Oxygen can have two effects: (1) it can repress the synthesis of denitrifying enzymes, and (2) it can inhibit their activity.'"'M lAlthough there are exceptions, as a general rule the presence of oxygen in the medium (and/or its active utilization as the terminal electron acceptor) represses the synthesis of the nitrate reducing enzyme system. When oxygen is absent, or is present in amounts that are insufficient to meet the needs of the culture, derepression occurs and the enzymes are synthesized. Complications occur, however, when the biomass is cycled between aerobic and anoxic conditions and this appears to alter the regulatory system so that some enzyme synthesis can continue at diminished rates even in the presence of dissolved oxygen.The effect of oxygen on the activity of the enzymes depends on the bacterial species involved. In some, the activities are diminished in the presence of oxygen, whereas in others they are not. Nevertheless, it appears that inhibition of enzyme activity by oxygen is the primary mechanism influencing nitrate reduction rates in systems in which the bacteria are continually cycled between aerobic and anoxic conditions,11" and that prior growth under anoxic conditions will provide enzymes which can function at diminished rate even in the presence of dissolved oxygen. One factor complicating the determination of the effects of oxygen on nitrate reduction in wastewater treatment systems is the necessity to grow the bacteria as flocculent cultures or as biofilms. Because diffusion is the only mechanism supplying oxygen to the bacteria in the interior of a floe particle or biofiim, some bacteria may be in an environment completely devoid of oxygen even when oxygen is present in the bulk liquid.,,K

Because of the complexity associated with the effects of dissolved oxygen on anoxic growth of heterotrophic bacteria, and because all effects have not been clearly defined, relatively simple models have been used to express them." ^" A popular approach has been to use Eq. 3.46 to depict the simultaneous effects of organic substrate (Ss), and nitrate (SNn) on p. and to add a third term which diminishes |x as the dissolved oxygen concentration, S,„ increases:

The third term is the function most commonly used to depict the effects of a classical noncompetitive inhibitor as modeled in enzyme kinetics.The parameter K,u is the inhibition coefficient for oxygen.

Implications of Multiple Nutrient Limitation. Biochemical operations are designed on the premise that there is a functional relationship between the specific growth rate of biomass and the concentration of the growth-limiting nutrient in a bioreactor. Because of that relationship, if engineering control can be exerted over the specific growth rate, it will be possible to control the concentration of the growth-limiting nutrient leaving the bioreactor. This can only be achieved, however, if the nutrient the engineer wishes to control is the growth-limiting one. If the design objective is the removal of soluble organic matter, then all other nutrients must be supplied in excess. Or, if the goal is to remove nitrate-N by allowing it to serve as

the terminal electron acceptor, then it should be made rate limiting at the appropriate place in the process. A clear definition of the objective to be met must be combined with knowledge of the concentrations of the various constituents in the wastewater to ensure that the resultant biochemical operation can indeed meet that objective.

Because oxygen is a gas of very low solubility, it must be supplied continuously to aerobic systems and the concentration in solution will depend on the relative rates of supply and utilization. Furthermore, because oxygen transfer is one of the major costs associated with aerobic wastewater treatment, it is uneconomic to oversize the oxygen delivery system. As a consequence, it is not uncommon for the oxygen concentration to decrease sufficiently to make S„/(K<, + Su) < 1.0 (where K<, is the half-saturation coefficient for dissolved oxygen). Thus, it would be instructive to examine the impact of this occurrence. Figure 3.3 illustrates the simultaneous limitation of the specific growth rate of autotrophic nitrifying bacteria by ammonia (the electron donor) and oxygen (the electron acceptor) using typical parameter values. These bacteria were chosen because they are more sensitive to dissolved oxygen concentration than heterotrophic bacteria, (K,,,, < K<, where the subscripts H and A signify heterotrophic and autotrophic bacteria, respectively). Examination of Figure 3.3 reveals two things. First, if we could operate a bioreactor in a way that maintained a constant specific growth rate, decreasing the oxygen concentration in the bioreactor would cause the ammonia concentration to increase. Second, decreasing the oxygen concentration is analogous to decreasing (1 for the bacteria. This can also be seen

Figure 3.3 Double Monod plot showing the effects of both ammonia nitrogen and dissolved oxygen concentrations on the specific growth rate of autotrophic nitrifying bacteria. The parameter values given were used to construct the curves with Eq. 3.46.

Ammonia Nitrogen Cone., mg/L

Figure 3.3 Double Monod plot showing the effects of both ammonia nitrogen and dissolved oxygen concentrations on the specific growth rate of autotrophic nitrifying bacteria. The parameter values given were used to construct the curves with Eq. 3.46.

by examining Eq. 3.46. The consequence of this is discussed in more detail in Chapter 6, but suffice it to say now that a decrease in (1 makes it more difficult for the autotrophic bacteria to compete for space in the bioreactor.

Both nitrogen and phosphorus are required for the synthesis of new biomass. If those proper quantities are not present, balanced biomass growth cannot occur and treatment performance will be impaired. Thus, care must be exercised to provide sufficient quantities. We have just seen, however, that if the concentrations of essential nutrients are very low in a bioreactor they can become rate limiting, which is undesirable when the treatment objective is removal of organic matter. This means that the concentration of nitrogen or phosphorus supplied to a bioreactor must be sufficiently high to meet the synthesis needs of the biomass as defined by stoichiometry while leaving enough residual in solution to prevent their concentrations from being rate limiting. Goel and Gaudy'"' determined that Ks for ammonia nitrogen during normal heterotrophic growth lies between 1.5 and 4.0 mg/L as N. Using 0.50 hr 1 as a representative value for |i, it can be shown that if the influent nitrogen concentration exceeds the stoichiometric requirement by 1.0 mg/L as N, nitrogen will not be rate limiting to heterotrophic biomass at the specific growth rates normally employed in wastewater treatment. Although some work has been done on kinetic limitation of heterotrophs by phosphorus, the results are not as clear as those with nitrogen. Attempts to measure the limiting phosphorus concentration in both pure and mixed microbial cultures found it to be too low to detect with the techniques available at the time.1" Consequently, if the concentration of phosphorus in the influent exceeds the stoichiometric amount by a few tenths of a mg/L as P, phosphorus should not be rate limiting. In some biochemical operations, the microorganisms pass through a growth cycle, and nutrients will be taken up in one phase and released in another. To prevent nutrient limitation during the phase of nutrient uptake, the amounts presented above should be in excess of the maximum quantity removed, not the net amount as determined by the final effluent.

3.2.10 Representative Kinetic Parameter Values for Major Microbial Groups

Aerobic Growth of Heterotrophic Bacteria. The values of the parameters |iH and Ks are very dependent on the organism and substrate employed. If an axenic bacterial culture is grown on each of several substrates under fixed environmental conditions, the values of (i„ and Ks will vary from substrate to substrate. Likewise, if the same substrate is fed to each of several pure cultures, the values of |iM and Ks will depend on the species of organism. This makes it very difficult to generalize about parameter values and care should be exercised in the use of values considered to be typical. It can be stated, however, that readily biodegradable substrates are characterized by high values of |x,, and low values of Ks, whereas slowly biodegradable substrates have low (1H values and high Ks values. For example, benzoic acid had jiH values between 0.61 and 0.64 hr 1 and Ks values between 4.2 and 5.8 mg/L as COD, whereas 2-chlorophenol had values of 0.020-0.025 hr 1 and 16-17 mg/L as COD for the two parameters.2' Even lower Ks values have been reported for very easily degradable substrates, such as biogenic materials like carbohydrates and amino acids, with values as low as 0.2 mg/L for galactose and 0.5 mg/L for glutamic acid."1 This means that degradation of many biogenic substrates may behave in a zero-order manner over a broad range of substrate concentrations.

Wastewaters usually contain complex mixtures of organic compounds and the total concentration of biodegradable soluble organic matter is commonly characterized by the COD concentration. When Ks is measured on such mixtures using the COD concentration, the values are generally one to two orders of magnitude higher than they are for single substrates expressed as COD. For example, poultry and soybean processing wastewater have been reported to have Ks values of 500 and 350 mg/L as five day biochemical oxygen demand (BODs), which is another measure of biodegradable organic matter."4 Thus, as a whole, overall removal of organic matter in wastewater treatment systems may behave in a lirst-order manner even though the removal of individual constituents may be zero-order.1"1

Domestic wastewater is perhaps the most common example of a complex substrate, and because of its ubiquity there has been considerable interest in characterizing its biodégradation kinetics. As one might expect from the discussion above, considerable variation in the parameter values has been reported, with |1H ranging from 0.12 to 0.55 hr ' and Ks from 10 to 180 mg/L as CODM" An important characteristic of domestic wastewater that has only recently been recognized is that the organic component can be divided into readily and slowly biodegradable fractions, greatly improving the ability of mathematical models to mimic process performance.2" Use of this division should decrease the range of values observed. As a consequence, values of 0.25 hr 1 and 20 mg/L as COD have been adopted as representative of the |iM and Ks values for the readily biodegradable fraction."

The microbial communities in wastewater treatment systems are complex, containing many microbial species, and the relative predominance of the species depends on the physical configuration of the system. Therefore, since the values of p.,, and Ks are species dependent, it follows that their values in mixed culture systems will depend on the bioreactor configuration. For example, reactors that subject the microorganisms to variations in substrate concentrations from very high to very low tend to select species that can grow rapidly (higher jiM), whereas reactors which maintain a low, uniform substrate concentration throughout select microorganisms that are good scavengers of substrate (low Ks).21This complicates kinetic analysis and requires that experiments to determine kinetic parameters be conducted with systems that mimic the physical configuration to be employed in the full-scale facility. This topic is discussed in more detail in Chapter 8.

The biodégradation kinetics for many xenobiotic compounds can best be characterized by the Andrews equation (Eq. 3.39). Dividing both the numerator and denominator by Ks yields:

Ss/Ks

Expressing the equation in this manner emphasizes that the degree of substrate inhibition is determined by the ratio of Ks/K„ rather than by K, alone, as we saw with Eq. 3.40. Furthermore, Eq. 3.49 also makes it easy to see that the larger the ratio, the more inhibitory the substrate. 1,3- and 1,4-Dichlorobenzene are both moderately inhibitory compounds and have ratios of 0.14 and 0.08, respectively.42

In Section 3.2.9, the undesirability of oxygen being rate limiting was discussed, suggesting that knowledge of the oxygen half-saturation coefficient for hcterotrophs,

K<u„ is important. In spite of that, relatively little work has been done to estimate K<1m values for mixed microbial cultures, probably because population shifts occur in the community in response to changes in the dissolved oxygen concentration, making estimation of the value difficult. Nevertheless, limited pure culture data suggests that K()H is very low. For example, values of 0.01, 0.08, and 0.15 mg 0:/L have been reported for Sphaerotiius natans7" (a filamentous bacterium), Candida utilis(a yeast), and Citrobacter sp.7" (a floc-forming bacterium), respectively. This suggests that dissolved oxygen concentrations must be very low before they have serious impacts on the growth of heterotrophic bacteria, although they may influence the competition between filamentous and floc-forming bacteria. For depicting the impacts of dissolved oxygen on general heterotrophic biomass growth, one group adopted a value of 0.2 mg 0,/L for K<U|.^

It will be recalled from Section 3.2.8 that many investigators use substrate removal, rather than biomass growth, as the primary event with which to characterize biochemical operations. In that case, the primary kinetic parameter is the maximum specific substrate removal rate, q, rather than the maximum specific growth rate. Equation 3.44 defined q as |i/Y- Thus, q will be influenced by variations in Y as well as variations in p.. Like |i, Y is influenced both by the substrate being degraded and the microorganism performing the degradation (see Section 2.4.1). It should be noted, however, that Y is a reflection of the energy available in a substrate whereas (1 is a reflection of how rapidly a microorganism can process that energy and grow. Because they represent different characteristics, there is no correlation between the two parameters. For example, some substrates that are degraded very slowly (low (1) provide more energy to the degrading culture (i.e., higher Y) than do substrates that are degraded rapidly.42 This suggests that deductions about the variability in q cannot be made from data on |i alone, and vice versa. Knowledge of the true growth yield is also important. Typical Y values are discussed in Section 2.4.1.

Anoxic Growth of Heterotrophic Bacteria. As seen in Chapter 2, the only difference between aerobic and anoxic growth of heterotrophic bacteria on many substrates, such as biogenic organic matter, is the nature of the terminal electron acceptor and its impact on the amount of ATP that the cells can generate. Thus, for substrates for which this is true, we might expect the kinetic parameters describing growth under the two conditions to be very similar, and that is exactly what has been observed. When mixed microbial cultures were grown with excess oxygen or nitrate as the terminal electron acceptors and peptone as the rate-limiting substrate, the values of (in and Ks were very similar, being 0.14 hr 1 and 67 mg/L as COD, respectively, under aerobic conditions and 0.13 hr 1 and 76 mg/L as COD under anoxic conditions.7" Furthermore, as expected from the lower potential ATP formation under anoxic conditions, the anoxic yield was lower, being only 0.39 mg biomass COD/mg substrate COD versus a value of 0.71 aerobically. Consequently, qM was almost twice as large under anoxic conditions. Although data directly comparing kinetic parameters under aerobic and anoxic conditions are limited, experience with treatment systems suggest that these findings are generally true.7"

Anoxic growth conditions are generally imposed in biochemical operations for the purpose of reducing the nitrate concentration to low levels. Thus, there is a possibility that the terminal electron acceptor concentration will become rate limiting. Proper modeling of this situation requires knowledge of KNO, the half-saturation coefficient for nitrate. As with oxygen, the half-saturation coefficient for nitrate as the terminal electron acceptor has been found to be low, with values around 0.1 to 0.2 mg/L as N being reported/" "" Consequently, values in that range have been adopted by investigators conducting modeling studies." "^

Another parameter required to fully define the kinetics of microbial growth under anoxic conditions is K1(), the oxygen inhibition coefficient used in Eq. 3.48. If the cells are growing in a dispersed state so that all are exposed to the oxygen concentration in the bulk liquid, it appears that they do not denitrify when the dissolved oxygen concentration is above 0.1 to 0.2 mg/L."" However, when they grow as aggregates or films, the requirement for oxygen transport by diffusion allows biomass in the interior to be free of oxygen even when the bulk liquid contains it. Consequently, anoxic growth will occur even when the dissolved oxygen concentration in the bulk liquid exceeds 0.2 mg/L."" Thus, modelers have assumed values for K„, ranging from 0.2" to 2.0" mg/L.

Aerobic Growth of Autotrophic Bacteria. The nitrifying bacteria are the most important aerobic autotrophs and for the nitrogen levels normally found in domestic wastewater the kinetics of their growth can be adequately represented by the Monod equation (Eq. 3.36). Because only a limited number of genera and species are involved, the variability in the values of the kinetic parameters is less than that associated with heterotrophs. The maximum specific growth rate coefficient for Sitro-somonas has been reported to lie between 0.014 1 and 0.092""' hr ', with a value of 0.032 hr 1 considered to be typical at 20°C.'": The half-saturation coefficient for ammonia has been reported to be between 0.06 and 5.6 mg/L as N,1"" but a commonly accepted value is 1.0 mg/L."1"' The maximum specific growth rate coefficient for Nitrobacter is similar to that for Nitrosomonas, having been reported to lie between 0.006 ' and 0.060"'" hr '. Likewise, the value considered to be typical,"" 0.034 hr ', is similar to that for Nitrosomonas. The reported range of the half-saturation coefficient for Nitrobacter is slightly larger than that for Nitrosomonas, being 0.06 to 8.4 mg/L as nitrite-N,'"" as is the value thought to be typical, 1.3 mg/L."': The maximum specific growth rate coefficients for the autotrophic bacteria are considerably less than those for heterotrophic bacteria, reflecting their more restricted energy yielding metabolism and the fact that they must synthesize all cell components from carbon dioxide. This suggests that special consideration must be given to their requirements during design of reactors in which both carbon oxidation and nitrification are to occur. Although the half-saturation coefficients for the autotrophs are less than the reported values for heterotrophs growing on complex substrates, they are similar to the values reported for heterotrophs growing on single organic compounds. As a consequence of their small size, the kinetics of nitrification will behave in a zero-order manner over a broad range of ammonia and nitrite concentrations. As will be seen later, this has a significant impact on bioreactor performance.

A major difference in the growth characteristics of heterotrophic and autotrophic biomass is the greater sensitivity of the latter to the concentration of dissolved oxygen. Whereas the value of the half-saturation coefficient for oxygen is very low for heterotrophs, the values for the two genera of autotrophs are sufficiently high in comparison to typical dissolved oxygen concentrations that dual nutrient limitation as expressed by Eq. 3.46 should be considered to be the norm. For example, values of K,,.a for both Nitrosomonas and Nitrobacter have been reported to lie between 0.3 and 1.3 mg/L."" Measurements which considered the effects of diffusional re sistance on half-saturation coefficients have suggested that the true values lie near the lower end of the range,'" and values of 0.50 and 0.68 have been adopted as typical for Nitrosomonas and Nitrobacter, respectively, in systems in which some diffusional resistance will occur."12

Another difference between heterotrophic and autotrophic biomass is the greater sensitivity of the latter to changes in pH. Although all bacteria grow poorly outside of the normal physiological pH range of 6.0 to 8.0, nitrifying bacteria are particularly sensitive to pH, especially Nitrosomonas, as shown in Figure 3.4.'"' There it can be seen that the rate reaches a maximum at a pH of about 8, and declines sharply for lower pH values. A wide range of pH optima has been reported,"" but most workers agree that as the pH becomes more acid the rate of ammonia oxidation declines. "1 Furthermore, if a culture is acclimated to a low pH the effect is less severe than if the pH is suddenly shifted. Siegrist and Gujer1" have modeled the effect in Figure 3.4 with Eq. 3.50:

|1A = jlAm[l + 10l""s "'"] ' (3.50)

where p.Am is the maximum specific growth rate at the optimum pH. It should be