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Application of Eq. 3.27 gives:

where [r] = mg COD/(Lhr). Thus, once rxlJ has been defined, the other rates are also known.

Similar equations can be written for the growth of heterotrophs with nitrate as the terminal electron acceptor and for the aerobic growth of autotrophs. The derivation of such equations is left as an exercise for the reader.

Bacteria divide by binary fission. Consequently, the reaction rate for bacterial growth can be expressed as first order with respect to the active biomass concentration:

'S represents soluble constituents and X represents particulate constituents, with the subscript denoting the particular constituent involved.

where p. is the specific growth rate coefficient (hr '). It is referred to as a specific-rate coefficient because it defines the rate of biomass growth in terms of the concentration of active biomass present, i.e., the mass of biomass COD formed per unit time per unit of active biomass COD present. Equation 3.35 holds for any type of bacterial growth, regardless of the nature of the electron donor or acceptor, although much of the following is written in terms of heterotrophic biomass growth on an organic substrate. Consequently, subscripts are not used at this point to distinguish between heterotrophic and autotrophic biomass. although they will be used later when it is necessary to make that distinction. Substitution of Eq. 3.35 into Fq 3.34 defines the rates of substrate removal and oxygen (electron acceptor) utilization associated with biomass growth. It is important to note that the equation for oxygen utilization is also true for other electron acceptors, such as nitrate, as long as the quantity is expressed in oxygen equivalents.

3.2.7 Effect of Substrate Concentration on |x

The Monod Equation. Originally, exponential growth of bacteria was considered to be possible only when all nutrients, including the substrate, were present in high concentration. In the early 1940s, however, it was found that bacteria grow exponentially even when one nutrient is present only in limited amount."' Furthermore, the value of the specific growth rate coefficient. p.. was found to depend on the concentration of that limiting nutrient, which can be the carbon source, the electron donor, the electron acceptor, nitrogen, or any other factor needed by the organisms for growth. Since that time, the generality of this observation has been substantiated often, so that it can now be considered to be a basic concept of microbial kinetics." Let us first consider the situation when only an organic substrate is growth limiting.

Figure 3.1 illustrates the relationship that is obtained when p. is measured as a function of a single limiting substrate concentration. A number of different types of experiments can be performed to develop such a relationship and they are discussed in Chapter 8. The important thing to note at this time is that p. initially rises rapidly as the substrate concentration is increased, but then asymptotically approaches a maximum, which is called the maximum specific growth rate. p..

The question of the best mathematical formula to express the relationship shown in Figure 3.1 has been the subject of much debate. No one yet knows enough about the mechanisms of biomass growth to propose a mechanistic equation that will characterize growth exactly. Instead, experimenters have observed the effects of v arious factors on growth and have then attempted to fit empirical equations to their observations. Consequently, all equations that have been proposed are curv e-fits and the only valid arguments for use of one over another are goodness of tit. mathematical utility, and broad acceptance.

The equation with historical precedence and greatest acceptance is the one proposed by Monod (mo no')."' Although his original work was done in batch reactors. it was later extended and refined by workers using continuous cullures of single bacterial species growing on defined media and it was concluded that the curve could be approximated adequately by the equation for a rectangular hyperbola." Consequently, Monod proposed the equation:

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