3.2.1 Generalized Equation for Biomass Growth
It will be recalled from Section 2.4.1 that biomass growth and substrate utilization are coupled. Furthermore, we see in Section 2.4.2 that environmental engineers account for maintenance energy needs through the decay reaction. This means that as long as the production of soluble microbial products is negligible, the only use of substrate is for biomass growth. Consequently, when a stoichiometric equation for biomass growth is written with the substrate as the basis, the stoichiometric coefficient for the biomass term will be the biomass true growth yield. With this in mind, the generalized equation for microbial growth can be written as:
Carbon source + energy source + electron acceptor + nutrients biomass + CO- + reduced acceptor + end products
For modeling purposes, it would be desirable to be able to write a quantitative equation in the same form for any situation, no matter what the carbon source, energy source, or electron acceptor. Using the concept of half-reactions, McCarty has devised a technique whereby this may be done.
Half-Reaction Approach. In the absence of significant soluble microbial product formation, all nonphotosynthetic microbial growth reactions consist of two components, one for synthesis and one for energy. The carbon in the synthesis component ends up in biomass, whereas any carbon associated with the energy component becomes carbon dioxide* Such reactions are also oxidation-reduction reactions and thus involve the transfer of electrons from a donor to an acceptor. For heterotrophic growth the electron donor is an organic substrate, whereas for autotrophic growth the electron donor is inorganic. To allow consideration of all of these factors, McCarty has written three types of half-reactions: one for cell material (RL), one for the electron donor (R,,), and one for the electron acceptor (R.,). These are presented in Table 3.2 for a variety of substances. Reactions 1 and 2 represent R, for the formation of biomass. Both are based on the empirical formula C<H-0:N, but one uses ammonia nitrogen as the nitrogen source whereas the other uses nitrate. Reactions 3-6 are half-reactions R, for the electron acceptors oxygen, nitrate, sulfate, and carbon dioxide, respectively. Reactions 7-17 are half-reactions R,, for organic electron donors. The first of these represents the general composition of domestic wastewater, while the next three are for wastes composed primarily of proteins, carbohydrates, and lipids, respectively. Reactions 11-17 are for specific organic compounds of interest in some biochemical operations. The last nine reactions represent possible autotrophic electron donors. Reactions 19-21 are for nitrification. To facilitate their combination, the half-reactions are all written on an electron equivalent basis, with the electrons on the right side.
The overall stoichiometric equation (R) is the sum of the half-reactions:
The minus terms mean that half-reactions R., and R, must be inverted before use. This is done by switching the left and right sides. The term f, represents the fraction of the electron donor that is coupled with the electron acceptor, i.e., the portion used for energy, hence the subscript e, and f„ represents the fraction captured through synthesis. As such, they quantify the endpoint of the reaction. Furthermore, in order for Eq. 3.14 to balance:
This equation is equivalent to stating that all electrons originally in the electron donor end up either in the biomass synthesized (f_) or in the electron acceptor (ft.). This is an important fundamental concept that we will return to later.
Empirical Formulas for Use in Stoichiometric Equations. As can be seen by examining Table 3.2, it was necessary to assume empirical formulas for biomass and alternative organic electron donors in order to write the half-reactions.
Various empirical formulas have been proposed to represent the organic composition of microbial cells. One of the oldest and most widely accepted in the field
Table 3.2 Oxidation Half-Reactions
Reaction number Half-reactions
Reactions for bacterial cell synthesis (R,)
Ammonia as nitrogen source:
1. — C\H-ON + — H O = - CO + — HCO, + — NH, + H 4 e
Nitrate as nitrogen source: 1 11
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