## Parallel Evolution Of Decay Equations

FIGURE 2.24 Energy circuit diagram of decay.

In hazardous, municipal, or industrial waste treatment, bioreactors are used primarily to reduce the concentration of contaminants in incoming wastewaters to acceptably low levels" (Armenante, 1993). Equations have been developed which describe the process of reduction of contaminants in bioreactors (see Figure 2.16) and design is based on these equations. This approach has evolved over time and can be traced by examining various editions of textbooks on sanitary engineering. For example, Leonard Metcalf and Harrison Eddy produced a standard text that has spanned the design evolution of wastewater treatment. Early versions of their text contained essentially no equations, and design was based on practical experience with the available systems, such as trickling filters or activated sludge units (Metcalf and Eddy, 1916, 1930). This text evolved with the field with revisions by George Tchobanoglous and, by the 1970s, it was filled with equations (Metcalf and Eddy, 1979). These equations are quantitative expressions of the practical experience developed over time by engineers. In many ways the equations are the heart of the engineering method because of their role in design. A simple but fundamental equation is reviewed below to demonstrate this approach to design.

Perhaps the simplest equation used to describe the biodegradation of organic materials (BOD) in domestic wastewaters is the first-order reaction:

where

Ce = effluent BOD, in mg/l Co = influent BOD, in mg/l k = first-order reaction rate constant, in 1/day t = time of flow through the system or hydraulic residence time, in days

This equation is the integrated form of the model shown in Figure 2.24. Basically, an initial amount of organic materials (Co) is degraded over a given amount of time (t) according to the rate constant (k) that depends on the temperature at which the reaction occurs. It has been used to describe many kinds of wastewater treatment systems, and it is used as a starting point for considering BOD removal in treatment wetlands by Reed et al. (1995) and Crites and Tchobanoglous (1998). It was first used in sanitary engineering to describe the BOD concept and to develop the test procedure (Gaudy, 1972). Early sanitary engineers struggled with standardizing BOD tests and much literature on the subject can be found in the Sewage Works

Journal of the 1930s (see, for example, Hoskins, 1933). Eventually the BOD definition was standardized as the amount of oxygen consumed in a 250-ml glass bottle filled with wastewater over a 5-day period, with suitable dilutions. As an aside, it is interesting that they termed this demand, which suggests a connection with the law of supply and demand in economics. Oxygen is "supplied" by various forms of reaeration (diffusion and primary productivity) and it is "demanded" by microorganisms that degrade the organic materials through aerobic respiration. This overall conception is described by the Streeter-Phelps equation mentioned earlier in terms of wastewater disposal in rivers (Figure 2.4). It is also interesting to note that this very basic equation had its origin in a method of measuring biodegradation, i.e., putting wastewater in a bottle and measuring the oxygen concentration decrease. The real object of concern was the organic materials in the water, but oxygen was recorded because it was relatively easy to measure and its concentration changed in direct proportion to the change in organic materials.

Engineering design equations for wastewater treatment are often stated with the form given above in order to show effectiveness of biodegradation (Ce/Co) on one side of the expression, essentially in terms of percent removal. Design criteria are often given in these terms. For example, it might be required that 90% of the influent BOD be degraded by the treatment system in order to meet a regulatory requirement. Design to meet this requirement is done by sizing the treatment system. For this step, the original equation is expanded with an expression for the hydraulic residence time:

where t = hydraulic residence time, in days L = length of the treatment system, in meters W = width of the treatment system, in meters D = depth of the treatment system, in meters Q = average flow rate, in m3/day

Plugging this expression into the original equation explicitly places dimensions that can be altered by design into consideration:

Values of Ce and Co are given and form the design criteria. Values of k and Q are known for the particular situations being designed for. Design consists of finding combinations of L, W, and D (i.e., size of the treatment system) that match with the situation. In other words, the above equation is solved for size, knowing all of the other parameters. Much of wastewater treatment engineering involves creating and solving design equations for the size of the treatment system in a similar fashion. Many alternative treatment systems exist and many equations have been developed to describe them. Some of these equations use theoretical expressions for reaction rates while others use empirical relationships based on practical experience. Various extensions for recirculation are often required and are incorporated into equations. Kadlec and Knight's (1996) text provides the state of the art in treatment wetland design equations and all indications are that this knowledge will continue to be increased and refined, as was true for traditional sanitary engineering.

A form of parallel evolution has occurred in ecology with equations for decomposition. In this case the goal was to develop equations that describe the process of decay, rather than equations that can be manipulated to meet biodegradation criteria. The model shown in Figure 2.24 is the same model arrived at by ecologists to model biodegradation in their contexts. It was first used by Jenny et al. (1949) and is still the basic approach taken, at least as a starting point. Decomposition, as described by this equation, is as important in ecology as is primary productivity. Another parallel is that the origin of this model was largely method-based, as was true with wastewater treatment engineering in terms of BOD measurement. In this case ecol-ogists place known amounts of organic materials (usually leaf litter) into mesh bags and use them to measure mass loss over time. These bags are placed into the environment being evaluated, and the mesh material of the bags allows access by at least those decomposer organisms that are smaller than mesh size. A set of bags is placed in the environment at the beginning of the study, and a subset is picked up at intervals and weighed throughout the study. In this way mass loss is recorded and the decay constant, k, can be found as the slope of the curve of mass loss vs. time. This is called the litter bag method of studying decomposition (Shanks and Olson, 1961), and Olson (1963) set out an early mathematical description of the modelling, which includes an analog of the Streeter-Phelps equation. Although advancements have been discussed (Boulton and Boon, 1991; Wieder and Lang, 1982), this approach is still the foundation for understanding decomposition in ecology.

In essence then, the same kind of thinking occurred in ecology to describe decomposition as occurred in sanitary engineering to describe reduction in BOD of wastewaters. Both the scientists (ecologists) and engineers (sanitary engineers) studied the decay process in their particular systems and came up with the same equation. The engineers took one further step of being able to manipulate the equation for design, but this was not required for the scientists. The parallel evolution of thinking on this single topic provides a connection for understanding the new field which combines ecology and engineering.