## Theoretical Modeling and Design of Biological Reactors

Theoretical modeling of biological wastewater reactors using mathematical equations allows engineers and designers to test their strategies and evaluate their treatment options virtually, therefore reducing the amounts of time and money as well as the potential hazardous incidents that could happen to an actual experimentation. In the existing system, a robust model can be used to optimize the operational strategies. The development of the model often involves the selection of suitable equations that accurately describe fluid flow in the reactor and biochemical reactions in the form of microbial growth on organic and inorganic materials in the reactor. Many equations derived hereafter are more or less simplified equations of the generic reactor types. This approach has its own advantages: first, it acknowledges that the most biological reactors in use for wastewater treatment are quite similar to the generic reactors described below; second, the methodologies of derivation of the equations for the generic reactors are valid for more "realistic" or complex reactors. Some of those equations related to reaction kinetics—mass balance, stoichiom-etry, and chemical thermodynamics—have been explained previously. The overriding goal of this section is to combine fluid flow with kinetics in several geometrical environments of the generic reactor types to derive the reaction rate expressions and concentration profiles of substrates in the reactors. For the sake of simplicity, we focus our attention initially on single reactions occurring in the liquid phase of constant density in single reactors.

### Batch reactors

In a batch reactor, at any given time after the reactor starts, there is feed neither coming in nor coming out. The mass balance of a batch reactor from Equation 1.11 will be the following (Equation 1.24):

For a constant volume, the above equation is the following (Equation 1.25):

This may be integrated from the initial concentration of A, CA0 to the final concentration CAf, i.e., the following (Equation 1.26):

The exact relationship between rA and CA (kinetics) needs to be known in order to solve Equation 1.26 to establish the concentration history of re-actant A.

For zero-order reactions (order = 0), r = k, so Equation 1.25 develops into the following (Equation 1.27):

For first-order reactions (order = 1), -r = kCA, so Equation 1.25 becomes the following (Equation 1.28):

For second-order reactions (order = 2), — r = k(CB0— CA0 + CA)CA, so Equation 1.25 turns into the following (Equation 1.29):

where CBf = CAf — CA0 + CB0 and CB0 is the initial concentration of re-actant B.

If CA0 = CB0, — r = kC2A, and Equation 1.25 will yield the following (Equation 1.30):

Continuously stirred tank reactors (CSTRs)

Continuously stirred tank reactors (CSTRs) are widely used in biological wastewater treatment processes and can be schematically viewed as tanks with input and output while a mechanical or pneumatic device provides the means of thoroughly mixing the liquid phase in tanks. In CSTRs, the liquid inside the reactor is completely mixed. The mixing is provided through an impeller, rising gas bubbles (usually oxygen), or both. The most characteristic feature of a CSTR is that it is assumed that the mixing is uniform and complete so that the concentrations in any phase do not change with position within the reactor.

The dissolved oxygen in the tank is the same throughout the bulk liquid phase. Because of this uniformity of oxygen distribution in the reactor, a CSTR for wastewater treatment operations has the advantage of decoupling aerator or stirrer from the reaction as long as oxygen is well provided for (no need to consider pesky fluid mechanics), thus simplifying process design and optimization. Under the steady state, where all concentrations within the reactor are independent of time, we can apply the following materials balance on the reactor (Equation 1.31):

Rate of addition to isactor

Rate of accumulation within reactor

Rate of removal from reactor

Replacing the statements in the above expression with mathematical symbols leads to the following (Equation 1.32):

where F is volumetric flow rate of feed and effluent liquid streams.

Rearrangement of Equation 1.32 yields the following (Equation 1.33):

where D = F/VR and is called dilution rate. The term characterizes the holding time or processing rate of the reactor under steady state condition. It is the number of tankful volumes passing through the reactor tank per unit time and equal to the reciprocal of the mean holding time of the reactor.

Because of lack of time dependence of concentrations in CSTR and thus differential form of reactor analysis as in a batch reactor, CSTRs have the advantage of being well-defined, easily reproducible reactors and are used frequently in many cell growth kinetics studies, despite the relatively high cost and long time for achieving steady state. Batch reactors, which can be as simple as a sealed beaker or flask and used in large numbers simultaneously in an incubator shaker, are still widely used for their inexpensive, quick, and unbridled benefits. No matter what type of reactor is used, the goal of studying cell growth kinetics should be based on the intended application and scope of the use of the kinetics. Only then, the experimental design and implementation may be formulated.

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