## Idealized Biochemical Reactors

Almost all bioreactors used for kinetics studies, whether batch or continuous reactors are the types of well-mixed reactors. Two well-mixed reactors are briefly described in the next several sections: the ideal batch reactor and the ideal continuous-flow stirred-tank reactor (CSTR).

The ideal batch reactor

In a batch reactor, the concentrations of nutrients, substrates, products, and bacterial cells vary with time as the microbial growth proceeds.

A molar material balance on component A in a batch reactor yields the following relation (Equation 4.11):

d_ dt culture volume culture volume

^ f molar concentration^

of component A

moles A formed by reaction v unit culture volume • unit time j

122 Food and Agricultural Wastewater Utilization and Treatment Or, mathematically the following (Equation 4.12):

where VR is culture volume, CA is molar concentration, and rA is reaction rate. The volume of the batch reactor is not the culture volume, VR, unless the reactor is full. VR is often considered as constant as long as there is not addition or removal of liquids from the reactor. This fact leads to the following simplification (Equation 4.13):

In order to calculate the change in concentration of component A, Equation 4.13 needs to be integrated over the time period, t-t0 (Equation 4.14):

CA0 A t0

where CA0 is the concentration of A at time, t0. The function that relates rA to t can be obtained based on the order of the reaction; the kinetic equations of biochemical reactions described in Chapter 1 can be used to solve Equation 4.14.

### The ideal plug-flow reactor

The plug-flow reactor is also called a piston-flow reactor. All materials leaving the plug-flow reactor will have been in the reactor for the same length of time. It is assumed that there is no longitudinal mixing or diffusion. A mathematical derivation of a plug-flow reactor, as shown in Fig. 4.3, can be made based on material balance for an element of reactor volume, dV for a component A (Equation 4.15):

input — output = disappearance due to reaction since input = F-, output = Fa + dFA

where FA is the flow rate of material A. Because d = d[FA0(1-XA)] = - FAodXA

So Equation 4.15 becomes

dV dXA

This can be integrated to produce the governing equation for a plug-flow reactor,

A Ca0 Jo

where CA0 = FA0/V and rA can be a zero, first, or second order reaction.

Ideal continuously stirred tank reactor (CSTR)

Fig. 4.4 schematically depicts a CSTR type of reactor. 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 thorough and complete such that the concentrations in any phase do not change with position within the reactor.

As indicated in Fig. 4.4, 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 the 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 4.20):

Rate of addition to reactor

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 4.21):

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

Rearrangement of Equation 4.21 yields the following (Equation 4.22):

where D = F/VR and is called the 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 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 number 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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