## F x x

Figure 5.1 Schematic diagram of a CSTR. The biomass separator returns biomass from the effluent to the reactor. The stream F„ removes biomass from the reactor at a concentration equal to that in the reactor.

microbial debris at concentration X,, is also present. Two effluent streams are discharged from the bioreactor, but because the bioreactor is completely mixed, the concentrations of all soluble constituents in them are the same as the concentrations in the bioreactor. One stream, with flow rate Fw, flows directly from the bioreactor and carries biomass and cell debris at concentrations equal to those in the bioreactor. The other, with flow F F„, passes through a biomass separator before discharge, making it free of suspended material. All particulate material removed by that separator is returned to the bioreactor.

### 5.1.1 Definitions of Residence Times

A residence time defines the average amount of time a constituent stays in a system. Two types of constituents are present in the CSTR in Figure 5.1: (1) soluble, denoted by the symbol S, and (2) particulate, denoted by the symbol X. Because their residence times are not necessarily the same, one must be defined for each.

Dissolved constituents are intimately associated with the fluid and cannot be easily separated from it. Thus, their residence time in a reactor is equal to the mean hydraulic residence time, which is defined by Eq. 4.15:

Particulate constituents can be separated from a fluid by physical means, such as filtration or settling, and engineers use this characteristic to control their discharge from a bioreactor. The bioreactor shown in Figure 5.1 has two discharge streams, one containing particulate material at a concentration equal to that in the bioreactor and one free of it. The stream containing particulate material, F„, is called the wast t = V/F

age stream because it provides the means for wasting particulate material from (he bioreactor. The other stream, called the main effluent stream, is free of particulate material because it has passed through a separator such as a filter or settler. Although in reality, such streams will contain small concentrations of particulate material due to inefficiencies in the separation device, we will consider them to be totally free of such material for now. In Part III, we consider how to account for the presence of particulate material in the effluent from real systems.

The importance of the biomass separator is that it makes the residence time of particulate materials greater than the residence time of soluble materials. Thus, we must define a second residence time, called the solids retention time or mean cell residence time, which represents the average length of time a particulate constituent stays in a bioreactor. It will be given the symbol (H\ for use in equations and the acronym SRT for use in the text. By analogy to Eq. 4.15, the SRT is defined as the mass of a particulate constituent contained in the bioreactor divided by the mass discharged from the bioreactor per unit time:

For the case illustrated in Figure 5.1, in which the concentration of particulate material in the wastage stream, X„, is equal to the concentration in the bioreactor. X, Eq. 5.1 may be simplified to:

It should be remembered, however, that the basic definition of SRT involves mass flow rates rather than volumetric flow rates. Comparison of Eq. 5.2 to Eq. 4.15 reveals that:

In other words, the closer F„ approaches F, the closer the SRT approaches the HRT, so that in the limiting case where no biomass separator is employed, the SRT and the HRT are the same.

### 5.1.2 Format for Model Presentation

To describe any reactor, mass balance equations must be written around a control volume for all constituents of importance. As seen in Section 4.3.1, the appropriate control volume for a CSTR is the entire reactor, since it is homogeneous throughout. If we let CA represent the mass-based concentration of constituent A in the reactor in Figure 5.1 and consider the reactor to have constant volume, the mass balance equation is:

(dCA/dt)V = F CA() - F„ CA - (F - FJC + rA-V (5.4)

where CA() is the concentration of A in the influent stream and CA is its concentration in the stream passing through the biomass separator. For soluble constituents, CA is the same as the concentration in the CSTR, whereas for particulate constituents. CA is zero. The reaction term rA represents the sum of all reactions in which constituent A participates, as seen in Eq. 3.12. Furthermore, as seen in Chapter 3, rA may be a function of the concentrations of several constituents. If so, it will be necessary to solve several mass balance equations simultaneously to determine the concentration of any single constituent in a bioreactor.

For the situation depicted in Figure 5.1, mass balance equations must be written for at least three constituents: Ss, X„ M, and X,). In addition, we will be interested in the amount of oxygen that must be supplied through an oxygen transfer system. We will also need a mass balance equation for it, making a total of four.

Consideration of the number of mass balance equations required for this simple situation, and reflection on the number of events and constituents that could be considered, as discussed in Chapters 2 and 3, makes it clear that a system is needed for providing the required information. The matrix format4 " discussed in Section 3.1.3 provides such a system.

Table 5.1 presents all of the information required to compile the reaction rate terms, r„ for insertion into the mass balance equations for the situation described above. It contains the information in Eq. 3.11, with the entries in the body of the table representing the stoichiometric coefficients in COD units for each constituent participating in each reaction (recall that oxygen is negative COD). The entries in the right column represent the process rates for the reactions, and the subscript H on the coefficients signifies that they are applicable to heterotrophic biomass. Only two reactions are considered in this case, growth and decay. The stoichiometric equation for growth with active biomass as the reference constituent was given in COD units by Eq. 3.33, and examination of it reveals that the coefficients in Table 5.1 correspond to the coefficients in it, and fheir sum should equal zero, as it does. The term, process rate, r„ in Table 5.1 refers to the generalized reaction rate for process j as defined in Eq. 3.10. For the growth process it can be obtained by substituting Eq. 3.35 into Eq. 3.34. As we saw in Section 3.2.2, the specific growth rate coefficient, is a function of the substrate concentration, Ss, but to simplify upcoming explanations, that substitution will be made later. The stoichiometric equation for the traditional approach of modeling decay is given by Eq. 3.53 and inspection of it reveals its correspondence to the coefficients in Table 5.1. Again, summing the coefficients reveals that continuity has been maintained. The process rate expression for decay may be obtained by multiplying Eq. 3.56 by the stoichiometric coefficient for biomass in Table 5.1, giving the result shown in the table.

As stated by Eq. 3.12, the overall rate expression for each constituent to be inserted into its mass balance equation is obtained by summing the products of the process rate expressions times the stoichiometric coefficients appearing in the column

Process |

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