Fed-batch equation

It has already been established that, at equilibrium, the fed-batch equation reduces to the CSTR equation. Thus, the value of a in the fed-batch may be computed from the CSTR residence time by... 

Multiphase reactors can be batch, fed-batch, or continuous. Most of the design equations derived in this chapter are general and apply to any of the operating modes. Unsteady operation of nominally continuous processes is treated in Chapter 14. 

We want this concentration to be achieved at the end of the fed-batch interval when t = tfuii = VfuulQin- Equate the concentrations in Equations (14.4) and (14.5) and solve for Qi . The solution is numerical. 

Equation (20-70) is the unsteady-state component mass balance for fed-batch concentration at constant retentate volume. Integration yields the equations for concentration and yield in Table 20-19. 

In this case, there is a continuous supply of nutrients and a continuous withdrawal of the culture broth including the submerged free cells. The governing equations for continuous cultures are the same as the ones for fed-batch cultures (Equations 7.20-7.22). The only difference is that feed flowrate is normally equal to the effluent flowrate (Fm=Fout=F) and hence the volume. V, stays constant throughout the culture. 

As a second example let us consider the fed-batch bioreactor used by Ka-logerakis and Luus (1984) to illustrate sequential experimental design methods for dynamic systems. The governing differential equations are (Lim et al., 1977) ... 

For fed-batch fermentation, the model equations need to include the continuous feeding of sterile substrate to the fermenter, but zero outflow. The increase in volume (total accumulation of mass) that occurs in the fermenter due to the feeding is represented by a total mass balance relationship. 

The analytical model developed for network C assumes that the reactions take place in a fed-batch reactor and is a variation of the model developed for network B (see Section 4.2.2.2). Equations (57) to (59), written for networkB, are valid here as well. In addition, the mass balances of the cofactors A and B are given by... 

The distinguishing feature of the fed-batch reactor is that there is only an ingoing flow and no outgoing flow (Figure 11.13). Assuming ideal mixing equation (11.5) thus... 

Thus, if D is made equal to fi, the cell concentration would not vary with time. For a given substrate concentration in the fermentor C, fi is given by the Monod equation (i.e.. Equation 4.6). Alternatively, we can adjust the substrate concentration Cj for a given value of fi. Practical operation usually starts as batch culture and, when an appropriate cell concentration is reached, the operation is switched to a fed-batch culture. 

The characteristics of the fed-batch culture, shown by Equations 12.21 and 12.23, make it possible to keep the concentrations of the substrate and/or the cell at the desired values. For example, after a batch culture, a feed medium that contains the substrate at a high concentration can be fed, either continuously or intermittently, to the fermentor under a fed-batch operation. The values of the dilution rate and the substrate concentration in the feed medium can be determined using Equation 12.23. Thus, by using the fed-batch operation, the yield and/or productivity can be greatly improved in a variety of areas of biotechnology by controlling the concentrations of substrate and cell. Some examples of where the fed-batch operation can be effectively used are as follows. 

Bioreactors that use enzymes but not microbial cells could be regarded as fermentors in the broadest sense. Although their modes of operation are similar to those of microbial fermentors, fed-batch operation is seldom practiced for enzyme reactors. The basic equations for batch and continuous reactors for... 

Limited pH changes may occur if water electrolysis reactions (Equations 3 and 4) occur at the same rate and efficiency. In a completely mixed reactor, the proton produced at the anode should neutralize the hydroxyl ion produced at the cathode. However, the results indicated that the pH decreased to less than 5.5 even under completely mixed conditions in fed-batch reactors. The pH drop indicate less hydroxyl production at the cathode, either because different electrolysis reactions occurred (other than Equation 4) or because of biochemical reactions in the reactor. The type and concentrations of ions in the solution will impact the pH changes and require further investigation. Sodium bicarbonate was used and was effective in buffering the system for the range of electric field strengths studied. 

The equations used for these simulations of fed-batch reactors are similar to Eqs. (4.3)-(4.6), but the reactor volume is time-varying and a feed term is present ... 

A simple mathematical model is used for quantitative description of the process and consists of a set of equations relating inputs, outputs, and key parameters of the system. The model for an alcoholic fermentation fed-batch process developed by Mayer (10) and adapted with the Ghose and Tyagi (11) linear inhibition term by the product was used as the starting point for the development of a model-based substrate sensor with product (ethanol) and biomass on-line measurements. 

The following equations describe a general form of the model for the fed-batch process and total medium, having only substrate as feed flow, and include mass balances for substrate, product, biomass, and kinetic relations, respectively. 

This section discusses treatment of experimental data, especially for conditions where state variables change over time. These are the most difficult data to treat and correspond to cultures from batch, fed-batch, or any continuous transient phase. In continuous steady state, the state variables and rates values do not alter with time, and the rate calculation results from the algebraic equation solution. 

Equations 4 to 6 present the mass balances for cells, substrate, and product in a fed-batch culture ... 

The fed batch reactor (FBR) is a reactor where fresh nutrients are added to replace those already used. The rate of the feed flow u may be variable, and there is no outlet flowrate from the fermenter. As a consequence of feeding, the reactor volume changes with respect to time. Figure 11-22 illustrates a simple fed-batch reactor. The balance equations are ... 

In this case F = F = F=Q. Reactor samples (i.e. F 0) do not affect the analysis because they reduce the culture volume without altering any of the concentrations. However, acid or base additions do affect concentrations. If the volume added is significant the reactor effectively becomes a fed-batch reactor. In this case Equations 4.2.11 and 4.2.12 (below) would apply. For negligible cell lysis the expressions for fx pp, x and for batch culture can be obtained from Equations 4.2.7-4.2.9 by setting D = 0 ... 

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