Concentration profiles for the autocatalytic reactions

The Monod equation has the same form as the Michaelis-Menten[29] equation for enzyme kinetics, but differs in that it is empirical while the latter is based on theoretical considerations. For the simple enzyme reaction given below, the Michaelis-Menten equations are provided below. 

The following example presents manufacturing of tequila in a batch reac- 

Example 3.1 Recently Harrera et al. [30] presented a model for tequila fermentation based on experimental data. This model is described below. If the fermentation is carried out in a batch reactor, find the concentration profiles for substrate, biomass, and product from time, t = 0 to t = 100 hrs. 

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FIG. 7-5 Product concentration profile for the autocatalytic reaction A + P => 2P with rate r = kCaCv. 

Figure 7-5 illustrates the dimensionless concentration profile for the reactant A and product P, C,/(Ca0 + Cpo), for C o/Cp0 = 2, indicative of a maximum rate at the inflexion point (maximum in slope of the concentration-time curve), typical to autocatalytic reactions. 

Figure 3.5a shows the concentration profile for this reaction with k = 0.5 time, Cao = 3 moles/volume, and Cro = 1.0 moles/volume and Figure 3.5b shows the reaction rate. As can be seen the reaction rate is parabolic, it is slower initially and goes through a maximum at Ca = Cr and again decreases. For an autocatalytic reactor, some product R here) should be present to start the reaction. 

Fig. 9.3. Stationary-state concentration profiles aS5(p) for a reaction-diffusion cell with a single cubic autocatalytic reaction (a) D = 0.1157, only small extents of reactant consumption arise (b) D = 0.0633, a higher extent of reactant consumption occurs, particularly towards the centre...

The solutions of conductivity problems shown in the previous sections were obtained for zero-order kinetics. When the approximation by zero-order kinetics is not justified, which is the case, especially for autocatalytic reactions, a numerical solution is required. Here the use of finite elements is particularly efficient. The geometry of the container is described by a mesh of cells and the heat balance is established for each of these cells (Figure 13.5). The problem is then solved by iterations. As an example, a sphere can be described by a succession of concentric shells (like onion skins). In each cell, a mass and a heat balance are established. This gives access to the temperature profile if one considers the temperature of the different cells, or the temperature and conversion may be obtained as a function of time. 

Figure 3 Computed concentration versus time profiles for various values of catalytic efficiencies e for (a) an autocatalytic system with reaction order p = 0.5 and (b) an autocatalytic system with the reaction order p = 1. The sigmoidal shape of the profile is only visible above a cerfain value for e.

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