Mixing-Cell Sequences

The mass and energy balances for an individual unit in the mixing-cell sequence may be written in general form as 

If we specify the concentration in the unit, then the reaction rate is fixed according to (-r) = (C.-C) = (6-5) 

Solving for C from equation (6-la) gives, in the usual fashion, 

Replacing this in equation (6-2a) yields the general relationship to be satisfied by temperature and concentration 

The problem of backwardness alluded to above resides in the fact that k in both of these equations depends on 6, not 6q. 

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Figure 5.4 Mixing-cell sequence with micromixing effects.

Figure 6.3 Mixing-cell sequence in the model of Deans and Lapidus. [After H. A. Deans and L. Lapidus, Amer. Inst. Chem. Eng. J., 6, 656, with permission of the American Institute of Chemical Engineers, (I960).]...

To simulate this type of operation a mixing-cell sequence is quite useful [J.B. Butt and D.M. Rohan, Chem. Eng. Sci., 23,489 (1968)]. Let us start with the familiar case of first-order kinetics for the main reaction and first-order kinetics for the deactivation, where coke deposition occurs via a reactant precursor mechanism [see scheme (XXVII) in Chapter 3). For the main reaction we may write... 

In Chapter 4 it was pointed out that the performance of a CSTR sequence approached that of a single PFR of equivalent total residence time as the number of units in a sequence approached infinity. This result is also obeyed by the F 6) and E 6) curves computed from the mixing-cell model reported in Figure 5.3. Since the plug-flow model represents one limit of the dispersion model (that when D 0), it is reasonable to assume that there is an interrelationship between mixing-cell and dispersion models that can be set forth for the more general case of finite values... 

Mixing-cell models were discussed extensively in Chapter 4 under the guise of the analysis of CSTR sequences. It is a good time to revisit some of this analysis from the specific point of view of modeling nonideal reactors. 

Use of the CSTR sequence as a model for nonideal reactors has been criticized on the basis that it lacks certain aspects of physical reality, such as the absence of backward communication between the individual mixing cell units. Such may be the case nonetheless the mathematical simplicity of the approach makes it very attractive, particularly for systems with complex kinetics, nonisothermal effects, or other complicating factors. 

Microorganisms exhibit nutritional preferences. The enzymes for common substrates such as glucose are usually constitutive, as are the enzymes for common or essential metabohc pathways. Furthermore, the synthesis of enzymes for attack on less common substrates such as lactose is repressed by the presence of appreciable amounts of common substrates or metabolites. This is logical for cells to consei ve their resources for enzyme synthesis as long as their usual substrates are readily available. If presented with mixed substrates, those that are in the main metabolic pathways are consumed first, while the other substrates are consumed later after the common substrates are depleted. This results in diauxic behavior. A diauxic growth cui ve exhibits an intermediate growth plateau while the enzymes needed for the uncommon substrates are synthesized (see Fig. 24-2). There may also be preferences for the less common substrates such that a mixture shows a sequence of each being exhausted before the start of metabolism of the next. 

Mixing of the electrode products causes hydrolytic precipitation of the nickel and, after separation of the nickel hydroxide, the filtrate was returned to the cells. The sequence of the electrolytic purification steps is outlined in Figure 6.28. Nickel hydroxide slurry is first added to the anolyte for the purpose of raising the pH to 3.7 (2 H+ + Ni(OH) = Ni2+ + 2 H20), and iron(II) is oxidized by introducing chlorine. This causes hydrolytic precipitation of the iron(III) and corrects the nickel ion deficiency by the low anodic current efficiency. The iron(III) hydroxide is removed by filteration. The clarified solution is then treated with nickel carbonate and further chlorine to oxidize the cobalt(II) and allow its separation as cobalt(I II) hydroxide. 

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