CSTRs With Variable Holdups

If the previous example is modified slightly to permit the volumes in each reactor to vary with time, both total and component continuity equations are required for each reactor. To show the effects of higher-order kinetics, assume the reaction is now nth-order in reactant A. 

Our mathematical model now contains six first-order nonlinear ordinary differential equations. Parameters that must be known are k kj, 3, n. 

Initial conditions for all the dependent variables that are to be integrated must be given Cai, Ca2, Ca3, Pi, P2. and Pj. The forcing functions Caoio and Fo(, must 

Let us now check the degrees of freedom of this system. There are six equations. But there are nine unknowns Cai, a3 Pi P2 P3, Fy, F, and F3. Clearly this system is not sufficiently specified and a solution could not be obtained. 

What have we missed in our modeling A good plant operator could take one look at the system and see what the problem is. We have not specified how the flows out of the tanks are to be set. Physically there would probably be control valves in the outlet lines to regulate the flows. How are these control valves to be set A common configuration is to have the level in the tank controlled by the outflow, i.e., a level controller opens the control valve on the exit 

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In the reactors studied so far, we have shown the effects of variable holdups, variable densities, and higher-order kinetics on the total and component continuity equations. Energy equations were not needed because we assumed isothermal operations. Let us now consider a system in which temperature can change with time. An irreversible, exothermic reaction is carried out in a single perfectly mixed CSTR as shown in Fig. 3.3. 

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