CSTR variable-density system

We will begin the discussion of continuous reactors with the ideal CSTR, first with a constant-density example and then with a variable-density example. The ideal PFR then will be treated, for the constant-density and then the variable-density case. To point out the differences between constant- and variable-density systems, and the differences between how the different reactors are treated, all of our analysis will be based on Reaction (4-B) and the rate equation given by Eqn. (4-13). 

Example 4.3 represents the simplest possible example of a variable-density CSTR. The reaction is isothermal, first-order, irreversible, and the density is a linear function of reactant concentration. This simplest system is about the most complicated one for which an analytical solution is possible. Realistic variable-density problems, whether in liquid or gas systems, require numerical solutions. These numerical solutions use the method of false transients and involve sets of first-order ODEs with various auxiliary functions. The solution methodology is similar to but simpler than that used for piston flow reactors in Chapter 3. Temperature is known and constant in the reactors described in this chapter. An ODE for temperature wiU be added in Chapter 5. Its addition does not change the basic methodology. 

The classical CRE model for a perfectly macromixed reactor is the continuous stirred tank reactor (CSTR). Thus, to fix our ideas, let us consider a stirred tank with two inlet streams and one outlet stream. The CFD model for this system would compute the flow field inside of the stirred tank given the inlet flow velocities and concentrations, the geometry of the reactor (including baffles and impellers), and the angular velocity of the stirrer. For liquid-phase flow with uniform density, the CFD model for the flow field can be developed independently from the mixing model. For simplicity, we will consider this case. Nevertheless, the SGS models are easily extendable to flows with variable density. 

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. 

The total mass balance gives the rate of change of volume with inlet and outlet flow rates for a well-mixed constant-density system. A fed batch is a special case of a variable-volume CSTR operation It has been defined as a bioreactor with inflowing substrate but without outflow. For this system, the equation becomes... 

Now the mass density varies with conversion because 1 mole of A is converted into 2 moles of C when the reaction system goes to completion. Therefore, we cannot use the CSTR mass-balance equations, Cjo — Cj = X to use the variable-... 

Consider the control of a jacketed, continuous, stirred-tank reactor (CSTR) in which the exothermic reaction A — B is carried out. This system can be described by 10 variables, as shown in Figure 20.7 h, T, Ca, Cai, T Fi, F Fc, T and T oy diree of which are considered to be externally defined C I and Tco- Its model involves four equations, assuming constant fluid density. 

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