CSTR constant-density system

VR/n is the volume of the CSTR used in the model Equations of this form were treated in Section 8.3.2.2. For a constant density system,... 

We derive the kinetics consequences for this scheme for reaction in a constant-volume batch reactor, the results also being applicable to a PFR for a constant-density system. The results for a CSTR differ from this, and are explored in Example 18-4. 

These rate laws are coupled through the concentrations. When combined with the material-balance equations in the context of a particular reactor, they lead to uncoupled equations for calculating the product distribution. For a constant-density system in a CSTR operated at steady-state, they lead to algebraic equations, and in a BR or a PFR at steady-state, to simultaneous nonlinear ordinary differential equations. We demonstrate here the results for the CSTR case. 

The following example illustrates a simple case of optimal operation of a multistage CSTR to minimize the total volume. We continue to assume a constant-density system with isothermal operation. 

A performance comparison between a BR and a CSTR may be made in terms of the size of vessel required in each case to achieve the same rate of production for the same fractional conversion, with the BR operating isothermally at the same temperature as that in the CSTR. Since both batch reactors and CSTRs are most commonly used for constant-density systems, we restrict attention to this case, and to a reaction represented by... 

Table 18.1 Comparison of PFR and CSTR for series-reaction network A -4 B -+ C (isothermal, constant-density system K = kz/ki)...

In CSTR reactors, both phases are considered to be in complete mixed and continuous-flow condition. In the general case where reactants can be gases and liquids, the following material balances can be applied (for simplicity we consider constant-density systems) (Hopper et al., 2001)... 

If the volumetric flowrates entering and leaving the CSTR are constant (this is equivalent to a constant density system), the above equation becomes... 

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... 

The design equation for a CSTR has a graphical interpretation that makes the existence of more than one steady state easier to understand. For a constant-density system, die design equation can be rearranged to... 

Consider a reactor system made up to two vessels in series a PFR of volume Vpp and a CSTR of volume EST, as shown in Figure 17.5. In Figure 17.5(a), the PFR is followed by the CSTR, and in Figure 17.5(b), the sequence is reversed. Derive E(d) for case (a) and for case (b). Assume constant-density isothermal behavior. 

Table 7-2 and Figs. 7-3 and 7-4 show the analytical solution of the integrals for two simple first-order reaction systems in an isothermal constant-volume batch reactor or plug flow reactor. Table 7-3 shows the analytical solution for the same reaction systems in an isothermal constant-density CSTR. 

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). 

Material Balance on A The material balance on A is just the CSTR design equation, written for reactant A. Since the total number of moles does not change as the reaction proceeds, and since a CSTR is isothermal by definition, the density of the system is constant. The constant-density form of the design equation is... 

Equations (4.1) or (4.2) are a set of N simultaneous equations in iV+1 unknowns, the unknowns being the N outlet concentrations aout,bout, , and the one volumetric flow rate Qout- Note that Qom is evaluated at the conditions within the reactor. If the mass density of the fluid is constant, as is approximately true for liquid systems, then Qout=Qm- This allows Equations (4.1) to be solved for the outlet compositions. If Qout is unknown, then the component balances must be supplemented by an equation of state for the system. Perhaps surprisingly, the algebraic equations governing the steady-state performance of a CSTR are usually more difficult to solve than the sets of simultaneous, first-order ODEs encountered in Chapters 2 and 3. We start with an example that is easy but important. 

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 system is sketched in Fig. 3.1 and is a simple extension of the CSTR considered in Example 2.3. Product B is produced and reactant A is consumed in each of the three perfectly mixed reactors by a first-order reaction occurring in the liquid. For the moment let us assume that the temperatures and holdups (volumes) of the three tanks can be different, but both temperatures and the liquid volumes are assumed to be constant (isothermal and constant holdup). Density is assumed constant throughout the system, which is a binary mixture of A and B. 

We will take a closer look at one of the simplest systems conceivable, a constant-volume and -density, cooled CSTR with a first-order, irreversible reaction A - B. While this model is quite simple it still contains most of the relevant issues surrounding an open-loop, nonlinear reactor. Referring to Fig. 4.5, this system can be described by one component balance and one energy balance ... 

Equations (8.3.26) and (8.3.27) are generally applicable to all types of CSTR cascades. If one recognizes that the use of such cascades is almost invariably restricted to liquid systems and that in such systems density changes caused by reaction or thermal effects are usually quite small, then additional relations or simplifications can be developed from these starting equations. In particular, this situation implies that at steady state the volumetric flow rate between stages is substantially constant. It also implies that for each reactor, 7, = t,, and that the following relation between concentration and fraction conversion is appropriate ... 

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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