CSTRs with Variable Density

The design equations for a CSTR do not require that the reacting mixture has constant physical properties or that operating conditions such as temperature and pressure be the same for the inlet and outlet environments. It is required, however, that these variables be known. Pressure in a CSTR is usually determined or controlled independently of the extent of reaction. Temperatures can also be set arbitrarily in small, laboratory equipment because of excellent heat transfer at the small scale. It is sometimes possible to predetermine the temperature in industrial-scale reactors for example, if the heat of reaction is small or if the contents are boiling. This chapter considers the case where both Pout and Tout are known. Density and Q ut wiU not be known if they depend on composition. A steady-state material balance gives 

An equation of state is needed to determine the mass density at the reactor outlet. Pout- Then, Qout can be calculated. 

Example 4.2 applied the method of false transients to a CSTR to find the steady-state output. A set of algebraic equations was converted to a set of ODEs. Chapter 16 shows how the method can be applied to PDEs by converting them to sets of ODEs. The method of false transients can also be used to find the equilibrium concentrations resulting from a set of batch chemical reactions. Formulate the ODEs for a batch reactor and integrate until the concentrations stop changing. Irreversible reactions go to completion. Reversible reactions reach equilibrium concentrations. This is illustrated in Problem 4.6(b). Section 11.1.1 shows how the method of false transients can be used to determine physical or chemical equilibria in multiphase systems. 

The design equations for a CSTR, Equations 4.1, do not require that the reacting mixture have constant physical properties or that operating conditions such as temperature and pressure be the same for the inlet and outlet environments. 

The fraction unreacted for the general case of any flow reactor is 

Note that Ya is a ratio of moles, and will be a ratio of concentrations only if density is constant. 

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What, if anything can be said about the residence time distribution in a nonisothermal (i.e., 7) / Tout) CSTR with variable density (i.e.. Pin Pout Rnd Qjfi Qout) ... 

While this equation is correct for the steady-state CSTR with variable density, it does not give a correct description of the transient CSTR for variable density because Na (meaning the number of moles of species A in the reactor) is not given by the preceding expression unless the density is constant. 

CSTRs with Variable Density 139 Code for Example 4.4... 

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. 

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. 

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 method of false transients converts a steady-state problem into a time-dependent problem. Equations 4.1 govern the steady-state performance of a CSTR. How does a reactor reach the steady state There must be a startup transient that eventually evolves into the steady state, and a simulation of that transient will also evolve to the steady state. The simulation need not be physically exact. Any startup trajectory that is mathematically convenient can be used even if it does not duplicate the actual startup. It is in this sense that the transient can be false. Suppose at time f = 0 the reactor is instantaneously filled with fluid of initial concentrations ao, bo, — The initial concentrations are usually set equal to the inlet concentrations, ai , , but other values can be used. The simulation begins with gin set to its steady-state value. For constant-density cases, gout is set to the same value, and V is constant. The variable-density case is treated in Section 4.3. 

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). 

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-... 

Equations (14.1)—(14.3) are a set of simultaneous ODEs that govern the performance of an unsteady CSTR. The minimum set is just Equation (14.2), which governs the reaction of a single component with time-varying inlet concentration. The maximum set has separate ODEs for each of the variables V, Hout, aout, bout, These are the state variables. The ODEs must be supplemented by a set of initial conditions and by any thermodynamic relations needed to determine dependent properties such as density and temperature. 

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... 

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