Transients in Isothermal CSTRs

Unsteady behavior in an isothermal perfect mixer is governed by a maximum of -I- 1 ordinary differential equations. Except for highly complicated reactions such as polymerizations (where N is theoretically infinite), solutions are usually straightforward. Numerical methods for unsteady CSTRs are similar to those used for steady-state PFRs, and analytical solutions are usually possible when the reaction is first order. 

Example 14.1 Consider a first-order reaction occurring in a CSTR where the inlet concentration of reactant has been held constant at uq for f 0. At time f = 0, the inlet concentration is changed to Up Find the outlet response for t 0 assuming isothermal, constant-volume, constant-density operation. 

Solution The solution uses a simplified version of Equation (14.2). 

FIGURE 14.1 Dynamic response of a CSTR to changes in inlet concentration of a component reacting with first-order kinetics. 

Stability. The first consideration is stability. Is there a stable steady state The answer is usually yes for isothermal systems. 

Example 14.1 shows how an isothermal CSTR with first-order reaction responds to an abrupt change in inlet concentration. The outlet concentration moves from an initial steady state to a final steady state in a gradual fashion. If the inlet concentration is returned to its original value, the outlet concentration returns to its original value. If the time period for an input disturbance is small, the outlet response is small. The magnitude of the outlet disturbance will never be larger than the magnitude of the inlet 

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The steady-state design equations (i.e., Equations (14.1)-(14.3) with the accumulation terms zero) can be solved to find one or more steady states. However, the solution provides no direct information about stability. On the other hand, if a transient solution reaches a steady state, then that steady state is stable and physically achievable from the initial composition used in the calculations. If the same steady state is found for all possible initial compositions, then that steady state is unique and globally stable. This is the usual case for isothermal reactions in a CSTR. Example 14.2 and Problem 14.6 show that isothermal systems can have multiple steady states or may never achieve a steady state, but the chemistry of these examples is contrived. Multiple steady states are more common in nonisothermal reactors, although at least one steady state is usually stable. Systems with stable steady states may oscillate or be chaotic for some initial conditions. Example 14.9 gives an experimentally verified example. 

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 principal advantage of continuous reaction vessels is that they operate (after an initial transient period) under steady-state conditions that are conducive to the formation of a highly uniform and well-regulated product. In this section, we shall confine the discussion to continuous stirred-tank reactors (CSTRs). These reactors are characterized by isothermal, spatially uniform operation. 

In an ideal continuous stirred tank reactor, composition and temperature are uniform throughout just as in the ideal batch reactor. But this reactor also has a continuous feed of reactants and a continuous withdrawal of products and unconverted reactants, and the effluent composition and temperature are the same as those in the tank (Fig. 7-fb). A CSTR can be operated under transient conditions (due to variation in feed composition, temperature, cooling rate, etc., with time), or it can be operated under steady-state conditions. In this section we limit the discussion to isothermal conditions. This eliminates the need to consider energy balance equations, and due to the uniform composition the component material balances are simple ordinary differential equations with time as the independent variable ... 

Analyze the transient startup behavior of a train of two liquid-phase CSTRs that operate isothermally at the same temperature. Four components participate in two independent chemical reactions. In the first independent elementary reaction, 1 mol of reactant A and 2 mol of reactant B reversibly produce 1 mol of intermediate product D ... 

It is desired to find the transient response for an operating CSTR undergoing forcing by time variation in the inlet composition. Assume isothermal behavior and linear rate of consumption of species A according to... 

The development of practical methods [56] for the systematic design of new oscillating reactions in continuous stirred tank reactors (CSTR) lead to the discovery of several dozens of different isothermal oscillating systems, including the CIMA reaction [57]. This reaction is one of the very few to also exhibit transient oscillatory behavior in batch conditions. This and the fact that it does not exhibit marked excitability character like the well-known Belousov-Zhabotinsky reaction [5], lead us to select the CIMA reaction for systematic research on stationary spatial structures in open spatial reactors [14]. 

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