The Nonisothermal Batch Reactor

In practice, it is not always possible, or even desirable, to carry out a reaaion under isothermal conditions. In this situation, both the energy and mass balances must be solved simultaneously  

Therefore, in this case T can be substituted from Eq. 8.2-5 into Eq. 8.1-3, which then becomes a single differential equation in Xa (or can be substituted into Eq. 8.2-1). This is done by utilizing Eq. 8.1-4, where the integral is evaluated by choosing increments of x and the corresponding T(xa) from Eq. 8.2-5. Again, the reactor residence time, 6, can be represented by the area under the curve 

Some analytical solutions are even possible for simple-order rate forms—they are given for the analogous situation for plug flow reactors in Chapter 9. Finally, the maximum adiabatic temperature change is found for x = 1.0, and then (for Xao = 0)  

More general situations require numerical solutions of the combined mass and heat balances. 

The temperature is constant or a prescribed function of time, T(0)—here the mass balance Eq. 8.1-3 can be solved alone as a differential equation  

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The description of the nonisothermal batch reactor then involves Equation (9.3.1) and either Equation (9.3.9) or (9.3.11) for nonisothermal operation or Equation (9.3.12)... 

The temperature varies with the reaction time in the nonisothermal batch reactor. To perform the energy balance, we use the same energy balance equation 14.67, annulling the molar flow terms, but considering the variation of sensible heat with temperature and time. Then,... 

THE NONISOTHERMAL BATCH REACTOR 397 time 9 can be represented by the area under the curve Nao / VrA xA, T x ) versus... 

The design equations for a nonisothermal batch reactor include A-fl DDEs, one for each component and one for energy. These DDEs are coupled by the temperature and compositional dependence of 91/. They may also be weakly coupled through the temperature and compositional dependence of physical properties such as density and heat capacity, but the strong coupling is through the reaction rate. 

A relatively simple example of a confounded reactor is a nonisothermal batch reactor where the assumption of perfect mixing is reasonable but the temperature varies with time or axial position. The experimental data are fit to a model using Equation (7.8), but the model now requires a heat balance to be solved simultaneously with the component balances. For a batch reactor. 

The design equations for a chemical reactor contain several parameters that are functions of temperature. Equation (7.17) applies to a nonisothermal batch reactor and is exemplary of the physical property variations that can be important even for ideal reactors. Note that the word ideal has three uses in this chapter. In connection with reactors, ideal refers to the quality of mixing in the vessel. Ideal batch reactors and CSTRs have perfect internal mixing. Ideal PFRs are perfectly mixed in the radial direction and have no mixing in the axial direction. These ideal reactors may be nonisothermal and may have physical properties that vary with temperature, pressure, and composition. 

For a single reaction in a nonisothermal batch reactor we can write the species and energy-balance equations... 

A more quantitative analysis of the batch reactor is obtained by means of mathematical modeling. The mathematical model of the ideal batch reactor consists of mass and energy balances, which provide a set of ordinary differential equations that, in most cases, have to be solved numerically. Analytical integration is, however, still possible in isothermal systems and with reference to simple reaction schemes and rate expressions, so that some general assessments of the reactor behavior can be formulated when basic kinetic schemes are considered. This is the case of the discussion in the coming Sect. 2.3.1, whereas nonisothermal operations and energy balances are addressed in Sect. 2.3.2. 

Consider accomplishing the reaction A + B C in nonisothermal batch reactor. The reaction occurs in the liquid phase. Find the time necessary to reach 80 percent conversion if the coolant supply is sufficient to maintain the reactor wall at 300 K. 

The design formulation of nonisothermal batch reactors consists of + 1 nonlinear first-order differential equations whose initial values are specified. The solutions of these equations provide Z s and 6 as functions of t. The examples below illustrate the design of nonisothermal ideal batch reactors. 

Chapter 1 treated single, elementary reactions in ideal reactors. Chapter 2 broadens the kinetics to include multiple and nonelementary reactions. Attention is restricted to batch reactors, but the method for formulating the kinetics of complex reactions will also be used for the flow reactors of Chapters 3 and 4 and for the nonisothermal reactors of Chapter 5. 

Most kinetic experiments are run in batch reactors for the simple reason that they are the easiest reactor to operate on a small, laboratory scale. Piston flow reactors are essentially equivalent and are implicitly included in the present treatment. This treatment is confined to constant-density, isothermal reactions, with nonisothermal and other more complicated cases being treated in Section 7.1.4. The batch equation for component A is... 

In this paper we formulated and solved the time optimal problem for a batch reactor in its final stage for isothermal and nonisothermal policies. The effect of initiator concentration, initiator half-life and activation energy on optimum temperature and optimum time was studied. It was shown that the optimum isothermal policy was influenced by two factors the equilibrium monomer concentration, and the dead end polymerization caused by the depletion of the initiator. When values determine optimum temperature, a faster initiator or higher initiator concentration should be used to reduce reaction time. 

In general, when designing a batch reactor, it will be necessary to solve simultaneously one form of the material balance equation and one form of the energy balance equation (equations 10.2.1 and 10.2.5 or equations derived therefrom). Since the reaction rate depends both on temperature and extent of reaction, closed form solutions can be obtained only when the system is isothermal. One must normally employ numerical methods of solution when dealing with nonisothermal systems. 

In this chapter, we first consider uses of batch reactors, and their advantages and disadvantages compared with continuous-flow reactors. After considering what the essential features of process design are, we then develop design or performance equations for both isothermal and nonisothermal operation. The latter requires the energy balance, in addition to the material balance. We continue with an example of optimal performance of a batch reactor, and conclude with a discussion of semibatch and semi-continuous operation. We restrict attention to simple systems, deferring treatment of complex systems to Chapter 18. 

If the batch reactor operation is both nonadiabatic and nonisothermal, the complete energy balance of equation 12.3-16 must be used together with the iiaterial balance of equation 2.2-4. These constitute a set of two simultaneous, nonlincmr, first-flijer ordinary differential equations with T and fA as dependent variables and I as Iidependent variable. The two boundary conditions are T = T0 and fA = fAo (usually 0) at I = 0. These two equations usually must be solved by a numerical procedure. (See problem 12-9, which may be solved using the E-Z Solve software.)... 

Figure 5.2 Graphical representation of the performance equations for batch reactors, isothermal or nonisothermal.

In practice the heat effects associated with chemical reactions result in nonisothermal conditions. In the case of a batch reactor the temperature changes as a function of time, whereas an axial temperature profile is established in a plug flow reactor. The application of the law of conservation of energy, in a similar... 

Plot the fractional conversion and temperature as a function of time for the batch reactor system described in Example 9.3.3 if the reactor is now adiabatic (U = 0). Compare your results to those for the nonisothermal situation given in Figure 9.3.3. How much energy is removed from the reactor when it is operated nonisothermally ... 

We shall recapitulate the governing equations in the next section and discuss the economic operation in the one following. The results on optimal control are essentially a reinterpretation of the optimal design for the tubular reactor. We shall not attempt a full derivation but hope that the qualitative description will be sufficiently convincing. The isothermal operation of a batch reactor is completely covered by the discussion in Chap. 5 of the integration of the rate equations at constant temperature. The simplest form of nonisothermal operation occurs when the reactor is insulated and the reaction follows an adiabatic path the behavior of the reactor is then entirely similar to that discussed in Chap. 8. 

Nonuniform temperatures, or a temperature level different from that of the surroundings, are common in operating reactors. The temperature may be varied deliberately to achieve optimum rates of reaction, or high heats of reaction and limited heat-transfer rates may cause unintended nonisothermal conditions. Reactor design is usually sensitive to small temperature changes because of the exponential effect of temperature on the rate (the Arrhenius equation). The temperature profile, or history, in a reactor is established by an energy balance such as those presented in Chap. 3 for ideal batch and flow reactors. 

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