Energy balance, batch reactor general

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. 

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. 

In batch reactors, for thermally simple types of reactions, that is, ones that can be attributed to a single reaction step, generally applicable to the propagation step of polymerization reactions, we can write the following thermal energy balance (6)... 

The semibatch reactor is a cross between an ordinary batch reactor and a continuous-stirred tank reactor. The reactor has continuous input of reactant through the course of the batch run with no output stream. Another possibility for semibatch operation is continuous withdrawal of product with no addition of reactant. Due to the crossover between the other ideal reactor types, the semibatch uses all of the terms in the general energy and material balances. This results in more complex mathematical expressions. Since the single continuous stream may be either an input or an output, the form of the equations depends upon the particular mode of operation. 

As has been seen in Sect. 3, the equations of mass, energy and momentum balances for batch and plug flow reactors generally constitute a system of ordinary differential equations, with initial values. It is convenient to write such a system in a compact vector form, viz. 

For multiple reactions occurring in either a semibatch or batch reactor, Equation (9-18) can be generalized in the same manner as the steady-state energy balance, to give... 

Hint 5. In general, it is possible to operate a CSTR at a higher production rate than a comparable batch (or tubular) process (the beneficial effecf of cold inlef (monomer) feed allows for a higher specific polymerization rate than in batch). Typical calculations with the energy balance equations of Chapter 13 would indicate that for the same operating conditions, one can run a CSTR at about twice the rate of a batch reactor of the same volume (and heat removal capacity). 

The use of the energy balance to analyze or size an ideal CSTR is not as complex mathematically as for an ideal batch reactor or an ideal PFR. Therefore, in this section, we will not limit ourselves to an adiabatic CSTR. The adiabatic case will be covered as part of a more general treatment. 

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