The Ideal Batch Reactor

For the ideal batch reactor it is assumed that at a certain point in time (r = 0) a reaction starts, in a mixture containing all the reactants. If the density of the reaction mixture does not change, the concentration of the reactant A changes with time in the following manner  

This equation is a mass balance for component A. In order to find the change of the concentration of A with time we have to substitute the kinetic equation. Let us first take the simplest form, eq. (3.1), and substitute this in eq. (3.12)  

Note that this equation applies only when there are no streams entering or leaving the reactor, and when the volume of the reaction mixture does not change, i.e., for constant density. Consequently the symbol (- dc/dt) may not be used to indicate the reaction rate in general Still one finds this faulty notation in many books on chemical kinetics. 

Integration of eq. (3.13) gives the concentration of A as a function of time, for for a first order reaction, in an ideal batch reactor, and for constant densityy (p = p ) when the original concentration c at time r = 0 is given 

Note that for first order reactions, in ideal batch reactors, the concentration of reactant A decreases exponentially with time see figure 3.1. 

Heat and mass transfer limitations are rarely important in the laboratory but may emerge upon scaleup. Batch reactors with internal variations in temperature or composition are difficult to analyze and remain a challenge to the chemical reaction engineer. Tests for such problems are considered in Section 1.5. For now, assume an ideal batch reactor with the following characteristics  

Reactants are quickly charged, mixed, and brought to temperature at the beginning of the reaction cycle. 

Mixing and heat transfer are sufficient to assure that the batch remains completely uniform throughout the reaction cycle. 

A batch reactor has no input or output of mass after the initial charging. The amounts of individual components may change due to reaction but not due to flow into or out of the system. The component balance for component A, Equation (1.6), reduces to 

The ideal, constant-volume batch reactor satisfies the following component balance  

Moreover, branching reaction mechanisms can take place when at least one reaction leads to multiplication of radicals, such as 

In this case, the fast increase of concentration of radicalic species can result in the loss of control of the reaction (runaway) and in the explosion of the system. This radicalic runaway may be strongly enhanced by linked thermal effects that are discussed in more details in Chap. 4. 

Kinetic mechanisms involving multiple reactions are by far more frequently encountered than single reactions. In the simplest cases, this leads to reaction schemes in series (at least one component acts as a reactant in one reaction and as a product in another, as in (2.7)-(2.8)), or in parallel (at least one component acts as a reactant or as a product in more than one reaction), or to a combination series-parallel. More complex systems can have up to hundreds or even thousands of intermediates and possible reactions, as in the case of biological processes [12], or of free-radical reactions (combustion [16], polymerization [4]), and simple reaction pathways cannot always be recognized. In these cases, the true reaction mechanism mostly remains an ideal matter of principle that can be only approximated by reduced kinetic models. Moreover, the values of the relevant kinetic parameters are mostly unknown or, at best, very uncertain. 

The model reduction procedure must be adapted to the use of the simplified models and to the availability of experimental data needed to evaluate the unknown parameters, as discussed in Chap. 3. In general, more complex models are used for the design of the reactor and for the simulation of the entire process, whereas more simplified models are best fit for feedback control. In the following chapters it is shown that fairly accurate results are obtained when a strongly simplified kinetic model is used for control and fault diagnosis purposes. 

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. 

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The feed is charged all at once to a batch reactor, and the products are removed together, with the mass in the system being held constant during the reaction step. Such reactors usually operate at nearly constant volume. The reason for this is that most batch reactors are liquid-phase reactors, and liquid densities tend to be insensitive to composition. The ideal batch reactor considered so far is perfectly mixed, isothermal, and operates at constant density. We now relax the assumption of constant density but retain the other simplifying assumptions of perfect mixing and isothermal operation. 

Figure 2.4. Schematic drawings of a cylindrical flow reactor and a batch reactor. In the ideal case the flow reactor operates as a plug-flow reactor in which the gas moves as a piston down through the tube, whereas the ideal batch reactor is a well-mixed Tank Reactor...

Substituting into the ideal batch reactor equation gives... 

Consider the ideal batch reactor illustrated in Figure 3.2.1. If it is assumed that the contents of the reactor are perfectly mixed, a material balance on the reactor can be written for a species i as ... 

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 ... 

Equation (7-60) is identical to that of the ideal batch reactor, Eq. (7-47), and the two reactor systems can be modeled in identical fashion. 

Note that here too, the summation on the left-hand side is over the independent reactions only, whereas the summation on the right-hand side is over all the reactions that take place in the reactor. We follow the same procedure as in the case of the ideal batch reactor. We first write the. summation on the right as two sums over dependent reactions and independent reactions. Next, we express the stoichiometric coefficient of species j in the kth-dependent reaction, (Sj)k, in terms of the stoichiometric coefficients of species j in the independent reactions, (Sj)m, using Eq. 2.4.9, and then switch the order of the. summations to obtain... 

Clearly, space-time Vr/V in the ideal tubular reactor is the same as residence time in the ideal batch reactor but this is true, even in the case of ideal operation, only if the reaction is not accompanied by a volume change. Otherwise, the space-time calculated with volumetric flow rate at entrance is not equal to residence time. The latter depends on degree of conversion and is therefore not a useful concept. In general, with flow reactors, residence times should be used with caution since there may be a distribution of them or they may depend on conversion. 

Figure 4.1. The three types of ideal chemical reactors. The ideal batch reactor (BR) is well mixed but closed to mass transfer. The ideal mixed flow reactor (MFR) is well mixed and subject to continuous mass transfer.The fluid in an ideal plug flow reactor (PFR) moves as slugs, which are closed to mass transfer with each other and therefore act as batch reactors moving through space.

In this chapter the most important operation modes of reactors are considered. Models are developed by combining simple reaction kinetics for single-phase reactions with mass balances for five ideal model reactors the ideal batch reactor the semi-batch reactor the plug flow reactor the perfectly mixed continuous reactor and the cascade of perfectly mixed reactors. For isothermal conditions, conversions can be calculated on the basis of chemical kinetics only. 

In all three models the rate of mixing of the reactants is assumed not to influence the rate of the chemical reactions. In the ideal batch reactor, the reactants are mixed before they react in the plug flow reactor the reactants are mixed immediately with each other, and in the perfectly mixed CSTR the entering reactants are mixed immediately with the reactor contents. In addition we present two other reactor models that approach reactor types that are frequently used in practice ... 

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