The Batch Reactor

We will write all reactor mass and heat balances as 

The number of moles of species j in a batch reactor is simply the reactor volume V times the concentration 

If the reactor is at constant volume, then we can divide each term by F to yield 

Let us immediately apply this equation to the first-order irreversible reaction 

The mass-balance equation on species A in a constant-density batch reactor is 

Here V is the volume of the reactor, p is the density, cp is the mean specific heat of the reactor contents (kj/kg K) and rQ is the rate of generation of heat by reaction (kj/s m3). 

A batch reactor is charged with reactant, the required conversion takes place, and the reactor is emptied. One consequence is that the concentration of the reactant and products in the reactor are a function of time. There are two types of electrochemical batch reactors those without electrolyte recirculation (Fig. 4.1a), and those with recirculation (Fig. 4.1b). The former are suitable as laboratory devices for obtaining performance data as a function of electrode potential, reactant concentration, and hence conversion as described in Section 3.2.2.2. The reactor is often run in a potentiostatic 

FIGURE 4.1. Schematic representation of batch reactors, (a) Without recirculation (PV= working electrode, C = counter electrode, S = separator) (b) with recirculation. (Vg = reactor volume, V, = reservoir volume). 

Chemical kinetics plays a major role in modeling the ideal chemical batch reactor hence, a basic introduction to chemical kinetics is given in the chapter. Simplified kinetic models are often adopted to obtain analytical solutions for the time evolution of concentrations of reactants and products, while more complex kinetics can be considered if numerical solutions are allowed for. 

Since complex systems may involve up to several hundreds (and even thousands) of chemical species and reactions, simple reaction pathways cannot always be recognized. In these cases, the true reaction mechanism remains an ideal matter of principle, which can be only approximated by reduced reaction networks. Also in simpler cases, reduced networks are more suitable for most practical purposes. Moreover, the relevant kinetic parameters are mostly unknown or, at best, very uncertain, so that they must be evaluated by exploiting adequate experimental campaigns. With the aim of presenting an example of the problems related to chemical kinetics, a case study is introduced and discussed in detail in the next subsection. 

The mathematical model of the batch reactor consists of the equations of conservation for mass and energy. An independent mass balance can be written for each chemical component of the reacting mixture, whereas, when the potential energy stored in chemical bonds is transformed into sensible heat, very large thermal effects may be produced. 

In Chaps. 2 to 6, a case study is developed in order to apply and test the methods developed along the whole book. To this purpose, the reaction between phenol and formaldehyde for the production of a prepolymer of phenolic resins has been chosen for several reasons. In fact, this reactive system is widely used in different forms for the production of different polymers moreover, it is characterized by a noticeable production of heat and by a complex kinetic behavior. Such features represent strong challenges for controlling and monitoring tasks. 

Two different classes of chemical reactions are singled out, namely the reactions of addition of formaldehyde to the aromatic ring, which introduce a methylol group as a substituent, and the reactions of condensation, which produce components with higher molecular weight. In the presence of an alkaline catalyst, the reactions of addition are strongly oriented in the -orto and -para positions of the aromatic ring, whereas the reactions of condensation occur both between two substituted positions 

It is assumed that all the tank-type reactors, covered in this and the immediately following sections, are at all times perfectly mixed, such that concentration and temperature conditions are uniform throughout the tanks contents. Fig. 3.8 shows a batch reactor with a cooling jacket. Since there are no flows into the reactor or from the reactor, the total material balance tells us that the total mass, within the reactor, remains constant. 

Rate of accumulation (Rate of production of A of reactant A J by chemical reaction 

This method of operation is ideal for small-scale production where the reaction time is relatively long and it is often used where a number of different products have to be manufactured using the same equipment, as for example, in the fine chemicals, pharmaceuticals and dye-stuffs industries. Batch operation is often preferred where undesirable side-reactions 

The capital cost of a batch reactor and its auxiliary equipment is generally low, but operating costs tend to be high because of the need to have a plant operator in attendance, especially when the reactor is being emptied, cleaned and recharged with a fresh batch of material. The product from a batch reactor is also subject to unpredictable variations in quality from one batch to another. 

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. 

The term semibatch reactor has sometimes been used for the continual operation of a reactor in the transient state. Such would be the case if a stirred reactor were started up and shut down on a repeated cycle, with a significant proportion of the production contributed from periods during 

A batch reactor generally consists of a closed vessel provided with a means of stirring and with temperature control. It may be held at constant pressure or it can be entirely enclosed at a constant volume. If the R simultaneous 

If the reaction takes place at constant pressure or if the volume is changed in some prescribed manner it is often preferable to work in terms of the total number of moles present or the gross extent of reaction, X, It will be recalled that the definition of the intensive rate of reaction 2 = 0 is 

In semibatch operation it may also be preferable to work with the total moles present instead of concentration, for the volume will be changing. In this case, we have 

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The hquid-phase chlorination of benzene is an ideal example of a set of sequential reactions with varying rates from the single-chlorinated molecule to the completely chlorinated molecule containing six chlorines. Classical papers have modeled the chlorination of benzene through the dichlorobenzenes (14,15). A reactor system may be simulated with the relative rate equations and flow equation. The batch reactor gives the minimum ratio of... 

Unsteady material and energy balances are formulated with the conservation law, Eq. (7-68). The sink term of a material balance is and the accumulation term is the time derivative of the content of reactant in the vessel, or 3(V C )/3t, where both and depend on the time. An unsteady condition in the sense used in this section always has an accumulation term. This sense of unsteadiness excludes the batch reactor where conditions do change with time but are taken account of in the sink term. Startup and shutdown periods of batch reactors, however, are classified as unsteady their equations are developed in the Batch Reactors subsection. For a semibatch operation in which some of the reactants are preloaded and the others are fed in gradually, equations are developed in Example 11, following. 

A semi-batch reactor has the same disadvantages as the batch reactor. However, it has the advantages of good temperature control and the capability of minimizing unwanted side reactions by maintaining a low concentration of one of the reactants. Semi-batch reactors are also of value when parallel reactions of different orders occur, where it may be more profitable to use semi-batch rather than batch operations. In many applications semi-batch reactors involve a substantial increase in the volume of reaction mixture during a processing cycle (i.e., emulsion polymerization). 

A continuous flow stirred tank reactor (CFSTR) differs from the batch reactor in that the feed mixture continuously enters and the outlet mixture is continuously withdrawn. There is intense mixing in the reactor to destroy any concentration and temperature differences. Heat transfer must be extremely efficient to keep the temperature of the reaction mixture equal to the temperature of the heat transfer medium. The CFSTR can either be used alone or as part of a series of battery CFSTRs as shown in Figure 4-5. If several vessels are used in series, the net effect is partial backmixing. 

Time must be included for emptying, cleaning, and filling the batch reactor (=30 min). 

The batch reactor initially contains 227 kg of acetyiated castor and die initial temperature is 613 K. Complete hydrolysis yields 0.156 kg acetic acid per kg of ester. Eor diis reaction, die specific reaction rate constant k is... 

Example 2.2 Derive the batch reactor design equations for the reaction set in Example 2.1. Assume a liquid-phase system with constant density. 

Chapter 1 treated the simplest type of piston flow reactor, one with constant density and constant reactor cross section. The reactor design equations for this type of piston flow reactor are directly analogous to the design equations for a constant-density batch reactor. What happens in time in the batch reactor happens in space in the piston flow reactor, and the transformation t = z/u converts one design equation to the other. For component A,... 

Chapter 2 treated multiple and complex reactions in an ideal batch reactor. The reactor was ideal in the sense that mixing was assumed to be instantaneous and complete throughout the vessel. Real batch reactors will approximate ideal behavior when the characteristic time for mixing is short compared with the reaction half-life. Industrial batch reactors have inlet and outlet ports and an agitation system. The same hardware can be converted to continuous operation. To do this, just feed and discharge continuously. If the reactor is well mixed in the batch mode, it is likely to remain so in the continuous mode, as least for the same reaction. The assumption of instantaneous and perfect mixing remains a reasonable approximation, but the batch reactor has become a continuous-flow stirred tank. 

Example 12.8 The batch reactor in Example 12.7 has been converted to a CSTR. Determine its steady-state performance at a mean residence time of 4 h. Ignore product inhibition. 

Figure 14.2 shows the numerical solution. Except for a continuous input of ten rabbits and one lynx per unit time, the parameter values and initial conditions are the same as used for Figure 2.6. The batch reactor has been converted to a CSTR. The oscillations in the CSTR are smaller and have a higher frequency than those in the batch reactor, but a steady state is not achieved.

Aqueous Phase Hass Balances. The usual material balances for the active species in the aqueous solution are considered. With respect to the case of homopolymerization (4) the conplexity of the resulting equations is increased because of the cross propagation and termination terms. For the batch reactor considered in this wortt, the following equations arise ... 

Table 12.10 summarizes the geometrical parameters and the heat exchanged per unit of volume of the batch reactors in the same reaction conditions as the HEX... 

The question of how to deal with the kinetics of such coupled processes depends on the sort of reactor employed. We will distinguish two cases, namely the flow reactor and the batch reactor (Fig. 2.4). 

The batch reactor is generally used in the production of fine chemicals. At the start of the process the reactor is filled with reactants, which gradually convert into products. As a consequence, the rate of reaction and the concentrations of all participants in the reaction vary with time. We will first discuss the kinetics of coupled reactions in the steady state regime. 

In order to illustrate how the mode of operation can positively modify selectivity for a large reactor of poor heat-transfer characteristics, simulations of the reactions specified in Example 5.3.1.4 carried out in a semibatch reactor were performed. The reaction data and process conditions are essentially the same as those for the batch reactor, except that the initial concentration of A was decreased to cao = 0.46 mol litre, and the remaining amount of A is dosed (1) either for the whole reaction time of 1.5 h with a rate of 0.1 mol m s", or (2) starting after 0.5 h with a rate of 0.15 mol m " s". It is assumed that the volume of the reaction mixture and its physical properties do not change during dosing. The results of these simulations are shown in Fig. 5.3-15. The results of calculation for reactors of both types are summarized in Table 5.3-3. 

Figure 3.10. The batch reactor with heat transfer.

For the batch reactor with these complex kinetics, the model is as follows = -(ki-bk2)CA = "1 CA-k3CB... 

Compare the batch reactor performance at constant Cp with that for variable Cp. Do this by setting both Cp values constant at a temperature Tj... 

Figure 5.63. Temperature profiles for the batch reactor, Tl, and for the jacket TC, which lags behind.

Figure 5.66. Space-time yields, SPTYB, profiles in the batch reactor as a result of varying TIMEON (A - 1000, B - 8, C - 9, D -15).

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