Energy balances for a batch reactor

Equations 14.2-3 and 14.2-4 bear a striking resemblance to the mass and energy balances for a batch reactor, Eqs. 14.1-13 and 14. There is, in fact, good physical reason why these equations should look very much alike. Our model of a plug-flow reactor, which neglects diffusion and does not allow for velocity gradients, assumes that each element of fluid travels through the reactor with no interaction with the fluid elements before or after it Therefore, if we could follow a small fluid element in a tubular reactor, we would find that it had precisely the same behavior in time as is found in a batch reactor. This similarity in the physical situation is mirrored in the similarity of the descriptive equations. 

Energy Balance Energy balances are needed solely because the rate of the chemical reaction may be a strong function of temperature (Arrhenius equation. Chap. 2). The purpose of the energy balance is to describe the temperature at each point in the reactor (or at each time for a batch reactor), so that the proper rate may be assigned to that point. 

This is the energy balance equation for a batch reactor, in which T is the temperature of reaction, Tj is the temperature of the cooling/heating coil, and To is the initial temperature of the system. All other parameters have already been defined. 

The operating point of an adiabatic CSTR at steady state must lie somewhere on the line that represents the adiabatic energy balance. For an adiabatic PFR at steady state, or for an adiabatic batch reactor, the energy balance line describes the path of the reaction, including the exit condition for a PFR and the final condition for a batch reactor. For any type of adiabatic reactor, if a given point (x, T) does not lie on the line, the energy balance is not satisfied. 

There are a variety of limiting forms of equation 8.0.3 that are appropriate for use with different types of reactors and different modes of operation. For stirred tanks the reactor contents are uniform in temperature and composition throughout, and it is possible to write the energy balance over the entire reactor. In the case of a batch reactor, only the first two terms need be retained. For continuous flow systems operating at steady state, the accumulation term disappears. For adiabatic operation in the absence of shaft work effects the energy transfer term is omitted. For the case of semibatch operation it may be necessary to retain all four terms. For tubular flow reactors neither the composition nor the temperature need be independent of position, and the energy balance must be written on a differential element of reactor volume. The resultant differential equation must then be solved in conjunction with the differential equation describing the material balance on the differential element. 

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. 

We can also obtain these expressions from the energy-balance equation for the steady-state PFTR by simply transforming dzju dt with A,/ V replacing Pw/At. The solutions of these equations for the batch reactor are mathematically identical to those in the PFTR, although the physical interpretations are quite different. 

The nonlinear dynamic model of this fed-batch reactor consists of a total mass balance, component balances for three components, an energy balance for the liquid in the reactor, and an energy balance for the cooling water in the jacket ... 

Numerous reactions are performed by feeding the reactants continuously to cylindrical tubes, either empty or packed with catalyst, with a length which is 10 to 1000 times larger than the diameter. The mixture of unconverted reactants and reaction products is continuously withdrawn at the reactor exit. Hence, constant concentration profiles of reactants and products, as well as a temperature profile are established between the inlet and the outlet of the tubular reactor, see Fig. 7.1. This requires, in contrast to the batch reactor, the application of the law of conservation of mass over an infinitesimal volume element, dV, of the reactor. In contrast to a batch reactor the existence of a temperature profile does not allow us to consider the mass balances for the reacting components and the energy balance separately. Such a separation can only be performed for isothermal tubular reactors. 

The formal similarity allows us to carry over the equations for mass and energy balances in the tubular reactor, Eqs. (3.4.11)-(3.4.14). The momentum equation has no meaning. Care must be taken however to distinguish between a batch reactor working at constant volume and one that works at constant pressure. The latter has the Eqs. (3.4.12) or (3.4.14) which were derived from an enthalpy balance. In the former case the heat added would be equated to the internal energy change. Thus in this case c should replace Cp and the internal energy of reaction replace the heat of reaction. These... 

In contrast to a batch reactor, the existence of a temperature profile does not allow us to consider the mass balances for the reacting components and the energy balance separately. Such a separation can only be performed for isothermal tubular reactors. 

Below, we apply the energy balances for macroscopic systems. First, we derive the energy balance equation for closed systems (batch reactors) and then for open systems (flow reactors). Microscopic energy balances, used to describe point-to-point temperature variations inside a chemical reactor, are outside the scope of this book. 

For convenience. Tables A.3a and A.3b in Appendix A provide the design equation and the auxiliary relations for ideal batch reactors. Table A.4 provides the energy balance equation. 

This is the form of the energy balance that is usually used for preliminary calculations. Equation 5.23 does not require that u be constant, but if it is constant, we can set dz = udt and 2/R = Aext/Ac to make Equation 5.23 similar to Equation 5.18. A PER with constant velocity and constant physical properties behaves like a batch reactor with constant volume and constant physical properties. The curves in Figure 5.2 could apply to a PFR as well as to the batch reactor analyzed in Example 5.5. 

The coefficient of dNi/dt in the summation of (6-4) is defined as the partial molar internal energy of species i, because differentiation with respect to mole numbers of component i is performed at constant T, p and mole numbers of all other species in the mixture. The unsteady-state mass balance for species i in a batch reactor,... 

If a batch reactor is completely insulated from the surroundings and there is only one chemical reaction, then the mass and thermal energy balances can be combined analytically to yield the maximum temperature rise for exothermic reactions. The same procedure provides an estimate of the maximum temperature drop if the reaction is endothermic. If pressure effects are negligible, in accord with the previous analyses, coupled heat and mass transfer yield (see equation 6-15) ... 

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). 

Laboratory-sized batch reactors are seldom heat transfer rate limited. Commercial-sized batch reactors are seldom not heat transfer rate limited. This conundrum arises from the energy balance for batch reactors, which, for a perfectly mixed reaction fluid, is... 

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. 

For a perfectly mixed batch reactor, the energy balance is... 

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

For a safe operation, the runaway boundaries of the phenol-formaldehyde reaction must be determined. This is done here with reference to an isoperibolic batch reactor (while the temperature-controlled case is addressed in Sect. 5.8). As shown in Sect. 2.4, the complex kinetics of this system is described by 89 reactions involving 13 different chemical species. The model of the system consists of the already introduced mass (2.27) and energy (2.30) balances in the reactor. Given the system complexity, dimensionless variables are not introduced. 

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)... 

---