Energy Balance on Batch Reactors

A batch reactor is usually well mixed, so that we may neglect spatial variations in the temperature and species concentration. The energy balance on batch reactors is found by setting the inlet flows = 0 in Equation (13-9), yielding 

Equation (13-10) is the preferred form of the energy balance when the number of moles. A//, is used in the mole balance, rather than the conversion, X. The number of motes of species i at any X is 

Consequently, in terms of conversion, the energy balance becomes 

Equation (13-11) must be coupled with the mole balance 

However, we note that the units of the product of mass flow rate and mass heat capacities would still be (he same as the product of molar flow and molar heat capacities, (e.g., cal/s K). respectively. 

---

The Unsteady-State Energy Balance 591 Energy Balance on Batch Reactors 594... 

Provided that gaseous reactants and products behave ideally and the specific volumes of liquid and solid reactants and products are negligible compared with the specific volumes of the gases, the internal energy of reaction may be calculated from Equation 9.1-5. (lliis quantity is required for energy balances on constant-volume batch reactors.)... 

The key assumption on which the design analysis of a batch reactor is based is that the degree of agitation is sufficient to ensure that the composition and temperature of the contents are uniform throughout the reaction vessel. Under these conditions one may write the material and energy balances on the entire contents of the reaction vessel. 

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. 

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 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. 

The condition for the practical implementation of such a feed control is the availability of a computer controlled feed system and of an on-line measurement of the accumulation. The later condition can be achieved either by an on-line measurement of the reactant concentration, using analytical methods or indirectly, by using a heat balance of the reactor. The amount of reactant fed to the reactor corresponds to a certain energy of reaction and can be compared to the heat removed from the reaction mass by the heat exchange system. For such a measurement, the required data are the mass flow rate of the cooling medium, its inlet temperature, and its outlet temperature. The feed profile can also be simplified into three constant feed rates, which approximate the ideal profile. This kind of semi-batch process shortens the time-cycle of the process and maintains safe conditions during the whole process time. This procedure was shown to work with different reaction schemes [16, 19, 20], as long as the fed compound B does not enter parallel reactions. 

To describe this process, material and energy balances are required. Recall that the mass balance on a batch reactor can be written as [refer to Equation (3.2.1)] ... 

Up to now we have focused on the steady-state operation of nonisothermal reactors. In this section the unsteady-state energy balance wtU be developed and then applied to CSTRs, plug-flow reactors, and well-mixed batch and semibateh reactors. 

This method of writing the mass balance, like Eq. (3-10) for batch operation of a tank reactor, separates the extensive variables F and F and relates them to an integral dependent on the intensive conditions in the reaction mixture. It is worthwhile to note the similarity between Eq. (3-13) and the more familiar design equation for heat-transfer equipment based on an energy balance. This may be written... 

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. 

Closure. After completing this chapter, the reader should be able to appi the unsteady-state energy balance to CSTRs, semibatch and batch reactor The reader should be able to discuss reactor safety using two examples on a case study of an explosion and the other the use of the ARSST to hel prevent explosions. Included in the reader s discussion should be how t start up a reactor so as not to exceed the practical stability liniit. After reac ing these examples, the reader should be able to describe how to operat reactors in a safe maimer for both single and multiple reactions. 

---