Energy balances unsteady-state operation

The input and output terms of equation 1.5-1 may each have more than one contribution. The input of a species may be by convective (bulk) flow, by diffusion of some kind across the entry point(s), and by formation by chemical reaction(s) within the control volume. The output of a species may include consumption by reaction(s) within the control volume. There are also corresponding terms in the energy balance (e.g., generation or consumption of enthalpy by reaction), and in addition there is heat transfer (2), which does not involve material flow. The accumulation term on the right side of equation 1.5-1 is the net result of the inputs and outputs for steady-state operation, it is zero, and for unsteady-state operation, it is nonzero. 

With respect to open systems, we make the additional distinction between steady-state, and unsteady-state operation. At steady state the accumulation terms, dU /dt, dS /dt and dM /dt, are zero. This reduces the energy balance from a differential equation into an algebraic equation. For this reason, steady-state processes are much simpler to calculate compared to unsteady-state processes. 

The energy balance (3.301) is applicable for catalysis, adsorption, and ion exchange. More specifically, in catalysis, where the steady-state condition exists, frequently the accumulation term is zero. In contrast, adsorption and ion exchange operate under unsteady-state condition. The analysis of the energy balance equation for catalytic fixed beds is presented in detail in Section 5.3.4. 

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. 

Figure 6.1 illustrates the fact that for various ranges of kinetic and reactor parameters it is possible for the mass and energy conservation relations for a CSTR to be in stable balance at more than one condition. This may imply that there are other balance conditions that are unstable the point needs to be examined. Which of the stable balances is attained in actual operation may be dependent on the details of startup procedure, for example, which are not subject to the control of the designer. Thus, it is important to investigate reactor stability using unsteady-state rather than steady-state models. 

Since the reactor does not operate isothermally, these five coupled mass balances must be solved in conjunction with the unsteady-state thermal energy balance for an adiabatic reactor, where (d Q/dt)jnpyit = 0. If pressure effects are negligible,... 

Batch operation is essentially unsteady-state, so the energy balance is drawn up with respect to conditions obtaining at some instant t. If the reactor is run isothermally the cooling requirements at t will be given by ... 

As we saw in Chapter 2, batch operation is essentially unsteady-state. The energy balance is drawn up at some instant t the average cooling rate over the entire batch Q is given by integration with respect to time ... 

Closure. After completing this chapter, the reader should be able to af ly the unsteady-state energy balance to CSTRs, semibatch and batch reactors. The reader should be able to discuss reactor safety using the ONCB and the T2 Laboratories case studies of explosions to help prevent future accidents. Included in the reader s discussion should be how to start up a reactor so as not to exceed the practical stability limit. After studying these examples, the reader should be able to describe how to operate reactors in a safe manner for both single and multiple reactions. 

Let s analyze the response to a small, positive change in temperature, ST, as we did with point U in Figure 8-7. Both G(T) and R(T) increase. However, the increase in G(T), SG, now is smaller than the increase in R(J), SR. Therefore, R(J) is greater than G(7), and the unsteady-state energy balance requires that the temperature decrease toward the temperature of the original operating point, SI. 

The question of whether the reactor operates at SI or S2 depends on how it is started up. In order to determine the effect of startup conditions, the unsteady-state energy and material balances must be solved simultaneously. This task is beyond the scope of this chapter. 

It is important to remember that the examples in this section are restricted to cases when steady-state can be applied. If we are interested in start-up or shutdown of these processes, or the case where there are fluctuations in feed or operating conditions, we must use the unsteady form of the energy balance. 

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