Batch reactor, adiabatic operation design equations

Although semi-analytical solutions are available in some cases [5], these are cumbersome and it is more usual to employ a numerical method. A simple example is presented below which illustrates the solution of the design equation for a batch reactor operated isothermally the adiabatic operation of the same system is then examined. 

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. 

It may be noted that temperature T increases linearly with conversion for exothermic reaction AHj is negative) and decreases with conversion for the endothermic reaction AHp is positive). For complete conversion (x = 1), net change (increase or decrease) in temperature is the maximum, which is equal to AT d. Thus, AT d is defined as the maximum change in temperature attained in an adiabatic reactor. The batch time Gg for adiabatic operation is calculated using the design equation... 

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