Analytical solution of the Euler–Lagrange equation in an optimization problem for a batch reactor with deactivating catalyst

A reaction carried out in a batch reactor in the presence of deactivating catalyst has been analysed. The considered model of the process encompasses all the cases analysed in the references. After formulating an optimization criterion and application of the necessary Euler–Lagrange condition to the resulting expression, an analytical method of solution of the second-order differential equation has been suggested. It enabled one to obtain a dependence between the catalyst activity and the substrate concentration at the optimal conditions. This then made possible to get a general relationship describing the optimal time of the chemical reaction with deactivation and the optimal trajectories of the substrate concentrations, catalyst activity and temperature.