An optimal catalyst activation policy for poisoning problems

Abstract

Pontryagins minimum principle is used to calculate the optimum distribution of active material throughout a single pellet that is uniformly experiencing activity decay. The definition of optimality is based upon balancing catalyst costs against the net return from reactant conversion. The optimal policy is full activation to a fractional depth with an inert core, the depth depending upon the Thiele parameter, poisoning time constant, operating time, and an economic parameter. An isothermal, first order, irreversible reaction is considered. Auxiliary calculations of the effectiveness factor are given for the reaction rate in the nonisothermal pellet with the catalyst distribution found to be optimal in the isothermal case. The results of the pellet calculations are utilized in numerically calculating a uniform activation policy that is optimal for a homogeneously poisoned bed. The definition of optimality is on the same basis as that for the single pellet and the results depend upon the same physical parameters in addition to the time constant for convection in the tube relative to that for diffusion in the pellets. Four regimes of behavior arise depending upon catalyst costs: (i) the reactor cannot be operated economically, (ii) only partial activation policies are economical, a single one being optimal, (iii) both partial and full activation schemes are economical, with one of the former being optimal, and (iv) full activation is optimal but the reactor can be operated economically with partially activated pellets. The results for both the single pellet and the packed tube are viewed as the homogeneous poisoning limit of many practical problems and are believed to reflect the major characteristics of more complicated poisoning mechanisms.