The dynamic unsteady-state reactor process in a cocurrent tubular reactor with single-run reagents (continuous phase) and with multi-run catalyst (dispersed phase) has been investigated. For the temperature-dependent catalyst deactivation the reactor process has been considered in which after optimal number B of catalyst residences in the reactor the whole amount of catalyst leaves the system; then the fresh catalyst is directed to the reactor and the next B-runs cycle of the optimal process starts. The optimization problem has been formulated in which a maximum of an average (for one cycle) process profit flux is achieved by a best choice of number B of catalyst residences in the reactor and best choice of temperature profiles along tubular reactor for each catalyst run, b = 1....B, respectively. The set of parallel-consecutive reactions, A + B-->R and R + B-->S, with desired product R has been taken into account. The algorithms of maximum principle have been used for optimization. Optimization procedure allows for finding an optimal number of catalyst runs B for which an average process profit flux reaches a maximum for a concrete value of fresh catalyst price L Optimal value of B increases with L A shape of optimal temperature profile constitutes the effect of compromise between the overall production rate of desired reagent R (production rate in the first reaction minus disappearance rate in the second one) and the necessary savings of catalyst. The optimal solutions show that the most important influence on the optimal temperature profile is due to the need of catalyst saving; low temperatures save catalyst during its initial runs. Moreover, the optimal temperature profile is independent of fresh catalyst price,. If optimal temperature profile for B-run reaction process is known, then, to obtain optimal profile for (B-r)-run process, it is enough to cut the initial part of optimal profile corresponding to the first r runs. (C) 2004 Elsevier B.V. All rights reserved.