We treat the irreversible extension of the classical problem of maximum mechanical work that may be obtained from a system composed of: a resource fluid at flow, a set of sequentially arranged engines, and an infinite bath. In the engine mode the fluid's temperature T decreases along the path, thus tending to the bath temperature T-e, and the system delivers work. In a related classical problem the process rates vanish due to the reversibility; here, however, finite rates and consistent losses of the work potential are admitted. The variational calculus leads to a finite-rate generalization of the maximum-work potential called the finite-rate exergy. This finite-rate exergy is a function of the usual thermal coordinates and the overall number of transfer units T or a rate index h, which is, in fact the Hamiltonian of optimal, active (energy-generating) relaxation process to equilibrium. The resulting bounds on the work delivered or supplied are stronger than the classical reversible bounds. (c) 2005 Elsevier Ltd. All rights reserved.