Optimizing Cascades of Endo-Reversible Heat Engines

Abstract

The endo-reversible (or finite time) model of thermal power plant is extended to a cascade. A first optimization gives the condition for maximizing the power output from a given design and is a generalization of previous results for one unit. A second optimization similarly generalizes results for one unit over the allotment of the irreversible heat transfer components either side of the Carnot reversible engine. This optimization itself is generalized from conditions of fixed total heat transfer capability to allow for varying cost and varying effectiveness in the most general case. Lagrange multipliers are given that show the change in optimized output for a change in constraints of area, cost or effectiveness. Explicit results are given for the temperature distribution in the two-component cascade which indicates the optimum distribution of power: two parts from the topping unit, one part from the bottoming unit. It is shown that in a given case one should seek therefore to operate with as few units as possible and in the ratio 2:1 for two units. The two-unit results are readily extended to provide explicit temperature distributions for any finite cascade, second optimized or not.