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Many types of equipment have two costs: the cost of the work potential dissipated by the equipment in operation, and its initial capital cost. The correct design minimizes the total cost. A mathematical technique has recently been developed for obtaining algebraic expressions for the minimum cost, as well as for the optimum design parameters in terms of the physical parameters of the system and of various unit costs, such as electric power and fuel costs. The present paper illustrates this mathematical optimization technique through its application to the design of a counter-flow heat exchanger. Algebraic expressions are obtained for the following design parameters: flow velocities and surface area per unit rate of heat exchanged. This mathematical technique is equally applicable to design problems where properties other than cost are to be minimized, such as weight.