An adiabatic device is to be designed that will take in two steady flows of gas at given states 1 and 2, with T₂ > T₁, and discharge the gas at a given pressure P₃ and undetermined temperature T₃. The pressure of all three states will be the same (P₁ = P₂ = P₃). The purpose of the device is to generate useful (shaft) power.

Kinetic and potential energy of the flow at states 1?3 may be neglected, one-dimensional flow may be assumed at states 1?3, steady state may be assumed, and the gas may be assumed to be an ideal gas with constant specific heats, for which the entropy is given by s_2?? s₁ = c_P ln(T₂/T₁) ? R ln(P₂/P₁).Using an energy balance, show that the shaft power output W? for given mass flow rates, states 1 and 2, and P₃ decreases linearly with increasing T₃, and hence that the maximum power output will be obtained for a design that gives the lowest possible T₃. Then, using an entropy balance, develop the expression for the lowest possible T₃ in terms of T₁, T₂, and the mass flow rate ratio f ? M?₁/M?₃. List any idealizations that you make other than those given above.

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M₁ M₂ W device. 3