The furnace of a particular power plant can be considered

Chapter 15, Problem 15112

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The furnace of a particular power plant can be considered to consist of two chambers: an adiabatic combustion chamber where the fuel is burned completely and adiabatically, and a heat exchanger where heat is transferred to a Carnot heat engine isothermally. The combustion gases in the heat exchanger are well-mixed so that the heat exchanger is at a uniform temperature at all times that is equal to the temperature of the exiting product gases, Tp. The work output of the Carnot heat engine can be expressed as W 5 QhC 5 Qa1 2 T0 Tp b where Q is the magnitude of the heat transfer to the heat engine and T0 is the temperature of the environment. The work output of the Carnot engine will be zero either when Tp 5 Taf (which means the product gases will enter and exit the heat exchanger at the adiabatic flame temperature Taf, and thus Q 5 0) or when Tp 5 T0 (which means the temperature of the product gases in the heat exchanger will be T0, and thus hC 5 0), and will reach a maximum somewhere in between. Treating the combustion products as ideal gases with constant specific heats and assuming no change in their composition in the heat exchanger, show that the work output of the Carnot heat engine will be maximum when Tp 5 "TafT0 Also, show that the maximum work output of the Carnot engine in this case becomes Wmax 5 CTafa1 2 T0 Taf b 2 where C is a constant whose value depends on the composition of the product gases and their specific heats.