Solved: A Stirling-Cycle Engine. The Stirling cycle is

Chapter 20, Problem 20.52

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A Stirling-Cycle Engine. The Stirling cycle is similar to the Otto cycle, except that the compression and expansion of the gas are done at constant temperature, not adiabatically as in the Otto cycle. The Stirling cycle is used in external combustion engines (in fact, burning fuel is not necessary; any way of producing a temperature difference will dosolar, geothermal, ocean temperature gradient, etc.), which means that the gas inside the cylinder is not used in the combustion process. Heat is supplied by burning fuel steadily outside the cylinder, instead of explosively inside the cylinder as in the Otto cycle. For this reason Stirling-cycle engines are quieter than Otto-cycle engines, since there are no intake and exhaust valves (a major source of engine noise). While small Stirling engines are used for a variety of purposes, Stirling engines for automobiles have not been successful because they are larger, heavier, and more expensive than conventional automobile engines. In the cycle, the working fluid goes through the following sequence of steps (Fig. P20.52): (i) Compressed isothermally at temperature from the initial state a to state b, with a compression ratio r. (ii) Heated at constant volume to state c at temperature (iii) Expanded isothermally at to state d. (iv) Cooled at constant volume back to the initial state a. Assume that the working fluid is n moles of an ideal gas (for which is independent of temperature). (a) Calculate Q, W, and for each of the processes and (b) In the Stirling cycle, the heat transfers in the processes and do not involve external heat sources but rather use regeneration: The same substance that transfers heat to the gas inside the cylinder in the process also absorbs heat back from the gas in the process Hence the heat transfers and do not play a role in determining the efficiency of the engine. Explain this last statement by comparing the expressions for and calculated in part (a). (c) Calculate the efficiency of a Stirlingcycle engine in terms of the temperatures and How does this compare to the efficiency of a Carnot-cycle engine operating between these same two temperatures? (Historically, the Stirling cycle was devised before the Carnot cycle.) Does this result violate the second law of thermodynamics? Explain. Unfortunately, actual Stirling-cycle engines cannot achieve this efficiency due to problems with the heat-transfer processes and pressure losses in the engine.