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 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 do-solar, 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):
\begin{equation}\begin{array}{l}{\text { (i) Compressed isothermally at
temperature } T_{1} \text { from the ir }} \\ {\text { state } a \text { to
state } b, \text { with a compression ratio } r .} \\ {\text { (ii) Heated at
constant volume to state } c \text { at temperature } T_{2}} \\ {\text {
(iii) Expanded isothermally at } T_{2} \text { to state } d .} \\ {\text {
(iv) Cooled at constant volume back to the initial state } a \text { .
}}\end{array}\end{equation}
Assume that the working fluid is \(n\) moles of an ideal gas (for which \(C_{V}\)
is independent of temperature). (a) Calculate \(Q, W,\) and \(\Delta U\) for each
of the processes \(a \rightarrow b, b \rightarrow c, c \rightarrow d,\) and \(d
\rightarrow a\) . (b) In the Stirling cycle, the heat transfers in the
processes \(b \rightarrow c\) and \(d \rightarrow a\) 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 \(b \rightarrow c\) also absorbs heat
back from the gas in the process \(d \rightarrow a .\) Hence the heat transfers
\(Q_{b \rightarrow c}\) and \(Q_{d \rightarrow a}\) do not play a role in
determining the efficiency of the engine. Explain this last statement by
comparing the expressions for \(Q_{b \rightarrow c}\) and \(Q_{d \rightarrow a}\)
calculated in part (a). (c) Calculate the efficiency of a Stirlingcycle engine
in terms of the temperatures \(T_{1}\) and \(T_{2} .\) 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.
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