The ingenious Stirling engine is a true heat engine that absorbs heat from an external source. The working substance can be air or any other gas. The engine consists of two cylinders with pistons, one in thermal contact with each reservoir (see Figure 4.7). The pistons are connected to a crankshaft in a complicated way that we'll ignore and let the engineers worry about. Between the two cylinders is a passageway where the gas flows past a regenerator: a temporary heat reservoir, typically made of wire mesh, whose temperature varies

gradually from the hot side to the cold side. The heat capacity of the regenerator is very large, so its temperature is affected very little by the gas flowing past. The four steps of the engine's (idealized) cycle are as follows:
i. Power stroke. While in the hot cylinder at temperature T_h, the gas absorbs heat and expands isothermally, pushing the hot piston outward. The piston in the cold cylinder remains at rest, all the way inward as shown in the figure.
ii. Transfer to the cold cylinder. The hot piston moves in while the cold piston moves out, transferring the gas to the cold cylinder at constant volume. While on its way, the gas flows past the regenerator, giving up heat and cooling to T_c.
iii. Compression stroke. The cold piston moves in, isothermally compressing the gas back to its original volume as the gas gives up heat to the cold reservoir. The hot piston remains at rest, all the way in.
iv. Transfer to hot cylinder. The cold piston moves the rest of the way in while the hot piston moves out, transferring the gas back to the hot cylinder at constant volume. While on its way, the gas flows past the regenerator, absorbing heat until it is again at T_h.

(a) Draw a PV diagram for this idealized Stirling cycle.
(b) Forget about the regenerator for the moment. Then, during step 2, the gas will give up heat to the cold reservoir instead of to the regenerator; during step 4 , the gas will absorb heat from the hot reservoir. Calculate the efficiency of the engine in this case, assuming that the gas is ideal. Express your answer in terms of the temperature ratio T_c / T_h and the compression ratio (the ratio of the maximum and minimum volumes). Show that the efficiency is less than that of a Carnot engine operating between the same temperatures. Work out a numerical example.
(c) Now put the regenerator back. Argue that, if it works perfectly, the efficiency of a Stirling engine is the same as that of a Carnot engine.
(d) Discuss, in some detail, the various advantages and disadvantages of a Stirling engine, compared to other engines.

i. Power stroke. While in the hot cylinder at temperature T_h, the gas absorbs heat and expands isothermally, pushing the hot piston outward. The piston in the cold cylinder remains at rest, all the way inward as shown in the figure.
ii. Transfer to the cold cylinder. The hot piston moves in while the cold piston moves out, transferring the gas to the cold cylinder at constant volume. While on its way, the gas flows past the regenerator, giving up heat and cooling to T_c.
iii. Compression stroke. The cold piston moves in, isothermally compressing the gas back to its original volume as the gas gives up heat to the cold reservoir. The hot piston remains at rest, all the way in.
iv. Transfer to hot cylinder. The cold piston moves the rest of the way in while the hot piston moves out, transferring the gas back to the hot cylinder at constant volume. While on its way, the gas flows past the regenerator, absorbing heat until it is again at T_h.

Draw PV diagram.

In a Stirling cycle, first extracting heat from a hot reservoir, then doing work and thereafter transferring waste heat to cold reservoir.
Therefore, the engine works through four stages.

i. Power stroke. While in the hot cylinder at temperature T_h, the gas absorbs heat and expands isothermally, pushing the hot piston outward. The piston in the cold cylinder remains at rest, all the way inward as shown in the figure.
ii. Transfer to the cold cylinder. The hot piston moves in while the cold piston moves out, transferring the gas to the cold cylinder at constant volume. While on its way, the gas flows past the regenerator, giving up heat and cooling to T_c.
iii. Compression stroke. The cold piston moves in, isothermally compressing the gas back to its original volume as the gas gives up heat to the cold reservoir. The hot piston remains at rest, all the way in.
iv. Transfer to hot cylinder. The cold piston moves the rest of the way in while the hot piston moves out, transferring the gas back to the hot cylinder at constant volume. While on its way, the gas flows past the regenerator, absorbing heat until it is again at T_h.

Prove that the efficiency is less than that of Carnot cycle.

From equipartition theorem, we can write the expression of absorbed heat Q_V at constant volume as below

$Q_{\mathrm{V}}=\left(\frac{F}{2}Nk\left(T_3-T_2\right)\right)$

Where F is degrees of freedom per molecule, N is number of molecules, k is Boltzmann constant, T₃ is temperature at point 3 and T₄ is temperature at point 4 of the PV diagram.

Now the expression of the absorbed heat Q_T along isothermal path is written as

$Q_{\mathrm{T}}=\left(Nk{T}_3\ln \left(\frac{V_1}{V_2}\right)\right)$

where V₁ and V₂ are volumes at point 1 and 2 of the diagram .

So the total the total absorbed heat is

$Q_{\mathrm{H}}=Q_{\mathrm{V}}+Q_{\mathrm{T}}$

Substitute the values we get

$Q_{\mathrm{H}}=\left(\left(\frac{F}{2}Nk\left(T_3-T_2\right)\right)+\left(Nk{T}_3\ln \left(\frac{V_1}{V_2}\right)\right)\right)$

Expelled heat Q'_V at constant volume from equipartition theorem is written as

$Q_{\mathrm{V}}^{\text{'}}=\left(\frac{F}{2}Nk\left(T_3-T_2\right)\right)$

Expelled heat Q'_T at constant volume in isothermal path is written as

$Q_{\mathrm{T}}^{\text{'}}=\left(Nk{T}_2\ln \left(\frac{V_1}{V_2}\right)\right)$

where V₁ and V₂ are volumes at point 1 and 2 of the diagram .

Total expelled heat $Q_{\mathrm{C}}=Q_{\mathrm{V}}^{\text{'}}+Q_{\mathrm{T}}^{\text{'}}$

On substitution we get

$Q_{\mathrm{C}}=\left(\left(\frac{F}{2}Nk\left(T_3-T_2\right)\right)+\left(Nk{T}_2\ln \left(\frac{V_1}{V_2}\right)\right)\right)$

Efficiency of Stirling engine is given as

$e=1-\frac{Q_{\mathrm{C}}}{Q_{\mathrm{H}}}\dots \dots ..\left(1\right)$

Substitute values of Q_C and Q_H in equation (1) we get

$e=1-\frac{\left(\frac{F}{2}Nk\left(T_3-T_2\right)+Nk{T}_2\ln \frac{V_1}{V_2}\right)}{\left(\frac{F}{2}Nk\left(T_3-T_2\right)+Nk{T}_3\ln \frac{V_1}{V_2}\right)}$

Upon simplification we get

$e=\left(1-\left(\frac{T_2}{T_3}\right)\frac{\ln \left(\frac{V_1}{V_2}\right)+\frac{F}{2}\left(\frac{T_3}{T_2}-1\right)}{\ln \left(\frac{V_1}{V_2}\right)+\frac{F}{2}\left(1-\frac{T_2}{T_3}\right)}\right)\dots \dots ..\left(2\right)$

Efficiency of Carnot engine is written as

$e_{\mathrm{C}}=\left(1-\frac{T_2}{T_3}\right)\dots \dots ..\text{(3)}$

As $1<\frac{\ln \left(\frac{V_1}{V_2}\right)+\frac{F}{2}\left(\frac{T_3}{T_2}-1\right)}{\ln \left(\frac{V_1}{V_2}\right)+\frac{F}{2}\left(1-\frac{T_2}{T_3}\right)}$, So $\left(1-\frac{T_2}{T_3}\right)>1-\left(\frac{T_2}{T_3}\right)\frac{\ln \left(\frac{V_1}{V_2}\right)+\frac{F}{2}\left(\frac{T_3}{T_2}-1\right)}{\ln \left(\frac{V_1}{V_2}\right)+\frac{F}{2}\left(1-\frac{T_2}{T_3}\right)}$

This means $e_{\mathrm{C}}>e$

Substitute $\left(\frac{V_1}{V_2}\right)=2,\left(\frac{T_2}{T_3}\right)=(3/4),F=5$in expression (2), we get

$$\begin{gathered} e=\left(1-\left(\frac{3}{4}\right)\frac{\ln \left(2\right)+\left(\frac{5}{2}\right)\left(\frac{4}{3}-1\right)}{\ln \left(2\right)+\left(\frac{5}{2}\right)\left(1-\frac{3}{4}\right)}\right) \\ =0.13 \end{gathered}$$

And,

$$\begin{gathered} e_{\mathrm{C}}=\left(1-\frac{3}{4}\right) \\ =0.25 \end{gathered}$$

As 0.13 < 0.25 So $e<e_{\mathrm{C}}$.

i. Power stroke. While in the hot cylinder at temperature T_h, the gas absorbs heat and expands isothermally, pushing the hot piston outward. The piston in the cold cylinder remains at rest, all the way inward as shown in the figure.
ii. Transfer to the cold cylinder. The hot piston moves in while the cold piston moves out, transferring the gas to the cold cylinder at constant volume. While on its way, the gas flows past the regenerator, giving up heat and cooling to T_c.
iii. Compression stroke. The cold piston moves in, isothermally compressing the gas back to its original volume as the gas gives up heat to the cold reservoir. The hot piston remains at rest, all the way in.
iv. Transfer to hot cylinder. The cold piston moves the rest of the way in while the hot piston moves out, transferring the gas back to the hot cylinder at constant volume. While on its way, the gas flows past the regenerator, absorbing heat until it is again at T_h.

Prove that the efficiency is less than that of Carnot cycle.

Show that efficiency for a perfect sterling engine is same as Carnot engine.

Total absorbed heat in this case can be written as

$Q_{\mathrm{H}}=\left(Nk{T}_3\ln \left(\frac{V_1}{V_2}\right)\right)$

and Total expelled heat is

$Q_{\mathrm{C}}=\left(Nk{T}_2\ln \left(\frac{V_1}{V_2}\right)\right)$

Upon substitution for efficiency in equation (1) we get

$$\begin{gathered} e=1-\frac{\left(Nk{T}_2\ln \left(\frac{v_1}{v_2}\right)\right)}{\left(Nk{T}_3\ln \left(\frac{v_1}{v_2}\right)\right)} \\ =\left(1-\frac{T_2}{T_3}\right) \end{gathered}$$

Now substitute e_C for $\left(1-\frac{T_2}{T_3}\right)$from equation (3) we get

e_C =e

i. Power stroke. While in the hot cylinder at temperature T_h, the gas absorbs heat and expands isothermally, pushing the hot piston outward. The piston in the cold cylinder remains at rest, all the way inward as shown in the figure.
ii. Transfer to the cold cylinder. The hot piston moves in while the cold piston moves out, transferring the gas to the cold cylinder at constant volume. While on its way, the gas flows past the regenerator, giving up heat and cooling to T_c.
iii. Compression stroke. The cold piston moves in, isothermally compressing the gas back to its original volume as the gas gives up heat to the cold reservoir. The hot piston remains at rest, all the way in.
iv. Transfer to hot cylinder. The cold piston moves the rest of the way in while the hot piston moves out, transferring the gas back to the hot cylinder at constant volume. While on its way, the gas flows past the regenerator, absorbing heat until it is again at T_h.

Prove that the efficiency is less than that of Carnot cycle.

Advantages:
The Stirling engine is a real heat engine which absorbs heat from external source.
The Stirling engine can use air as well as any other gas as its working substance.
The Stirling engine can function more properly (without knocking) in cold environment compare to other engines.
It has a simpler mechanism as compared to other reciprocating engines such that it is easily balanced generating only a few vibrations.
Outside the cylinders, the combustion is continuous and no atmospheric expansion occurs unlike ICE (internal combustion engines).
Disadvantages:
The Stirling engine is to be warmed up which requires high temperature differentials for functioning efficiently.
The Stirling engine needs heat exchangers for heat output as well as heat input.