We consider a three-stroke-cycle that is composed of a sequence of isothermal expansion followed by an isobaric compression completing the cycle via an isochoric compression. The cycle is shown in the V-P diagram, Figure 2.

In Figure 2 the corners of the cycle are labeled {1, 2, 3}. Each corner represents the state of the gas according to {V, P, T} format. For ideal gases these three quantities are related via Equation of state. The equation of state for one mole of an ideal gas is, PV = RT, R is the ideal gas constant. In Figure 2 specific numeric values are used. For instance, vertex 1 is labeled {10, 300, 361}; these stand for 10 lit volume, 300 kPa pressure and 361 K temperature, respectfully. These values may be adjusted on a need basis. The net mechanical work for the shown cycle is,

W 1231 = W 12 + W 23

where, the volumetric mechanical work is, W 12 = ∫ 1 2 P d V . Utilizing the ideal gas equation of state yields, W₁₂ = RT₁ln(V₂/V₁).

Similarly, for the change of state from 2 to 3 we have, W₂₃ = P₂(V₃ − V₂). The net volumetric work for the cycle is,

W 1231 = P 1 V 1 ln ( V 2 V 1 ) + ( P 3 V 3 − P 2 V 2 ) , (1)

The cycle requires two different intake heats, Q ← 12 and Q ← 31 . For the isothermal and isochoric legs, we have

Q 12 = P 1 V 1 ln ( V 2 / V 1 ) , and Q 31 = C v Δ T ≡ f R / 2 ( T 1 − T 3 )

where for the molar heat capacity of the ideal gas we applied C_v = fR/2, f is the degrees of the freedom. Hence the net intake heat is,

Q 1231 = P 1 V 1 ln ( V 2 V 1 ) + 1 2 f ( P 1 V 1 − P 3 V 3 ) , (2)

Efficiency of the cycle is e = W₁₂₃₁/Q₁₂₃₁. Utilizing (1) and (2) we arrive at,

e = ln ( V 2 V 1 ) + P 3 P 1 ( 1 − V 2 V 1 ) ln ( V 2 V 1 ) + 1 2 f ( 1 − P 3 P 1 ) . (3)

That is the efficiency, it is written as an explicit function of the degrees of freedom and the relevant numeric parameters that can be read off the V-P diagram, Figure 2. As an example, utilizing numeric values of the vertices in Figure 2 for two different values of degrees of freedom, monatomic and polyatomic with degrees of freedom, f = 3 and 8, respectively Equation (3) yields, e_f=3 = 20.5%, and e_f=8 = 11.5%. That is the working materials with polyatomic gases with higher degrees of freedom result less efficiencies. This is because polyatomic gases with higher degrees of freedom do require greater intake heat per degree temperature change. One also notes that the contour of the cycle and consequentially the value of the mechanical work is independent of the degrees of freedom but the intake heat depends on f.

We consider a three-stroke-cycle that is composed of a sequence of adiabatic expansion followed by an isobaric compression completing the cycle with an isochoric compression. The cycle is shown in the V-P diagram, Figure 3.

There are differences between the cycles shown in Figure 3 vs. the one shown in Figure 2. For the former state 1 and 2 are connected via an isothermal process where for the latter the states are connected adiabatically. As noted the isothermal process is independent of the degrees of freedom, where the adiabatic

process depends on f. Here we evaluate the volumetric value of the mechanical work done via adiabatic process. For ideal gases subject to adiabatic processes it is known [1] the states are related via PV^γ = const, here γ is the adiabatic index and is, γ = C_P/C_V i.e. the ratio of the molar heat capacity at constant pressure to the molar heat capacity at constant volume. Applying Mayer [1] relationship, C_P − C_V = R, and knowing that, C_V = R(1/2f), one gets the γ in terms of the degrees of freedom namely, γ = 1 + 2/f. Applying these relationships in conjunction with equation of state of ideal gases for states related via adiabatic process one

obtains additional useful relationships such as: T V 1 − γ = C R and T P 1 γ − 1 = c 1 γ R . For the volumetric work w 12 = ∫ 1 2 P d V applying above equations we arrive at, w 12 = C 1 − γ ( T 2 − T 1 ) . Utilizing w₂₃ from Equation (1) the net volumetric work for the cycle shown in Figure 3 is written as,

w 1231 = 1 2 f ( P V ) 1 + ( P V ) 3 − ( 1 2 f + 1 ) ( P V ) 2 , (4)

On the other hand, the intake heat of the cycle is given by Equation (2). Utilizing these we arrive at the efficiency of the cycle,

e = 1 P 1 − P 3 [ P 1 + 2 f P 3 − ( 1 + 2 f ) P 2 ( V 2 V 1 ) ] , (5)

As one expects the volumetric mechanical work is a function of the degrees of freedom, yet the intake heat is independent of f. Consequently, depending on value of f the contour of the cycle changes accordingly. Two such contours are depicted in Figure 3. The steep dashed curve between state (1) and (2) corresponds to f = 3, the soft solid curve connecting state (1) to (4) corresponds to f = 8. For two values of degrees of freedom such as, f = 3 and 8 Equation (5) yields two values of efficiencies, e_f=3 = 22.5% and e_f=8 = 12.5%.

By comparing the efficiencies of the cycles depicted in Figure 2 and Figure 3 we realize their comparative values. It is interesting noting that the impact of the value of f in these two cycles are inverted. For the former the contour of the cycle i.e the volumetric work is f independent, where for the latter it dependents on f. On the other hand, the intake heat has baring on the former and has no impact on the latter.

For comprehensive understanding one needs to compare Figure 2 and Figure 3.

We consider a three-stroke-cycle that is composed of a sequence of adiabatic expansion followed by an isothermal compression closing the cycle via an isochoric compression. One such cycle is shown in the V-P diagram, Figure 4.

Evaluation of the efficiency of the cycle shown in Figure 4 as a function of degrees of freedom is straight forward. The volumetric mechanical work and intake heat of each process leg of the cycle is given in the previous cases. Utilizing these expressions, the efficiency of the given cycle yields,

e = ( P V ) 1 − ( P V ) 2 + 2 f ( P V ) 2 ln ( V 3 V 2 ) V 1 ( P 1 − P 3 ) . (6)

Applying Equation (6) for the numeric values of the vertices shown in Figure 4, we arrive at the efficiencies, namely, e_f=3 = 21.0% and e_f=8 = 12.4%.