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Fluids

The energy balance

As indicated in the diagram at the previous page, in all four steps of the cycle heat is exchanged, whereas in two of the steps work is delivered. Let us collect all information in a table.

process	heat	kinetic energy	work
(V1,P1)→(V1,P2)	3(P2-P1)V1/2	3(P2-P1)V1/2	0
(V1,P2)→(V2,P2)	5P2(V2-V1)/2	3P2(V2-V1)/2	P2(V2-V1)
(V2,P2)→(V2,P1)	-3(P2-P1)V2/2	-3(P2-P1)V2/2	0
(V2,P1)→(V1,P1)	-5P1(V2-V1)/2	-3P1(V2-V1)/2	-P1(V2-V1)
sum	ΔP ΔV	0	ΔP ΔV

We first observe that in each of the separate processes of the cycle total energy is conserved, as expressed by

Heat = kinetic energy + work

In the last line of the table we have collected the total heat which is consumed by the gas, the kinetic energy of the gas and the total work delivered by the gas in the complete cycle, where ΔP=P₂-P₁ and ΔV=V₂-V₁.
We find that the net work of the cycle is positive and equal to the area ΔP ΔV encircled by the cycle. This is generally true for clockwise cycles in the pressure-volume diagram, whatever their shape.

Heat capacity