[cseil]

Hello everyone,
I'm doing an exercise which asks me to demonstrate that the efficiency of an imaginary cycle performed with an ideal gas is:

η = 1 - γ[(V1/V2)-1]/[(P3/P2)-1]

I'm referring to the P6.3 image of the attachment.

I know that η is work obtained / heat absorbed.
But in this case what is the heat absorbed by the system? I have 1-2 isobaric (compression of an ideal gas, so increase of temperature) and 2-3 isochroic (increase of pressure and temperature). The 3-1 is an adiabatic with no heat exchanged.

Please could you give me some advice?
Thanks

[mjc123]

Isobaric compression means temperature decrease; T/V is constant.

[cseil]

That was stupid!
Thank you.

η = Q1-Q2/Q1 = 1 - Q2/Q1

Q1 is the heat of the isobaric compression, Q2 the isochoric one.

Q2 = Cvln(T3/T2)
Q1 = Cpln(T2/T1)

T3 is p3V2/R
T2 is p2V2/R
T1 is p1V1/R

So T3/T2 = p3/p2 and T2/T1 is V2/V1.

η = 1- (1/γ)[(p3/p2)/(V2/V1)]

I don't know how to proceed now.
I could invert the 1/γ etc.. but I don't know how to have V1/V2 -1 and p2/p3 -1..

[mjc123]

Surely Q2 = C_v(T₃-T₂)
and Q1 = C_p(T₂-T₁)
Your desired expression is then equal to 1 + Q1/Q2 (remembering that Q1 is negative). Is that right?

[cseil]

It is.
Thank you.

That was not so difficult, but I did two mistakes.
I considered a temperature increase during the isobaric compression and considered Q2 = Cv(T3/T2) not Cv(T3-T2).

Thank you for the help.

[cseil]

I am sorry but I've another doubt about cycles.
I am always considering the P6.3 cycle.

1-2 is an isobaric compression, there's a temperature decrease.
Is there an outgoing flux of heat, right?
The system gives heat to the ambient.

But when there's a compression of a gas and a temperature increase, or generally just a temperature increase,
should I consider the heat flux incoming (so the Q arrow entering the graph) or outgoing (the arrow going out)?

If the gas compresses, there's no flux of heat from the external, but the opposite.
Temperature increase and the system gives heat to the ambient. In this case do we have another outgoing flux of heat?

I really have some problems with it.

[mjc123]

1-2 is an isobaric compression, there's a temperature decrease.
Is there an outgoing flux of heat, right?
The system gives heat to the ambient.

Yes

But when there's a compression of a gas and a temperature increase, or generally just a temperature increase,
should I consider the heat flux incoming (so the Q arrow entering the graph) or outgoing (the arrow going out)?

This is more complicated. Compressing the gas does work on it; what happens to that energy? In the isothermal limit, it is all lost to the surroundings as heat (heat flux out), and T does not change. In the adiabatic limit, there is no heat flow, and the energy goes into raising the temperature. For an isovolumetric temperature rise, the heat must come from outside (heat flux in).

If the gas compresses, there's no flux of heat from the external, but the opposite.
Temperature increase and the system gives heat to the ambient. In this case do we have another outgoing flux of heat?

I'm sorry, this makes no sense.