Engine thermal efficiency and Volume ratios

[sandpants]

The question:

A perfect gas undergoes the following cyclic processes:
State 1 to 2 cooling at constant pressure.
State 2 to 3 heating at constant volume.
State 3 to 1 adiabatic expansion.

Deduce an expression for the thermal efficiency of the cycle in terms
of r the volume compression ratio (r=V₁/V₂) and γ (where γ = ratio of specific heats Cp/Cv)

η = 1 - γ(r-1)/(r^γ-1)

My attempt at the solution:
First I tried sketching the cycle

Bare with me as I present you the silly symbol art.

P

3.
^'.
|..|
|...\
|...'-.
|...'-._
2<---------':.1 v

I'd like to work in specific terms

As it is a perfect gas
P₁v₁= RT₁
P₁v₂= RT₂
P₂v₂= RT₃

Heats from 1->2, 2->3, 3->1
Q_1->2=C_p(T₂-T₁)
Q_2->3=C_v(T₃-T₂)
Q_3->1 = 0 ; adiabatic.

Also, polytropic relations
v₂/v₁ = (P₁/P₂)^1/n
as r = v₁/v₂⇔ r = (P₂/P₁)^1/n
∴ rⁿ = P₂/P₁ and
1/rⁿ = P₁/P₂

Substituting Ideal Gas expressions in terms of T_n
Q_1->2=C_p((P₁v₂-P₁v₁)/R)
Q_2->3=C_v((P₂v₂-P₁v₂)/R)

Thermal efficiency
This is what I am unsure off. I begin assuming quite a few things. First I assume that heat in the cooling process is the equivalent of heat escaping to a cold reservoir; coincidentally, heat from the pressurization is the heat INPUT from the hot reservoir. As such:

η = [Q_2->3 - Q_1->2]/Q_2->3

η = 1 - Q_1->2/Q_2->3

∴ η = 1 - C_p(P₁v₂-P₁v₁)/C_v(P₂v₂-P₁v₂)

η = 1 - γ((P₁v₂-P₁v₁)/(P₂v₂-P₁v₂))

From here:

P₁v₂/P₂v₂-P₁v₂ = P₁/(P₂-P₁) = 1/rⁿ-1

And
P₁v₁/P₂v₂-P₁v₂ = r/(rⁿ-1)

My Result
∴
η = γ(1-r)/(rⁿ-1) =/= η = 1 - γ(r-1)/(r^γ-1)

Can I assume n=γ in this situation? only 1 process is adiabatic.

 

[Andrew Mason]

Apply the adiabatic condition from 3→1.

AM

 

[sandpants]

Apply the adiabatic condition from 3→1.

AM

Can you be more specific? Apply where?

If it's adiabatic there is no heat - I do not understand how the process can be related to thermal efficiency.

 

[Andrew Mason]

Can you be more specific? Apply where?

If it's adiabatic there is no heat - I do not understand how the process can be related to thermal efficiency.

What you need is the relationship between P1 and P2 in terms of V1 and V2. That is determined by the adiabatic condition PV^γ = K.

AM

 

[sandpants]

What you need is the relationship between P1 and P2 in terms of V1 and V2. That is determined by the adiabatic condition PV^γ = K.

AM

Indeed, the ratios match up and allow you to express them with n=γ. Thermodynamics is always like that - an answer under your nose at all times.

But another issue is that the numerator does not match up.
The expected form is r-1 when I get 1-r despite getting the same denominator.

 

[Andrew Mason]

Indeed, the ratios match up and allow you to express them with n=γ. Thermodynamics is always like that - an answer under your nose at all times.

But another issue is that the numerator does not match up.
The expected form is r-1 when I get 1-r despite getting the same denominator.

You are using the absolute values of Qh and Qc so you have to make sure that the original equation reflects that. For example, Qc = |Cp(T2-T1)| = Cp(T1-T2)

AM