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sandpants
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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)/(rγ-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
Q1->2=Cp(T2-T1)
Q2->3=Cv(T3-T2)
Q3->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 Tn
Q1->2=Cp((P1v2-P1v1)/R)
Q2->3=Cv((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 = [Q2->3 - Q1->2]/Q2->3
η = 1 - Q1->2/Q2->3
∴ η = 1 - Cp(P1v2-P1v1)/Cv(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)/(rγ-1)
Can I assume n=γ in this situation? only 1 process is adiabatic.
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)/(rγ-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
Q1->2=Cp(T2-T1)
Q2->3=Cv(T3-T2)
Q3->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 Tn
Q1->2=Cp((P1v2-P1v1)/R)
Q2->3=Cv((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 = [Q2->3 - Q1->2]/Q2->3
η = 1 - Q1->2/Q2->3
∴ η = 1 - Cp(P1v2-P1v1)/Cv(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)/(rγ-1)
Can I assume n=γ in this situation? only 1 process is adiabatic.
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