(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

The figure represents a simplified PV diagram of the Joule ideal-gas cycle. All processes are quasi-static, and [tex] C_P[/tex] is constant. Prove that the thermal efficiency of an engine performing this cycle is

[tex] 1 - \left(\frac{P_1}{P_2}\right)^\frac{\gamma - 1}{\gamma} [/tex]

http://img50.imageshack.us/img50/7734/thermo1ym3.jpg [Broken]

http://g.imageshack.us/img50/thermo1ym3.jpg/1/ [Broken]

2. Relevant equations

[tex] PV = RT [/tex]

[tex] \gamma = \frac{C_P}{C_V} [/tex]

[tex] dE = dq + dw [/tex] (should have strokes through the d's on dq and dw, but I don't know how to latex inexact differentials)

[tex] \eta = 1 - \frac{|q_{out}|}{|q_{in}|} [/tex]

On adiabatic processes,

[tex] TV^{\gamma - 1} = [/tex] constant

[tex] PV^{\gamma} = [/tex] constant

3. The attempt at a solution

First of all, [tex] \eta = 1 - \frac{|q_{out}|}{|q_{in}|} [/tex]

[tex]q_{in} [/tex] is only path 2->3 and [tex] q_{out} [/tex] is only path 4->1.

2->3

Since we have an ideal gas,

[tex] dq = C_{P}dT [/tex]

[tex] q_{in} = \int^{T_3}_{T_2} C_{P}dT [/tex]

However,

[tex] dT = \left(\frac{\partial T}{\partial P}\right)_{V} dP + \left(\frac{\partial T}{\partial V}\right)_{P} dV [/tex]

since dP = 0 in the 2->3 process, we have:

[tex] dT = \left(\frac{\partial T}{\partial V}\right)_{P} dV [/tex]

Now, using the ideal-gas equation of state and solving for T:

[tex] T = \frac{PV}{R} [/tex]

Differentiating:

[tex] \left(\frac{\partial T}{\partial V}\right)_{P} = \frac{P}{R} [/tex]

Now, substituting into the above expression, we get:

[tex] q_{in} = \frac{C_{P} P_{2}}{R} \int^{V_{3}}_{V_{2}}dV [/tex]

So,

[tex] |q_{in}| = \frac{C_{P} P_{2}}{R} (V_{3} - V_{2}) [/tex]

Now, considering process 4->1

4->1

We have the same process as above, but with different pressures and volumes. Therefore,

[tex] q_{out} = \frac{C_{P} P_{1}}{R} \int^{V_{1}}_{V_{4}} dV [/tex]

Since q_{out} is negative, we switch signs,

[tex] q_{out} = \frac{C_{P} P_{1}}{R} (V_{4} - V_{1}) [/tex]

Plugging into the efficiency formula [tex]\frac{C_{P} P_{1}}{R} [/tex] cancels, and we get:

[tex] \eta = 1 - \frac{P_{1} (V_{4} - V_{1})}{P_{2} (V_{3} - V_{2})} [/tex]

Now, I'm pretty sure I have to use the identities:

[tex] P_{1} V^{\gamma}_{4} = P_{2} V^{\gamma}_{3} [/tex]

[tex] P_{1} V^{\gamma}_{1} = P_{2} V^{\gamma}_{2} [/tex]

I have tried dividing these two equations so that all P's cancel.

I have also tried subtracting the equations.

I can't, for the life of me, get my efficiency in the form that the problem asks me to put it in.

Any help would be greatly appreciated. Thanks.

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# Homework Help: Efficiency of an engine performing the Joule ideal-gas cycle.

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