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

http://img134.imageshack.us/img134/4534/thermo3eo8.jpg [Broken]

http://g.imageshack.us/img134/thermo3eo8.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 = \frac{|W|}{|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 = \frac{|W|}{|q_{in}|} [/tex]

1->2

We have [tex] dP = 0 [/tex] and [tex] P = constant [/tex]

[tex] W = \int^{V_{2}}_{V_{1}} dV [/tex]

[tex] W = -P_{2} (V_{2} - V_{1}) [/tex]

Since this is an ideal gas, we know:

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

Therefore,

[tex] q = C_{P} (T_{2} - T{1}) [/tex]

2->3

We have [tex] dV = 0 [/tex] and [tex] dw = 0 [/tex] since the process is isochoric.

Therefore,

[tex] dE = dq [/tex]

Since this is an ideal gas [tex] dE = C_{V} dT [/tex]

Therefore,

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

[tex] q = C_{V} (T_{3} - T{2}) [/tex]

By hypothesis,

[tex] w = 0 [/tex]

3->1

This is an adiabatic process, so [tex] dq = 0 [/tex]

Therefore,

[tex] dE = dw [/tex]

Since this is an ideal gas [tex] dE = C_{V} dT [/tex]

Therefore,

[tex] W = \int^{T_{1}}_{T_{3}} C_{V} dT [/tex]

[tex] W = C_{V} (T_{1} - T_{3}) [/tex]

Also, by hypothesis,

[tex] q = 0 [/tex]

Plugging all this into the equation for [tex] \eta [/tex], we get:

[tex] \eta = \frac{P_{2} (V_{2} - V_{1} ) - C_{V} (T_{1} - T_{3})}{C_{P} (T_{2} - T_{1}) + C_{V} (T_{3} - T_{2})} [/tex]

Now, I don't know how to manipulate this to get it in the form that the problem asked for.

Any help would be greatly appreciated. Thanks.

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# Homework Help: Efficiency of an imaginary ideal-gas engine cycle

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