- #1
Jacobpm64
- 239
- 0
Homework Statement
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
http://g.imageshack.us/img134/thermo3eo8.jpg/1/
Homework 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
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.
Last edited by a moderator: