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Heat engine undergoing an elliptical cycle

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21
Problem Statement
Evaluate the efficiency of this engine
Relevant Equations
The equation for an ellipse:

##\frac{(x-x_o)^2}{a^2} + \frac{(y-y_o)^2}{b^2} = 1##
An ideal diatomic gas undergoes an elliptic cyclic process characterized by the following points in a ##PV## diagram:

$$(3/2P_1, V1)$$
$$(2P_1, (V1+V2)/2)$$
$$(3/2P_1, V2)$$
$$(P_1, (V1+V2)/2)$$


This system is used as a heat engine (converting the added heat into mechanical work).

Evaluate the efficiency of this engine setting ##P_1=1## and ##P_2= 2P_1##

We know that the efficiency is defined as the benefit/cost ratio:

$$e = \frac{W}{Q_h}$$

Let's focus first on the work done by the engine; the work done by the working substance is the area under the ##PV## graph. Then:

$$W = \pi (P_2 - P_1)(V_2 - V_1)$$

$$W = \pi P_1(V_2 - V_1)$$

My problems come when calculating ##Q_h##; I have been told an analytic method: https://chemistry.stackexchange.com/questions/116757/how-to-get-the-efficiency-of-a-heat-engine-which-undergoes-an-elliptical-cycle . But I am convinced there has to be an easier one...

I have been thinking I have been thinking about how I could make an analogy with the same problem but with a rectangular shape (which is much easier to solve).
 

vela

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The rectangle that circumscribes the ellipse has area ##\Delta P \Delta V##. Your expression for ##W## is ##\pi## times that. That can't possibly be correct.
 
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The rectangle that circumscribes the ellipse has area ##\Delta P \Delta V##. Your expression for ##W## is ##\pi## times that. That can't possibly be correct.
$$W = \pi (P_2 - P_1)(V_2 - V_1)$$

It is for the ellipse (my question). I have just mentioned the rectangle as an example. However, the exercise is about an ellipse cycle
 

vela

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You missed the point.
 
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Sorry I realized that:

$$W = \pi (P_2 - P_1)(V_2 - V_1)$$

It 's wrong. Instead we have:

$$W=\text{Area}=\pi\left(\frac{s-1}2V_1\right)\left(\frac{r-1}2P_1\right)$$

Where ##s## and ##r ## are the volume and pressure compression ratios respectively.
 
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I have been thinking I have been thinking about how I could make an analogy with the same problem but with a rectangular shape (which is much easier to solve).
Once you solve the problem properly for the elliptical cycle, it will be possible to identify what would be considered an equivalent rectangular cycle with the same efficiency and the same center (for whatever that might be worth).
 

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