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 21
 Problem Statement
 Evaluate the efficiency of this engine
 Relevant Equations

The equation for an ellipse:
##\frac{(xx_o)^2}{a^2} + \frac{(yy_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/howtogettheefficiencyofaheatenginewhichundergoesanellipticalcycle . 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).
$$(3/2P_1, V1)$$
$$(2P_1, (V1+V2)/2)$$
$$(3/2P_1, V2)$$
$$(P_1, (V1+V2)/2)$$
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/howtogettheefficiencyofaheatenginewhichundergoesanellipticalcycle . 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).