What is the Efficiency of a Heat Engine?

[TFM]

Okay so:

Work = Q_1_2 - Q_3_1

Q_{12) = (C_V (\frac{p_1v_3}{nR} - \frac{p_1v_1}{nR}) + (V_3 - V_1)P_1

Okay so Q23

Q = U + W along path 23

W = 0

\Delta U = C_P \Delta T (V const)

using Ideal gas law, T = PV/nR

\Delta T = \frac{P_3V_3}{nR} - \frac{P_1V_3}{nR}

\Delta U = C_P \frac{P_3V_3}{nR} - \frac{P_1V_3}{nR}

Q_{23} = C_P \frac{P_3V_3}{nR} - \frac{P_1V_3}{nR}


So:

e = \frac{Work}{Q_{12}}

Work = \left(C_V (\frac{p_1v_3}{nR} - \frac{p_1v_1}{nR}) + (V_3 - V_1)P_1\right) - \left(C_P \frac{P_3V_3}{nR} - \frac{P_1V_3}{nR}\right)

e = \frac{\left(C_V (\frac{p_1v_3}{nR} - \frac{p_1v_1}{nR}) + (V_3 - V_1)P_1\right) - \left(C_P \frac{P_3V_3}{nR} - \frac{P_1V_3}{nR}\right)}{(C_V (\frac{p_1v_3}{nR} - \frac{p_1v_1}{nR}) + (V_3 - V_1)P_1}


So does this look okay now?

 

[Andrew Mason]

...
So:

e = \frac{Work}{Q_{12}}

Work = \left(C_V (\frac{p_1v_3}{nR} - \frac{p_1v_1}{nR}) + (V_3 - V_1)P_1\right) - \left(C_P \frac{P_3V_3}{nR} - \frac{P_1V_3}{nR}\right)

e = \frac{\left(C_V (\frac{p_1v_3}{nR} - \frac{p_1v_1}{nR}) + (V_3 - V_1)P_1\right) - \left(C_P \frac{P_3V_3}{nR} - \frac{P_1V_3}{nR}\right)}{(C_V (\frac{p_1v_3}{nR} - \frac{p_1v_1}{nR}) + (V_3 - V_1)P_1}

Try:

Q_{12} = n\gamma C_v(T_2-T_1) and Q_{23} = nC_v(T_3-T_2)

\eta = W/Q_h = \frac{Q_{12} - Q_{23}}{Q_{12}} = \frac{(\gamma(T_2-T_1) - (T_3 - T_2)}{\gamma(T_2-T_1)}

Work out T1 and T3 using:

TV^{\gamma-1} = const.

ie.

T_3 = T_1\left(\frac{V_1}{V_3}\right)^{\gamma-1} = \frac{P_1V_1}{nR}\left(\frac{V_1}{V_3}\right)^{\gamma-1}

T_2 = P_1V_3/nR

AM

 

[TFM]

I see. So:

Q = n C_P \Delta T

From the expression for gamma,

C_p = f C_V (I am using f because gamma doesn't show well on latex)

so

Q = n f C_V \Delta T


Q_{12} = n f C_V (T_2 - T_1)

Q_{23} = n C_V (T_3 - T_2)


e = W/Q_h = \frac{Q_{12} - Q_{23}}{Q_{12}}

\frac{Q_{12} - Q_{23}}{Q_{12}} = \frac{nf C_V (T_2 - T_1) - n C_V (T_3 - T_2)}{n f C_V (T_2 - T_1)}

Cancels to:

\frac{Q_{12} - Q_{23}}{Q_{12}} = \frac{f(T_2 - T_1) - (T_3 - T_2)}{f(T_2 - T_1)}

so using adiabatic law:

TV^f = c

T_{1}V_{1}^{f -1}= T_{3}V_{3}^{f-1}

T_{1}= T_{3}\frac{V_{3}^{f-1}}{V_{1}^{f -1}}

Use ideal gas law:

T = PV/nR

\T_1= \frac{P_3V_3}{nR}\frac{V_{3}^{f-1}}{V_{1}^{f -1}}

T_2 = \frac{P_1V_3}{nR}

T_3 = \frac{P_1V_1}{nR}\frac{V_{1}^{f-1}}{V_{3}^{f -1}}

so:

\frac{(f(\frac{P_1V_3} - \frac{P_3V_3}{nR}\frac{V_{3}^{f-1}}{V_{1}^{f -1}}) - (\frac{P_1V_1}{nR}\frac{V_{1}^{f-1}}{V_{3}^{f -1}} - \frac{P_1V_3})}{f(\frac{P_1V_3} - \frac{P_3V_3}{nR}\frac{V_{3}^{f-1}}{V_{1}^{f -1}})}

Okay so far?

 

[Andrew Mason]

I
\frac{(f(\frac{P_1V_3} - \frac{P_3V_3}{nR}\frac{V_{3}^{f-1}}{V_{1}^{f -1}}) - (\frac{P_1V_1}{nR}\frac{V_{1}^{f-1}}{V_{3}^{f -1}} - \frac{P_1V_3})}{f(\frac{P_1V_3} - \frac{P_3V_3}{nR}\frac{V_{3}^{f-1}}{V_{1}^{f -1}})}

Okay so far?

Notice that:

\eta = \frac{(\gamma(T_2-T_1) - (T_3 - T_2)}{\gamma(T_2-T_1)}

reduces to:

\eta = 1 - \frac{1}{\gamma}\frac{(T_3 - T_2)}{(T_2-T_1)}

So you just have to show that:

\frac{(T_3 - T_2)}{(T_2-T_1)} = \frac{(1-\frac{P_3}{P_1})}{(1-\frac{V_3}{V_1})}

AM

 

[TFM]

Notice that:

\eta = \frac{(\gamma(T_2-T_1) - (T_3 - T_2)}{\gamma(T_2-T_1)}

reduces to:

\eta = 1 - \frac{1}{\gamma}\frac{(T_3 - T_2)}{(T_2-T_1)}

I see that now.

\eta = \frac{(\gamma(T_2-T_1) - (T_3 - T_2)}{\gamma(T_2-T_1)}

is the same as:

\eta = \frac{(\gamma(T_2-T_1)}{\gamma(T_2-T_1)} - \frac{(T_3 - T_2)}{\gamma(T_2-T_1)}

Which cancels down to:

\eta = 1 - \frac{1}{\gamma}\frac{(T_3 - T_2)}{(T_2-T_1)}

Okay so:

\frac{(T_3 - T_2)}{(T_2-T_1)}

T_1= \frac{P_3V_3}{nR}\frac{V_{3}^{f-1}}{V_{1}^{f -1}}

T_2 = \frac{P_1V_3}{nR}

T_3 = \frac{P_1V_1}{nR}\frac{V_{1}^{f-1}}{V_{3}^{f -1}}

So:

\frac{(\frac{P_1V_1}{nR}\frac{V_{1}^{f-1}}{V_{3}^{f -1}} - \frac{P_1V_3}{nR})}{(\frac{P_1V_3}{nR} -\frac{P_3V_3}{nR}\frac{V_{3}^{f-1}}{V_{1}^{f -1}})}

Factorise out:

\frac{(\frac{P_1V_1}{nR}\frac{V_{1}^{f-1}}{V_{3}^{f -1}} - \frac{P_1V_3}{nR})}{\frac{V_3}{nR}(P_1 - P_3 \frac{V_3^{\gamma - 1}}{v_1^{\gamma - 1}})}

and:

\frac{\frac{p_1}{nR}(v_1\frac{v_1^{\gamma - 1}}{v_3^{\gamma - 1}} - V_3)}{\frac{V_3}{nR}(P_1 - P_3 \frac{V_3^{\gamma - 1}}{v_1^{\gamma - 1}})}


Does this look okay so far?

 

[TFM]

I am also assuming that the nR can cancel, so:

P_1(v_1\frac{v_1^{\gamma - 1}}{v_3^{\gamma - 1}} - V_3)}{V_3(P_1 - P_3 \frac{V_3^{\gamma - 1}}{v_1^{\gamma - 1}})}

Does this look right?

 

[Redbelly98]

To be honest I have lost track. However, it would be desirable to simplify the term

v₁ v₁^γ-1​

The idea is, once you have an expression for the efficiency, for you to manipulate and simplify it algebraically and get

<br /> e = 1 - \frac{1}{\gamma}\left(\frac{1 - \frac{p_3}{p_1}}{1 - \frac{v_1}{v_3}}\right) <br />​

which, as I had mentioned earlier, has a typo corrected from what was given in Post #1 (v1/v3 instead of v3/v1).

 

[TFM]

Oops, my Latex went a bit wrong. it should have been:

\frac{P_1(v_1\frac{v_1^{\gamma - 1}}{v_3^{\gamma - 1}} - V_3)}{V_3(P_1 - P_3 \frac{V_3^{\gamma - 1}}{v_1^{\gamma - 1}})}