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

$$e = 1 - \frac{1}{\gamma}\left(\frac{1 - \frac{p_3}{p_1}}{1 - \frac{v_1}{v_3}}\right)$$​

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:

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