- #1

Toby_phys

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1. From an initial state (##p_1##, ##V_1##) the gas is cooled at constant pressure to (##p_1##, ##V_2##); Let's call the start and end temperature ##T_1## and ##T_2##

2.The gas is heated at constant volume to (##p_2##, ##V_2##);Lets call the start and end temperature ##T_2## and ##T_3##

3.The gas expands adiabatically back to (##p_1##, ##V_1##). Let's call the start and end temperature ##T_3## and ##T_1##

Assuming constant heat capacities, show that the thermal efficiency η is

$$

\eta=1-\gamma\frac{V_2/V_1 -1}{p_2/p_1-1}

$$Efficiencey is defined as: $$\eta=\frac{W}{Q_h}$$ the work done over the heat entered. The heat enters at stage 2 (and some leaves at stage 1 but that doesn't matter). So I need to find the heat entered at stage 2 and the work done.

**Stage 1:**

From the ideal gas equation we get:

$$

p_1V_1=nRT_1, \ \ \ \ p_2V_2=nRT_2 \implies \frac{T_2}{T_1}=\frac{V_2}{V_1}

$$

The work done is just force times distance which is pressure times change in volume:

$$

\Delta W=-p_1\Delta V=-p_1(V_2-V_1)

$$

**Stage 2:**

It doesn't change in volume and so no work is done. However heat is put into the system, increasing the pressure. We need to find this heat.

##\Delta U= Q_h##

For an ideal gas we have:

$$

\Delta U= C_v\Delta T=C_v(T_3-T_2)

$$

Where ##C_v## is heat capacity at constant volume.

**Stage 3:**

Stage 3 is adiabatic so ##\Delta U=\Delta W=C_v(T_1-T_3)##

We also have, using the ideal gas law:

$$

T_3=\frac{p_2V_2}{p_1V_1}T_1

$$Let us sub this into the efficiency:

$$

\eta=\frac{C_v(T_1-T_3)-p_1(V_2-V_1)}{C_v(T_3-T_2)}

$$

If we get ##T_3## and ##T_2## in terms of ##T_1## and sub these in we get:

$$

\eta=-1-\frac{p_1(V_2-V_1)}{C_vT_1(\frac{p_2V_2}{p_1V_1}-\frac{V_2}{V_1})}

$$

And with the ideal gas law, with ##n=1## for simplicity we get ##T_1=\frac{p_1V_1}{R}##

$$

\implies\eta=-1-\frac{R(V_2/V_1-1)}{C_vT_1(\frac{p_2V_2}{p_1V_1}-\frac{V_2}{V_1})}

$$

##R=C_p-C_v## and ##\gamma=C_p/C_v##

$$

\implies\eta=-1-\frac{(\gamma-1)(V_2/V_1-1)}{C_vT_1(\frac{p_2}{p_1}-1)\frac{V_2}{V_1}}

$$

I have no real clue really if this is right or wrong.