Cyclic processes

A cyclic process is a series of transformations that take the gas back to its original state. These form a closed loop on a \(pV\) diagram.


Key Concepts

Heat engine

A simple cyclic process involves:

  1. Isobaric expansion - work is done on the surroundings
  2. Isovolumetric cooling
  3. Isobaric compression - work is done on the gas
  4. Isovolumetric heating

Let's consider the cycle for a heat engine.

Since \(W=p\Delta V\), the area between a line on the \(pV\) diagram and the \(V\) axis gives work done. The net work done during a cyclic process is given by the enclosed area.

We can show the net work done using a Sankey diagram.

Refrigerator

A refrigerator completes the opposite process.

Essentials

Thermal efficiency

The dimensionless quantity, thermal efficiency, of any mechanical process is defined as the ratio of the useful work done to the total energy input:

\(\eta={\textrm{useful work done}\over \textrm{energy input}}\)

For a heat engine:

  • Energy input is \(Q_\textrm{H}\)
  • Work done is \(W_\textrm{out}=Q_\textrm{H}-Q_\textrm{C}\)

\(\eta={Q_\textrm{H}-Q_\textrm{C}\over Q_\textrm{H}}=1-{Q_\textrm{C}\over Q_\textrm{H}}\)

Carnot cycle

The most efficient cyclic process is known as the Carnot cycle. This involves:

  1. Isothermal expansion - work is done on the surroundings
  2. Adiabatic expansion - work is done on the surroundings
  3. Isothermal compression - work is done on the gas
  4. Adiabatic compression - work is done on the gas 

Note that you may be required to calculate the work done during any part of the process using the 'counting squares' method. This is a little arduous!

  • Calculate the size represented by one square using the bottom left square on the axes
  • Count the squares enclosed in the area required and multiply

To calculate the thermal efficiency we must consider entropy.

The total change in entropy, \(\Delta S=\Delta S_1+\Delta S_2+\Delta S_3+\Delta S_4\)

Since stages 2 and 4 are adiabatic, \(Q=0\) and so \(\Delta S _2 = \Delta S_4=0\)

Stages 1 and 3 take place at constant temperature, so \(\Delta S_1 = {Q_\textrm{H} \over T_\textrm{H}}\) and \(\Delta S_3 = {Q_\textrm{C} \over T_\textrm{C}}\)

Since the Carnot cycle is a reversible process, \(\Delta S = {Q_\textrm{H} \over T_\textrm{H}} - {Q_\textrm{C} \over T_\textrm{C}} = 0\)

\( {Q_\textrm{H} \over T_\textrm{H}} = {Q_\textrm{C} \over T_\textrm{C}} \Rightarrow {Q_\textrm{C} \over Q_\textrm{H}} = {T_\textrm{C} \over T_\textrm{H}}\)

Substituting into the general equation for thermal efficiency of a heat engine:

\(\eta_\textrm{Carnot}=1-{T_\textrm{cold}\over T_\textrm{hot}}\)

The equation for thermal efficiency for a Carnot cycle makes clear that we can approach but never attain 100% efficiency. To do so we would need to have a high temperature thermal energy store approaching infinity or a low temperature thermal energy sink approaching absolute zero:

  • If \(T_\textrm{hot} \rightarrow \infty\)\(\eta \rightarrow1\)
  • If \(T_\textrm{cold}\rightarrow 0 \textrm{K}\)\(\eta \rightarrow1\)

At the operating temperatures provided, the Carnot cycle is the most efficient process possible.

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