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Engines and cycles

Heat and matter · level 8, lesson 4 of 6

A question to start

A Stirling engine on a Mars reactor takes the same gas round the same loop, again and again. How much work does it get from each trip round the loop?

The idea

An engine takes a gas through a cycle: it heats, expands, cools and is squeezed, and ends where it started. Drawn on a graph of pressure \(p\) against volume \(V\), the cycle is a closed loop.

The work the gas does as it expands is the area under the curve; the work done on it as it is squeezed back is the area under the return path. So the net work per cycle is the area inside the loop. Going round clockwise, it is work out: an engine.

300 kPa100 kPa1 L2 L work per cycleVp

Real engines use smooth curves rather than a rectangle, and the efficiency is the net work divided by the heat taken in. The Carnot limit, \(1 - T_c/T_h\), caps it for any cycle.

Worked examples

1. The rectangle. Height \(300 - 100 = 200\) kPa, width \(2 - 1 = 1\) L \(= 0.001\) m³. \(W = 200{,}000 \times 0.001 = 200\) J per cycle.

2. Power. At 30 cycles a second: \(200 \times 30 = 6{,}000\) W, 6 kW.

3. Efficiency. Suppose the gas takes in 800 J of heat per cycle. Efficiency \(= 200/800 = 25\%\), the figure the energy topic assumed for a reactor's engines.

Try it

  1. What does the area inside a cycle on a \(p\)–\(V\) diagram tell you?
  2. A rectangular cycle runs between 50 and 250 kPa and between 2 and 5 litres. Find the work per cycle.
  3. What is the work if the gas goes round the loop anticlockwise?
  4. An engine does 150 J per cycle from 600 J of heat. Find its efficiency and the heat it rejects.
Answers
  1. The net work done by the gas in one cycle.
  2. \(200{,}000 \times 0.003 = 600\) J.
  3. The same size, but done on the gas: it is then a fridge or heat pump, not an engine.
  4. \(150/600 = 25\%\); it rejects \(600 - 150 = 450\) J.

Common mistake

Using litres or kPa in \(W = p\Delta V\). 1 litre is \(10^{-3}\) m³ and 1 kPa is 1,000 Pa. In SI units the answer comes out in joules.

Where it leads