The Physics
Opus in profectus


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The word "engine" is a bit old fashioned. At one time, an engine was any kind of mechanical contrivance. For example…

With the exception of the last example, the original meaning of the word engine is now pretty much obsolete. This can be traced back to the development of the steam engine in the middle of the 19th century. Nowadays, an engine generally refers to a device that transforms heat into mechanical energy. Technically, such a device is a heat engine, but in the current era, the adjective "heat" is generally dropped.

Although the two devices are often confused, an engine is not the same as a motor. An electric motor (often just called a motor) is a device for transforming electric energy into mechanical energy.

What about compressed air motors?

This gives rise to several linguistic problems. Why are automobiles sometimes called "motorcars"? Many cars do have motors in them, but the device responsible for propelling them isn't a motor. It's an engine. At the dawn of the 21st century cars use motors to drive the wiper blades, open and close the windows, adjust the seat and side view mirrors, and spin CDs; but they only use an engine to drive the wheels (although this is likely to change). The largest auto maker in the US may be called General Motors, but what they really sell are automobiles driven by engines.

The description of an engine is simple, but it refers to a wide variety of different devices. Engines can be found in cars, trucks, motorcycles, planes, boats, ships, trains, lawnmowers, chain saws, model airplanes, portable generators, cranes, augers, drills, and rockets (to give but a few examples). Engines can be classified according to one or more of several schemes.

otto cycle

Start with piston diagrams.


Describe each part of the cycle

Intake Stroke

  • intake valve open
  • piston moving down
  • volume increases
  • constant pressure (atmospheric)
  • temperature decreases
  • isobaric
  • "suck" in air and fuel (via carburetor or fuel injector)

Compression Stroke

  • both valves closed
  • piston moving up
  • volume decreases
  • pressure increases
  • temperature increases
  • adiabatic
  • "squeeze" the fuel air mixture, but keep it below the ignition temperature to prevent preignition


  • both valves closed
  • piston at topmost position
  • pressure increases
  • constant volume
  • temperature increases
  • isochoric
  • "bang" goes the sparkplug

Power Stroke

  • both valves closed
  • piston moving down
  • pressure decreases
  • volume increases
  • temperature decreases
  • adiabatic
  • "push" the piston down as the fuel air mixture expands outward
  • one useful stroke out of four

Valve Exhaust

  • exhaust valve open
  • piston at bottommost position
  • pressure drops to atmospheric
  • constant volume
  • temperature decreases
  • isochoric
  • "pop" open the exhaust valve

Exhaust Stroke

  • exhaust valve open
  • piston moving up
  • constant pressure (atmospheric)
  • volume decreases
  • temperature increases
  • isobaric
  • "blow" the combustion products out of the cylinder

pressure-volume diagrams

Then move on to PV diagrams (indicator diagrams).

James Watt called them indicator diagrams rather than pressure-volume graphs since the word graph hadn't been invented yet.

A steam-indicator is a device for plotting the pressure in the cylinder of a steam engine as a function of the phase of the engine's working cycle. The shape of this diagram reveals possible faults of the machine. With a planimeter one can determine the mean effective steam pressure in the engine or, when the stroke and diameter of the cylinder and the number of revolutions per minute are known, the power of the engine,

A history lesson from The Canadian Museum of Making

Watt & Southern, c. 1796 The indicator was soon adapted to provide a written record of each individual application instead of merely a transient observation. This was a tremendous analytical breakthrough, allowing, as it did, an accurate picture to be formed of the pressure of steam at any time during the movement of the piston. The inspiration was due to John Southern (1758-1815), Watt's draughtsman, who recorded in a letter dated 14th March 1796 that he had 'contrived an instrument that shall tell accurately what power any engine exerts'.

Annotated line graph

  1. intake stroke
    The piston starts out at the top of the cylinder and moves downward. The volume above the piston expands, drawing air and fuel into the combustion chamber. Since the intake valve is open during this stroke, the pressure inside the cylinder will be constant and roughly equal to the pressure of the environment. This segment of the cycle is thus represented by a horizontal line (an isobar) running from left to right (from minimum to maximum volume). Assuming the amount of gas within the cylinder increases in proportion to its volume, there is effectively no change in its temperature.
  2. compression stroke
    Both valves are closed and the gas is squeezed rapidly inside the cylinder. Since the process is rapid, there is essentially no time for heat to be lost to the environment. This segment of the cycle is thus represented by an adiabat running from right to left. Since adiabats are steeper than isotherms, this segment crosses a few isotherms; which is in agreement with our expectation that the temperature should increase. Since the curve runs "backward", the area under it is negative; that is, work is being done on the gas.
  3. ignition
    Igniting the fuel-air mixture inside the cylinder does two obvious things: it raises the temperature and it raises the pressure above the piston. It occurs so rapidly, however, that there isn't a lot of time for the piston to react and the volume above it is effectively constant. This segment of the cycle is thus represented by a vertical line (an isochor) running from bottom to top. The curve crosses several isotherms (since the fuel-air mixture's heat of combustion is dumped into the system), but no work is done on or by the gas (since the area under a vertical line is zero).
  4. power stroke
    The piston is pushed down by the intense pressure following ignition. Volume increases but it happens so quickly that the heat lost to the environment is minimal. This segment of the cycle is thus represented by an adiabat running from left to right. Since this curve runs "forward" the area under it is positive and work is done on the environment. That's why it's called the "power stroke". This is the only part of the cycle that results in any useful work being done. Since the curve of the power stroke is higher than the curve of the compression stroke, the net work of the gas is positive. Thus (overall) each cycle does work on the environment. This makes sense as engines are a devices for doing work. In addition, the curve crosses several isotherms. Closer inspection of the indicator diagram shows that more isotherms are crossed during the power stroke than were crossed during the compression stroke. This is an important observation that sets us up for the next part of the cycle.
  5. valve exhaust
    The valve is popped open at the end of the power stroke, which reduces the pressure inside the cylinder to the pressure of the environment. It occurs so rapidly, however, that the volume above the piston does not effectively change. This segment of the cycle is thus represented by a vertical line (an isochor) running from top to bottom. Since the graph is vertical, no work is done on or by the gas at this time. The line does cross several isotherms, however, which means that internal energy is decreasing. Since no work is done, this decrease must be as a result of heat exhausted to the environment. In accordance with the work-energy theorem, this heat lost is less than the heat gained during combustion (the difference between them being equal to the net work done by the gas).
  6. exhaust stroke
    The last part of the cycle returns the system to its initial conditions. Since the exhaust valve is open during this portion of the cycle, the pressure inside the cylinder is effectively equal to that of the environment, but since the piston is moving upward, the volume above it decreases. This segment of the cycle is thus represented by a horizontal line (an isobar) running from right to left (from maximum to minimum volume). Assuming the amount of gas within the cylinder decreases in proportion to its volume, there is effectively no change in its temperature. In addition, the area under this segment is equal and opposite the area under the intake stroke. Thus (in the ideal world, at least) no work is done on or by the gas as it is drawn into and forced out of the cylinder.

the essence of engines

And lastly, the essence of engines.

The waterwheel-heat engine analogy
aspect waterwheel heat engine
working fluid water heat (caloric)
gradient height difference between
high and low water reservoirs
temperature difference between
hot and cold thermal reservoirs

Add bridge text.

Carnot diagram of engines with increasing efficiency

Efficiency in general vs. efficiency in engines. What symbol would you prefer: η [eta] or ℰ [script capital e] or e [lowercase e]?

efficiency =  work out
energy in
efficiency =  Qhot − Qcold
efficiency =  1 −  Qcold
ηreal = 1 −  QC
ηideal = 1 −  TC

Efficiency applies to more than just engines.

Selected efficiencies (smallest to largest)
efficiency (%) device, event, phenomenon, process
2 human, swimming, surface
4 human, swimming, underwater
3 human, shoveling
17 steam engine
25 US electrical system, 1950
25 human, cycling
30 gasoline engine, typical
35 US electrical system, 2020
40 diesel engine, typical
40 natural gas turbine
40–50 human, running
41 gasoline engine, greatest
50 diesel engine, greatest
50 muscle contractions
60 fuel cell
80 electric motor

other cycles