Physics · Molecular physics and thermodynamics · Grade 9

Heat engine: the Otto cycle

What fraction of the petrol's heat turns the wheels, and why not all of it?

The situation

You filled the car's tank with petrol. Only part of the heat released when petrol burns turns the wheels — the rest escapes into the air through the radiator and the exhaust gases. Inside the engine four strokes repeat dozens of times a second: intake, compression, power and exhaust. Slow a cut-away engine down 50 times and watch: why does the efficiency rise when the compression ratio rises, and why can even an ideal engine not turn all the heat into work?

Topic

Laws of thermodynamics, heat engines. Car engine: the Otto cycle and efficiency

Key ideas

First and second laws of thermodynamics
p–V diagram and work
Efficiency and the Carnot limit

Formulas

Q = Δ U + A

The first law of thermodynamics: the heat given to a gas increases its internal energy and does work.

p V^{γ} = const, T V^{γ - 1} = const

Adiabatic compression and expansion (no heat exchange). $γ = c_{p} / c_{V}$ , for air ≈ 1.4.

η_{O t t o} = 1 - \frac{1}{r^{γ - 1}}, r = \frac{V_{1}}{V_{2}}

The efficiency of the ideal Otto cycle depends only on the compression ratio $r$ and on $γ$ .

η_{K} = 1 - \frac{T_{m i n}}{T_{m a x}}

The Carnot efficiency is the upper limit for any engine working between these temperatures.

A = \oint p d V = Q_{1} - Q_{2}, P = A \frac{n}{2}

The work of one cycle is the area of the closed curve on the p–V diagram. In a four-stroke engine one cycle takes 2 revolutions of the crankshaft ( $n$ is in rev/s).

Theory

The four strokes

Intake: the intake valve is open and the piston moves down, drawing in the fuel–air mixture. Compression: the valves are closed and the mixture is compressed $r$ times and heats up. When the piston is at top dead centre the spark plug fires: the mixture burns almost instantly and the pressure and temperature rise sharply. Power stroke: the hot gas expands and pushes the piston — only in this stroke does the gas do work. Exhaust: the exhaust valve opens and the spent gas is pushed out.

p–V diagram and work

During compression the piston does work on the gas, and during expansion the gas does work on the piston — because the expansion curve is higher, positive work remains. It equals the area bounded by the closed curve. The intake and exhaust lines (p ≈ 100 kPa) coincide and cancel each other. The crankshaft and flywheel carry the energy gained in the power stroke through the other three strokes.

Why does the compression ratio matter?

Compressed gas is hotter, so heat is supplied at a higher temperature: $η = 1 - r^{1 - γ}$ rises. But if $r$ is too large, the petrol–air mixture ignites by itself without a spark (knocking) — that is why $r$ is limited in petrol engines but large in diesels.

The second law and the Carnot cycle

Every heat engine has three parts: a heater (here the burning petrol, $T_{m a x}$ ), a working substance (the gas in the cylinder) and a cooler (the atmosphere, $T_{m i n}$ ). The working substance takes $Q_{1}$ from the heater, does work $A$ and gives the remaining $Q_{2} = Q_{1} - A$ to the cooler. The second law (Thomson's, or Lord Kelvin's, statement): $Q_{2} = 0$ is impossible — there is no perpetual motion machine of the second kind that turns heat taken from a single source completely into work. It would not violate energy conservation — for example, a ship driven by the heat of ocean water — but it violates the second law. The Carnot cycle is a reversible cycle of two isotherms and two adiabats. Carnot's theorem: all reversible engines working between $T_{m a x}$ and $T_{m i n}$ have the same efficiency, $η_{K} = 1 - T_{m i n} / T_{m a x}$ ; an irreversible engine has less. In the Otto cycle heat is supplied over the range $T_{2} \dots T_{3}$ and removed over $T_{4} \dots T_{1}$ , so its efficiency is always lower than the Carnot value. The model is the ideal air cycle: $γ$ is constant, combustion is instantaneous, and there is no friction or heat transfer to the walls; the efficiency of real petrol engines is about 25–35 %.

The T–S diagram and entropy

In a reversible process the fundamental equation of thermodynamics is $T d S = d U + p d V$ , that is, $δ Q = T d S$ . So on a T–S diagram the area under a curve is heat, and the area of a closed cycle is the work of the cycle: $A = \oint T d S$ . Adiabats ( $Q = 0$ ) are vertical lines here, and the Carnot cycle is a rectangle. For the same change of entropy the Otto cycle stays inside the Carnot rectangle — this shows why its efficiency is lower. For an irreversible cycle $\oint δ Q / T < 0$ (the Clausius inequality): friction and heat losses generate entropy and reduce the efficiency. For $T_{m i n} = 0$ K we would have $η_{K} = 1$ , but by the third law absolute zero cannot be reached. The statistical meaning of entropy — in the “Entropy and the arrow of time” experiment.

Heat engines and the environment

The lower heating value of petrol is about 43 MJ/kg. If the efficiency is $η$ , then to do 1 kW·h (3.6 MJ) of work $m = 3.6 / (43 η)$ kg of petrol is burned, and burning 1 kg of petrol produces about 3.1 kg of CO₂ (see the panel). In a real petrol engine ( $η \approx 0.3$ ) this is about 280 g of petrol and 860 g of CO₂. CO₂ is a greenhouse gas; the exhaust gases also contain carbon monoxide (CO), nitrogen oxides and soot — a catalytic converter reduces them. The $Q_{2}$ given to the cooler heats the surroundings (thermal pollution). That is why raising efficiency, saving fuel and public transport are important ways to protect nature. If we run the cycle in reverse, then, at the expense of external work, heat is moved from a cold body to a hot one — this is a refrigerator and a heat pump (in the “Refrigerator and heat pump” experiment).

Did you know?

In 1876 the German engineer Nikolaus Otto built the first practical four-stroke internal combustion engine — which is why this cycle bears his name.
In modern petrol engines the compression ratio is about 9–12. Above that, the mixture ignites by itself without a spark and knocking begins — that is why high-compression engines need high-octane petrol.

Inside the experiment

The experiment starts from a real-life situation: you predict the result, change the controls, run the experiment and draw a conclusion. Inside are missions, theory, a quiz and a guided tour in which the camera moves and explains the key points.

More experiments in this section

Melting, boiling and evaporation — Grade 7
Calorimeter: heat balance — Grade 7
Tandoor and sandal: heat — Grade 7
Gas under a piston — Grade 9
Humidity and dew — Grade 9
Entropy and the arrow of time — Grade 9
Brownian motion and diffusion — Grade 9
Surface tension and capillarity — Grade 9
Crystalline and amorphous solids — Grade 9
Phase transitions — Grade 10