Thermodynamics and engines (AQA A-Level Physics): Revision Notes

Power and efficiency of an engine

Understanding how engines convert fuel into useful work is essential for analysing engine performance. When evaluating an engine, we consider different types of power measurements and efficiency indicators to assess how well the engine operates.

Types of power in engines

Engines involve several stages of energy conversion, each with its own power measurement. Three key power indicators help us understand engine performance at different stages of the energy conversion process.

Input power

Input power represents the rate at which energy is supplied to the engine through fuel combustion. It depends on two factors: the energy content of the fuel and how quickly the fuel is consumed.

The formula for input power is:

$P_{\text{input}} = \text{calorific value of fuel (J} \cdot \text{kg}^{-1}\text{)} \times \text{fuel flow rate (kg} \cdot \text{s}^{-1}\text{)}$

Input power is measured in watts (W), kilowatts (kW), or megawatts (MW).

The calorific value of a fuel measures its energy density, indicating how much energy is released per kilogram of fuel burned.

Input power varies depending on several factors including engine type, vehicle speed (which affects air resistance), driving style, and tyre condition. It represents the total energy being consumed by the engine per unit time.

Indicated power

Indicated power represents the theoretical maximum power output that an engine could deliver, assuming no frictional losses. This measurement is based on the work done in each engine cycle, as shown by the area enclosed in a pressure-volume (p-V) diagram.

The formula for indicated power is:

$P_{\text{ind}} = \text{area of p–V loop (J)} \times \text{number of cycles per second (s}^{-1}\text{)} \times \text{number of cylinders}$

This calculation assumes frictionless motion and represents an idealized upper limit for engine performance. To find the area of the p-V loop, you may need to count squares on a graph and multiply by a scaling factor that converts the graph area into joules.

Output power (brake power)

Output power, also called brake power, measures the actual power delivered to the engine's crankshaft (the engine's flywheel). This is the useful power available after accounting for losses from the gearbox, alternator, water pump, and other auxiliary components.

Output power can be calculated using:

$P_{\text{out}} = T\omega$

where:

• $T$ is the output torque in newton-metres (N m)
• $\omega$ is the angular velocity in radians per second (rad s⁻¹)

When working with engines, angular velocity is often given in revolutions per minute (rpm), which needs conversion to rad s⁻¹ using:

$\omega = \frac{\text{rpm} \times 2\pi}{60}$

For example, an engine idling at 800 rpm has an angular velocity of:

$\omega = \frac{800 \times 2\pi}{60} = 84 \text{ rad s}^{-1}$

Therefore, a car rated at 250 bhp has an output power of:

$250 \times 746 \text{ W} = 187 \text{ kW}$

Frictional power

Frictional power represents the power lost due to friction in the engine and transmission system. It is the difference between the theoretical maximum power (indicated power) and the actual power delivered (output power).

The formula for frictional power is:

$P_{\text{friction}} = P_{\text{ind}} - P_{\text{out}}$

or equivalently:

$P_{\text{friction}} = \text{indicated power} - \text{output (brake) power}$

This lost power is dissipated as heat through various frictional processes occurring within the engine components and the transmission system.