Efficiency

In this lesson you will learn how to calculate efficiency, interpret Sankey diagrams, and suggest ways to improve the efficiency of devices. This is part of AQA GCSE Combined Science Trilogy (8464), Section 6.1.

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What Is Efficiency?

Efficiency is a measure of how much of the total input energy (or power) is usefully transferred. No device is 100% efficient because some energy is always dissipated.

The Equations

$\text{Efficiency} = \frac{\text{useful output energy transfer}}{\text{total input energy transfer}}$Efficiency=total input energy transferuseful output energy transfer​

$\text{Efficiency} = \frac{\text{useful output power}}{\text{total input power}}$Efficiency=total input poweruseful output power​

Efficiency is a ratio with no units. It can be expressed as a decimal (0 to 1) or as a percentage (0% to 100%).

To convert a decimal to a percentage, multiply by 100.

Exam Tip: Read the question carefully to see whether it asks for a decimal or a percentage. AQA accepts both but you must be consistent.

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Worked Examples

Worked Example 1

A motor uses 800 J of electrical energy and does 600 J of useful mechanical work. Calculate its efficiency.

$\text{Efficiency} = \frac{600}{800} = 0.75 = 75\%$Efficiency=800600​=0.75=75%

Worked Example 2

An LED bulb has an input power of 10 W and produces 8.5 W of useful light output. What is its efficiency?

$\text{Efficiency} = \frac{8.5}{10} = 0.85 = 85\%$Efficiency=108.5​=0.85=85%

Worked Example 3

A filament bulb is 5% efficient and has an input power of 60 W. How much useful light power does it produce?

$\text{Useful output} = \text{Efficiency} \times \text{Total input} = 0.05 \times 60 = 3 \text{ W}$Useful output=Efficiency×Total input=0.05×60=3 W

Worked Example 4

A machine does 450 J of useful work. Its efficiency is 0.60. What was the total energy input?

$\text{Total input} = \frac{\text{useful output}}{\text{efficiency}} = \frac{450}{0.60} = 750 \text{ J}$Total input=efficiencyuseful output​=0.60450​=750 J

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Why Efficiency Cannot Exceed 100%

Because of the conservation of energy, the useful output can never be greater than the total input. Some energy is always dissipated as internal (thermal) energy. Therefore efficiency is always less than 1 (or less than 100%).

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Sankey Diagrams

A Sankey diagram is an arrow diagram that shows energy transfers, with the width of each arrow proportional to the amount of energy.

How to Read a Sankey Diagram

• The left arrow represents the total input energy.
• The arrow that continues to the right represents the useful output.
• The arrow(s) that branch downward (or upward) represent the wasted energy (dissipated).

graph LR
    A["Total input energy\n(400 J)"] --> B["Useful output\n(100 J)"]
    A --> C["Wasted energy\n(300 J)\n(internal store of surroundings)"]
    style C fill:#ffcccc,stroke:#cc0000

In this example: Efficiency = 100 / 400 = 0.25 = 25%.

Note: In a real Sankey diagram the widths are drawn to scale. In Mermaid we represent the concept diagrammatically — on paper, draw the arrow widths proportional to the energy values.

How to Draw a Sankey Diagram

1. Calculate the useful output and wasted energy.
2. Draw the input arrow on the left — its width represents total input energy.
3. Split the arrow: the useful output continues right, the wasted energy branches downward.
4. Make the widths proportional to the energy values.
5. Label each arrow with the energy value and store.

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Comparing Efficiencies

Device	Typical Efficiency
Electric heater	~100% (all energy becomes heat, which is useful)
LED bulb	80–90%
Electric motor	70–90%
Gas boiler (condensing)	90–95%
Petrol car engine	25–30%
Filament light bulb	5–10%
Coal power station	30–40%

Exam Tip: An electric heater can be close to 100% efficient because the "useful" output is heat, which is the same as the wasted output. AQA may test whether you understand this subtlety.

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Improving Efficiency

To improve efficiency, reduce the amount of energy that is dissipated:

Strategy	How It Helps
Lubricate moving parts	Reduces friction → less energy dissipated as heat
Use thermal insulation	Reduces heat loss to surroundings
Streamline moving objects	Reduces air resistance
Use more efficient components	e.g., LEDs instead of filament bulbs

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Efficiency Chains

When multiple devices are connected in series, the overall efficiency is the product of each individual efficiency.

$\text{Overall efficiency} = \eta_1 \times \eta_2 \times \eta_3 \times \ldots$Overall efficiency=η1​×η2​×η3​×…

Worked Example 5

A power station converts chemical energy to electricity at 40% efficiency. Transmission lines have 95% efficiency. What is the overall efficiency?

$\text{Overall} = 0.40 \times 0.95 = 0.38 = 38\%$Overall=0.40×0.95=0.38=38%

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Common Mistakes

Mistake	Correction
Calculating efficiency > 100%	Check your fraction — useful output should be the numerator
Confusing input and output	Input is the total energy going in; output is the useful energy coming out
Forgetting to convert percentage to decimal (or vice versa)	Multiply by 100 for %, divide by 100 for decimal
Thinking an electric heater wastes energy	If heating is the purpose, then the "wasted" heat is actually useful

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Summary

• Efficiency = useful output / total input (energy or power).
• Efficiency is a ratio (decimal 0–1 or percentage 0–100%).
• Efficiency can never exceed 100% due to conservation of energy.
• Sankey diagrams represent energy transfers with proportional arrow widths.
• Improve efficiency by reducing dissipation: lubrication, insulation, streamlining.
• Overall efficiency of a chain = product of individual efficiencies.
• This topic is assessed in AQA GCSE Combined Science Trilogy (8464), Section 6.1.

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Extended Worked Examples

Worked Example 6 — Deriving Wasted Energy from Efficiency

A pump is 65% efficient and consumes 4,000 J of electrical energy. How much energy is usefully transferred and how much is dissipated?

$E_{useful} = 0.65 \times 4000 = 2600 \text{ J}$Euseful​=0.65×4000=2600 J

$E_{wasted} = 4000 - 2600 = 1400 \text{ J}$Ewasted​=4000−2600=1400 J

The wasted 1,400 J ends up in the internal (thermal) store of the surroundings, mainly by heating of the pump windings and casing.

Worked Example 7 — Comparing Two Light Bulbs

A 60 W filament bulb is 5% efficient; a 10 W LED bulb is 85% efficient. Compare their useful light output.

Filament: $P_{useful} = 0.05 \times 60 = 3.0$Puseful​=0.05×60=3.0 W.

LED: $P_{useful} = 0.85 \times 10 = 8.5$Puseful​=0.85×10=8.5 W.

The LED uses one-sixth of the electrical power but produces almost three times the useful light.

Worked Example 8 — Chain of Devices

A wind turbine captures 35% of the kinetic energy of the wind. The generator is 95% efficient. Cables are 98% efficient. Calculate the overall efficiency from wind to grid.

$\eta_{total} = 0.35 \times 0.95 \times 0.98 = 0.326 = 32.6\%$ηtotal​=0.35×0.95×0.98=0.326=32.6%

Worked Example 9 — Reverse Problem

A motor is required to deliver 540 J of useful mechanical energy. It is 72% efficient. What is the minimum electrical input?