🔥 Filament lamp
The useful arrow is a thin strip only 1 square wide. The wasted arrow is 19 squares wide, so nearly the whole input arrow curves away. Efficiency = (5 ÷ 100) × 100% = 5%.
Energy Transfers » Sankey Diagrams
What you'll learn this session
Study time: 30 minutes
Edexcel spec: 4.5
- How a Sankey diagram shows where the input energy of a device goes
- How to read energy values from a Sankey diagram
- How to draw a Sankey diagram to scale on squared paper
- How to find the efficiency of a lamp, a motor, a car engine and a power station from a diagram
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Unlock This CourseWhat a Sankey diagram shows
In Conservation of Energy and Efficiency you split the energy supplied to a device into useful energy and wasted energy. A Sankey diagram is a picture of that split. It is one wide arrow that divides into smaller arrows, and it lets you see at a glance how efficient a device is.
The one rule that matters is this: the width of each arrow is proportional to the amount of energy it represents. An arrow that is twice as wide stands for twice as much energy. The length of an arrow means nothing at all.
Here is how to picture one. It is read from left to right.
- A wide arrow comes in from the left. This is the input arrow. Its width shows the total energy input.
- Part of the arrow carries straight on to the right. This is the useful arrow. Its width shows the useful energy output.
- The rest peels off and curves away, usually downwards. This is the wasted arrow. Its width shows the wasted energy. If energy is wasted in two ways, there are two wasted arrows.
Energy is conserved, so no width can appear or vanish. The widths of all the output arrows always add up to the width of the input arrow.
Key terms:
- Sankey diagram: an energy transfer diagram in which the width of each arrow is proportional to the energy it represents.
- Input arrow: the single wide arrow on the left that shows the total energy input.
- Useful arrow: the arrow that carries straight on and shows the useful energy output.
- Wasted arrow: an arrow that peels off to the side and shows wasted energy.
- Scale: how much energy each square (or each millimetre) of width stands for.
Comparing two lamps
A lamp is supplied with energy electrically. The useful output is light. The wasted energy heats the lamp and the room. The figures below are typical for the two kinds of lamp.
| Lamp | Total energy input | Useful energy output (light) | Wasted energy (heating) |
|---|---|---|---|
| Filament lamp | 100 J | 5 J | 95 J |
| LED lamp | 100 J | 40 J | 60 J |
Now picture both Sankey diagrams drawn with the same scale, 1 square = 5 J. Each input arrow is 20 squares wide.
💡 LED lamp
The useful arrow is 8 squares wide and the wasted arrow is 12 squares wide. Much more of the arrow carries straight on. Efficiency = (40 ÷ 100) × 100% = 40%.
This is the big strength of a Sankey diagram. You do not need to read a single number to see which lamp is more efficient. The wider the straight-on arrow compared with the input arrow, the more efficient the device.
Drawing a Sankey diagram to scale
The electric motor of a garage door is supplied with 400 J. It transfers 280 J usefully to lift the door. It wastes 100 J heating the motor and the air, and 20 J as sound.
Worked example 1: drawing the diagram
- Check the numbers. 280 + 100 + 20 = 400 J. The outputs add up to the input, so the diagram will work.
- Choose a scale. Pick one that gives easy widths. With 1 square = 40 J, the input arrow is 400 ÷ 40 = 10 squares wide.
- Work out every width. Useful: 280 ÷ 40 = 7 squares. Heating: 100 ÷ 40 = 2.5 squares. Sound: 20 ÷ 40 = 0.5 squares. Check: 7 + 2.5 + 0.5 = 10 squares.
- Draw the input arrow. On the left of the squared paper, draw two horizontal lines 10 squares apart.
- Draw the useful arrow. Carry the top 7 squares straight on to the right and finish with an arrowhead.
- Draw the wasted arrows. Split off the bottom 3 squares and curve them downwards as two arrows, one 2.5 squares wide and one 0.5 squares wide. Keep each arrow the same width all the way along.
- Label every arrow with what it is and its energy: 'electrical input 400 J', 'useful, lifting the door 280 J', 'wasted, heating 100 J', 'wasted, sound 20 J'. Write the scale next to the diagram.
Reading values from a Sankey diagram
Exam diagrams are often drawn on a grid with only one value given. Find the scale first, then count squares.
Worked example 2: a car engine
A Sankey diagram for a petrol car engine has an input arrow 10 squares wide, labelled 1000 J. The straight-on arrow, labelled 'useful, to the kinetic store of the car', is 3 squares wide. One wasted arrow curves downwards, labelled 'heating the engine, the exhaust gases and the surroundings'.
Scale: 1000 J ÷ 10 squares = 100 J for each square
Useful energy output = 3 × 100 = 300 J
Width of wasted arrow = 10 − 3 = 7 squares, so wasted energy = 7 × 100 = 700 J
Efficiency = (useful energy output ÷ total energy output) × 100% = (300 ÷ 1000) × 100% = 30%
If the diagram is not on a grid, measure the widths with a ruler in millimetres and use them in the same way, for example 1 mm = 20 J.
You can even find the efficiency without knowing any energy values, because the widths are in the same proportion as the energies.
Worked example 3: a power station
A Sankey diagram for a power station that burns fuel has an input arrow 20 squares wide. The straight-on arrow, labelled 'useful, transferred electrically', is 7 squares wide. The wasted arrow, labelled 'heating the surroundings', is 13 squares wide.
Efficiency = (width of useful arrow ÷ width of input arrow) × 100%
Efficiency = (7 ÷ 20) × 100% = 0.35 × 100% = 35%
So for every 1000 J from the fuel, 350 J is transferred usefully and 650 J is wasted.
How power stations work is covered later, in Electricity from Sun, Fossil Fuels and Nuclear.
The devices side by side
| Device | Total energy input | Useful energy output | Wasted energy | Efficiency |
|---|---|---|---|---|
| Filament lamp | 100 J | 5 J (light) | 95 J (heating) | 5% |
| LED lamp | 100 J | 40 J (light) | 60 J (heating) | 40% |
| Garage door motor | 400 J | 280 J (lifting the door) | 120 J (heating and sound) | 70% |
| Petrol car engine | 1000 J | 300 J (kinetic store of the car) | 700 J (heating) | 30% |
| Power station | 1000 J | 350 J (transferred electrically) | 650 J (heating) | 35% |
Each row is a Sankey diagram written as numbers. In every row, useful + wasted = input. If an exam question gives you a table like this, each energy value becomes one arrow: the input, the useful output and each wasted amount. The efficiency is not an arrow.
Common mistakes
- Using the length of an arrow. Only the width counts. A long thin arrow shows less energy than a short wide one.
- Output widths that do not add up. The output arrows together must be exactly as wide as the input arrow. Check by adding before you draw.
- Arrows that change width. Each arrow stays the same width along its whole length, even round the curve.
- Mixing up the arrows. Useful energy goes straight on. Wasted energy peels off to the side.
- No labels. An unlabelled arrow can lose marks. Label every one with what it is, and its energy value if you know it.
- Counting squares along the arrow. Count squares across the arrow, at right angles to the way it points.
Exam-style question
A student draws a Sankey diagram on squared paper for an electric leaf blower. The input arrow is 20 squares wide and is labelled 'energy supplied electrically, 2000 J'. Three arrows leave the diagram. The straight-on arrow is 12 squares wide and is labelled 'kinetic store of the moving air'. A second arrow, 7 squares wide, curves downwards and is labelled 'heating the blower and the air'. A third, narrower arrow curves downwards and is labelled 'sound'.
(a) State what the width of an arrow on a Sankey diagram represents. [1 mark]
(b) Calculate the useful energy output of the leaf blower. [2 marks]
(c) Determine the energy wasted as sound. [2 marks]
(d) Calculate the efficiency of the leaf blower. [3 marks]
(e) A second leaf blower is supplied with the same energy but is more efficient. Describe two ways its Sankey diagram would be different, when drawn to the same scale. [2 marks]
Model answer
(a) The amount of energy transferred (the width is proportional to the energy) (1).
(b) Scale: 2000 ÷ 20 = 100 J for each square (1). Useful energy output = 12 × 100 = 1200 J (1).
(c) Width of sound arrow = 20 − 12 − 7 = 1 square (1). Energy wasted as sound = 1 × 100 = 100 J (1).
(d) efficiency = (useful energy output ÷ total energy output) × 100% (1) = (1200 ÷ 2000) × 100% (1) = 60% (1).
(e) The straight-on (useful) arrow would be wider (1). The wasted arrows would be narrower, with the input arrow the same width (1).
Exam tip
In part (c) the diagram does not tell you the width of the sound arrow, and you do not need it. The three output arrows must add up to the 20 squares of the input arrow, so subtract the two widths you know. Then check your answers: 1200 + 700 + 100 = 2000 J.