Interpret and complete quantitative Sankey diagrams
Use a stated common arrow-width scale and balance all branches of an energy transfer.
Section P1.7.1 · Energy
Cross-referenced to the 2028–2029 0653 section using original teaching from the related single-science courses. A 0653-specific objective-by-objective and Core/Supplement review is in progress. Written practice is self-checked and does not demonstrate supervised laboratory competence or award an official Cambridge grade.
Before you begin
- Balance input and distinct outputs.
- Use proportional lengths on one stated scale.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
What other output completes 200 J input and 120 J useful output?
What you will learn
- Interpret proportional branch widths.
- Find a missing output and check a consistent energy scale.
Widths represent amounts on a common scale
A Sankey diagram represents energy transfers with arrow widths proportional to the transferred amount, or power if all labels consistently describe a rate. In the original energy example, 200 J input has a width of 100 drawing units, useful 120 J has width 60, and other 80 J has width 40. The scale is 0.50 drawing unit per joule; 60 + 40 = 100 and 120 + 80 = 200.
Measure across an arrow rather than along its length: a long route through a device does not imply a greater energy amount. At a split, the input width equals the sum of output widths when the drawing uses the same scale and the account includes every relevant destination and no omitted store change.
Check numbers and geometry together
If 300 J input divides into 210 J useful output and one other branch, that branch is 90 J. At 0.40 drawing unit per joule the widths are 120, 84 and 36 units. Using a different scale on the unwanted branch would make the geometry misleading even if the written totals happen to balance.
A branch called wasted still transfers conserved energy, often into more dispersed internal stores. Useful output depends on the stated task. Joule amounts alone do not determine a transfer rate, and the lengths of bent arrows do not encode time. A Sankey is an energy account, not a claim that its arrows are physical trajectories of objects or that energy is destroyed at a downward branch.
Sources for this lesson
Pause and explain
Does a longer Sankey arrow necessarily carry more energy?
Worked example
A 200 J input splits into useful 120 J and one other branch. The scale is 0.50 drawing unit per joule. Find the remaining amount and all widths.
Show the worked solution
- Other amount = 200 − 120 = 80 J.
- Widths are 200 × 0.50 = 100, 120 × 0.50 = 60 and 80 × 0.50 = 40.
- Both amount and width sums balance.
Answer 80 J; widths 100, 60 and 40 drawing units.
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Choose Easy, Normal or Difficult. Save your answers and open worked review after finishing.
Test this topic →From guided practice to a new situation
Write your answer and reasoning first. Use a hint only if you are stuck. The model solution is for self-checking; your written work is not automatically marked.
Complete a 200 J input split with useful 120 J, at 0.50 drawing unit per joule.
Give me a hint
Subtract the amount and use the same width factor three times.
Compare my reasoning
- Other output = 80 J.
- Input width = 100 and useful width = 60.
- Other width = 40; outputs total input width 100.
80 J; widths 100, 60, 40.
Look for these in your work
- Balance amounts.
- Balance widths on the common scale.
A 300 J input produces useful 210 J and one other output. Find the remaining output and widths at 0.40 drawing unit per J.
Give me a hint
Use the exhaustive account and a single proportional scale.
Compare my reasoning
- Remaining output = 90 J.
- Widths for input and useful output are 120 and 84.
- Other width is 36, and 84 + 36 = 120.
90 J; widths 120, 84 and 36 drawing units.
Look for these in your work
- Use one scale.
- Check the split quantitatively.
A 200 J input is drawn 100 units wide, useful 120 J is 60 wide, but other 80 J is 80 wide. Identify the error even though the labels add to 200 J.
Give me a hint
Compare each width divided by its labelled amount.
Compare my reasoning
- Input and useful branches use 0.50 unit per J.
- The other branch uses 1 unit per J instead.
- It should be 40 units wide; 60 + 40 would balance the input width.
The other branch has a different scale; correct its width to 40 drawing units.
Look for these in your work
- Check geometric as well as numerical consistency.
- Use cross-arrow width rather than route length.
Common mistakes
- Use width, not arrow length.
- Use one scale for all branches.
- A wasted branch does not destroy energy.
What can you explain now?
At 0.40 drawing unit per J, how wide is a 90 J branch?
Compare with the explanation
36 drawing units.
Multiply the energy amount by the common width scale.
After trying it yourself, choose your next review. This is your self-assessment.
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Make it stick
Explain the Sankey width scale, balance a missing branch and identify a drawing whose labels balance but widths do not.
If you needed the solution, close it and solve the problem again from a blank page. A correct answer is stronger when you can explain the reason for each step.