What Is a Sankey Diagram Used For?

At both SL and HL, a Sankey diagram is used to represent energy transfers in a system. The width of each arrow is proportional to the amount of energy transferred, allowing useful and dissipated energy to be compared visually.

How a Sankey Diagram Works

A Sankey diagram applies the principle of conservation of energy from Topic A.3, Work, energy and power. The total energy entering the system must equal the total energy leaving it:

$$E_{\text{input}} = E_{\text{useful}} + E_{\text{dissipated}}$$

The main arrow shows the input energy. It then divides into arrows representing different energy transfers.

For example, suppose a motor receives $500\ \text{J}$ of electrical energy. It transfers $350\ \text{J}$ into useful kinetic energy and dissipates $150\ \text{J}$ as thermal energy and sound. The useful arrow should therefore occupy 70% of the original width, while the dissipated arrow occupies 30%.

To draw this example to scale, choose a convenient conversion such as $1\ \text{cm}$ representing $100\ \text{J}$. The input arrow is then $5.0\ \text{cm}$ wide, the useful output is $3.5\ \text{cm}$ wide, and the dissipated output is $1.5\ \text{cm}$ wide. Because , the widths visibly confirm conservation of energy.

The diagram can also be used to calculate efficiency:

$$\eta = \frac{E_{\text{useful}}}{E_{\text{input}}} = \frac{350}{500} = 0.70 = 70\%$$

A common misconception is that the sideways arrow represents energy that has been “lost.” Energy is not destroyed; it is dissipated to the surroundings and becomes less useful.

Exam Technique

In an IB response, ensure arrow widths are proportional and satisfy energy conservation. If asked to calculate efficiency from a Sankey diagram, read the useful and total input values carefully, show the substitution, and express the answer as either a decimal or percentage.