Sankey Diagrams
A Sankey diagram is a specialised flow diagram that uses arrows — called 'flows' — whose widths are proportional to the quantity they represent. In physics, they are used to track how energy moves through a system, showing at a glance how much useful energy is produced and how much is wasted as heat, sound, or other forms of degraded energy. The total width of all output arrows always equals the total width of the input arrow, visually enforcing the principle of conservation of energy. Think of a Sankey diagram like a river splitting into tributaries: the main river (input energy) divides into smaller streams, some of which flow usefully to a destination (useful output) while others disappear into marshland (wasted energy). The key insight is that no water — and no energy — is created or destroyed; it simply changes direction or form. This makes Sankey diagrams an exceptionally powerful tool for identifying where energy losses occur in machines, power stations, or any engineering system. Sankey diagrams are widely used by engineers and energy analysts to compare the efficiencies of different systems. A highly efficient system has one thick output arrow and very thin waste arrows, whereas an inefficient system disperses energy into many thick waste flows. Reading a Sankey diagram correctly allows you to calculate the efficiency of any device instantly.
- The Input Arrow
- Output Arrows: Useful and Wasted
- Width is Proportional to Energy
- Conservation of Energy Check
- Reading Efficiency from the Diagram
The Input Arrow
Every Sankey diagram begins on the left with a single horizontal arrow representing the total energy input to the system. The width of this arrow is drawn to scale — for example, 1 mm of width might represent 10 J of energy. If a motor receives 100 J of electrical energy, the input arrow is drawn 10 mm wide. All other arrows in the diagram must together have the same total width, because energy is conserved.
Output Arrows: Useful and Wasted
The input arrow splits into two or more output arrows. The arrow continuing straight ahead (or slightly upward) conventionally represents the useful energy output — the energy that does the intended job. Arrows that branch downward (or sometimes upward) represent wasted energy, typically dissipated as thermal energy due to friction or resistance, or as sound. Each waste arrow is labelled with the form of energy it represents.
Width is Proportional to Energy
The critical rule is that arrow widths are strictly proportional to energy values. If a light bulb converts 100 J of electrical energy into 10 J of light and 90 J of heat, the 'light' arrow is drawn one-tenth as wide as the input arrow, and the 'heat' arrow is nine-tenths as wide. This proportionality is what makes Sankey diagrams quantitatively meaningful, not just illustrative.
Conservation of Energy Check
Because energy cannot be created or destroyed (First Law of Thermodynamics), the widths of all output arrows must sum to exactly the width of the input arrow. This is a built-in self-check: if your arrows do not balance, you have made an arithmetic error. Mathematically: E_input = E_useful + E_wasted. A Sankey diagram that does not balance is physically incorrect.
Reading Efficiency from the Diagram
Efficiency is defined as the ratio of useful energy output to total energy input, expressed as a percentage. On a Sankey diagram, efficiency is read visually as the fraction of the total input width occupied by the useful output arrow. A thick useful arrow relative to the input arrow indicates high efficiency. A system where most of the input arrow is consumed by waste arrows is clearly inefficient, making problems with energy losses immediately obvious to any observer.
The Key Equation
- \eta — Efficiency (%): The percentage of input energy that is converted into useful output energy; dimensionless when expressed as a decimal.
- E_{\text{useful}} — Useful energy output (J): The portion of input energy that performs the intended task, such as kinetic energy from a motor or light from a bulb.
- E_{\text{input}} — Total energy input (J): The total amount of energy supplied to the system; represented by the full width of the input arrow on a Sankey diagram.
Worked Example
Problem. A petrol car engine receives 2000 J of chemical energy from fuel. It produces 500 J of kinetic energy (useful), 1200 J of heat energy from exhaust gases, and 300 J of heat energy from friction. Draw the proportional widths for a Sankey diagram and calculate the engine's efficiency.
Answer. The engine has an efficiency of 25%. The Sankey diagram has a 20 mm input arrow splitting into a 5 mm kinetic arrow, a 12 mm exhaust heat arrow, and a 3 mm friction heat arrow.
Origins · 1 minMatthew Henry Phineas Riall Sankey and how this was discovered
Captain H. R. Sankey was an Irish-born British Royal Engineers officer and energy engineer who invented the flow diagram that bears his name. In 1898, he published a landmark paper on the energy efficiency of steam engines, introducing a diagram in which arrow widths were proportional to energy flows. This innovation allowed engineers to visualise energy losses quantitatively for the first time, revolutionising the analysis of thermal machines and industrial processes. Sankey originally created his diagram to compare the energy efficiency of the steam engine against an ideal Carnot engine — revealing starkly just how much energy was wasted in Victorian-era technology. His visual approach was so intuitive that it is now used in fields ranging from national energy statistics to metabolic biology.
Related Concepts
- Conservation of Energy — Sankey diagrams are a direct visual representation of the conservation of energy principle — the total width of input and output arrows must always be equal.
- Energy Efficiency — The efficiency equation η = E_useful / E_input × 100% is the key calculation performed using information read directly from a Sankey diagram.
- Kinetic Energy — Kinetic energy is a common useful output shown on Sankey diagrams for mechanical systems such as engines and motors.
- Power — Sankey diagrams can represent power flows (in watts) as well as energy flows, connecting directly to the concept of power as the rate of energy transfer.