4. Energy Flow Diagrams

Learning outcomes

I can construct energy flow diagrams that show how energy is transferred between stores within a system.
I can identify the energy stores and transfer pathways involved in a variety of physical situations.
I can distinguish between useful energy transfers and energy dissipated to the surroundings.
I can interpret and construct Sankey diagrams to represent energy transfers and transformations.
I can use energy flow diagrams and Sankey diagrams to demonstrate the conservation of energy in real-world systems.

Energy Flow Diagrams

Energy flow diagrams help us follow energy as it moves through a physical system. They show where energy begins, how it is transferred, where useful energy ends up, and where energy is dissipated to the surroundings.

A more quantitative type of energy flow diagram is called a Sankey diagram. In a Sankey diagram, the widths of the arrows represent the amounts of energy being transferred.

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https://www.edplace.com/userfiles/image/sankey%20diagram%201.png

Learning Targets

I can construct energy flow diagrams that show how energy is transferred between stores within a system.
I can identify the energy stores and transfer pathways involved in a variety of physical situations.
I can distinguish between useful energy transfers and energy dissipated to the surroundings.
I can interpret and construct Sankey diagrams to represent energy transfers and transformations.
I can use energy flow diagrams and Sankey diagrams to demonstrate the conservation of energy in real-world systems.

1. Following Energy Through a System

An energy flow diagram is a visual way of showing what happens to energy in a system.

A simple energy flow can be represented as:

Initial energy store → transfer pathway → final energy store

For example, when a person lifts a box:

Chemical store in body → mechanical work → gravitational potential store of box

This tells us three important things:

Where the energy started
How the energy was transferred
Where the energy ended up

Energy is measured in joules (J).

2. Energy Stores

An energy store describes where energy is held in a system.

Important stores include:

Kinetic
Thermal
Chemical
Gravitational potential
Elastic potential
Magnetic
Electrostatic
Nuclear

For example, consider a stretched elastic band.

Before release:

Elastic potential store

After release:

Kinetic store

The energy is transferred mechanically:

Elastic potential store → mechanical work → kinetic store

3. Transfer Pathways

Stores should not be confused with transfer pathways.

The four main energy transfer pathways are:

Mechanical work – a force acts through a distance.
Electrical work – energy is transferred by moving electric charges.
Heating – energy is transferred because of a temperature difference.
Waves – energy is transferred by electromagnetic or mechanical waves.

A battery, for example, contains a chemical store.

When the battery powers a motor, energy is transferred electrically.

Therefore:

Chemical store → electrical work → kinetic store

4. Constructing an Energy Flow Diagram

Suppose a battery powers a small electric motor.

Start by identifying the initial store:

Chemical store in battery

Next, identify the pathway:

Electrical work

Then identify the useful final store:

Kinetic store of motor

A simple diagram is:

Chemical store
in battery
     ↓
Electrical work
     ↓
Kinetic store
of motor

However, this is not the whole story.

The motor also becomes warm and produces sound.

A more realistic diagram is:

                    ┌──→ Kinetic store
Chemical store      │     of motor
in battery ─────────┤
                    │
                    └──→ Thermal stores
                         of surroundings

The diagram now shows that the energy has been distributed among several stores.

5. Useful Energy Transfers

The useful energy transfer is the transfer that produces the effect we want from a device.

Consider an electric fan.

We want the fan blades to move.

Therefore, the increase in the kinetic store of the fan and moving air is useful.

Similarly:

Device	Intended useful transfer
Lamp	Energy transferred by visible light
Electric heater	Increase in thermal store of room
Motor	Increase in kinetic store
Speaker	Energy transferred by sound waves
Crane	Increase in gravitational potential store
Phone charger	Increase in chemical store of battery

What counts as "useful" depends on the purpose of the system.

For example, heating is useful in an electric heater but usually unwanted in an electric motor.

6. Dissipated Energy

Real devices do not transfer all their input energy into the desired store or pathway.

Some energy becomes dissipated to the surroundings.

Common unwanted transfers include:

Heating caused by friction
Heating of electrical components
Sound produced by machines
Vibrations
Unwanted light

For example, an electric motor might receive 100 J of energy.

Suppose:

70 J → useful kinetic transfer

30 J → thermal stores of motor and surroundings

The 30 J has not disappeared. It has been dissipated into the surroundings.

7. Energy Is Not "Lost"

You will often hear the phrase wasted energy.

This does not mean the energy has been destroyed.

It means that energy has been transferred somewhere that is not useful for the intended purpose.

For example, friction in a motor causes components to become warm.

Energy has been transferred into the thermal stores of:

The motor
The surrounding air
Other nearby materials

Once energy has spread through the surroundings, it becomes more difficult to use for further useful work. This is called energy dissipation.

A better scientific statement than "energy was lost" is:

Energy was dissipated to the surroundings.

8. Worked Example — Electric Kettle

Consider an electric kettle.

Energy is transferred electrically to the heating element.

The heating element transfers energy to the water.

The intended result is an increase in the thermal store of the water.

Some energy also increases the thermal stores of:

The kettle
The surrounding air
The work surface

A simplified flow diagram is:

Electrical transfer
        ↓
      Kettle
     ↙      ↘
Useful      Dissipated
heating     heating
   ↓            ↓
Thermal      Thermal stores
store of      of kettle and
water         surroundings

9. Introducing Sankey Diagrams

A Sankey diagram is a special energy flow diagram.

The key feature is:

The width of each arrow represents the amount of energy being transferred.

A large energy transfer has a wide arrow.

A small energy transfer has a narrow arrow.

Usually:

Input energy enters from the left.
Useful output continues forwards.
Dissipated energy branches away, commonly downward.

10. Reading a Sankey Diagram

Suppose a lamp receives:

100 J input energy

It transfers:

40 J by useful light

and:

60 J by heating to the surroundings

The Sankey diagram would have an input arrow representing 100 J.

That arrow divides into:

100 J INPUT ━━━━━━━━━━┳━━━━→ 40 J useful
                      ┃
                      ┃
                      ┃
                      ↓
                     60 J
                  dissipated

The 60 J arrow must be wider than the 40 J arrow because more energy is being transferred by heating.

The exact arrow widths should be proportional to the energy quantities.

11. Sankey Diagrams and Conservation of Energy

Sankey diagrams provide a visual demonstration of the law of conservation of energy.

Energy cannot be created or destroyed.

Therefore:

\text{Total energy input}=\text{Total energy output}

If there are useful and dissipated outputs:

\text{Input energy}= \text{Useful energy}+\text{Dissipated energy}

This relationship is reflected in the widths of the arrows: the widths of all output branches together correspond to the width of the input arrow.

12. Worked Example — Lamp

A lamp receives 200 J of energy.

It transfers:

50 J usefully by light
The remainder by heating

First calculate the dissipated energy:

E_{\text{dissipated}}=200-50

E_{\text{dissipated}}=150\text{ J}

Check conservation:

200=50+150

Therefore:

Input = 200 J

Useful = 50 J

Dissipated = 150 J

The dissipated branch of the Sankey diagram should be three times as wide as the useful branch.

13. Arrow Width Matters

Consider two machines that each receive 100 J.

Machine A

Useful = 80 J
Dissipated = 20 J

Machine B

Useful = 30 J
Dissipated = 70 J

Machine A's Sankey diagram would have a large forward useful arrow and a relatively narrow dissipated branch.

Machine B would have a much smaller useful arrow and a much wider dissipated branch.

This allows us to compare devices visually without immediately performing calculations.

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14. Drawing a Sankey Diagram to Scale

Suppose a device receives 100 J and produces:

70 J useful energy
30 J dissipated energy

We could choose the scale:

1 mm = 10 J

The input arrow would therefore be:

100\div10=10\text{ mm}

The useful arrow would be:

70\div10=7\text{ mm}

The dissipated arrow would be:

30\div10=3\text{ mm}

Notice:

7+3=10\text{ mm}

The arrow widths therefore demonstrate conservation of energy.

15. Worked Example — Hair Dryer

A hair dryer receives 500 J of energy.

Suppose:

350 J is transferred usefully to heating and moving the air.
150 J is dissipated through unwanted heating and sound.

Check:

350+150=500\text{ J}

A Sankey diagram would show:

500 J INPUT
━━━━━━━━━━━━━━━━━━━━━━━┳━━━━━━━━━━━━━━━→ 350 J useful
                       ┃
                       ┃
                       ┗━━━━━━↓
                         150 J dissipated

Because 70% of the input is useful, the useful arrow should represent 70% of the original width.

16. Worked Example — Electric Motor

An electric motor receives 100 J.

Suppose:

70 J increases the useful kinetic store.
30 J is dissipated into thermal stores.

The energy balance is:

100=70+30

The Sankey diagram clearly shows that the useful output is larger than the dissipated output.

This diagram also connects directly to efficiency:

\text{Efficiency} = \frac{\text{useful energy output}} {\text{total energy input}}

Therefore:

\text{Efficiency} = \frac{70}{100} =0.70

or:

70\%

17. Worked Example — Car

Fuel contains a chemical energy store.

When the engine operates, some energy is transferred mechanically and increases the car's kinetic store.

However, energy is also dissipated through:

Heating of the engine
Heating caused by friction
Air resistance
Sound
Heating of tyres and road

A simplified flow is:

Chemical store
in fuel
    ↓
Mechanical transfers
   ↙              ↘
Kinetic           Thermal stores
store of car      + sound
USEFUL            DISSIPATED

A Sankey diagram could then represent the relative amounts of these transfers.

18. Worked Example — Falling Ball

A ball held above the ground has energy in its gravitational potential store.

As it falls:

Gravitational potential store → mechanical transfer → kinetic store

Immediately before impact, much of the transferred energy is associated with the ball's kinetic store.

When the ball strikes the ground, energy is transferred again.

Some may enter:

Elastic stores as the ball deforms
Thermal stores of the ball and floor
Sound waves
Kinetic store if the ball rebounds

An energy flow diagram can therefore become more complex as a system changes over time.

19. Real-World Example — Hydroelectric Power

Water stored behind a dam has a gravitational potential energy store.

When released:

Gravitational potential store → kinetic store of water

The moving water turns a turbine:

Kinetic store → mechanical transfer → kinetic store of turbine

The turbine drives a generator:

Mechanical transfer → electrical transfer

Some energy is dissipated through:

Friction
Heating
Sound
Turbulence

A Sankey diagram can show how much of the original energy ultimately becomes useful electrical output.

20. Flow Diagrams vs Sankey Diagrams

The two diagrams are related, but they serve slightly different purposes.

Energy Flow Diagram	Sankey Diagram
Shows the sequence of energy transfers	Shows energy transfers quantitatively
Identifies stores	Shows amounts of energy
Identifies pathways	Arrow width represents energy
Can show several stages	Clearly separates useful and dissipated outputs
Useful for explaining a system	Useful for analyzing conservation and efficiency

In many situations, using both diagrams together gives the clearest picture.

21. Using Sankey Diagrams to Find Missing Energy

Sankey diagrams can also be used to solve problems.

Suppose:

Input = 800 J

Useful output = 600 J

The dissipated energy must be:

E_{\text{dissipated}} = E_{\text{input}}-E_{\text{useful}}

E_{\text{dissipated}} = 800-600

\boxed{E_{\text{dissipated}}=200\text{ J}}

Check:

600+200=800\text{ J}

The energy balance confirms conservation of energy.

22. Conservation in Complex Systems

There may be more than two outputs.

Suppose an electric machine receives 1000 J and transfers:

650 J into useful kinetic output
200 J into thermal stores
100 J through sound
50 J through unwanted vibration

Then:

650+200+100+50=1000\text{ J}

Therefore:

E_{\text{input}}=E_{\text{total output}}

Even though the energy has been divided among several pathways and stores, none has disappeared.

This is one of the major advantages of Sankey diagrams: they make conservation of energy visually obvious.

23. Comparing Efficient and Inefficient Devices

Consider two lamps that each receive 100 J.

Lamp A

75 J useful
25 J dissipated

Lamp B

10 J useful
90 J dissipated

Their Sankey diagrams would look dramatically different because the relative widths of the useful and dissipated arrows would be different. This makes Sankey diagrams particularly useful for comparing devices.

24. How to Construct a Sankey Diagram

When drawing your own Sankey diagram:

Step 1: Identify the total input energy.

Step 2: Identify all useful outputs.

Step 3: Identify all dissipated outputs.

Step 4: Check conservation:

E_{\text{input}}=E_{\text{all outputs}}

Step 5: Choose a suitable scale.

For example:

1\text{ mm}=10\text{ J}

Step 6: Draw the input arrow to scale.

Step 7: Split the arrow into useful and dissipated branches.

Step 8: Label every transfer and include units.

25. Common Mistakes

Mistake 1 — Making all arrows the same width

In a Sankey diagram, arrow width represents energy.

More energy → wider arrow.

Less energy → narrower arrow.

Mistake 2 — Saying dissipated energy disappears

Energy cannot disappear.

It has been transferred to other stores, often thermal stores in the surroundings.

Mistake 3 — Confusing stores and pathways

Thermal store = where energy is stored.

Heating = how energy is transferred.

Mistake 4 — Forgetting conservation

If:

E_{\text{input}}\neq E_{\text{outputs}}

something is missing from your diagram or calculation.

Mistake 5 — Using arrow length instead of width

It is the width, not the length, of a Sankey arrow that represents the amount of energy.

Mistake 6 — Assuming heating is always wasted

Heating is unwanted in a motor but useful in a heater or kettle.

"Useful" depends on what the device is intended to do.

26. Key Terms

Energy flow diagram – a diagram showing energy stores and transfers within a system.

Sankey diagram – an energy-transfer diagram in which arrow widths represent quantities of energy.

Input energy – energy supplied to a system.

Useful energy – energy transferred in the way intended.

Dissipated energy – energy transferred into less useful stores, often thermal stores of the surroundings.

Energy store – a way energy is held in a system.

Transfer pathway – a process through which energy moves between stores.

Conservation of energy – the principle that energy cannot be created or destroyed.

Efficiency – the proportion of input energy transferred usefully.

27. Check Your Understanding

1. What information does an energy flow diagram show?

2. Name the four main energy transfer pathways.

3. A battery powers a motor. Identify the initial store, transfer pathway and useful final store.

4. What is meant by dissipated energy?

5. Why should we avoid saying that energy has been "lost"?

6. What does the width of an arrow represent in a Sankey diagram?

7. A machine receives 500 J and transfers 350 J usefully. How much energy is dissipated?

8. A lamp receives 100 J and transfers 20 J usefully by light. Sketch an appropriate Sankey diagram.

9. Explain how a Sankey diagram demonstrates conservation of energy.

10. A motor receives 600 J. It transfers 420 J to useful kinetic output, 150 J to thermal stores and 30 J through sound. Show that energy has been conserved.

Key Takeaways

Energy flow diagrams show where energy starts, how it is transferred and where it ends up.
Energy transfers may be useful or dissipated, depending on the intended purpose of the system.
Dissipated energy has not disappeared; it has been transferred to other stores.
Sankey diagrams represent energy transfers using arrows of proportional width.
Wider arrows represent larger energy transfers.
Sankey diagrams can show useful and dissipated energy at the same time.
The total width of the output branches corresponds to the original input because energy is conserved.
Therefore:

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

Energy flow diagrams and Sankey diagrams allow us to analyze real machines, appliances, vehicles and energy-generation systems.

Did You Know?

Sankey diagrams are named after Matthew Henry Phineas Riall Sankey, who used this style of diagram in the late 19th century to illustrate energy flows and the efficiency of a steam engine. Today, Sankey diagrams are used far beyond school physics: engineers and energy analysts use them to visualize flows of electricity, fuel, heat, materials and even resources through entire countries and industries.