Efficiency can be written as a decimal or percentage.

For example:

efficiency = 0.80

is the same as:

efficiency = 80%

Worked Example: Electric Motor

A motor receives:

500 J

of electrical energy.

It produces:

400 J

of useful mechanical energy.

Efficiency:

efficiency = 400/500

efficiency = 0.80

As a percentage:

0.80 × 100 = 80%

The motor is 80% efficient.

Finding Wasted Energy

Because energy is conserved:

input energy = useful energy + wasted energy

Therefore:

wasted energy = input energy − useful energy

For the previous motor:

wasted energy = 500 − 400

wasted energy = 100 J

So:

500 J input → 400 J useful + 100 J wasted

Worked Example: Lamp

A lamp receives:

200 J

of electrical energy.

It transfers:

60 J

as useful light.

Efficiency:

efficiency = 60/200 × 100

efficiency = 30%

Wasted energy:

200 − 60 = 140 J

Therefore:

200 J electrical → 60 J useful light + 140 J thermal energy

The lamp is 30% efficient at producing the desired light output.

Efficiency Using Power

Efficiency can also be calculated using power.

When input and output are measured over the same time interval:

efficiency = useful output power / total input power

As a percentage:

efficiency (%) = useful output power / total input power × 100

This is especially useful for machines operating continuously.

Worked Example: Motor Power

An electric motor receives:

2500 W

of electrical power.

It provides:

2000 W

of useful mechanical power.

Efficiency:

efficiency = 2000/2500 × 100

efficiency = 80%

Power dissipated:

2500 − 2000 = 500 W

This means energy is being dissipated at a rate of:

500 J/s

Why Energy and Power Give the Same Efficiency

Suppose the motor operates for 10 s.

Input energy:

E = Pt

E = 2500 × 10

E = 25,000 J

Useful output energy:

E = 2000 × 10

E = 20,000 J

Efficiency:

efficiency = 20,000/25,000

efficiency = 0.80

or:

80%

The same result is obtained using either energy or power.

Rearranging the Efficiency Equation

If efficiency and input energy are known:

useful output = efficiency × input

Remember to convert a percentage to a decimal.

For example:

75% = 0.75

Therefore:

useful output = 0.75 × input

If useful output and efficiency are known:

input = useful output / efficiency

Worked Example: Finding Useful Energy

A machine receives:

2400 J

and is:

75% efficient

Convert:

75% = 0.75

Useful energy:

useful output = 0.75 × 2400

useful output = 1800 J

Wasted energy:

2400 − 1800 = 600 J

Worked Example: Finding Input Energy

A machine must produce:

3600 J

of useful energy.

It is:

80% efficient

Convert:

80% = 0.80

Input energy:

input = 3600/0.80

input = 4500 J

Therefore:

900 J

is transferred into unwanted forms.

Worked Example: Finding Useful Power

A motor receives:

10 kW

of electrical power.

It is:

85% efficient

Useful mechanical power:

Puseful = 0.85 × 10

Puseful = 8.5 kW

Wasted power:

10 − 8.5 = 1.5 kW

Why Real Systems Are Not 100% Efficient

In real systems, interactions between components cause some energy to be transferred into unwanted forms.

Common causes include:

• friction
• electrical resistance
• air resistance
• sound
• vibration
• deformation
• turbulence
• heating of components

These processes transfer energy into the surroundings.

Because these transfers cannot normally be eliminated completely:

real systems are not 100% efficient

Friction

Friction is one of the most common causes of energy dissipation in mechanical systems.

When surfaces move against one another, mechanical energy is transferred into thermal energy.

Examples include:

• gears
• bearings
• tyres
• engine components
• conveyor systems
• bicycle chains

You may notice machinery becoming warm during operation because of these transfers.

 

 

 

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Reducing Friction

Engineers can reduce friction using:

• lubricants
• ball bearings
• smoother surfaces
• appropriate materials
• improved alignment

Reducing friction means less mechanical energy is transferred into unwanted thermal energy.

This can increase efficiency.

Electrical Resistance

Electrical devices also dissipate energy because conductors have resistance.

When current passes through resistance:

electrical energy → thermal energy

This is useful in devices designed for heating.

Examples include:

• kettles
• heaters
• toasters

But it is usually unwanted in:

• motors
• power cables
• computers
• transformers
• generators

Electrical components becoming warm is evidence that some energy is being transferred thermally.

Useful or Wasted Depends on Purpose

An important point is that whether an energy transfer is considered useful depends on the purpose of the device.

Consider an electric heater.

Electrical energy becoming thermal energy is:

useful

But in an electric motor, excessive heating is:

wasted

Similarly, light from a lamp is useful, while light from a heating element may not be.

Efficiency must therefore always be considered in relation to the intended output.

Sound and Vibration

Machines often produce sound and vibration.

These transfers usually represent energy that is not contributing to the intended output.

Examples include:

• noisy engines
• vibrating motors
• rattling machinery
• tyre noise
• electrical transformers

Reducing unwanted vibration and sound can improve efficiency as well as make machinery quieter and more reliable.

Air Resistance

Moving objects transfer energy to the surrounding air.

This is particularly important for:

• cars
• trains
• aircraft
• bicycles

Air resistance generally increases as speed increases.

Therefore, high-speed vehicles may transfer large amounts of energy into the surrounding air.

Streamlining can reduce these unwanted energy transfers.

 

 

 

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Improving Mechanical Efficiency

Mechanical systems can often be made more efficient by:

• reducing friction
• using lubrication
• using bearings
• reducing unnecessary mass
• improving aerodynamics
• reducing vibration
• improving gear design
• using suitable materials

However, completely eliminating unwanted transfers is usually impossible.

Improving Electrical Efficiency

Electrical systems can be improved through:

• lower-resistance conductors
• efficient motors
• improved insulation
• better electronic control systems
• reduced standby consumption
• efficient lighting
• improved cooling systems
• appropriate operating voltages

The goal is to increase the fraction of input energy transferred into the intended output.

Comparing Devices

Suppose three motors receive the same input energy:

1000 J

Motor C is the most efficient because the greatest fraction of its input energy becomes useful output.

It also dissipates the least energy.

Comparing Devices with Different Inputs

Efficiency is a ratio, so devices can be compared even when their input energies are different.

Device A

Input = 500 J
Useful = 400 J

efficiency = 400/500 × 100 = 80%

Device B

Input = 2000 J
Useful = 1500 J

efficiency = 1500/2000 × 100 = 75%

Device B produces more useful energy overall, but Device A is more efficient.

This is an important distinction.

Efficiency Does Not Mean Power

A device can be highly efficient but have low power.

Another device can be less efficient but have much greater power.

For example:

Device A:

input power = 100 W

useful power = 90 W

Efficiency:

90%

Device B:

input power = 2000 W

useful power = 1600 W

Efficiency:

80%

Device B provides much more useful power, but Device A is more efficient.

Sankey Diagrams

A Sankey diagram visually represents energy transfers.

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

For example:

100 J input

might divide into:

70 J useful

and:

30 J wasted

Because:

70 + 30 = 100

the total width of the output arrows equals the width of the input arrow.

Sankey diagrams make it easy to compare the useful and wasted energy transfers in different systems.

Reading a Sankey Diagram

Suppose a motor receives:

500 J

The diagram shows:

350 J useful mechanical energy

and:

150 J thermal and sound energy

Efficiency:

efficiency = 350/500 × 100

efficiency = 70%

The wasted fraction is:

150/500 × 100

= 30%

Therefore:

70% useful + 30% wasted = 100%

Light Bulbs

Different lighting technologies can provide the same useful function while transferring different amounts of energy into unwanted thermal energy.

An efficient lamp transfers a greater proportion of its electrical input into useful light and a smaller proportion into unwanted heating.

This means less electrical energy is required to produce a particular useful light output.

 

 

 

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Worked Comparison: Lamps

Suppose two lamps each receive:

100 J

of electrical energy.

Lamp A produces:

15 J useful light

Lamp B produces:

80 J useful light

Lamp A:

efficiency = 15/100 × 100

efficiency = 15%

Lamp B:

efficiency = 80/100 × 100

efficiency = 80%

Lamp B converts a much greater fraction of the electrical input into the desired output.

Electric Motors

Electric motors convert:

electrical energy → mechanical energy

Some energy is also transferred through:

• heating of wires
• friction in bearings
• sound
• vibration

A more efficient motor provides more mechanical output for the same electrical input.

This can reduce:

• electricity consumption
• operating costs
• unwanted heating
• environmental impacts associated with energy production

Vehicle Engines

Vehicle engines and motors convert stored energy into mechanical energy.

In combustion engines, significant energy can be transferred into:

• hot exhaust gases
• cooling systems
• friction
• sound

Only part of the input energy becomes useful mechanical output.

Electric motors generally have different energy-loss pathways, including electrical resistance, magnetic losses, friction, and heating.

 

 

 

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Worked Example: Vehicle

An engine receives:

50 MJ

of chemical energy.

It transfers:

18 MJ

into useful mechanical energy.

Efficiency:

efficiency = 18/50 × 100

efficiency = 36%

Energy dissipated:

50 − 18 = 32 MJ

The remaining energy has been transferred mainly into thermal energy and other unwanted outputs.

Power Station Efficiency

Efficiency is also important in electricity generation.

Suppose a power station receives:

1000 MW

of energy per second from its fuel source and produces:

400 MW

of electrical power.

Efficiency:

efficiency = 400/1000 × 100

efficiency = 40%

The remaining:

600 MW

is transferred into other forms, especially thermal energy.

Efficiency Through Several Stages

Many real systems involve several energy conversions.

For example:

fuel → electricity → motor → motion

Suppose:

Power station efficiency = 40%

Motor efficiency = 80%

Start with:

100 J

After the power station:

100 × 0.40 = 40 J

After the motor:

40 × 0.80 = 32 J

Overall useful output:

32 J

Therefore:

overall efficiency = 32%

For several stages, the overall efficiency can be found by multiplying the decimal efficiencies:

overall efficiency = efficiency₁ × efficiency₂ × efficiency₃ ...

Worked Example: Multi-Stage System

A system contains:

Generator efficiency = 90%

Motor efficiency = 80%

Machine efficiency = 75%

Convert to decimals:

0.90 × 0.80 × 0.75

= 0.54

Therefore:

overall efficiency = 54%

Only 54% of the original input reaches the final useful output.

This demonstrates how losses at several stages can significantly reduce overall efficiency.

Efficiency and Sustainability

Improving efficiency can reduce the amount of energy required to perform a task.

This can lead to:

• lower energy consumption
• lower operating costs
• reduced fuel use
• reduced waste heat
• lower demand for energy resources
• reduced environmental impacts

For example, if two motors perform the same useful task but one is more efficient, the more efficient motor requires less input energy.

Worked Example: Energy Savings

Machine A is 60% efficient.

Machine B is 80% efficient.

Both must provide:

2400 J

of useful energy.

Machine A

input = 2400/0.60

input = 4000 J

Machine B

input = 2400/0.80

input = 3000 J

Machine B requires:

1000 J less input energy

to perform the same useful task.

Can Efficiency Be 100%?

In an idealized model, we sometimes assume:

efficiency = 100%

This simplifies calculations.

Real machines, however, always have some unwanted energy transfers.

For example:

• wires have resistance
• surfaces experience friction
• objects interact with air
• machines vibrate
• components become warm

Therefore, real energy-transfer systems do not achieve perfect efficiency.

Can Efficiency Be Greater Than 100%?

No.

An efficiency greater than 100% would mean:

useful output energy > input energy

That would require energy to be created.

This violates conservation of energy.

Therefore:

0% ≤ efficiency ≤ 100%

for a correctly defined system.

If a calculation gives more than 100%, check:

• the data
• the units
• which quantity is input
• which quantity is useful output

Efficiency Versus Energy Conservation

These ideas should not be confused.

Conservation of Energy

Total energy is always conserved.

input energy = total output energy

Efficiency

Efficiency asks what fraction of that output is useful.

For example:

100 J input → 70 J useful + 30 J thermal

Energy conservation:

100 J = 70 J + 30 J

Efficiency:

70%

The system is only 70% efficient, but 100% of the energy still exists.

Practical Example: Electric Elevator

An elevator motor receives:

50 kW

of electrical power.

The elevator receives:

42 kW

of useful mechanical power.

Efficiency:

efficiency = 42/50 × 100

efficiency = 84%

Power dissipated:

50 − 42 = 8 kW

This means:

8000 J

is transferred into other forms every second.

Practical Example: Pump

A water pump receives:

3000 W

of electrical power.

It transfers:

2250 W

usefully to the water.

Efficiency:

efficiency = 2250/3000 × 100

efficiency = 75%

Wasted power:

3000 − 2250 = 750 W

Improving Efficiency

Engineers try to increase efficiency by reducing unwanted energy transfers.

Possible approaches include:

Mechanical Systems

• lubrication
• bearings
• aerodynamic design
• reducing unnecessary vibration
• improving alignment
• reducing friction

Electrical Systems

• low-resistance conductors
• efficient motors
• improved electronic components
• appropriate cooling
• improved circuit design

Buildings

• insulation
• efficient lighting
• improved heating and cooling systems
• reducing unwanted air leakage

Transportation

• aerodynamic shapes
• lightweight materials
• efficient motors
• regenerative braking
• reduced rolling resistance

Efficiency improvements often involve engineering trade-offs involving cost, reliability, safety, materials, and performance.

Common Mistakes

Saying Energy Is "Used Up"

Energy is not destroyed.

It is transferred or dissipated.

Saying Wasted Energy Disappears

Wasted energy still exists.

It has been transferred into forms that are less useful for the intended purpose.

Dividing Input by Output

The correct relationship is:

useful output ÷ total input

not the other way around.

Forgetting to Multiply by 100

If the question asks for percentage efficiency:

0.75 = 75%

Using 80 Instead of 0.80

When using an efficiency in another calculation:

80% = 0.80

not 80.

Assuming the Largest Output Means the Highest Efficiency

Efficiency is a ratio.

A machine producing more useful energy may still be less efficient if it requires much more input energy.

Confusing Power with Energy

Efficiency can be calculated using either energy or power, but the quantities must be compared appropriately.

Use:

useful energy / input energy

or:

useful power / input power

Assuming Thermal Energy Is Always Wasted

Thermal energy from a heater is useful.

Thermal energy from a motor is generally unwanted.

The intended purpose of the device determines what counts as useful.

Assuming 100% Efficiency Is Realistic

Idealized calculations may assume 100%, but real systems experience energy dissipation.

Check Your Understanding

1. Define efficiency.
2. Explain the difference between useful and wasted energy transfers.
3. Why is the term dissipated energy often more scientifically useful than saying energy is "lost"?
4. A machine receives 500 J and produces 400 J of useful energy. Calculate its efficiency.
5. How much energy is wasted in Question 4?
6. A motor receives 2000 J and produces 1500 J of useful mechanical energy. Calculate its efficiency.
7. A lamp receives 100 J and transfers 25 J as useful light. Calculate its efficiency.
8. A device receives 500 W and provides 400 W of useful output power. Calculate its efficiency.
9. How much power is dissipated in Question 8?
10. A motor is 80% efficient and receives 5000 J. Calculate its useful output.
11. Calculate the wasted energy in Question 10.
12. A machine produces 3000 J of useful output and is 75% efficient. Calculate its required input energy.
13. A motor receives 10 kW and is 85% efficient. Calculate its useful mechanical power.
14. Explain why friction reduces mechanical efficiency.
15. Explain why electrical resistance can reduce the efficiency of a motor.
16. Give three examples of unwanted energy transfers in machines.
17. Why can thermal energy be useful in one device but wasted in another?
18. Machine A receives 1000 J and produces 700 J usefully. Machine B receives 500 J and produces 400 J usefully. Which is more efficient?
19. Explain why the device producing the greatest useful output is not necessarily the most efficient.
20. A power station is 40% efficient and receives 2000 MW. Calculate its electrical output power.
21. A 90%-efficient motor receives 5 kW. Calculate its useful output power.
22. A system contains two stages with efficiencies of 80% and 75%. Calculate the overall efficiency.
23. A system contains three stages with efficiencies of 90%, 80%, and 70%. Calculate the overall efficiency.
24. Explain why several efficient devices connected together can have a noticeably lower overall efficiency.
25. Describe three ways engineers can improve the efficiency of a mechanical system.
26. Describe two ways engineers can reduce unwanted electrical energy transfers.
27. Explain why improving efficiency can reduce environmental impacts.
28. A machine must produce 6000 J of useful output. Compare the required input energy if it is 60% efficient and if it is 80% efficient.
29. Explain why an efficiency of 120% is impossible.
30. Explain how a system can be only 60% efficient while still obeying the law of conservation of energy.

Key Terms

Efficiency — Fraction of the total input energy or power transferred into the intended useful output.

Useful energy — Energy transferred in the form required for the intended purpose.

Wasted energy — Energy transferred in forms that are not useful for the intended purpose.

Energy dissipation — Transfer and spreading of energy into less useful stores, often thermal energy in the surroundings.

Input energy — Total energy supplied to a system.

Output energy — Energy transferred from a system.

Useful output — Portion of the output that performs the intended function.

Input power — Rate at which energy is supplied to a system.

Useful output power — Rate at which useful energy is transferred.

Watt (W) — Unit of power equal to one joule per second.

Friction — Force opposing motion that can transfer mechanical energy into thermal energy.

Electrical resistance — Opposition to electric current that causes electrical energy to be transferred thermally.

Air resistance — Resistive force caused by motion through air.

Sankey diagram — Diagram showing energy transfers using arrows whose widths represent energy quantities.

Overall efficiency — Efficiency of a system containing several energy-transfer stages.

Conservation of energy — Principle stating that energy cannot be created or destroyed.

Key Takeaways

• Efficiency measures the fraction of input energy transferred into the desired useful output.
• Efficiency can be expressed as a decimal or percentage.
• Efficiency can be calculated using either energy values or power values.
• Useful output is determined by the intended purpose of the device.
• Wasted energy has not disappeared; it has been transferred into less useful forms.
• Total energy is always conserved.
• Real systems are less than 100% efficient because some energy is dissipated.
• Common causes of energy dissipation include friction, electrical resistance, air resistance, sound, vibration, and deformation.
• Mechanical efficiency can often be improved by reducing friction and resistance.
• Electrical efficiency can often be improved by reducing unwanted heating and resistance.
• A more efficient device requires less input energy to produce the same useful output.
• A device with a greater useful output is not necessarily more efficient.
• Efficiency and power are different quantities.
• Several energy conversions can reduce the overall efficiency of a system.
• For multi-stage systems, decimal efficiencies can be multiplied to determine overall efficiency.
• Sankey diagrams show how input energy divides into useful and wasted outputs.
• Improving efficiency can reduce energy consumption, operating costs, resource use, and environmental impacts.
• An efficiency greater than 100% is impossible because it would violate conservation of energy.

The central idea is:

Energy is never destroyed — but not all transferred energy remains useful.

A more efficient system transfers a greater fraction of its input energy into the output we actually want.

Energy Resources and Energy Density

Different energy resources store very different amounts of energy. A small mass of one fuel may contain far more usable energy than the same mass of another.

Energy density describes the amount of energy stored or available per unit mass or per unit volume.

Energy density is important because energy resources must often be:

• stored
• transported
• carried by vehicles
• delivered to power stations
• contained safely
• converted into useful energy

A resource with a high energy density can provide a large amount of energy without requiring as much mass or volume.

 

 

 

6

Energy Density by Mass

Energy density by mass tells us how much energy is available from each unit of mass.

It can be calculated using:

energy density = energy stored / mass

Common units include:

J/kg

kJ/kg

MJ/kg

For fuels, MJ/kg is particularly common.

For example, if 2 kg of a fuel stores 80 MJ:

energy density = 80/2

energy density = 40 MJ/kg

This means each kilogram contains approximately 40 MJ of energy.

Energy Density by Volume

Sometimes volume is more important than mass.

Volumetric energy density can be calculated using:

energy density = energy stored / volume

Possible units include:

J/m³

MJ/m³

MJ/L

kWh/L

This is particularly important when storage space is limited.

For example, fuel tanks in:

• cars
• aircraft
• ships
• spacecraft

have limited volume.

A fuel with high energy per litre can therefore be extremely useful even if another resource has a similar energy density per kilogram.

Gravimetric and Volumetric Energy Density

Two terms are useful when comparing energy-storage systems.

Gravimetric energy density means energy stored per unit mass.

Volumetric energy density means energy stored per unit volume.

These can lead to different conclusions.

A resource may have:

• high energy per kilogram
• but low energy per litre

Hydrogen is an important example. Hydrogen has a very high energy content per kilogram, but hydrogen gas occupies a large volume unless it is compressed or liquefied.

Therefore, both mass and volume must sometimes be considered.

Comparing Energy Resources

Approximate energy densities by mass for some fuels are:

Values vary depending on composition and measurement conditions.

The important idea is the comparison, not memorizing every value.

Hydrogen has a very high energy density by mass, while gasoline and diesel also contain large amounts of energy for their mass.

 

 

 

6

Worked Example: Comparing Fuels

Suppose Fuel A has an energy density of:

20 MJ/kg

and Fuel B has:

45 MJ/kg

For 10 kg of Fuel A:

E = 20 × 10

E = 200 MJ

For 10 kg of Fuel B:

E = 45 × 10

E = 450 MJ

The same mass of Fuel B stores more than twice as much energy.

This can be a major advantage when mass must be minimized.

Finding Energy from Energy Density

The relationship can be rearranged:

energy stored = energy density × mass

For example, gasoline has an approximate energy density of:

46 MJ/kg

If a vehicle carries:

30 kg

of gasoline:

E = 46 × 30

E = 1380 MJ

or:

1.38 GJ

This represents the approximate chemical energy stored in the fuel, not the useful mechanical energy ultimately delivered by the vehicle.

Energy Density and Efficiency

Energy density and efficiency are different concepts.

Energy density tells us how much energy a resource contains per unit mass or volume.

Efficiency tells us how much of that input energy becomes useful output.

A fuel may have a high energy density but be used in a relatively inefficient system.

For example, suppose:

Fuel energy = 1000 MJ

Engine efficiency = 30%

Useful mechanical energy:

useful energy = 0.30 × 1000

useful energy = 300 MJ

The fuel contained 1000 MJ, but only 300 MJ became the intended mechanical output.

Energy Density of Batteries

Batteries store energy chemically and release it electrically.

Battery energy density is often expressed as:

Wh/kg

or:

Wh/L

Recall:

1 Wh = 3600 J

Therefore:

1 kWh = 3.6 MJ

Modern rechargeable batteries store much less energy per kilogram than liquid fuels when comparing the raw stored-energy values.

However, electric motors can convert stored electrical energy into mechanical energy very efficiently.

This means energy density alone does not determine overall vehicle performance.

 

 

 

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Worked Example: Battery

A battery has an energy density of:

250 Wh/kg

and has a mass of:

400 kg

Stored energy:

E = 250 × 400

E = 100,000 Wh

Therefore:

E = 100 kWh

Convert to joules:

100 kWh × 3.6 MJ/kWh = 360 MJ

The battery stores approximately:

360 MJ

of energy.

Comparing Battery and Gasoline Energy

Suppose:

Battery system = 250 Wh/kg

Convert this to MJ/kg:

250 Wh/kg × 3600 J/Wh

= 900,000 J/kg

= 0.90 MJ/kg

Gasoline ≈ 46 MJ/kg

By raw stored energy per kilogram, gasoline contains far more energy.

However, this does not mean a gasoline vehicle is automatically more energy efficient.

Electric motors typically convert a greater fraction of their input energy into useful mechanical energy than combustion engines.

Therefore, meaningful comparisons must consider:

energy density + efficiency + system mass + storage requirements + environmental effects

Why Energy Density Matters in Transportation

Transportation systems must carry their energy source with them.

Every kilogram of fuel or battery adds mass.

Greater vehicle mass can require additional energy to:

• accelerate
• climb hills
• support the vehicle
• transport the energy-storage system itself

High energy density is therefore particularly valuable for:

• aircraft
• spacecraft
• long-distance vehicles
• ships
• portable equipment

 

 

 

4

Aviation and Energy Density

Aircraft provide a clear example of why energy density matters.

An aircraft must lift:

• passengers
• cargo
• engines
• structure
• fuel or energy storage

If the energy-storage system becomes much heavier, additional lift and energy are required.

This makes energy per kilogram extremely important.

Liquid hydrocarbon fuels have historically been useful for aviation partly because they store large amounts of energy for their mass and volume.

Battery-powered aircraft face a major engineering challenge because batteries generally store considerably less energy per kilogram.

Volume Matters Too

Mass is not the only limitation.

Consider two fuels:

Fuel A:

40 MJ/kg

Fuel B:

50 MJ/kg

Fuel B appears better by mass.

But suppose Fuel B requires a very large storage tank.

A vehicle may not have enough physical space.

Engineers therefore consider:

energy per kilogram

and:

energy per litre

A practical energy source must often perform well in both categories.

Hydrogen

Hydrogen demonstrates why the distinction is important.

Hydrogen has a very high energy content by mass.

Approximately:

120 MJ/kg

This is much higher than gasoline.

However, hydrogen gas has a low density under ordinary conditions.

To store useful quantities, hydrogen may need to be:

• compressed to high pressure
• liquefied at extremely low temperature
• stored chemically in another material

These storage methods require additional equipment and energy.

 

 

 

6

Hydrogen Example

Suppose 5 kg of hydrogen stores approximately:

120 MJ/kg

Total energy:

E = 120 × 5

E = 600 MJ

That is a large amount of energy for only 5 kg of hydrogen.

However, the complete storage system includes:

• pressure tanks or cryogenic tanks
• valves
• insulation
• safety equipment
• fuel delivery systems

Therefore, the energy density of the whole system is lower than that of the hydrogen alone.

This is an important engineering principle:

Always consider the complete system, not just the energy-storage material.

Nuclear Fuel and Energy Density

Nuclear fuels have vastly greater energy density than chemical fuels because nuclear reactions release energy from changes within atomic nuclei rather than ordinary chemical bonds.

A relatively small mass of nuclear fuel can therefore release enormous amounts of energy.

This reduces the quantity of fuel that must be transported to a power station.

 

 

 

7

Nuclear Energy: Important Trade-Off

Very high energy density does not automatically make an energy resource the best choice.

Nuclear energy also requires consideration of:

• radioactive materials
• radioactive waste
• reactor safety
• construction costs
• decommissioning
• security
• long-term waste management

Therefore:

energy density is only one criterion for evaluating an energy resource.

Fossil Fuels

Fossil fuels include:

• coal
• oil
• natural gas

They have been widely used partly because they:

• contain substantial energy
• can be stored
• can be transported
• provide energy when needed
• have established infrastructure

However, their combustion releases carbon dioxide.

Other environmental concerns can include:

• air pollution
• extraction impacts
• habitat disruption
• spills
• methane leakage

Their energy density must therefore be considered alongside their environmental effects.

Renewable Energy Resources

Some renewable resources do not use a stored fuel in the same way.

Examples include:

• solar
• wind
• hydroelectric
• tidal
• wave energy

For these systems, fuel energy density may not be the most useful comparison.

Instead, engineers may consider:

• power per unit area
• availability
• storage requirements
• capacity factor
• location
• infrastructure
• environmental effects
• cost

 

 

 

4

Solar Energy

Solar energy does not require fuel to be transported to a solar panel.

This is a major advantage.

However, sunlight is spread over a large area.

Producing large amounts of power may therefore require:

• large areas of solar panels
• suitable locations
• energy storage
• electrical transmission infrastructure

Solar energy illustrates an important point:

low concentration of an energy resource does not necessarily make it unsuitable.

It simply creates different engineering requirements.

Wind Energy

Wind turbines extract kinetic energy from moving air.

There is no fuel tank and therefore no conventional fuel energy density.

Advantages include:

• renewable resource
• no fuel combustion during operation
• no direct carbon dioxide emissions from fuel combustion

Challenges include:

• variable wind
• suitable locations
• energy storage or grid balancing
• land or offshore infrastructure

Again, energy-resource decisions require more than one measurement.

Hydroelectric Energy

Hydroelectric systems store energy as gravitational potential energy in elevated water.

Recall:

ΔEₚ = mgΔh

The amount of stored energy depends on:

• mass of water
• gravitational field strength
• height difference

A reservoir may therefore require an enormous volume of water to store large amounts of energy.

However, hydroelectric systems can provide substantial power and can be highly useful for grid-scale energy storage and electricity production where geography allows.

Pumped Hydroelectric Storage

Pumped-storage systems use excess electrical energy to pump water uphill.

The energy transformation is approximately:

electrical → mechanical → gravitational potential

When electricity is needed:

gravitational potential → kinetic → mechanical → electrical

This system may require a very large volume of water, but it can store enormous total quantities of energy.

This shows that low energy density does not necessarily mean low total energy storage.

Large-scale systems can compensate by using large masses or volumes.

Energy Density and Portable Technology

Energy density is also important in:

• smartphones
• laptops
• cameras
• drones
• medical devices
• power tools
• wearable electronics

A battery with greater energy per kilogram can provide longer operating time without making the device excessively heavy.

A battery with greater energy per litre can store more energy without making the device excessively large.

For portable devices, engineers often want both:

high gravimetric energy density

and:

high volumetric energy density

Energy Density Versus Power Density

Energy density should not be confused with power density.

Energy density tells us:

how much energy can be stored

Power density tells us:

how quickly energy can be delivered relative to mass or volume

A storage system might hold a large amount of energy but be unable to release it very quickly.

Another might store less energy but deliver extremely high power for a short period.

This distinction matters for:

• vehicles
• power tools
• grid storage
• electronics
• acceleration systems

Worked Comparison

Suppose Battery A stores:

200 Wh/kg

Battery B stores:

300 Wh/kg

A device requires:

60,000 Wh

Battery A mass:

mass = energy / energy density

mass = 60,000/200

mass = 300 kg

Battery B:

mass = 60,000/300

mass = 200 kg

Using Battery B reduces the required battery mass by:

100 kg

This could have a major effect on vehicle performance.

Comparing Fuels Using Required Mass

Suppose a system needs:

900 MJ

of stored chemical energy.

Fuel A:

30 MJ/kg

Required mass:

m = 900/30

m = 30 kg

Fuel B:

45 MJ/kg

Required mass:

m = 900/45

m = 20 kg

Fuel B requires 10 kg less fuel to store the same amount of energy.

Including Efficiency in the Comparison

Suppose a vehicle needs:

300 MJ

of useful mechanical energy.

System A

Fuel energy density:

45 MJ/kg

Efficiency:

30%

Required input energy:

300/0.30 = 1000 MJ

Fuel mass:

1000/45 ≈ 22.2 kg

System B

Energy storage density:

1 MJ/kg

Efficiency:

90%

Required input energy:

300/0.90 ≈ 333 MJ

Storage mass:

333/1 ≈ 333 kg

System B is much more efficient but requires a heavier energy-storage system in this simplified comparison.

This demonstrates why transportation engineering involves trade-offs.

Evaluating Energy Resources

Energy density is important, but it should never be the only factor considered.

A complete comparison might include:

Transportation Decisions

Consider choosing an energy system for a vehicle.

A designer might ask:

• How far must the vehicle travel?
• How heavy can the energy-storage system be?
• How much space is available?
• How quickly must the vehicle refuel or recharge?
• How much power must be delivered?
• What infrastructure exists?
• What emissions are produced?
• How efficient is the drivetrain?
• How safe is the storage system?

Different vehicles can therefore favor different technologies.

Cars

Cars can potentially use:

• gasoline
• diesel
• batteries
• hydrogen
• biofuels

Liquid fuels offer high energy density and rapid refueling.

Battery systems can offer high drivetrain efficiency and avoid direct exhaust emissions.

Hydrogen can offer high energy per kilogram of fuel but requires specialized storage.

The best comparison depends on the whole system and intended use.

Aircraft

Aircraft place especially strong limits on mass.

This makes gravimetric energy density extremely important.

An energy-storage system that is practical for a stationary building or road vehicle may be much more difficult to use in an aircraft because the aircraft must continually carry the storage mass.

Ships

Ships can carry much greater masses than aircraft.

Therefore, mass energy density may be less restrictive.

However, ships travel long distances and require very large total quantities of energy.

Designers must consider:

• fuel volume
• fuel mass
• safety
• emissions
• cost
• refueling infrastructure

Grid Energy Storage

Stationary energy storage has different priorities.

A grid battery or pumped-storage facility does not have to carry its own energy-storage system while moving.

Therefore, mass may be less important.

Engineers may prioritize:

• cost
• lifespan
• efficiency
• safety
• response time
• total storage capacity
• environmental impact

This demonstrates why the "best" energy technology depends strongly on its application.

Technological Trade-Offs

Imagine two storage technologies.

Technology A

• very high energy density
• expensive
• difficult to recycle
• high power output

Technology B

• lower energy density
• inexpensive
• long lifespan
• easy to recycle

There is no automatically correct choice.

For an aircraft, Technology A's low mass may be extremely important.

For a stationary storage facility, Technology B may be preferable.

Engineering decisions involve balancing competing requirements.

Environmental Trade-Offs

Energy density can indirectly affect environmental impacts.

A higher-energy-density fuel may require:

• less material extraction
• less transport mass
• smaller storage systems

But it might also:

• produce greenhouse gases
• create hazardous waste
• require environmentally damaging extraction
• create safety risks

Therefore, high energy density does not automatically mean environmentally preferable.

Life-Cycle Thinking

A complete environmental comparison considers the entire life cycle of an energy technology.

This may include:

raw materials → manufacturing → transportation → operation → maintenance → disposal/recycling

For example, evaluating a battery might include:

• mineral extraction
• battery manufacturing
• electricity used for charging
• operating lifetime
• recycling
• disposal

Similarly, evaluating gasoline should include:

• oil extraction
• refining
• transportation
• combustion
• emissions

This produces a more meaningful comparison than considering energy density alone.

Energy Density and Sustainability

A sustainable energy system must balance many factors.

These include:

• sufficient energy supply
• acceptable environmental impact
• affordability
• reliability
• resource availability
• safety
• efficiency
• appropriate energy density

Energy density can influence which technologies are practical, but sustainability requires a broader evaluation.

Common Mistakes

Confusing Energy Density with Material Density

Material density is:

density = mass / volume

Energy density describes:

energy / mass

or:

energy / volume

They are different quantities.

Confusing Energy Density with Power

Energy density tells us how much energy is stored.

Power tells us how quickly energy is transferred.

Confusing Energy Density with Efficiency

A system can have:

high energy density + low efficiency

or:

low energy density + high efficiency

These quantities describe different characteristics.

Assuming the Highest Energy Density Is Always Best

Energy density is only one criterion.

Other factors include:

• safety
• emissions
• cost
• renewability
• efficiency
• storage
• infrastructure

Comparing Different Units Directly

Do not directly compare:

MJ/kg

with:

MJ/L

One is energy per mass.

The other is energy per volume.

They answer different questions.

Ignoring the Storage System

The energy-storage material may require:

• tanks
• containers
• cooling
• insulation
• electronics
• structural support

These components add mass and volume.

Ignoring Efficiency

Stored energy is not necessarily equal to useful energy.

If a system is:

30% efficient

only 30% of the stored energy becomes the desired useful output.

Assuming Renewable Means High Energy Density

Renewability and energy density are separate characteristics.

Wind and sunlight are renewable but are not stored fuels like gasoline.

Check Your Understanding

1. Define energy density.
2. Explain the difference between gravimetric and volumetric energy density.
3. State two common units for energy density.
4. A fuel contains 100 MJ in 4 kg. Calculate its energy density.
5. A fuel has an energy density of 40 MJ/kg. How much energy is stored in 15 kg?
6. A system requires 600 MJ. Calculate the mass of a fuel with an energy density of 30 MJ/kg.
7. Repeat Question 6 for a fuel with an energy density of 50 MJ/kg.
8. Which fuel requires less mass?
9. Explain why high energy density is particularly important for aircraft.
10. Explain why energy density may be less important for stationary energy storage.
11. What is the difference between energy density and power density?
12. Explain why hydrogen can have a high gravimetric energy density but present volumetric storage challenges.
13. Why must the mass of a hydrogen tank be considered when evaluating the complete system?
14. A battery has an energy density of 200 Wh/kg and a mass of 300 kg. Calculate its stored energy in kWh.
15. Convert the answer to Question 14 into MJ.
16. Explain why gasoline's high energy density does not automatically make a gasoline vehicle more efficient than an electric vehicle.
17. What is meant by the efficiency of an energy-conversion system?
18. A fuel contains 1000 MJ and is used in a 35%-efficient engine. Calculate the useful mechanical energy.
19. Explain why nuclear fuels are described as having extremely high energy density.
20. Give two disadvantages that must still be considered when evaluating nuclear energy.
21. Explain why solar and wind resources cannot be evaluated simply by comparing fuel energy density.
22. Describe how pumped hydroelectric storage stores energy.
23. Explain why a low-energy-density storage system can still store a very large total amount of energy.
24. Why is volumetric energy density important for vehicles with limited storage space?
25. Explain how energy density influences the design of portable electronic devices.
26. List five factors other than energy density that should be considered when comparing energy resources.
27. Explain how life-cycle analysis can improve comparisons between energy technologies.
28. Why might the best energy-storage technology for an aircraft differ from the best technology for an electrical grid?
29. Explain how energy density can influence environmental impacts.
30. Evaluate the statement: "The energy source with the highest energy density is always the best energy source."

Key Terms

Energy density — Amount of energy stored or available per unit mass or volume.

Gravimetric energy density — Energy stored per unit mass.

Volumetric energy density — Energy stored per unit volume.

Energy resource — Source from which usable energy can be obtained.

Fuel — Substance that can release stored energy through a physical, chemical, or nuclear process.

Energy storage — Retaining energy so that it can be used later.

Battery — Device that stores chemical energy and can transfer it electrically.

Power density — Rate at which energy can be delivered relative to mass or volume.

Efficiency — Fraction of input energy transferred into useful output.

Renewable resource — Energy resource replenished naturally on a human timescale.

Non-renewable resource — Resource consumed faster than natural processes replace it.

Fossil fuel — Carbon-rich fuel such as coal, oil, or natural gas formed from ancient biological material.

Hydrogen — Energy carrier with high energy content per unit mass but challenging volumetric storage requirements.

Nuclear fuel — Material capable of releasing energy through nuclear reactions.

Pumped storage — Energy-storage system that stores gravitational potential energy by moving water to a higher elevation.

Life-cycle analysis — Evaluation of environmental impacts throughout production, use, and disposal.

Energy carrier — Substance or system used to transport or deliver energy from one location or process to another.

Key Takeaways

• Energy density describes how much energy is stored per unit mass or volume.
• Gravimetric energy density measures energy per unit mass.
• Volumetric energy density measures energy per unit volume.
• High energy density allows large amounts of energy to be carried using less mass or space.
• Energy density is particularly important in transportation and portable technology.
• Aircraft are especially sensitive to energy-storage mass.
• Batteries generally have lower gravimetric energy density than liquid hydrocarbon fuels, although electric motors can be highly efficient.
• Hydrogen has very high energy content per kilogram but presents significant storage challenges because of its low density.
• The complete storage system must be considered, not just the fuel or battery material.
• Nuclear fuels have extremely high energy densities compared with chemical fuels.
• High energy density does not automatically make an energy resource safe, clean, economical, or sustainable.
• Renewable resources such as wind and solar require different comparisons because they do not use conventional stored fuels.
• Energy density is different from efficiency.
• Energy density is different from power density.
• Energy density by mass and energy density by volume answer different engineering questions.
• Energy resources should be compared using multiple criteria, including efficiency, cost, emissions, storage, safety, reliability, infrastructure, renewability, and environmental impact.
• Life-cycle analysis provides a broader way to compare energy technologies.
• The best energy resource depends strongly on the intended application.

The central idea is:

energy density = energy stored / mass or volume

But choosing an energy technology requires a much broader question:

How much energy can it provide, how efficiently can we use it, what does it take to store and deliver it, and what are the consequences of doing so?