Useful output vs wasted energy Β· The efficiency equation Β· Sankey diagrams Β· Making devices more efficient
β‘
Energy transfers
Identify the useful and wasted energy transfers in everyday devices
π’
The efficiency equation
Calculate efficiency as a decimal or percentage using useful Γ· total energy or power
π
Sankey diagrams
Interpret and sketch Sankey diagrams to represent energy flow in a device
π‘οΈ
Wasted energy
Explain why energy is wasted (mostly as thermal energy) and how this relates to efficiency
π§
Improving efficiency
Describe methods engineers use to reduce energy waste in real devices
π
Conservation of energy
Use the fact that total energy in = total energy out to check calculations
β‘ Useful Energy and Wasted Energy
Whenever energy is transferred by a device, not all of it does what we want. We split the output energy into two types:
Useful energy transfer: Energy transferred to the intended store or pathway β the energy we actually want the device to produce.
Wasted energy: Energy transferred to an unintended store β energy that is dissipated (spread out) and no longer useful. It is almost always thermal energy.
The Law of Conservation of Energy tells us that:
Total energy input = Useful energy output + Wasted energy output
This means energy is never created or destroyed β it just changes form. Some of those forms are useful to us, and some are not.
Examples of energy transfers
Device
Useful output
Wasted output(s)
Electric light bulb
Light (EM radiation)
Thermal energy (heat)
Electric motor
Kinetic energy
Thermal energy, sound
Petrol engine
Kinetic energy
Thermal energy, sound
Loudspeaker
Sound energy
Thermal energy
Solar cell
Electrical energy
Thermal energy
LED bulb
Light (EM radiation)
Thermal energy (small)
Wasted energy is eventually dissipated to the thermal energy store of the surroundings, making the environment slightly warmer.
π’ The Efficiency Equation
Efficiency tells us what fraction of the total input energy is actually transferred usefully. A perfectly efficient device would have an efficiency of 1 (or 100%), but this is impossible in practice.
efficiency = useful energy output (J) Γ· total energy input (J)
efficiency = useful power output (W) Γ· total power input (W)
Efficiency is a dimensionless ratio between 0 and 1. To convert to a percentage, multiply by 100:
% efficiency = (useful energy output Γ· total energy input) Γ 100%
Symbol table
Symbol
Quantity
Unit
Euseful
Useful energy output
Joules (J)
Etotal
Total energy input
Joules (J)
Puseful
Useful power output
Watts (W)
Ptotal
Total power input
Watts (W)
Ξ· (eta)
Efficiency
None (ratio) or %
Example: old filament bulb β 5% efficient
5%
95% wasted
Example: LED bulb β 85% efficient
85% useful
15%
Useful energy
Wasted energy
Efficiency can never be greater than 1 (or 100%) because that would mean creating energy β violating the law of conservation of energy.
You can also rearrange to find useful output or total input:
Useful energy output = efficiency Γ total energy input
Total energy input = useful energy output Γ· efficiency
π Sankey Diagrams
A Sankey diagram is a flow diagram that shows energy transfers in a device. The width of each arrow is proportional to the amount of energy it represents.
Sankey diagram: A scale diagram where arrow widths represent the amount of energy transferred. The single input arrow splits into useful (straight) and wasted (bent downward) arrows.
How to read a Sankey diagram
The input arrow enters from the left β its width represents total energy input.
The useful output arrow continues straight to the right.
The wasted output arrow(s) bend downward (usually labelled "thermal energy to surroundings").
The widths of all output arrows must add up to the width of the input arrow.
In the diagram above, 100 J enters the device. 60 J leaves as useful energy, and 40 J is wasted as heat. The efficiency = 60 Γ· 100 = 0.6 (60%).
Key rules for drawing Sankey diagrams
Arrow widths must be drawn to scale (e.g., 1 mm per joule).
Label every arrow with the energy type and value.
Wasted energy arrows always point downward.
All output widths sum to equal the input width.
A more efficient device has a thicker useful arrow and a thinner wasted arrow. A perfectly efficient device would have no downward arrows at all.
π§ Improving Efficiency of Devices
Engineers constantly look for ways to reduce wasted energy and make devices more efficient. Since most wasted energy becomes thermal energy, strategies often focus on reducing unwanted heating or friction.
π’οΈ Lubrication
Oil or grease between moving parts reduces friction, so less kinetic energy is converted to unwanted thermal energy. Used in engines, gearboxes, bicycle chains.
π Thermal insulation
Insulating materials (foam, fibreglass, double glazing) slow down thermal energy transfer to the surroundings. Used in buildings, ovens, pipes.
π‘ Better components
Replacing old technology with newer designs β e.g., LED bulbs instead of filament bulbs, or brushless motors instead of brushed motors β dramatically reduces losses.
π Streamlining
Aerodynamic shapes reduce air resistance, so vehicles waste less energy pushing air out of the way. Used in cars, trains, aircraft.
βοΈ Regenerative braking
In electric vehicles, braking recovers kinetic energy as electrical energy rather than wasting it as heat in brake pads.
π Heat exchangers
Capture thermal energy that would otherwise escape and use it to pre-heat incoming materials β very common in industrial processes and condensing boilers.
No device can ever be 100% efficient. There will always be some energy dissipated to the thermal store of the surroundings. The goal is to minimise this waste.
Why efficiency matters
Higher efficiency means:
Less fuel or electricity is needed to do the same job β lower running costs
Fewer fossil fuels are burned β less COβ and greenhouse gas emissions
Less waste heat released into the environment β reduced thermal pollution
Longer-lasting devices (less heat damage to components)
βοΈ Worked Examples
Example 1: A filament light bulb has a total power input of 60 W. It transfers 3 W as useful light energy. Calculate the efficiency of the bulb as a decimal and as a percentage. State how much power is wasted.
1
Identify the values given:
Total power input = 60 W
Useful power output = 3 W
2
Write the equation:
efficiency = useful power output Γ· total power input
3
Substitute and calculate:
efficiency = 3 Γ· 60 = 0.05
Find wasted power:
Wasted power = 60 β 3 = 57 W (as thermal energy/heat)
Efficiency = 0.05 (5%) | Wasted power = 57 W
Example 2: A electric motor takes in 2400 J of electrical energy. Its efficiency is 0.75. Calculate the useful kinetic energy output and the energy wasted as heat.
1
Identify the values given:
Total energy input = 2400 J
Efficiency = 0.75
2
Rearrange the equation for useful energy output:
Useful energy output = efficiency Γ total energy input
3
Substitute and calculate:
Useful energy output = 0.75 Γ 2400 = 1800 J
Useful kinetic energy = 1800 J | Wasted thermal energy = 600 J
Example 3: A gas boiler transfers 12,000 J of chemical energy from burning gas. The useful thermal energy delivered to the home is 9,600 J. Calculate the efficiency and the wasted energy. Describe how a condensing boiler reduces waste.
1
Identify the values given:
Total energy input = 12,000 J
Useful energy output = 9,600 J
Describe how efficiency can be improved:
A condensing boiler contains a heat exchanger that captures thermal energy from the waste exhaust gases (which would otherwise escape up the flue). This captured energy is used to pre-heat the cold water returning to the boiler, so less chemical energy is needed to heat it β increasing efficiency to around 90β95%.
Efficiency = 0.80 (80%) | Wasted energy = 2,400 J | Condensing boilers use a heat exchanger to recover waste heat from exhaust gases.
Example 4 (Sankey diagram): A car engine receives 500 J of chemical energy from fuel. It produces 150 J of kinetic energy, 275 J of thermal energy from the exhaust, and 75 J of sound and other losses. Draw the figures on a Sankey diagram and calculate efficiency.
Sankey diagram values (scale: 1 mm per 10 J, so 50 mm input):
Input arrow: 50 mm wide
Useful (kinetic) arrow: 15 mm wide
Exhaust heat arrow: 27.5 mm wide
Sound/friction arrow: 7.5 mm wide
Efficiency = 0.30 (30%) | Most energy (55%) is wasted as exhaust heat; only 30% becomes useful kinetic energy.
π² Practice Questions
Q1. A kettle has a total power input of 2000 W. It wastes 200 W as thermal energy to the surroundings. What is its efficiency?
Q2. A solar panel receives 5000 J of light energy from the Sun and produces 850 J of electrical energy. Calculate the efficiency as a percentage. Give your answer to 2 significant figures.
Q3. In a Sankey diagram, wasted energy arrows are always drawn pointing in which direction?
Q4. An electric motor has an efficiency of 0.65 and a total power input of 400 W. Calculate the useful power output in watts.
Q5. Which of the following changes would increase the efficiency of a car engine?
π Challenge Questions
Exam-style questions β attempt each one before revealing the answer.
C1. A power station burns coal and produces 900 MW of total power. It delivers 360 MW of electrical power to the National Grid. The remaining energy is wasted as thermal energy to a nearby river.
(a) Calculate the efficiency of the power station.
(b) Calculate the power wasted as thermal energy to the river.
(c) Explain one environmental consequence of this thermal energy being released into the river.
(a) efficiency = 360 Γ· 900 = 0.40 (40%)
(b) Wasted power = 900 β 360 = 540 MW
(c) The thermal energy heats the river water, reducing the dissolved oxygen content. This can harm or kill aquatic organisms (fish, invertebrates) that rely on dissolved oxygen to respire. This is called thermal pollution.
C2. A student tests two light bulbs using a joulemeter:
Bulb
Energy input (J)
Useful light energy (J)
Filament bulb
600
30
LED bulb
600
510
(a) Calculate the efficiency of each bulb.
(b) For the same light output, suggest how much more electrical energy the filament bulb uses compared to the LED. Show your reasoning.
(c) Describe what happens to the wasted energy in each case.
(b) To produce 30 J of light, the filament bulb needs 600 J input. The LED only needs 30 Γ· 0.85 β 35.3 J input. So the filament bulb uses approximately 600 Γ· 35.3 β 17 times more energy for the same light output.
(c) In both bulbs, the wasted energy is transferred to the thermal energy store of the surroundings β it heats the air around the bulb. Because the filament bulb wastes 95%, it becomes very hot to touch. The LED only wastes 15%, so it stays much cooler.
C3. A student draws a Sankey diagram for a petrol car engine. The input is 1000 J of chemical energy. The engine produces 280 J of kinetic energy, 620 J of thermal energy (exhaust and engine heat), and 100 J of sound energy.
(a) Verify that energy is conserved.
(b) Calculate the efficiency of the engine.
(c) An engineer proposes fitting a heat exchanger to recover 200 J of the exhaust thermal energy and use it to heat the passenger cabin. Explain whether this increases the engine's efficiency, and calculate the new efficiency if the kinetic energy output remains unchanged but the 200 J of heat for the cabin is now counted as useful output.
(a) Total output = 280 + 620 + 100 = 1000 J β Energy is conserved.
(b) efficiency = 280 Γ· 1000 = 0.28 (28%)
(c) Using a heat exchanger does not change the total energy input or the laws of thermodynamics β but it does change what we count as "useful." If the 200 J of cabin heating is now useful, then:
Useful output = 280 (kinetic) + 200 (cabin heat) = 480 J
New efficiency = 480 Γ· 1000 = 0.48 (48%)
This is correct β the engine itself has not become more mechanically efficient, but the system as a whole is using more of the available energy usefully, so the overall efficiency of the vehicle system is higher.
C4 (Extended writing): Explain how the efficiency of a household heating system can be improved, and evaluate the environmental and economic benefits of doing so. [6 marks]
Model answer (6-mark level):
Methods of improvement:
β’ Installing a condensing boiler: uses a heat exchanger to recover thermal energy from exhaust gases that would otherwise escape, increasing efficiency from ~80% to ~90β95%.
β’ Adding wall and loft insulation: reduces the rate of thermal energy loss from the building, so the boiler does not need to work as hard.
β’ Fitting a smart thermostat/programmer: ensures heating only runs when needed, reducing total energy input required.
β’ Double or triple glazing: reduces conduction and convection losses through windows.
Environmental benefits:
β’ Less fuel (often natural gas) is burned for the same heat output, so fewer greenhouse gases (COβ, CHβ) are emitted, reducing the contribution to climate change.
β’ Reduced demand on fossil fuel reserves.
Economic benefits:
β’ Lower fuel bills for the household β less energy needed means less money spent.
β’ Although installation costs exist (e.g., new boiler, insulation), long-term savings outweigh initial costs.
β’ Government incentives (grants) often help offset initial costs.
A good answer links each method to the physics principle (reducing energy dissipation) and gives specific, quantified or detailed reasoning.