Examples, calculations and Sankey diagrams

Doc Brown's Physics Revision Notes

Suitable for GCSE/IGCSE Physics/Science courses or their equivalent

This page will answer many questions e.g.

What do we mean by efficiency in energy transfer?

How do you calculate efficiency? What is the efficiency formula?

What does a Sankey diagram tell you?

How do you draw a Sankey diagram?

Energy transfer and efficiency

  • a) Know and understand that energy can be transferred usefully from one form to another, or stored, or dissipated, but energy cannot be created or destroyed.

    • This is the law of conservation of energy.

    • However, energy is only useful if it can be converted from one form to another.

    • Examples - from a suitable energy source ==> useful form (plus waste in most cases)

      • When a gun fires chemical energy is converted into heat energy, sound energy, light energy and mainly kinetic energy. When the bullet embeds itself into some material the kinetic energy of movement is converted into some sound energy, but mainly heat energy.

      • Photovoltaic solar panels convert light energy into electrical energy.

      • We use a large number of electrical devices in the home eg

        • TV converts electrical energy into useful light and sound, but some waste heat

        • A charged mobile phone battery converts chemical energy into electrical energy, which in turn is converted into useful light and sound energy.

      • Wind turbines convert kinetic energy into electrical energy

  • b) Know and understand that when energy is transferred only part of it may be usefully transferred, the rest is ‘wasted’.

  • c) Know and understand that wasted energy is eventually transferred to the surroundings, which will become warmer.

    • Appreciate that the wasted energy becomes increasingly spread out and so becomes less useful.

  • d) Be able to calculate the efficiency of a device using the equations:

    • % efficiency = useful energy out x 100 / total energy in

    • % efficiency = useful power out x 100 / total power in

    • With the time 100 the efficiency has a value between 0% and 100%.

    • Omitting the 100 gives you the efficiency as a decimal fraction between zero (0) and one (1).

    • You must be able to calculate efficiency as a decimal fraction (ie omit the x 100) or as a percentage (0-100%).

    • The efficiency formula triangle if you need to rearrange the equation to calculate energy in or energy out.

    • It should be pointed out that virtually no device is 100% efficient, there is no such device as a perfect machine.


  • The Law of conservation of energy states that energy cannot be created but only changes from one form to another.

  • Know that appliances transfer energy but they rarely transfer all of the energy to the place we want, but whatever happens to the energy, its neither lost nor gained, but not all of it is usefully transferred.

    • The Sankey diagram above illustrates how you can represent what happens to the energy when some kind of energy consuming device is in operation.

    • The energy quantity indicated by the purple arrow must equal the energy content of the blue arrow plus the energy content of the red arrow.

  • Appreciate that we need to know the efficiency of appliances so that we can choose between them, including how cost effective they are, and how to improve them.

    • Devices eg industrial machines, household appliances, light bulbs etc. can only be useful if they can efficiently transform one form of 'source' energy ('total input') into an appreciable percentage of a 'useful' energy ('useful output').

    • Such devices should be designed to 'waste' as little energy as possible, the less waste energy, the greater the efficiency of the device.

    • See the energy flow diagram and efficiency formula 'triangle' above and the calculation of efficiency and Sankey diagram further down the page.

    • It should be pointed out that virtually no device is 100% efficient, there is no such device as a perfect machine.

      • One of the most efficient 'energy converters' is an electric heater were nearly all the input electrical energy is converted into useful output heat energy

        • It is sometimes stated that an electrical heat is 100% efficient, BUT if the electric bar is glowing and visible, then there must be some energy loss as light.

    • Whether of not the energy outputs are useful or waste, most of the energy input in a device ultimately ends up as heat energy.

    • The most useful energy sources are in a sense 'highly concentrated' like a battery or a fuel like petrol, but as the energy is used you cannot recover the waste energy - it has been dissipated to the surroundings and become useless heat energy in this 'diluted' form.

  • You are expected to use your skills, knowledge and understanding ...

    • Be able to compare the efficiency and cost effectiveness of methods used to reduce ‘energy consumption’,

      • You should know what the term ‘pay-back time’ means - the time it takes to recover your energy investment from the savings you make from eg installing insulation, low energy light bulbs, any new-replacement appliance etc.

      • A general formula to calculate payback time

      •   cost of installation/device etc. (£)
        payback time (years)   = -------------------------------------------------------
          savings per year in energy/fuel costs (£)
      • cost of installation (£) ÷ savings per year in fuel costs (£)

      • You should be able to make judgements about the cost effectiveness of different methods of reducing energy consumption over a set period of time.

        • This is not restricted to a consideration of building insulation but may include:

          • low energy light bulbs and LED lighting, these are 4-10 times more efficient than the old filament bulbs in terms of useful output of light energy. They are more expensive but are designed to last a lot longer and these lighting devices are cost effective with a payback time of months. If an LED bulb cost £5 and saved £15 a year on the electricity bill, the payback time is 4 months.

            • payback time = £5/£15 = 1/3 year (4 months)

            • LED bulbs are more costly than low energy bulbs but can provide even greater savings.

          • replacing old appliances with energy efficient ones

          • ways in which ‘waste’ energy can be useful, eg heat exchangers. Heat exchanges are a means of using potentially waste heat. To extract the heat from a device or industrial process, a cooler fluid (gas/liquid) is brought into contact with the heat source and so heats up via a heat exchanger. The now hotter fluid can now be passed through another heat exchange system to re-release the heat to some useful purpose.

        • Example of heater exchangers

          • Some of the heat from a car engine is passed into a heat exchanger and released through the car's interior heating system.

          • In some industrial processes which involve an exothermic (heat releasing) reaction eg manufacture of ammonia from hydrogen and nitrogen, some of the heat from the reaction is used to heat the incoming reactant gases to the correct high temperature for the reaction.

    • Be able to describe the energy transfers and the main energy wastages that occur with a range of appliances.

      • You should be familiar with common electrical appliances found in the home as these will be examined on.

      • Examples will not be limited to electrical appliances; however, in this case all the information would be given in the question.

      • Modern appliances are much better designed these days to be 'greener' and waste much less energy, but this does come at a price when you come to buy your more expensive replacement.

        • You payback time will depend very much on the cost effectiveness of your purchase!

    • Be able to interpret and draw a Sankey diagram.

      • You should be able to use a Sankey diagram to calculate the efficiency of an appliance.

      • From a Sankey diagram you can see quite clearly in a visual way what happens to the energy input into a device ie what proportion of energy was useful and how much energy was wasted.

      • The greater the width of the 'arrow' the greater proportion of energy it represents.

      • An electric motor illustrates the idea of a Sankey diagram.

      • You will find an electrical motor in devices such as an electric drill, washing machine, food mixer, electric car etc. etc. A lot of our lives runs on electrical power!

      • The Sankey diagram above analyses what happens to every 100 J of electrical energy that is used by the electric motor.

      • total energy in = total energy out = T% = 100% = K% + S% + H%

      • This electric motor would have an efficiency of 56% useful energy out as kinetic energy with losses of 17% sound from friction-vibration and 27% heat energy loss from moving parts friction or warm electrical wiring in the motor.

      • In this example the numbers are easy, but whatever the numbers are, from the Sankey diagram, you need to be able to get to the proportion of useful energy and covert it to a percentage of the total energy input.

Energy resources and electrical power supply revision notes index

Types of energy - a comparison with examples explained

Energy resources and their uses - a general survey

Renewable energy (1) Wind power and solar power, advantages and disadvantages

Renewable energy (2) Hydroelectric power and geothermal power, advantages and disadvantages

Renewable energy (3) Wave power and tidal power, advantages and disadvantages

Biofuels, renewables and non-renewables, advantages and disadvantages

Generating electricity and the 'National Grid' power supply, mention of small scale supplies

Energy transfer and efficiency - calculations and Sankey diagrams

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