4.4-4.5 - Efficiency and Sankey diagrams

What efficiency means

A device is useful because it makes a wanted energy transfer. A lamp is useful because some electrical energy is transferred by radiation as light. A motor is useful because some input energy is transferred mechanically to a kinetic energy store. A speaker is useful because some energy is transferred by radiation as sound.

The useful output is the part of the output energy that does the job the device is meant to do. The wasted output is the part that is transferred in other ways. Wasted energy is not destroyed; it has usually spread out to the surroundings and is harder to use.

The total energy output is all the output pathways added together:

When no energy accumulates in the device, conservation of energy means total input equals total output. If its own stores change, include that change in the energy account. Edexcel states the efficiency relationship using useful energy output and total energy output. So if a question gives the total energy input instead, use that value as the total energy output. You must remember this equation: it is not given in the exam.

An efficiency of $100\%$ would mean every joule of output energy is useful. Efficiency cannot exceed 100%, because the useful output is only part of the total output. If a calculation gives more than $100\%$, check that the useful output is on top and the total output is underneath. A device with unwanted outputs has a lower efficiency; an ideal heater can be treated as 100% efficient if all its output heats the intended space.

Using the equation

The efficiency equation is a ratio, so the two energy values must be in the same unit. If the useful output is in joules, the total output must also be in joules. If the values are both in kilojoules, the units cancel in the same way. Because the units cancel, efficiency has no unit. Without the $\times 100\%$, the same calculation gives efficiency as a decimal, such as $0.18$; both forms are correct, and a decimal efficiency is never more than $1$.

Worked example: a lamp transfers $100 J$ of energy in total. $18 J$ is transferred usefully as light. Calculate the efficiency of the lamp.

The remaining $82 J$ is wasted output, mostly transferred as thermal energy to the surroundings. The calculation uses $18 J$, not $82 J$, because efficiency is about the useful part.

If efficiency is given and you need the useful output, change the percentage into a decimal multiplier. For example, $65\% = 0.65$.

Worked example: a device has a total energy output of $240 J$ and an efficiency of $65\%$. Find the useful output.

Then the wasted output is:

The answer is sensible because the useful and wasted outputs add back to the total output: $156 J + 84 J = 240 J$.

If a motor transfers 140 J usefully and 60 J is wasted, first add the outputs: $140+60=200\,\mathrm{J}$. Efficiency is $140/200\times100\%=70\%$. Dividing by 60 J would compare useful with wasted energy, not with the total.

To find the total output from the useful output and the efficiency, divide by the efficiency as a decimal. For example, a motor that is $40\%$ efficient gives $300 J$ of useful output:

Reading Sankey diagrams

A Sankey diagram represents energy transfers using arrows. The arrow on the left usually shows the total input energy. Arrows leaving the device show the output energy pathways.

The important feature is arrow width. A wider arrow represents a larger amount of energy. The length of an arrow is not the quantity being measured.