D. Energy and Society · Work, Energy and Power

Efficiency and Thermal Energy

D3.3 D3.4 D2.1 D3.1About 70 minutes

Learning goals

I can calculate percent efficiency from useful output and total input, as energy or power.
I can explain why no real device is 100% efficient, and why the wasted energy is usually thermal energy.
I can describe the relationship between efficiency and thermal energy transfer to the surroundings.
I can draw and interpret a Sankey diagram and find the overall efficiency of a chain of devices.

Words to know

efficiency
The fraction of the input energy that becomes useful output: efficiency = (useful output energy ÷ total input energy) × 100%.

useful output energy
The energy in the form the device is meant to produce (for example, kinetic energy from a motor).

wasted (non-useful) energy
Input energy that becomes forms we did not want, usually thermal energy and sound that spread into the surroundings.

Sankey diagram
A flow diagram in which the width of each arrow is proportional to the amount of energy it carries.

heat engine
A device that uses thermal energy to do mechanical work, such as a car engine or a steam turbine. It must release part of its input as thermal energy to cooler surroundings.

Where does the fuel go?

Fill a car with gasoline and drive it. Only part of the fuel's chemical energy ends up moving the car. The rest warms the engine, the exhaust, the radiator and the air. That is why a car engine needs a cooling system, and why the hood is warm after a drive. Efficiency tells us what share of the energy put in comes out as the form we want. It is one of the most important numbers in engineering, and it matters for household budgets and for the climate.

Calculating efficiency

efficiency = (useful output energy ÷ total input energy) × 100%

Because power is energy per second, you can also use power, measured over the same time:

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

Other ways to write the same idea:

useful output = efficiency × input
input = useful output ÷ efficiency
wasted energy = input − useful output

Use a decimal (0.75) inside calculations and a percent (75%) in final answers. Efficiency has no units, because the joules cancel.

Worked example

Example 1: Efficiency of a hoist motor

An electric hoist motor uses 8000 J of electrical energy to lift a 50 kg crate 12 m. Find (a) the useful output energy, (b) the efficiency, (c) the energy wasted.

(a) Useful output = gain in Eg = mgΔh = (50)(9.8)(12) = 5880 J.
(b) Efficiency = 5880 ÷ 8000 × 100% = 73.5% ≈ 74%.
(c) Wasted = 8000 − 5880 = 2120 J, mostly thermal energy in the motor windings, bearings and cable, plus some sound.

5880 J useful; about 74% efficient; 2120 J (2.1 × 10³ J) wasted.

Worked example

Example 2: Efficiency of a ramp

A worker pushes a 30 kg box 4.0 m up a ramp, using a force of 140 N along the ramp. The ramp rises 1.2 m. Find the efficiency of the ramp and the thermal energy produced by friction.

Input work = FΔd = (140 N)(4.0 m) = 560 J.
Useful output = mgΔh = (30)(9.8)(1.2) = 352.8 J.
Efficiency = 352.8 ÷ 560 × 100% = 63%.
Thermal energy = 560 − 352.8 ≈ 207 J (assuming the box ends at the same speed it started).

The ramp is about 63% efficient; about 2.1 × 10² J becomes thermal energy.

Why is nothing 100% efficient?

In every real device, some energy is transformed into forms we did not intend:

Friction in moving parts (axles, gears, bearings) and air resistance turn kinetic energy into thermal energy.
Electrical resistance in wires and motor coils turns electrical energy into thermal energy.
Vibration and sound carry energy away, and they end up as thermal energy too.

Almost all wasted energy ends up as thermal energy spread out in the surroundings at a temperature only a little above room temperature. Energy that is spread out like this is very hard to collect and use again. So even though energy is conserved, the useful energy available keeps going down.

A Sankey diagram for a car engine (example values). Arrow widths are proportional to energy: 100 J of chemical energy becomes 25 J of useful kinetic energy and 75 J of thermal energy. The efficiency is 25%.

Efficiency and thermal energy transfer

The link between efficiency and thermal energy is direct: the more of the input that ends up transferred to the surroundings as thermal energy, the lower the efficiency.

Heat engines (car engines, and the steam turbines in thermal and nuclear generating stations) turn thermal energy into mechanical work. To keep running, they must release a large share of their input as thermal energy to cooler surroundings (the radiator and exhaust, or a lake or cooling tower). This limits their efficiency, often to well under 50%. The Generating Electrical Energy lesson compares real stations.
Electrical devices waste energy mostly through resistance and friction. Electric motors are often 85% to 95% efficient; old incandescent bulbs turned only a few percent of their electrical energy into light.
When heating is the goal, the "waste" becomes the useful output. An electric baseboard heater or a kettle turns nearly all of its electrical energy into thermal energy where you want it.

Improving efficiency usually means reducing unwanted thermal energy transfer (lubricating moving parts, using thicker wires, insulating hot parts) or using the waste thermal energy for something else (a car heater uses engine heat in winter; some power plants supply waste heat to warm buildings).

Worked example

Example 3: Car engine power

A car engine takes in chemical energy from fuel at a rate of 120 kW, and delivers 30 kW to turn the wheels. (a) What is its efficiency? (b) At what rate is thermal energy released to the surroundings? (c) How much thermal energy is released in a 15 min drive at this rate?

(a) Efficiency = 30 ÷ 120 × 100% = 25%.
(b) Wasted power = 120 − 30 = 90 kW, mostly thermal energy.
(c) ΔE = PΔt = (90 000 W)(900 s) = 8.1 × 10⁷ J (81 MJ).

25% efficient; 90 kW of thermal power; 8.1 × 10⁷ J in 15 minutes.

Worked example

Example 4: Efficiency of a chain

Electrical energy is produced at a generating station with 35% efficiency, sent through transmission lines that are 94% efficient, and used by an electric motor that is 90% efficient. What is the overall efficiency from fuel to motor output?

The output of each stage is the input of the next, so the efficiencies multiply (as decimals).
Overall = 0.35 × 0.94 × 0.90 = 0.2961
Overall efficiency ≈ 30%

About 30%. Of every 100 J in the fuel, only about 30 J becomes useful motor output.

Efficiencies in a chain multiply. Each stage passes on only part of what it receives, so the overall efficiency (30%) is lower than that of any single stage.

Just for fun

An old incandescent light bulb is really a small electric heater that happens to glow a little. That's why it is so hot to the touch, and why LED bulbs, which waste much less energy as thermal energy, replaced them.

Hands-on activity

Waste-heat detective

You need: notebook, pencil, a timer and household devices that are in normal use.

Grown-up note: check for warmth by holding the back of your hand near a surface, not by touching it, and never near a bulb, stove, toaster, iron or hair dryer element. Do not unplug or open anything, and keep hands dry.

Choose five devices that have been running for at least 15 minutes: for example a laptop, a phone charger, a TV, a fridge (feel the air near the grille at the bottom or back) and a game console.
Rank them from "warmest nearby" to "coolest nearby".
For each, write the useful output and the wasted output.
Draw a rough Sankey diagram for two of them, making the arrow widths match your estimate of how much energy is wasted.
Suggest one way to improve the efficiency of each device, or to use its waste thermal energy.

What to look for: sample: a laptop (useful: light and sound from the screen and speakers, plus computing; most ends up as thermal energy, which is why it has a fan); a fridge (it moves thermal energy out of the food compartment and releases it, plus the motor's wasted energy, into the kitchen, so the air near the grille is warm). A device that feels noticeably warm is turning a lot of its input into thermal energy.

Your turn

A motor uses 1500 J of electrical energy and produces 1200 J of kinetic energy. Find its efficiency.

A 60% efficient engine produces 900 J of useful work. How much energy did it take in?

A pump uses 2.0 kW of electrical power and lifts water at a useful rate of 1.3 kW. Find its efficiency and the wasted power.

A 40 kg box is lifted 2.5 m by a winch that uses 1600 J. Find the efficiency.

A student claims a machine is 110% efficient. What is wrong?

Find the overall efficiency of a 90% efficient generator powering an 80% efficient motor.

Why is an electric kettle nearly 100% efficient, while a car engine is not?

More practice: warm-up

Write the formula for efficiency.

Input 200 J, useful output 50 J. Efficiency?

What are the units of efficiency?

Into what form does most wasted energy end up?

In a Sankey diagram, what does the width of an arrow show?

More practice: on your own

A crane motor is 70% efficient. How much electrical energy does it need to lift a 600 kg beam 15 m?

A 75 kg cyclist (with bike) rides up a 40 m hill. Her body transforms 1.5 × 10⁵ J of chemical energy. Ignoring friction, find the efficiency of her body plus bike.

A 1500 W hair dryer delivers 1000 W as thermal energy in the air stream, 50 W as kinetic energy of the air, and the rest as thermal energy in the dryer body and sound. If the useful output is the warm, moving air, find the efficiency.

Explain, using thermal energy, why lubricating the chain of a bicycle increases its efficiency.

A three-stage system has efficiencies 80%, 75% and 90%. What overall input is needed for a useful output of 540 J?

Challenge

A ramp 5.0 m long rises 1.5 m. A 40 kg crate is pushed up it at constant speed. The ramp's efficiency is 70%. Find (a) the work input, (b) the push force along the ramp, (c) the friction force.

An electric car's battery delivers 60 kW to a motor that is 92% efficient, while the car climbs a hill at constant speed. Air resistance and rolling friction need 25 kW. How fast is the car gaining Eg (in kW)?

Explain why a car engine is often more "useful" overall in a Canadian winter than in summer, even though its efficiency at turning chemical energy into kinetic energy is unchanged.

Check your understanding

Quiz

A device takes in 400 J and gives out 300 J of useful energy. Its efficiency is:

Most of the energy wasted by machines ends up as:

A motor is 80% efficient and must deliver 400 J of useful work. The input needed is:

Two devices in a chain are 50% and 60% efficient. The overall efficiency is:

Which statement best describes the relationship between efficiency and thermal energy transfer?

Why is efficiency never greater than 100%?

For parents and teachers

Big idea: Efficiency measures how much input becomes useful output (D3.3), and the waste is usually thermal energy transferred to the surroundings (D3.4). This builds on the Grade 8 efficiency of machines and Grade 9 Sankey diagrams.

How to help: Compare EnerGuide labels on appliances, or talk about why the car heater works in winter. Ask "What is the useful output here, and where does the rest go?"

Common difficulties:

Dividing input by output (giving answers above 100%).
Adding efficiencies in a chain instead of multiplying them.
Using percent values (35 × 94) instead of decimals (0.35 × 0.94) when multiplying.
Thinking thermal energy is always waste (it is the useful output of heaters and kettles).

Safety: the waste-heat activity is done by holding a hand near devices, never touching hot surfaces or bulbs.