D3.4 Relate efficiency to thermal energy transfer

Ontario Grade 11 Physics

Energy and Society

How useful energy output compares with energy input

A device receives energy and transfers energy to its surroundings or to another part of a system. Efficiency tells us what fraction of the input becomes the output we want. Some energy may be transferred as thermal energy, meaning energy transferred because of a temperature difference. That transfer can be unwanted, as when a motor warms its surroundings, or wanted, as when a heater warms a room. The purpose of the device determines which energy transfer is useful.

What you will learn

Define efficiency as a comparison between useful energy output and total energy input.
Explain how thermal energy transfer can reduce efficiency or be the useful output, depending on the purpose of a device.
Calculate efficiency and identify the energy transferred usefully and as thermal energy.
Check that an efficiency result, its units, and its interpretation make sense.

1. Prerequisites: energy, systems, and direction

Energy is a scalar quantity. A scalar has magnitude but no direction. Energy is measured in joules, with the symbol

\mathrm{J}

. Power is the rate of energy transfer. It is measured in watts, where

1\ \mathrm{W}=1\ \mathrm{J/s}

. Efficiency can be calculated using energy values or power values, as long as the input and output refer to the same time period.

A system is the object or group of objects being studied. Before discussing energy transfer, identify the system and what counts as its surroundings. For example, if the system is a motor, energy transferred from the motor to the surroundings as thermal energy has left the system. For an energy-flow description, we will call energy entering the system positive and energy leaving it negative. This sign choice helps describe direction; the efficiency calculation itself uses positive input and output magnitudes.

P=\frac{E}{t}

Energy is a scalar and is measured in joules.
Power is energy transferred per second and is measured in watts.
State the system before deciding whether thermal transfer is entering or leaving.

2. Efficiency and thermal energy transfer

Efficiency compares the useful energy output with the total energy input. Useful means that the output serves the device's intended purpose. Efficiency is a ratio, so it has no unit. It is often written as a percentage by multiplying the ratio by 100%. A device cannot deliver more useful energy than the total energy supplied to it, so its efficiency cannot be greater than 100% in this model.

Not all input energy necessarily becomes useful output. Some may be transferred to the surroundings as thermal energy. For a motor intended to turn a shaft, energy transferred as unwanted warming is not useful output for that purpose. If the intended purpose is to heat a room, energy transferred as thermal energy to the room is useful. Do not decide that thermal energy is automatically a loss: decide whether that transfer matches the system's purpose.

For an energy accounting model with no other output categories, the input energy is divided between useful output energy and energy transferred as thermal energy that is not useful. All quantities in this comparison are magnitudes in joules. The transfer direction still matters in the physical description: energy may leave the device and warm its surroundings, or enter a room's heating system and warm the room.

A labelled energy-flow sketch can make the comparison clear: input energy enters the device; useful output goes toward the intended task; and unwanted thermal transfer goes to the surroundings. The arrows describe energy transfers, not moving objects. No motion direction or vector calculation is needed for this efficiency relationship.

\eta=\frac{E_{\mathrm{useful}}}{E_{\mathrm{input}}}\times100\%

Efficiency is useful output divided by total input.
Thermal transfer is useful when it serves the intended purpose and a loss when it does not.
For the stated two-output model, input equals useful output plus unwanted thermal transfer.

3. A reliable calculation method

First name the system and its intended purpose. Then list the known input and useful output, including units, and identify the unknown. If the question gives unwanted thermal transfer instead of useful output, find the useful energy by subtracting that transfer from the input in the two-output model.

Use the efficiency relationship and keep units through the energy calculation. The numerator and denominator must refer to comparable quantities for the same device and interval. When using power instead of energy, compare useful output power with input power for the same interval.

Finally, interpret the result. A value such as

0.72

means that 72% of the input is counted as useful output under the chosen purpose. Check that the ratio has no unit, that the percentage is between 0% and 100%, and that the thermal transfer has been classified according to the device's purpose.

E_{\mathrm{input}}=E_{\mathrm{useful}}+E_{\mathrm{thermal,unwanted}}

Choose what counts as useful before calculating.
Use consistent units and the same time interval.
Check the result and explain what the percentage means.

4. Comparing devices and interpreting results

A higher efficiency means a larger fraction of the input is useful for the stated purpose. It does not automatically mean that a device transfers more total energy. For instance, a device with a smaller input could have a higher efficiency but still provide less useful energy than a device with a larger input.

Comparisons are fair only when the intended purpose is clear. A heater may have thermal energy as its useful output, while a motor may treat warming of its surroundings as unwanted. If the purpose changes, the useful-output category can change too. The physical energy transfer has not changed; its classification has.

Efficiency also does not tell us where thermal energy goes unless the situation states the direction. Describe it in words: for example, energy leaves the motor and warms the surroundings. Keep this transfer description separate from the efficiency ratio, which uses positive magnitudes.

Efficiency is a fraction of input, not a measure of total energy by itself.
The intended purpose determines what counts as useful.
Describe the direction of thermal transfer in words.

Worked example

Finding a motor's efficiency

A motor receives

500\ \mathrm{J}

of energy. It transfers

140\ \mathrm{J}

as unwanted thermal energy to its surroundings. In the two-output model, determine the motor's efficiency.

Set the system and direction
The system is the motor. Its intended purpose is to produce useful output, not to warm the surroundings. The unwanted thermal energy leaves the motor, so it is an energy transfer out of the system. Use positive magnitudes in the calculation.
Find useful output
The input is divided between useful output and unwanted thermal transfer. Subtract the thermal transfer from the input.
$E_{\mathrm{useful}}=500\ \mathrm{J}-140\ \mathrm{J}=360\ \mathrm{J}$
Calculate efficiency
Divide useful output energy by input energy, then convert the ratio to a percentage.
$\eta=\frac{360\ \mathrm{J}}{500\ \mathrm{J}}\times100\%=72\%$

Answer: The motor's efficiency is 72%.

Check: The joules cancel in the ratio, leaving no unit. The answer is between 0% and 100%. The remaining 28% corresponds to

140\ \mathrm{J}

transferred as unwanted thermal energy, so the result agrees with the stated energy accounting.

Worked example

A heater with useful thermal output

A room heater receives

2.40\times10^5\ \mathrm{J}

of energy during a stated interval. It transfers

2.04\times10^5\ \mathrm{J}

as thermal energy to the room. Treat this transfer as the useful output. Calculate the efficiency.

Set the system and purpose
The system is the heater. Its purpose is to warm the room, so thermal energy transferred from the heater to the room is useful. The energy leaves the heater and enters the room.
Identify the energy values
The input energy is $2.40\times10^5\ \mathrm{J}$ , and the useful thermal output is $2.04\times10^5\ \mathrm{J}$ . Both values describe the same interval.
Calculate efficiency
Use useful output divided by input. The joule units cancel.
$\eta=\frac{2.04\times10^5\ \mathrm{J}}{2.40\times10^5\ \mathrm{J}}\times100\%=85.0\%$

Answer: The heater's efficiency for warming the room is 85.0%.

Check: The ratio is unitless and is below 100%. The unused portion is 15.0% of the input, which corresponds to

3.60\times10^4\ \mathrm{J}

not counted as useful room-heating output in this comparison. Thermal energy is useful here because it reaches the intended place.

Worked example

Using power to compare energy transfer

A device takes in energy at a rate of

80.0\ \mathrm{W}

. Its useful output power is

52.0\ \mathrm{W}

. Determine its efficiency and the power not counted as useful output.

Define the system and outputs
The system is the device, and its intended purpose is represented by the useful output power. The input rate is $80.0\ \mathrm{J/s}$ and the useful output rate is $52.0\ \mathrm{J/s}$ . Any remaining power is transferred in other ways, including possible unwanted thermal transfer.
Calculate efficiency
Power values can be compared directly because they refer to the same device and interval. Divide useful output power by input power.
$\eta=\frac{52.0\ \mathrm{W}}{80.0\ \mathrm{W}}\times100\%=65.0\%$
Find the remaining power
Subtract useful output power from input power. The difference is the total power not counted as useful output in this comparison; do not label all of it thermal unless the situation says so.
$P_{\mathrm{not\ useful}}=80.0\ \mathrm{W}-52.0\ \mathrm{W}=28.0\ \mathrm{W}$

Answer: The efficiency is 65.0%, and

28.0\ \mathrm{W}

is not counted as useful output.

Check: The watt units cancel in the efficiency ratio. The result is below 100%, and the remaining power is positive. The values add back to the input power, as expected.

Common mistakes and how to avoid them

Treating every thermal energy transfer as wasted energy.

Correction: Decide whether the thermal transfer serves the device's stated purpose. Heating a room can be useful thermal output.

Dividing input energy by useful output energy.

Correction: Efficiency uses useful output divided by total input.

Reporting a percentage greater than 100% without checking the energy values.

Correction: For this model, useful output cannot exceed total input. Recheck the values, units, and which quantity is the input.

Calling all energy that is not useful output thermal energy.

Correction: Only identify the remainder as thermal when the situation supports that claim. Otherwise call it energy not counted as useful output.

Lesson summary

Efficiency compares useful output with total input.
Thermal energy transfer may be useful or unwanted, depending on the device's purpose.
Energy input equals useful output plus unwanted thermal transfer only when those are the two output categories in the model.
Use consistent units, state the transfer direction, and check that the efficiency is sensible.

Check your understanding

Question 1

A device takes in

600\ \mathrm{J}

and provides

450\ \mathrm{J}

of useful output. What is its efficiency?

25%
75%
133%
correctIndex

Show answer and explanation

75%

Useful output divided by input is

450\ \mathrm{J}/600\ \mathrm{J}=0.75

, or 75%. The units cancel, and the value is below 100%.

Question 2

A room heater transfers thermal energy into the room it is meant to warm. How should this transfer be classified for the heater's purpose?

Useful output
Always wasted output
Input energy to the heater
correctIndex

Show answer and explanation

Useful output

The heater is intended to warm the room, so thermal energy transferred to the room is useful output.

Key terms

Efficiency: The fraction or percentage of total input energy that becomes useful output.
Thermal energy transfer: Energy transferred because of a temperature difference.
System: The object or group of objects selected for study.
Useful output: Energy transferred in a way that serves the system's intended purpose.
Power: The amount of energy transferred per second.

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Published by DoAssignment. This AI-assisted lesson follows Ontario Grade 11 Physics (SPH3U), expectation D3.4. It is a study resource, not an official curriculum publication.

Before publication, the draft is checked for structure, mathematical or chemical notation, calculations, course boundaries, and readability, and then requires administrator approval. Errors can still occur, so corrections are welcomed.

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D3.4 · Relate efficiency to thermal energy transfer

Ontario Grade 11 Physics

How useful energy output compares with energy input

What you will learn

1. Prerequisites: energy, systems, and direction

2. Efficiency and thermal energy transfer

3. A reliable calculation method

4. Comparing devices and interpreting results

Finding a motor's efficiency

A heater with useful thermal output

Using power to compare energy transfer

Common mistakes and how to avoid them

Lesson summary

Check your understanding

Question 1

Question 2

Key terms

Continue through SPH3U

About this lesson and its review