Physics 001.001.019 Efficiency in Systems

Alignment

Learning Intentions

By the end of the lesson, students will be able to:

• Use the first law of thermodynamics, , to account for energy transfers involving heat and work.
• Calculate efficiency using .
• Identify useful energy output and wasted energy in heat transfer systems.
• Solve multi-step problems involving heat transfer, work, internal energy and efficiency.

Success Criteria

By the end of the lesson, students have successfully:

• Identified whether energy is transferred by heating, work, or both.
• Applied the sign convention for and correctly.
• Calculated , useful energy output, wasted energy and percentage efficiency.
• Explained why no real heat transfer system is efficient.
• Communicated solutions using correct units, substitutions and significant figures.

Syllabus Reference

• Unit 1: Thermal, Nuclear and Electrical Physics
• Topic 1: Heating Processes
• Solve problems involving the efficiency of heat transfers using and

Phenomenon

A car engine burns fuel to release thermal energy. Some of this energy is transformed into useful mechanical work that moves the car, but much of it is transferred as heat to the engine block, coolant, exhaust gases and surroundings.

Even though energy is conserved, not all energy remains useful. The total energy is still accounted for, but the useful output is always less than the input.

Key Idea

Efficiency compares the useful energy output of a system to the total energy input. In heat transfer systems, some energy is always transferred to the surroundings as less useful thermal energy.

Concept

The first law of thermodynamics states that the change in internal energy of a system is equal to the energy transferred by heating plus the work done on or by the system:

Efficiency is calculated using:

For a heat engine or energy transfer device:

So:

Convention

The sign convention for is:

• : heat is transferred into the system.
• : heat is transferred out of the system.
• : work is done on the system.
• : work is done by the system.
• : internal energy increases.
• : internal energy decreases.

Efficiency should be expressed as a percentage and must be between and for real systems.

Misconceptions

Common misconceptions students have regarding the concept when applying to various situations and solving problems. It could be a conceptual, mathematical or logical misconception.

• Energy is “lost” rather than transferred into less useful forms.
• Efficiency means total energy output divided by total energy input, rather than useful output divided by total input.
• Heat and temperature are the same thing.
• A system can be more than efficient.
• Work done by the system should be treated as positive in all situations, rather than following the stated sign convention.

Further Reading

• Thermal efficiency of engines
• First law of thermodynamics
• Heat engines and refrigerators
• Energy conservation and energy degradation
• Sustainable energy technologies

Explicit Instruction

Efficiency problems usually require two ideas:

1. Conservation of energy using the first law:

2. Usefulness of energy transfer using efficiency:

A good problem-solving process is:

1. Identify the system.
2. Identify the energy input.
3. Identify the useful energy output.
4. Identify wasted energy if needed.
5. Apply if heat and work are involved.
6. Apply .
7. Check that the answer is physically reasonable.

Worked Examples

Worked Example 1

A heat engine receives of thermal energy from burning fuel. It does of useful work. Calculate the efficiency of the engine.

Therefore, the engine is efficient.

The wasted energy is:

So, is transferred to the surroundings as less useful energy, mostly heat and sound.

Worked Example 2

A gas absorbs of heat energy. It expands and does of work on the surroundings.

Using the convention that work done by the system is negative:

Calculate the change in internal energy of the gas.

The internal energy of the gas increases by .

This means entered the gas by heating, but was transferred out as useful mechanical work.

Worked Example 3

A thermal system receives of energy. It does of useful work and its internal energy increases by .

a) Calculate the energy transferred to the surroundings as waste heat.

b) Calculate the efficiency of the system.

Therefore, the system is efficient.

Check for Understanding

Check 1

A device receives of energy and produces of useful output.

Calculate its efficiency.

Expected answer:

Check 2

A gas has of heat transferred into it and does of work on the surroundings.

Calculate .

Expected answer:

Check 3

A system has an efficiency of . Its useful energy output is .

Calculate the energy input.

Expected answer:

Investigation (Alternative to Explicit)

Hypothesis

If energy is transferred to heat water using an electrical heater, then the useful thermal energy gained by the water will be less than the electrical energy supplied because some energy will be transferred to the surroundings.

Data Collection

Students measure:

• Mass of water,
• Initial temperature,
• Final temperature,
• Voltage across the heater,
• Current through the heater,
• Heating time,

Useful energy gained by the water:

Electrical energy input:

Efficiency:

Analysis

Students calculate:

• Useful thermal energy gained by the water
• Electrical energy supplied to the heater
• Efficiency of the heating process
• Wasted energy transferred to the beaker, heater and surroundings

Evaluation

Students discuss:

• Sources of heat loss to the environment
• Measurement uncertainty in temperature, time, mass, current and voltage
• Whether insulation would improve efficiency
• Why the calculated efficiency may be lower than expected
• Whether the system was closed or open

Problems

The following problems are designed to practise solving efficiency and first law of thermodynamics questions.

1. A heat engine receives of thermal energy and does of useful work. Calculate its efficiency.

2. A motor receives of electrical energy and produces of useful mechanical energy. Calculate:

   • a) its efficiency
   • b) the energy wasted

3. A gas absorbs of heat and does of work on the surroundings. Calculate .

4. A gas loses of heat to the surroundings while of work is done on it. Calculate .

5. A system receives of energy. Its efficiency is . Calculate the useful energy output.

6. A heat engine has an efficiency of and produces of useful work. Calculate the energy input.

7. A thermal system receives of energy. It does of useful work and its internal energy increases by . Calculate the wasted energy.

8. A gas has and . Calculate and explain what happens to the internal energy of the gas.

9. A gas has and . Calculate and explain whether the internal energy increases or decreases.

10. A heat transfer system has an energy input of and wastes . Calculate:

    • the useful energy output
    • the efficiency

Followup

Self-check

Students should be able to answer the following questions:

• Can I identify the system being analysed?
• Can I distinguish between energy input, useful output and wasted energy?
• Can I apply the sign convention for and ?
• Can I calculate using ?
• Can I calculate efficiency using ?
• Can I explain why real systems are never perfectly efficient?

Next Topic

The next topic is the application of heating processes to energy technologies, including how improvements in efficiency affect sustainable energy use and the design of modern thermal systems.