Heat Engines

A heat engine is a device that converts part of the thermal energy transferred from a high-temperature source into useful mechanical work.

All heat engines operate using the same basic energy pathway:

hot reservoir → heat engine → useful work + cold reservoir

A heat engine cannot convert all of the thermal energy it receives into useful work. Some energy must be transferred to a lower-temperature region.

This limitation is a consequence of the Second Law of Thermodynamics.

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The Basic Parts of a Heat Engine

A heat engine requires three main components:

• hot reservoir
• engine or working substance
• cold reservoir

The engine operates repeatedly in a cycle, taking energy from the hot reservoir, converting some of it into work, and rejecting the remainder to the cold reservoir.

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The Hot Reservoir

The hot reservoir is the high-temperature energy source.

It supplies thermal energy to the engine.

Examples include:

• burning fuel
• hot combustion gases
• steam produced in a boiler
• a nuclear reactor
• geothermal heat
• concentrated solar thermal energy

Its temperature is usually represented by:

T_h

The energy transferred from it is often represented by:

Q_h

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The Cold Reservoir

The cold reservoir receives the thermal energy that the engine cannot convert into useful work.

Its temperature is represented by:

T_c

and the rejected thermal energy is:

Q_c

Possible cold reservoirs include:

• atmosphere
• cooling water
• rivers
• lakes
• oceans
• cooling towers

The cold reservoir must have:

T_c < T_h

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Energy Flow Through a Heat Engine

The basic process is:

Q_h enters engine

Some becomes:

W

and the remainder leaves as:

Q_c

Therefore:

Q_h = W + Q_c

This is an application of the:

First Law of Thermodynamics

Energy is conserved.

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Heat Engine Energy Equation

The energy relationship is:

Q_h = W + Q_c

Rearranging:

W = Q_h − Q_c

where:

• Q_h = thermal energy absorbed from hot reservoir
• Q_c = thermal energy rejected to cold reservoir
• W = useful work produced

All quantities can be measured in:

joules (J)

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Worked Example: Energy Flow

A heat engine absorbs:

Q_h = 2000 J

and transfers:

Q_c = 1300 J

to the cold reservoir.

Find the work output.

Use:

W = Q_h − Q_c

W = 2000 − 1300

W = 700 J

Therefore, the engine produces:

700 J of useful work

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What Happens to the Energy?

For the previous example:

2000 J enters

Then:

700 J → useful work

1300 J → cold reservoir

Check:

700 + 1300 = 2000

Therefore:

energy in = energy out

The First Law is satisfied.

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Heat Engine Efficiency

The efficiency of a heat engine describes the fraction of input thermal energy converted into useful work.

Efficiency is represented by:

η

The equation is:

η = W/Q_h

where:

• η = efficiency
• W = useful work output
• Q_h = thermal energy supplied by the hot reservoir

Efficiency has no unit.

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Percentage Efficiency

Efficiency is often expressed as a percentage:

η = (W/Q_h) × 100%

For example, if an engine receives 1000 J and produces 300 J of useful work:

η = (300/1000) × 100%

η = 30%

This means 30% of the input thermal energy became useful work.

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Worked Example: Engine Efficiency

An engine receives:

Q_h = 2500 J

and produces:

W = 900 J

Efficiency:

η = W/Q_h

η = 900/2500

η = 0.36

Therefore:

η = 36%

The remaining 64% is transferred elsewhere, primarily as rejected thermal energy.

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Calculating Rejected Energy

An engine receives:

5000 J

and has an efficiency of:

40%

First find useful work:

W = ηQ_h

W = 0.40 × 5000

W = 2000 J

Now:

Q_c = Q_h − W

Q_c = 5000 − 2000

Q_c = 3000 J

Therefore:

• input = 5000 J
• useful work = 2000 J
• rejected heat = 3000 J

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Efficiency Using Q_h and Q_c

Since:

W = Q_h − Q_c

and:

η = W/Q_h

we can substitute:

η = (Q_h − Q_c)/Q_h

Therefore:

η = 1 − Q_c/Q_h

This provides another useful efficiency equation.

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Worked Example

A heat engine absorbs:

Q_h = 4000 J

and rejects:

Q_c = 2800 J

Efficiency:

η = 1 − Q_c/Q_h

η = 1 − 2800/4000

η = 1 − 0.70

η = 0.30

Therefore:

efficiency = 30%

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Why Heat Engines Need Two Reservoirs

A heat engine requires a temperature difference.

Energy enters from:

high temperature

and some energy is ultimately transferred toward:

lower temperature

This temperature difference allows the engine to produce work.

Without a temperature difference:

T_h = T_c

there would be no spontaneous net heat flow available to drive the engine.

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The Second Law

The First Law tells us:

Q_h = W + Q_c

But the First Law alone does not prevent:

Q_c = 0

which would imply:

W = Q_h

and therefore:

η = 100%

The Second Law of Thermodynamics tells us why this cannot happen for a cyclic heat engine operating between finite-temperature reservoirs.

Some thermal energy must be rejected to the colder reservoir.

Therefore:

Q_c > 0

and:

η < 100%

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Why 100% Efficiency Is Impossible

A heat engine cannot completely convert thermal energy from a single reservoir into work during a cycle.

Some energy must be transferred to the colder surroundings.

This is sometimes expressed through the Kelvin-Planck statement of the Second Law.

In simple terms:

no heat engine can be 100% efficient

This is a fundamental physical limitation, not merely a failure of engineering.

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Reversible Heat Engines

A perfectly reversible heat engine is an idealized engine with no entropy-producing processes such as:

• friction
• turbulence
• uncontrolled expansion
• heat transfer across finite temperature differences

Such an engine would have the:

maximum possible efficiency

between its two operating temperatures.

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The Carnot Engine

The Carnot engine is an ideal reversible heat engine.

Its maximum theoretical efficiency depends only on the temperatures of its hot and cold reservoirs.

The Carnot efficiency is:

η_max = 1 − T_c/T_h

where:

• T_h = hot reservoir temperature
• T_c = cold reservoir temperature

Both temperatures must be measured in:

kelvin

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Worked Example: Carnot Efficiency

An ideal heat engine operates between:

T_h = 600 K

and:

T_c = 300 K

Use:

η_max = 1 − T_c/T_h

η_max = 1 − 300/600

η_max = 1 − 0.50

η_max = 0.50

Therefore:

maximum efficiency = 50%

No heat engine operating between these two temperatures can exceed 50% efficiency.

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Another Carnot Example

Suppose:

T_h = 800 K

T_c = 320 K

η_max = 1 − 320/800

η_max = 1 − 0.40

η_max = 0.60

Therefore:

maximum theoretical efficiency = 60%

A real engine would have:

η_real < 60%

because real engines contain irreversible processes.

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Temperature Difference and Efficiency

From:

η_max = 1 − T_c/T_h

we can see that maximum efficiency increases when:

T_h increases

or:

T_c decreases

Therefore, a larger difference between hot and cold reservoir temperatures can allow greater theoretical efficiency.

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Why Kelvin Must Be Used

Suppose:

T_h = 500°C

and:

T_c = 100°C

Do not calculate:

1 − 100/500

because Carnot efficiency requires absolute temperatures.

Convert:

T_h = 500 + 273

T_h = 773 K

T_c = 100 + 273

T_c = 373 K

Then:

η_max = 1 − 373/773

η_max ≈ 0.517

or approximately:

51.7%

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Real Engines

Real heat engines have efficiencies lower than the Carnot limit because they contain:

• friction
• turbulence
• heat leakage
• electrical resistance
• finite temperature differences
• finite pressure differences
• incomplete combustion
• mechanical losses

These effects are:

irreversible

and produce entropy.

Therefore:

η_real < η_max

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Internal Combustion Engines

Gasoline and diesel engines are examples of:

internal combustion engines

Fuel burns inside the engine.

The combustion produces hot gases.

These gases expand and push pistons.

The piston motion turns a:

crankshaft

which can ultimately drive the wheels.

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Four-Stroke Gasoline Engine

A common gasoline engine uses four strokes:

Intake

Fuel-air mixture enters the cylinder.

Compression

The piston compresses the mixture.

Power

The mixture ignites.

Hot expanding gases push the piston downward.

Exhaust

Combustion products leave the cylinder.

The cycle then repeats.

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Energy Flow in a Car Engine

A simplified pathway is:

chemical energy in fuel

↓

thermal energy from combustion

↓

expanding gases

↓

mechanical work on pistons

↓

rotational mechanical energy

↓

vehicle motion

However, substantial energy is also transferred through:

• hot exhaust
• engine cooling
• friction
• sound

Therefore only part of the fuel's energy becomes useful mechanical work.

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Diesel Engines

Diesel engines are also internal combustion heat engines.

Unlike conventional gasoline engines, they ignite fuel primarily through:

high compression and resulting high temperature

rather than a spark plug.

Diesel engines are widely used in:

• trucks
• buses
• ships
• generators
• heavy machinery

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Steam Engines

A steam engine is an example of an:

external combustion heat engine

Fuel burns outside the engine's working cylinder.

Thermal energy heats water to produce steam.

The steam expands and performs mechanical work.

Historically, steam engines were central to:

the Industrial Revolution

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Steam Turbines

Modern thermal power stations generally use:

steam turbines

rather than traditional piston steam engines.

Water is heated to produce high-pressure steam.

The steam flows through turbine blades, causing the turbine to rotate.

The turbine drives an:

electrical generator

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Thermal Power Stations

A simplified power-station energy pathway is:

fuel or nuclear energy

↓

thermal energy

↓

steam

↓

turbine rotation

↓

generator

↓

electrical energy

The steam must eventually release energy to a colder reservoir.

This commonly occurs in a:

condenser

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The Condenser

After passing through a turbine, steam enters a condenser.

The condenser removes thermal energy from the steam.

The steam:

condenses back into liquid water

The water can then return to the boiler.

The condenser therefore helps establish the engine's:

cold side

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Cooling Towers

Cooling towers are often associated with thermal power stations.

They help transfer waste thermal energy to the atmosphere.

The large white clouds commonly visible above cooling towers are mainly:

condensed water droplets

rather than smoke.

Cooling systems are necessary partly because a heat engine must reject thermal energy to a colder reservoir.

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Fossil-Fuel Power Stations

In a fossil-fuel power station:

chemical energy → thermal energy → mechanical energy → electrical energy

Coal, natural gas or another fuel supplies the original energy.

Combustion provides the high-temperature reservoir.

Cooling systems provide the low-temperature reservoir.

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Nuclear Power Stations

A nuclear power station is also based on a heat-engine cycle.

The major difference is the source of thermal energy.

Instead of combustion:

nuclear fission

produces the energy.

That energy is used to produce steam and operate turbines.

Therefore:

nuclear reactor → heat source

steam turbine → heat engine

cooling system → cold reservoir

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Gas Turbines

A gas turbine is another type of heat engine.

Air is compressed.

Fuel is burned.

The resulting hot gases expand through turbine blades.

Part of the turbine's output drives the compressor, while the remainder can provide useful mechanical or electrical power.

Gas turbines are used in:

• power stations
• aircraft
• ships

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Jet Engines

Jet engines are heat engines.

A simplified sequence is:

air intake → compression → combustion → expansion → exhaust

Fuel combustion produces high-temperature gases.

These gases expand and ultimately produce thrust.

Jet engines demonstrate the conversion:

chemical energy → thermal energy → mechanical energy

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Combined-Cycle Power Plants

A gas turbine produces extremely hot exhaust gases.

Instead of immediately releasing this energy, a combined-cycle power plant uses the hot exhaust to generate steam.

The steam then powers a second turbine.

Therefore:

gas turbine + steam turbine

extract more useful energy from the same fuel.

This improves overall efficiency.

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Heat Engines in Nature

Not every heat engine needs to be a manufactured piston or turbine.

Any system that converts part of an energy flow driven by temperature differences into mechanical work can have heat-engine-like characteristics.

Atmospheric circulation, for example, is driven partly by unequal solar heating.

Temperature differences help produce:

• winds
• convection
• storms

The Earth's atmosphere can therefore be studied using thermodynamic concepts related to heat engines.

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Human Body: Not a Simple Heat Engine

The human body converts chemical energy into mechanical work and thermal energy.

However, it is not accurately modeled as a conventional heat engine operating between hot and cold reservoirs.

Muscles obtain useful work primarily through:

chemical free-energy transformations

rather than by operating a thermal cycle between two temperatures.

This distinction is important when identifying genuine heat engines.

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Heat Engine vs Electric Motor

An electric motor is not normally classified as a heat engine.

An electric motor converts:

electrical energy → mechanical energy

A heat engine converts energy supplied through a high-temperature thermal process into:

mechanical work

Examples of heat engines include:

• gasoline engine
• diesel engine
• steam engine
• steam turbine
• gas turbine
• jet engine

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Heat Engine vs Refrigerator

A heat engine and refrigerator are related but operate differently.

Heat engine

Natural heat flow helps produce work:

hot → engine → cold

with:

work output

Refrigerator

Work is supplied to move thermal energy against its spontaneous direction:

cold → hot

with:

work input

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Comparing the Two

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Efficiency and Energy Conservation

Suppose an engine is:

35% efficient

This does not mean 65% of the energy disappears.

It means:

35% → useful work

and approximately:

65% → other energy transfers

such as rejected thermal energy.

Energy is always conserved.

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Useful vs Rejected Energy

Imagine:

Q_h = 10,000 J

and:

η = 30%

Then:

W = 0.30 × 10,000

W = 3000 J

Therefore:

Q_c = 10,000 − 3000

Q_c = 7000 J

Energy accounting:

10,000 J = 3000 J + 7000 J

Nothing has disappeared.

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Efficiency Cannot Exceed 1

Because:

η = W/Q_h

and:

W < Q_h

then:

η < 1

or:

η < 100%

If a calculation gives:

η > 100%

there must be an error in:

• calculation
• measurements
• interpretation of the energy transfers

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Improving Engine Efficiency

Engineers improve real heat engines by reducing unnecessary irreversibility.

Strategies include:

• reducing friction
• improving lubrication
• improving combustion
• reducing heat leakage
• recovering exhaust energy
• improving turbine design
• reducing turbulence
• operating at higher hot-reservoir temperatures where materials permit
• improving cooling systems
• using combined cycles

However:

no engineering improvement can violate the Second Law

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Waste Heat Recovery

Many engines release hot exhaust.

Instead of simply allowing this energy to enter the environment, some systems use:

waste heat recovery

For example:

hot exhaust → heat exchanger → additional useful energy

Combined-cycle power plants use this principle.

Recovering waste heat improves overall efficiency but still cannot produce 100% conversion.

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Why Power Stations Need Cooling

A thermal power station cannot simply convert all boiler energy into electricity.

Some energy must be rejected to a colder reservoir.

Cooling systems provide this function.

Possible cold reservoirs include:

• cooling towers
• rivers
• lakes
• oceans
• atmospheric air

This requirement comes directly from the Second Law.

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Heat Engines and Entropy

Heat engines operate because energy moves from:

higher temperature → lower temperature

The engine intercepts part of this energy flow and converts some of it into work.

However, entropy considerations require some energy to continue toward the cold reservoir.

Therefore heat-engine operation connects:

• First Law
• Second Law
• entropy
• temperature
• energy conservation
• efficiency

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First Law and Heat Engines

The First Law gives:

Q_h = W + Q_c

It tells us:

energy is conserved

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Second Law and Heat Engines

The Second Law tells us:

Q_c cannot simply be eliminated

for a cyclic heat engine operating between finite-temperature reservoirs.

Therefore:

η < 100%

Together the laws explain both:

where the energy goes

and:

why there is a limit to how much becomes useful work

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Worked Example: Complete Engine Problem

A heat engine receives:

Q_h = 6000 J

and rejects:

Q_c = 3900 J

Find the work.

W = Q_h − Q_c

W = 6000 − 3900

W = 2100 J

Find the efficiency.

η = W/Q_h

η = 2100/6000

η = 0.35

η = 35%

Therefore:

• input energy = 6000 J
• work output = 2100 J
• rejected energy = 3900 J
• efficiency = 35%

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Worked Example: Finding Input Energy

An engine is:

25% efficient

and produces:

500 J

of work.

Use:

η = W/Q_h

Rearrange:

Q_h = W/η

Q_h = 500/0.25

Q_h = 2000 J

Therefore the engine must absorb:

2000 J

from the hot reservoir.

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Worked Example: Finding Rejected Energy

From the previous example:

Q_h = 2000 J

W = 500 J

Use:

Q_c = Q_h − W

Q_c = 2000 − 500

Q_c = 1500 J

Therefore 1500 J is transferred to the cold reservoir.

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Worked Example: Comparing Engines

Engine A:

Q_h = 5000 J

W = 1500 J

η_A = 1500/5000

η_A = 30%

Engine B:

Q_h = 4000 J

W = 1600 J

η_B = 1600/4000

η_B = 40%

Therefore:

Engine B is more efficient

even though it receives less total energy.

Efficiency compares the fraction converted into useful work.

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Worked Example: Real vs Maximum Efficiency

A real engine operates between:

T_h = 900 K

and:

T_c = 300 K

Maximum theoretical efficiency:

η_max = 1 − 300/900

η_max = 1 − 0.333

η_max ≈ 66.7%

Suppose the actual engine is:

40% efficient

This is physically reasonable because:

40% < 66.7%

If someone claimed the engine were 75% efficient, that would exceed the Carnot limit and would therefore be impossible between those reservoir temperatures.

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Common Mistakes

Mistake: Saying a heat engine converts all heat into work.
Some energy must be rejected to a cold reservoir.

Mistake: Saying rejected energy disappears.
Energy is conserved and transferred to the surroundings.

Mistake: Confusing Q_h and Q_c.
Q_h enters from the hot reservoir; Q_c leaves toward the cold reservoir.

Mistake: Using Q_c as the input in η = W/Q_h.
Efficiency compares useful work with energy received from the hot reservoir.

Mistake: Forgetting to convert decimal efficiency to a percentage.
0.35 = 35%.

Mistake: Writing 35% as 35 in calculations.
Use 0.35.

Mistake: Saying a reversible engine is 100% efficient.
It has the maximum possible efficiency between its reservoir temperatures, not necessarily 100%.

Mistake: Using Celsius in η_max = 1 − T_c/T_h.
Use kelvin.

Mistake: Assuming an electric motor is a heat engine.
An electric motor primarily converts electrical energy directly into mechanical work.

Mistake: Saying power-station cooling towers are simply wasting energy because of poor engineering.
Real systems have avoidable losses, but the Second Law also imposes a fundamental requirement to reject thermal energy.

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Key Terms

Heat engine — device operating cyclically that converts part of an energy transfer from a high-temperature source into useful work while rejecting energy to a colder reservoir.

Hot reservoir — high-temperature source supplying thermal energy to an engine.

Cold reservoir — lower-temperature region receiving rejected thermal energy.

Working substance — material, such as gas or steam, that undergoes thermodynamic changes inside an engine.

Q_h — thermal energy absorbed from the hot reservoir.

Q_c — thermal energy rejected to the cold reservoir.

Work (W) — useful mechanical energy transferred by the engine.

Efficiency (η) — fraction of input thermal energy converted into useful work.

Carnot engine — ideal reversible heat engine used to establish the maximum theoretical efficiency.

Carnot efficiency — maximum theoretical efficiency between two temperatures: η_max = 1 − T_c/T_h.

Internal combustion engine — engine in which fuel burns within the engine itself.

External combustion engine — engine in which heating occurs outside the working cylinder or turbine system.

Steam turbine — device in which expanding steam produces rotational mechanical work.

Gas turbine — heat engine using expanding hot gases to rotate turbine blades.

Condenser — component that removes thermal energy from vapor, causing it to condense.

Waste heat — thermal energy transferred away rather than converted into the desired useful output.

Combined-cycle power plant — plant that uses more than one thermodynamic cycle to recover additional useful energy.

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Key Takeaways

• A heat engine converts part of an energy flow from a high-temperature source into useful work.
• Every heat engine requires a hot reservoir and a colder reservoir.
• Energy from the hot reservoir is represented by Q_h.
• Rejected energy is represented by Q_c.
• Useful work is represented by W.
• Energy conservation gives Q_h = W + Q_c.
• Therefore W = Q_h − Q_c.
• Heat-engine efficiency is η = W/Q_h.
• Percentage efficiency is η = (W/Q_h) × 100%.
• Efficiency can also be written as η = 1 − Q_c/Q_h.
• A heat engine cannot convert all its input thermal energy into work.
• Some energy must be rejected to a colder reservoir.
• Therefore no cyclic heat engine operating between finite-temperature reservoirs can be 100% efficient.
• The First Law explains energy conservation within the engine.
• The Second Law explains the fundamental efficiency limit.
• A reversible heat engine provides the maximum possible efficiency between two temperatures.
• The Carnot efficiency is η_max = 1 − T_c/T_h.
• Carnot temperatures must be measured in kelvin.
• Increasing T_h or decreasing T_c can increase the maximum theoretical efficiency.
• Real engines operate below the Carnot limit because of irreversibility.
• Friction, turbulence, heat leakage and finite temperature differences reduce real engine efficiency.
• Internal combustion engines include gasoline and diesel engines.
• Steam engines and steam turbines are examples of external-heating heat-engine systems.
• Gas turbines and jet engines are also heat engines.
• Thermal power stations use heat engines to convert thermal energy into mechanical and then electrical energy.
• Nuclear power plants also operate thermal heat-engine cycles; nuclear fission supplies the thermal energy.
• Cooling systems are essential because heat engines must reject energy to a colder reservoir.
• Combined-cycle systems improve efficiency by recovering energy that would otherwise leave in hot exhaust.
• An electric motor is not normally classified as a heat engine because its primary conversion is electrical energy to mechanical energy.
• Energy rejected by an engine is not destroyed; it is transferred to the surroundings.
• Heat-engine theory provides a practical connection between energy conservation, entropy, the Second Law and engineering efficiency.

Check Your Understanding

1. Define a heat engine.
2. Identify the three main parts of a basic heat-engine system.
3. What is the role of the hot reservoir?
4. What is the role of the cold reservoir?
5. What do Q_h, Q_c and W represent?
6. Write the energy-conservation equation for a heat engine.
7. An engine absorbs 3000 J and rejects 1900 J. Calculate its work output.
8. Write the equation for heat-engine efficiency.
9. An engine absorbs 5000 J and produces 1750 J of work. Calculate its efficiency.
10. An engine is 30% efficient and receives 8000 J. Calculate its useful work output.
11. For the engine in Question 10, calculate the energy rejected to the cold reservoir.
12. Explain why a heat engine cannot be 100% efficient.
13. What is a Carnot engine?
14. Write the equation for maximum Carnot efficiency.
15. Why must temperatures in the Carnot equation be measured in kelvin?
16. An ideal engine operates between 750 K and 300 K. Calculate its maximum theoretical efficiency.
17. Identify three examples of real heat engines.
18. Describe the main energy transformations occurring in a thermal power station.
19. Explain why thermal power stations require cooling systems.
20. Explain how the First and Second Laws of Thermodynamics together determine the operation and limitations of a heat engine.