5.5 Efficiency

In physics and engineering, efficiency is a measure of how well a system or device converts input energy into useful output work or energy. No real-world machine is perfectly efficient — some input energy is always lost to non-useful forms such as heat due to friction. The Carnot cycle establishes the fundamental upper limit on how efficient any heat engine can be.

Efficiency

Definition: Efficiency ($\eta$) is the ratio of the useful energy output of a device to the total energy input. It is a dimensionless quantity, often expressed as a percentage.

Formulas:

$\eta =\frac{\text{Useful Energy Output}}{\text{Total Energy Input}}$

$\eta =\frac{\text{Useful Work Output}}{\text{Total Work Input}}$

To express as a percentage:

$\%\text{ Efficiency}=\frac{\text{Useful Work Output}}{\text{Total Work Input}}\times 100\%$

• Input Energy: The total energy supplied to the machine (e.g., electrical energy for a motor, chemical energy from fuel for an engine).
• Useful Output Energy: The energy that accomplishes the machine's intended task (e.g., kinetic energy of a moving car, light from a bulb).
• Lost Energy: The difference between input and output energy, converted into non-useful forms like heat, sound, or vibration.

$\text{Energy Input}=\text{Useful Energy Output}+\text{Energy Lost}$

Ideal vs. Real Machines

• Ideal Machine: A theoretical machine with no energy losses. Work input equals work output; efficiency = 100%.
• Real Machine: All practical machines experience energy losses due to friction, air resistance, and other dissipative forces. Therefore, useful output is always less than input, and efficiency is always less than 100%.

Efficiency in Terms of Power

Since power is energy per unit time ($P=W/t$), efficiency can also be expressed as:

$\eta =\frac{\text{Useful Power Output}}{\text{Total Power Input}}$

Worked Example: A motor consumes 1000 W of electrical energy and produces 800 W of mechanical energy.

$\eta =\frac{800\text{ W}}{1000\text{ W}}=0.8\Rightarrow 80\%$

The remaining 200 W are dissipated primarily as heat.

Carnot Engine Efficiency

The Carnot cycle sets a fundamental upper limit on the efficiency of any heat engine operating between two temperature reservoirs. No real heat engine can exceed the efficiency of an ideal Carnot engine operating between the same temperatures. This is a direct consequence of the Second Law of Thermodynamics.

The efficiency of a Carnot engine is:

$\eta =1-\frac{T_{\text{sink}}}{T_{\text{source}}}$

Where:

• $T_{\text{source}}$ = absolute temperature of the hot reservoir (in Kelvin)
• $T_{\text{sink}}$ = absolute temperature of the cold reservoir (in Kelvin)

Key implications:

• Efficiency increases when $T_{\text{source}}$ is raised or $T_{\text{sink}}$ is lowered.
• 100% efficiency would require $T_{\text{sink}}=0\text{ K}$ (absolute zero), which is unattainable.
• All real engines operate below the Carnot efficiency for their temperature range.
• This result is a consequence of the Second Law of Thermodynamics.

Worked Example: A Carnot engine operates between 0°C and 100°C.

$T_{\text{source}}=373\text{ K},T_{\text{sink}}=273\text{ K}$

$\eta =1-\frac{273}{373}\approx 0.268=26.8\%$

Even between the freezing and boiling points of water, the maximum possible efficiency is only about 27%.

Summary Table

References

• Efficiency and Power (Video Lecture)