What you'll learn

• Why the First Law of Thermodynamics alone is not enough to explain how engines work.
• The definition of the Second Law of Thermodynamics in terms of heat engines.
• How to calculate the actual efficiency and the maximum theoretical efficiency of an engine.
• Why practical engines fall short of maximum efficiency, and how Combined Heat and Power (CHP) schemes make better use of energy.

Why the First Law isn't enough

The First Law of Thermodynamics is simply the conservation of energy: energy cannot be created or destroyed, only transferred or transformed.

If we applied only the First Law to a machine, it would theoretically be completely fine to build an engine that takes in 1000 joules of heat and turns it into exactly 1000 joules of useful work. Energy is conserved, so the First Law is happy.

However, over centuries of engineering, nobody has ever been able to build an engine that turns 100% of its heat input into work. There is always some "waste" heat left over. This real-world limitation tells us that another fundamental rule is at play: the Second Law of Thermodynamics.

The Second Law and Heat Engines

To understand the Second Law, we first need to define the components of a theoretical engine.

The Second Law of Thermodynamics dictates that heat will naturally flow from hot to cold, never the reverse without external work. Because of this, an engine can only extract work from the flow of heat between two regions of different temperatures.

This means an engine with 100% efficiency is physically impossible. You must always throw some heat away to the cold sink.

In the diagram above:

• $Q_H$ is the heat energy transferred into the engine from the source.
• $W$ is the useful mechanical work done by the engine.
• $Q_C$ is the waste heat energy rejected from the engine to the sink.

Because energy must still be conserved (the First Law still applies!), the total energy entering the engine must equal the total energy leaving it:

$$Q_H=W+Q_C$$

Rearranging this gives us the work done:

$$W=Q_H-Q_C$$

Calculating Engine Efficiency

The efficiency of any system is the useful output divided by the total input. For a heat engine, the useful output is the work $W$, and the energy we had to "pay for" is the heat from the hot source $Q_H$.

$$efficiency=\frac{W}{Q_H}$$

By substituting $W=Q_H-Q_C$ into the equation, we can also write it purely in terms of the heat transfers:

$$efficiency=\frac{Q_H-Q_C}{Q_H}$$

(Note: Efficiency is a ratio and has no units. It can be expressed as a decimal or multiplied by 100 to give a percentage).

Maximum Theoretical Efficiency

In the 19th century, Sadi Carnot proved that even a perfectly frictionless, ideal engine has a strict efficiency limit. This maximum limit depends entirely on the absolute temperatures of the hot source and the cold sink.

$$maximum\; theoretical\; efficiency=\frac{T_H-T_C}{T_H}$$

This is often called the Carnot efficiency. It represents the absolute best-case scenario for an engine operating between those two temperatures. No real engine can ever exceed, or even perfectly reach, this value.

To increase the maximum theoretical efficiency, you must either:

• Increase the temperature of the hot source ($T_H$).
• Decrease the temperature of the cold sink ($T_C$).

Real Engines and CHP Schemes

In the previous example, the absolute best the turbine could do was about $63\%$. In reality, the actual efficiency of that turbine would likely be closer to $40\%$.

Why do practical engines have much lower efficiencies than the theoretical maximum?

• Friction: Moving parts (like pistons and bearings) rub against each other, converting some useful work back into unrecoverable heat.
• Heat losses: Heat escapes through the engine casing to the surroundings before it can be used to do work.
• Non-ideal processes: The Carnot formula assumes the engine cycle happens infinitely slowly and reversibly, which is impossible in a machine designed to deliver power quickly.

Combined Heat and Power (CHP)

We know from the Second Law that we must reject heat ($Q_C$) to a cold sink. Usually, this heat is just dumped into the atmosphere or a nearby river—it is completely wasted.

We can maximise our use of the energy by capturing this waste heat. In a Combined Heat and Power (CHP) scheme, a power plant generates electricity (the work, $W$), but instead of venting the waste heat ($Q_C$) from the steam, it pipes that hot water into local homes, hospitals, or factories to provide central heating.