Heat engines, refrigerators and the Carnot engine

A heat engine

A car engine, a power station and a steam engine all do the same three things in each cycle:

1. take heat $Q_h$ from a hot reservoir at $T_h$;
2. do work $W$;
3. give the rest, $Q_c$, to a cold reservoir at $T_c$, usually the air or a river.

Over a cycle $\Delta U = 0$, so $W = Q_h - Q_c$, and the efficiency is

The picture labels the hot side $T_1$, $Q_1$ and the cold side $T_2$, $Q_2$, as NCERT does. We write $h$ for hot and $c$ for cold, which is easier to remember.

A refrigerator

A refrigerator is an engine run backwards. Work $W$ moves heat $Q_c$ out of the cold inside and gives $Q_h = Q_c + W$ to the warm kitchen. How well it works is its coefficient of performance:

A good one has a COP of $3$ to $6$: it moves several joules of heat for each joule of work.

The Carnot engine

Carnot imagined an engine with no losses at all. Its gas goes round two isotherms and two adiabats, every step reversible.

For this engine $\dfrac{Q_c}{Q_h} = \dfrac{T_c}{T_h}$, so

Carnot's theorem says no engine working between the same two temperatures can do better. The best refrigerator between them has $\text{COP} = \dfrac{T_c}{T_h - T_c}$.

You should find that a hotter source helps an engine, while a small gap between the temperatures helps a refrigerator.

Watch out. Use kelvin. Between $100\ ^\circ\text{C}$ and $0\ ^\circ\text{C}$ the best efficiency is $1 - \dfrac{273\ \text{K}}{373\ \text{K}}$, about $27\%$, not $100\%$.

A worked example

A Carnot engine works between $600\ \text{K}$ and $290\ \text{K}$ and takes in $1800\ \text{J}$ of heat per cycle. Find its efficiency, its work and the heat it gives to the sink. Then find the best COP for a freezer at $263\ \text{K}$ in a room at $303\ \text{K}$.

1. Efficiency. $\eta = 1 - \dfrac{290\ \text{K}}{600\ \text{K}} = 0.517$, or $51.7\%$.
2. Work. $W = 0.517 \times 1800\ \text{J} = 930\ \text{J}$, so $Q_c = 1800\ \text{J} - 930\ \text{J} = 870\ \text{J}$.
3. Freezer. $\text{COP} = \dfrac{263\ \text{K}}{303\ \text{K} - 263\ \text{K}} = 6.58$.

Practise with new numbers

Try it in Python

The program works out the engine and the freezer of the example.

Remember these

• Engine: $W = Q_h - Q_c$ and $\eta = \dfrac{W}{Q_h}$.
• Refrigerator: $\text{COP} = \dfrac{Q_c}{W}$ and $Q_h = Q_c + W$.
• Carnot, the best: $\eta = 1 - \dfrac{T_c}{T_h}$, in kelvin.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. An engine works between $457\ ^\circ\text{C}$ and $37\ ^\circ\text{C}$. What is the best efficiency it could have?

2. An engine takes in $1700\ \text{J}$ per cycle and gives $1190\ \text{J}$ to its sink. Find its work and its efficiency.

3. Why can no real engine reach the Carnot efficiency?

4. A refrigerator with a COP of $3.5$ uses $160\ \text{J}$ of work. How much heat does it take from its inside, and how much reaches the kitchen?