Conduction

Heat passed along

Hold one end of a metal rod in a flame and the other end soon gets hot, though the metal does not move. Hot, jiggling particles pass energy to their cooler neighbours. In metals, free electrons carry it fast. This is conduction.

In the steady state, when the temperatures no longer change, the heat flowing each second through a slab is

where $A$ is the area, $L$ the thickness, $\Delta T$ the difference of temperature across it, and $k$ the thermal conductivity.

Thermal resistance

Write $H = \dfrac{\Delta T}{R}$, with $R = \dfrac{L}{kA}$: the slab's thermal resistance. Layers one after another carry the same heat, and their resistances add, like resistors in series.

You should find that a thin layer of wool cuts the heat flow by more than four times, and that most of the temperature drop happens across the wool.

Watch out. A good conductor feels cold to the touch because it carries heat away from your hand quickly, not because it is colder.

A worked example

A house wall of $12\ \text{m}^2$ is $22\ \text{cm}$ of brick, lined inside with $5.0\ \text{cm}$ of mineral wool. Inside is $20\ ^\circ\text{C}$, outside $0\ ^\circ\text{C}$. Find the heat flow.

1. Resistances. Wool: $\dfrac{0.050\ \text{m}}{0.04\ \text{W m}^{-1}\text{K}^{-1} \times 12\ \text{m}^2} = 0.104\ \text{K/W}$. Brick: $\dfrac{0.22\ \text{m}}{0.60\ \text{W m}^{-1}\text{K}^{-1} \times 12\ \text{m}^2} = 0.031\ \text{K/W}$.
2. Heat flow. $\dfrac{20\ \text{K}}{0.135\ \text{K/W}} = 148\ \text{W}$, against $654\ \text{W}$ for the brick alone.
3. Between the layers. $20\ ^\circ\text{C} - 148\ \text{W} \times 0.104\ \text{K/W} = 4.5\ ^\circ\text{C}$: the wool takes most of the drop.

Practise with new numbers

Try it in Python

The program works out the lined wall of the example.

Remember these

• $H = \dfrac{kA\,\Delta T}{L}$ in the steady state.
• Thermal resistance $\dfrac{L}{kA}$; layers in series add.
• Metals conduct well; still air and wool conduct badly.

Check yourself

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

1. Heat flows through a $3.0\ \text{m}^2$ steel plate $1.0\ \text{cm}$ thick with its faces $2.0\ \text{K}$ apart. How many watts?

2. Why do two thin blankets keep you warmer than one thick one of the same total weight?

3. Why does a metal spoon in hot tea get hot, but a wooden one does not?

4. Two slabs of the same size, of conductivities $k$ and $3k$, are put face to face. Across which is the larger temperature drop?