You met conduction in Conduction, Convection and Radiation: heat passing through a material from molecule to molecule. Now we ask how fast. Builders, cooks and engineers all need the rate, in joules per second, or watts, and often need to solve the formula for a thickness or an area.
Why it works
Think of heat crossing a slab of thickness $L$ and area $A$. Twice the area is like two slabs side by side, so twice the rate: $P \propto A$. A bigger temperature difference drives heat harder: $P \propto \Delta T$. A thicker slab spreads the same temperature difference over more distance, so each layer passes on less: $P \propto 1/L$. What is left is the material, through its thermal conductivity $k$. In metals, free electrons carry energy quickly, so $k$ is large; in foam, trapped air carries it slowly.
Key idea
$$P = \frac{Q}{t} = \frac{kA\,\Delta T}{L}$$ Thin, wide slabs with big temperature differences pass heat fastest. A temperature difference in $^\circ\text{C}$ is the same number in kelvins.
| Material | $k$ in $\text{W/(m}\cdot\text{K)}$ |
|---|---|
| Copper | about 390 |
| Aluminum | about 220 |
| Glass, brick | about 0.8 |
| Water | about 0.6 |
| Wood | about 0.12 |
| Still air | about 0.025 |
| Styrofoam | about 0.010 |
Worked examples
Worked example: A window
A glass window ($k = 0.84\,\text{W/(m}\cdot\text{K)}$) is $1.2\,\text{m}^2$ in area and $4.0\,\text{mm}$ thick. The inner surface is $15\,^\circ\text{C}$ warmer than the outer surface. Find the rate of heat flow.
- Thickness in meters: $L = 0.0040\,\text{m}$. $\Delta T = 15\,\text{K}$.
- $P = \dfrac{(0.84)(1.2)(15)}{0.0040} = 3780\,\text{W}$, about $3.8\,\text{kW}$.
Answer: $3.78 \times 10^{3}\,\text{W}$
Worked example: A foam cooler
A styrofoam cooler ($k = 0.010$) has walls $2.0\,\text{cm}$ thick with a total area of $0.80\,\text{m}^2$. Outside is $25\,^\circ\text{C}$ warmer than inside. How fast does heat leak in, and how long until it has delivered enough heat to melt $1.0\,\text{kg}$ of ice ($3.34 \times 10^{5}\,\text{J}$)?
- $P = \dfrac{(0.010)(0.80)(25)}{0.020} = 10\,\text{W}$.
- $t = \dfrac{Q}{P} = \dfrac{3.34 \times 10^{5}\,\text{J}}{10\,\text{J/s}} = 3.34 \times 10^{4}\,\text{s}$, a bit over nine hours.
Answer: $10\,\text{W}$; about $9.3\,\text{h}$
Common mistake
Thinking a good conductor must be hot. A good conductor is a material that passes heat along quickly, at any temperature. A metal bench and a wooden bench in the same shade are at the same temperature, but the metal feels colder because its large $k$ pulls heat out of your skin faster. Check: temperature tells you which way heat flows; $k$ tells you how fast.
A real single-pane window loses heat more slowly than the first example says, because thin films of still air cling to both sides of the glass and add their own insulation.