14.5 Conduction

Heat Conduction

Thermal Conductivity Calculation and Interpretation

Thermal conductivity ($k$) measures how well a material transfers heat. A high $k$ means heat flows through the material easily; a low $k$ means the material resists heat flow. The units are W/(m·K) or equivalently W/(m·°C).

The core equation for conduction is Fourier's law:

$\frac{Q}{t} = kA\frac{\Delta T}{d}$

• $Q/t$: rate of heat transfer (watts, W)
• $k$: thermal conductivity of the material, W/(m·K)
• $A$: cross-sectional area the heat flows through (m²)
• $\Delta T$: temperature difference between the hot side and the cold side (K or °C)
• $d$: thickness of the material (m)

A few things to notice about this equation. The heat flow rate increases when the temperature difference is larger, the area is larger, or the material is thinner. That should match your intuition: a thin copper pan heats food faster than a thick wooden cutting board.

If you need to solve for $k$ itself, rearrange:

$k = \frac{(Q/t) \cdot d}{A \cdot \Delta T}$

You'll need the heat transfer rate, the material's thickness, the cross-sectional area, and the temperature difference across the material.

Molecular Basis of Heat Conduction

At the microscopic level, conduction is about energy passing from faster-vibrating particles to slower ones.

• Molecules in a hotter region have more kinetic energy and vibrate more vigorously.
• When they collide with cooler neighboring molecules, they transfer some of that energy.
• This chain of collisions continues through the material, moving thermal energy from the hot side to the cold side without the material itself flowing anywhere.

In metals, there's an additional mechanism: free electrons. Metals have a "sea" of electrons that aren't bound to individual atoms. These electrons move freely and carry kinetic energy from hot regions to cold regions very efficiently. This is the main reason metals like silver and copper are such good conductors.

In non-metallic solids (like ceramics or diamond), conduction happens primarily through lattice vibrations. Atoms in a crystal lattice are connected like masses on springs, and vibrations propagate through the structure. These quantized vibrations are called phonons. Materials with stiff, orderly lattices (like diamond) can actually conduct heat very well through phonons alone.

Comparison of Thermal Conductivities

The range of $k$ values across common materials is enormous, spanning roughly four orders of magnitude:

Metals (high $k$):

• Silver: 429 W/(m·K)
• Copper: 401 W/(m·K)
• Aluminum: 237 W/(m·K)

These are used wherever you want heat to flow quickly: cookware, heat sinks on electronics, heat exchangers.

Non-metals and insulators (low $k$):

• Glass: ~0.8 W/(m·K)
• Wood: 0.04–0.4 W/(m·K)
• Fiberglass insulation: ~0.04 W/(m·K)
• Air: 0.024 W/(m·K)

These are used wherever you want to block heat flow: building insulation, oven mitts, styrofoam cups. Notice that air has an extremely low $k$. Many insulating materials (fiberglass, down feathers, wool) work by trapping small pockets of air so that conduction through the air is minimized and convection currents can't develop.

Liquids and gases generally have lower thermal conductivities than solids. Water, for example, has $k \approx 0.6$ W/(m·K). In fluids, convection (bulk fluid motion) usually transfers far more heat than conduction alone, which is why stirring a pot heats food more evenly.

Additional Concepts in Thermal Conduction

A few related ideas that connect conduction to other topics:

• Thermal resistance ($R$): the opposition to heat flow through a material, defined as $R = d/(kA)$. This is directly analogous to electrical resistance, and just like resistors, thermal resistances add in series. This is how engineers calculate heat loss through layered walls.
• Thermal diffusivity: measures how quickly temperature changes spread through a material. It equals $k/(\rho c)$, where $\rho$ is density and $c$ is specific heat. A material can have high conductivity but also high heat capacity, meaning it conducts well but changes temperature slowly.
• Wiedemann-Franz law: in metals, thermal conductivity and electrical conductivity are proportional, because free electrons carry both heat and charge. This is why good electrical conductors (copper, silver) are also good thermal conductors.