Thermal Conduction and Conductivity: Law, Formula, Uses

Thermal conduction is the transfer of heat through a material without the material itself moving as a whole. In a metal rod heated at one end, energetic particles collide with neighbours and pass energy along until the far end warms up.

Understanding conduction helps us design cookware, building insulation, electronics cooling, and winter clothing. The rate depends on the material, area, temperature difference, and thickness, which is summed up in the law of thermal conduction.

Key Concepts

Temperature gradient is change in temperature per unit length, unit kelvin per metre. Thermal conductivity k is a material property measuring how well it conducts heat, unit watt per metre per kelvin. Metals have high k: copper about 400, aluminium about 237, steel about 50. Insulators have low k: wood about 0.1, glass wool about 0.04, air about 0.026. Heat current H is energy per second, unit watt.

Steady state means temperatures at each point stop changing with time, though heat keeps flowing. Conduction occurs mainly by free electrons in metals and by lattice vibrations called phonons in non-metals.

Formulas

Fourier law: H = k A (T_hot - T_cold) / L, where H is heat per second in watt, k in watt per metre per kelvin, A is cross-sectional area in square metre, temperature difference in kelvin, L is length or thickness in metre. Thermal resistance: R = L / (k A), unit kelvin per watt, and H = delta T / R. Assumptions: steady state, uniform material, one-dimensional flow, no heat loss from sides.

Worked Example

A copper rod has length 0.50 metre, area 2.0 x 10^-4 square metre, hot end 100 degree Celsius, cold end 0 degree Celsius. k for copper = 400 watt per metre per kelvin. Find heat current.

Step 1: delta T = 100 kelvin. Step 2: H = 400 x 2.0 x 10^-4 x 100 / 0.50. Step 3: numerator = 400 x 2.0 x 10^-4 x 100 = 8.0. Step 4: divide by 0.50 gives 16 watt. So 16 joule per second flows. In 60 seconds energy = 16 x 60 = 960 joule.

Common Misconceptions

• Cold flows from cold end: Wrong. Only heat flows; cold is absence of heat.
• Thicker metal always heats slower end to end: Not necessarily. Larger area increases flow, longer length decreases it.
• All solids conduct equally: Wrong. Diamond and copper conduct hundreds of times better than wood or plastic.

Real-World Applications

Solid pans use aluminium or copper bases for even heating, while handles use plastic or wood insulators. Double-glazed windows trap air to reduce conduction. Heat sinks in computers use finned aluminium to spread heat. Building walls use foam and cavity insulation, and thermos flasks minimise conduction with vacuum and foam supports.

Conduction moves heat by particle collisions, fastest in metals. Fourier law H = k A delta T / L predicts the rate and guides choices of conductors for heating and insulators for saving energy.