Heat guide
Conduction: Heat Transfer Through Matter
Conduction transfers thermal energy through matter because neighbouring regions are at different temperatures.
How conduction transfers energy
Microscopic interactions transfer energy from higher-energy regions toward lower-energy regions. In metals, mobile electrons contribute strongly to thermal conduction. In non-metals, lattice vibrations and molecular interactions dominate.
Fourier’s law
For one-dimensional steady conduction through a plane layer with constant thermal conductivity, the heat-transfer rate can be written Q̇ = kAΔT/L. Thermal conductivity k, area A and temperature difference increase the rate, while greater thickness L reduces it.
Heat flux
Dividing heat-transfer rate by area gives heat flux. For the same simplified plane layer, q″ = kΔT/L. Heat flux is useful when comparing thermal loading independently of total surface area.
Steady state is an assumption
The plane-wall equation assumes the temperature profile is no longer changing with time. During warm-up or cool-down, thermal storage matters and the temperature field changes continuously.
Real geometry can require more detailed models
Curved walls, contact resistances, multidimensional heat flow and temperature-dependent properties require modified equations or numerical analysis. The simple model remains valuable because it shows how the main variables influence conduction.
Steady and transient conduction answer different problems
Fourier law relates conductive heat flux to a temperature gradient, but many practical problems change with time. A cold object placed in a warm environment initially develops internal temperature gradients that evolve as energy diffuses through it.
Whether a steady approximation is reasonable depends on the timescale of interest, geometry and thermal diffusivity. Transient conduction is essential when warm-up, cool-down, thermal shock or short heating pulses matter.
Contact resistance can dominate otherwise conductive paths
Two solids pressed together rarely touch over their entire apparent area. Microscopic roughness leaves voids that can contain air or another interstitial material, creating a contact resistance between the two bodies. Increasing contact pressure, improving flatness or adding a suitable interface material can reduce that resistance.
This effect explains why a highly conductive heat sink can still perform poorly if its interface with a component is inadequate. In building assemblies, imperfect contact between layers can likewise alter measured performance. The conductivity of the bulk materials is only one part of the complete conductive path.
A worked plane-wall example
Consider a 0.10 m thick layer with thermal conductivity 0.50 W/(m·K), area 2.0 m² and a 20 K temperature difference across it. Under the one-dimensional steady-state model, Q̇ = kAΔT/L = 0.50 × 2.0 × 20 / 0.10 = 200 W. Doubling the thickness would halve this conductive rate if the other quantities remained unchanged.
The result applies to the layer itself under the stated assumptions. Surface convection, thermal bridges, contact resistances and changing temperatures can alter the heat flow through a real assembly.
Why temperature profiles are useful
For a uniform plane layer with constant conductivity, no internal heat generation and steady one-dimensional conduction, temperature changes linearly through the thickness. In a multilayer wall, the temperature drop across each layer is proportional to that layer’s thermal resistance.
A large temperature drop across an insulation layer therefore does not mean the insulation is failing. It is often evidence that the high-resistance layer is doing much of the work of limiting heat flow.
Conduction can also be radial
Heat flowing through pipe insulation or a cylindrical wall spreads through changing area as radius changes. The plane-wall expression is therefore not exact for cylinders or spheres. Radial conduction equations contain logarithmic or radius-dependent terms that account for the changing heat-transfer area.
This geometry distinction matters when insulation thickness is not small compared with pipe diameter, or when analysing vessels, wires and spherical systems.