Heat Conduction Calculator
Calculate the steady-state heat transfer rate through a flat wall, cylindrical pipe, or spherical shell using Fourier's law. Used by engineers sizing insulation, heat exchangers, and building envelopes.
About this calculator
Fourier's law states that heat flows from hot to cold at a rate proportional to the temperature gradient and the material's thermal conductivity k (W/m·K). For a plane wall: Q = k × A × ΔT / L, where A is area (m²), ΔT is temperature difference (K), and L is thickness (m). For a hollow cylinder of inner radius r₁ and outer radius r₂: Q = 2π × k × L × ΔT / ln(r₂/r₁), where this calculator uses r₂ = area + thickness and r₁ = area. For a hollow sphere: Q = 4π × k × ΔT / (1/r₁ − 1/r₂). The thermal resistance analogy — R = L / (k × A) for a plane wall — lets engineers treat multi-layer systems like electrical resistors in series, summing resistances to find total heat loss. Higher k materials (e.g., copper, 400 W/m·K) conduct far more heat than insulators (e.g., mineral wool, 0.04 W/m·K).
How to use
Plane-wall example: a concrete wall (k = 1.7 W/m·K), area A = 10 m², thickness L = 0.2 m, temperature difference ΔT = 20 K. Step 1 — Apply Fourier's law: Q = k × A × ΔT / L = 1.7 × 10 × 20 / 0.2. Step 2 — Numerator: 1.7 × 10 × 20 = 340 W·m. Step 3 — Divide by thickness: 340 / 0.2 = 1 700 W. The wall conducts 1 700 W (1.7 kW) of heat. To halve that loss, you could double the wall thickness to 0.4 m or add insulation in series. Select 'plane' geometry, enter your values, and the calculator shows Q instantly.
Frequently asked questions
What is Fourier's law of heat conduction and when does it apply?
Fourier's law states that the conductive heat flux through a material is proportional to the negative temperature gradient: q = −k × dT/dx W/m². In one-dimensional steady-state form it simplifies to Q = k × A × ΔT / L for a plane wall. It applies whenever heat transfer is dominated by molecular diffusion through a solid or stationary fluid — for example, through building walls, pipe insulation, heat-exchanger walls, and electronic circuit boards. It does not account for convection or radiation, which typically must be added as additional thermal resistances when calculating total heat loss from a surface.
How do I calculate heat loss through a cylindrical pipe using this calculator?
Select 'cylindrical' geometry and enter the inner radius as the 'area' field (in metres), the insulation thickness as 'thickness', the pipe length as a separate scaling factor if needed, and the thermal conductivity of the insulating material. The formula Q = 2π × k × ΔT / ln(r₂/r₁) gives heat loss per unit length; multiply by pipe length to get total Q. This geometry is common for steam pipes, district heating networks, and refrigerant lines, where the logarithmic area increase from inner to outer radius significantly affects the result compared to a flat-wall approximation.
Why is thermal conductivity so important in heat conduction calculations?
Thermal conductivity k determines how readily a material transfers heat by conduction; it ranges from about 0.02 W/m·K for aerogel insulation to over 400 W/m·K for copper. Because Q is directly proportional to k in Fourier's law, substituting aerogel for copper reduces heat transfer by a factor of roughly 20 000 for the same geometry and temperature difference. In building design, choosing insulation with low k minimises heating and cooling loads; in heat exchangers, high-k materials maximise heat transfer area utilisation. Engineers must also remember that k varies with temperature and moisture content, so published values should match the actual operating conditions.