What is Fourier's law of heat conduction?

Fourier's law states that the steady-state heat flow rate (Q/t, in watts) through a material equals k·A·ΔT/d, where k is thermal conductivity (W/(m·K)), A is the cross-sectional area (m²), ΔT is the temperature difference (K or °C), and d is the thickness (m). A higher k means the material conducts heat more readily.

What is thermal conductivity (k)?

Thermal conductivity is a material property that quantifies how well it conducts heat. Metals like copper (~385 W/(m·K)) and aluminum (~205 W/(m·K)) are excellent conductors. Insulation materials like mineral wool (~0.04 W/(m·K)) and wood (~0.15 W/(m·K)) resist heat flow.

Does the formula assume steady-state or transient conditions?

Fourier's law in this form gives the steady-state heat flow rate — when temperatures on both sides are held constant and the rate is no longer changing with time. For transient (time-varying) problems you need the full heat equation with time derivatives.

Can I use Fahrenheit temperatures?

Yes. Select °F from the temperature-unit options. The tool converts both temperatures to Celsius before computing ΔT. A difference of 1 °F equals 5/9 °C, so the conversion is applied automatically.

What does a negative result mean?

If T_hot is lower than T_cold, ΔT is negative and Q/t is negative, meaning heat flows in the opposite direction (from the "cold" side to the "hot" side as labeled). In practice just swap the two temperature inputs to get a positive flow rate.

What is the R-value and how does it relate?

Thermal resistance R = d/k (in m²·K/W for a unit area). The heat flow rate Q/t = A·ΔT/R. A higher R-value means more resistance to heat flow — the same idea as insulation ratings on building products.

Heat Transfer Conduction Calculator

Enter material conductivity, area, thickness, and temperature difference to find the steady-state heat flow rate using Fourier's law.

Inputs

Material (thermal conductivity k)

Cross-sectional Area (A)

Thickness (d)

Temperature Unit

°C °F K

Hot-side Temperature (T_hot)

°C

Cold-side Temperature (T_cold)

°C

Fill in the inputs and click Calculate Heat Flow Rate.

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Summary

This calculator applies Fourier's law of heat conduction: Q/t = k·A·ΔT/d, where k is thermal conductivity, A is the cross-sectional area, ΔT is the temperature difference across the material, and d is the thickness. Select a material preset or enter a custom conductivity value. Results are shown in watts (W), and a step-by-step substitution confirms each calculation.

How it works

Select a material from the presets — its thermal conductivity (k) fills automatically — or choose "Custom" and enter k in W/(m·K).
Enter the cross-sectional area (A) of the surface through which heat flows.
Enter the thickness (d) of the material layer.
Enter the hot-side and cold-side temperatures; pick °C, °F, or K.
Click Calculate — the tool evaluates Q/t = k·A·ΔT/d and displays the heat flow rate in watts.
A step-by-step breakdown shows each substitution so you can verify the arithmetic.

Use cases

Estimating heat loss through a wall, window, or insulation panel.
Choosing the right insulation material for a building or industrial pipe.
Physics and engineering homework on Fourier's law of conduction.
Comparing how fast heat flows through copper vs. wood vs. glass.
HVAC design — calculating heat gain or loss through structural elements.
Lab reports and thermal-engineering coursework requiring unit-correct answers.

Frequently Asked Questions

Last updated: 2026-06-11 · Reviewed by Nham Vu