Heat Conduction Calculator (Fourier’s Law)
Work out how much heat flows through a wall, a window or any layer of material. Solve Fourier’s law Q = kAΔT ÷ d, add up layers into an R-value and U-value, see the temperature at every layer, and convert SI and US R-values.
How to Use
- Pick Layered wall for a wall, roof or window made of layers, Single layer for Fourier’s law on one material, or R ↔ U to convert R-values and U-values.
- For a wall, choose a material for each layer (it fills a typical conductivity) and enter its thickness. Leave unused layers blank.
- Enter the area and the inside and outside temperatures. Keep the ISO 6946 surface resistances on for a real wall or window in air.
- Add an energy price per kWh if you want the running cost per day and per 30 days.
- Read the heat loss, U-value, R-value and energy per day, and see the temperature at every layer on the profile and in Show Work.
Worked Example
An insulated cavity wall. 10 m² of 102.5 mm brick (k 0.7), 100 mm mineral wool (k 0.04) and 102.5 mm brick, 20 °C inside and 0 °C outside. The resistances are 0.13 (inside air) + 0.1464 + 2.5 + 0.1464 + 0.04 (outside air) = 2.963 m²·K/W, or R-16.8 in US units. U = 1 ÷ 2.963 = 0.3375 W/(m²·K), so the wall loses 0.3375 × 10 × 20 = 67.5 W, 1.62 kWh a day.
Where the temperature drops. The heat flux is 6.75 W/m². Each layer takes a share of the 20 °C in proportion to its resistance: 0.88 °C across the inside air film, 0.99 °C across the inner brick and 16.9 °C across the mineral wool, so the room-side surface stays at 19.1 °C.
The common mistake: leaving out the surface air films. Fourier’s law on 4 mm of glass alone gives 1.0 × 1 × 20 ÷ 0.004 = 5,000 W through 1 m². The still air clinging to each face adds 0.13 + 0.04 = 0.17 m²·K/W, over 40 times the glass’s own 0.004, and the real loss is 114.9 W. For thin or highly conductive layers the surfaces, not the material, set the heat loss.
Show Work
Formulas
Typical thermal conductivities near room temperature (handbook class values, rounded; CRC Handbook for the pure metals and still air). Building products vary with density, moisture and grade, so use the declared value on the product for design.
| Material | k, W/(m·K) (typical) | Thickness for R = 1 m²·K/W |
|---|---|---|
| Copper | about 400 | 400 m |
| Aluminium (pure; alloys lower) | 205–237 | about 237 m |
| Carbon steel | 45–50 | about 50 m |
| Glass | 0.8–1.0 | about 1 m |
| Brick | 0.6–1.0 (0.7 used) | 0.7 m |
| Softwood timber | about 0.13 | 130 mm |
| Mineral wool | 0.035–0.04 | 35–40 mm |
| Still air | about 0.026 | 26 mm (if it stayed perfectly still) |
Fourier and the Flow of Heat
In the 1780s the Dutch scientist Jan Ingenhousz coated rods of different metals with wax, heated one end of each and watched how far the melting spread, an early side-by-side comparison of how well materials conduct heat. Silver and copper melted the wax furthest; that ranking still holds.
The law itself is Joseph Fourier’s. He presented a memoir on the propagation of heat to the Paris Academy in 1807 and published the full theory in Théorie analytique de la chaleur in 1822. Heat flow is proportional to the temperature gradient, and solving his heat equation led him to write functions as sums of sines and cosines, the Fourier series used everywhere in science today.
Building science turned the law into R-values and U-values, which add up layer by layer, and standards such as ISO 6946 fixed the extra resistance of the thin air films on each surface. To work out how much insulation to buy for walls or an attic to reach a target R-value, see the Insulation Calculator.
About This Tool
This calculator applies Fourier’s law of conduction in three ways. The layered wall adds up to four materials in series with the standard surface resistances, giving the R-value, U-value, heat loss, energy and cost per day, and the temperature at every layer, drawn as a profile through the wall. The single-layer mode solves Q = kAΔT ÷ d for any of its five values, and R ↔ U converts between R-values and U-values in SI and US units, with the thickness of common materials that would give the same resistance. It covers steady conduction only; air leaks and radiation through glass are not included.
Everything runs in your browser; nothing you enter is sent anywhere.
Related tools: Insulation Calculator, Thermal Expansion Calculator, and Specific Heat Calculator.
Frequently Asked Questions
What is Fourier’s law of heat conduction?
The heat flow through a layer is Q = k × A × ΔT ÷ d: conductivity times area times temperature difference, divided by thickness. 100 mm of mineral wool (k ≈ 0.04 W/(m·K)) over 10 m² with 20 °C across it passes 0.04 × 10 × 20 ÷ 0.1 = 80 W.
How do I calculate the U-value of a wall with several layers?
Add the resistance of each layer, R = d ÷ k, plus the surface resistances (0.13 inside and 0.04 outside, m²·K/W), then U = 1 ÷ R. For 102.5 mm brick, 100 mm mineral wool and 102.5 mm brick: R = 0.13 + 0.1464 + 2.5 + 0.1464 + 0.04 = 2.963, so U = 0.3375 W/(m²·K), and 10 m² at a 20 °C difference loses 67.5 W.
How do I convert R-values between US and SI units?
Multiply an SI R-value (m²·K/W) by 5.678 to get the US R-value (ft²·°F·h/BTU), or divide the other way. An R-13 wall batt is 13 ÷ 5.678 = 2.289 m²·K/W, a U-value of 0.4368 W/(m²·K); R-19 is 3.346 m²·K/W.
Why doesn’t Fourier’s law work for a single pane of glass?
4 mm of glass on its own (k ≈ 1.0 W/(m·K)) would pass 1.0 × 1 × 20 ÷ 0.004 = 5,000 W per square metre at a 20 °C difference. In reality the thin layers of still air on each face add 0.17 m²·K/W of resistance, far more than the glass’s 0.004, so the real loss is about 114.9 W (U ≈ 5.75).
How do I find the temperature inside a wall?
The temperature drops across each layer in proportion to its resistance. In the insulated brick wall above, from 20 °C inside to 0 °C outside, the inner surface sits at 19.1 °C and most of the fall happens across the mineral wool; an uninsulated 225 mm brick wall has an inner surface of only 14.7 °C, which feels cold and invites condensation.
How do I use the Heat Conduction Calculator (Fourier’s Law)?
Simply type your numbers and read the result, which refreshes the instant you change something. There is nothing to submit and nothing to wait for.
Do I need to install or sign up for anything?
Not at all — it runs in the browser with nothing to install and no account. After it loads once, it even works without an internet connection.
Is my information private?
Yes. Everything happens in your browser. Nothing you type is sent to a server or saved anywhere.
Common Use Cases
Insulating a wall
Filling the cavity of a 10 m² brick wall cuts the loss at a 20 °C difference from 407 W (solid 225 mm brick) to 67.5 W, saving 8.147 kWh a day.
Sizing insulation
To hold the loss through 10 m² to 50 W at a 20 °C difference, mineral wool (k 0.04) needs to be 160 mm thick.
Windows
A single-glazed 1 m² window loses about 114.9 W at 20 °C inside and 0 °C out, more than the whole 10 m² insulated wall.
Running costs
At 0.25 per kWh, the uninsulated 10 m² brick wall costs 2.44 a day to heat through at a steady 20 °C difference.
Cookware and heat sinks
A 3 mm copper pan base (0.02 m²) passes 2,667 W for every 1 °C across it, which is why copper spreads heat so evenly.
Reading product labels
An R-13 batt is 2.289 m²·K/W, the same resistance as 91.6 mm of mineral wool or 1.6 m of brick.
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