Fourier's Law of Heat Conduction
Compute conductive heat flux q = −k · (dT/dx) through a slab, wall, or solid layer. Solve for heat flux, thermal conductivity, temperature gradient, or wall thickness, with a live temperature profile, heat-flow schematic, and full thermal-resistance metrics — all in one place.
The minus sign matters
Fourier's law is q = −k · dT/dx. The minus sign encodes the fact that heat flows down the temperature gradient, from hot to cold. In a plane wall with T₁ > T₂ and x pointing from hot to cold, dT/dx is negative, so q is positive — heat flows in the +x direction. If you drop the minus sign, you get the flux magnitude rather than the signed vector quantity.
Conductivity spans six orders of magnitude
Thermal conductivity ranges from ~0.02 W/(m·K) for aerogels and still air up to ~400 W/(m·K) for copper, with diamond at ~2000 W/(m·K) at the very top. This enormous range is why a millimetre of copper conducts as much heat as a metre of insulation. Metals conduct via free electrons, while insulators conduct via lattice vibrations (phonons), and the mechanism determines how k changes with temperature.
Steady state and one-dimensional flow
The simple q = k·ΔT/L form assumes steady state (temperatures do not change with time), one-dimensional conduction (no lateral heat loss), constant k (no temperature dependence), and a plane wall (no curvature). For cylindrical or spherical geometry, transient problems, or temperature-dependent k, the full partial differential equation ∂T/∂t = α∇²T must be solved, usually numerically.
Thermal resistance analogy
For a plane wall, the heat rate is Q = ΔT / R where R = L / (k·A) is the thermal resistance, exactly analogous to Ohm's law V = I·R. This lets you analyse composite walls by adding resistances in series — brick + insulation + plasterboard gives R_total = R₁ + R₂ + R₃, and the total heat flow is ΔT_overall / R_total. The same series/parallel algebra used in circuits applies directly to thermal networks.
How to Use This Calculator
Fourier's law relates four quantities: the heat flux q, the thermal conductivity k, the temperature difference ΔT, and the wall thickness L. Pick which one to solve for and fill in the others.
1. Choose what to solve for
Select q (heat flux), k (thermal conductivity), ΔT (temperature difference), or L (thickness). The calculator rearranges q = k·ΔT/L for whichever quantity is unknown and greys out that field.
2. Pick a preset for context
The presets cover the most common materials — brick, copper, glass, insulation, steel, still air, and aluminum. Each preset loads realistic k, T₁, T₂, L, and A values so you can see typical magnitudes immediately.
3. Set the geometry
Thickness L and cross-sectional area A define the geometry of the conduction problem. Heat flux q depends only on ΔT, k, and L; total heat rate Q depends additionally on the area A.
4. Read the result and verdict
The result card shows q, Q, R (thermal resistance in K/W), and R'' (R-value per unit area in m²·K/W). A colour-coded verdict classifies the flux magnitude from very low (insulation) through moderate (brick) to very high (metal).
5. Study the diagrams
The plane-wall diagram shows the temperature profile through the slab with the hot face on the left and the cold face on the right, along with a heat flux arrow from hot to cold. The comparison chart shows how the same ΔT and L would produce very different fluxes for different materials.
6. Switch to Advanced for depth
Advanced mode adds a q-vs-gradient chart, a q-vs-thickness chart, a sensitivity panel showing how q reacts to ±10% changes in k, ΔT, and L, and a full companion statistics table with the mean temperature, thermal resistance, and material classification.
What This Tool Offers
Built around the classic Fourier's law of heat conduction q = −k · (dT/dx), the foundation of all conduction calculations.
Four-way solving
Solve for heat flux q, thermal conductivity k, temperature difference ΔT, or wall thickness L — the calculator rearranges Fourier's law for whichever quantity you select.
Plane-wall diagram
A large annotated schematic showing the slab, the hot and cold faces, the linear temperature profile, and the heat flux arrow from hot to cold.
Temperature profile chart
T(x) plotted across the wall with T₁ and T₂ marked at the boundaries and the mean temperature indicated on the line.
q vs gradient & thickness
Two advanced-mode charts: heat flux proportional to the temperature gradient at fixed k, and inverse-proportional to thickness at fixed ΔT.
Material comparison
Heat flux computed for seven common materials at the same ΔT and L — see how brick, copper, steel, glass, insulation and air compare at a glance.
Reference tables
Thermal conductivity values, thermal resistance formulas, and a full variable reference table with unit conversions.
Reference Tables & Extra Tools
A thermal-resistance calculator plus lookup tables for conductivity values and composite wall analysis.
Composite wall thermal resistance
Fourier's Law Formulas at a Glance
For one-dimensional steady-state conduction in a plane wall.
| Quantity | Formula | Units | Notes |
|---|---|---|---|
| Differential form | q = −k · (dT/dx) | W/m² | Local heat flux, any geometry |
| Plane wall | q = k · (T₁ − T₂) / L | W/m² | Steady state, constant k |
| Total heat rate | Q = q · A | W | Multiply flux by area |
| Thermal resistance | R = L / (k · A) | K/W | Analogous to Ohm's law |
| R-value (per area) | R'' = L / k | m²·K/W | Building insulation metric |
| Series resistances | R_total = Σ Rᵢ | K/W | For composite walls |
| Parallel resistances | 1/R = Σ 1/Rᵢ | K/W | For heat flow through parallel paths |
| Thermal diffusivity | α = k / (ρ · cp) | m²/s | Governs transient behaviour |
| Cylindrical shell | R = ln(r₂/r₁) / (2πkL) | K/W | For pipes and tubes |
| Spherical shell | R = (1/r₁ − 1/r₂) / (4πk) | K/W | For spherical vessels |
Typical Thermal Conductivity Values
At 25 °C unless otherwise noted. Values vary with temperature and material grade.
| Material | k (W/(m·K)) | Class | Typical application |
|---|---|---|---|
| Diamond (type IIa) | 2000 – 2200 | Ultra-high | Heat spreaders, high-power electronics |
| Copper (pure) | 385 – 401 | Metal | Heat exchangers, bus bars, pipework |
| Aluminum (pure) | 205 – 237 | Metal | Heat sinks, fins, cookware |
| Brass (70Cu-30Zn) | ~110 | Metal | Fittings, low-cost heat conduction |
| Carbon steel | 45 – 55 | Metal | Structural steel, boiler tubes |
| Stainless steel (304) | 14 – 17 | Metal | Food processing, chemical plant |
| Water (liquid) | 0.60 | Liquid | Cooling, thermal storage |
| Glass (soda-lime) | 0.8 – 1.0 | Ceramic | Windows, labware |
| Brick (common red) | 0.6 – 0.9 | Ceramic | Building walls, fireplaces |
| Concrete | 1.0 – 1.8 | Ceramic | Floors, foundations, slab-on-grade |
| Wood (oak) | 0.16 – 0.22 | Insulator | Furniture, structural timber |
| Glass-fiber insulation | 0.035 – 0.045 | Insulator | Attic, wall cavity insulation |
| Polyurethane foam | 0.022 – 0.030 | Insulator | Refrigerators, pipe insulation |
| Still air (25 °C) | 0.026 | Insulator | Reference for convective boundary layers |
| Aerogel (silica) | 0.012 – 0.020 | Ultra-insulator | Aerospace, specialty insulation |
Variable Reference
| Symbol | Meaning | Typical range | Units |
|---|---|---|---|
| q | Heat flux (per unit area) | 0.01 – 10⁷ | W/m² |
| Q | Total heat rate | 0.01 – 10⁹ | W |
| k | Thermal conductivity | 0.01 – 400 | W/(m·K) |
| T₁, T₂ | Hot/cold face temperatures | -100 – 2000 | °C or K |
| ΔT | Temperature difference T₁ − T₂ | 0.1 – 1500 | K |
| L | Wall thickness in direction of flow | 10⁻⁵ – 5 | m |
| A | Cross-sectional area | 10⁻⁴ – 10⁴ | m² |
| dT/dx | Temperature gradient | -10⁷ – 10⁷ | K/m |
| R | Thermal resistance | 10⁻⁵ – 100 | K/W |
| R'' | R-value (per unit area) | 10⁻⁵ – 10 | m²·K/W |
| α | Thermal diffusivity | 10⁻⁸ – 10⁻³ | m²/s |
| ρ | Density | 0.1 – 22000 | kg/m³ |
| cp | Specific heat capacity | 100 – 5000 | J/(kg·K) |
What Does q = −k · (dT/dx) Measure?
Fourier's law of heat conduction states that the heat flux through a material is proportional to the negative of the temperature gradient. In plain terms: heat flows from hot regions to cold regions, and the rate of flow depends on how steeply the temperature changes with position. The proportionality constant is the material's thermal conductivity k, which measures how readily the material transports heat. This single equation is the foundation of all steady-state and transient conduction analysis, from house insulation to rocket nose cones.
The law was proposed by Joseph Fourier in 1822, and it is the thermal analogue of Ohm's law and Fick's law. Just as voltage drives current through a resistor, temperature difference drives heat through a thermal resistance. The three-transport analogy — charge, mass, and heat — is one of the most powerful ideas in physics: the same mathematical structure describes electrical current, diffusion of molecules, and conduction of heat, so techniques from one field transfer directly to the others.
Fourier's Law Formula
The differential form (for any geometry) is:
For a plane wall in steady state with T₁ at x = 0 and T₂ at x = L:
The total heat rate (multiply by area) and the thermal resistance are:
R = L / ( k · A ) · R'' = L / k
For composite walls in series, resistances add:
Fourier's Law Variables
| Symbol | Meaning | Notes |
|---|---|---|
| q | Heat flux (W/m²) | Vector quantity; signed by the direction of flow |
| k | Thermal conductivity (W/(m·K)) | Material property; can vary with temperature |
| dT/dx | Temperature gradient (K/m) | Negative when T decreases along x |
| ΔT | Temperature difference (K) | T₁ − T₂ for a plane wall |
| L | Wall thickness (m) | Distance over which ΔT is applied |
| A | Cross-sectional area (m²) | Converts flux q into total rate Q |
| R | Thermal resistance (K/W) | R = L / (k·A); series/parallel rules apply |
| R'' | R-value per area (m²·K/W) | Building-industry insulation metric |
| α | Thermal diffusivity (m²/s) | k / (ρ·cp); governs transient response |
How Each Part Affects the Heat Flux
Heat flux is linear in both k and ΔT, and inversely proportional to L. Doubling the conductivity doubles the flux; doubling the temperature difference doubles the flux; doubling the wall thickness halves the flux. These three effects are independent, so designing a thermal system means choosing a material (k), a geometry (L and A), and an operating condition (ΔT) to hit a target heat transfer rate. Adding insulation means picking a low-k material and/or making L larger; the effect is multiplicative, so a 10× thicker wall and a 10× lower-k material together reduce heat loss by 100×.
The temperature gradient |dT/dx| = ΔT/L captures both effects in a single number. A thin wall with a moderate ΔT can have a steeper gradient — and therefore higher flux — than a thick wall with a large ΔT. In building science the R-value R'' = L/k summarises insulation performance: a wall with R'' = 3.0 m²·K/W loses heat at half the rate of a wall with R'' = 1.5, all else equal. In electronics cooling, the thermal resistance R = L/(k·A) determines how much junction temperature rises for a given power dissipation, and is the parameter engineers quote on heatsink datasheets.
Fourier's Law Calculation Examples
k = 0.7 W/(m·K), T₁ = 20 °C, T₂ = 0 °C, L = 0.2 m, A = 10 m²
ΔT = 20 K · |dT/dx| = 100 K/m
q = 0.7 × 20 / 0.2 = 70 W/m²
Q = 70 × 10 = 700 W · R = 0.2/(0.7×10) = 0.2857 K/W
k = 400 W/(m·K), ΔT = 80 K, L = 0.01 m, A = 0.01 m²
q = 400 × 80 / 0.01 = 3,200,000 W/m²
Q = 3.2 × 10⁶ × 0.01 = 32,000 W
k = 0.9 W/(m·K), ΔT = 20 K, L = 0.005 m, A = 1.5 m²
q = 0.9 × 20 / 0.005 = 3,600 W/m²
Q = 3,600 × 1.5 = 5,400 W
k = 0.04 W/(m·K), ΔT = 15 K, L = 0.1 m, A = 10 m²
q = 0.04 × 15 / 0.1 = 6 W/m²
Q = 6 × 10 = 60 W · R'' = 0.1/0.04 = 2.5 m²·K/W
Frequently Asked Questions
Fourier's law states that the heat flux q through a material is proportional to the negative of the temperature gradient dT/dx, with the proportionality constant being the material's thermal conductivity k. In one dimension: q = −k · dT/dx. The negative sign encodes the physically obvious fact that heat flows from hot to cold. This single equation is the foundation of all conduction analysis — steady or transient, simple geometry or complex, one material or many.
The minus sign makes the heat flux a vector quantity that points in the direction of actual heat flow. In a coordinate system where x increases from the hot side to the cold side, the temperature decreases with x, so dT/dx is negative. Multiplying by −k (with k positive) gives a positive q, meaning heat flows in the +x direction, from hot to cold. If you drop the minus sign, q becomes a magnitude and you lose the directional information — which matters when combining heat flows from different directions or writing the full energy equation.
Thermal conductivity spans about six orders of magnitude. Metals are the best ordinary conductors: copper ~400 W/(m·K), aluminium ~237, brass ~110, carbon steel ~50, stainless steel ~15. Non-metals are much lower: water ~0.6, glass ~0.9, brick ~0.7, concrete ~1.5, wood ~0.2. Insulators are below 0.1: fibreglass ~0.04, polyurethane foam ~0.025, still air ~0.026. Aerogels reach ~0.015, and diamond peaks around 2000 W/(m·K) for pure single crystals. The huge range is why material choice dominates thermal design.
Thermal resistance R = L/(k·A) is the thermal analogue of electrical resistance. Just as V = I·R relates voltage, current, and resistance, the thermal equation Q = ΔT/R relates heat rate, temperature difference, and thermal resistance. The real power of the analogy is that resistances combine exactly like electrical resistors: in series (heat flowing through multiple layers) they add, R_total = R₁ + R₂ + …; in parallel (heat flowing through alternative paths) they combine as reciprocals, 1/R = Σ 1/Rᵢ. This lets you analyse a composite wall, a finned heat sink, or a building envelope using simple circuit arithmetic.
The simple plane-wall form assumes steady state, one-dimensional conduction, constant k, and a plane geometry. It breaks down for transient problems (where the temperature at each point changes with time), for cylindrical or spherical geometries (where the area changes with radius), for temperature-dependent k (which requires integration), for very short time or length scales comparable to the phonon mean free path (where the continuum assumption fails), and for very high heat fluxes (where the finite speed of heat propagation becomes relevant). The full Fourier equation ∂T/∂t = α∇²T covers most of these cases when solved numerically.
For a wall made of multiple layers in series, compute the thermal resistance of each layer separately (Rᵢ = Lᵢ/(kᵢ·A)), then add them to get the total resistance. The heat flux through the whole stack is q = ΔT_overall / (A · R_total), and it is the same through every layer in steady state (energy conservation). The temperature drop across any individual layer is ΔTᵢ = q · Lᵢ / kᵢ, which lets you find the interface temperatures. The same procedure works for cylindrical shells (pipe insulation) using R = ln(r₂/r₁)/(2πkL) and spherical shells using R = (1/r₁ − 1/r₂)/(4πk).
The R-value is the thermal resistance per unit area, R'' = L/k, with units m²·K/W (or in US customary units, ft²·°F·h/Btu). It is the standard metric for building insulation: higher R-value means better insulation. To convert to SI, 1 h·ft²·°F/Btu ≈ 0.176 m²·K/W. The relationship R'' = L/k shows that R-value grows linearly with thickness and inversely with conductivity, so doubling the thickness of an insulation layer doubles its R-value. A typical house wall has R-13 to R-20 in US units (R'' ≈ 2.3 to 3.5 m²·K/W); attic insulation is typically R-30 to R-60.
Variables
| Symbol | Quantity | Unit |
|---|---|---|
| q | Heat flux | W/m2 |
| k | Thermal conductivity | W/m.K |
| dT/dx | Temperature gradient | K/m |
What it means
This is identical to id=56. Fourier’s law is the basic equation for conduction heat transfer. It states that the heat flux (q) is proportional to the negative temperature gradient. The thermal conductivity k is a material property that measures the ability to conduct heat. This law is used to compute heat loss through walls, insulate pipes, cool electronics, and design heat exchangers. In solids, heat conduction is due to lattice vibrations and free electrons. Understanding Fourier’s law is crucial for engineers in building, chemical, mechanical, and aerospace disciplines to manage energy and maintain safe operating temperatures.
Worked example
Fourier's Law – Two Examples
Real‑World| Parameter | Value |
|---|---|
| k | 401 W/m·K |
| dT/dx | −100 K/m |
| Parameter | Value |
|---|---|
| k | 0.04 W/m·K |
| dT/dx | −20 K/m |
Common mistakes
- Same as ID 56 – see that entry.
- Note: For multi‑dimensional heat flow, use the full heat equation.
Applications
Fourier's law (q = −k·dT/dx) is the fundamental equation for conductive heat transfer. It is used in design of thermal insulation, heat exchangers, and thermal management systems. Engineers determine heat flux, temperature gradients, and thermal conductivity. The law is critical for sizing cooling systems in electronics, for designing furnaces and ovens, and for evaluating building energy efficiency. By applying Fourier's law, engineers can predict temperature profiles, ensure safe operating temperatures, and optimise the thermal performance of devices and systems. It is also the basis for thermal conductivity measurement techniques.
- Design of heat sinks, heat spreaders, and thermal interface materials
- Insulation design for buildings, pipes, and cold chains
- Thermal modelling of power electronics and LED lighting
- Furnace and reactor thermal design
- Thermal property characterisation of materials
Related calculators · Thermal Properties
Recently updated formulas
Suggest an improvement
Found a mistake? Have a better example? Let us know.
You may send up to 3 suggestions per 24 hours.