Heat Conduction Calculator

Calculate conductive heat transfer using Q/t = kAΔT/d (Fourier's Law).

🔥 Thermodynamics📐 Q/t = kAΔT/d🌡️ Fourier's Law
Thermal conductivity k (W/m·K)
Area (A) m²
Temperature difference (ΔT) K or °C
Thickness (d) m

k (W/m·K): Air=0.026, Fiberglass insulation=0.04, Wood=0.12, Brick=0.8, Concrete=1.7, Glass=1.0, Steel=50, Copper=401, Diamond=2000

⚠️ Enter valid positive numbers.

What Is Heat Conduction (Fourier's Law)?

Heat conduction is the transfer of thermal energy through a material without bulk motion of the material. Fourier's Law: Q/t = k·A·ΔT/d, where Q/t is heat flow rate (watts), k is thermal conductivity (W/m·K), A is cross-sectional area (m²), ΔT is temperature difference (K or °C), and d is thickness (m). Heat flows from hot to cold.

Thermal conductivity k spans many orders of magnitude: Diamond (k = 2,000 W/m·K) is the best solid conductor; aerogel (k = 0.015 W/m·K) is among the best insulators. Air (k = 0.026) insulates because it conducts poorly — the key to double-pane windows, fiberglass insulation, and down jackets (trapping air in gaps).

For materials in series (layers), the thermal resistances add: R_total = Σ(dᵢ/kᵢ) per unit area. The combined heat flow: Q/t = A·ΔT_total/R_total. In building science, the R-value = d/k (m²·K/W) quantifies insulation performance — higher R-value = better insulation. US R-values use customary units.

In the Newton's law of cooling approximation (for surfaces): Q/t = h·A·ΔT, where h is the convective heat transfer coefficient (W/m²·K). Real heat transfer combines conduction through materials and convection at surfaces. The overall heat transfer coefficient U = 1/(Σ d/k + 1/h_inner + 1/h_outer) combines both effects.

Formula Reference Table

Solve ForFormulaNotes
Heat flow rateQ/t = k·A·ΔT/dWatts = J/s
Thermal conductivityk = (Q/t)·d/(A·ΔT)W/m·K
Thicknessd = k·A·ΔT/(Q/t)m
Thermal resistanceR = d/(k·A) = ΔT/(Q/t)K/W
R-value (per area)R'' = d/km²·K/W; US: °F·ft²·hr/Btu
Layers in seriesQ/t = A·ΔT_total/Σ(dᵢ/kᵢ)Resistances add

3 Worked Examples

Example 1
Home Wall Insulation

150 mm fiberglass (k=0.04), area = 100 m², ΔT = 25°C.

  • Q/t = 0.04 × 100 × 25 / 0.15
  • Q/t = 100/0.15 = 666.7 W
  • Annual energy loss = 666.7 × 365×24 × 3600 / 3.6×10⁶ = 5,840 kWh/yr
✓ 666.7 W heat loss through insulated wall
Example 2
Find k of Material

Material: 50 mm thick, A = 1 m², ΔT = 100 K, Q/t = 120 W.

  • k = (Q/t)×d/(A×ΔT) = 120×0.05/(1×100)
  • k = 6/100 = 0.06 W/m·K
  • This is close to wood (k≈0.12) or foam insulation
✓ k = 0.06 W/m·K
Example 3
Double Pane Window

Inner glass 4 mm (k=1.0) + air gap 12 mm (k=0.026) + outer glass 4 mm (k=1.0). A=1.5 m², ΔT=20 K.

  • R_total = (0.004/1.0 + 0.012/0.026 + 0.004/1.0) = 0.004+0.462+0.004 = 0.470 m²·K/W
  • Q/t = A×ΔT/R_total = 1.5×20/0.470 = 63.8 W
  • Single pane (4 mm): Q/t = 1.5×20/0.004 = 7,500 W — 117× worse!
✓ Double pane: 63.8 W; single pane: 7,500 W (117× more loss)

Real-World Applications

🏠
Building Insulation
Fiberglass (k=0.04) vs spray foam (k=0.026) vs aerogel (k=0.015): conductivities determine wall U-values and heating/cooling costs. Double R-value → halve heat loss → halve energy bills.
⚡
Electronic Cooling
CPU heat spreaders use copper (k=401) or vapor chambers. Thermal interface materials (k=1–10 W/m·K) fill gaps. Heatsink fins (aluminum, k=237) maximize A to lower total thermal resistance.
🏥
Cryogenic Storage
Vacuum-insulated cryogenic vessels use vacuum + multi-layer insulation (MLI) to achieve effective k < 0.0001 W/m·K. Liquid nitrogen (−196°C) can be stored for weeks in dewars.
🌡️
Geothermal Energy
Earth's geothermal gradient (~30°C/km in crust): Q/t = k×A×ΔT/d. With k_crust ≈ 2.5 W/m·K: heat flow ≈ 65 mW/m² worldwide average.
🔧
Heat Pipes
Copper-water heat pipes have effective k = 10,000–100,000 W/m·K due to phase-change convection, used in laptops and CPUs to spread heat from hot spots.

Common Mistakes to Avoid

⚠️
Using d in cm or mm instead of meters

d must be in meters: 150 mm = 0.150 m. Using 150 gives k values 1,000× too small.

⚠️
Confusing heat flow rate (W) with heat energy (J)

Q/t is power in Watts. Total heat Q = (Q/t) × time in seconds (Joules).

⚠️
Applying to convection without adding convection resistance

Fourier's law covers conduction only. Surfaces also have convection resistance 1/(h·A). Total resistance = Σ(d/k) + 1/h_in + 1/h_out.

⚠️
Wrong k for mixed materials

Composite materials: use geometric or harmonic mean k depending on layering. Parallel layers: k_eff = (Σkᵢ·dᵢ)/Σdᵢ. Series layers: 1/k_eff = Σ(1/kᵢ) (resistances add).

⚠️
Forgetting radiation at high temperatures

At high temperatures (furnaces, combustion), radiation Q = εσAT⁴ dominates over conduction. Fourier's law alone underestimates heat transfer when surfaces exceed ~500°C.

Frequently Asked Questions

What is thermal conductivity k? ▾
k (W/m·K) measures how readily a material conducts heat. Diamond (k=2000): best solid conductor (phonon transfer). Metals: good (silver=429, copper=401, aluminum=237, steel=50). Ceramics: intermediate (concrete=1.7, glass=1.0). Polymers and wood: poor (0.1–0.3). Air (k=0.026) and aerogel (0.015): excellent insulators.
How does double-pane window insulation work? ▾
Fourier's law: Q/t = k·A·ΔT/d. The thin air gap (k=0.026, d=12 mm) has much higher thermal resistance than glass (k=1.0, d=4 mm): R_air = 0.012/0.026 = 0.46 vs R_glass = 0.004. The air gap provides >98% of total insulation. Argon (k=0.018) further improves performance.
What is the R-value of insulation? ▾
R-value = d/k (m²·K/W in SI; ft²·°F·hr/BTU in US customary). Higher R-value = better insulation. US fiberglass batt R-19 ≈ 3.3 m²·K/W (d=140 mm, k=0.04). To convert: SI R-value × 5.68 = US R-value.
How is thermal resistance calculated for layers? ▾
In series (perpendicular to heat flow): R_total = Σ(dᵢ/kᵢ) per unit area. Heat flows through each layer with the same heat flux. Q/t = A×ΔT_total/R_total. In parallel (layers side by side): Q/t_total = Σ(Q/t)_each. Series adds resistances; parallel adds conductances.
What is the Fourier number? ▾
Fo = k·t/(ρ·Cp·L²) = α·t/L² (dimensionless), where α = k/(ρCp) is thermal diffusivity. Fo characterizes transient heat conduction: Fo << 1 means the interior hasn't 'felt' the surface temperature change yet; Fo > 1 means the object is nearly at a new equilibrium. Used in baking, casting, and quenching analysis.
How does conduction differ from convection and radiation? ▾
Conduction: heat flows through static material via atomic vibrations and electron motion — requires material contact. Convection: heat is carried by bulk fluid motion — requires fluid (gas or liquid). Radiation: electromagnetic energy transfer, requires no medium. Most real systems involve all three; which dominates depends on material, geometry, and temperature difference.
Why do metals feel colder than wood at the same temperature? ▾
Both are at room temperature, but metal has k ≈ 50 W/m·K vs wood ≈ 0.12 W/m·K. When you touch metal, heat flows rapidly from your warm skin (k_skin ≈ 0.5 W/m·K) to the metal — high Q/t feels cold. Wood's low k means slower heat flow — feels 'warmer' even at the same temperature. You're feeling k, not T.
How is heat conduction modeled in computers? ▾
Finite element analysis (FEA) solves ∇·(k∇T) = q (steady state) or ρCp·∂T/∂t = ∇·(k∇T) + q (transient). The domain (chip, heat sink, PCB) is meshed into thousands of elements; each has its own k. Boundary conditions: known T, known Q/t, or convection. FEA gives complete 3D temperature maps.

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