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Mechanical

Heat conduction through a wall

Heat flow through a flat layer with a temperature difference.

Ref: MEC-07
Rev: v1
Field: Mechanical
Result: Q̇ [W]

Q̇ = \frac{k_{t} A (T_{h} - T_{c})}{L}

Q̇ in watts, k_t thermal conductivity in W/(m·K), A area in m², T_h and T_c the hot and cold face temperatures, L the thickness in metres.

Heat flows through a wall faster when the wall is large, the material conducts well, and the temperature difference is big. It slows as the wall gets thicker.

Calculator

Solve for

Q̇Heat flowComputed

k_tThermal conductivity

W/(m·K)

0.02400

AArea

m²

0.1100

T_hHot face temperature

°C

1060

T_cCold face temperature

°C

−205

LWall thickness

0.021

Heat flux q = Q/A64 W/m²
Thermal resistance L/(kA)0.03125 K/W

What drives the result

Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.

Heat flow (W) current value 640 W

k_tThermal conductivityKey driver

16320,000

LWall thickness

1608,000

AArea

6.46,400

T_hHot face temperature

3201,920

T_cCold face temperature

4801,280

Heat flow is the temperature difference divided by a thermal resistance, exactly like Ohm’s law with L/(kA) as the resistance.
Insulation works by having a very small k; doubling the thickness helps as much as halving k.
Thermal conductivity spans four orders of magnitude between foam and copper, so the material choice dominates.

Chart data as tables

Ranked variables

Variable	Typical range	Lowest result	Highest result	Swing (W)	Local step	Change in result
k_t Thermal conductivity	0.02 to 400 W/(m·K)	16	320,000	320,000	+10%	+10 %
L Wall thickness	0.02 to 1 m	160	8,000	7,840	+10%	−9.09 %
A Area	0.1 to 100 m²	6.4	6,400	6,394	+10%	+10 %
T_h Hot face temperature	10 to 60 °C	320	1,920	1,600	+10%	+10 %
T_c Cold face temperature	−20 to 5 °C	480	1,280	800	+10%	0 %

Sweep of Wall thickness

Wall thickness (m)	Heat flow (W)
0.02	8,000
0.03261	4,906
0.05318	3,008
0.08672	1,845
0.1414	1,131
0.2306	693.8
0.3761	425.5
0.6132	260.9
1	160

Sweep one variable

Vary

Heat flow (W)Hover, touch, or use the arrow keys on the chart

Variables and how each one contributes

Q̇

Heat flow

Result · W

What the equation gives you. Everything else feeds it.

k_t

Thermal conductivity

W/(m·K) · typical 0.02 to 400

How well the material conducts heat (brick 0.8, steel 50, copper 400, foam 0.03). Directly proportional.

Area

m² · typical 0.1 to 100

Area normal to the heat flow. Directly proportional.

T_h

Hot face temperature

°C · typical 10 to 60

Temperature of the warmer surface. Linear through the difference T_h − T_c.

T_c

Cold face temperature

°C · typical −20 to 5

Temperature of the colder surface. Linear, in the opposite direction to T_h.

Wall thickness

m · typical 0.02 to 1

Distance along the heat flow. Inversely proportional: doubling the thickness halves the heat flow.

Worked example with the values above

Temperature difference: 20 − 0 = 20 K.
Q = kAΔT/L = 0.8 × 10 × 20 / 0.25.
Q = 640 W.

Assumptions and limits

Steady state, one-dimensional flow through a plane wall.
Constant conductivity, no heat generation inside.
Face temperatures known.

Common mistakes

Using the air temperatures instead of the surface temperatures.
Thickness in millimetres or centimetres.
Treating a multi-layer wall as one layer: add the resistances.

Statistics and uncertainty

Conductivity values for building and insulating materials often carry ±10–20% uncertainty, and that usually dominates: δQ/Q ≈ √[(δk/k)² + (δA/A)² + (δL/L)² + (δ(ΔT)/ΔT)²].

Where it is used

Heat loss through walls, hulls and enclosures.
Cooling of electronics through a heat-sink base.
Sizing insulation.

Maths you need

Temperature difference: heat flows from hot to cold
Ohm’s law analogy: flow = driving difference ÷ resistance

Related formulas

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