Math 2.0

Classical thermodynamics, heat transfer Definition

Thermal conductivity

Builds on Thermal conductionHeat fluxTemperatureGradient

Definition

Let a homogeneous isotropic material conduct heat, with temperature field TT and heat flux πͺ\mathbf{q} of conduction. Assume that πͺ\mathbf{q} is proportional to the gradient βˆ‡T\nabla T at each point. The thermal conductivity of the material is the number kk such that πͺ=βˆ’kβ€‰βˆ‡T.\mathbf{q} = -k \, \nabla T. Its unit is W mβˆ’1 Kβˆ’1\mathrm{W \, m^{-1} \, K^{-1}}. The reciprocal 1/k1/k is the thermal resistivity.

Remarks

Example: a slab of thickness L=0.1L = 0.1 metres conducts a heat flux of 200200 watts per square metre when its faces differ by 2020 kelvin. Then βˆ£βˆ‡T∣=20/0.1=200\abs{\nabla T} = 20 / 0.1 = 200 kelvin per metre, so k=200/200=1k = 200 / 200 = 1 watt per metre per kelvin.

Non-example: the ratio Q/Ξ”TQ / \Delta T of a whole wall, in watts per kelvin, is not the conductivity. It depends on the size and shape of the wall; it is the thermal conductance.

The second law of thermodynamics forces kβ‰₯0k \ge 0. In general kk depends on temperature, and in an anisotropic material it is replaced by a tensor.

Reading path

To understand this page from the ground up, read these 11 pages in order. The foundations come first.

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Euclidean space, object, no evidence levelEuclidean spaceobjectFunction, concept, no evidence levelFunctionconceptHeat flux, concept, no evidence levelHeat fluxconceptTemperature, concept, no evidence levelTemperatureconceptThermal conduction, concept, no evidence levelThermal conductionconceptTotal derivative, concept, no evidence levelTotal derivativeconceptFunction of several variables, concept, no evidence levelFunction of several v…conceptDifferentiable function of several variables, concept, no evidence levelDifferentiable functi…conceptPartial derivative, concept, no evidence levelPartial derivativeconceptGradient, concept, no evidence levelGradientconceptThermal conductivity, definition, no evidence levelThermal conductivitydefinition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
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  1. Euclidean space object
  2. Function concept
  3. Heat flux concept
  4. Temperature concept
  5. Thermal conduction concept
  6. Total derivative concept
  7. Function of several variables concept
  8. Differentiable function of several variables concept
  9. Partial derivative concept
  10. Gradient concept
  11. Thermal conductivity definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Thermal_conductivity_and_resistivity
About this page
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Kinddefinition
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MSC80A19, 80A17
DefinesThermal conductivity
UsesThermal conduction, Heat flux, Temperature, Gradient
Changed3 November 2026

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