Math 2.0

Classical thermodynamics, heat transfer Definition

Heat flux

Builds on HeatVector fieldFlux through a surface

Definition

Let a body occupy an open set Ω⊆ℝ3\Omega \subseteq \R^3, and let II be a time interval. The heat flux is a vector field 𝐪 ⁣:Ω×I→ℝ3\mathbf{q} \colon \Omega \times I \to \R^3 with this property. Let S⊆ΩS \subseteq \Omega be any oriented smooth surface with unit normal 𝐧\mathbf{n}, and let t∈It \in I. Then the rate at which heat crosses SS in the direction of 𝐧\mathbf{n} is the flux ∫S𝐪(x,t)⋅𝐧(x) dA(x).\int_S \mathbf{q}(x, t) \cdot \mathbf{n}(x) \, dA(x). Its unit is the watt per square metre.

Remarks

Example: in a slab 0<x1<L0 < x_1 < L whose faces are held at temperatures T1>T2T_1 > T_2, at steady state with constant conductivity kk, the flux is 𝐪=(k(T1−T2)/L,0,0)\mathbf{q} = \left(k (T_1 - T_2)/L, 0, 0\right). Take a square of area AA parallel to the faces, with 𝐧=(1,0,0)\mathbf{n} = (1, 0, 0). The heat rate through it is kA(T1−T2)/L>0k A (T_1 - T_2)/L > 0, so heat goes from the hot face to the cold face.

Non-example: the total heat rate QQ through a wall, in watts, is not a heat flux. It is the integral of the normal component of the flux, and it does not depend on a point.

Some texts call the scalar qn=𝐪⋅𝐧q_n = \mathbf{q} \cdot \mathbf{n} on a given surface the heat flux. The vector is also written 𝐪′′\mathbf{q}'' or ϕ\boldsymbol{\phi}.

Reading path

To understand this page from the ground up, read these 12 pages in order. The foundations come first.

Show the path as a graph
Euclidean space, object, no evidence levelEuclidean spaceobjectFunction, concept, no evidence levelFunctionconceptHeat, concept, no evidence levelHeatconceptParametrised surface, concept, no evidence levelParametrised surfaceconceptVector-valued function, concept, no evidence levelVector-valued functionconceptSurface area, concept, no evidence levelSurface areaconceptTangent plane and normal vector of a surface, concept, no evidence levelTangent plane and nor…conceptVector field, concept, no evidence levelVector fieldconceptOrientable surface, concept, no evidence levelOrientable surfaceconceptSurface integral, concept, no evidence levelSurface integralconceptFlux through a surface, concept, no evidence levelFlux through a surfaceconceptHeat flux, definition, no evidence levelHeat fluxdefinition
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.
  • no evidence level
  1. Euclidean space object
  2. Function concept
  3. Heat concept
  4. Parametrised surface concept
  5. Vector-valued function concept
  6. Surface area concept
  7. Tangent plane and normal vector of a surface concept
  8. Vector field concept
  9. Orientable surface concept
  10. Surface integral concept
  11. Flux through a surface concept
  12. Heat flux definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Heat_flux
About this page
Tag8ZCHRMTZN4
URIm2://8ZCHRMTZN4
Kinddefinition
WritingW0 stub
MSC80A19, 80A17
DefinesHeat flux
UsesHeat, Vector field, Flux through a surface
Changed3 November 2026

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