Math 2.0

Classical thermodynamics, heat transfer Definition

Robin boundary condition for convective cooling

Builds on Robin boundary conditionFourier's law of heat conductionHeat transfer coefficientThermal conductivityNormal vector

Definition

Let a body occupy a bounded open set Ω⊆ℝ3\Omega \subseteq \R^3 with C1C^1 boundary ∂Ω\partial \Omega and outward unit normal ν\nu. Let k>0k > 0 be its thermal conductivity, and let TT be its temperature, continuously differentiable up to ∂Ω\partial \Omega. Let the body be surrounded by a fluid at temperature T∞T_\infty, with heat transfer coefficient h>0h > 0. The convective boundary condition is the Robin condition −k ∂T∂ν=h(T−T∞)on ∂Ω.-k \, \frac{\partial T}{\partial \nu} = h \left(T - T_\infty\right) \quad \text{on } \partial \Omega.

Remarks

The left side is the outward normal component of the conductive flux from Fourier's law. The right side is the flux carried away by the fluid.

Example: a slab 0<x<L0 < x < L with T(0)=T0T(0) = T_0 and the convective condition at x=Lx = L, in steady state. Then T(x)=T0−qx/kT(x) = T_0 - q x / k with q=(T0−T∞)/(L/k+1/h)q = (T_0 - T_\infty) / (L / k + 1 / h). Check: −kT′(L)=q-k T'(L) = q, and h(T(L)−T∞)=h(T0−T∞−qL/k)=qh (T(L) - T_\infty) = h (T_0 - T_\infty - q L / k) = q, since T0−T∞=q(L/k+1/h)T_0 - T_\infty = q (L / k + 1 / h).

Non-example: the condition T=T∞T = T_\infty on ∂Ω\partial \Omega is not of this type. It is the Dirichlet condition, which is the limit h→∞h \to \infty. The limit h→0h \to 0 gives the insulated Neumann condition ∂T/∂ν=0\partial T / \partial \nu = 0.

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Euclidean space, object, no evidence levelEuclidean spaceobjectFunction, concept, no evidence levelFunctionconceptHeat flux, concept, no evidence levelHeat fluxconceptHeat transfer coefficient, concept, no evidence levelHeat transfer coeffic…conceptNormal vector, concept, no evidence levelNormal vectorconceptRobin boundary condition, concept, no evidence levelRobin boundary condit…conceptTemperature, concept, no evidence levelTemperatureconceptThermal conduction, concept, no evidence levelThermal conductionconceptThermal conductivity, concept, no evidence levelThermal conductivityconceptTotal 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 levelGradientconceptFourier's law of heat conduction, statement, no evidence levelFourier's law of heat…statementRobin boundary condition for convective cooling, definition, no evidence levelRobin boundary condit…definition
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 flux concept
  4. Heat transfer coefficient concept
  5. Normal vector concept
  6. Robin boundary condition concept
  7. Temperature concept
  8. Thermal conduction concept
  9. Thermal conductivity concept
  10. Total derivative concept
  11. Function of several variables concept
  12. Differentiable function of several variables concept
  13. Partial derivative concept
  14. Gradient concept
  15. Fourier's law of heat conduction statement
  16. Robin boundary condition for convective cooling definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Robin_boundary_condition
About this page
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MSC80A19, 35Q79
DefinesRobin boundary condition for convective cooling
UsesRobin boundary condition, Fourier's law of heat conduction, Heat transfer coefficient, Thermal conductivity, Normal vector
Changed3 November 2026

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