Math 2.0

Classical thermodynamics, heat transfer Definition

Fin heat transfer

Builds on Steady-state heat conductionFourier's law of heat conductionHeat transfer coefficientThermal conductivityHyperbolic functions

Definition

Consider a straight fin of length LL with constant cross-section area AcA_c and perimeter PP, attached at x=0x = 0 to a base at temperature TbT_b. The fin has thermal conductivity kk and lies in a fluid at temperature T∞T_\infty with heat transfer coefficient hh. Assume steady state, temperature T(x)T(x) uniform over each cross-section, and constant kk and hh. Put θ(x)=T(x)−T∞\theta(x) = T(x) - T_\infty and m=hP/(kAc)m = \sqrt{h P / (k A_c)}. The fin equation is θ′′(x)−m2θ(x)=0,0<x<L,θ(0)=Tb−T∞,\theta''(x) - m^2 \theta(x) = 0, \qquad 0 < x < L, \qquad \theta(0) = T_b - T_\infty, completed by a condition at the tip x=Lx = L.

Remarks

The equation is the energy balance of a thin slice: conduction by Fourier's law gives kAcθ′′k A_c \theta'', and the loss from the sides is hPθh P \theta.

Example: with an insulated tip, θ′(L)=0\theta'(L) = 0, the solution is θ(x)=θ(0)cosh⁡(m(L−x))/cosh⁡(mL)\theta(x) = \theta(0) \cosh(m (L - x)) / \cosh(m L), using hyperbolic functions. Check: θ′′=m2θ\theta'' = m^2 \theta, and θ′(L)=−mθ(0)sinh⁡(0)/cosh⁡(mL)=0\theta'(L) = -m \theta(0) \sinh(0) / \cosh(m L) = 0. The heat removed is Q=−kAcθ′(0)=hPkAc θ(0)tanh⁡(mL)Q = -k A_c \theta'(0) = \sqrt{h P k A_c} \, \theta(0) \tanh(m L).

Non-example: a fin whose cross-section shrinks towards the tip does not satisfy this equation, since AcA_c and PP depend on xx. Its equation has variable coefficients.

For a very long fin, θ(x)=θ(0)e−mx\theta(x) = \theta(0) e^{-m x} and Q=hPkAc θ(0)Q = \sqrt{h P k A_c} \, \theta(0). The fin efficiency with insulated tip is η=tanh⁡(mL)/(mL)\eta = \tanh(m L) / (m L).

Reading path

To understand this page from the ground up, read these 16 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 fluxconceptHeat transfer coefficient, concept, no evidence levelHeat transfer coeffic…conceptHyperbolic functions, concept, no evidence levelHyperbolic functionsconceptSteady-state heat conduction, concept, no evidence levelSteady-state heat con…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…statementFin heat transfer, definition, no evidence levelFin heat transferdefinition
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. Hyperbolic functions concept
  6. Steady-state heat conduction 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. Fin heat transfer definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Fin_(extended_surface)
About this page
TagPA463VMZ82
URIm2://PA463VMZ82
Kinddefinition
WritingW0 stub
MSC80A19, 34B05
DefinesFin heat transfer
UsesSteady-state heat conduction, Fourier's law of heat conduction, Heat transfer coefficient, Thermal conductivity, Hyperbolic functions
Changed3 November 2026

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