A convecting fin combines conduction inside a solid with convection from its surface to the surrounding fluid. This case checks that fixed-temperature, adiabatic, and convective boundary conditions are assembled consistently with the conduction constitutive law and the overall energy balance.

What this case verifies

• Coupling of Fourier conduction and Newton's law of cooling
• Fixed-temperature base, convective sides, and adiabatic tip
• Surface integration of the convection coefficient and exposed perimeter
• The full axial temperature profile, not only a single nodal value

Problem definition

The straight fin has constant cross-sectional area $A_c$ and perimeter $P$ . Its base is held at $T_b=400$ K. The four side faces use $h=169$ W/m²K and $T_\infty=300$ K, while the tip has zero heat flux. Thermal conductivity $k$ and the convection coefficient are constant, and radiation is neglected.

Reference solution

Defining excess temperature as $\theta=T-T_\infty$ gives the steady one-dimensional fin equation

$$\frac{d^2\theta}{dx^2}-m^2\theta=0, \qquad m=\sqrt{\frac{hP}{kA_c}}$$

Applying $\theta(0)=T_b-T_\infty$ and $d\theta/dx|_{x=L}=0$ yields

$$\frac{T(x)-T_\infty}{T_b-T_\infty} =\frac{\cosh[m(L-x)]}{\cosh(mL)}$$

The adiabatic-tip temperature is therefore

$$T(L)=T_\infty+\frac{T_b-T_\infty}{\cosh(mL)}$$

Neither a correct conduction implementation nor a correct convection implementation alone is sufficient to reproduce this solution; the test exercises both physics contributions and their boundary integration.

Results

The tip temperature is effectively exact, and the entire axial profile agrees with the hyperbolic-cosine solution to within 0.1%. Agreement over the profile demonstrates that the result is not a coincidental match at one point.

Engineering significance

The case underpins thermal analysis of motor housings, cooling fins, and heat sinks. Real natural or forced convection can have a spatially varying $h$ ; this constant-coefficient case deliberately isolates the thermal solver before that modeling uncertainty is introduced.

Reference

• MIT Unified Engineering, Heat Transfer From a Fin — adiabatic-tip temperature distribution.

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