For steady multi-dimensional heat conduction with constant thermal conductivity and no heat generation, the temperature field satisfies the Laplace equation:
\[\nabla^2 T = \frac{\partial^2 T}{\partial x^2} + \frac{\partial^2 T}{\partial y^2} = 0\]
Conduction Shape Factor (\(S\))
While analytical series or numerical methods can solve this, engineering designs often use the Conduction Shape Factor \(S\) to compute heat transfer rates directly:
\[q = S k (T_1 - T_2)\]
Where \(k\) is the solid thermal conductivity, and \(T_1, T_2\) are boundary temperatures.
Buried Cylinder Case
For an isothermal cylinder of diameter \(D\) and length \(L\) buried horizontally at depth \(z\) in a semi-infinite medium (with surface temperature \(T_2\)), the shape factor is:
\[S = \frac{2 \pi L}{\cosh^{-1}(2z/D)}\]
References
- Hahne, E., & Grigull, U. (1975). A shape factor database for steady-state heat conduction. International Journal of Heat and Mass Transfer, 18(6), 751-767.
