Embedded Isothermal Cylinder
Calculates the shape factor S and heat transfer rate Q for an isothermal cylinder of length Lc and diameter D buried at depth d (to the cylinder centre) in a semi-infinite medium of conductivity k, below an isothermal surface at temperature T2.
The governing relationship is:
Q=S⋅k⋅(T1−T2)where T1 is the cylinder temperature and S is selected from three cases.
Shape Factor Cases
Case 1 — Long cylinder (Lc≥10r), depth ≥ length (d≥Lc):
S=cosh−1(d/r)2πLcCase 2 — Long cylinder (Lc≥10r), deep burial (d>3r):
S=ln(2d/r)2πLcThis is an approximation valid when d/r≫1, where cosh−1(d/r)≈ln(2d/r).
Case 3 — Deep slender burial (d≥10r, Lc≥10d): not implemented (formula was image-only in source).
Thermal Resistance
R=k⋅S1[K/W]Reference
Holman, J. P., Heat Transfer, 7th ed., McGraw-Hill, 1990, pp. 74–83.