Heat transfer coefficient wall to soil

By using a conduction shape factor for a horizontal cylinder buried in a semi-infinite medium, the heat transfer coefficient for a buried pipeline or cable can be expressed as follows. For the case of laying depth L larger than half of the diameter D, the cosh can be simplified using the logarithmic equation.

In the method of Carslaw & Jaegers reference book on conduction from 1959, the external heat transfer coefficient is called $h_{soil+amb}$ and is combining the heat transfer by conduction through the surroundings and the heat transfer by convection above the soil surface. The quantity $e_{soil}$ stands for the thickness of an extra thin soil layer modeling the thermal resistance resulting from convection at the soil surface. This thickness can be determined from the continuity equation of the heat flux rate at the sea/soil interface.

Symbol
$h_{buried}$
Unit
W/(K.m$^2$)
Formulae
 $\frac{2k_4}{D_{ref} \cosh^{-1}\left(\frac{2\left(H+e_{soil}\right)}{D_{ext}}\right)}$ Carslaw & Jaeger $\frac{\frac{2k_4}{D_{ref}} \mathrm{Bi}_p}{\left(1+{\mathrm{Bi}_p}^2 {\alpha_0}^2+\frac{2\mathrm{Bi}_p \alpha_0{\cdot}1}{tanh\left(\alpha_0\right)}\right)^{0.5}}$ Morud & Simonsen $\frac{\frac{2k_4}{D_{ref}} \mathrm{Bi}_p sinh\left(\alpha_0\right)}{\left(\left(cosh\left(\alpha_0\right)+\mathrm{Bi}_p \alpha_0 sinh\left(\alpha_0\right)+\frac{\mathrm{Bi}_p}{\mathrm{Bi}_g}\right)^2-\left(1+\frac{\mathrm{Bi}_p}{\mathrm{Bi}_g}\right)^2\right)^{0.5}}$ Ovuworie / OTC 23033
Related
$D_{ext}$
$D_{ref}$
$e_{soil}$
Used in
$U_{buried}$