Constant depending on the burial depth

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. This expression is appropriate for deeply buried pipes. When the top of line is close the soil surface (i.e. when the pipe is just barely buried), the burial depth approaches the radius and $\alpha_0$ tends to zero since $cos^{-1}(x)=0$ when x → 1.

$\operatorname{acosh}{\left(\frac{2 H}{D_{ext}} \right)}$hyperbolic functions
$\ln{\left(\sqrt{-1 + \frac{4 H^{2}}{D_{ext}^{2}}} + \frac{2 H}{D_{ext}} \right)}$natural logarithm
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