Heat transfer coefficient buried part of wall to soil

The heat transfer coefficient of ground is combining the inside film coefficient, heat transfer coefficient of pipe wall and heat transfer coefficient of soil.

The equation acc. Carslaw & Jaegers is used for $h_{buried}$ reaching a limit when $H$ → $D_{ext}/2$ using the quantity $e_{limit}$.

Symbol
$h_{ground}$
Unit
W/(K.m$^2$)
Formulae
$\frac{2 k_{4}}{D_{ref} \operatorname{acosh}{\left(1 + \frac{2 e_{limit}}{D_{ext}} \right)}}$Carslaw & Jaeger
$\frac{4 C_{g1} k_{4} \left(\frac{\pi}{2} - \operatorname{atan}{\left(\sqrt{\frac{C_{g2} + 1}{C_{g2} - 1}} \tan{\left(\frac{\beta_{b}}{2} \right)} \right)}\right)}{D_{ref} \beta_{b} \sqrt{C_{g2}^{2} - 1} \left(- \beta_{b} + \pi\right)}$Morud & Simonsen $C_{g2} > 1$
$\frac{2 C_{g1} k_{4} \ln{\left(\frac{\sqrt{\frac{1 - C_{g2}}{C_{g2} + 1}} + \tan{\left(\frac{\beta_{b}}{2} \right)}}{- \sqrt{\frac{1 - C_{g2}}{C_{g2} + 1}} + \tan{\left(\frac{\beta_{b}}{2} \right)}} \right)}}{D_{ref} \beta_{b} \sqrt{1 - C_{g2}^{2}} \left(- \beta_{b} + \pi\right)}$Morud & Simonsen $C_{g2} ≤ 1$
$\frac{4 \mathrm{Bi}_{p} k_{4} \operatorname{atan}{\left(\sqrt{\frac{1 - K_{par}}{K_{par} + 1}} \right)} \sin{\left(\beta_{0} \right)}}{D_{ref} \pi \sqrt{1 - K_{par}^{2}} \left(1 + \frac{\mathrm{Bi}_{p}}{\mathrm{Bi}_{g}}\right)}$Ovuworie $|K_{par}| < 1$
$\frac{2 \mathrm{Bi}_{p} k_{4} \sin{\left(\beta_{0} \right)}}{D_{ref} \pi \left(1 + \frac{\mathrm{Bi}_{p}}{\mathrm{Bi}_{g}}\right)}$Ovuworie $K_{par} = 1$
$\frac{4 \mathrm{Bi}_{p} k_{4} \operatorname{atan}{\left(\sqrt{\frac{K_{par} - 1}{K_{par} + 1}} \right)} \sin{\left(\beta_{0} \right)}}{D_{ref} \pi \left(1 + \frac{\mathrm{Bi}_{p}}{\mathrm{Bi}_{g}}\right) \sqrt{K_{par}^{2} - 1}}$Ovuworie $K_{par} > 1$
$\frac{2 \mathrm{Bi}_{p} k_{4}}{D_{ref} \left(\left(1 + \frac{\mathrm{Bi}_{p}}{\mathrm{Bi}_{g}}\right) \left(2 \mathrm{Bi}_{p} + 1\right)\right)^{0.5}}$OTC 23033
$\mathrm{Bi}_{g}$
$\mathrm{Bi}_{p}$
Used in
$U_{ground}$