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
$\frac{2k_4}{D_{ref} \cosh^{-1}\left(1+\frac{2e_{limit}}{D_{ext}}\right)}$Carslaw & Jaeger
$\frac{4k_4 C_{g1}}{D_{ref} \beta_b \left(\pi-\beta_b\right) \sqrt{{C_{g2}}^2-1}} \left(\frac{\pi}{2}-\arctan\left(\sqrt{\frac{C_{g2}+1}{C_{g2}-1}} tan\left(\frac{\beta_b}{2}\right)\right)\right)$Morud & Simonsen $C_{g2}$ > 1
$\frac{2k_4 C_{g1}}{D_{ref} \beta_b \left(\pi-\beta_b\right) \sqrt{1-{C_{g2}}^2}} \ln\left(\frac{tan\left(\frac{\beta_b}{2}\right)+\sqrt{\frac{1-C_{g2}}{1+C_{g2}}}}{tan\left(\frac{\beta_b}{2}\right)-\sqrt{\frac{1-C_{g2}}{1+C_{g2}}}}\right)$Morud & Simonsen $C_{g2}$ ≤ 1
$\frac{4k_4 \mathrm{Bi}_p sin\left(\beta_0\right)}{D_{ref} \pi \left(1+\frac{\mathrm{Bi}_p}{\mathrm{Bi}_g}\right) \sqrt{1-{K_{par}}^2}} \arctan\left(\sqrt{\frac{1-K_{par}}{1+K_{par}}}\right)$Ovuworie $|K_{par}|$ < 1
$\frac{2k_4 \mathrm{Bi}_p sin\left(\beta_0\right)}{D_{ref} \pi \left(1+\frac{\mathrm{Bi}_p}{\mathrm{Bi}_g}\right)}$Ovuworie $K_{par}$ = 1
$\frac{4k_4 \mathrm{Bi}_p sin\left(\beta_0\right)}{D_{ref} \pi \left(1+\frac{\mathrm{Bi}_p}{\mathrm{Bi}_g}\right) \sqrt{{K_{par}}^2-1}} \arctan\left(\sqrt{\frac{K_{par}-1}{K_{par}+1}}\right)$Ovuworie $K_{par}$ > 1
$\frac{2k_4 \mathrm{Bi}_p}{D_{ref} \left(\left(1+\frac{\mathrm{Bi}_p}{\mathrm{Bi}_g}\right) \left(1+2\mathrm{Bi}_p\right)\right)^{0.5}}$OTC 23033
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