The first formula is based on Kuehn and Goldstein 'Correlating equations for natural convection heat transfer between horizontal circular cylinders', 1976 who presented a correlation for natural convection heat transfer from a horizontal cylinder which is valid at any Rayleigh and Prandtl number.
The second formula is based on J. Lis and P.O. Kellard 'Measurements of the thermal conductivity of sulphur hexafluoride and a 50 percent (volume) mixture of SF6 and nitrogen', 1965. The value of the exponent indicates that the heat is transferred by turbulent convection.
The third formula is based on J. Vermeer: 'A simple formula for the calculation of the convective heat transfer between conductor and sheath in compressed gas insulated (CGI) cables', 1983 as published in Elektra 87. This is a simplified version of the formula by Lis & Kellard qhich yields results that differ less than 1% for the whole range of conditions encountered in compressed gas insulated cables which is comparable to GIL.
$0.079 \left(\mathrm{Gr}_{\mathrm{c}} \mathrm{Pr}_{\mathrm{gas}} \left(- \frac{D_{\mathrm{c}}}{D_{\mathrm{encl}}} + 1\right)^{6.5}\right)^{0.333}$ | Vermeer1983 |
$0.087 \left(\mathrm{Gr}_{\mathrm{c}} \mathrm{Pr}_{\mathrm{gas}} \left(- \frac{D_{\mathrm{c}}}{D_{\mathrm{encl}}} + 1\right)^{6.5}\right)^{0.329}$ | Lis&Kellard1965 |
$\frac{2}{\ln{\left (\frac{2}{\left(\frac{0.0015264073619969 \mathrm{Ra}_{\mathrm{c}}^{3.75}}{\left(0.705418889231171 \left(\frac{1}{\mathrm{Pr}_{\mathrm{gas}}}\right)^{0.6} + 1\right)^{6.25}} + 1.54070215745864 \cdot 10^{-14} \mathrm{Ra}_{\mathrm{c}}^{4.995}\right)^{0.0667}} + 1 \right )}}$ | Kuehn&Goldstein1976 |