Nusselt number, conductor to gas

The convective heat transfer in SF6, N2 and mixtures of SF6 and N2 between horizontal coaxial cylinders of various different dimensions was studied by J. Lis: 'Experimental investigation of natural convection heat transfer in simple and obstructed horizontal annuli', 1966. He carried out measurements of the Nu number and used the thermal conductivities of SF6 and its mixtures with N2 were taken from the work of 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.

According to the work of Lis, the Nu number can be expressed as a function of the product of Gr and Pr and the geometrical factor 1, which contains the ratio of the diameters of the cylinders. For conditions prevailing in compressed gas insulated (CGI) cables, that is for $10^{9}$ < $Gr \cdot Pr$ < $5 \cdot 10^{10}$, the formula published by R. Giblin: 'Transmission de la chaleur par convection naturelle', 1974 holds. The value of the exponent indicates that the heat is transferred by turbulent convection.

In order to keep the mathematical operations as simple as possible, the formula was rewritten without significant loss of accuracy by J. Vermeer: 'A simple formula for the calculation of the convective heat transfer between conductor and sheath in compressed gas insulated (CGI) cables', Elektra 87, 1983. CGI cables are comparable to gas insulated lines (GIL) and the Cigre Guide 218 'Gas Insulated Transmission Lines (GIL)', 2003 is based on this work.

The first two formulae are not used directly but an approximation is derived using a constant $c_{gas}$ depending on the gas concerned. Vermeer1983 published values of $c_{gas}$ for SF6 and N2, we derived values for Air, CO2 and O2 the same way for the Cableizer method.

Kuehn and Goldstein published in 'Correlating equations for natural convection heat transfer between horizontal circular cylinders', 1976 a correlation for natural convection heat transfer from a horizontal cylinder which is valid at any Rayleigh and Prandtl number. This equation is used directly in the method by Eteiba2002.

Symbol
$\mathrm{Nu}_c$
Formulae
 $0.087\left(\mathrm{Gr}_c \mathrm{Pr}_{gas} \left(1-\frac{D_c}{D_{encl}}\right)^{6.5}\right)^{0.329}$ Giblin1974 based on Lis&Kellard1965 (not used) $0.079\left(\mathrm{Gr}_c \mathrm{Pr}_{gas} \left(1-\frac{D_c}{D_{encl}}\right)^{6.5}\right)^{0.333}$ Vermeer1983 (not used directly) $\frac{2}{\ln\left(1+\frac{2}{\left(\left(0.649{\mathrm{Ra}_c}^{\frac{1}{4}} \left(1+\left(\frac{0.559}{\mathrm{Pr}_{gas}}\right)^{\frac{3}{5}}\right)^{\frac{-5}{12}}\right)^{15}+\left(0.12{\mathrm{Ra}_c}^{0.333}\right)^{15}\right)^{0.0667}}\right)}$ Kuehn&Goldstein1976 used in Eteiba2002
Related
$D_c$
$D_{encl}$
$\mathrm{Pr}_{gas}$
Used in
$h_{conv,c}$