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Grashof number cable—gas

Symbol
$\mathrm{Gr}_{og}$
Formulas
$\frac{g \beta_{gas} {L_{d}}^3}{{\nu_{gas}}^2} \left(\theta_{e}-\theta_{di}\right)$Riser closed at both ends (Hartlein & Black)
$\frac{g \beta_{gas} {L_{d}}^3}{{\nu_{gas}}^2} \left(\theta_{e}-\theta_{di}\right)$Riser open at both ends, 133 ≤ $Ra$ ≤ 7000 (Hartlein & Black IIa)
$\frac{g \beta_{gas} {L_{d}}^3}{{\nu_{gas}}^2} \left(\theta_{e}-\theta_{air}\right)$Riser open at both ends, $Ra$ > 7000 (Hartlein & Black IIb)
$\frac{g \beta_{gas} {L_{d}}^3}{{\nu_{gas}}^2} \left(\theta_{e}-\frac{\theta_{e}+\theta_{di}+\theta_{air}}{3}\right)$Riser open at top and closed at bottom (Hartlein & Black)
$\frac{g \beta_{gas} {\delta_{d}}^3}{{\nu_{gas}}^2} \left(\theta_{e}-\theta_{di}\right)$Riser closed at both ends (Anders)
$\frac{g \beta_{gas} {\delta_{d}}^4}{L_{d} {\nu_{gas}}^2} \left(\theta_{e}-\theta_{air}\right)$Riser open at both ends (Anders/Hartlein & Black)
$\frac{g \beta_{gas} {L_{d}}^3}{{\nu_{gas}}^2} \left(\theta_{e}-\theta_{gas}\right)$Riser open at top and closed at bottom (Anders)
Related
$\beta_{gas}$
Volumetric thermal expansion coefficient gas [1/K]
$\delta_{d}$
Distance cable—duct [m]
$g$
Standard acceleration of gravity [m/s$^2$]
$L_{d}$
Length duct [m]
$\nu_{gas}$
Kinematic viscosity gas [m$^2$/s]
$\theta_{air}$
Ambient air temperature [°C]
$\theta_{di}$
Temperature duct inner surface [°C]
$\theta_{e}$
External temperature object [°C]
$\theta_{gas}$
Gas temperature [°C]
Used in
$\mathrm{Nu}_{og}$
Nusselt number cable—gas
$\mathrm{Ra}_{int}$
Rayleigh number gas→riser