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Grashof number, gas to duct

Symbol
$\mathrm{Gr}_{gd}$
Formulae
$none$Riser closed at both ends (Hartlein & Black)
$none$Riser open at both ends, $Ra$ >= 10^5 (Hartlein & Black IIa)
$\frac{g \beta_{gas} \left(\frac{D_{di}}{2}\right)^4}{{\nu_{gas}}^2 L_d} |\theta_{air}-\theta_{di}|$Riser open at both ends, 0.1 ≤ $Ra$ < 10^5 (Hartlein & Black IIb)
$\frac{g \beta_{gas} \left(\frac{D_{di}}{2}\right)^3}{{\nu_{gas}}^2} |\theta_{air}-\theta_{di}|$Riser open at top and closed at bottom (Hartlein & Black)
$none$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)
$\frac{g \beta_{gas} \left(\frac{D_{di}}{2}\right)^4}{L_d {\nu_{gas}}^2} \left(\theta_{dm}-\theta_{di}\right)$Riser open at top and closed at bottom (Anders)
Related
$\beta_{gas}$
Volumetric thermal expansion coefficient for gas [1/K]
$D_{di}$
Inner diameter of duct [m]
$\delta_d$
Distance between cable and duct [m]
$g$
Standard acceleration of gravity [m/s$^2$]
$L_d$
Length of duct [m]
$\nu_{gas}$
Kinematic viscosity for gas [m$^2$/s]
$\theta_{air}$
Ambient air temperature [°C]
$\theta_{di}$
Temperature of duct inner wall [°C]
$\theta_{dm}$
Mean temperature of the medium in the duct [°C]
$\theta_e$
External temperature of the object [°C]
Used in
$\mathrm{Nu}_{gd}$
Nusselt number, gas to duct
$\mathrm{Ra}_{int}$
Rayleigh number inside riser