Dimensionless multiplier correcting the Conductive Annulus Heat Transfer Coefficient to account for the Natural Thermal Convection effects in the Annulus:

U = \epsilon_a  \cdot U^* = \epsilon  \cdot \frac{ \lambda_a}{d_t \cdot \ln (d_{ci}/d_t) }

The most popular empirical correlations are:

\epsilon = \begin{cases} 
1, & \mbox{if  } \ {\rm Ra} < 10^3
\\ 
0.18 \cdot {\rm Ra}^{0.25}, & \mbox{if  } \ {\rm Ra} > 10^3
\end{cases}

\epsilon = \begin{cases} 
1, & \mbox{if  } \ {\rm Ra} < 10^3
\\
0.105 \cdot {\rm Ra}^{0.3}, & \mbox{if  } \ 10^3 < {\rm Ra} < 10^6
\\ 
0.4 \cdot {\rm Ra}^{0.2}, & \mbox{if  } \ {\rm Ra} > 10^6
\end{cases}

See also