# Correlation

The studied domain was limited to a single isolated cylinder tube in a pool of static propane as demonstrated in 'MESH' section. The $\Delta T$ is less than 6K for natural convection regime, hence the variation of density remains small and can be calculated with Boussinesq approximation (Equation 2). Considering the value of Rayleigh number for this maximum $\Delta T$, calculated in 'BIBLIOGRAPHIC RESEARCH Boussinesq approximation' section, laminar regime is adopted.

In the literature, the Correlations by Morgan (1975) [10] and by Churchill and Chu (1975) [11] are found to be well adapted for the range of Rayleigh number $Ra_D$ obtained for $\Delta T$ lying between 1 - 6K.

Correlation by Morgan(1975) :

$Nu_D= \frac{hD}{k}=CRa_D^n$   (37)

in which C=0.125, n=0.333, for $Ra_D~10⁷ - 10^{12}$

Correlation by Churchill and Chu (1975) :

$Nu_D=(0.60+\frac{0.387Ra_D^{\frac{1}{6}}}{(1+(\frac{0.559}{Pr})^{\frac{9}{16}})^{\frac{8}{27}}})^2$   (38)

for $Ra_D<10^{12}$

And considering also the classical equation of heat transfer

$Q_{wall}=h\Delta T$   (39)