Natural convection occurs due to density differences of the fluid
Non Dimensional parameters
$Gr_L$ is [[Grashof number]]
$Ra_L$ is [[Rayleigh Number]]
$Pr$ is [[Prandtl number]]
$Nu$ is [[Nusselts number]]
## 1 Vertical Plate
### 1.1 Rough calculation
$\bar{Nu_L}=CRa_L^n $
For laminar flow ($Ra_L < 10^9$):
C = 0.59 and n = $\frac{1}{4}$
For turbulent flow ($Ra_L \ge 10^9$):
C = 0.1 and n = $\frac{1}{3}$
Can also be used for vertical cylinders if
$\frac{D}{L}\ge \frac{35}{Gr_L^{\frac{1}{4}}}$
### 1.2 Churchill and Chu
$Nu=0.68 + \frac{0.663 Ra_L^{1/4}}{[1+(0.492/Pr)^{9/16}]^{4/9}} \ \ \ \ \ \ while\ Ra_L \le 10^8$
## 2 Horizontal Plate
For horizontal plates the characteristic length must be calculated as
$L = \frac{A}{P}$
### 2.1 Upper surface of hot plate or lower surface of cold plate
Top surface of a hot object in a cold environment
or bottom surface of a cold object in a hot environment
$\bar{Nu_L}=0.54Ra_L^{1/4}\ \ \ \ \ \ (10^4\le Ra_L \le 10^7)$
$\bar{Nu_L}=0.15Ra_L^{1/3}\ \ \ \ \ \ (10^7\le Ra_L \le 10^{11})$
### 2.2 Lower surface of hot plate or upper surface of cold plate
for the bottom surface of a hot object in a colder environment
or top surface of a cold object in a hotter environment
$\bar{Nu_L}=0.27Ra_L^{1/4}\ \ \ \ \ \ (10^5\le Ra_L \le 10^{10})$
Newer books have this equation
$\bar{Nu_L}=0.52Ra_L^{1/5}\ \ \ \ \ \ (10^5\le Ra_L \le 10^{10})$
## 3 Horizontal Cylinder
### 3.1 Churchill - Chu
$\bar{Nu_D}=\left[0.60+\frac{0.387Ra_D^{1/6}}{[1+(0.559/Pr)^{9/16}]^{8/27}}\right]^2\ \ \ \ \ Ra_D \le 10^{12}$
## 4 Rectangular Box
From Incropera & Dewitt equation for Nusselts number a s a function of Rayleigh number is
$\bar{Nu_L}=0.22\left( \frac{Pr}{0.2+Pr}Ra_L \right)^{0.28}\left(\frac{H}{L}\right)^{-(1/4)}$
for
$\begin{bmatrix}
2<\frac{H}{L}<10 \\
Pr<10^5 \\
10^3 < Ra_L <10^{10}
\end{bmatrix}
$
$\bar{Nu_L}=0.18\left( \frac{Pr}{0.2+Pr}Ra_L \right)^{0.29}$
for
$\begin{bmatrix}
1<\frac{H}{L}<2 \\
<10^{-3}Pr<10^5 \\
10^3 < \frac{Ra_L Pr}{0.2+Pr}
\end{bmatrix}
$
for larger aspect ratios H/L >10 please see book for another equation