Heat conduction in a cooling fan
Input(s)
z: Distance from Start of Fan (cm)
L: Length of Fan \((\mathrm{cm})\)
B: Breadth of Fan \((\mathrm{cm})\)
\(\mathrm{h}\): Heat Transfer Coefficient \(\left(\mathrm{J} / \mathrm{s} \mathrm{m}^{2} \mathrm{~K}\right)\)
k: Thermal Conductivity (W/cm K)
\(T_{w}\): Temperature of Wall (K)
\(T_{a}\): Ambient Temperature of Air (K)
Output(s)
\(\xi\): Dimensionless Distance (dimensionless)
N: Dimensionless Heat Transfer Coefficient (dimensionless)
\(\Theta\): Dimensionless Temperature (dimensionless)
T: Temperature at Distance \(\mathrm{Z}\) from Wall at the Fan \((\mathrm{K})\)
Formula(s)
\[
\begin{gathered}
\xi=\frac{\mathrm{z}}{\mathrm{L}} \\
\mathrm{N}=\left(\frac{\mathrm{h} *\left(\mathrm{~L}^{2}\right)}{\mathrm{k} * \mathrm{~B}}\right)^{0.5} \\
\Theta=\frac{\cosh (\mathrm{N} *(1-\xi))}{\cosh (\mathrm{N})} \\
\mathrm{T}=\mathrm{T}_{\mathrm{a}}+\Theta *\left(\mathrm{~T}_{\mathrm{w}}-\mathrm{T}_{\mathrm{a}}\right)
\end{gathered}
\]
Reference(s)
Bird, R.B., Stewart, W.E. and Lightfoot, E.N. (2002). Transport Phenomena (Second Ed.). John Wiley & Sons, Chapter: 10, Pages: 308, 309.