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.