Electrical heating of a pipe

Input(s)

$\mathrm{R}$: Radius $(\mathrm{m})$

$\kappa$: Ratio of Inner Radius to Outer Radius $\left(\mathrm{m}^{2} \mathrm{~kg} \mathrm{~s}^{-2} \mathrm{~K}^{-1}\right)$

L: Length $(\mathrm{m})$

$\mathrm{T}_{\kappa}$: Desired Temperature $(\mathrm{K})$

$\mathrm{T}_{\mathrm{a}}$: Ambient Air Temperature (K)

$\mathrm{k}$: Thermal Conductivity $(\mathrm{W} / \mathrm{m} \mathrm{K})$

h: Heat Transfer Coefficient $\left(\mathrm{W} /\left(\mathrm{m}^{2} \mathrm{~K}\right)\right)$

Output(s)

P: Electrical Power (Watt)

Formula(s)

$$ \mathrm{P}=\frac{\pi * \mathrm{R}^{2} *\left(1-\kappa^{2}\right) * \mathrm{~L} *\left(\mathrm{~T}_{\kappa}-\mathrm{T}_{\mathrm{a}}\right)}{\left(1-\kappa^{2}\right) * \frac{\mathrm{R}}{2 * \mathrm{~h}}-\frac{(\kappa * \mathrm{R})^{2}}{4 * \mathrm{k}} *\left(1-\frac{1}{\kappa^{2}}-2 * \ln (\kappa)\right)} $$

Reference(s)

Bird, R.B., Stewart, W.E. and Lightfoot, E.N. (2002). Transport Phenomena (Second Ed.). John Wiley \& Sons, Chapter: 10, Page: 325.