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.