Surface temperature of a heating coil

Input(s)

$T_{o}$: Initial Surface Temperature $(\mathrm{K})$

$\mu$: Viscosity (Pa s)

$\rho$: Density $\left(\mathrm{g} / \mathrm{m}^{3}\right)$

g: Gravitational Acceleration $\left(\mathrm{ft} / \mathrm{s}^{2}\right)$

$\beta$: Velocity Flux (1/s)

D: Diameter (m)

Q: Net Energy Input Rate $(\mathrm{J} / \mathrm{s})$

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

Output(s)

$T_{1}-T_{o}$: Final Temperature of Surface (K)

Formula(s)

$$ \mathrm{T}_{1}-\mathrm{T}_{\mathrm{o}}=\frac{\mu^{2}}{\left(\rho^{2}\right) * \mathrm{~g} * \beta *\left(\mathrm{D}^{3}\right)} * \varnothing\left(\frac{\mathrm{Q} *\left(\rho^{2}\right) * \mathrm{~g} * \beta *\left(\mathrm{D}^{2}\right)}{\mathrm{k} *\left(\mu^{2}\right)}\right) $$

Reference(s)

Bird, R.B., Stewart, W.E. and Lightfoot, E.N. (2002). Transport Phenomena (Second Ed.). John Wiley \& Sons, Chapter: 11, Page: 361.