Consider a thin$-$walled circular tube with a diameter $D=0\mathrm{.}05m,$ maintained at the surface temperature of $T_s=100C$ by embedded electric heating cables. Water flows through the tube at a mass flow rate $m^{}=0\mathrm{.}2k\frac{g}{s}.$ The convection coefficient of quiescent air outside the tube is $\frac{?}{\text{b}\text{a}\text{r}}(h{)}_{\text{a}\text{i}\text{r}}=7[\frac{W}{m^2}*(K)].$

a$)[30$pt$]$ Assuming the average mean temperature $\frac{?}{\text{b}\text{a}\text{r}}(T{)}_m=\frac{T_{m,i}+T_{m,o}}{2}=37[C],$ determine the average convection coefficient associated with the water flow, $\frac{?}{\text{b}\text{a}\text{r}}(h{)}_{\text{w}\text{a}\text{t}\text{e}\text{r}}.$

Use Eqn $8\mathrm{.}62,{}_D=\frac{(\frac{f}{8})(R_D-1000)Pr}{1+12\mathrm{.}7(\frac{f}{8}{)}^{0\mathrm{.}5}(P^{\frac{2}{3}}-1)}$

b$)10pt$ Determine the mean temperature at the exit of the tube, $T_{m,o}.$ Also, evaluate whether additional iterations are necessary based on your results.

c$)[5$pt$]$ Determine the log mean temperature difference, $T_{lm},$ and the heat transfer rate associated with the water flow in the tube, $q_{\text{w}\text{a}\text{t}\text{e}\text{r}}.$

d$)[5$pt$]$ Determine the electric power that the heating cable requires to maintain the tube's surface temperature at $100C,$ accounting for the convection heat transfer rates to the water flow and the surrounding air. Assume radiation effects are negligible.

For water flowing through the tube,

$T_{m,i}=15[C]$

Properties at $\frac{?}{\text{b}\text{a}\text{r}}(T{)}_m=37[C]($assumed$),$

$c_p=4178[\frac{J}{k}g*K]$

$k=0\mathrm{.}628[\frac{W}{m}*K]$

$=695{10}^{-6}[(N)*\frac{s}{m^2}]$

$Pr=4\mathrm{.}62$

$=361\mathrm{.}9{10}^{-6}[K^{-1}]$