Liquid flows through an annulus which has a solid metal core of diameter $\backslash ($ D$\_\{$i$\}\backslash ).$ The liquid is initially at a mean temperature $\backslash ($ T$\_\{0\}\backslash ),$ and heat is transferred from the flowing fluid to the inner metal core at a known constant flux $\backslash ($ q $\backslash )$ "inner. Meanwhile, the outer surface of the annulus is at a fixed temperature $\backslash ($ T$\_\{\backslash$mathrm$\{$s$\}\}\backslash )$ which is also less than $\backslash ($ T$\_\{0\}\backslash ),$ and the Nusselt number $($based on the outer diameter $\backslash ($ D$\_\{\backslash$mathrm$\{$o$\}\}\backslash ))$ for heat transfer from the liquid to the outer surface is a constant value of $3.$

ire $\backslash ($ T$\_\{$s$\}\backslash )$

a$.$ Following the energy balance approach for pipe flow, in which the liquid temperature is approximated as a mean temperature $\backslash ($ T$\_\{\backslash$mathrm$\{$m$\}\}\backslash ),$ which varies only with $\backslash ($ x $\backslash )($the distance along the pipe, from $0$ to $\backslash ($ L $\backslash )),$ derive an expression for $\backslash ($ d T$\_\{\backslash$mathrm$\{$m$\}\}/$ d x $\backslash ),$ if the liquid volumetric flow rate is through the annulus is $\backslash ($ Q $\backslash ).$ You may neglect conduction in the direction of the flow.

b$.$ Using an appropriate boundary condition, solve the equation obtained in part $($a$)$ to determine the mean liquid temperature at $\backslash ($ x$=$L $\backslash ).$