Question$1$:

Consider fully developed laminar flow of water in a circular tube $($radius$,r_o=15mm)$ which has a constant heat flux, $q=5000\frac{W}{m^2}$ maintained. We will attempt to take into account the effect of temperature on the flow profile as shown in Figure $1$ below.

Given the velocity profile of the water in the tube is $u(r)=0\mathrm{.}1[1-(\frac{r}{r_o}{)}^2].$

Figure $1.$ The flow profile of laminar flow.

a$.$ Neglecting axial conduction, perform an energy balance on the differential element and show that the differential equation governing the system is:

$u\frac{dT}{dx}=\frac{}{r}\frac{\text{d}\text{e}\text{l}}{\text{d}\text{e}\text{l}\text{r}}(r\frac{\text{d}\text{e}\text{l}\text{T}}{\text{d}\text{e}\text{l}\text{r}})$

Hint: Take the annular control element of length $x$ and thickness of $r.$

b$.$ Find the mean velocity of the water, $u_m(\frac{m}{s}).$

c$.$ Use the properties of water at $50C$ and show that the temperature profile of the water can be approximated as:

$T=T_w-87+116(\frac{r}{r_o}{)}^2-29(\frac{r}{r_o}{)}^4$

Where $T_w$ is the wall temperature $(C).$

d$.$ At a particular axial position of the tube, the wall temperature, $T_w=400C.$

i$.$ Find the mean $($or bulk$)$ temperature at this position.

ii$.$ Plot the temperature distribution, does your value of mean temperature appear reasonable?