PROBLEM NO$.1$

If the expressions for the velocity and temperature in a pipe for the flow of a certain fluid in the pipe are given respectively as:

$u=U_m=C_1$

$T=T_w+C_2[1-(\frac{r}{r_o}{)}^2]$

where $U_m$ is the mean velocity at that location, $T_w$ is temperature at the wall surface, $r_o$ is radius of the pipe, and $C_l$ and $C_2$ are constants,

$($a$)$ Obtain the expression for the convective heat transfer coefficient $h.$ Simplify you expression as much as possible.

$($b$)$ Show that the Nusselt number $({}_D)$ is equal to $8.$

$[30$ points$]$

PROBLEM NO$.2$

The mean Nusselt number $({}_m)$ for laminar flow heat transfer in the thermal entrance region of a circular duct for the case of constant surface temperature $(T_w=$ Constant$)$ is

${}_m=\frac{1}{2}ln[\frac{1}{8{\displaystyle \underset{n=0}{\overset{x}{}}}(\frac{A_n}{{}_n^2})exp(-{}_n^2)}]$

The associated table of constants for this expression are given in Table $6\mathrm{.}3$ of the textbook. The table is also in your notes.

If the expressions for the velocity and temperature in a pipe for the flow of a certain fluid in the pipe are given respectively as:

$u=U_m=C_1$

$T=T_w+C_2[1-(\frac{r}{r_o}{)}^2]$

where $U_m$ is the mean velocity at that location, $T_w$ is temperature at the wall surface, $r_o$ is radius of the pipe, and $C_l$ and $C_2$ are constants,

$($a$)$ Obtain the expression for the convective heat transfer coefficient $h.$ Simplify you expression as much as possible.

$($b$)$ Show that the Nusselt number $({}_D)$ is equal to $8.$