Problem $3\mathrm{.}1(20$ points$)$

A chemical engineer is studying the convective heat transfer in a pipe. The heat transfer coefficient $(h)$ is influenced by the following variables:

Fluid velocity $(v),$

Fluid density $(),$

Fluid viscosity $(),$

Pipe diameter $(d),$

Thermal conductivity of the fluid $(k).$

Using the Buckingham $-$theorem, derive a dimensionless relationship to describe the dependence of the heat transfer coefficient $(h)$ on the given variables.

The resulting dimensionless groups will include:

Nusselt number $()$ :

$=\frac{hd}{k}$

Reynolds number $($Re$)$:

$Re=\frac{vd}{}$

Prandtl number $(Pr)$ :

$Pr=\frac{cp}{k}$

The final relationship will be:

$=f(Re,Pr)$

where $f$ is a function determined empirically or theoretically.

$2.$ Explain the physical significance of the numerator and denominator of each dimensionless group $,$ and Pr