Note: In your submitted report you need to write and explain all equations and correlations you use. You should also include a copy of the computer code that you have used for the parametric calculations.

Hot water flows through a long and horizontal AISI $1010$ Carbon Steel pipe whose inner and outer diameters are $50$ and $60$ mm $,$ respectively. At a location along the pipe the mean water temperature is $90C,$ and mean water velocity is $4\frac{m}{s}.$ The surrounding air is stagnant and has a temperature of $25C.$

a$)$ Assume that the pipe is covered by an insulation layer with a thermal conductivity of $0\mathrm{.}03\frac{W}{m}.K.$ Neglecting radiation heat transfer, calculate and plot the rate of heat loss, per unit length, for insulation layer thicknesses in the range of $10$ to $30$ mm $.$

b$)$ For a fixed insulation layer thickness of $20$ mm $,$ and assuming that the surface emissivity of the outer surface of the insulator layer is $0\mathrm{.}85,$ calculate the rate of heat loss, this time considering free convection as well as radiation heat transfer.

c$)$ Repeat parts a and b$,$ this time assuming the air is not stagnant, and flows with a velocity of $1\frac{m}{s}$ across the horizontal pipe.