In an atmospheric environment with a temperature of $25C$ and a heat transfer coefficient of $5\frac{W}{m^2}K,$ blood flows through a pipe with an

outer diameter of $6cm.$ Assuming the external surface temperature of the pipe is steady at $220C$ under all the following conditions:

Find the heat flux in the uninsulated case. $(15$p$)$

Insulation up to the critical radius thickness around the pipe with a plastic material with a heat conduction coefficient of $0\mathrm{.}2\frac{W}{m}K.$ If this is done, what percentage will be the difference in the heat flux compared to the uninsulated case in option a$?(20$p$)$

Show with a graph whether the insulation increases heat transfer. Finding the minimum insulation thickness find it if necessary. $(30$p$)$

Insulate the pipe with $5cm$ thick glass wool with a thermal conductivity coefficient of $0\mathrm{.}04\frac{W}{m}K$ instead of plastic material. Comment on the situation by drawing a graph in terms of insulation? $(15$p$)$

What percentage is the difference in heat flux, taking into account the insulated case in and the uninsulated case in $a(20p)$