A hot water pipe with outside radius $r1=10cm$ has a temperature $T1=77C.$ A thick insulation,

$(k=0\mathrm{.}01\frac{W}{m}.K)$ applied to reduce the heat loss, has an outer radius $r2=15cm$ and temperature

$T2=32C.$ Assume one$-$dimensional, steady$-$state heat transfer with constant properties and no

internal heat generations and neglect radiation interactions with surroundings. Calculate and plot

$($about $10$ data points from r $1$ to r $2)$ the temperature distribution in the insulation in $T-r$

coordinates using :

a$.$ Fourier equation in radial coordinates

b$.$ The heat equation in radial coordinates

c$.$ Calculate the total heat transfer per meter of length of the pipe $(q_r$ in W$)$

d$.$ Calculate the heat transfer per unit area $(q^{\text{'}\text{'}}$ r in $\frac{W}{m^2})$