$25/$ Consider a cylindrical pipe, as illustrated in Figure below. Initially the pipe is at a uniform

temperature of $T_1.$ At time $t=0$a uniform heat flux of $q_0{?}_0$ is applied to the inner surface. The heat

is removed from the outer surface by convection, which is characterized by a heat transfer coefficient

$h$ and an ambient temperature $T.$ Find the temperature distribution within the cylinder as a function

of time and radial position. On a dimensional basis, the problem is

$\frac{\text{d}\text{e}\text{l}\text{T}}{\text{d}\text{e}\text{l}\text{t}}=\frac{k}{r}\frac{\text{d}\text{e}\text{l}}{\text{d}\text{e}\text{l}\text{r}}r\frac{\text{d}\text{e}\text{l}\text{T}}{\text{d}\text{e}\text{l}\text{r}}$

$-k\frac{\text{d}\text{e}\text{l}\text{T}}{\text{d}\text{e}\text{l}\text{r}}{|}_{R_i}=q_0^{\text{'}\text{'}}$

$-k\frac{\text{d}\text{e}\text{l}\text{T}}{\text{d}\text{e}\text{l}\text{r}}{|}_{R_o}=h(T(r=R_o,t)-T_{})$

$T(r,t=0)=T_1$