In an industrial fabrication process, a fluid, with density $=800k\frac{g}{m^3}$ and specific heat capacity $c=5000\frac{J}{k}g*C,$ emerges from a tank at a temperature, $T_i=400C.$ The fluid then enters a metal pipe with inner radius $a=2\mathrm{.}0cm$ and outer radius $b=3\mathrm{.}0cm$ and thermal conductivity $=180\frac{W}{m}*C.$ Outside the pipe the temperature is fixed at $T_{\text{o}\text{u}\text{t}}=15C.$

If the fluid flows at speed $v=8\mathrm{.}0\frac{m}{s}$ and the length of the pipe is $L=25m,$ what is the temperature of the fluid at the end of the pipe? $($Answer: $83C)$

Consider the situation described in the first problem. If the temperature of the fluid at the end of the pipe must be no more than $T_{\text{m}\text{a}\text{x}}=60C$ and the fluid flows at speed $v=8\mathrm{.}0\frac{m}{s},$ what is the minimum length of the pipe? $($Answer: $31$ m $)$

Consider the situation described in the first problem. If the temperature of the fluid at the end of the pipe must be no more than $T_{\text{m}\text{a}\text{x}}=60C$ and the length of the pipe is $L=25m,$ what is the maximum flow speed of the fluid? $($Answer: $6\mathrm{.}5\frac{m}{s})$