Chapter 4, Problem 4.29P

Hot water is transported from a cogeneration power station to commercial and industrial users through steel pipes of diameter $D=150\text{mm,}$ with each pipe centered in concrete $\left(k=1.4\text{W/m}\cdot \text{K}\right)$ of square cross section $\left(w=300\text{mm}\right).$ The outer surfaces of the concrete are exposed to ambient air for which $T_{\infty }=0°\text{C}$ and $h=25{\text{W/m}}^2\cdot \text{K}\text{.}$

1. If the inlet temperature of water flowing through the pipe is $T_i=90°\text{C,}$ what is the heat loss per unit length of pipe in proximity to the inlet? The temperature of the pipe $T_1$ may be assumed to be that of the inlet water.
2. If the difference between the inlet and outlet temperatures of water flowing through a 100-m-long pipe is not to exceed 5°C, estimate the minimum allowable flow rate $\stackrel{.}{m}.$ A value of $c=4207\text{J/kg}\cdot \text{K}$ may be used for the specific heat of the water.