## Related questions with answers

Pressurized steam at 450 K flows through a long, thin-walled pipe of 0.5-m diameter. The pipe is enclosed in a concrete casing that is of square cross section and 1.5 m on a side. The axis of the pipe is centered in the casing, and the outer surfaces of the casing are maintained at 300 K. What is the heat loss per unit length of pipe?

Solution

Verified$\textbf{Known:}$

- pressurized steam temperature, $T_1=450$K
- thin-walled pipe diameter, $D=0.5$m
- concrete casing square cross section, $w=1.5$m
- casing outer surface temperature, $T_2=300$K
- concrete thermal conductivity found using thermodynamic tables, $k=1.4\frac{\text{W}}{\text{mK}}$

Problem of this case represents $\textbf{Case 6}$ from the $\textbf{Table 4.1}$. Heat flow between pipes is given by $\textbf{Equation 4.20}$ (divide it by length) and the shape factor is found from the table.

$\begin{align*} &q=Sk(T_1-T_2)\\ &q'=\frac{q}{L}\\ &S=\frac{2 \pi L}{\ln (1.08 \frac{w}{D})} \\ \end{align*}$

Substitute the shape factor expression into the heat flow per length expression and apply the given parameters to find the estimated value:

$\begin{align*} &\frac{q}{L}=\frac{2 \pi }{\ln (1.08 \frac{w}{D})} k(T_1-T_2)\\ \\ &\frac{q}{L}=\frac{2 \pi }{\ln (1.08 \frac{1.5}{0.5})} 1.4 \cdot (450-300)\\ \\ &\boxed{q'=1122.40 \frac{\text{W}}{\text{m}}} \end{align*}$

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