Related questions with answers

Radioactive wastes (knw=20W/mK)\left(k_{\mathrm{nw}}=20 \mathrm{W} / \mathrm{m} \cdot \mathrm{K}\right) are stored in a spherical, stainless steel (kss=15W/mK)\left(k_{\mathrm{ss}}=15 \mathrm{W} / \mathrm{m} \cdot \mathrm{K}\right) container of inner and outer radii equal to ri=0.5mr_{i}=0.5 \mathrm{m} and ro=0.6mr_{o}=0.6 \mathrm{m}. Heat is generated volumetrically within the wastes at a uniform rate of q˙=105\dot{q}=10^{5}, and the outer surface of the container is exposed to a water flow for which h=1000W/m2K1000 \mathrm{W} / \mathrm{m}^{2} \cdot \mathrm{K} and T=25CT_{\infty}=25^{\circ} \mathrm{C}. (a) Evaluate the steady-state outer surface temperature, Ts,oT_{s, o}. b) Evaluate the steady-state inner surface temperature, Ts,iT_{s, i}. (c) Obtain an expression for the temperature distribution, T(r), in the radioactive wastes. Express your result in terms of ri,Ts,i,krwr_{i}, T_{s, i}, k_{\mathrm{rw}}, and q˙\dot{q}. Evaluate the temperature at r=0. (d) A proposed extension of the foregoing design involves storing waste materials having the same thermal conductivity but twice the heat generation (q˙=2×105W/m3)\left(\dot{q}=2 \times 10^{5} \mathrm{W} / \mathrm{m}^{3}\right) in a stainless steel container of equivalent inner radius (ri=0.5m)\left(r_{i}=0.5 \mathrm{m}\right). Safety considerations dictate that the maximum system temperature not exceed 475C475^{\circ} \mathrm{C} and that the container wall thickness be no less than t=0.04 m and preferably at or close to the original design (t=0.1 m). Assess the effect of varying the outside convection coefficient to a maximum achievable value of h=5000W/m2Kh=5000 \mathrm{W} / \mathrm{m}^{2} \cdot \mathrm{K} (by increasing the water velocity) and the container wall thickness. Is the proposed extension feasible? If so, recommend suitable operating and design conditions for h and t, respectively.

Question

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

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Known:\textbf{Known:}

  • pressurized steam temperature, T1=450T_1=450K
  • thin-walled pipe diameter, D=0.5D=0.5m
  • concrete casing square cross section, w=1.5w=1.5m
  • casing outer surface temperature, T2=300T_2=300K
  • concrete thermal conductivity found using thermodynamic tables, k=1.4WmKk=1.4\frac{\text{W}}{\text{mK}}

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

q=Sk(T1T2)q=qLS=2πLln(1.08wD)\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:

qL=2πln(1.08wD)k(T1T2)qL=2πln(1.081.50.5)1.4(450300)q=1122.40Wm\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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