Steam at \(250^{\circ} \mathrm{C}\) flows in a stainless steel pipe \((k=15 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K})\) whose inner and outer diameters are \(4 \mathrm{~cm}\) and \(4.6 \mathrm{~cm}\), respectively. The pipe is covered with \(3.5-\mathrm{cm}-\) thick glass wool insulation \((k=0.038 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K})\) whose outer surface has an emissivity of \(0.3\). Heat is lost to the surrounding air and surfaces at \(3^{\circ} \mathrm{C}\) by convection and radiation. Taking the heat transfer coefficient inside the pipe to be \(80 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\), determine the rate of heat loss from the steam per unit length of the pipe when air is flowing across the pipe at \(4 \mathrm{~m} / \mathrm{s}\). Evaluate the air properties at a film temperature of \(10^{\circ} \mathrm{C}\) and \(1 \mathrm{~atm}\).

#Answer# Based on the outlined steps, first, we calculate the thermal resistance of each layer: For the stainless steel pipe: $$R_{pipe}=\frac{\ln(0.023 / 0.02)}{2 \pi \times 15 \times 1} \approx 0.00399\, \mathrm{K/W}$$ For the glass wool insulation: $$R_{insulation}=\frac{\ln(0.056 / 0.023)}{2 \pi \times 0.038 \times 1} \approx 0.834\, \mathrm{K/W}$$ Next, we find the Nusselt number and the heat transfer coefficient for the air: Suppose we have the following air properties: \(\rho = 1.2\, \mathrm{kg/m^3}\), \(\mu = 1.81 \times 10^{-5}\, \mathrm{kg/m\cdot s}\), \(k_{air} = 0.024\, \mathrm{W/m\cdot K}\), and \(Pr = 0.7\). Then calculate the Reynolds number: $$Re = \frac{(1.2)(4)(0.056)}{1.81 \times 10^{-5}} \approx 15,\!212$$ Calculate the Nusselt number: $$Nu = 0.023 \times Re^{0.8} \times Pr^{0.4} \approx 68.8$$ Determine the air heat transfer coefficient: $$h_{air} = \frac{Nu \times k_{air}}{D} \approx 29.9\, \mathrm{W/m^2K}$$ Then, use the assumption that the overall heat transfer coefficient is approximately equal to \(h_{air}\) (\(U_{overall} = h_{air}\)) to calculate the heat transfer rate per unit length: $$q = \frac{T_{steam} - T_{\infty}}{R_{pipe} + R_{insulation} + \frac{1}{U_{overall}}} = \frac{373 - 273}{0.00399 + 0.834 + \frac{1}{29.9}} \approx 117.8\, \mathrm{W/m}$$ Thus, the rate of heat loss per unit length from the steam flowing inside a stainless steel pipe with glass wool insulation is approximately \(117.8\, \mathrm{W/m}\).