A diffuse, gray radiation shield \(90 \mathrm{~mm}\) in diameter with emissivities of \(\varepsilon_{2, i}^{r}\) and \(\varepsilon_{2, o}^{r}\) on its inner and outer surfaces respectively, is concentric with a long tube transporting a hot process fluid. The situation is shown in Figure P15.20. The tube surface is diffuse and gray with an emissivity of \(\varepsilon_{t}^{r}\) and a diameter of \(50 \mathrm{~mm}\). The area between the tube and the radiation shield is evacuated. The outer surface of the shield is exposed to a very large room whose walls are at \(17^{\circ} \mathrm{C}\) and the shield experiences convection with the surrounding air at \(37^{\circ} \mathrm{C}\) and with a heat-transfer coefficient \(h=10 \mathrm{~W} / \mathrm{m}^{2} \mathrm{~K}\). Determine the steady-state operating temperature of the inner tube surface if the shield is held at \(75^{\circ} \mathrm{C}\). What resistance(s) control(s) the rate of heat transfer?

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FIGURE P15.20 T = = 37C air 25 mm -T wall = 17 C 20=0.1 E = 0.025 Shield T = 75 C To 45 mm = 0.85