1. Consider a large plane wall of thickness L = 0.2 m, thermal conductivity k = 1.2 W/m.°C, and the surface area A-15 m². The two sides of the wall are maintained at constant temperatures of T, = 120°C and T, -50°C, respectively, as shown in the figure. 0 Plane wall Determine (a) The variation of temperature within the wall in symbolic equation (don't substitute the numbers). State your assumptions. (b) Sketch the temperature profile in the wall (c) The value of temperature at x=0.1 m and x = 0.15 m. (d) The rate of heat transfer through the wall at steady conditions at x=0.1 m and x = 0.15 m. (e) The heat flux at x=0.1 m and x 0.15 m. (f) Repeat part a and b with heat generated within the wall is considered. (g) Comment on the difference in the values will be obtained for the heat rate and flux compared to those obtained in part d and e. Justify your answer. 2. An aluminum plate 4 mm thick is mounted in a horizontal position, and its bottom surface is well insulated. A special, thin coating is applied to the top surface such that it absorbs 80% of any incident solar radiation, while having an emissivity of 0.25. The density and specific heat of aluminum are known to be 2700 kg/m' and 900 J/kg. K, respectively. (a) Consider conditions for which the plate is at a temperature of 25°C and its top surface is suddenly exposed to ambient air at 7-20 °C and to solar radiation that provides an incident flux of 900 W/m². The convection heat transfer coefficient between the surface and the air is h=_20 W/m². K. What is the initial rate of change of the plate temperature? (b) What will be the equilibrium temperature of the plate when steady-state conditions are reached? 3. a) The general heat conduction equation for spherical system is: 1 a k 69 11()+(0) + 3 (* si 6 ) + à = pc, ³/ sin sin Assuming steady state, 1-dimensional, constant properties and no heat generation, obtain a general relation for the temperature distribution when the sphere is solid. Are the heat flux and heat rate independent or dependent on r ? Justify your answer mathematically for both cases. b) Repeat part (a), for steady state, 1-dimensional, constant properties and with heat generation conditions with appropriate boundary conditions. c) Consider a metallic hollow sphere with high thermal conductivity and radius r. The sphere is covered with a layer of insulation. The inside wall temperature is T, and is exposed to an environment at T. Does adding more insulation with a thermal conductivity of k will decrease/increase the heat transfer rate? 4. Aluminum fins 1.5 cm wide and 1.0mm thick are placed on a 2.5-cm-diameter tube to dissipate the heat. The tube surface temperature is 170°C, and the ambient-fluid temperature is 25°C. Calculate the heat loss per fin for h=130 W/m². C. Assume k =200 W/m. °C for aluminum.