Problem 5. 2D USS Heat Conduction in a Long Square Bar A long square bar (0.2 m x 0.2 m) initially with a uniform temperature of Ti = 500 K is subjected to cooling by air at Te = 300 K at all four side surfaces. The thermophysical properties of this bar are given as follows: p = 1000 kg/m', Cp - 3000 J/kg-K, and k - 30 W/mK. The convection coefficient for air is h. We are interested in the temperature distribution within the square bar at a specified cooling time. a) If h = 30 W/mK, estimate the temperature at the center of the square bar at t = 10 minutes. [Note: use lumped capacitance method, if possible) (Ans: 477.4 K) b) If h = 100 W/mK, estimate the highest (center) and lowest (corner) temperatures in (of) the square bar at += 10 minutes. (Note: use the graphical solution, if possible] (Ans; 454.9K, 414.5 K) c) If two adjacent side surfaces (bottom and right) are now maintained at a constant temperature T. = 500 K and the Tin other two (top and left) surfaces are subjected to Air convection cooling (T. = 300 K, h = 100 W/mK) (see the diagram on the right), determine the temperature distribution in the square bar at 1 = 10 minutes and at steady state, using the finite difference method. [Note: you need to determine T1, T2...., T, in the diagram and the temperature at the center of the square bar. First, list all necessary nodal temperature equations. Then, show your procedure and solve for these nodal temperatures, You may use computer to find the solutions for this problem if you wish.] d) Heat must be provided to the bottom and the right surfaces TS to maintain them at the constant temperature T. Estimate the heat rate entering these two surfaces per unit length of the square bar. (Ans: 4170 W/m) Problem 5. 2D USS Heat Conduction in a Long Square Bar A long square bar (0.2 m x 0.2 m) initially with a uniform temperature of Ti = 500 K is subjected to cooling by air at Te = 300 K at all four side surfaces. The thermophysical properties of this bar are given as follows: p = 1000 kg/m', Cp - 3000 J/kg-K, and k - 30 W/mK. The convection coefficient for air is h. We are interested in the temperature distribution within the square bar at a specified cooling time. a) If h = 30 W/mK, estimate the temperature at the center of the square bar at t = 10 minutes. [Note: use lumped capacitance method, if possible) (Ans: 477.4 K) b) If h = 100 W/mK, estimate the highest (center) and lowest (corner) temperatures in (of) the square bar at += 10 minutes. (Note: use the graphical solution, if possible] (Ans; 454.9K, 414.5 K) c) If two adjacent side surfaces (bottom and right) are now maintained at a constant temperature T. = 500 K and the Tin other two (top and left) surfaces are subjected to Air convection cooling (T. = 300 K, h = 100 W/mK) (see the diagram on the right), determine the temperature distribution in the square bar at 1 = 10 minutes and at steady state, using the finite difference method. [Note: you need to determine T1, T2...., T, in the diagram and the temperature at the center of the square bar. First, list all necessary nodal temperature equations. Then, show your procedure and solve for these nodal temperatures, You may use computer to find the solutions for this problem if you wish.] d) Heat must be provided to the bottom and the right surfaces TS to maintain them at the constant temperature T. Estimate the heat rate entering these two surfaces per unit length of the square bar. (Ans: 4170 W/m)
