Problem 1-Consider a 5-m-long constantan block (k = 23 W/mK)
30 cm high and 50 cm wide. The block is completely submerged in iced water at 0 °C that is well stirred, and the heat transfer coefficient is so high that the temperatures on both sides of the block can be taken to be 0°C. The heat transfer through the bottom surface is negligible. The top surface of the block is heated uniformly by a 6 kW resistance heater. Using the finite difference method with a mesh size of Δx = Δy = 10 cm and taking advantage of symmetry,
(a) obtain the finite difference formulation of this problem for steady two-dimensional heat transfer,
(b) determine the unknown nodal temperatures by solving those equations, and
(c) determine the rate of heat transfer from the block to the iced water.
Problem 2
Consider an aluminum alloy fin (k = 180 W/mK, = 0.9) of triangular cross section whose length is L = 5 cm, base thickness is b = 1 cm, and width w in the direction normal to the plane of paper is very large. The base of the fin is maintained at a temperature of T0 = 180°C. The fin is losing heat by convection to the ambient air at T∞= 25°C with a heat transfer coefficient of h = 25 W/m2K and by radiation to the surrounding surfaces at an average temperature of Tsurr= 290 K. Using the finite difference method with six equally spaced nodes along the fin in the x- direction, determine
(a) the temperatures at the nodes and
(b) the rate of heat transfer from the fin for w = 1 m. Assume steady one- dimensional heat transfer in the fin. Note that Δx is varying.
Problem 3 -Two 3-m-long and 0.4-cm-thick cast iron (k = 52 W/mK, ε = 0.8) steam pipes of outer diameter 10 cm are connected to each other through two 1-cm-thick flanges of outer diameter 20 cm, as shown in the figure. The steam flows inside the pipe at an average temperature of 200 °C with a heat transfer coefficient of 180 W/m2K. The outer surface of the pipe is exposed to convection with ambient air at 8 °C with a heat transfer coefficient of 25 W/m2K as well as radiation with the surrounding surfaces at an average temperature of Tsurr = 290K.
Assuming steady one-dimensional heat conduction along the flanges and taking the nodal spacing to be 1 cm along the flange
(a) obtain the finite difference formulation for all nodes,
(b) determine the temperature at the tip of the flange by solving those equations, and
(c) determine the rate of heat transfer from the exposed surfaces of the flange.
Instructions: You have to choose two methods of analysis for each problem, so that you can compare the results. Your choices are ANSYS, MATLAB, and Excel. You must use ANSYS at least for one problem of your choice.