Question

Assume that we are working with an aluminum alloy (k = 180 W/moC) triangular fin with a length, L = 5 cm, base thickness, b = 1 cm, a very large width, w = 1 m. The base of the fin is maintained at a temperature of T0 = 200oC (at the left boundary node). The fin is losing heat to the surrounding air/medium at T? = 25oC with a heat transfer coefficient of h = 15 W/m2oC. Using the finite difference numerical method described previously, perform the following steps. 1- Rewrite the system of equations described by equations (4), (5), and (7) in the format AT = b where T is the temperature vector T(1:M). Start with 6 nodes which is the same number of nodes in Example 5-2 for comparison. 2- Solve this system using MATLAB and determine the temperature field T(1:6) and compare your results with that in example 5-2 of reference [1]. To check your answers, your resulting temperatures should match those seen in Example 5-2 in reference [1] provided in the supplementary background material on UBlearns. 3- Determine the fin rate of heat transfer ??? ?????? and the fin efficiency ???????? assuming a width, w = 1 m. 4- In order to determine a more precise temperature field you need to increase the number of nodes to M nodes, where M can be any integer number > 1. Rewrite the general system of equations for M nodes into the format AT = b, noting that equations (5) and (7) for boundary nodes will stay the same and equation (4) is M-2 equations. 5- Write a script file PP1P1.m that does the following a. Define all variables given in the statement above such as L, w, k, etc. with the same values given in the example. b. Preallocate matrix A and vector b using the zeros function. c. Generate the entries of matrix A and vector b for rows [2:M-1] using for loops. d. Entries for row 1 and row M can be entered manually. e. Solve the system AT = b for M = 11, 21, and 101. f. For M=101, plot the temperature T vs x where x is measured from the fin base, x(1) = 0. x can be determined using node location and should have corresponding units associated with it. Compare with the temperature calculated from analytical solution, equation (11), for the same x, and plot the analytical T on the same plot of the numerical T. Use different line styles, line width and Fully annotate your plot (titles, axis labels, legends, etc). Note: Use function besseli() in MATLAB to determine the modified Bessel function ??0 and ??1 (Hint: besseli(nu,Z) computes the modified Bessel function of the first kind, I?(z), for each element of the array Z. The order nu need not be an integer, but must be real. The argument Z can be complex. The result is real where Z is positive.) g. Find the rate of heat transfer ??? ?????? and the fin efficiency ???????? and compare with that of the analytical solution.

Answer 1