Matlab (heat conduction equations)

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nnnnooo1

Engineering

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can you solve calculus

heat conduction equations

using matlab.

i need quality work plase

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Task 2: Consider the steady-state temperature, u(I), of a copper bar of length L = 5 m where the left end of the bar is fixed at u= -5°C and the right end has a heat flux, q = ku'(L), of -20 W/m² (i.e. outward) where[] k = 400 W/mK is the thermal conductivity of copper. There is an energy density of f = (L - 1)(L – 2x) kW/m2 along the length of the bar. (i) Give the differential equation and boundary conditions for this problem. Make sure to indicate the values of 1 for which each of them hold. (ii) Use a uniform grid spacing of h = L/N for N = 1,2,3..., and derive a second order accurate finite difference method for this heat conduction problem. Fully specify the system of equations in the case N = 9. (iii) Submit a matlab file that allows the user to select N = 1,2,3,4,... and then solves the problem, plots a labelled graph of the temperature along the bar and prints the tem- perature at the right hand end to the screen. (You may use the code Heat1D_extras.m, as developed in lectures, as a starting point.) Task 3: Obtain the file fem2D-50Lines.zip from Blackboard Learn, save it to an area on your H drive (or somewhere equally safe) and unzip it to obtain a folder called fem2D. Inside this folder is the set of codes that we used in the lecture for the 2D finite element approximation of heat conduction in the L, X Ly metre rectangle shown on the left in the figure. Existing Problem: Assignment Problem: u given du = B ön om given -V?u=f on given u= a -V?u=f dm = B with f given with f = y(ar + By) given u given u= a (i) Run this heat transfer problem for a 20 x 20 mesh and give the global node number that corresponds to the midpoint (Lx/2, Ly/2) of the computational domain. Explain your working. Then give the computed value of the temperature at the midpoint (Lc/2, L,/2). (The supplied values Le = 3 and Ly = 2 should not be altered.) (ii) Take your student ID, add together the digits, call the result a and then set B = a + 2 where y is the largest single digit in your ID as used above (without the slash)]. Alter your code to solve the problem on the right of the figure above, with f(x, y) = y(ar + By), and again give the computed temperature at (Lx/2, Ly/2) for a 20 x 20 mesh. Include also in your report a surface plot of the temperature field and full details of how you arrived at a, ß and . END 1 See e.g. http://www.engineeringtoolbox.com/thermal-conductivity-d_429.html 2 From https://www.math.hu-berlin.de/-cc/cc_homepage/software/software.shtml 3For example, for the ID 1702518/2 you would use 1702518 and get a=1+7+2+5+1+8=24, n = 8 and B=a+y=24 +8= 32.
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