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Assignment_7

# Assignment_7 - y x T numerically and plot the results...

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ME 303 ADVANCED ENGINEERING MATHEMATICS Assignment 7 Suggestion: use Excel as described in handouts 5.1 and 5.2. Depending on how you enter the discretized PDE and BC formulas, Excel sometimes has trouble starting the Gauss-Seidel iteration. If you have problems, enter initial guesses first in all interior cells (say zeros), then replace the guesses with the formulas. Question 1. A long bar has an L-shaped cross-section, as shown in the sketch below. Three sides of the bar are insulated, while other two sides are maintained at 100ºC and 0ºC. The steady-state temperature () y x T , in the bar cross-section is governed by the Laplace equation, subject to the boundary conditions shown below. Choose cm x 0 . 1 = Δ , cm y 5 . 0 = Δ . Use second-order, central differences to discretize the Laplace equation and first-order differences to discretize the BCs, where required. Hence, solve for
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Unformatted text preview: ( ) y x T , numerically and plot the results. Question 2. A circular rod is 12 cm long and 2 cm in diameter. As shown on the sketch below, the rod is insulated on all surfaces and is initially at 15ºC. For ≥ t , the = x end is heated by an electrical device that inputs 20 watts of power to the rod. The temperature ( ) y x T , of the rod is governed by the 1-D diffusion equation subject to Neumann BCs at = x and cm x 12 = . Choose cm x 2 = Δ and 5 = Δ t seconds. Solve for ( ) y x T , numerically for 150 ≤ ≤ t seconds by applying each of : (i) the explicit scheme (ii) the fully implicit scheme (iii) Crank-Nicolson scheme Material properties of the rod are: C m W k ° = 200 , s m 2 5 10 5 − ⋅ = α For comparison, plot T versus x at s t 150 = based on the results from (i), (ii), and (iii). 2 cm diameter...
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