The cross section of a rectangular bar with aspect ratio a has temperature distribution T(x, y, t). The bar is sufficiently long, and has uniformity along its z-dimension, so it can be idealized as a two-dimensional problem. Three sides of the bar are kept at a uniform temperature To. The fourth side of the bar is subject to a temperature profile f(x). 49 To L f(x) This temperature profile is governed by the partial differential equation 80 T(x, y, t) = A,, sin To aL n=1 nTX ("T) ƏT Ət where k is the coefficient of thermal conductivity. This equation has the analytical solution for T(x, y,t) = T(x, y, t) - To of k To 8²T 8²T + მე2 მy2 |sinh | n# x (nr (a - 4)) - Σ B₁,m sin ( B₁.,m sin (my) exp (=k aL m=1 P (= kx² (n²³o ²³ + m²) t a²L² ¹²)]. 2(1-(-1)") and B₁,m nã sinh(nao) m(exp(nña)-exp(-nπa)) *(n²a²+m²) where the coefficients for f(x) = 1+To are An f Set up the finite difference equation using the second order central difference method in the spatial dimensions and the explicit Euler method in the temporal dimension. Solve the equation with Ar = om, Ay = om, L=1m, a = 1, k = 294m², and the initial temper- ature profile of T(x, y) = To. Compare the numerical solution to the analytical solution at t = 1.79 x 10-6s, 3.06 x 10-6s, 1.770 x 10-5s, 2.2520 x 10-4s, and 4.5845 x 10-4s. =

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Transcribed Image Text:The cross section of a rectangular bar with aspect ratio a has temperature distribution T(x, y, t). The bar is sufficiently long, and has uniformity along its z-dimension, so it can be idealized as a two-dimensional problem. Three sides of the bar are kept at a uniform temperature To. The fourth side of the bar is subject to a temperature profile f(x). Y To To aL f(x) This temperature profile is governed by the partial differential equation * = *(+) T*(x,y,t) = Σ. An sin ( n=1 მე-2 To x where k is the coefficient of thermal conductivity. This equation has the analytical solution for T*(x, y, t) = T(x, y,t) - To of (25) sinh ( (--Đ))- En("E). Bn, Sin m aL m-1 exp 2(1-(-1)) and Bn.m nāsinh(nĩa) - m²))]. _kπ² (n²a² + m²) = m(exp(nña)-exp(-nña)) π(n²a²+m²) where the coefficients for f(x) = 1+T are An Set up the finite difference equation using the second order central difference method in the spatial dimensions and the explicit Euler method in the temporal dimension. Solve the equation with Ar=m, Ay=1m, L=1m, a = 1, k = 294m², and the initial temper- ature profile of T(x, y) = To. Compare the numerical solution to the analytical solution at t = 1.79 x 10-6s, 3.06 x 10-6s, 1.770 x 10-5s, 2.2520 x 10-4s, and 4.5845 x 10-4s.

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