Question

A two-dimensional rectangular plate is subjected to prescribed boundary conditions. Using the results of the exact solution for the heat equation presented in Sec- tion 4.2, calculate the temperatures along the mid-plane of the plate (x m at y 0.25, 0.5, and 0.75 m by considering the first five nonzero terms of the infinite series. Assess the error resulting from using only the first three terms of the infinite series y (m) T2-1 50°C T1 = 50°C T 50°C r (m) 0 0 2 T1 50°C

Answer 1

SCHEMATIC: y(m) W 1 T = 150 oC T. = 50 oC T, = 50 OC 0 T1 = 50 °C ASSUMPTIONS: (1) Two-dimensional, steady-state conduction, (2) Constant properties ANALYSIS: From Section 4.2, the temperature distribution is n+1 T-T 2 T2-Ti T sin L ) sinh (nrTW/L) L sinh (nTW/L Considering now the point (x.y) - (1.0,0.5) and recognizing x/L - 1/2, y/L 1/4 and W/L-1/2 θ (1,0.5)-T-互-3 (-1)"+1 + 1 1,05)-T +sinsinh(nt/4) 2丿sinh (nr/2) Sinj T-T n=1 When n is even (2, 4, 6 ), the corresponding term is zero; hence we need only consider n and 9 as the first five non-zero terms 1, 3, 5, 7 (π )sinh (T/4)+-sin(3t )sinh (3r/4) + (2Jsinh (T/2) θ (1.0.5)- 2sin π| Sinj- 3 (2Jsinh (3t/2) 5 ( 2 Įsinh (5π/2) 2 7 ( 2 Įsinh (7r/2) 9 ( 2 sinh (9m2) θ (1, 0.5)-10.755-0.063 + 0.008-0.001 +0.0001-0.445 T(1, 0.5)-0(1,0.5)(T2-Tİ)-1-0.445 (150-50)+50-94.5° C.

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