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s2_95c

# s2_95c - ACM 95/100c April 9 2004 Problem Set II 1(20 pts...

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ACM 95/100c April 9, 2004 Problem Set II 1.- (20 pts) The heat transfer in a one-dimensional homogeneous bar in 0 < x < L is described by the equation ∂T ∂t - κ 2 T ∂x 2 = 0 , where κ is a constant. The temperatures of the ends of the bar are fixed (by attaching each end to a large reservoir) such that T (0 , t ) = T 0 , T ( L, t ) = T 1 where T 0 and T 1 are constant temperatures. (a) Show that using the transformation T = w ( x, t ) + 1 L ( T 1 - T 0 ) x + T 0 an equation for w ( x, t ) can be obtained with homogeneous end conditions. (b) Solve the equation for w by a series-solution method with initial temperature distribution given by T ( x, 0) = f ( x ) , 0 < x < L. Note: This is an initial condition for T , not w . (c) What is the temperature distribution in the bar when t → ∞ ? 2.- (40 pts) A rod with insulated sides occupies the portion 1 < x < 2 of the x -axis. The thermal conductivity of the rod material is not constant but depends on x in such a way that the rod temperature T ( x, t ) satisfies the heat equation ∂T ∂t - A 2 ∂x x 2 ∂T ∂x = 0 where A is a constant. The boundary and initial conditions are

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s2_95c - ACM 95/100c April 9 2004 Problem Set II 1(20 pts...

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