051407-132a-hw 4

# 051407-132a-hw 4 - insulating – no heat can ²ow o³ oF...

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C HEMICAL E NGINEERING 132A Professor Todd Squires Spring Quarter, 2007 Homework 4 Due Friday, May 18 Problem 1. A bar is located between x = 0 and x = 2, and has an inital temperature profile T 0 ( x ) = 0 , 0 x < 1 / 2 (1) T 0 ( x ) = 100 , 1 / 2 x 3 / 2 (2) T 0 ( x ) = 0 , 3 / 2 x 2 (3) At t = 0, the rod is placed between two ice blocks to maintain T = 0 at both ends ( x = 0 and x = 2), for all time. Solve the heat equation ∂T ∂t = k 2 T ∂x 2 (4) to find the evolution of the temperature. Show all steps – the separable solution, Fourier Sine Series, etc. As t → ∞ , what happens to the temperature in the bar? Problem 2. Consider the same problem as above, except now the ends are
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Unformatted text preview: insulating – no heat can ²ow o³ oF them. Remember this means ∂T/∂x = 0 For all times at x = 0 and x = 2. As t → ∞ , what happens to the temperature in the bar? How is this di³erent than in Problem 1, and why? Problems 3a-d. p503, probs 1, 2, 5, 8 Problems 4a-d. p503, probs. 10, 16, 21, 27 (these analyze your solutions above; not so involved) Problem 5a-c. p504, probs 32, 33, 36. Comment on what these boundary conditions correspond to physically....
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