051707-132a-clab 7

# 051707-132a-clab 7 - C HEMICAL E NGINEERING 132A Professor...

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Unformatted text preview: C HEMICAL E NGINEERING 132A Professor Todd Squires Spring Quarter, 2007 Computer Lab Assignment 7 New Commands: <<Graphics‘Animation‘ Animate[Plot[f[x,t],{x,0,1},PlotRange->{0,1}],{t,0,10,.5}] as = Table[a[n],{n,0,50}] a20 = as[[20]] Today we’re going to solve the heat equation and animate the solutions. As you may remember from class, the solution to the heat equation ∂T ∂t = K ∂ 2 T ∂x 2 (1) is in general c ( x, t ) = summationdisplay β ( a b eta cos βx + b β sin βx ) e- β 2 Kt . (2) This follows from using a separable solution, solving for each of the two resulting ODEs, and then summing over all possible solutions. To find the specific solution to the problem we are solving, we must impose initial and boundary conditions. As in class, say we have a rod whose ends (located at x = 0 and x = L ) are held in ice water, so that T (0 , t ) = 0 (3) T ( L, t ) = 0 (4) Initially, at t=0, the rod is hot in the middle: T ( x, 0) = 0 , ≤ x < L/ 4 , (5) = 100 , L/ 4 ≤ x ≤ 3 L/ 4 , (6) = 0 3 L/ 4 < x ≤ L. (7) Recall from class how we solved this. We first used the boundary conditions (3-4) to constrain which functions weRecall from class how we solved this....
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051707-132a-clab 7 - C HEMICAL E NGINEERING 132A Professor...

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