final - C(x) = x . 2] Consider steady 2-dimensional heat...

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Engineering Analysis Final Exam 1] Consider a rod with a uniform internal heat source that is kept at the background temperature at one end and insulated on the other, i.e. 2 T x 2 C = 1 2 T t for t 0,0 x L where 2 and C are positive constants, with boundary conditions T x = 0, t = 0, T x x = L , t = 0, T x, t = 0 = f x (arbitrary differentiable function). (a) [10] Find the temperature distribution approached at large t . Sketch the graph of this distribution as a function of x , taking a system of units where C and L are both 1. (b) [15] Obtain a PDE and boundary conditions for the transient component of the temperature distribution. (c) [15] Solve for the transient temperature distribution. What is the timescale for the steady-state temperature distribution to be approached along the entire rod? (d) [10] Solve for the temperature distribution if the heat source is not uniform, but given by
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Unformatted text preview: C(x) = x . 2] Consider steady 2-dimensional heat conduction in the half-plane defined, in polar coordinates, by ,r 1. Laplace's equation in polar coordinates is 2 T r 2 1 r T r 1 r 2 2 T 2 = 0. The boundary conditions are T = 0, r = 0, T = ,r = 0, T , r = 1 = g (arbitrary differentiable function). Also, the temperature is bounded as r . (a) [10] Using separation of variables, decompose the PDE into ODEs in each of the two coordinates. (b) [10] Solve the ODEs. (c) [15] Find product solutions that satisfy the homogenous boundary conditions. (d) [15] Find a superposition of these solutions that also satisfies the remaining boundary condition....
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This note was uploaded on 08/30/2011 for the course CGN 2200 taught by Professor Glagola during the Spring '11 term at University of Florida.

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