BVP_Solutions

BVP_Solutions - NONLINEAR BOUNDARY VALUE PROBLEM SOLUTIONS...

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1 NONLINEAR BOUNDARY VALUE PROBLEM SOLUTIONS The temperature distribution of the rectangular fin shown in Figure 1(a) below, considering conduction and radiation heat transfer, is given by ) ( 4 4 2 2 = T T kA P dx T d σε (1) where T = temperature, x = location along the fin, k = thermal conductivity, bd A = = cross sectional area, ) ( 2 d b P + = = perimeter, T = surrounding temperature, σ = Stefan-Boltzman constant, and ε = emissivity. The values for the constants are given in the table below, where are values are reported in a consistent set of units: Constant Value Units k 42 W/m – º b 0.5 m d 0.2 m T 500 º K l 2 m 0.1 dimensionless 8 10 7 . 5 × W/m 2 – º K 4 The boundary conditions are given as 1000 ) 0 ( = = x T º K 350 ) ( = = l x T º K The temperature at three equally spaced interior node points 1, 2, and 3, as shown in Figure 1(b), is to be determined using three different methods. This distance h between each pair of nodes is 0.5 m.

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2 a) Using Eq. (1), derive a nonlinear set of finite difference equations for the temperature at interior nodes 1, 2, and 3. Use central differencing to approximate 2 2 dx T d at each node. Then write a Matlab program to solve the nonlinear system of equations for the temperature at the three nodes. In your program, call the “fsolve” rootfinding function available in Matlab using the default convergence tolerances and an intial guess of 675º K for all three nodes. (Note: “fsolve” is only available in the Optimization Toolbox, so after coding your Matlab .m
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This note was uploaded on 09/05/2011 for the course EGM 3344 taught by Professor Raphaelhaftka during the Spring '09 term at University of Florida.

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BVP_Solutions - NONLINEAR BOUNDARY VALUE PROBLEM SOLUTIONS...

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