Ch E 310 - Fall 10 - Lecture 25

Ch E 310 - Fall 10 - Lecture 25 - Lecture 25 December 2...

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Lecture 25 – December 2, 2010 Agenda: Boundary Value Problems (solving ODEs) Shooting Method Derivative Boundary Conditions Solving Non-linear ODEs
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What is a Boundary Value Problem? ODEs can only be solved given additional information These are called “boundary conditions” If the boundary conditions are specified at the same value of the independent variable (e.g., t ) the problem is considered to be an initial value problem Some boundary conditions are specified at different values of the independent variable (e.g., initial and final time points) – these are boundary value problems
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Boundary Value Problems: Graphically Initial value problem: Boundary value problem:
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Boundary value problems arise commonly in science and engineering We most often see them for problems where conditions are specified at different locations in space (rather than time) Example: steady-state temperature distribution along a rod between two plates Boundary Value Problems: Examples
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Boundary Value Problems: Example Derivation : differential element of thickness D x accumulation = heat in – heat out Divide by element’s volume ( p r 2 D x ): Take the limit as D x → 0 Fourier’s law: ( k = thermal conductivity)            T T x r h r x x q r x q D D p 2 0 2 2       T T r h x x x q x q D D 2 0   T T r h dx dq 2 0 dx dT k q
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Boundary Value Problems: Example Plug Fourier’s law into the heat balance equation and divide through by k : where h' is an effective heat transfer coefficient [ m -2 ] Second-order ODE subject to boundary conditions: Known parameters: L = 10 m, h' = 0.05 m -2 , T = 200 K, T (0) = 300 K, T (10) = 400 K   T T h dx T d 2 2 0     b a T L T T T , 0
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This note was uploaded on 09/21/2011 for the course CH E 310 taught by Professor Staff during the Spring '08 term at Iowa State.

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Ch E 310 - Fall 10 - Lecture 25 - Lecture 25 December 2...

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