Heat Equation Special Solution - Tools in Mechanical Engineering - Lecture Slides, Slides of Mechanical Engineering

Download Heat Equation Special Solution - Tools in Mechanical Engineering - Lecture Slides and more Slides Mechanical Engineering in PDF only on Docsity! Heat Equation Special Solution 2 2 2 1 2 1 2 One-dimensional unsteady heat equation: A bar of length L with both ends at constant temperatures and , respectively ( 0, ) , u( , ) Also, the initial temperature distribution u uc t x T T u x t T x L t T ∂ ∂ = ∂ ∂ = = = = 1 2 1 2 inside the bar is given: ( , 0) ( ) for 0 If 0, the solution is given in chapter 11.5, say ( , ) ( , ) is the homogeneous solution. If 0, we can define a new distribution function h O u x t f x x L T T u x t U x t T T T = = ≤ ≤ = = = = = ≠ ( , ) ( , ) ( , )h O Ou x t u x t T U x t T= + = + Docsity.com 2 2 2 2 2 2 , the new function also satisfies the heat equation and matches both boundary conditions ( 0, ) ( 0, ) 0 Also ( , ) ( , ) 0 Therefore the solution is: O O O O O O u U U uc c t t x x u x t U x t T T T u x L t U L t T T T ∂ ∂ ∂ ∂ = = = ∂ ∂ ∂ ∂ = = = + = + = = = + = + = 1 2 2 2 2 2 2 2 2 2 2 2 ( , ) ( , ) where ( , ) is the homogeneous solution. If Introduce a new solution u(x,t)=U(x,t)+ψ(x) ( ) In order for ( , ) to satisfy the heat equatio Ou x t U x t T U x t T T u U U uc c u c c t t x x x u x t ψ ψ = + ≠ ⇒ ∂ ∂ ∂ ∂ ∂ ′′= = = − = − ∂ ∂ ∂ ∂ ∂ n 0 ( )x ax b ψ ψ ′′⇒ = = + Docsity.com 2 2 2 2 2 2 0 The original problem: (0, ) 0 , (1, ) 100 , ( ,0) ( ) The new problem: (0, ) 0 , (1, ) 0 , ( ,0) ( ) 100 The solution ( , ) ( , ) 100 2From chapter 11.5: ( ,0)n u uc t x u t C u t C u x f x U Uc t x U t C U t C U x f x x u x t U x t x C U x L ∂ ∂ = ∂ ∂ = ° = ° = ∂ ∂ = ∂ ∂ = ° = ° = − = + = ( ) ( ) 2 1 0 0.000986 1 sin 1002 (50 100 )sin [( 1) 1] 100( , ) [( 1) 1]sin L n n n t n n x dx L x n x dx U x t n x e n π π π π π ∞ − =       = − = − − = − + ∫ ∫ ∑ Docsity.com ( ) 20.000986 1 ( , ) ( , ) 100 100 [( 1) 1]sin 100n n t n u x t U x t x n x e x n π π ∞ − = = + = − + +∑ 0 0.2 0.4 0.6 0.8 10 20 40 60 80 100100 0 u0 x( ) u10 x( ) u50 x( ) u200 x( ) u500 x( ) 10 x Docsity.com