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3661199134 - Lecture Notes ME4906 Mechanical Engineering...

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Lecture Notes ME4906, Mechanical Engineering, PolyU 2008/2009 Chapter 5 Partial Differential Equations and Finite Difference Methods 5.2 PDE of 1D Heat Transfer Problem 5.2.1 Governing Equation and Analytical Solution Consider an element (materials) of width dx , density ρ , mass dm = ρ dx , heat capacity c v , conductivity k , internal source intensity (per unit volume) ( ) , q x t & , temperature distribution T(x,t) . Define x T k Q = as heat flux. The heat flux changes into Q+dQ after passing though the materials with a width of dx . Figure 2. Energy balance in a 1D element Energy conservation leads to the following heat transfer equations which are typical parabolic PDEs. ( ) ( ) 2 2 / / 0 2 2 0 0 dx v v dm dx Q k T x steady state t v source free q T dQ T Q Q dQ qdx dm c q c t dx t T T T k q c x t x ρ ρ ρ ÷ = =− ∂ ∂ ∂ = = + + = →− + = → + = → = & & & & dx T Q k x = − 2 2 dQ T T Q dx k k dx dx x x + = + q &
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In 3-D, the general equation is 2 2 2 2 2 2 2 2 , v T k T q c t x y z ρ + = = + + & For steady state (temperature does not change with time) without source ( 0 = q ), the equation for heat transfer (temperature distribution) is 2 2 2 3D : 0, 1D : 0, T d T T ax b dx
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