3661199134 - Lecture Notes ME4906 Mechanical Engineering PolyU 2008/2009 Chapter 5 Partial Differential Equations and Finite Difference Methods 5.2

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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 = dx , heat capacity c v , conductivity k , internal source intensity (per unit volume) ( ) , qxt & , 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. () ( ) 22 // 0 0 0 dx vv dm dx Q k T x steady state t v source free q Td Q T Q Q dQ qdx dm c q c td x t TT T kq c xt x ÷ = =− ∂ ∂ ∂ ∂ = = −+ + = + = ∂∂ → + =  →= & && & dx T Qk x =− 2 2 dQ T T Qd x k kd x dx x x  += +   q &
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In 3-D, the general equation is 222 22 , v T kTqc tx 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 dT Ta xb dx ∇= = =+ , where a and b are determined by boundary conditions .
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This note was uploaded on 08/27/2010 for the course ME ME4906 taught by Professor Dr.g.p.zheng during the Spring '10 term at NYU Poly.

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3661199134 - Lecture Notes ME4906 Mechanical Engineering PolyU 2008/2009 Chapter 5 Partial Differential Equations and Finite Difference Methods 5.2

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