lecture23 - 2.57 Nano-to-Macro Transport Processes Fall...

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1 2.57 Nano-to-Macro Transport Processes Fall 2004 Lecture 23 In the last lecture, we talked about the energy flux along a thin film as q = ∑∑∑ f v = ω . x x V k x k y k z y d x θ v Temperature Gradient v x v y ϕ v v x v y θ v z Or Electrical Field (a) (b) Now let us consider the conduction of gas molecules between two plates. Similarly, the Boltzmann equation is JG G G F τ v ⋅∇ r G g + g = v G f 0 + v G f 0 , r m F where the bulk force m v G f 0 = 0 for phonons, g=f-f 0 . Noticing v G G g v y g (d<<x) r y and G f 0 = df 0 dT , the x direction component gives r dT dx x g + v y g y = v df 0 d T = S 0 () . x d d x We can solve g first and then substitute the expression f = g+f 0 into any flux equation. Under the diffuse assumption, we obtain the following figure for the conductivities of the material. σ k k = ξ = 1 bulk bulk d Λ In the y direction, we have 2.57 Fall 2004 – Lecture 23 1
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τ v g y g y += v df 0 . y d y Since no heat is generated in the volume, we finally obtain q y () = const . The () = T ( η ) T 2 normalized temperature θη obeys TT 2 1 ξ ) = E 2 ( ) + θ ( ') E 1 (| '|) d ' , 2( 0 λξ = d /, where = y λλ / d = Knudsen number, may also represent normalized 4 4 uy 2 2 blackbody emissive power u = T ( y ) T . This is the linear, nonhomogeneous, uu 2 T 1 4 T 2 4 1 Fredholm integral equation of the second kind. The function E 1 (| η - η ’|) is called the kernel. The Fredholm integral equation does not have an analytical solution, although approximate solution methods have been developed. To solve the equation numerically, first we discretize the equation with 2 dT T i + 1 2 T i + T = i 1 dx 2 2 x 2 . 0 ξ Half trapezium at both ends The integration is calculated by dividing the area into many trapezia. We have n 1 ( E (| ' |) d 0 E E 1 ( ) + i 1 i ' = 1 + E (| |) , 1 2 2 1 0 which gives totally n+1 equations (n is the section number). Therefore, we get a set of linear equations for to solve. i Another way to conduct the integration is to use the Gauss- Legendre method, which 1 n () ) generates sections in (0, ξ ) with varying width. Then we can use f xdx = f ( x W i i 0 i = 1 to find the approximating value, where W i is the weight of different f ( x ) .
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lecture23 - 2.57 Nano-to-Macro Transport Processes Fall...

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