lecture22 - 2.57 Nano-to-Macro Transport Processes Fall...

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2.57 Fall 2004 – Lecture 22 1 2.57 Nano-to-Macro Transport Processes Fall 2004 Lecture 22 We have talked about the heat flux as 1 xyz xx kkk qf v V ω = ∑∑∑ = . The Boltzmann equation is 00 0 rr v F vg g v f f S m ττ ⎛⎞ ⋅∇ + =− + = ⎜⎟ ⎝⎠ GG G JG , where 0 0 v F f m = G for phonons, g=f-f 0 . Noticing y r g v y G G (d<<x) and 0 0 r df dT f dT dx ∇= G , the x direction component gives () 0 0 yx df gd T gv v S x yd T d x += = , the solution of which is 0 exp y y gS C v τ −= . One boundary condition is required to determine C. Assuming both top and bottom of the film diffusely scatter phonons, we have 0 0 0, , 0 for 0, 2 , , 0 for , 2 yf f g f fg π θ == = ⎪⎝ = . Finally we get At y = 0, 0, 2 , 0 C = -S , 0 ,1 e x p cos y gy S v =−− , y d x θ v Temperature Gradient Or Electrical Field v x v y ϕ v v x v y θ v z (a) (b)
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2.57 Fall 2004 – Lecture 22 2 At y = d, , 2 π θ ⎛⎞ ⎜⎟ ⎝⎠ , ( ) 0 C = -S exp / cos dv τ , () 0 ,1 e x p cos dy gy S v =− . Note: cos y v is the ratio between the traveled distance and the mean free path of phonons. ( ) max 2 00 0 1 sin 4 xyz xx x vvv D q y vf d d d V ωπ ω ωϕ == ∑∑∑ ∫∫ , where sin cos x vv ϕ = according to our spherical coordinate system, f=g+f 0 . Thus () ( ) ( ) max 2 0 2 0 [ cos sin cos sin exp 1 sin cos 4 x D df dT y qy
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This note was uploaded on 01/12/2011 for the course ME 305 taught by Professor Wright,j during the Spring '10 term at Birla Institute of Technology & Science, Pilani - Hyderabad.

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lecture22 - 2.57 Nano-to-Macro Transport Processes Fall...

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