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homework3solution - Solution to Homework#3 Problem 1...

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Solution to Homework #3 Problem 1 According to the phase relationship between the austenite and ferrite, the schematic concentration profile is given by c wt% 0.02 0.24 0.8 x (mm) The boundary conditions for c (x) are % wt 8 . 0 C ) t , 0 ( C 0 = = % wt 02 . 0 ) t , ( C = ξ α % wt 24 . 0 ) t , ( C = ξ γ 0 ) t , ( C = At T =1073K, s m RT D s m RT D / 10 12 . 2 1073 314 . 8 133900 exp 10 7 ) 133900 exp( 10 7 / 10 61 . 1 1073 314 . 8 84100 exp 10 2 ) 84100 exp( 10 2 2 12 6 6 2 10 6 6 × = × × = × = × = × × = × = γ α Since in each phase, the concentration profile is described by ) 2 / ( t D x erf B A c α α α α + = ) 2 / ( t D x erf B A c γ γ γ γ + = Assuming t D k 2 i = ξ , by using the boundary conditions, we have 0 ) , 0 ( C A A t c = = γ γ ( ) ) ( ) ( ) 2 / ( ) , ( 0 γ γ γ γ γ γ γ γ γ γ ξ ξ k erf C C B k erf B A t D erf B A t c = + = + = α α α α A B B A t c = = + = 0 ) , ( ) ( / ) ( ) ( ) 2 / ( ) , ( α α α α α α α α α α α α ξ ξ k erfc C A k erfc A k erf A A t D erf B A t c = = = + = Therefore, we have
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= t D x erf k erfc C k erfc C t x c α α α α
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