soln09 - Homework 9 Problem1 1 Let us rst derive an...

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± ± Homework 9 November 13, 2006 Problem1 1. Let us ﬁrst derive an analytical espression for S model ( λ, t, θ )us ing both the models mentioned in the problem statement. Mode l1 : Sequen t ia lReac t ionMode l According this model C is formed from B through an intermediate species X, we can write the rate of formation of B, X and C as shown in Equation(1). In writing the rate rules in Equation(1) we have made use of the fact that the volume is constant. d [ B ] = k 1 [ B ] dt d [ X ] = k 1 [ B ] k 2 [ X ] dt d [ C ] = k 2 [ X ] ( 1 ) dt We can solve the equations simultaneously to get the expressions for the concentrations of [ B ], [ X ]and [ C ] shown in Equation(2). [ B ]=[ B ] 0 e k 1 t k 1 t k 2 t e e [ X B ] 0 k 1 + k 2 k 1 k 1 k 2 k 2 e k 1 t k 1 e k 2 t [ C B ] 0 1+ + (2) k 1 k 2 k 2 k 1 If we look at the absoption spectra of B and C in Figure 1 as given by SVD we realise that B has two distinct peaks and C has 1 Cite as: William Green, Jr., course materials for 10.34 Numerical Methods Applied to Chemical Engineering, Fall 2006. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].

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± ± ± −0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 The normalized absorption SVD−absorption of B SVD−absorption of C The absoption spectrum of species B and C 430 440 450 460 470 480 490 The wavelength (nm) Figure 1: Normalized Absoption Spectrum of B and C obtained from SVD only one peak, which suggests that B can be written as a sum of two diﬀerent gaussians and C can be written as a sum of only one gaussian as shown in Equation(3). ( λ λ B, 1 ) 2 ( λ λ 2 ) 2 A B ( λ )= A 1 exp + A 2 exp 2 2 w 1 w 2 ( λ λ C, 1 ) 2 A C ( λ A C, 1 exp 2 (3) w C, 1 Putting everything together, we can get an analytical expression for S model ( λ, t, θ ) as shown in Equation(4). ± ±± k 1 t ( λ λ 1 ) 2 ( λ λ 2 ) 2 S model ( λ, t, θ )=[ B ] 0 e A 1 exp 2 + A 2 exp 2 w 1 w 2 ± ±± k 2 e k 1 t k 1 e k 2 t ( λ λ C, 1 ) 2 +[ B ] 0 1+ + A C, 1 exp 2 (4) w k 1 k 2 k 2 k 1 C, 1 As is mentioned in the problem statement we cannnot determine [ B ] 0 or the magnitude of absoptions independently and we have 2 Cite as: William Green, Jr., course materials for 10.34 Numerical Methods Applied to Chemical Engineering, Fall 2006. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
± ± ² ² ³³ ± ² ² ³³ ´ to club some of the parameters together. If we do that we get Equation(5) ² ² ³ ² ³³ k 1 t ( λ λ B, 1 ) 2 ( λ λ 2 ) 2 S model ( λ, t, θ )= e K 1 exp 2 + K 2 exp 2 w 1 w 2 ² ³² ² ³³ k 2 e k 1 t k 1 e k 2 t ( λ λ C, 1 ) 2 + 1+ + K C, 1 exp 2 (5) w k 1 k 2 k 2 k 1 C, 1 The parameters that we need to determined in the above model are k 1 , k 2 , K 1 , K 2 , K C, 1 , λ 1 , λ 2 , λ C, 1 , w 1 , w 2 and w C, 1 .

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soln09 - Homework 9 Problem1 1 Let us rst derive an...

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