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HW/Tutorial Week #10 WWWR Chapters 27, ID Chapter 14 Tutorial #10 WWWR # 27.6 & 27.22 To be discussed on March 31, 2015. By either volunteer or class list.

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Unsteady-State Diffusion Transient diffusion, when concentration at a given point changes with time Partial differential equations, complex processes and solutions Solutions for simple geometries and boundary conditions

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Fick’s second law of diffusion 1-dimensional, no bulk contribution, no reaction Solution has 2 standard forms, by Laplace transforms or by separation of variables

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Transient diffusion in semi-infinite medium uniform initial concentration C Ao constant surface concentration C As –Initial condition, t = 0, C A (z,0) = C Ao for all z –First boundary condition: at z = 0, c A (0,t) = C As for t > 0 –Second boundary condition: at z = , c A ( ,t) = C Ao for all t –Using Laplace transform, making the boundary conditions homogeneous

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–Thus, the P.D.E. becomes: –with (z,0) = 0 (0,t) = c As – c Ao ( ,t) = 0 –Laplace transformation yields which becomes an O.D.E.

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–Transformed boundary conditions: –General analytical solution: –With the boundary conditions, reduces to –The inverse Laplace transform is then

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–As dimensionless concentration change, With respect to initial concentration With respect to surface concentration –The error function is generally defined by

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–The error is approximated by If 0.5 If 1 –For the diffusive flux into semi-infinite medium, differentiating with chain rule to the error function and finally,

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Transient diffusion in a finite medium, with negligible surface resistance –Initial concentration c Ao subjected to sudden change which brings the surface concentration c As –For example, diffusion of molecules through a solid slab of uniform thickness –As diffusion is slow, the concentration profile satisfy the P.D.E.

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–Initial and boundary conditions of c A = c Ao at t = 0for 0 z L c A = c As at z = 0for t > 0 c A = c As at z = Lfor t > 0 –Simplify by dimensionless concentration change –Changing the P.D.E. to Y = Y o at t = 0for 0 z L Y = 0at z = 0for t > 0 Y = 0at z = Lfor t > 0

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–Assuming a product solution, Y(z,t) = T(t) Z(z) –The partial derivatives will be –Substitute into P.D.E. divide by DAB, T, Z to

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–Separating the variables to equal - 2, the general solutions are –Thus, the product solution is: –For n = 1, 2, 3…,

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–The complete solution is: where L = sheet thickness and –If the sheet has uniform initial concentration, for n = 1, 3, 5… –And the flux at z and t is

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Example 1

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Example 2

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Concentration-Time charts

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Example 3

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© meg/aol ‘02 Solutions To The Linear Diffusion Equation Martin Eden Glicksman Afina Lupulescu Rensselaer Polytechnic Institute Troy, NY, 12180 USA.

© meg/aol ‘02 Solutions To The Linear Diffusion Equation Martin Eden Glicksman Afina Lupulescu Rensselaer Polytechnic Institute Troy, NY, 12180 USA.

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