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

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

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

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

–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.

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

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

–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,

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.

–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

–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

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

–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

Example 1

Example 2

Concentration-Time charts

Example 3