Transient Diffusion in Semi-Infinite and Finite Media: Analytical Solutions

1. Tutorial #10 WWWR # 27.6 & 27.22 To be discussed on April 1, 2014. By either volunteer or class list. HW/Tutorial Week #10WWWR Chapters 27, ID Chapter 14

2. 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

3. 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

4. Transient diffusion in semi-infinite medium uniform initial concentration CAo constant surface concentration CAs • Initial condition, t = 0, CA(z,0) = CAo for all z • First boundary condition: at z = 0, cA(0,t) = CAs for t > 0 • Second boundary condition: at z = , cA(,t) = CAo for all t • Using Laplace transform, making the boundary conditions homogeneous

5. Thus, the P.D.E. becomes: • with • (z,0) = 0 • (0,t) = cAs – cAo • (,t) = 0 • Laplace transformation yields which becomes an O.D.E.

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

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

8. 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,

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

10. Initial and boundary conditions of • cA = cAo at t = 0 for 0  z  L • cA = cAs at z = 0 for t > 0 • cA = cAs at z = L for t > 0 • Simplify by dimensionless concentration change • Changing the P.D.E. to Y = Yo at t = 0 for 0  z  L Y = 0 at z = 0 for t > 0 Y = 0 at z = L for t > 0

11. 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

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

13. 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

14. Example 1

21. Example 2

29. Concentration-Time charts

32. Example 3