1 / 35

HW/Tutorial Week #10 WWWR Chapters 27, ID Chapter 14

Tutorial #10 WWWR # 27.6 & 27.22 To be discussed on April 1, 2014. By either volunteer or class list. HW/Tutorial Week #10 WWWR Chapters 27, ID Chapter 14. Unsteady-State Diffusion. Transient diffusion, when concentration at a given point changes with time

chailyn
Download Presentation

HW/Tutorial Week #10 WWWR Chapters 27, ID Chapter 14

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  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

  15. Example 2

  16. Concentration-Time charts

  17. Example 3

More Related