1. Louisiana Tech University Ruston, LA 71272 Slide 1 Mass Transport & Boundary Layers Steven A. Jones BIEN 501 Friday, May 2, 2008

2. Louisiana Tech University Ruston, LA 71272 Slide 2 Mass Transport Major Learning Objectives: 1.Describe combined diffusion and convection.

3. Louisiana Tech University Ruston, LA 71272 Slide 3 Mass Transport Minor Learning Objectives: 1.Present the mass transfer equations with convection.

4. Louisiana Tech University Ruston, LA 71272 Slide 4 A Motivating Example For the NO synthesis response, consider a layer of endothelial cells. These cells will be subjected to convection by the flow of blood. NO x z

5. Louisiana Tech University Ruston, LA 71272 Slide 5 A Motivating Example A simplified diagram of the initial and boundary conditions is as follows: x z

6. Louisiana Tech University Ruston, LA 71272 Slide 6 Partial Differential Equation The partial differential equation governing combined diffusion and convection is:

7. Louisiana Tech University Ruston, LA 71272 Slide 7 Comment on Example 9.6.1 However, the example in section 9.6.1 uses a constant concentration boundary condition and the example described in class uses a constant flux boundary condition. NO 1 st order reaction Constant Concentration

8. Louisiana Tech University Ruston, LA 71272 Slide 8 Comment on Example 9.6.1 Compare the form of the solutions. Constant flux: Constant concentration:

9. Louisiana Tech University Ruston, LA 71272 Slide 9 Comment on Example 9.6.1 Slattery rewrites the solution as: Question 1: Why is this form “More Useful?”

10. Louisiana Tech University Ruston, LA 71272 Slide 10 Comment on Example 9.6.1 Slattery rewrites the solution as: Question 1: Why is this form “More Useful?” Answer: The complementary error function (erfc) is a tabulated function. Most software, like MatLab, has a subroutine already available to calculate. Numerical integration is not necessary.

11. Louisiana Tech University Ruston, LA 71272 Slide 11 Comment on Example 9.6.1 Slattery rewrites the solution as: Question 2: The constant flux solution is similar in form, is it possible to write it in terms of the complementary error function?

12. Louisiana Tech University Ruston, LA 71272 Slide 12 Comment on Example 9.6.1 Slattery rewrites the solution as: Question 2: The constant flux solution is similar in form, is it possible to write it in terms of the complementary error function? Answer: I don’t know, but I suspect it is, and if you work it out and give me the answer I will have something good to say if I write a letter of recommendation for you.

13. Louisiana Tech University Ruston, LA 71272 Slide 13 Platelet Synthesis Response NO is synthesized by platelets and released continuously, as in the previous example. NO 1 st order reaction Constant Flux

14. Louisiana Tech University Ruston, LA 71272 Slide 14 Platelet Synthesis Response OK. I’ll admit that nothing happens continuously in physiology. The point is that NO release is “relatively” constant. NO Flux Time (minutes) 0123 Delay Activation

15. Louisiana Tech University Ruston, LA 71272 Slide 15 Platelet Release Response In contrast, compounds like ADP and serotonin (5-HT) are stored in platelet granules and released “instantaneously” upon activation. No Reaction Zero Flux

16. Louisiana Tech University Ruston, LA 71272 Slide 16 Platelet Release Response In contrast, compounds like ADP and serotonin (5-HT) are stored in platelet granules and released “instantaneously” upon activation. How will concentrations differ from the synthesis response? Zero Flux

17. Louisiana Tech University Ruston, LA 71272 Slide 17 Synthesis/Release Comparison ConditionSynthesis (NO)Release (ADP) At the surfaceConstant FluxZero Flux At infinityInitially Zero Initial Condition ReactionPseudo 1 st orderNone

18. Louisiana Tech University Ruston, LA 71272 Slide 18 Conservation of Mass Rate of increase Conservation of Mass: Diffusive transport

19. Louisiana Tech University Ruston, LA 71272 Slide 19 Transformed Governing Equation Use the Fourier transform to transform the spatial variable in the governing equations and boundary conditions. The governing PDE transforms from: To: Or:

20. Louisiana Tech University Ruston, LA 71272 Slide 20 Solution to Transformed Equation With: Because:

21. Louisiana Tech University Ruston, LA 71272 Slide 21 Transformed BC’s 2.The concentration must go to zero for large values of z. 3.All of the mass is initially concentrated at the wall. Where we recall that: 1

22. Louisiana Tech University Ruston, LA 71272 Slide 22 Evaluation of C From the general solution: And the transformed boundary condition at the surface: We have: So:

23. Louisiana Tech University Ruston, LA 71272 Slide 23 Invert the Fourier Transform The inverse Fourier transform is: Which simplifies to:

24. Louisiana Tech University Ruston, LA 71272 Slide 24 Invert the Fourier Transform Yeah, ok, the “simplification” looks more complicated than the original equation. The point is that the “simplified” version has that error function look to it. e to the something squared.

25. Louisiana Tech University Ruston, LA 71272 Slide 25 Invert the Fourier Transform The trick is to complete the square in the exponent:

26. Louisiana Tech University Ruston, LA 71272 Slide 26 Invert the Laplace Transform Now apply the transformation: To: And get:

27. Louisiana Tech University Ruston, LA 71272 Slide 27 Agonist Release Model Therefore the solution to the agonist release model is:

28. Louisiana Tech University Ruston, LA 71272 Slide 28 Example Concentration Profiles Increasing time 10 s 0.1 s

29. Louisiana Tech University Ruston, LA 71272 Slide 29 NO Concentration Profiles Increasing time 5 s