1. Louisiana Tech University Ruston, LA 71272 Slide 1 Co-Current and Counter-Current Exchange in Dialysis Steven A. Jones BIEN 501 Monday, May 5, 2008

2. Louisiana Tech University Ruston, LA 71272 Slide 2 Co-Current Exchange Conservation of Mass Membrane Diffusion:

3. Louisiana Tech University Ruston, LA 71272 Slide 3 Different Notation I personally do not find the notation to be completely intuitive. For me, a better way is to talk about the mass flux per unit area along the z -direction. The conservation of mass then becomes:

4. Louisiana Tech University Ruston, LA 71272 Slide 4 Definitions VariableDefinition Flow rate of blood Mass flux per unit area of membrane from blood to dialysate Concentration of the solute in the blood Concentration of the solute in the dialysate Width of the membrane Height of the membrane

5. Louisiana Tech University Ruston, LA 71272 Slide 5 The Differential Equation With: Divide through by dz Take the limit as dz goes to 0.

6. Louisiana Tech University Ruston, LA 71272 Slide 6 Interpretation This equation has the simple interpretation that concentration decreases along the length of the membrane as a result of mass loss through the membrane. Because it uses well- defined derivatives, its meaning is much more clear than the equation in the book: Consider:

7. Louisiana Tech University Ruston, LA 71272 Slide 7 Dialysate Side, Mass Conservation On the dialysate side, the amount of mass coming in is equal to the amount of mass leaving the blood. Thus, And these two equations can be combined to obtain:

8. Louisiana Tech University Ruston, LA 71272 Slide 8 The Two Basic Equations Conservation of Mass is then: And Membrane Diffusion is: Where:

9. Louisiana Tech University Ruston, LA 71272 Slide 9 Compare to the Book’s Notation A comparison of our membrane equation: To the equation in the book: Clarifies what the book means by, for if we divide by dA m : Thus, is the rate of change of total mass flux with membrane area, and it is equivalent to the mass flux per unit area as we have defined it.

10. Louisiana Tech University Ruston, LA 71272 Slide 10 Combine the Two Equations Blood SideDialysate Side Divide the blood side by Q b h and divide the dialysate side by Q d h

11. Louisiana Tech University Ruston, LA 71272 Slide 11 Subtract With: So:

12. Louisiana Tech University Ruston, LA 71272 Slide 12 Combine the Two Equations We need to determine the transport across the membrane. We will integrate the equation below with respect to z: From

13. Louisiana Tech University Ruston, LA 71272 Slide 13 Combine the Two Equations

14. Louisiana Tech University Ruston, LA 71272 Slide 14 Eq. 7.8.10 In our notation, the book’s equation is:

15. Louisiana Tech University Ruston, LA 71272 Slide 15 Compare Co- and Counter-Current Co-Current Counter-Current The concentration differences (inlet vs. outlet) are closer to one another for counter-current exchange. What about the net mass transfer?

16. Louisiana Tech University Ruston, LA 71272 Slide 16 Total Mass Transfer From Equations 7.8.1 and 7.8.2: And similarly: Note sign change.

17. Louisiana Tech University Ruston, LA 71272 Slide 17 Total Mass Transfer Combine to eliminate flow rates (co-exchange):

18. Louisiana Tech University Ruston, LA 71272 Slide 18 Total Mass Transfer

19. Louisiana Tech University Ruston, LA 71272 Slide 19 Total Mass Transfer Combine to eliminate flow rates (counter-exchange):

20. Louisiana Tech University Ruston, LA 71272 Slide 20 Total Mass Transfer Sort of a cross- difference w.r.t 0 and L.

21. Louisiana Tech University Ruston, LA 71272 Slide 21 Look at the Plots z=0 z=Lz=L z=Lz=L Look at

22. Louisiana Tech University Ruston, LA 71272 Slide 22 Clearance ( K )

23. Louisiana Tech University Ruston, LA 71272 Slide 23 Efficiency Co-current Counter-current Where Are the same for co- and counter-current.

24. Louisiana Tech University Ruston, LA 71272 Slide 24 Questions Co-current Counter-current Where What do these functions look like? If Q B = Q D, do you get infinite efficiency? For what value of Z is E a maximum?

25. Louisiana Tech University Ruston, LA 71272 Slide 25 Efficiency as a function of Z

26. Louisiana Tech University Ruston, LA 71272 Slide 26 Question Does it help to have a large Q D ?

27. Louisiana Tech University Ruston, LA 71272 Slide 27 Counter-Current Exchange Conservation of Mass

28. Louisiana Tech University Ruston, LA 71272 Slide 28 Compare Co- and Counter-Current Co-Current Counter-Current

29. Louisiana Tech University Ruston, LA 71272 Slide 29 Integrate w.r.t. z