CoCurrent and CounterCurrent Exchange in Dialysis Steven A

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

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

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: Louisiana Tech University Ruston, LA 71272 3

Definitions Variable Definition 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 Louisiana Tech University Ruston, LA 71272 4

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

Interpretation Consider: 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 welldefined derivatives, its meaning is much more clear than the equation in the book: Louisiana Tech University Ruston, LA 71272 6

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: Louisiana Tech University Ruston, LA 71272 7

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

Compare to the Book’s Notation A comparison of our membrane equation: To the equation in the book: Clarifies what the book means by divide by d. Am: , for if we 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. Louisiana Tech University Ruston, LA 71272 9

Combine the Two Equations Blood Side Dialysate Side Divide the blood side by Qbh and divide the dialysate side by Qdh Louisiana Tech University Ruston, LA 71272 10

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

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

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

Eq. 7. 8. 10 In our notation, the book’s equation is: Louisiana Tech University Ruston, LA 71272 14

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

Total Mass Transfer From Equations 7. 8. 1 and 7. 8. 2: And similarly: Louisiana Tech University Ruston, LA 71272 Note sign change. 16

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

Total Mass Transfer Louisiana Tech University Ruston, LA 71272 18

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

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

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

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

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

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

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

Question Does it help to have a large QD? Louisiana Tech University Ruston, LA 71272 26

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

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

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