Details

Following are the details of the mathematical model.

The system is at steady state.

The physical properties are constant.

There is fully developed laminar flow in the blood side and plug flow in the dialyzate side.

All fibers have the same diameter, fluid flow and mass transfer characteristics.

The equation for continuity of the solute for the blood side is:

subject to the boundary conditions:

,

and

.

The equation for continuity of the solute for the dialyzate side is:

,

with boundary condition

.

The average (cup) concentrations of the solute in the blood and the dialyzate are:

,

and

,

respectively.

Here:

and are the average velocities of the blood and the dialyzate, respectively

and are the axial and radial coordinates

and are the inner and outer diameters of the fiber

is the number of fibers

is the length of the fiber

is the diffusivity of the solute in the blood

is a mass transfer coefficient

is the Sherwood number, defined as the ratio of convective mass transfer rate to diffusion rate

Reference

[1] C. Ronco, A. Brendolan, C. Crepaldi, M. Rodighiero and M. Scabardi, "Blood and Dialysate Flow Distributions in Hollow-Fiber Hemodialyzer Analyzed by Computerized Helical Scanning Technique," Journal of the American Society of Nephrology, 13(Suppl 1), 2002 S53–S61. jasn.asnjournals.org/content/13/suppl_1/S53.