Countercurrent Flow Hemodialyzer

Initializing live version
Download to Desktop

Requires a Wolfram Notebook System

Interact on desktop, mobile and cloud with the free Wolfram Player or other Wolfram Language products.

This Demonstration models a countercurrent hemodialyzer. In a typical hemodialyzer, blood flows through a bundle of approximately 10,000 hollow fibers, each with an inner diameter of about 200 micrometers, wall thickness of 20 micrometers and a length of 16 to 25 cm. The walls of the hollow fibers function as the dialysis membrane; dialyzate flows countercurrently in the shell side [1].


Solutes are removed from the blood at different rates depending on their molecular weight and diffusivity; thus the blood is recirculated until all solutes reach a desirable level in the patient's blood. Use the sliders to vary the solute diffusion coefficient and the average velocities of the blood and dialyzate to observe the effects on the dialyzate concentrations.


Contributed by: Clay Gruesbeck (December 2019)
Open content licensed under CC BY-NC-SA


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:




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 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


[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.


Feedback (field required)
Email (field required) Name
Occupation Organization
Note: Your message & contact information may be shared with the author of any specific Demonstration for which you give feedback.