Modified Ergun equation for dialysate flow in a hollow fiber dialyzer

A hollow-fiber dialyzer contains thousands of hollow fibers packed inside a cylindrical module, also called a hollow-fiber membrane module (HFMM). Typically, the fiber diameter around 200 µm, while the module’s diameter and length are 3-4 cm and 15-25 cm respectively. In this study, we focus on the flow of dialysate outside the fibers (in the shell region). Prior studies model the shell region as a porous medium, estimating its Darcy’s permeability either experimentally or through the Ergun equation. Using a 3D computational model that solves the full Navier-Stokes and continuity equations for dialysate flow through commercial dialyzers geometries, we calculate pressure drops between the inlet and outlet for a wide range of dialysate flow rates (0.0001 to 1000 mL/min) and packing densities (0.4 and 0.6). We validate our model against experimental flow-field data from four commercial dialyzers (Medtronic PHYLTER® 15SD and 17SD, Baxter CT190G and Fresenius F6HPS) reported in the literature. When characterizing the HFMM as a packed column, we find that the Blake-Kozeny constant (A) in the Ergun equation does not have a constant value of 150, as has been previously reported, but varies linearly with packing density. Similarly, the Burke-Plummer constant (B), often cited as 1.75, also varies with packing density and a modified Reynold’s number (Re’) based on the dialysate flow. We observe that B approaches a constant value after Re’ ≈ 1, indicating the onset of inertial (or weakly turbulent) flow behavior, analogous to transitions seen in the Moody diagram. In summary, we present a modified Ergun equation that accounts for packing density and Re’, enabling accurate, low-cost porous media simulations of dialysate flow in hollow-fiber dialyzers without needing empirical permeability inputs. This formulation provides a generalized framework for modeling shell-side flow in HFMMs including medical devices like hemodialyzers and extracorporeal membrane oxygenators.