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TECHNICAL PAPERS: Fluids/Heat/Transport

A Numerical and Experimental Study of Mass Transfer in the Artificial Kidney

[+] Author and Article Information
Zhijie Liao, Churn K. Poh, Zhongping Huang

Department of Mechanical Engineering, University of Kentucky, Lexington, KY 40506

Peter A. Hardy

Center for Biomedical Engineering, University of Kentucky, Lexington, KY 40506

William R. Clark

Baxter Healthcare Corp., Renal Division, McGaw Park, IL 60085School of Medicine, Indiana University, Indianapolis, IN 46202

Dayong Gao

Department of Mechanical Engineering and Center for Biomedical Engineering, University of Kentucky, Lexington, KY 40506

J Biomech Eng 125(4), 472-480 (Aug 01, 2003) (9 pages) doi:10.1115/1.1589776 History: Received August 13, 2002; Revised February 11, 2003; Online August 01, 2003
Copyright © 2003 by ASME
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References

National Institutes of Health, National Institute of Diabetes and Digestive and Kidney Diseases, 2001, USRDS (U.S. Renal Data System) 2001 Annual Data Report: Atlas of End-Stage Renal Disease in the United States, Bethesda, MD.
Mujais, S.K., and Schmidt, B., 1995, “Operating Characteristics of Hollow Fiber Dialyzers,” in Clinical Dialysis, edited by Nissenson, A.R., et al., Appleton & Lange, Norwalk, Connecticut, pp. 75–92.
Ross, S. M., Uvelli, D. A., and Babb, A. L., 1973, “A One-dimensional Mathematical Model of Transmembrane Diffusional and Convective Mass Transfer in a Hemodialyzer,” ASME paper no. 73-WA/Bio-14.
Chang,  Y. L., and Lee,  C. J., 1988, “Solute Transport Characteristics in Hemodiafiltration,” J. Membr. Sci., 39, pp. 99–111.
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Poh, C. K., 2001, “Characterization of Fluid Transport in Artificial Kidney: a MRI Approach,” Master Thesis, University of Kentucky, Lexington, KY.
Lemanski,  J., and Lipscomb,  G. G., 1995, “Effect of Shell-side Flows on Hollow-Fiber Membrane Device Performance,” AIChE J., 41, pp. 2322–2326.
Osuga,  T., Obata,  T., Ikehira,  H., Tanada,  S., Sasaki,  Y., and Naito,  H., 1998, “Dialysate Pressure Isobars in a Hollow-fiber Dialyzer Determined from Magnetic Resonance Imaging and Numerical Simulation of Dialysate Flow,” Artif. Organs, 22, pp. 907–909.
Happel, J., and Brenner H., 1965, Low Reynolds Number Hydrodynamics, Prentice-Hall, Englewood Cliffs, NJ.
Skartsis,  L., Kardos,  J. L., and Khomami,  B., 1992, “Resin Flow Through Fiber Beds During Composite Manufacturing Process. Part I: Review of Newtonian flow Through Fiber Beds,” Polym. Eng. Sci., 32, pp. 221–230.
Liao,  Z., Poh,  C. K., Huang,  Z., Hardy,  P. A., Gao,  D., and Clark,  W. R., 2002, “Measurement of Hollow-Fiber-Membrane Transport Properties in Hemodialysis” [Abstract], ASAIO J., 48, p. 179.
Liao,  Z., Poh,  C. K., Li,  B., Huang,  Z., Clark,  W. R., Hardy,  P. A., and Gao,  D., 2001, “Modeling of Dialysate Flow in an Artificial Kidney: A Porous Media Model” [Abstract], ASAIO J., 47, p. 160.
Liao, Z., 2002, “Numerical and Experimental Studies of Mass Transfer in Artificial Kidney and Hemodialysis,” Ph.D. thesis, University of Kentucky, Lexington, KY.
Frank,  A., Lipscomb,  G. G., and Dennis,  M., 2000, “Visualization of Concentration Fields in Hemodialyzers by Computed Tomography,” J. Membr. Sci., 175, pp. 239–251.
Bird, R. B., Stewart, W. E., and Lightfoot, E. N., 1960, Transport Phenomena, Wiley, New York.
Kedem,  O., and Katchalsky,  A., 1958, “Thermodynamics Analysis of the Permeability of Biological Membranes to Non-electrolytes,” Biochim. Biophys. Acta, 27, pp. 229–246.
Patankar, S., 1980, Numerical Heat Transfer and Fluid Flow, Hemisphere Publishing Co., New York.
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Figures

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(a) Schematic of the configuration of an artificial kidney structure. There are thousands of hollow fibers inside the shell (housing). The hollow fiber is very tiny with an inner diameter of 200 μm and a porous membrane wall thickness of 15–50 μm. (b) Computational domain of numerical model (details described further in the text). The hollow fibers contained in the tubesheet are not counted in the model because they do not contribute to the mass transfer between blood and dialysate sides. Computational dialysate inlet and outlet are the outer surface of cylindrical fiber bundle which is exposed in the raised collar and where the dialysate comes into and goes out from fiber bundle.
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Schematic of computational domain in the lumen side. The velocity profile in the inlet is assumed as the Hagen-Poiseuille flow profile. Js and Jv are calculated from K-K equations.
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Mass transport of solute through membrane. Black spots indicate the solute molecules. Cbs and Pbs are solute concentration and hydraulic pressure in the lumen-side/blood-side membrane surface, respectively, Cds and Pds are solute concentration and hydraulic pressure in the shell-side/dialysate-side membrane surface, respectively. Transmembrane flux includes solution transmembrane flux and solute transmembrane flux.
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Geometry configuration near the dialysate inlet (reprinted from Journal of Membrane Science, Vol. 175, A. Frank, G. G. Lipscomb, and M. Dennis, Visualization of Concentration fields in Hemodialyzers by Computed Tomography, pp. 239–251, copyright (2000), by permission from Elsevier Science with minor modification).
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Schematic of computational domain in the shell side. Flow distribution (velocity profile) is assumed uniform in computational dialysate inlet and outlet.
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Fluid control volume. A denotes cross-sectional area.
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Computational meshes for shell-side flow. Each mesh still contains many hollow fibers, which are indicated by black spots in the figure. Shell-side flow is coupled with lumen-side flow through the source terms in Darcy equations.
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Flow chart of computer algorithm. After N-S equations and Darcy equations calculate the distribution of velocity, concentration, and pressure in the lumen and shell sides, respectively, the pressure and concentration in the both membrane surfaces of lumen and shell sides will be transferred to K-K equations as boundary conditions to calculate the transmembrane flux of solution and solute, which are then used to calculate the solution and solute source term in the Darcy equations, and used as the boundary conditions in the N-S equations to continue to solve N-S and Darcy equations in the next iteration. The convergence criterion is set as |urn+1−urn|/|urn|<10−6,|uzn+1−uzn|/|uzn|<10−6, and |Cn+1−Cn|/|Cn|<10−6 for lumen and shell side, the superscript “n” means the current iteration and “n+1” means the next iteration.
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Clearance measurement: experimental setup
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Schematic of concentric annual rings in shell side. Each ring has the same area. Each experiment was performed on the dialyzer with one ring open and other two rings sealed.
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Numerical and experimental results of local urea clearance in different annular ring with different dialysate flow rate. “Inner” in the graph denotes the inner ring open only with two other rings sealed, “middle” denotes the middle ring open only, and “outer” denotes the outer ring open only. “exp.” in the graph means experimental result, and “num.” means numerical result. Error bar indicates MEAN±STD, sample size n=3.
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Velocity field in shell side of CT190G (the axial length is 24.0 cm and the radius is 1.7 cm). Dialysate flow rate Qd=500 ml/min. The radial velocity dominates in the dialysate inlet and outlet, while axial velocity prevails in the middle range. The velocity vector depicts both magnitude and direction of velocity at the location of the starting point of the arrow.
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Distribution of pressure in shell side of CT190G (the axial length is 24.0 cm and the radius is 1.7 cm). Dialysate flow rate Qd=500 ml/min. The pressure in the dialysate outlet is set to zero.
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Distribution of concentration in shell side of CT190G (the axial length is 24.0 cm and the radius is 1.7 cm). Dialysate flow rate Qd=500 ml/min. The urea concentration in the dialysate inlet is set to zero.
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Axial velocity distributions in different cross sections in the dialysate side. r=0.0 is in the centerline; r=0.156 is in the periphery; Qd=500 ml/min;Qb=500 ml/min; dialyzer: CT190G; location (1) is three quarters away from dialysate inlet; location (2) is in the middle cross section; location (3) is one quarter away from dialysate inlet.

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