Numerical Analysis of Fluid Flow within Hollow Fiber Membranes of the Total Artificial Lung

Transcription

1 International Conerence on Trends in Industrial and Mechanical Engineering (ICTIME'01) March 4-5, 01 Dubai Numerical Analysis o Fluid Flow within Hollow Fiber Membranes o the Total Artiicial Lung Khalil Khanaer 1 and Keith Cook Abstract A numerical study was conducted to analyze luid low within hollow iber membranes o the artiicial lungs. The hollow iber bundle was approximated using a porous media model. In addition, the transport equations were solved using the inite element ormulation based on the Galerkin method o weighted residuals. Comparisons with previously published work on the basis o special cases were perormed and ound to be in excellent agreement. A Newtonian viscous luid model or the blood was used. Dierent low models or porous media such, as Brinkman-extended Darcy model, Darcy s law model, and the generalized low model were considered. Results were obtained in terms o streamlines, velocity vectors, and pressure distribution or various Reynolds number and Darcy number. The results rom this investigation showed that the pressure drop increases signiicantly or small Darcy numbers. Moreover, the deinition o porous medium is ound to have a signiicant eect on the low characteristics within the TAL. Keywords Artiicial Lung, Blood oxygenation, Porous Medium, Hollow iber membrane.. T I. INTRODUCTION RANSPORT phenomena through porous media have been the subject o various studies due to an increasing need or a better understanding o the associated transport processes. Some aspects o transport in porous media were discussed in recent monographs by Nield and Bejan [1], Vaai [, 3], Hadim and Vaai [4] and Vaai and Hadim [5]. Signiicant advances have been accomplished in applying porous media theory in modeling biomedical applications. Examples include computational biology, tissue replacement production, drug delivery, advanced medical imaging, porous scaolds or tissue engineering and eective tissue replacement to alleviate organ shortages, and transport in biological tissues [6-10]. Vaai and Tien [11, 13] presented an in-depth analysis o the generalized transport through porous media. They developed a set o governing equations utilizing the local volume-averaging technique. An important application o porous media includes modeling low characteristics through the iber bundle o the 1 Vascular Mechanics Laboratory, Biomedical Engineering Department, University o Michigan. 0 Lurie Biomedical Engineering Department, 1101 Beal Ave, Ann Arbor, MI USA; phone: ; khanaer@umich.edu Department o Surgery, University o Michigan, Ann Arbor, MI 48109, USA artiicial lung due to the diiculty in describing the geometry o the individual hollow iber membranes and also the density and total number o nodal points required to capture the characteristics o the low. Lung disease is the third leading cause o death and responsible or one in seven atalities in the United States according to the American Lung Association, totaling close to 335,000 Americans each year [14]. Developing eicient and biocompatible artiicial lungs has received considerable attention due to high atality rates because o acute and chronic diseases o the lung [15,16]. The goal o the artiicial lungs is to provide oxygen and carbon dioxide transer suicient to support patients with acute respiratory insuiciency or end-stage pulmonary disease [17- ]. Although the current methods such as extracorporeal membrane oxygenation, or ECMO, or supporting patients with lung disease are occasionally successul as a bridge to transplant, ECMO requires multiple transusions and is complex, labor intensive, time limited, costly, non ambulatory, and prone to inection. Moreover, Blood low through the artiicial lung is driven by the right ventricle while a mechanical pump is required in ECMO circuit which may causes blood cell trauma [19]. Artiicial lungs are usually designed to meet gas exchange requirements or the patient. To achieve eicient gas exchange, artiicial lungs use crosslow principle and consist o layers o microporous hollow ibers (diameters are around 300 microns) through which air lows. As the oxygen-poor blood lows across the hollow iber bundle, it generates transverse mixing that is essential or enhanced gas exchange between oxygen inside the iber and the carbon dioxide rom the blood lowing over the bundle. To the best o the authors knowledge, no attention has been paid to investigate low characteristics and pressure drop within an artiicial lung using various models o porous medium. Thereore, the main objective o the present study was to examine the momentum transport process inside an artiicial lung using Darcy s model, Brinkman-extended Darcy, and the generalized model. Another challenge in artiicial lung design is minimizing thrombus ormation in the blood lowing around the ibers. Thereore, low ield within the simpliied model o an artiicial lung was analyzed under various pertinent parameters such as Reynolds number and Darcy number to address the likelihood o platelet activation and thrombosis. 133

2 International Conerence on Trends in Industrial and Mechanical Engineering (ICTIME'01) March 4-5, 01 Dubai II. MATHEMATICAL FORMULATION The geometry o the simpliied model used or numerical analysis in this investigation is similar to MC3 s BioLung prototype (Michigan Critical Care Consultant, Ann Arbor, MI) which uses radial blood perusion through a concentrically wound hollow iber abric (Fig. 1). subscript reers to the luid phase. The permeability o the hollow iber bundle K and the geometric unction F can be represented as in Ergun [6] and Vaai [7, 8]: 3 ε d HFM 1.75 K =, F = (5) 150(1 ε ) 3 150ε The void raction o the iber bundle region is given as N iberd HFM ε = 1 (6) D D o i Fig. 1 Schematic diagram o the D artiicial lung and boundary conditions A Newtonian viscous luid model or the blood was used, with a viscosity o Pa.s. The Newtonian assumption was considered a good approximation due to the relatively high shear rate o blood low within the pulmonary artery (PA) and anastomoses. The hollow iber bundle was approximated using a porous medium approximation and the transport equations commonly known as the generalized model were solved to determine low characteristics. Moreover, the porous medium is viewed as a continuum with the solid and luid phases in thermal equilibrium, isotropic, homogeneous, and saturated with an incompressible luid. Hence, the porous medium has a unique porosity ε and permeability K values. By incorporating the above points, the system o the governing equations or the iber bundle can be expressed in vectorial orms based on the volume average technique [8, 11-1, 3-5] as: Continuity Equation < V >= 0 (1) Momentum Equation ρ < V > + < ( V ) V > = < > + < > t P V ε ε < V > Fε [ < V > < V > ] J K K () The luid motion outside the iber bundle is governed by the Navier-Stokes equations with constant density and luid properties, together with the continuity equation. In a Cartesian coordinate with a ixed reerence rame, the conservation o mass and momentum equations or transient, laminar low without body orces are given by: V = 0 (3) ρ V + ( V V) = + ρ P V (4) t Where ρ is the blood density, is the blood viscosity, P is the pressure, V is the velocity vector, J = V / V is a unit vector oriented along the pore velocity vector and the Where N iber is the total number o ibers, d HFM is the hollow iber membrane outer diameter, D o and D i are outer and inner diameter o the iber bundle, respectively. Employing the hollow iber membrane diameter o 300 μm and the measured eective blood oxygenator rontal area o about, A, = 100 cm, the iber bundle properties using Eqs. (5, 6) were: void raction, ε, = 0.75; length, L, = 3. cm, and permeability, K, = m. Boundary conditions were spatially uniorm low at the inlet o the artiicial lung. These boundary conditions can be summarized as: Inlet port: U = U o,v = 0 U Exit port: = V = 0 X Walls o the artiicial lung: U=V=0 At the interace between luid and porous: V = Vporous V Vporous = e n n where ρ U ε (7a) (7b) (7c) (7d) e = and Reynolds number (Re) is deined as H o Re = (8) where U o is the inlet velocity and H is the height o the inlet port. The Darcy number is expressed by K Da = (9) H III. NUMERICAL SCHEME A inite element ormulation based on the Galerkin method is utilized to solve the governing equations. The application o this technique is well documented by Taylor and Hood [8] and Gresho et al. [9]. The objective o the inite-element method (FEM) is to reduce the continuum problem (ininite number o degrees o reedom) to a discrete problem (inite number o degrees o reedom) described by a system o algebraic equations. In the current investigation, the continuum domain is divided into a set o non-overlapping regions called elements. Nine node quadrilateral elements with bi-quadratic interpolation unctions are utilized to 134

3 International Conerence on Trends in Industrial and Mechanical Engineering (ICTIME'01) March 4-5, 01 Dubai discretize the physical domain. Moreover, interpolation unctions in terms o local normalized element coordinates are implemented to approximate the dependent variables within each element. Subsequently, substitution o the approximations into the system o the governing equations and boundary conditions yields a residual or each o the conservation equations. These residuals are then reduced to zero in a weighted sense over each element volume using Galerkin method. A variable grid-size system is employed in the present investigation to capture the rapid changes in the dependent variables (Fig. ). the eect o Darcy number on the pressure drop within the artiicial lung. Fig. 4 illustrates that the pressure drop is signiicantly large when Darcy number is small. Fig. 3 Eect o varying Darcy number on the streamlines (Re = 00, ε = 0.75) Fig. Grid system used in the present study Each side o the model was divided into 80 nodes. Extensive numerical experimentation is perormed to attain grid-independent results. Steady state solution was declared when the relative change in the dependent variable between two consecutive iterations is satisied by the ollowing criterion: γ + 1 γ ϕi, j ϕi, j γ + (10) ϕ Where i, j γ ϕ i, j stands or the dependent variables at iteration γ. IV. RESULTS AND DISCUSSION The characteristics o the low and pressure ields within an artiicial lung were examined by exploring the eects o Darcy number and various models o porous medium. Such ield variables were examined by outlaying the steady state version o the streamline, pressure ield, and velocity vectors. In the current numerical investigation, the ollowing parametric domains o the dimensionless groups were considered: 1 Re 00 and10 6 Da 10. The eect o porosity on the low characteristics was not considered in the present work since its inluence is well documented in the literature. As such ε = 0.75 was assumed in this study. Eect o Darcy number The eect o Darcy number on the streamlines and pressure distribution is illustrated in Figs. 3 and 4. Figure 3 shows that Darcy number has a signiicant eect on the streamlines. As Darcy number decreases, the activities within the enclosure decrease signiicantly. For small values o Darcy numbers, the porous layer is considered less permeable to luid penetration and consequently the luid experiences a pronounced large resistance as it lows through the porous matrix. This results in hindering low activities in the porous region as depicted in Fig. 3. This eect is more pronounced in Fig. 4 which shows Fig. 4 Eect o varying Darcy number on the pressure (Re = 00, ε = 0.75) Eect o Varying the Flow Model or Porous Media on the Streamlines and Pressure Drop Figure 5 illustrate the eect o using dierent low models or porous media such as Darcy s law model, Brinkman s extension, and the generalized model on the streamlines and pressure drop distribution. It is interesting to note rom Fig. 5 that Brinkman s extension o the Darcy model and the generalized model are very close or the conditions used. Figure 5 shows that Darcy s law model exhibits a vortex within the artiicial lung and a lower pressure drop (Fig. 5b) compared with other models. Since the industrial artiicial lungs are characterized by a small Darcy number and low Reynolds number, Darcy s law model may be suicient to solve or the low characteristics within the artiicial lungs. 135

4 International Conerence on Trends in Industrial and Mechanical Engineering (ICTIME'01) March 4-5, 01 Dubai (a) (b) Fig. 5 Eect o varying the low model o porous medium on streamlines and pressure (Re = 00, ε = 0.75) V. CONCLUSION A computational model o an artiicial lung device was developed in this study. The transport equations were solved using the inite element ormulation based on the Galerkin method o weighted residuals. Signiicant dierences in the pressure drop across the artiicial lung and the streamlines were ound in comparing dierent models o porous medium. The impedance (or pressure drop) was ound to be smaller large Darcy number. The numerical results reported in this work can provide inormation on the low properties at low Reynolds number, which is the low condition inside the human body. Moreover, porous media theory permits the study o luid motion across small spaces o variable and complex geometry. ACKNOWLEDGMENTS This work was supported by Frankel Vascular Research Fund and NIH-NHLBI R01HL A. REFERENCES [1] Nield D.A., Bejan A. (1995) Convection in Porous Media, nd Ed., Springer-Verlag, NY. [] Vaai K. (000) Handbook o Porous Media, 1st Ed., Marcel Dekker, Inc., NY. [3] Vaai K. (005) Handbook o Porous Media, nd Ed., Taylor and Francis Group, NY. [4] Hadim H., Vaai K. (000) Overview o Current Computational Studies o Heat Transer in Porous Media and Their Applications- Forced Convection and Multiphase Transport. Advances in Numerical Heat Transer ; , Taylor and Francis, NY. [5] Vaai K., Hadim H. (000), Overview o Current Computational Studies o Heat Transer in Porous Media and Their Applications- Natural Convection and Mixed Convection. Advances in Numerical Heat Transer ; , Taylor and Francis, NY. [6] Yang N., Vaai K. (006) Modeling o Low-Density Lipoprotein (LDL) Transport in the Artery- Eects o Hypertension. In press or Int. J. o Heat and Mass Transer. [7] Ai L., Vaai K. (006) A Coupling Model or Macromolecule Transport in a Stenosed Arterial Wall. In press or Int. J. o Heat and Mass Transer. [8] Khanaer K., Vaai K., Kangarlu A. (003) Computational Modeling o Cerebral Diusion-Application to Stroke Imaging. Magnetic Resonance Imaging 1; [9] Khanaer K., Vaai K., Kangarlu K. (003) Water Diusion in Biomedical Systems as Related to Magnetic Resonance Imaging. Magnetic Resonance Imaging 1; [10] Khaled A.-R A., Vaai K. (003) The Role o Porous Media on Modeling Flow and Heat Transer in Biological Tissues. Int. J. Heat Mass Transer 46; [11] Vaai K., Tien C.L. (1981) Boundary and Inertia Eects on Flow and Heat Transer in Porous Media. Int. J. Heat Mass Transer 4; [1] Vaai K., C.L. Tien (198) Boundary and Inertia Eects on Convective Mass Transer in Porous Media. Int. J. Heat Mass Transer 5; [13] Khanaer, K., Berguer, R., Schlicht, M., Bull, J.: Numerical Modeling o Coil Compaction in the Treatment o Cerebral Aneurysms Using Porous Media Theory. J. 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Prog., 48, pp [6] Vaai, K., 1984, Convective Flow and Heat Transer in Variable- Porosity Media, J. Fluid Mech., 147, pp [7] Vaai, K., 1986, Analysis o the Channeling Eect in Variable Porosity Media, ASME J. Energy Resour. Technol., 108, pp [8] C. Taylor, P. Hood, A numerical solution o the Navier-Stokes equations using inite-element technique, Comput. Fluids 1 (1973) [9] P.M. Gresho, R.L. Lee, R.L. Sani, On the time-dependent solution o the incompressible Navier-Stokes equations in two and three dimensions, in: Recent Adv. Num. Methods in Fluids, Pineridge, Swansea, UK, Khalil Khanaer is an Assistant Research Scientist in the Biomedical Engineering Department at University o Michigan. He has several years o research experience in the ield o computational luid dynamics (CFD) and extensive experience in sotware and algorithm development, including the 136

5 International Conerence on Trends in Industrial and Mechanical Engineering (ICTIME'01) March 4-5, 01 Dubai use o commercial CFD packages. He is currently an associate editor or Special Topics and Reviews in Porous Media Journal and in the Editorial Board o Annals o Vascular Surgery Keith Cook is a Research Assistant Proessor in the University o Michigan Departments o Surgery and Biomedical Engineering. He has 17 years o experience in the design, development, and testing o respiratory support devices. He received the 003 Young Investigator Award and the 007 Medorte Innovation Fellowship rom the American Society o Artiicial Internal Organs or this work. He is currently the editor o the respiratory support section o the American Society o Artiicial Internal Organs Journal. 137