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Journal of Biomechanical Engineering | Volume 128 | Issue 1 | TECHNICAL PAPERS
TECHNICAL PAPERS: Fluids/Heat/Transport

An Axisymmetric Single-Path Model for Gas Transport in the Conducting Airways

[+] Author and Article Information
Srinath Madasu1

Department of Chemical Engineering,  The Pennsylvania State University, University Park, PA 16802

Ali Borhan, James S. Ultman

Department of Chemical Engineering,  The Pennsylvania State University, University Park, PA 16802

1

E-mail: sum13@psu.edu

J Biomech Eng 128(1), 69-75 (Aug 03, 2005) (7 pages) doi:10.1115/1.2133762 History: Received April 21, 2005; Revised August 03, 2005
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In conventional one-dimensional single-path models, radially averaged concentration is calculated as a function of time and longitudinal position in the lungs, and coupled convection and diffusion are accounted for with a dispersion coefficient. The axisymmetric single-path model developed in this paper is a two-dimensional model that incorporates convective-diffusion processes in a more fundamental manner by simultaneously solving the Navier-Stokes and continuity equations with the convection-diffusion equation. A single airway path was represented by a series of straight tube segments interconnected by leaky transition regions that provide for flow loss at the airway bifurcations. As a sample application, the model equations were solved by a finite element method to predict the unsteady state dispersion of an inhaled pulse of inert gas along an airway path having dimensions consistent with Weibel’s symmetric airway geometry. Assuming steady, incompressible, and laminar flow, a finite element analysis was used to solve for the axisymmetric pressure, velocity and concentration fields. The dispersion calculated from these numerical solutions exhibited good qualitative agreement with the experimental values, but quantitatively was in error by 20%–30% due to the assumption of axial symmetry and the inability of the model to capture the complex recirculatory flows near bifurcations.
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Copyright © 2006 by American Society of Mechanical Engineers
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References

1
Cumming, G., Crank, J., Horsefield, J. K., and Parker, I., 1966, “Gaseous Diffusion in the Airways of the Human Lung,” Respir. Physiol., 1 , pp. 58–74.
2
La Force, R. C., and Lewis, B. J., 1970, “Diffusional Transport in the Human Lung,” J. Appl. Physiol., 28 , pp. 291–298.
3
Weibel, E. R., 1963, "Morphometry of the Human Lung", Academic Press, New York.
4
Scherer, P. W., Shendalman, L. H., Greene, N. M., and Bouhuys, A., 1975, “Measurement of Axial Diffusivities in a Model of the Bronchial Airways,” J. Appl. Physiol., 38 , pp. 719–723.
5
Engel, L. A., and Paiva, M., 1985, "Gas Mixing and Distribution in the Lung", Marcel Dekker, Inc., New York.
6
Paiva, M., Lacquet, L. M., and Vander Linden, L. P., 1976, “Gas Transport in a Model Derived From Hansen-Ampaya Anatomical Data of the Human Lung,” J. Appl. Physiol., 41 , pp. 115–119.
7
Miller, F. J., Overton, J. H., Jaskot, R. J., and Menzel, D. B., 1985, “A Model of the Regional Uptake of Gaseous Pollutants in the Lung. I. The Sensitivity of the Uptake of Ozone in the Human Lung to Lower Respiratory Tract,” Toxicol. Appl. Pharmacol., 79 , pp. 11–27.
8
Zhao, Y., Brunskill, C. T., and Lieber, B. B., 1997, “Inspiratory and Expiratory Steady Flow Analysis in a Model Symmetrically Bifurcating Airway,” ASME J. Biomech. Eng., 119 , pp. 52–58.
9
Oden, J. T., and Carey., G. F., 1983, "Finite Elements, Mathematical Aspects", Prentice Hall, Englewood Cliffs, NJ, Vol. IV .
10
Brooks, A. N., and Hughes, T. J. R., 1982, “Streamline Upwind∕Petrov-Galerkin Formulations for Convection Dominated Flows With Particular Emphasis on the Incompressible Navier-Stokes Equations,” Comput. Methods Appl. Mech. Eng.  [CrossRef], 32 , pp. 199–259.
11
Belytschko, T., Liu, W. K., and Moran, B., 2000, "Nonlinear Finite Elements for Continua and Structures", John Wiley & Sons, New York.
12
Donea, J., 1984, “Recent Advances in Computational Methods for Steady and Transient Transport Problems,” Nucl. Eng. Des., 80 , pp. 141–162.
13
Kundert, K. S., and Vincentelli., A. S., 1988, A Sparse Linear Equation Solver , University of California, Berkeley, Version 1.3a.
14
Simone, A. F., Ultman, J. S., and Ben Jebria, A., 1988, “Bronchial Distribution of Gas Mixing in a Model of the Upper and Central Airways,” J. Appl. Physiol., 65 , pp. 1693–1702.
15
Taylor, G. I., 1953, “Dispersion of Soluble Matter in Solvent Flowing Slowly Through a Pipe,” Proc. R. Soc. London, Ser. A, 219A , pp. 186–203.
16
Gill, W. N., and Sankarasubramanian, R., 1970, “Dispersion of Soluble Matter in Solvent Flowing Slowly Through a Pipe,” Proc. R. Soc. London, Ser. A, 316A , pp. 341–350.

Figures

Grahic Jump Location
+Schematic of a model for the lung anatomy
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Schematic of a model for the lung anatomy
Figure 1

Schematic of a model for the lung anatomy

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+Physical domain (a) and computational domain (b) of the ASPM. Ri is the radius of a cylindrical branch i, and Ti indicates a conical transition region i located between branch i−1 and branch i.
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Physical domain (a) and computational domain (b) of the ASPM. Ri is the radius of a cylindrical branch i, and Ti indicates a conical transition region i located between branch i−1 and branch i.
Figure 2

Physical domain (a) and computational domain (b) of the ASPM. Ri is the radius of a cylindrical branch i, and Ti indicates a conical transition region i located between branch i−1 and branch i.

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+Distribution of nodes on a rectangular element used in the computation of the physical variables; solid circles represent the velocity, concentration nodes and open circles represent the pressure nodes
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Distribution of nodes on a rectangular element used in the computation of the physical variables; solid circles represent the velocity, concentration nodes and open circles represent the pressure nodes
Figure 3

Distribution of nodes on a rectangular element used in the computation of the physical variables; solid circles represent the velocity, concentration nodes and open circles represent the pressure nodes

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+Mesh of quadrilateral elements used in the mesh refinement study. This ASPM geometry corresponds to the trachea and first generation of the three-generation physical model used by Simone (14)
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Mesh of quadrilateral elements used in the mesh refinement study. This ASPM geometry corresponds to the trachea and first generation of the three-generation physical model used by Simone (14)
Figure 4

Mesh of quadrilateral elements used in the mesh refinement study. This ASPM geometry corresponds to the trachea and first generation of the three-generation physical model used by Simone (14)

Grahic Jump Location
+Contour plots of axial velocity (a), radial velocity (b), pressure (c), and stream function (d) near the first leak of the five-generation ASPM for Re=365
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Contour plots of axial velocity (a), radial velocity (b), pressure (c), and stream function (d) near the first leak of the five-generation ASPM for Re=365
Figure 5

Contour plots of axial velocity (a), radial velocity (b), pressure (c), and stream function (d) near the first leak of the five-generation ASPM for Re=365

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+Axial distribution of pressure at the symmetry line of the five-generation ASPM for Re=365. Vertical lines indicate the location of the transition regions. Axial distance and pressure are plotted as dimensionless quantities.
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Axial distribution of pressure at the symmetry line of the five-generation ASPM for Re=365. Vertical lines indicate the location of the transition regions. Axial distance and pressure are plotted as dimensionless quantities.
Figure 6

Axial distribution of pressure at the symmetry line of the five-generation ASPM for Re=365. Vertical lines indicate the location of the transition regions. Axial distance and pressure are plotted as dimensionless quantities.

Grahic Jump Location
+Radially averaged concentration profiles corresponding to the experiments of Scherer (4) at Re=182 and Pe=306. The inlet concentration curve at port 2 (a) and output concentration profile at port 3 (in the middle of the fifth generation branch) (b) are shown when a rectangular pulse of 0.3‐second width is introduced at port 2.
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Radially averaged concentration profiles corresponding to the experiments of Scherer (4) at Re=182 and Pe=306. The inlet concentration curve at port 2 (a) and output concentration profile at port 3 (in the middle of the fifth generation branch) (b) are shown when a rectangular pulse of 0.3‐second width is introduced at port 2.
Figure 7

Radially averaged concentration profiles corresponding to the experiments of Scherer (4) at Re=182 and Pe=306. The inlet concentration curve at port 2 (a) and output concentration profile at port 3 (in the middle of the fifth generation branch) (b) are shown when a rectangular pulse of 0.3‐second width is introduced at port 2.

Grahic Jump Location
+Comparison of ASPM predictions to experimental data for benzene-inert gas dispersion (4), using two different leak concentrations at the transition region wall in the simulations, the bulk average concentration and the wall concentration
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Comparison of ASPM predictions to experimental data for benzene-inert gas dispersion (4), using two different leak concentrations at the transition region wall in the simulations, the bulk average concentration and the wall concentration
Figure 8

Comparison of ASPM predictions to experimental data for benzene-inert gas dispersion (4), using two different leak concentrations at the transition region wall in the simulations, the bulk average concentration and the wall concentration

Grahic Jump Location
+Comparison of ASPM predictions to experimental data for helium-inert gas dispersion (14) in terms of the difference in residence times (a) and variances (b) between the trachea and third-generation branches
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Comparison of ASPM predictions to experimental data for helium-inert gas dispersion (14) in terms of the difference in residence times (a) and variances (b) between the trachea and third-generation branches
Figure 9

Comparison of ASPM predictions to experimental data for helium-inert gas dispersion (14) in terms of the difference in residence times (a) and variances (b) between the trachea and third-generation branches

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