Advertisement

Transport in Porous Media

, Volume 93, Issue 3, pp 517–541 | Cite as

Knudsen Diffusion Through Cylindrical Tubes of Varying Radii: Theory and Monte Carlo Simulations

  • Yong Shi
  • Yong Taek Lee
  • Albert S. KimEmail author
Article

Abstract

In this study, Knudsen diffusion of low-pressure gases of infinite mean free path through various tubes is studied using the integral equation theory (IET), standard diffusion theory, and Monte Carlo (MC) simulations. We investigated the transmission probabilities (TPs) of linearly diverging–converging, sinusoidally bulging, and periodic tubes as compared with TPs of conventional straight cylinders. An exact analytic solution for the TP through the straight cylindrical tube was developed using the standard diffusion theory with a linear concentration approximation. IET for the TPs through the diverging–converging and bulging tubes were developed. MC simulation techniques were applied to calculate TPs through all the tube types azimuthal symmetry of which was held with tube radius changing only along the axial coordinate (z). The linearly diverging–converging and sinusoidally bulging tubes provide noticeably higher TPs than those of the equivalent straight tubes. Periodic tubes show that if the tube length scaled by the equivalent diameter is of an order of or greater than the periodicity coefficient (equal to the number of peaks on the tube wall), then the TP of the periodic tube is larger than that of the equivalent straight tube.

Keywords

Knudsen transport Periodic tube Transmission probability Integral equation theory Monte Carlo simulation 

List of Symbols

\({\bar{v}}\)

Mean molecular speed

β

Angle at the inlet and outlet of the diverging–converging tube in magnitude

\({\varepsilon}\)

Peak amplitude of tube, i.e., a dimensionless distance from the imaginary tube surface of radius r 0 at the mid-point (at z = L/2) to the peak of the diverging–converging or bulging tube

\({\eta_{{\rm dir}^{\rm C}}}\)

Direct TP of Clausing’s analytic solution, taken from Walsh’s work

\({\eta_{\rm dir}^{{\rm SKL}, \, {\rm MC}}}\)

Direct TP calculated using Monte Carlo simulations in this study

\({\eta_{\rm ind}^{{\rm SKL},\,{\rm MC}}}\)

Indirect TP calculated using Monte Carlo simulations in this study

\({\eta_{\rm tot}^{{\rm SKL}}}\)

Total TP as a summation of direct TP by Clausing’s integral equation theory and indirect TP by our theory developed in this study

\({\eta_{{\rm dir}}^{{\rm PP}}}\)

Direct TP of Pollard & Present’s theory for the first-order correction

\({\eta_{{\rm dir}}^{{{\rm W}}, \, {\rm Asym}}}\)

Direct TP of Walsh’s asymptotic form

\({\eta_{{\rm ind}}^{{{\rm C}} \, {\rm , Asym}}}\)

Indirect TP of Clausing’s asymptotic form

\({\eta_{{\rm ind}}^{{\rm K}}}\)

Indirect TP corresponding to the original Knudsen diffusivity, D K

\({\eta_{{\rm ind}}^{{\rm PP}}}\)

Indirect TP of Pollard & Present’s theory for the first-order correction

\({\eta_{{\rm ind}}^{{\rm SKL}}}\)

Indirect TP analytically derived in this study

\({\eta_{{\rm tot}}^{{\rm PP}}}\)

Total TP of Pollard & Present’s theory for the first-order correction

λ

Mean free length of molecules

ω (z)

Escape probability that the molecule at z will pass the (right) outlet without returning to the left reservoir

a

Radius of a straight cylindrical tube

d

Diameter of a straight cylindrical tube

\({d_{\rm eq}^{\rm bulging}}\)

Equivalent diameter of this bulging tube

deq

Equivalent diameter of a tube of an axially varying diameter having the identical void volume and length of a straight tube

\({d_{{\rm eq}}^{{\rm div-conv}}}\)

Equivalent diameter of the conically diverging–converging tube of Eq. 10

DK

Knudsen diffusivity for a semi-infinitely long tube of diameter d

dp (z)

Diameter of a tube as a function of the axial position z

L

Length of a tube

L

Half length of a tube

Lcross/deq

Dimensionless crossing distance at which the TPs of periodic and straight tubes are identical

M

Molecular weight

n (z)

Number concentration of molecules in the tube

n1

Number concentration of molecules at the tube inlet

n2

Number concentration of molecules at the tube outlet

Nw

Number of molecules passing through the central cross-sectional area per unit time

r(z)bulging

Radius of the bulging (sinusoidally diverging–converging) tube

r(z)div-conv

Radius of the conically diverging–converging tube

r

Distance of molecules, traveling without collisions

r0

Inlet and outlet radius of diverging–converting and bulging tubes

r1 (z)

Radius at the left side of the diverging–converting/bulging tube

r2 (z)

Radius at the right side of the diverging–converting/bulging tube

T

Temperature

V

Volume of void spaces in a tube

z

Axial coordinate of a tube

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Allen M.P., Tildesley D.J.: Computer Simulation of Liquids. Oxford University Press, Oxford (1987)Google Scholar
  2. Bhatia S., Nicholson D.: Some pitfalls in the use of the Knudsen equation in modelling diffusion in nanoporous materials. Chem. Eng. Sci. 66(3), 284–293 (2011)CrossRefGoogle Scholar
  3. Bhatia S.K., Jepps O., Nicholson D.: Tractable molecular theory of transport of Lennard–Jones fluids in nanopores. J. Chem. Phys. 120(9), 4472 (2004)CrossRefGoogle Scholar
  4. Bosanquet, C.H.B.: British TA Rept, pp BR–507 (1944)Google Scholar
  5. Brunauer S., Emmett P.H., Teller E.: Adsorption of gases in multimolecular layers. J. Am. Chem. Soc. 60(2), 309–319 (1938)CrossRefGoogle Scholar
  6. Budd P.M., McKeown N.B., Fritsch D.: Free volume and intrinsic microporosity in polymers. J. Mater. Chem. 15, 1977–1986 (2005)CrossRefGoogle Scholar
  7. Calabro V., Jiao B., Drioli E.: Theoretical and experimental study on membrane distillation in the concentration of orange juice. Ind. Eng. Chem. Res. 33(7), 1803–1808 (1994)CrossRefGoogle Scholar
  8. Clausing P.: Uber die stromung sehr verdunnter gase durch rohren von beliebiger lange. Ann. Phys. (Leipzig) 12, 961–989 (1932)Google Scholar
  9. Clausing P.: The flow of highly rarefied gases through tubes of arbitrary length. J. Vac. Sci. Technol. 8(5), 636–756 (1971)CrossRefGoogle Scholar
  10. Curcio E., Drioli E.: Membrane distillation and related operations—a review. Sep. Purif. Rev. 34(1), 35–86 (2005)CrossRefGoogle Scholar
  11. Davis D.H.: Monte Carlo calculation of molecular flow rates through a cylindrical elbow and pipes of other shapes. J. Appl. Phys. 31(7), 1169–1176 (1960)CrossRefGoogle Scholar
  12. Diban N., Voinea O., Urtiaga A., Ortiz I.: Vacuum membrane distillation of the main pear aroma compound: experimental study and mass transfer modeling. J. Membr. Sci. 326(1), 64–75 (2009)CrossRefGoogle Scholar
  13. Ermak D.L., Mccammon J.A.: Brownian dynamics with hydrodynamic interactions. J. Chem. Phys. 69(4), 1352–1360 (1978)CrossRefGoogle Scholar
  14. Evans R. III, Watson G., Mason E.: Gaseous diffusion in porous media at uniform pressure. J. Chem. Phys. 35(6), 2076–2083 (1961)CrossRefGoogle Scholar
  15. Evans R. III, Watson G., Mason E.: Gaseous diffusion in porous media. II. Effect of pressure gradients. J. Chem. Phys. 36(7), 1894–1902 (1962)CrossRefGoogle Scholar
  16. Flory C.A., Cutler L.S.: Integral equation solution of low-pressure transport of gases in capillary tubes. J. Appl. Phys. 73(4), 1561–1569 (1993)CrossRefGoogle Scholar
  17. Haddad O.M., Abuzaid M.M.: Effect of periodically oscillating driving force on basic microflows in porous media. J. Porous Media 9(7), 695–707 (2006)CrossRefGoogle Scholar
  18. Haddad O.M., Abuzaid M.M., Al-Nimr M.A.: Developing free-convection gas flow in a vertical open-ended microchannel filled with porous media. Numer. Heat Transf. A 48(7), 693–710 (2005)CrossRefGoogle Scholar
  19. Haddad O.M., Al-Nimr M.A., Taamneh Y.: Hydrodynamic and thermal behavior of gas flow in microchannels filled with porous media. J. Porous Media 9(5), 403–414 (2006)CrossRefGoogle Scholar
  20. Haddad O.M., Al-Nimr M., Al-Omary J.: Forced convection of gaseous slip-flow in porous micro-channels under Local Thermal Non-Equilibrium conditions. Transp. Porous Media 67(3), 453–471 (2007)CrossRefGoogle Scholar
  21. Iczkowski R.P., Iczkowski R.P., Margrave J.L., Margrave J.L., Robinson S.M., Robinson S.M.: Effusion of gases through conical orifices. J. Phys. Chem. 67(2), 229–233 (1963)CrossRefGoogle Scholar
  22. Karniadakis G., Beskok A.: Micro Flows—Fundamentals and Simulation. Springer, New York (2002)Google Scholar
  23. Knudsen M.: Die gesetze der molekularströmung und der inneren reibungsströmung der gase durch röhren. Ann. Phys. 333(1), 75–130 (1909)CrossRefGoogle Scholar
  24. Knudsen M., Fisher W.: The molecular and the frictional flow of gases in tubes. Phys. Rev. (Ser. I) 31(5), 586–588 (1910)CrossRefGoogle Scholar
  25. Kogan A.: Direct solar thermal splitting of water and on-site separation of the products–ii. experimental feasibility study. Int. J. Hydrogen Energy 23(2), 89–98 (1998)CrossRefGoogle Scholar
  26. Krishna R., van Baten J.M.: A molecular dynamics investigation of the unusual concentration dependencies of Fick diffusivities in silica mesopores. Micropor. Mesopor. Mater. 138(1-3), 228–234 (2011)CrossRefGoogle Scholar
  27. Lawson K.W., Lloyd D.R.: Membrane distillation. II. Direct contact MD. J. Membr. Sci. 120(1), 123–133 (1996)CrossRefGoogle Scholar
  28. Lim Y.I., Bhatia S.: Simulation of methane permeability in carbon slit pores. J. Membr. Sci. 369(1-2), 319–328 (2011)CrossRefGoogle Scholar
  29. Mason E., Malinauskas A., Evans R. III: Flow and diffusion of gases in porous media. J. Chem. Phys. 46(8), 3199–3216 (1967)CrossRefGoogle Scholar
  30. Mulder M.: Basic Principles of Membrane Technology. 2nd edn. Kluwer, Dordecht (1996)CrossRefGoogle Scholar
  31. Park H.B., Jung C.H., Lee Y.M., Hill A.J., Pas S.J., Mudie S.T., Wagner E.V., Freeman B.D., Cookson D.J.: Polymers with cavities tuned for fast selective transport of small molecules and ions. Science 318(5848), 254–258 (2007)CrossRefGoogle Scholar
  32. Pollard W.G., Present R.D.: On gaseous self-diffusion in long capillary tubes. Phys. Rev. 73(7), 762–774 (1948)CrossRefGoogle Scholar
  33. Present R.D., De Bethune A.J.: Separation of a gas mixture flowing through a long tube at low pressure. Phys. Rev. 75(7), 1050–1057 (1949)CrossRefGoogle Scholar
  34. Qtaishat M., Matsuura T., Kruczek B., Khayet M.: Heat and mass transfer analysis in direct contact membrane distillation. Desalination 219(1-3), 272–292 (2008)CrossRefGoogle Scholar
  35. Reif F.: Fundamentals of Statistical and Thermal Physics. McGraw-Hill, New York (1965)Google Scholar
  36. Reynolds, T.W., Richley, E.A.: Flux patterns resulting from free-molecule flow through converging and diverging slots. NASA Tech Note TN D-1864, pp 1–69 (1964)Google Scholar
  37. Reynolds, T.W., Richley, E.A.: Free molecule flow and surface diffusion through slots and tubes—a summary. NASA Technical Report TR R-255 (1967)Google Scholar
  38. Richley, E.A., Reynolds, T.W.: Numerical solutions of free-molecule flow in converging and diverging tubes and slots. NASA Tech Note TN D-2330, pp 1–45 (1964)Google Scholar
  39. Ruthven D.M., Desisto W.J., Higgins S.: Diffusion in a mesoporous silica membrane: validity of the Knudsen diffusion model. Chem. Eng. Sci. 64, 3201–3203 (2009)CrossRefGoogle Scholar
  40. Steckelmacher W.: Knudsen flow 75 years on: the current state of the art for flow of rarefied gases in tubes and systems. Rep. Prog. Phys. 49, 1083–1107 (1986)CrossRefGoogle Scholar
  41. Talley W.K., Whitaker S.: Monte Carlo analysis of Knudsen flow. J. Comput. Phys. 4(3), 389–410 (1969)CrossRefGoogle Scholar
  42. Villani S.: Isotope Separation. American Nuclear Society, Hinsdale (1976)Google Scholar
  43. Walsh J.W.T.: Radiation from a perfectly diffusing circular disc (part I). Proc. Phys. Soc. Lond. 32, 59–71 (1919)CrossRefGoogle Scholar
  44. Zhang J., Gray S., Li J.D.: Modelling heat and mass transfers in DCMD using compressible membranes. J. Membr. Sci. 387(388), 7–16 (2012)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2012

Authors and Affiliations

  1. 1.Civil and Environmental EngineeringUniversity of Hawaii at ManoaHonoluluUSA
  2. 2.Chemical Engineering, College of EngineeringKyung Hee UniversityGyeonggi-doSouth Korea

Personalised recommendations