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


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.


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

List of Symbols


Mean molecular speed


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


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


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


Radius of a straight cylindrical tube


Diameter of a straight cylindrical tube

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

Equivalent diameter of this bulging tube


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


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


Length of a tube


Half length of a tube


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


Molecular weight

n (z)

Number concentration of molecules in the tube


Number concentration of molecules at the tube inlet


Number concentration of molecules at the tube outlet


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


Radius of the bulging (sinusoidally diverging–converging) tube


Radius of the conically diverging–converging tube


Distance of molecules, traveling without collisions


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




Volume of void spaces in a tube


Axial coordinate of a tube


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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

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