Transport in Porous Media

, Volume 19, Issue 1, pp 15–36 | Cite as

The viscous air flow pattern in the Stefan diffusion tube

Application to the mutual diffusion in a porous media
  • M. Benkhalifa
  • G. Arnaud
  • J. -P. Fohr


In this paper, we consider the problem of the binary viscous diffusion of vapour through a Stefan tube, which is the model of an elementary capillary. While some preceding results in particular cases supposed parabolic velocity profiles and showed air recirculation, we treat here the general problem of a tube of finite length, submitted to a double viscous diffusion of vapour and air from a liquid surface. The movement of gas is expressed with conservation equations and ideal gas equations. The following added restrictions: constant temperature, no buyoancy effect, no inertial forces, are compatible with a capillary. A numerical solution based on the control volume method is obtained at every point in the tube. The results give the vapour and air flux, describe the circulation pattern and show that the vapour profile of concentration is level. In the lower part of the cylindrical tube space, over a distance of the length of a radius an important radial movement occurs, due to the recirculation of air which changes direction once it reaches the liquid surface. The velocity profile of the gas flow then becomes parabolic in the upper part of the tube.

In order to easily obtain a numerical solution, the system of dimensionless equations is expanded to a series and transformed into a set of sub-systems. The little parameter used for this expansion is tied to the vapour concentration on the liquid surface. The solution of the sub-system of order zero, which is easier to compute, represents a good approximation of the complete solution. These solutions are situated in comparison with the Stefan diffusion and show that the influence of the viscous effect on the vapour flux is limited to a few percents.

In order to apply the results to porous media where the pores are not so regular, we consider at last the diffusion in a tube including a contracted section in the middle of the tube. Since the diffusion paths are longer, the vapour flux is reduced, while the viscosity effect becomes more considerable. The reduction of the air flux is more significant than that of the vapour. This part of the study provides a better understanding of the diffusion through the pits at the wall fiber, and gives data for the air flux which permeates into the oak wood and produces tannin oxidation and thus discolouration.

Key words

Viscous binar diffusion Stefan tube air flux capillary with a contracted section 



ρv/ρ vapour concentration


vapour concentration in equilibrium with its liquid phase


coefficient of molecular diffusion of a vapour in air (m2/s)


vector density of mass flux of dry air (kg/m2s)


vector density of mass flux of vapour (kg/m2s)


Capillary length (m)


dry air molar mass (kg/mole)


Vapour molar mass (kg/mole)


atmospheric pressure


gas mixture total pressure (Pascal)


Ideal gas constant (J/mole K)




tube or capillary radius (m)


Temperature (K)


axial component ofV


gas mixture velocity vector (m/s)


radial component ofV

Greek Letters


density of gas (kg/m3)


gas mixture dynamic viscosity (kg/ms)

Numerical Values of Parameters


3×10−5m2/s (water vapour in air)










8.31 J/moleK




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Copyright information

© Kluwer Academic Publishers 1995

Authors and Affiliations

  • M. Benkhalifa
    • 1
  • G. Arnaud
    • 1
  • J. -P. Fohr
    • 1
  1. 1.Laboratoire d'Etudes ThermiquesPoitiersFrance

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