Velocity Profiles in Porous-Walled Ducts

Makiya (1970) has derived the following equations for capillary mass transfer of a nonelectrolyte based on irreversible thermodynamics : Solute d dz Specifying the initial conditions of concentration, volumetric 00w rate, and pressure, he solved these equations for selected membrane parameters using the IBM-CSMP program (1969). In this note simplifications of the above equations are made which permit analytical solutions in dimensionless form for ideal semipermeable membranes and for pure water transport.An ideal semipermeable membrane transfers no solute. This condition requires that the phenomenological coefficients be equal Lp = LPD = LD as shown by Katchalsky and Curran (1965). Equation ( 1 ) is consistent with this definition since it shows that QWCA is constant (= QwoCAo) down the capillary indicating no solute mass transfer in the radial direction.Since the kinetic energy of a fluid flowing in a capillary is small (low Reynolds number) compared to frictional effects, Equation ( 3 ) can be simplified toUpon substitution of Equation (5) for P and CA = (QwoCAO)/QW into Equation ( 2 ) one getsThe term 1/Qw is nonlinear, and no solutions were found. Therefore 1/Qw was expanded in a Taylor series, and the first two terms were taken to give tho following equation:

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“…Comparison of these results also shows that the arbitrary decay constant (uKsL of Kozinski et al (1970) can be expressed as…”

Section: Applicationsmentioning

confidence: 81%

“…Kozinski et al (1970) presented solutions for low Reynolds number flow of an incompressible Newtonian fluid in tubes and slits with permeation through the walls. They assumed the radial velocity at the wall decreased exponentially with axial distance.…”

Section: Applicationsmentioning

confidence: 99%

Capillary mass transfer

1972

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“…1) giving: (9) Upon integration of Eq. (9) from the wall of the tube (r = R; u = 0) to a distance r from the center where the axial velocity u is to be evaluated, gives: (10) Now, the overall volumetric fl ow rate of the fl uid can be obtained by integrating Eq. (10) as follows: Substituting Eq.…”

Section: Modeling Flow Through Tubular Hollow--fibersmentioning

confidence: 99%

“…Classical Papers of Berman 8 , Yuan and Finkelstein 9 , and Kozinski et al 10 are the basis of most of these CFD based analyses of fl ow through porous tube or duct. These authors presented analytical solutions based on perturbation treatment of fully developed laminar fl ow with the parabolic velocity profi le in rectangular and tubular channels (assuming constant fl uid properties) by solving steady state two-dimensional Navier-Stokes equation (i.e., the equation of motion with constant density and constant viscosity) along with the equation of continuity.…”

Section: Introductionmentioning

confidence: 99%

Flow behavior in weakly permeable micro-tube with varying viscosity near the wall

Gupta

Kumar

Chand

2017

Polish Journal of Chemical Technology

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Weakly permeable micro-tubes are employed in many applications involving heat and/or mass transfer. During these processes, either solute concentration builds up (mass transfer) or steep change in temperature (heat transfer) takes place near the permeable wall causing a change in the viscosity of the fl uid. Results of the present work suggest that such change in viscosity leads to a considerable alteration in the fl ow behavior, and the commonly assumed parabolic velocity profi le no longer exists. To solve the problem numerically, the equation of motion was simplifi ed to represent permeation of incompressible, Newtonian fl uid with changing viscosity through a micro-tube. Even after considerable simplifi cation, the accuracy of the results was the same as that obtained by previously reported results for some specifi c cases using rigorous formulation. The algorithm developed in the present work is found to be numerically robust and simple so that it can be easily integrated with other simulations.

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“…This was because the wall velocity is proportional to the trans-membrane pressure profile and increases dramatically along the axial direction. 53 Because Karode's analysis was based on fluid mass balance, as extended from the Hagen-Poiseuille equation, only mean fluid velocities were investigated without detailed analysis of the internal flow. Aforementioned theoretical studies (in comparison with this work) on quiescent fluid flows through porous ducts are summarized in Table 1 for various factors.…”

Section: Introductionmentioning

confidence: 99%

Laminar flow with injection through a long dead‐end cylindrical porous tube: Application to a hollow fiber membrane

Kim

Lee

2010

AIChE Journal

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Effects of membrane length and hydraulic resistance on the steady-state laminar flow of a fluid with injection in a dead-end cylindrical porous tube have been investigated using the perturbation approach of the Navier-Stokes and continuity equations. Analytic solutions of the dynamic equations, reduced to non linear differential equations, were obtained and applied to submerged, dead-end hollow fiber membranes of large resistances and small wall Reynolds numbers. The velocity and pressure profiles were solved using the method of separation of variables as with physical properties of the membrane and fluid and the wall velocity at the dead end. The constant flux approximation was found to be valid only for a short membrane with a large hydraulic resistance. The constant permeability approximation used in this study is universal for a membrane of an arbitrary length, through which the fluid velocities and pressure increase exponentially from the dead end to the open end.

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Velocity Profiles in Porous-Walled Ducts

Abstract: This equation is not intended as a general correlation for packing media, It is presented to stress the fact that the Ergun equation, advanced by several textbooks without restriction to flow range, may not be applicable for values of Re/ (1 -e) greater than about 500.

Cited by 89 publications

References 7 publications

Capillary mass transfer

Capillary mass transfer

Flow behavior in weakly permeable micro-tube with varying viscosity near the wall

Laminar flow with injection through a long dead‐end cylindrical porous tube: Application to a hollow fiber membrane

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