The effect on mass transfer of momentum and concentration boundary layers at the entrance region of a slit with a nanofiltration membrane wall

Geraldes, Vı́tor; Semião, Viriato; Pinho, Maria Norberta de

doi:10.1016/s0009-2509(01)00441-9

Cited by 56 publications

(47 citation statements)

References 17 publications

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“…Our initial motivation for considering junctions between porous and solid walls lay in the internal flow geometry of spiral wound membrane modules [1][2][3][4][5][6][7], where the pressure singularities uncovered here and in the preceding paper by Nitsche and Parthasarathi [8] would be expected to couple with osmotic effects and thereby affect the salt concentration profile in desalination by reverse osmosis. Having deployed an algebraic criterion for ascertaining the radial exponent of the leading inner similarity solution at arbitrary wedge angle, we then combined numerical analysis with asymptotic expansions to map out the flow field at all radii for three specific wedge angles: 45, 90, and 135 degrees.…”

Section: Discussionmentioning

confidence: 96%

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Stokes flow singularity at a corner joining solid and porous walls at arbitrary angle

Nitsche

2017

J Eng Math

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Motivated by the internal flow geometry of spiral-wound membrane modules with ladder-type spacers, we consider the Stokes flow singularity at a corner that joins porous and solid walls at arbitrary wedge angle Θ. Seepage flux through the porous wall is coupled to the pressure field by Darcy's law; slip is described by a variant of the Beavers-Joseph boundary condition. On a macroscopic, outer length scale, the singularity appears like a jump discontinuity in the normal velocity, characterized by a non-integrable 1/r divergence of the pressure. For arbitrary Θ, we develop an algebraic criterion to determine the admissible radial exponent(s) in a leading, inner similarity solution-which represents a weaker, integrable singularity in the pressure. A complete map of exponent versus Θ is provided for 0 < Θ < 2π : this has an intricate structure with infinitely many solution branches clustering around Θ = π and Θ = 2π . By generalizing the similarity form with a (ln r ) term and iterating on the slip and seepage conditions, we can carry the outer and inner power series to arbitrarily high order. Nevertheless, a numerical splice is required in between. For this purpose, we apply an iterative, numerical-asymptotic patching scheme described by Nitsche and Parthasarathi (J Fluid Mech 713:183-215, 2012). Detailed velocity and pressure profiles are calculated for three wedge angles (Θ = 3π/4, π/2, π/4) and two dimensionless slip lengths (σ = 20, 40). The general trends for decreasing wedge angle are (i) weakening of the pressure singularity, (ii) increasing magnitude of the radial component of velocity, and (iii) movement of the inner-outer transition farther from the corner. Wedges with Θ < π are seen to differ fundamentally from the flat wedge (Θ = π ) previously considered by Nitsche and Parthasarathi (2012).Keywords Low-Reynolds-number flow · Porous walls · Generalized similarity solutions · Boundary integral method Electronic Supplementary material The online version of this article

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Section: Discussionmentioning

confidence: 96%

“…1). Within spiral-wound membrane modules, such a geometry can be found at the junction between the membrane and (1) the sealed entry or exit section of the wall [1,2], or (2) a transverse filament of a ladder-type spacer [3][4][5][6][7]. These two cases are distinguished by the respective wedge angles Θ = π versus Θ = π/2.…”

Section: Introductionmentioning

confidence: 99%

Stokes flow singularity at a corner joining solid and porous walls at arbitrary angle

Nitsche

2017

J Eng Math

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“…The slit-type configuration represents the basic model to characterize the flow in industrial membrane modules, thus restricting the flow to laminar conditions and causing severe problems of CP. Geraldes et al [18] proposed a correlation between the concentration and the hydrodynamic boundary layer thickness which gives an insight into the mechanisms involved in the growth of the boundary layers in the nanofiltration (NF) processes. Wiley and Fletcher [19] proposed a general purpose CFD model of concentration polarization and fluid flow.…”

Section: Introductionmentioning

confidence: 99%

Visualization and modeling of the polarization layer in crossflow reverse osmosis in a slit-type channel

Salcedo‐Díaz

García-Algado

Garcia‐Rodriguez

et al. 2014

Journal of Membrane Science

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The evolution of the concentration polarization layer during crossflow reverse osmosis in a slit channel has been studied. Digital Holographic Interferometry allows visualizing the polarization layer as an interference fringe pattern. An especial module with four windows has been designed to see the development of the polarization layer along the membrane channel. Several experiments with a constant transmembrane pressure of 6 bar, three different feed concentrations (3, 6 and 9 kg/m 3 ) and three different Re (13, 38 and 111) had been carried out. The process has been modelled simulating the experimental conditions. The computed results were found to be consistent with the experimental ones. All the experiments show a continuous increase of the polarization layer along the channel, regardless of the crossflow velocity, except near the outlet of the cell due to an edge effect. This increase in the polarization layer is greater at lower Re, although it does not influence very much the permeate flux. In contrast, what substantially affects the permeate flux, much more than Re number, is the feed concentration due to its osmotic pressure.

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“…A more comprehensive model, which could predict the flow behavior in the vicinity of the porous surface with variable permeability (Das et al, 2002) was reported. A lot of work has been reported on impact of physical processes on the flow in membrane slits or spirals, where circulating flows are often in laminar range (Geraldes et al, 2002). A general purpose CFD model was proposed by Wiley and Fletcher (2003), which combined the concentration polarization phenomena with the hydrodynamics characteristics and was verified experimentally.…”

Section: Modelling and Simulation In Membrane Separation Processesmentioning

confidence: 99%

Fundamentals of Membrane Processes

Kumar¹

2012

Membrane Technology and Environmental Applications

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The effect on mass transfer of momentum and concentration boundary layers at the entrance region of a slit with a nanofiltration membrane wall

Cited by 56 publications

References 17 publications

Stokes flow singularity at a corner joining solid and porous walls at arbitrary angle

Stokes flow singularity at a corner joining solid and porous walls at arbitrary angle

Visualization and modeling of the polarization layer in crossflow reverse osmosis in a slit-type channel

Fundamentals of Membrane Processes

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