Simple Theoretical Results on Reversible Fouling in Cross-Flow Membrane Filtration
In cross-flow membrane filtration, fouling results from material deposit which clogs the membrane inner surface. This hinders filtration, which experiences the so-called limiting flux. Among the models proposed by the literature, we retain a simple one: a steady-state reversible fouling is modelled...
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| Format: | Article |
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MDPI AG
2019-04-01
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| Series: | Membranes |
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| Online Access: | https://www.mdpi.com/2077-0375/9/4/48 |
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| author | Pierre Haldenwang Braulio Bernales Pierrette Guichardon Nelson Ibaseta |
| author_facet | Pierre Haldenwang Braulio Bernales Pierrette Guichardon Nelson Ibaseta |
| author_sort | Pierre Haldenwang |
| collection | DOAJ |
| description | In cross-flow membrane filtration, fouling results from material deposit which clogs the membrane inner surface. This hinders filtration, which experiences the so-called limiting flux. Among the models proposed by the literature, we retain a simple one: a steady-state reversible fouling is modelled with the use of a single additional parameter, i.e., <inline-formula> <math display="inline"> <semantics> <msub> <mi>N</mi> <mi>d</mi> </msub> </semantics> </math> </inline-formula>, the ratio of the critical concentration for deposition to the feed concentration at inlet. To focus on fouling, viscous pressure drop and osmotic (counter-)pressure have been chosen low. It results in a minimal model of fouling. Solved thoroughly with the numerical means appropriate to enforce the nonlinear coupling between permeation and concentration polarization, the model delivers novel information. It first shows that permeation is utterly governed by solute transfer, the relevant non-dimensional quantities being hence limited to <inline-formula> <math display="inline"> <semantics> <msub> <mi>N</mi> <mi>d</mi> </msub> </semantics> </math> </inline-formula> and <inline-formula> <math display="inline"> <semantics> <mrow> <mi>P</mi> <msub> <mi>e</mi> <mrow> <mi>i</mi> <mi>n</mi> </mrow> </msub> </mrow> </semantics> </math> </inline-formula>, the transverse Péclet number. Furthermore, when the role played by <inline-formula> <math display="inline"> <semantics> <msub> <mi>N</mi> <mi>d</mi> </msub> </semantics> </math> </inline-formula> and moderate <inline-formula> <math display="inline"> <semantics> <mrow> <mi>P</mi> <msub> <mi>e</mi> <mrow> <mi>i</mi> <mi>n</mi> </mrow> </msub> </mrow> </semantics> </math> </inline-formula> (say <inline-formula> <math display="inline"> <semantics> <mrow> <mi>P</mi> <msub> <mi>e</mi> <mrow> <mi>i</mi> <mi>n</mi> </mrow> </msub> <mo><</mo> <mn>40</mn> </mrow> </semantics> </math> </inline-formula>) is investigated, all results can be interpreted with the use of a single non-dimensional parameter, <inline-formula> <math display="inline"> <semantics> <msub> <mi>F</mi> <mi>l</mi> </msub> </semantics> </math> </inline-formula>, the so-called fouling number, which simply reads <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mi>F</mi> <mi>l</mi> </msub> <mo>≡</mo> <mi>P</mi> <msub> <mi>e</mi> <mrow> <mi>i</mi> <mi>n</mi> </mrow> </msub> <msubsup> <mi>N</mi> <mi>d</mi> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msubsup> </mrow> </semantics> </math> </inline-formula>. Now rendered possible, the overall fit of the numerical data allows us to put forward analytical final expressions, which involve all the physical parameters and allow us to retrieve the experimental trends. |
| first_indexed | 2024-03-12T06:35:29Z |
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| id | doaj.art-d5a34ffc4a984ce0b803472509151cf1 |
| institution | Directory Open Access Journal |
| issn | 2077-0375 |
| language | English |
| last_indexed | 2024-03-12T06:35:29Z |
| publishDate | 2019-04-01 |
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| series | Membranes |
| spelling | doaj.art-d5a34ffc4a984ce0b803472509151cf12023-09-03T01:21:27ZengMDPI AGMembranes2077-03752019-04-01944810.3390/membranes9040048membranes9040048Simple Theoretical Results on Reversible Fouling in Cross-Flow Membrane FiltrationPierre Haldenwang0Braulio Bernales1Pierrette Guichardon2Nelson Ibaseta3Aix Marseille Univ, CNRS, Centrale Marseille, M2P2, 38 rue Joliot-Curie, 13451 Marseilles, FranceAix Marseille Univ, CNRS, Centrale Marseille, M2P2, 38 rue Joliot-Curie, 13451 Marseilles, FranceAix Marseille Univ, CNRS, Centrale Marseille, M2P2, 38 rue Joliot-Curie, 13451 Marseilles, FranceAix Marseille Univ, CNRS, Centrale Marseille, M2P2, 38 rue Joliot-Curie, 13451 Marseilles, FranceIn cross-flow membrane filtration, fouling results from material deposit which clogs the membrane inner surface. This hinders filtration, which experiences the so-called limiting flux. Among the models proposed by the literature, we retain a simple one: a steady-state reversible fouling is modelled with the use of a single additional parameter, i.e., <inline-formula> <math display="inline"> <semantics> <msub> <mi>N</mi> <mi>d</mi> </msub> </semantics> </math> </inline-formula>, the ratio of the critical concentration for deposition to the feed concentration at inlet. To focus on fouling, viscous pressure drop and osmotic (counter-)pressure have been chosen low. It results in a minimal model of fouling. Solved thoroughly with the numerical means appropriate to enforce the nonlinear coupling between permeation and concentration polarization, the model delivers novel information. It first shows that permeation is utterly governed by solute transfer, the relevant non-dimensional quantities being hence limited to <inline-formula> <math display="inline"> <semantics> <msub> <mi>N</mi> <mi>d</mi> </msub> </semantics> </math> </inline-formula> and <inline-formula> <math display="inline"> <semantics> <mrow> <mi>P</mi> <msub> <mi>e</mi> <mrow> <mi>i</mi> <mi>n</mi> </mrow> </msub> </mrow> </semantics> </math> </inline-formula>, the transverse Péclet number. Furthermore, when the role played by <inline-formula> <math display="inline"> <semantics> <msub> <mi>N</mi> <mi>d</mi> </msub> </semantics> </math> </inline-formula> and moderate <inline-formula> <math display="inline"> <semantics> <mrow> <mi>P</mi> <msub> <mi>e</mi> <mrow> <mi>i</mi> <mi>n</mi> </mrow> </msub> </mrow> </semantics> </math> </inline-formula> (say <inline-formula> <math display="inline"> <semantics> <mrow> <mi>P</mi> <msub> <mi>e</mi> <mrow> <mi>i</mi> <mi>n</mi> </mrow> </msub> <mo><</mo> <mn>40</mn> </mrow> </semantics> </math> </inline-formula>) is investigated, all results can be interpreted with the use of a single non-dimensional parameter, <inline-formula> <math display="inline"> <semantics> <msub> <mi>F</mi> <mi>l</mi> </msub> </semantics> </math> </inline-formula>, the so-called fouling number, which simply reads <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mi>F</mi> <mi>l</mi> </msub> <mo>≡</mo> <mi>P</mi> <msub> <mi>e</mi> <mrow> <mi>i</mi> <mi>n</mi> </mrow> </msub> <msubsup> <mi>N</mi> <mi>d</mi> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msubsup> </mrow> </semantics> </math> </inline-formula>. Now rendered possible, the overall fit of the numerical data allows us to put forward analytical final expressions, which involve all the physical parameters and allow us to retrieve the experimental trends.https://www.mdpi.com/2077-0375/9/4/48membrane separationcross-flow filtrationpolarization of concentrationlimiting fluxreversible foulingStarling–Darcy boundary conditions |
| spellingShingle | Pierre Haldenwang Braulio Bernales Pierrette Guichardon Nelson Ibaseta Simple Theoretical Results on Reversible Fouling in Cross-Flow Membrane Filtration Membranes membrane separation cross-flow filtration polarization of concentration limiting flux reversible fouling Starling–Darcy boundary conditions |
| title | Simple Theoretical Results on Reversible Fouling in Cross-Flow Membrane Filtration |
| title_full | Simple Theoretical Results on Reversible Fouling in Cross-Flow Membrane Filtration |
| title_fullStr | Simple Theoretical Results on Reversible Fouling in Cross-Flow Membrane Filtration |
| title_full_unstemmed | Simple Theoretical Results on Reversible Fouling in Cross-Flow Membrane Filtration |
| title_short | Simple Theoretical Results on Reversible Fouling in Cross-Flow Membrane Filtration |
| title_sort | simple theoretical results on reversible fouling in cross flow membrane filtration |
| topic | membrane separation cross-flow filtration polarization of concentration limiting flux reversible fouling Starling–Darcy boundary conditions |
| url | https://www.mdpi.com/2077-0375/9/4/48 |
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