Membrane filtration Pleated filters Pore morphology Outlook Models for membrane filtration Linda Cummings Department of Mathematical Sciences New Jersey Institute of Technology Faculty research talk, March 2016
Membrane filtration Pleated filters Pore morphology Outlook Overview Membrane filtration – applications & issues Focus on two key industrial challenges: Efficiency of pleated filter cartridges; Modeling internal membrane structure Modeling, results & implications Current & future modeling directions
Membrane filtration Pleated filters Pore morphology Outlook Membrane Filters Membrane filters: Thin layers of porous media, through which “feed solution”, carrying particles, passes. Designed to remove particles of a certain size range from the feed. Used in a huge number of applications, e.g.: Water purification; Cleaning of air or other gases (HEPA filters in A/C, vacuums); Treatment of radioactive sludge; Purification processes in the biotech industry; Beer clarification; Coffee; . . .
Membrane filtration Pleated filters Pore morphology Outlook Membrane Filters: Fouling During filtration the filter becomes fouled, which increases its resistance and lowers filtration efficiency. Several different modes of fouling: Deposition (adsorption) of small particles on the pore walls within membrane; Deposition of large particles on top of membrane (sieving, or blocking); Cake formation, which occurs in the late stages of filtration (think of a coffee filter).
Membrane filtration Pleated filters Pore morphology Outlook Fouling & efficiency: constant pressure Filtration generally takes place under one of two scenarios: constant pressure or constant flux. With constant pressure, as the fouling occurs, and system resistance increases, the flux decreases monotonically. Once flux falls below some threshold value the filter must be discarded (or cleaned). A key indicator of filter performance is provided by flux-throughput curves. Optimal performance would be to maintain the flux high for as long as possible.
Membrane filtration Pleated filters Pore morphology Outlook Fouling & efficiency: constant flux In the constant flux scenario a pump is used to maintain a constant rate of throughput. As fouling occurs, the operating pressure required to drive the pump increases. Once the driving pressure passes some threshold, the filter is deemed unsustainable and must be discarded (or cleaned). In this case a key performance characteristic is provided by the (inverse) pressure–throughput curve for the filter. 47mm disc - Filter 1 1.0 47mm disc - Filter 2 47mm disc - Filter 3 Initial Pressure/Instantaneous 1" laid over pleat cartridge- Filter 1 0.8 1" laid over pleat cartridge - Filter 2 1" laid over pleat cartridge - Filter 3 0.6 Pressure 0.4 0.2 0.0 0 40 80 120 160 Throughput [L/m 2 ]
Membrane filtration Pleated filters Pore morphology Outlook Background to the problem: MPI workshops MPI: “Mathematical Problems in Industry”. Workshops jointly sponsored by NSF, IMA and participating companies.
Membrane filtration Pleated filters Pore morphology Outlook Background to the problem: MPI workshops 2013/2014 Many major companies heavily invested in membrane filtration, e.g. W.L. Gore & Associates; Pall Corporation ($ 2.6bn). Multi-billion $$ industry, in the US alone. At 2013/2014 workshops both companies brought problems pertaining to better prediction of membrane filter performance. 2013: Pall asked “Why do our pleated membrane filters underperform?” 2014: Pall asked “How can we better predict membrane filter efficiency from known membrane characteristics?”
Membrane filtration Pleated filters Pore morphology Outlook Membrane filtration: Efficiency is critical Naive approach to filtration says: choose a filter with pores smaller than the particles you wish to remove. This is highly inefficient however – system resistance then very high and huge driving pressures required (expensive). In practice pores are larger (perhaps 10 × ) than most particles in the feed, and much of the filtration takes place within the membrane interior via adsorption (detailed mechanisms largely unknown). For large-scale filtration space can be an issue – may want to pack filters into a small volume. However, this can also lead to increases in system resistance, and efficiency losses. Also want to maximize throughput and filter lifetime. Modeling therefore has a key role to play in investigating efficient filtration scenarios.
Membrane filtration Pleated filters Pore morphology Outlook Pleated filter cartridges (Pall, MPI 2013) Commonly used in a wide range of applications. Pleated structure offers advantage of large filtration area, within a small volume. Membrane is sandwiched between much more porous “support” layers, before being pleated and packed into annular cylindrical cartridge.
Membrane filtration Pleated filters Pore morphology Outlook Pleated filter cartridges Cartridge placed inside external housing. The feed solution is driven from exterior to interior of the cylinder, passing across the filter membrane. The filtration efficiency is not what manufacturers would wish, however.
Membrane filtration Pleated filters Pore morphology Outlook Pleated filter inefficiency 1.4 Flat Disc Format Flux Ratio relative to flat disc 1.2 Pleated Cartridge 1.0 0.8 0.6 0.4 0.2 0.0 0.2µm PES/ 0.65µm PES/ 0.2µm PES/ 0.5µm PES/ 0.2µm PES/ 0.2µm PVDF 0.2µm PES 0.1µm PVDF 0.1µm PES 0.1µm PES LOP LOP LOP Fanpleat Fanpleat small-core standard-core small-core standard-core standard-core From Kumar, Martin & Kuriyel, 2015
Pleated filter: mathematical modeling Membrane filtration Simplify geometry to obtain tractable model. efficiency, and perhaps (ultimately) suggest remedies? Can mathematical modeling pinpoint the reasons for the low Pleat tip No flux Inflow Pleated filters y Membrane Membrane, thickness D L Pore morphology y= − H y=H Pleat valley Outflow No flux x Outlook
Membrane filtration Pleated filters Pore morphology Outlook Pleated filter: idealized geometry y Membrane, thickness D Pleat valley Inflow y=H No flux x Pleat tip y= − H Outflow No flux Membrane L Z Y X Flow from feed inlet Pleat valleys Cartridge Pleat tips Flow to permeate outlet (b)
Membrane filtration Pleated filters Pore morphology Outlook Pleated filter: Modeling assumptions y Membrane, thickness D Pleat valley Inflow y=H No flux x Pleat tip y= − H Outflow No flux Membrane L Periodic geometry – consider single pleat. Neglect curvature & axial variation and simplify to 2D rectangular geometry. Membrane thickness D , support layer thickness H , pleat length L : D ≪ H ≪ L . Neglect flow through pleat tips and valleys. Pores cylindrical, initial radius A 0 , and traverse membrane.
Membrane filtration Pleated filters Pore morphology Outlook Pleated filter: Modeling y Membrane, thickness D Y P Y =0 H Pleat valley P X =0 P=P Inflow y=H 0 No flux Membrane Y=D/2 x Pleat tip L X Y=−D/2 y= − H Outflow No flux P X =0 P=0 −H P =0 Membrane L Y Assume incompressible Darcy flow within support layers U = ( U , V ) = − K µ ∇ P , ∇ · U = 0 , ∇ = ( ∂ X , ∂ Y ) Pressure drop P 0 between inlet and outlet. Darcy flow through membrane, U m ; Y -component satisfies | V m | = K m � � P | Y = D / 2 − P | Y = − D / 2 µ D � � | V m | = K ∂ P = K ∂ P � � and � � µ ∂ Y µ ∂ Y � � Y = D / 2 Y = − D / 2
Membrane filtration Pleated filters Pore morphology Outlook Flux through membrane pores Support permeability K constant in time (no fouling) but may vary spatially: K ( X , Y ). Membrane permeability K m will vary in both space and time: K m ( X , T ). Account for membrane fouling by adsorption and blocking. Hagen-Poiseuille gives flux through unblocked pore as Q u , pore = 1 R u = 8 µ D ( P | Y = D / 2 − P | Y = − D / 2 ) , π A 4 . R u Assume blocking of pore introduces additional resistance in series and write �� A 0 � 4 � Q b , pore = 1 R b = 8 µ D ( P | Y = D / 2 − P | Y = − D / 2 ) , + ρ b . π A 4 R b A 0
Membrane filtration Pleated filters Pore morphology Outlook Blocking & membrane permeability Assume a bimodal distribution of particle sizes in feed: very small particles that are adsorbed within pores and shrink them; and large particles that can block pores from above. Let N ( X , T ) be number density of unblocked pores, with N 0 = N ( X , 0). Then flux per unit area of membrane is | V m | = NQ u , pore + ( N 0 − N ) Q b , pore Net membrane permeability given in terms of these quantities by K m = π A 4 � N N 0 − N � 0 ( A 0 / A ) 4 + ( A 0 / A ) 4 + ρ b ) 8 Close model by specifying evolution of N and A .
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