Microfiltration,Ultrafiltration, and

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1 HATER 3 Microfiltration,Ultrafiltration, and Nanofiltration ti Models for redicting Flux for MF, UF and NF 1) apillary Model 2) Film Theory Model 3) Flux aradox (back-transport) 4) Resistance in series Model

2 BAKGROUND ON MODELLING - Model flux is predicted d as a function of system operation parameters ( pressure, flow rate, etc.) and physicochemical properties (viscosity, density, charge, diffusivity, etc.). - Although h the operating techniques of MF, UF, NF, and RO are similar, the separation mechanisms should be different : i) MF and UF have most frequently been visualized as sieve filtration. ii) NF and RO are almost certainly not merely separation by size, but by other parameters (charge, solubility, diffusivity, i it size, etc.)

3 BAKGROUND ON MODELLING A number of mathematical models are available that attempt to describe the mechanism of transport through membrane None are wholly satisfactory : i) The major problem appears to be an inability to precisely model the phenomenon occurring near the membrane surface. ii) The contribution and importance of the chemical nature of the membrane to membrane processes is still not clearly identified.

4 1) apillary Model - The best description of fluid flow through microporous membrane in an ideal situation : i) uniformly distributed evenly sized pores in the membrane, ii) no fouling, iii) negligible concentration polarization, etc. - redicting Flux from pore statistics using Hagen-oiseuille Equation. J = f ( dp, ε, ΔT, η, Δx ) Feed F Δx skin Membrane d p Void Sublayer ermeate

5 1) apillary Model Hagen-oiseuille Model for Laminar flow through channels: J 2 d ΔT 2 = ε ( l / h m ) 32Δx η d : ore diameter ε : Surface porosity (total open area of pores per unit membrane surface) Δ T : Transmembrane pressure (Δ T = F ) η : Viscosity of solvent (usually water) Δ x : Thickness of membrane skin

6 1) apillary Model J = f ( dp, ε, ΔT, η, Δx ) J = 2 ε d Δ 32Δx η T ( l / h m 2 ) Feed F Δx skin Membrane d p Void Sublayer ermeate

7 Example of capillary model Example) Membrane ; XM 100A UF (Amicon) dp ; Mean pore diameter = 175 A N ; Number of pores/cm 2 = 3 x 10 9 Δ x ; Skin thickness = 0.2μmμ [Δ T = 100ka = 106g /cm sec2] [η = 1 centipoise = 10-2 g/cm sec ] What is the water flux, J( l/m 2 h ) through this membrane?

8 Direct microscopic observation of membrane surface

9 Example of capillary model Answer ) porosity(ε) = N (π/4) dp 2 = 3 x 10 9 x (π/4) x (175x10-8 ) 2 = x 10-3 cm 2 /cm % of the membrane surface is occupied by pores. J = x 10-3 x (175 x 10-8 ) 2 cm 2 x 10 6 g/cm sec 2 32 x 0.2 x 10-4 cm x 10-2 g/cm sec = 3.45 x 10-3 cm/sec = 3.45 x 10-3 cm/sec x (m/100cm) x (1000L/m 3 ) x (3600sec/hr) = L/m 2 h (LMH)

10 omparison of experimental flux with calculated flux

11 Remark: - Several factors causing the discrepancy between calculated and experimental flux: 1) Hagen oiseuille Equation assumes all pores are right circular cylinders, which is highly unlikely with those membranes. A tortuosity (actual pore length/cylinder length )factor should have been included in the equation. J ( l / h τ : m 2 pore ) 2 ε d ΔT = 32ΔxΔ η τ tortuosity 2) As the pores get smaller in diameter, the number of pores in various sizes increases, which makes estimation of pore size distribution and pore density more difficult. 3) Smaller pores could be more tortuous. t 4) The number of dead end pore can not be accounted for. 5) The contribution of the chemical nature of the membrane material is not taken into account.

12 Remark: - According to this model, flux is proportional to T, but inversely proportional to η J ( l / h m 2 2 As is controlled by Temp. and feed concentration, increasing T or Temp. increases flux. Δ ε d Δ T ) = 32Δx η τ Δ η - This equation is applicable under restricted conditions, where concentration polarization effects are negligible. The following conditions are required: ΔT i) should be low, ii) Feed concentration should be low, iii) Flow velocity should be high. η If not, concentration polarization develops and Hagen-oiseuille equation no longer adequately describes the MF/UF process and mass-transfer limited model should be used.

13 Remark: - ore size and Flow distribution For a single capillary of diameter dpi, the flow rate, d d J h L q T p i i Δ = π π ) / ( d d x d p T p i i Δ Δ = π η / If fi is the number fraction of pores with diameter dpi, x p d t pi Δ Δ = η π 128 / 4 the fraction of solvent passing through the pores, Fi, is 4 p i i d f = max 4 dp p i p i i i i d f d f F min dp

14 Remark: From Fig. 3.5 & Fig. 3.7 The flow is strongly biased to the larger pores, with 50% of the solvent flow passing through 20-50% of the pores. Membrane permeability will be very sensitive to the population of large pores, and their loss by plugging g or obstruction. referred flow through h a minority it of larger pores will produce heterogeneous flow patterns normal to the membrane.

15 Direct microscopic observation of membrane surface

16 Distribution of pore size and volume throughput

17 2) Film-Theory (Mass transfer) Model Fig.1 Schematic of the film-theory model permeate membrane boundary layer bulk solution J V J S w differential element v.c d D dy b As solution is filtered, solute is brought to the membrane surface by convective transport which results in concentration gradient (or concentration polarization). The resulting concentration gradient causes the solute to be transported back The resulting concentration gradient causes the solute to be transported back into the bulk of solution due to diffusional effects. δ O y

18 2) Film-Theory (Mass transfer) Model i) Longitudinal mass transport within the boundary layer is assumed negligible (mass transfer within the film is one dimensional) J V D( d/ dx) ii) At steady state, the solute flux is constant throughout the film [ ] and equal to the solute flux through the membrane [ J ]. V A material balance for the solute in a different element gives the equation. J S = J V = J V D( d / dx) Boundary ondition ; = b at x=0 W at x= δ b : bulk solute concentration ti δ : thickness of the boundary layer : permeate solute concentration w : solute concentration at the membrane surface D : diffusion coefficient of solute

19 Derivation of the equation ) ( = V J dx d D ) ln( 0 = = V b w V J D dx J d D dx w δ δ ln ln 0 = = w S w V b k D J b δ 0) ( ln = = w S V b b if k J δ b D/δ = k S k f ffi i k s : mass-transfer coefficient - If w g ) ( ln lim Model polarization Gel k J g S = b g : Gel oncentration J lim : limiting flux

20 2) Film-Theory (Mass transfer)model

21 Retention and oncentration olarization Observed retention ; S = 1 - p / b, p True retention ; R = 1 - p / w oncentration polarization ; M = w / b = 1 S + S exp(j V /k s ) So M can be calculated from the measurement of the retention and So, M can be calculated from the measurement of the retention and the permeate flux, when k s is known.

22 orrelation between operating parameters and flux D 0) ( ln ln = = p for k D J b w S b w V δ

23 Effect of feed velocity, temperature and bulk concentartion on flux

24 Film theory model for macromolecular solutions

25 Experiment vs. Theory : Flux aradox -For colloidal suspensions: Experimental flux values are often one to two orders of magnitude higher than those indicated by the Lévéque and Dittus-Boelter relationships. But the reason is not clear.

26 Film theory model for colloidal suspensions

27 Experiment vs. Theory : Flux aradox -In colloidal suspensions, the diffusion coefficient calculated from the ultrafiltrate flux using the Lévéque or Dittus-Boelter equations generally from one to three orders of magnitude higher than the theoretical Stokes-Einstein diffusivity. -Minor adjustments in molecular parameters such as diffusivity (D), kinematic Minor adjustments in molecular parameters such as diffusivity (D), kinematic viscosity (ν), or gel concentration ( g ) are incapable of resolving order of magnitude discrepancies.

28 Experiment vs. Theory : Flux aradox - Back-diffusive transport of colloidal particles away from the membrane surface into the bulk stream (D / x) is substantially augmented over that predicted by the Lévéque and Dittus-Boelter relationships. - For colloidal suspensions, mass transfer from the membrane into the bulk stream is driven by some force other than the concentration gradient. - M.. orter s opinion (1972) : Tubular inch Effect is responsible for this augmented mass transfer.

29 How to solve the flux paradox? harge repulsion Inertial lift Shear induced diffusion Diffusion Axial velocity profile Drag torque ermeation drag Sedimentation Axial drag van der Waals attraction Semipermeable membrane J J J Fig.Ⅳ-1. Forces and torques acting on a charged, spherical particle suspended in a viscous fluid undergoing laminar flow in the proximity of a flat porous surface.[modified from Wiesner(1992)]

30 Forces acting on particles in a viscos flow undergoing laminar flow

31 Back transport velocity of particles under laminar flow Back transport velocity = diffusion + shear induced diffusion + lateral migration

32 Back transport velocity of particles under laminar flow Back transport velocity = Diffusion + shear induced diffusion + lateral migration + interaction induced k transport ( m/sec ) Bac Total Interaction induced Shear induced Diffusion Lateral migration Back tr ransport ( L/m 2 /hr ) article size ( μm ) Fig. omparison of different models explaining a critical flux over a range of particle size.

33 Back transport velocity as a function of particle size and fluid velocity

34 article size distributions deposited on the membrane surface = f ( size, flux) 14 Initial size distribution article size ( μm ) Fig. Ⅳ-5. Depositing particle size distribution for each flux condition. Size distribution when flux is infinite means the initial particle size distribution measured experimentally (T = 298 K, Ψ 0 = 50 mv, A = 3.4 x J).

35 Remark on the back transport velocity 1) rossflow MF/UF are occasionally operated in the turbulent regime, whereas all of the models described are restricted to laminar flow. 2) Brownian and shear-induced diffusion may be considered simultaneously by adding the diffusion coefficients, although recent simulations have shown that the diffusion coefficients are not strictly additive. 3) The models described are based on idealized suspensions of equisized spheres, which do not irreversibly stick to the membrane or cake surfaces but rather are free to diffuse or lift away. Further experiments and models are needed to study Brownian, shear induced diffusion, and inertial lift in real suspensions of non-spherical, deformable particles having both narrow and broad size distributions.

36 Remark on the back transport velocity 4) onsiderable experimental and theoretical research remains to complete our understanding of crossflow microfiltration. For example, issues of direct membrane fouling by the attachment of particles and precipitates to the membrane pores and surface have not been adequately addressed. d

37 Definitions of various Fluxes Flux (J) i ) ritical Flux (J c ) ii ) Limiting Flux (J lim ) iii ) Steady-state Flux (J ss ) i) J ii ) J J c J lim iii ) J Δ c Δ Δ t J ss Δ c : critical transmembrane pressure time(t)

38 4 ) Resistance in series Model J = Δ T μ ( R + m R s ) (1) if R s J >> R m, Δ T = μ Rs (2 )

39 4 ) Resistance in series Model - If the solute is thought of as a cake of deposited particles, the gel(cake, deposit, etc) resistance can be obtained from conventional filtration theory. J R S = Δ T ( R m + R S α V ) η = b m / A m = α m s / A (1) (2) R s : solute resistance (1/m) R m : membrane resistance (1/m) V : cumulative solvent volume (m 3 ) b : Bulk solute concentration (kg/m 3 ) A m : membrane area (m 2 ) m s : mass of solute (kg) α : specific resistance (m/kg)

40 4 ) Resistance in series Model For unstirred conditions R s grows and combining equation (1) with (2) gives, (3) ) / ( 1 ) ( + Δ = = A V R dt dv A t J m b m m T η α (4 ).( 3) + = dv V dv R dt eqn From V b t V m m η α η (4 ) Δ + Δ = dv A dv A dt m m T T At constant pressure, integration of eqn. (4) gives, p, g q ( ) g, 2 m b V V R t + = η α η / 2 b m m m A V A R V t A A T T Δ + Δ = Δ Δ η α η (well-known filtration equation) 2 m m A A T T Δ Δ

41 t / V = R Δ m T η A m + α V 2 Δ T b η 2 A m t/v slope = = α b η 2 Δ T A Modified d m 2 Fouling Id Index (MFI) T V From a plot of t/v vs. V, α is obtained experimentally. If R m is negligible, gg V 2 = ( 2 ΔpT A 2 m / b α η ) t (5) Which predicts that filtrate (permeate) accumulates according to t 1/2

42 arman-kozeny eqn. for α - α, the specific resistance of the deposit can be related to article properties by the arman-kozeny relationship. ( 1 ε ) 180 α = (6 ) 2 3 ρ s d s ε ε : voidage (or porosity) ρ s : solute density D s : solute particle diameter - The effect of pressure on α is frequently expressed by the relationship, α = α o Δ s o α o : constant s : compressibility factor For compressible solids, typical values of s = 0.2~0.7

43 α, alculated vs. Experimental - omparison the values of α obtained from the arman-kozeny eqn. (eq.6) with those measured experimentally. Bovine Serum Albumine colloidal l silica calculated α = 3.0~7.0 x (m/kg) α = 3.0~6.0 x (depending on ε) experimental α = 1.0~7.0 x (m/kg) α = 5.0~15.0 x (depending di on Δ) - The similarity between calculated and experimental α values for these particulate solutes supports the use of a conventional filtration model for MF/UF.

44 5) Rejection Mechanisms in Nanofiltration During the last few decades, the drinking water industry has become increasingly concerned about the occurrence of microorganic pollutants in the source waters for the drinking water supply s : pesticides in surface waters. : Approach to tackle the problem 1) developing alternatives for the use of pesticides 2) implementing activated carbon filtration 3) installing NF/RO 1990 s : Endocrine disrupting compounds (EDs) and pharmaceutically active compounds (has) having negative effect on the hormonal system of human and animal life <example; Estradiol, NDMA (N-nitrosodimethylamine)> : The polar compound, NDMA can not be removed by activated carbon : New approach to tackle the problem 1) installing NF/RO -

45 5) Rejection Mechanisms in Nanofiltration - Three major solute membrane interactions affecting removal efficiency of organic pollutants in Nanofiltartion : 1) Steric Hindrance (Sieving effect) 2) Electrostatic repulsion 3) Hydrophobic-hydrophobic/ adsorptive interactions - These solute - membrane properties are determined by i) solute properties: molecular l weight/ size, charge, hydrophobicity (expressed by low K ow Values) ii) membrane properties: molecular weight cutoff (MWO)/pore size, surface charge (zeta-potential)) hydrophobicity (contact angle) iii) Operating conditions : pressure, flux, recovery iv) feed water composition: ph, temperature, t DO, inorganic i balance

46 Three major solute membrane interactions in NF 1) Steric Hindrance (Sieving effect) - It is mainly determined by the size of solute and the size of the membrane pores. - It generally leads to a typical S-shaped curve in function of the molecular weight. (rejection vs. molecular weight of solute). - Solutes with a molecular weight higher than the MWO of membrane are well rejected. S l i h l l i h l h h MWO f b il - Solutes with a molecular weight lower than the MWO of membrane can easily permeate through the membrane.

47 Three major solute membrane interactions in NF 2) Electrostatic repulsion - In the presence of electrostatic repulsion in NF, the flux may be described by extended Nernst-lanck equation. According to this eqation, the flux of charged solutes (or ions) through a charged membrane governed by several factors: i) onvection ii) Diffusion iii) Donnan potential -The effect of Donnan potential is to repel the co-ion having same charge of the fixed charge in the membrane) from th emmebrane, and because of elctroneutrality requirements, the counter-ion having opposite in charge of the fixed charge in the emnbrane) is also rejected. - This equation predicts that the solute rejection is a function of feed concentration and charge of the ion, but the equation includes the effects of convective and diffusional i fluxes.

48 Three major solute membrane interactions in NF 3) Hydrophobic-hydrophobic/ y p adsorptive interactions - These interactions are important factors in the rejection of uncharged organic molecules. - Log K ow ; logarithm of the octanol-water partition coefficient log K ow <1 hydrophilic, 1 < log K ow < 2 intermediate, log K ow > 2 hydrophobic - Hydrophobic solutes adsorb more to the membrane and are thus more easily dissolved in the membrane matrix. As a result, solution and consecutive diffusion of hydrophobic solutes in the membrane matrix leads to higher permeation. - Hydrophilic molecules are better rejected compared to hydrophobic molecules of similar molecular weight. It might be explained by hydration of hydrophil molecules. When a hydrophilic is hydrated, the effective molecular size might be larger compared to a less hydrated hydrophobic molecule of the same molecular size.

CHAPTER 3 Microfiltration,Ultrafiltration Models for Predicting Flux for MF, UF

CHAPTER 3 Microfiltration,Ultrafiltration Models for Predicting Flux for MF, UF HAPTER 3 Microfiltration,Ultrafiltration Models for Predicting Flux for MF, UF 1) apillary Model 2) Fil Theory Model 3) Flux Paradox (back-transport) 4) Resistance in series Model BAKGROUND ON MODELLING