Models for membrane filtration Linda Cummings Department of - - PowerPoint PPT Presentation

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.

0.0 0.2 0.4 0.6 0.8 1.0 40 80 120 160

Throughput [L/m2] Initial Pressure/Instantaneous Pressure

47mm disc - Filter 1 47mm disc - Filter 2 47mm disc - Filter 3 1" laid over pleat cartridge- Filter 1 1" laid over pleat cartridge - Filter 2 1" laid over pleat cartridge - Filter 3

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

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

0.2µm PES/ 0.2µm PVDF LOP small-core 0.65µm PES/ 0.2µm PES LOP standard-core 0.2µm PES/ 0.1µm PVDF LOP small-core 0.5µm PES/ 0.1µm PES Fanpleat standard-core 0.2µm PES/ 0.1µm PES Fanpleat standard-core

Flux Ratio relative to flat disc

Flat Disc Format Pleated Cartridge

From Kumar, Martin & Kuriyel, 2015

Membrane filtration Pleated filters Pore morphology Outlook

Pleated filter: mathematical modeling

Can mathematical modeling pinpoint the reasons for the low efficiency, and perhaps (ultimately) suggest remedies? Simplify geometry to obtain tractable model.

Membrane No flux Pleat tip y=H Inflow y=−H Outflow x Membrane, thickness D Pleat valley No flux y L

Membrane filtration Pleated filters Pore morphology Outlook

Pleated filter: idealized geometry

Membrane No flux Pleat tip y=H Inflow y=−H Outflow x Membrane, thickness D Pleat valley No flux y L

(b)

Flow to permeate outlet Cartridge Pleat valleys Pleat tips

Z Y

feed inlet Flow from

X

Membrane filtration Pleated filters Pore morphology Outlook

Pleated filter: Modeling assumptions

Membrane No flux Pleat tip y=H Inflow y=−H Outflow x Membrane, thickness D Pleat valley No flux y 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 A0, and traverse membrane.

Membrane filtration Pleated filters Pore morphology Outlook

Pleated filter: Modeling

Membrane No flux Pleat tip y=H Inflow y=−H Outflow x Membrane, thickness D Pleat valley No flux y L

Y=−D/2 Y −H H =0 PX =0 P =0 PX =0 L X

Y

PY P=P P=0 Membrane Y=D/2

Assume incompressible Darcy flow within support layers U = (U, V ) = −K µ ∇P, ∇ · U = 0, ∇ = (∂X, ∂Y ) Pressure drop P0 between inlet and outlet. Darcy flow through membrane, Um; Y -component satisfies |Vm| = Km µD

• P|Y =D/2 − P|Y =−D/2
• and

|Vm| = K µ ∂P ∂Y

• Y =D/2

= K µ ∂P ∂Y

• 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 Km will vary in both space and time: Km(X, T). Account for membrane fouling by adsorption and blocking.

Hagen-Poiseuille gives flux through unblocked pore as Qu,pore = 1 Ru (P|Y =D/2 − P|Y =−D/2), Ru = 8µD πA4 . Assume blocking of pore introduces additional resistance in series and write Qb,pore = 1 Rb (P|Y =D/2 − P|Y =−D/2), Rb = 8µD πA4 A0 A 4 + ρb

• .

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 N0 = N(X, 0). Then flux per unit area of membrane is |Vm| = NQu,pore + (N0 − N)Qb,pore Net membrane permeability given in terms of these quantities by Km = πA4 8

• N

(A0/A)4 + N0 − N (A0/A)4 + ρb)

• Close model by specifying evolution of N and A.

Membrane filtration Pleated filters Pore morphology Outlook

Permeability evolution

For large particles assume cumulative size distribution function G(S) (no. of particles per unit volume with radius < S). Then Probability per unit time that pore of radius A blocked

• = (G∞ − G(A))Qu,pore

⇒ ∂N ∂T = −N πA4 8µD (G∞ − G(A))(P|Y =D/2 − P|Y =−D/2).

We assume G(S) = G∞(1 − e−BS) throughout (B−1 then a typical large-particle size).

Adsorption: Pore radius A shrinks in time. Assume simplest possible law: uniform adsorption within pores (will consider more complicated laws later): ∂A ∂T = −E, A|T=0 = A0.

Membrane filtration Pleated filters Pore morphology Outlook

Asymptotic solution to model

Introduce scalings P = P0p, (X, Y ) = L(x, ǫy) (ǫ = H/L), T = TBt, K = Kavk, Km = Km0km, A(T) = A0a(t) (TB = 8µD/(πP0G∞A4

0) is pore-blocking timescale).

Leading order pressure in both support layers then independent of y: p+

0 (x), p− 0 (x).

Anticipate solution as asymptotic expansion in ǫ2. Flux continuity across membrane requires p+

y |y=δ/2 = p− y |y=−δ/2 = ǫ2Γkm

k [p+|y=δ/2 − p−|y=−δ/2], where Γ = Km0L2 KavHD = Km0 ǫ2δKav (δ = D/H) measures relative importance of resistance of membrane and support material.

Membrane filtration Pleated filters Pore morphology Outlook

Asymptotics: ǫ2 ≪ 1; δ ≪ 1

Seek solution as asymptotic expansion in ǫ2. Darcy model gives 2nd order coupled BVPs for p±

0 (need to

go to O(ǫ2) for solvability condition). ∂ ∂x

• k(x)∂p+

∂x

• = Γkm(x)(p+

0 − p− 0 ),

∂ ∂x

• k(x)∂p−

∂x

• = −Γkm(x)(p+

0 − p− 0 ),

p+

0 |x=0 = 1,

∂p+ ∂x

• x=1

= 0, ∂p− ∂x

• x=0

= 0, p−

0 |x=1 = 0.

Note that as Γ → 0 (membrane resistance dominates) solutions converge to p+

0 = 1, p− 0 = 0.

Membrane filtration Pleated filters Pore morphology Outlook

Iterative solution scheme

At t = 0 assign km(x, 0) = km0 = 1 and support permeability k(x). Then

1

Solve BVP for p±

0 ;

2

Use this solution and current membrane permeability and pore radius a(t) to solve for number of unblocked pores n(t): ∂n ∂t = −na4e−ba(p+|y=0+ − p−|y=0−), n|t=0 = 1.

3

Update pore radius via ∂a ∂t = −β, a|t=0 = 1, β = 8µED πA5

0P0G∞

≪ 1.

4

Return to 1.

Membrane filtration Pleated filters Pore morphology Outlook

Iterative solution scheme

Simulate model for relevant parameter values, investigating cases of uniform support permeability, and decreasing support permeability gradients k′(x) < 0.

(b)

Flow to permeate outlet Cartridge Pleat valleys Pleat tips

Z Y

feed inlet Flow from

X

Membrane filtration Pleated filters Pore morphology Outlook

Typical parameter values

Parameter Description Typical Value L Length of the pleat 1.3 cm H Support layer thickness 1 mm D Membrane thickness 300 µm A0 Initial pore radius 2 µm (very variable) B−1 Characteristic particle size 4 µm (very variable) E Adsorption coefficient within pores Unknown (depends on characteristics of membrane and feed solution) G∞ Total particle concentration Depends on application N0 Number of pores per unit area 7×1010 m−2 (very variable) P0 Pressure drop Depends on application Kav Average support layer permeability 10−11 m2 (very variable) Km0 Clean membrane permeability 5×10−13 m2 (very variable)

Membrane filtration Pleated filters Pore morphology Outlook

Results: Flow & membrane permeability

x y

0.2 0.4 0.6 0.8 1 −1 −0.8 −0.6 −0.4 −0.2 0.2 0.4 0.6 0.8 1 −1 −0.9 −0.8 −0.7 −0.6 −0.5 −0.4 −0.3 −0.2 −0.1 0.2 0.4 0.6 0.8 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

x km

.01tf .02tf .05tf .08tf .13tf .18tf .25tf tf

With uniform support permeability k(x) = 1 permeability remains symmetric about pleat centerline. Fouling occurs preferentially at the pleat ends. As total blocking occurs km → 0 and permeability gradients along pleat ultimately smooth out.

Membrane filtration Pleated filters Pore morphology Outlook

Results: Decreasing support permeability k′(x) < 0

0.2 0.4 0.6 0.8 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

x km

.01tf .02tf .05tf .08tf .13tf .18tf .25tf tf 0.2 0.4 0.6 0.8 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

x km

.01tf .02tf .05tf .08tf .13tf .18tf .25tf tf

Compare only profiles with same average support permeability (two examples shown). Membrane permeability now develops asymmetry. Fluid has an initially easy path through upper support layer, so less passes through leftmost part of membrane. Fouling occurs preferentially at the far pleat end.

Membrane filtration Pleated filters Pore morphology Outlook

Results: Membrane performance (Γ = 10)

Performance characterized by flux-throughput curves. Compare our pleated filter model with closest equivalent non-pleated filter:

1 2 3 4 5 6 7 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Throughput Total Flux

Flat k1 k2 k3

Flat filter significantly outperforms pleated filters. Among pleated filters, uniform support permeability “best”.

Membrane filtration Pleated filters Pore morphology Outlook

Parametric dependence: ρb

0.5 1 1.5 2 2.5 3 0.02 0.04 0.06 0.08 0.1 0.12 0.14

Throughput Total Flux

ρb=0.25 ρb=1 ρb=2 ρb=5 ρb=10

ρb measures relative increase in pore resistance when blocked upstream by large particle.

Membrane filtration Pleated filters Pore morphology Outlook

Parametric dependence: b

0.5 1 1.5 2 2.5 3 0.02 0.04 0.06 0.08 0.1 0.12 0.14

Throughput Total Flux

b=0.2 b=0.5 b=1 b=2 b=10

b measures ratio of initial pore radius, A0, to typical size of a large particle in our bimodal distribution. (When b ≪ 1 most large particles are sieved.)

Membrane filtration Pleated filters Pore morphology Outlook

Dependence on Γ = Km0/(ǫ2δKav)

Γ is key system parameter characterizing where primary system resistance arises. In limit Γ → 0 anticipate pleated model to approach “flat filter” solution. Simulations bear this out.

1 2 3 4 5 6 7 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Throughput Total Flux

Flat Pleated, Γ=0.1 Pleated, Γ=1 Pleated, Γ=10 Pleated, Γ=100

Membrane filtration Pleated filters Pore morphology Outlook

Conclusions (i)

Simple model of pleated filter, which we believe captures key features. Important factors for performance are permeability ratio Γ = Km0H/(L2DKav), and support permeability profile k(x). Performance converges to that of the equivalent flat filter as Γ decreases (use more permeable support material, or decrease pleat thickness H, or increase pleat length L). However, increasing L means either using bigger cylinders (bad), or decreasing inner cylinder radius, which effectively introduces negative permeability gradients k′(x) < 0, negatively impacting performance – tradeoff. More precise predictions require both accurate data, and more sophisticated modeling – work in progress.

Membrane filtration Pleated filters Pore morphology Outlook

Improved membrane description (Pall, MPI 2014)

Most filtration/fouling literature assumes identical cylindrical pores, which traverse membrane from top to bottom. Real membrane structure is usually much more complex – pores may be tortuous, may undergo branching, etc. Many membrane filters in commercial use also have permeability depth gradients, being typically more permeable

• n the upstream side.

Industry interested in improved mathematical descriptions of internal membrane structure and particle deposition, which can better predict membrane fouling and lifetime, and guide membrane design.

Membrane filtration Pleated filters Pore morphology Outlook

Model membrane geometry

Flat membrane in (Y , Z)-plane. Flow perpendicular to membrane, carrying bimodal distribution of particles: large particles which block pores upstream, and small particles at concentration C(X, T). Pores again traverse membrane, but are axisymmetric and slender, with variable radius A(X, T) (⇒ depth-dependent permeability). Pores arranged in repeating square lattice, period 2W .

C(X,T) 2W 2W X D A(X,T)

Membrane filtration Pleated filters Pore morphology Outlook

Flow within slender pore (no blocking)

Incompressible Darcy flow U = (U(X, T), 0, 0) within membrane, permeability K(X, T): U = −K(X, T) µ ∂P ∂X , ∂ ∂X

• K(X, T)∂P

∂X

• = 0.

Can relate pore radius to permeability: K(X, T) = φmKp, φm = πA2 (2W )2 , Kp = A2 8 , and (averaged) pore velocity Up related to Darcy velocity by U = φmUp Advection equation for small particles carried by flow: Up ∂C ∂X = −ΛAC, C(0, T) = C0 (cross-sectionally averaged, Pe ≫ 1, omitting details); Λ captures physics of attraction between particles and pore wall.

Membrane filtration Pleated filters Pore morphology Outlook

Membrane fouling (adsorption)

As before we consider two fouling modes: adsorption & blocking (cake formation can be bolted on later). For adsorption: Pore radius shrinks due to adsorption of small particles, ∂A ∂T = −ΛαC, A(X, 0) = A0(X). In absence of pore-blocking this closes the model on previous slide.

Membrane filtration Pleated filters Pore morphology Outlook

Membrane fouling (blocking)

To include blocking of pores by large particles also, follow pleated filter modeling, tracking number of unblocked pores per unit membrane area, N(T). ∂N ∂T = N πA4 8µ ∂P ∂X (G∞ − G(A)), N(0) = N0 = 1 (2W )2 . Here assume all large particles are bigger than pores: G(A) = 0. Modified incompressible Darcy equation U = −πA4 8µ ∂P ∂X

• N

(A0/A)4 + N0 − N (A0/A)4 + ρb

• ;

∂U ∂X = 0 (permeability K(X, T) implicit in this equation).

Membrane filtration Pleated filters Pore morphology Outlook

Parameter values

To investigate how pore shape and permeability gradients affect filtration & fouling, simulate filtration through pores of varying profiles (fix initial membrane porosity/permeability). Typical parameter values, where known:

Parameter Description Typical Value 2W Length of the square repeating lattice 4.5 µm (very variable) Λ Particle-wall attraction coefficient Unknown (depends on characteristics of membrane) D Membrane thickness 300 µm A0 Initial pore radius 2 µm (very variable) B−1 Characteristic particle size 4 µm (very variable) α Depends on the particle size Unknown (depends on characteristics of feed solution) G∞ Total particle concentration Depends on application N0 Number of pores per unit area 7×1010 m−2 (very variable) P0 Pressure drop Depends on application Qpore Flux through a single pore Depends on application C0 Intial particle concentration Depends on application

Membrane filtration Pleated filters Pore morphology Outlook

Parameter values

Where parameters are unknown, experimental data may be available to help estimate them, e.g., can estimate particle-wall attraction coefficient Λ by fitting model solutions for C(X, T) to fluorescence data: From Jackson et al., Biotechnol. Prog. 2014. Note however that this is not an example of an efficient filtration scenario!

Membrane filtration Pleated filters Pore morphology Outlook

Schematic geometry

Membrane

Membrane filtration Pleated filters Pore morphology Outlook

Example results: Initially uniform pore profile

Plot normalized pore radius a(x, t) =

1 W A(X, T), and particle

concentration c(x, t) =

1 C0 C(X, T), x = X/D, at various

timesteps t = T/TB. Though porosity & permeability are initially uniform, gradients quickly develop. Pore closes first at upstream side.

0.2 0.4 0.6 0.8 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

x Radius of pore and particle concentration

a(x,0) a(x,0.4tf) a(x,0.7tf) a(x,tf=9.05) c(x,0) c(x,0.4tf) c(x,0.7tf) c(x,tf)

λ = 8ΛµD2/(P0W ) = 2

Membrane filtration Pleated filters Pore morphology Outlook

Results: Comparing selected (normalized) pore profiles

0.2 0.4 0.6 0.8 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

x Radius of pore and particle concentration

a(x,0) a(x,0.4tf) a(x,0.7tf) a(x,tf=9.05) c(x,0) c(x,0.4tf) c(x,0.7tf) c(x,tf)

(a)

0.2 0.4 0.6 0.8 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

x Radius of pore and particle concentration

a(x,0) a(x,0.4tf) a(x,0.7tf) a(x,tf=8.2) c(x,0) c(x,0.4tf) c(x,0.7tf) c(x,tf)

(b)

0.2 0.4 0.6 0.8 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

x Radius of pore and particle concentration

a(x,0) a(x,0.4tf) a(x,0.7tf) a(x,tf=9.8) c(x,0) c(x,0.4tf) c(x,0.7tf) c(x,tf)

c)

0.2 0.4 0.6 0.8 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

x Radius of pore and particle concentration

a(x,0) a(x,0.4tf) a(x,0.7tf) a(x,tf=9.7) c(x,0) c(x,0.4tf) c(x,0.7tf) c(x,tf)

Membrane filtration Pleated filters Pore morphology Outlook

Results: Selected (normalized) pore profiles a1(x)–a4(x)

All pore profiles compared (a1-a4) have same initial membrane porosity πA2 1

0 a(x, 0)2dx/(2W )2D (results with same initial

membrane permeability are almost identical). In all cases shown pore closure occurs first at upstream membrane surface (flux then goes to zero). Can, by careful choice of initial pore profile, or by an unreasonably small choice of λ, obtain pore closure at internal points, or even the downstream end. The pore with linear decreasing initial profile stays open longest (longest filter lifetime). In addition to simply sustaining a nonzero flux, however, want to know which pore profiles give maximal total throughput

• ver the filter lifetime.

Membrane filtration Pleated filters Pore morphology Outlook

Results: Flux vs throughput

Compare 5 different pore profiles (a1-a4 on previous slide, plus

• ne other: uniform, linear increasing, linear decreasing,

concave, convex). Linear decreasing profile clearly “best”, among those considered.

e)

0.5 1 1.5 2 0.1 0.2 0.3 0.4 0.5 0.6 0.7

Throughput Flux

a1 a2 a3 a4 a5

Membrane filtration Pleated filters Pore morphology Outlook

Results: Particle concentration within pores

0.5 1 1.5 2 0.05 0.1 0.15 0.2 0.25 0.3 0.35

Throughput Particle concentration at pore outlet

a1 a2 a3 a4 a5

(a)

0.2 0.4 0.6 0.8 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

time Particle concentration at pore outlet

=0.1 =0.5 =1 =2

(b)

Dimensionless parameter λ = 8ΛµD2/(P0W ) measures strength of attraction between small particles & pore walls and governs deposition. Sample simulations show how the normalized concentration, c(1, t) = C(D, T)/C0, at pore outlet, varies both with pore profile, and with λ (λ = 1 in plot (a); a(x, 0) = a1(x), uniform pore, in plot (b)).

Membrane filtration Pleated filters Pore morphology Outlook

Results: Particle concentration at pore outlet

0.5 1 1.5 2 0.05 0.1 0.15 0.2 0.25 0.3 0.35

Throughput Particle concentration at pore outlet

a1 a2 a3 a4 a5

(a)

0.2 0.4 0.6 0.8 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

time Particle concentration at pore outlet

=0.1 =0.5 =1 =2

(b)

Fixed λ, (a): There is a clear distinction between performance

• f different pore profiles.

Again we see that the membrane with linear decreasing pore profile (a3(x)) gives the greatest throughput for any given particle concentration at the pore outlet. In practice the user has in mind a “tolerance” value of c(1, t): larger values of λ give better particle removal (plot (b)).

Membrane filtration Pleated filters Pore morphology Outlook

Results: Optimum permeability profile

Given that results depend rather strongly on pore profile, a good question to ask is: can we determine the optimum profile? Difficult question to address in general (work in progress!) but within selected function classes can answer. Here compare linear pore profiles, of equal averaged porosity

• r permeability (same values of

1

0 a(x, 0)2dx or

1

0 a(x, 0)4dx), and see which gives greatest total throughput

• ver filter lifetime.

Membrane filtration Pleated filters Pore morphology Outlook

Results: Optimum permeability profile

1 0.5 0.5 1 0.5 1 1.5 2 2.5 3

Pore Gradient Throughput

=.1 =.2 =.5 =1 =2 1 0.5 0.5 1 0.5 1 1.5 2 2.5 3 Pore Gradient Throughput k0=0.1 k0=0.2 k0=0.3 k0=0.4 k0=0.5 k0=0.6 k0=0.7 k0=0.8 k0=0.9

(b)

For fixed net permeability, how does the gradient of a linear pore affect net throughput over filter lifetime? Not all pore gradients may be allowable for a fixed

• permeability. Plot (a) shows results for a permeability value

for which a wide range of pore gradients is possible. Observe that as λ ↓, the pore gradient is less significant (less deposition, and hence less filtration, is occurring!).

Membrane filtration Pleated filters Pore morphology Outlook

Results: Optimum permeability profile

1 0.5 0.5 1 0.5 1 1.5 2 2.5 3

Pore Gradient Throughput

=.1 =.2 =.5 =1 =2 1 0.5 0.5 1 0.5 1 1.5 2 2.5 3 Pore Gradient Throughput k0=0.1 k0=0.2 k0=0.3 k0=0.4 k0=0.5 k0=0.6 k0=0.7 k0=0.8 k0=0.9

(b)

Note (plot (b)): as net permeability increases (larger pores), not all pore gradients are possible at fixed permeability. Real filter membranes don’t have the simple pore structure assumed here, but optimal pore gradient translates into a permeability gradient, which has meaning for any membrane.

Membrane filtration Pleated filters Pore morphology Outlook

Conclusions (ii)

Allowing (axisymmetric) variations in pore radius provides a simple way to model gradations in membrane structure. Simulation results bear out empirical findings that negative permeability gradients are preferable. Within a given class of pore shapes, and given appropriate data to fix model parameters, can address issue of which pore shape maximizes total throughput over filter lifetime (or other performance-related questions). Generalizing the formulation of this optimization is ongoing work.

Membrane filtration Pleated filters Pore morphology Outlook

Outlook

Membrane filtration is a complex process, but simple mathematical models can provide significant insight. Considered three specific modeling directions here, but many extensions are possible and should be explored. Interaction force (electrostatics) may vary as deposition

• ccurs – may need to allow deposition parameter Λ to vary

depending on concentration of particles already deposited. In slow filtration (or with very small particles) diffusion may be important within pores – more complicated model for particle transport (careful averaging needed). A particular area for future study is pore branching – models here all consider simple isolated pores that span membrane from upstream to downstream. Both branching and recombining of pores may occur in real membranes, and may be important for filtration dynamics.

Membrane filtration Pleated filters Pore morphology Outlook

Acknowledgements

NSF DMS-1261596 NSF DMS-1211713 Collaborators: Pejman Sanaei (NJIT) Giles Richardson (Southampton, UK) Tom Witelski (Duke) Anil Kumar (Pall Corporation, Westborough, MA)