models for membrane filtration

Membrane filtration Pleated filters Pore morphology Outlook

Models for membrane filtration

Linda CummingsDepartment of Mathematical Sciences

New Jersey Institute of Technology

Faculty research talk, March 2016


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 FiltersP. Apel / Radiation Measurements 34 (2001) 559–566 563

Fig. 2. A few examples of porous structures produced in thin polymeric !lms using various methods of irradiation and chemical treatment: (A)cross section of a polycarbonate TM with cylindrical non-parallel pore channels; (B) polypropylene TM with slightly conical (tapered towardsthe center) parallel pores; (C) polyethylene terephthalate TM with cigar-like pores; (D) polyethylene terephthalate TM with “bow-tie” pores.

pores can be modi!ed by covalent binding of charged groupsor by adsorption of ionic polyelectrolytes (Froehlich andWoermann, 1986). The immobilization of aminoacids to thePET track membranes based on the reactions of end carboxyland hydroxyl groups was reported (Marchand-Brynaertet al., 1995; Mougenot et al., 1996). However, the surfacedensity of the immobilized in this way species is ratherlow.The radiation-induced graft polymerization onto track

membranes is a process which has been studied in moredetail (Zhitariuk et al., 1989; Zhitariuk, 1993; Tischenkoet al., 1991; Shtanko and Zhitariuk, 1995). Styrene (St),methacrylic acid (MAA), N -vinyl pyrrolidone (VP),2-methyl 5-vinyl pyridine (2M5VP), N -isopropyl acryl-amide (NIPAAM) and some other monomers have beengrafted onto PET track membranes. Grafting of St in-creases the chemical resistance and makes the membranehydrophobic. MAA and VP were grafted onto TMs to in-crease wettability which is especially important when aque-ous solutions are !ltered through small-pore membranes.2M5VP was grafted with the aim to make the membranehydrophilic and change its surface charge from negative topositive. During the past decade the grafting of NIPAAMand other intelligent polymers were extensively studied inthe research work carried out at TRCRE (Takasaki) andGSI (Darmstadt) (Yoshida et al., 1993, 1997; Reber et al.,1995).

7. Applications

Applications of commercially produced track membranescan be categorized into three groups: (i) process !ltration;(ii) cell culture; (iii) laboratory !ltration. The process !l-tration implies the use of membranes mostly in the formof cartridges with a membrane area of at least 1 m2. Pu-ri!cation of deionized water in microelectronics, !ltrationof beverages, separation and concentration of various sus-pensions are typical examples. There is a strong competi-tion with other types of membranes available on the mar-ket. Casting membranes often provide a higher dirt load-ing capacity and a higher throughput. For this reason theuse of track membranes in this !eld is still limited (Brock,1984).In the recent years a series of products were de-

veloped for the use in the domain called cell and tis-sue culture (Stevenson et al., 1988; Sergent-Engelenet al., 1990; Peterson and Gruenhaupt, 1990; Roth-man and Orci, 1990). Adapted over the years to a va-riety of cell types, porous membrane !lters are nowrecognized as providing signi!cant advantages for cul-tivating cells and studying the cellular activities suchas transport, absorption and secretion (van Hinsberghet al., 1990). The use of permeable support systems basedon TMs has proven to be a valuable tool in the cell biology(Costar=Nuclepore Catalog, 1992).

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Figure 1: Magnified membrane with various pore distributions and sizes [7].membrane_photo

2 Darcy Flow Modeldarcymodel

The modeling throughout this section assumes that the membrane is flat and lies in the (Y, Z)-plane,with unidirectional Darcy flow through the membrane in the positive X-direction. The membraneproperties and flow are assumed homogeneous in the (Y, Z)-plane, but membrane structure mayvary internally in the X-direction (depth-dependent permeability) thus we seek a solution in whichproperties vary only in X and in time T . Throughout this section we use uppercase fonts todenote dimensional quantities; lowercase fonts, introduced in §sec:press2.1.1, §sec:flux2.1.2, §sec:modified_darcy_scaling2.2.1 and §sec:flux_modified2.2.2, willbe dimensionless.

The superficial Darcy velocity U = (U(X, T ), 0, 0) within the membrane is given in terms of thepressure P by

U = −K(X, T )

µ

∂P

∂X,

∂

∂X

(K(X, T )

∂P

∂X

)= 0, 0 ≤ X ≤ D, (1) eq:darcy

where K(X, T ) is the membrane permeability at depth X. We consider two driving mechanisms: (i)constant pressure drop across the membrane specified; and (ii) constant flux through the membranespecified. In the former case the flux will decrease in time as the membrane becomes fouled; in thelatter, the pressure drop required to sustain the constant flux will rise as fouling occurs. We willfocus primarily on case (i) in this paper, and so assume this in the following model description;our simulations for the constant flux scenario shown later require minor modifications to the theory(§sec:flux2.1.2). With constant pressure drop, the conditions applied are

P (0, T ) = P0, P (D, T ) = 0. (2) pressBC

The key modeling challenge is how to link the permeability K(X, T ) to measurable membranecharacteristics in order to obtain a predictive model. In this section we consider a simple model inwhich the membrane consists of a series of identical axisymmetric pores of variable radius A(X, T ),which traverse the entire membrane. The basic setup is schematized in Figure

pore-schem2: we consider

a filtrate, carrying some concentration C of particles, which are deposited within the pore. Wesuppose the pores to be arranged in a square repeating lattice, with period 2W .

Mass conservation shows that the pore velocity, Up (the cross-sectionally averaged axial velocitywithin each pore), satisfies

(πA2Up)X = 0, (3) pore

while Darcy’s law for the superficial velocity U within the pore gives

U = − πA4

8µ(2W )2

∂P

∂X= −φKp

µ

∂P

∂X, (4) darcy

3

Membrane filters: Thin layers of porous media, through which“feed solution”, carrying particles, passes. Designed to removeparticles 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 Filters: Fouling

During filtration the filter becomes fouled, which increases itsresistance and lowers filtration efficiency.

Several different modes of fouling:

Deposition (adsorption) of small particles on the pore wallswithin membrane;Deposition of large particles on top of membrane (sieving, orblocking);Cake formation, which occurs in the late stages of filtration(think of a coffee filter).

identical to that for the un-fouled membrane; the 50%decline in filtrate flux due to IgG fouling had no effect oneither the initial LRV or the rate of LRV decline during thesubsequent phage challenge. These data also demonstrate theneed to perform virus spiking studies over the full processthroughput, since the initial LRV after prefouling is unlikelyto be representative of the true process LRV.

The results in Figure 2 demonstrate that LRV for theDV20 membrane is strongly dependent on throughput of thevirus challenge, in contrast to the strong dependence on fluxdecline seen by Bolton et al.3 The different behaviorobserved in these studies is likely due to differences in themorphology of the virus filters; the DV20 membrane is rela-tively homogeneous while the Viresolve NFP membranesexamined by Bolton et al.3 are highly asymmetric with avery open structure (with micron-size pores) at the inletregion of the filter. It is also possible that the results areaffected by differences in the proteins; Bolton et al.3 prefil-tered their IgG through the Viresolve Prefilter media whilethe IgG used in this work was prefiltered only through a 0.2-mm sterilizing grade filter. SEC analysis of the IgG used inthis work showed no measurable aggregates >40 nm in size,suggesting that the protein fouling was not due to cake for-mation. Note that Khan et al.7 observed an increase in LRVwith increasing throughput due to protein cake formation,which is exactly opposite to the behavior seen in this study.

Confocal microscopy

Figure 3 shows a fluorescent image of a cross-sectionthrough a single layer of two DV20 membranes after filtra-

tion of 11 L/m2 (top panel) and 44 L/m2 (bottom panel) of asuspension containing approximately 108 pfu/mL of the fluo-rescently labeled PP7 bacteriophage. These cross-sectionalviews were constructed from 54 in-focus images of the x–yplanes inside the membrane. These images were stackedalong the z-axis and then sliced through an arbitrary x–zplane, approximately 1 mm thick, using the Olympus Fluo-ViewTM viewer software. The fluorescently labeled PP7 bac-teriophage are easily visible throughout the upper region ofthe DV20 membrane, extending approximately one-quarterof the way through the approximately 40 mm thick mem-brane. The depth of penetration was very similar at the twoloadings, with the image at 44 L/m2 showing a much greaterfluorescent intensity. Additional details on the confocalmicroscopy are provided by Bakhshayeshi et al.19

Independent experiments were performed to verify that themajority of filtered bacteriophage actually enter the DV20membrane. 100 mL of a solution containing a mixture of 2.83 107 pfu/mL PP7 and 2.5 3 104 pfu/mL PR772 bacterio-phages were filtered through a DV20 membrane. The filtratewas collected in four samples, each containing approximately24 mL, with approximately 4 mL of retentate left in the hous-ing above the filter. No PR772 phage were observed in the fil-trate samples and less than 0.01% of the PP7 phage passedthrough the membrane. The residual retentate was collectedby draining the housing, with any remaining phage collectedby circulating 200 mL of acetate buffer through the filterheadspace upstream of the membrane at a flow rate of approx-imately 40 mL/min. Approximately 20 6 15% of the PP7phage were recovered in the residual retentate plus bufferrinse, with the remainder being retained within the filter (con-sistent with the confocal image in Figure 3). In contrast,75 6 25% of the large PR772 bacteriophage were recoveredin the residual retentate plus the buffer rinse. The large errorbars on the calculated values of the percent recovery reflectthe inherent uncertainties in the plaque assay.

Internal polarization model

The confocal images in Figure 3 suggest that the retainedbacteriophage accumulate within the upper region of themembrane, which we refer to as the “reservoir zone” withinthe filter. This accumulation of retained virus would lead toa reduction in LRV as the “rejection zone” of the membraneis challenged by a continually increasing concentration ofvirus from the reservoir zone. As with classical membraneprocesses, the polarization occurs when the virus is selec-tively retained by the membrane so that the concentrationincreases near the surface of the rejection layer. In contrastto standard concentration polarization theory, rejection ininternal polarization occurs within the membrane after con-vective transfer of viruses through the reservoir zone. Thethickness of this internal concentration polarization layer isthus determined by the physical dimensions of the reservoirzone as opposed to the balance between convection and dif-fusion in a concentration polarization boundary layer.

The slightly asymmetric structure of the Ultipor VF GradeDV20 membrane, with a more open pore size in the reservoirzone and a tighter pore size near the filter exit, has beenobserved previously. In contrast to scanning electron micros-copy cross-sections showing relative homogeneity, dextranretention tests performed with the membrane oriented with theshiny-side up yielded contrasting results to tests with theshiny-side down.20 Further demonstration of this asymmetry

Figure 3. Confocal scanning image of the cross-section of aDV20 membrane after filtration of 11 L/m2 (toppanel) and 44 L/m2 (bottom panel) of a suspensioncontaining approximately 108 pfu/mL fluorescentlylabeled PP7 bacteriophage.

4 Biotechnol. Prog., 2014, Vol. 00, No. 00


Fouling & efficiency: constant pressure

Filtration generally takes place under one of two scenarios:constant pressure or constant flux. With constant pressure, as thefouling occurs, and system resistance increases, the flux decreasesmonotonically. Once flux falls below some threshold value the filtermust be discarded (or cleaned). A key indicator of filterperformance is provided by flux-throughput curves. Optimalperformance would be to maintain the flux high for as long aspossible.


Fouling & efficiency: constant flux

In the constant flux scenario a pump is used to maintain aconstant rate of throughput. As fouling occurs, the operatingpressure required to drive the pump increases. Once the drivingpressure passes some threshold, the filter is deemed unsustainableand must be discarded (or cleaned). In this case a key performancecharacteristic is provided by the (inverse) pressure–throughputcurve for the filter.

Figure 10. A plot for inverse of filtration resistance across various filter formats: (a) un-pre-

filtered feed, DMEM (13.4g/L) supplemented with 0.3g/L Hy-SoyT hydrolysate, and (b) shows

results of using DMEM (13.4g/L) supplemented with 0.3g/L Hy-SoyT hydrolysate, pre-filtered

through a 10µm rated filter. The fouling characteristic of a 1-inch LOP cartridge element,

dictated by the shape of the curve, is similar to disc for a 10 µm pre-filtered feed. Also, an

increase in throughput for pre-filtered feed indicates the presence of >10 µm size particles in the

initial feed solution. Filtration was performed at 500LMH operating flux using 0.2µm

PES/0.1µm PVDF membrane.

© 2014 Pall Corporation.

0.0

0.2

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0.8

1.0

0 40 80 120 160Throughput [L/m2]

Initi

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ress

ure/

Inst

anta

neou

s Pr

essu

re

47mm disc - Filter 147mm disc - Filter 247mm disc - Filter 31" laid over pleat cartridge- Filter 11" laid over pleat cartridge - Filter 21" laid over pleat cartridge - Filter 3

0.0

0.2

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0 50 100 150 200 250Throughput [L/m2]

Initi

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47mm disc - Filter 147mm disc - Filter 21" laid-over pleat cartridge - Filter 11" laid-over pleat cartridge - Filter 2


Background to the problem: MPI workshops

MPI: “Mathematical Problems in Industry”.

Workshops jointly sponsored by NSF, IMA and participatingcompanies.

3

The Problem

! Competing requirements: Fine separation with low power consumption

! Obvious Resolution: Use largest pore size and void fraction consistent with separation requirement

! Complication: Fouling depends on many parameters, not just pore size and void fraction.

! Challenge: Devise a common mathematical description of membrane morphology which

– Distinguishes common membrane types

– Connects to separation and fouling performance


Background to the problem: MPI workshops 2013/2014

Many major companies heavily invested in membranefiltration, e.g.

W.L. Gore & Associates;

Pall Corporation ($ 2.6bn).3

The Problem

! Competing requirements: Fine separation with low power consumption

! Obvious Resolution: Use largest pore size and void fraction consistent with separation requirement

! Complication: Fouling depends on many parameters, not just pore size and void fraction.

! Challenge: Devise a common mathematical description of membrane morphology which

– Distinguishes common membrane types

– Connects to separation and fouling performance

Multi-billion $$ industry, in the US alone.

At 2013/2014 workshops both companies brought problemspertaining to better prediction of membrane filterperformance.

2013: Pall asked “Why do our pleated membrane filtersunderperform?”

2014: Pall asked “How can we better predict membrane filterefficiency from known membrane characteristics?”


Membrane filtration: Efficiency is critical

Naive approach to filtration says: choose a filter with poressmaller than the particles you wish to remove.

This is highly inefficient however – system resistance then veryhigh and huge driving pressures required (expensive).

In practice pores are larger (perhaps 10×) than most particlesin the feed, and much of the filtration takes place within themembrane interior via adsorption (detailed mechanisms largelyunknown).

For large-scale filtration space can be an issue – may want topack filters into a small volume. However, this can also leadto increases in system resistance, and efficiency losses.

Also want to maximize throughput and filter lifetime.

Modeling therefore has a key role to play in investigatingefficient filtration scenarios.


Pleated filter cartridges (Pall, MPI 2013)

A.I. Brown et al. / Journal of Membrane Science 341 (2009) 76–83 77

Fig. 1. (a) Exploded view of a 1′′ cartridge (without housing), illustrating the wayin which the pleated membrane is installed inside the cartridge; in this case a PPDof 0.65 is shown. dC represents the diameter of the cartridge and LC the length. (b)Illustration of the different pleat densities and membrane configurations used inthis work and the total membrane surface area available per 1′′ cartridge.

to reduce the overall pressure drop by optimising the pleat countper unit length. It has been reported that pleat height also impactsupon overall pressure drop [9]. The most recent studies [5,6,14]have produced models utilising computational fluid dynamics toexplore the influence that pleat designhas onairfilterperformance.These include compression of the medium, pleat deformation andpleat crowding at high PPD values. When simulations were per-formed incorporating medium compression and area loss, therewas good agreement with experimental data. Examination of thedata presented in these studies suggested that of the two factorsincorporated into themodel, area loss appeared tohave the greatestimpact onperformance [5,6]. Airfilters are similar in basic design tothose used in the biopharmaceutical industry, but as noted aboveare generally optimised so as to reduce the pressure drop acrossthe pleat. The cartridges used in biopharmaceutical manufactureare optimised to maximise the filtration area and to handle higherviscosity fluids. Whilst findings generated for air filters may wellbear on the characteristics of their biopharmaceutical counterpartsthese key differences must be borne in mind.

For 0.45 m PVDF membrane filter cartridges used with liquidfeeds, pleating has been associated with a drop in cartridge per-formance when compared with flat sheet membranes [15]. Severalresearchers [16,17] have considered the effect of the permeabilityof the drainage material upon cartridge performance. Golan andParekh [16] developed permeability models to account for the dif-ferent process feeds that a cartridge may be required to filter. Adrainage permeability term to account for the compressibility ofthe drainagematerial,was introduced. This varied across the length

Fig. 2. (a) Photograph of a cross-section of 10′′ UEAV membrane with an Ultipleat®

pleat configuration and (b) photograph of a cross-section of a 10′′ EAV cartridgewitha Fan pleat configuration.

of the pleat,but no experimental datawas presented formodel vali-dation.Giglia and Yavorsky [17] built upon this work and comparedthe performance of a range of sterile cartridges consisting of differ-ent pore sizes and membrane materials. A difference in membranepermeabilitywas observedbetweenflat sheet discs and10′′ pleatedmembrane cartridges as has been reported previously [15,18].

This studypresents a systematic experimental investigation intothe effect that membrane pleating has upon the clean water fluxof a large scale cartridge (membrane area ∼1m2). By studying thechanges in clean water flux when transitioning from a flat sheetdisc of membrane to a large scale cartridge we aim to describe andto quantify the effect that pleat design has upon cartridge perfor-mance.Dilute yeast suspensions are alsoused as aprobe to visualiseand quantify the accessibility of micron-sized particles, such asprotein aggregates, into the membrane pleats.

2. Experimental

2.1. Fabrication and design of pleated membrane cartridges

In order to investigate the influence of pleat design and configu-ration a series of 2.54 cm(1′′) sections froma25.4 cm(10′′) cartridge(Pall Europe Ltd., Portsmouth, UK) were specially fabricated. Thesecartridges were designed to provide a range of pleat geometriesbetween that of 10′′ cartridges and flat sheet discs. Fully packed 1′′

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 intoannular cylindrical cartridge.


Pleated filter cartridgesA.I. Brown et al. / Journal of Membrane Science 341 (2009) 76–83 79

Fig. 3. (a) Schematic diagram of pleated membrane cartridge housing (not to scale). Length of cartridge, LC, varies depending upon whether a 10′′ or 1′′ cartridge is insertedinto the base unit. All other dimensions are the same, regardless of cartridge height. (b) Piping and instrumentation diagram illustrating the experimental rigs utilised forinvestigation of the different cartridge configurations of Supor® EAV 0.2 !m rated membrane. V = vessel, P = pump, HV = hand valve, PG = pressure gauge, FI = flow indicator.(c) Representation of the experimental set-up for small-scale flat sheet discs of 25 mm diameter. Rigs operated as described in Section 2.2.

The variation in the grey level at various points within the imagewas recorded. Grey scale variation data was plotted using a runningaverage method with a sampling proportion of 0.1.

3. Results and discussion

3.1. Effect of pleat characteristics upon clean water flux

Initial experiments aimed to demonstrate the differencebetween flat sheet and pleated membrane performance when nor-malised for membrane surface area. The clean water flux for both aflat sheet and a 10′′ pleated membrane cartridge were determinedusing a 0.2 !m rated Supor® EAV membrane. The average flux forthe two different configurations are given in Fig. 4 as a function ofapplied transmembrane pressure (TMP). The smaller area of the flatsheet disc lead to higher variation in the quantification of the flux(indicated by the large error bars) than for the 10′′ cartridge. Thisis in agreement with previous findings when working with smallareas of membrane [20]. From Fig. 4 it can be observed that at an

equivalent TMP, the permeate flux was considerably lower for the10′′ cartridge than for the flat sheet disc. The reduction in flux isabout 53% on average. This was identical to a measured flux reduc-tion of 53% for a PVDF sterilising grade membrane compared to aflat sheet [18], though is lower than a flux reduction of 70% for apleated glass fibre cartridge [5].

Average membrane resistances (RM) for the flat sheet discsand the 10′′ cartridge shown in Fig. 4 were calculated [21] as1.60 × 1010 m−1 and 3.43 × 1010 m−1, respectively. In principle themembrane resistance due to the porosity of the membrane and theresistance to flow that the pores create [22] should be identical sinceboth the cartridge and flat sheet are made from the same material.

In order to investigate the influence of membrane pleating on themeasured membrane resistances a series of 1′′ pleated membranecartridges were specially fabricated as described in Section 2.1. Theproperties of these are summarised in Table 1. The measured waterflux profiles for various specially fabricated 1′′ cartridges with aFan pleat and hP = 15 mm and varying PPD are shown in Fig. 5(a). Itcan clearly be seen that as the pleat structure becomes more open

Cartridge placed inside external housing.

The feed solution is driven from exterior to interior of thecylinder, passing across the filter membrane.

The filtration efficiency is not what manufacturers would wish,however.


Pleated filter inefficiency

Figure 3. Plot showing the flux ratios with various filter formats and various sterilizing -grade

media. The test fluid was deionized water. The flux ratios ranged from 0.50 to 0.85. Among

different filters tested, 0.2µm PES/0.2µm PVDF pleated cartridges had the lowest flux ratio.

LOP construction is referred as LOP. The error bar for both disc and pleated filter shows

variability observed among different runs.

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

0.2µm PES/0.2µm PVDF

LOPsmall-core

0.65µm PES/0.2µm PES

LOPstandard-core

0.2µm PES/0.1µm PVDF

LOPsmall-core

0.5µm PES/0.1µm PESFanpleat

standard-core

0.2µm PES/0.1µm PESFanpleat

standard-core

Flu

x R

ati

o r

ela

tive

to

fla

t d

isc Flat Disc Format

Pleated Cartridge

From Kumar, Martin & Kuriyel, 2015


Pleated filter: mathematical modeling

Can mathematical modeling pinpoint the reasons for the lowefficiency, and perhaps (ultimately) suggest remedies?

Simplify geometry to obtain tractable model.

A.I.

Brow

net

al./

Jour

nalo

fMem

bran

eScienc

e34

1 (2

009)

76–

8377

Fig.

1.(a

)Ex

plo

ded

view

ofa

1′′

cartridge

(withouthousing),i

llustra

tingth

ew

ayin

whichth

eple

ated

mem

bra

ne

isin

stal

led

insideth

eca

rtridge

;in

this

case

aPP

Dof0.6

5is

show

n.d

Cre

pre

sents

the

dia

met

erofth

eca

rtridge

and

L Cth

ele

ngt

h.(

b)

Illu

stra

tion

ofth

ediffere

ntple

atden

sities

and

mem

bra

ne

configu

rations

use

din

this

work

andth

eto

talm

embra

ne

surfac

ear

eaav

aila

ble

per

1′′ca

rtridge

.

tore

duce

the

over

allpre

ssure

dro

pby

optim

isin

gth

eple

atco

unt

per

unit

lengt

h.Ithas

bee

nre

ported

that

ple

athei

ghtal

soim

pac

tsupon

over

allpre

ssure

dro

p[9

].Th

em

ost

rece

nt

studie

s[5

,6,14]

hav

epro

duce

dm

odel

sutilising

com

putational

fluid

dyn

amic

sto

explore

thein

fluen

ceth

atple

atdes

ignhas

onairfilter

per

form

ance

.Th

ese

include

com

pre

ssio

nofth

em

ediu

m,p

leat

def

orm

atio

nan

dple

atcr

owdin

gat

hig

hPP

Dva

lues

.W

hen

sim

ulations

wer

eper

-fo

rmed

inco

rpora

ting

med

ium

com

pre

ssio

nan

dar

ealo

ss,th

ere

was

good

agre

emen

tw

ith

exper

imen

taldat

a.Ex

amin

atio

nofth

edat

apre

sente

din

thes

est

udie

ssu

gges

ted

that

ofth

etw

ofa

ctors

inco

rpora

ted

into

them

odel,a

realo

ssap

pea

redto

hav

eth

egr

eate

stim

pac

tonper

form

ance

[5,6

].Air

filter

sar

esim

ilar

inbas

icdes

ignto

those

use

din

the

bio

phar

mac

eutica

lin

dustry,b

utas

not

edab

ove

are

gener

ally

optim

ised

soas

tore

duce

the

pre

ssure

dro

pac

ross

the

ple

at.Th

eca

rtridge

suse

din

bio

phar

mac

eutica

lm

anufa

cture

are

optim

ised

tom

axim

iseth

efiltra

tion

area

andto

han

dle

hig

her

visc

osity

fluid

s.W

hilst

findin

gsge

ner

ated

forairfilter

sm

ayw

ell

bea

ronth

ech

arac

terist

icsoft

heirbio

phar

mac

eutica

lcounte

rpar

tsth

ese

key

differe

nce

sm

ust

be

born

ein

min

d.

For0.4

5 m

PVDF

mem

bra

ne

filter

cartridge

suse

dw

ith

liqu

idfe

eds,

ple

atin

ghas

bee

nas

soci

ated

with

adro

pin

cartridge

per

-fo

rman

cew

hen

com

par

edw

ith

flat

shee

tm

embra

nes

[15].Se

vera

lre

sear

cher

s[1

6,17]hav

eco

nsider

edth

eef

fect

ofth

eper

mea

bility

ofth

edra

inag

em

ater

ialupon

cartridge

per

form

ance

.Gola

nan

dPa

rekh

[16]dev

eloped

per

mea

bility

model

sto

acco

untfo

rth

edif-

fere

ntpro

cess

feed

sth

ata

cartridge

may

be

requ

ired

tofilter.A

dra

inag

eper

mea

bility

term

toac

count

forth

eco

mpre

ssib

ility

of

thedra

inag

em

ater

ial,w

asin

troduce

d.T

his

varied

acro

ssth

ele

ngt

h

Fig.

2.

(a)Ph

otogr

aph

ofa

cross

-sec

tion

of10

′′UEA

Vm

embra

new

ith

anUltip

leat

®

ple

atco

nfigu

ration

and

(b)p

hot

ogr

aph

ofa

cross

-sec

tion

ofa

10′′EA

Vca

rtridge

with

aFa

nple

atco

nfigu

ration.

oft

heple

at,b

utn

oex

per

imen

tald

ataw

aspre

sente

dfo

rm

odel

vali-

dat

ion.G

igliaan

dYa

vors

ky[1

7]b

uiltuponth

isw

ork

and

com

par

edth

eper

form

ance

ofa

range

ofs

terile

cartridge

sco

nsist

ing

ofd

iffer-

entpore

size

san

dm

embra

nem

ater

ials.A

differe

nce

inm

embra

ne

per

mea

bilityw

asobse

rved

bet

wee

nflat

shee

tdis

csan

d10

′′ple

ated

mem

bra

ne

cartridge

sas

has

bee

nre

ported

pre

viousl

y[1

5,18].

This

studypre

sents

asy

stem

atic

exper

imen

talinve

stig

atio

nin

toth

eef

fect

that

mem

bra

ne

ple

atin

ghas

upon

the

clea

nw

ater

flux

ofa

larg

esc

ale

cartridge

(mem

bra

ne

area

∼1m

2).By

studyi

ngth

ech

ange

sin

clea

nw

ater

flux

when

tran

sitionin

gfrom

aflat

shee

tdis

cofm

embra

neto

alarg

esc

ale

cartridge

weai

mto

des

crib

ean

dto

quan

tify

the

effe

ctth

atple

atdes

ign

has

upon

cartridge

per

for-

man

ce.D

ilute

yeas

tsusp

ensionsar

eal

souse

das

apro

beto

visu

alise

and

quan

tify

the

acce

ssib

ility

ofm

icro

n-s

ized

par

ticles

,su

chas

pro

tein

aggr

egat

es,into

them

embra

neple

ats.

2.Exp

erim

ental

2.1.

Fabricatio

nan

dde

sign

ofpl

eate

dm

embr

aneca

rtridg

es

Inord

erto

inve

stig

ateth

ein

fluen

ceofp

leat

des

ign

and

configu

-ra

tionase

ries

of2

.54

cm(1

′′ )se

ctio

nsfrom

a25.4

cm(1

0′′ )

cartridge

(Pal

lEuro

peLt

d.,Po

rtsm

outh

,UK)w

ere

spec

ially

fabrica

ted.T

hes

eca

rtridge

sw

ere

des

igned

topro

vide

ara

nge

ofple

atge

om

etries

bet

wee

nth

atof10

′′ca

rtridge

san

dflat

shee

tdis

cs.F

ully

pac

ked

1′′S.G

iglia

etal

./Jo

urna

lofM

embr

ane

Scie

nce

365

(201

0) 3

47–3

5535

1

Fig.

7.D

efine

dpe

rfor

man

cera

nge

for

smal

lsc

ale

devi

cere

duce

ssc

ale-

upun

cer-

tain

ty.

isde

fined

asth

esc

alin

gfa

ctor

unce

rtai

nty

rati

o(U

sf)

acco

rdin

gto

the

follo

win

gfo

rmul

a:

Usf

=F h

/Sl

F l/S

h=

F h F l

S h S l(2

)

whe

reF h

isth

efu

llsc

ale

devi

cehi

ghen

dpo

tent

ial

perf

orm

ance

,F l

isth

efu

llsc

ale

devi

celo

wen

dpo

tent

ialp

erfo

rman

ce,S

his

the

scal

ing

devi

cehi

ghen

dpo

tent

ialp

erfo

rman

ce,a

ndS l

isth

esc

alin

gde

vice

low

end

pote

ntia

lper

form

ance

.

4.2.

Redu

cing

mem

bran

eva

riab

ility

unce

rtai

nty

Bysp

ecif

ying

only

ana

rrow

rang

eof

the

dist

ribu

tion

for

scal

ing

devi

ces,

the

unce

rtai

nty

insc

alin

gfr

omsm

all

scal

eto

larg

esc

ale

devi

ces

ism

inim

ized

(lar

gesc

ale

devi

ces

can

cont

ain

any

qual

i-fie

dm

embr

ane,

soth

esy

stem

mus

tbe

size

dto

acco

mm

odat

eth

efu

llra

nge

ofpo

tent

ialm

embr

ane

perf

orm

ance

espe

cial

lyif

filte

rsar

ere

plac

edfo

reac

hba

tch

orar

eot

herw

ise

peri

odic

ally

repl

aced

).Fo

rex

ampl

e,if

only

the

mid

dle

thir

dof

the

dist

ribu

tion

rang

eis

sele

cted

fors

mal

lsca

lede

vice

s,as

illus

trat

edin

Fig.

7,th

enth

epe

r-fo

rman

ceof

the

smal

lsca

lede

vice

will

rang

efr

om0.

9to

1.1.

Sinc

eth

ela

rge

scal

ede

vice

sw

illra

nge

from

0.7

to1.

3,th

esc

alin

gsa

fety

fact

orw

illbe

(in

acco

rdan

cew

ith

Eq.(

1)),

(1.3

/0.9

)/(0

.7/1

.1)=

2.3,

whe

reS h

beco

mes

the

scal

ing

devi

cehi

ghen

dpo

tent

ial

perf

or-

man

cew

ithi

nth

esu

bset

ofth

edi

stri

buti

on,a

ndS l

beco

mes

the

scal

ing

devi

celo

wen

dpo

tent

ialp

erfo

rman

cew

ithi

nth

esu

bset

ofth

edi

stri

buti

on.I

nth

isex

ampl

e,th

ism

etho

dre

sult

sin

abou

ta

0

20406080

100

120

140

140

120

100

8060

4020

0Pr

edic

ted

Flow

Rat

e (lp

m)

Measured Flow Rate (lpm)

SHR

P-G

SHR

P-A

SHC

-GSH

C-A

SHF-

ASH

F-G

SHR

-ASH

R-G

Fig.

8.W

ater

flow

rate

scal

ing

pred

icti

ons,

excl

udin

gho

usin

gpr

essu

relo

sses

and

plea

ting

effe

cts.

0246810

020

4060

8010

012

014

016

0

Flow

Rat

e (L

PM)

Housing ΔP (kPa)

0123

040

8012

016

0Fl

ow R

ate

(LPM

)

SQRT(Housing ΔP (kPa))

Fig.

9.N

on-m

embr

ane

pres

sure

loss

esas

soci

ated

wit

hho

usin

gan

dde

vice

con-

stru

ctio

n.In

sets

how

sex

pect

edlin

earr

elat

ions

hip

betw

een

squa

rero

otof

pres

sure

drop

and

flow

rate

.

35%

savi

ngs

insc

ale-

upsi

zing

requ

irem

ents

com

pare

dto

conv

en-

tion

alra

ndom

mem

bran

ese

lect

ion

used

fors

calin

gde

vice

s.It

isno

tne

cess

ary

that

the

mem

bran

ein

the

scal

ing

devi

ceor

igin

ate

from

the

cent

erpo

rtio

nof

the

mem

bran

epo

pula

tion

.Any

defin

edpo

r-ti

onof

the

mem

bran

epo

pula

tion

will

redu

cesc

alin

gun

cert

aint

yin

acco

rdan

cew

ith

Eq.(

1).

Fig.

10.

Plea

ted

stru

ctur

es.

Membrane

No fluxPleat tip

y=HInflow

y=−H Outflow

x

Membrane, thickness D

Pleat valleyNo flux

y

L


Pleated filter: idealized geometry

Membrane

No fluxPleat tip

y=HInflow

y=−H Outflow

x


Pleat valleyNo flux

y

LA.I. Brown et al. / Journal of Membrane Science 341 (2009) 76–83 79







Average membrane resistances (RM) for the flat sheet discsand the 10′′ cartridge shown in Fig. 4 were calculated [21] as1.60 × 1010 m−1 and 3.43 × 1010 m−1, respectively. In principle themembrane resistance due to the porosity of the membrane and theresistance to flow that the pores create[22] should be identical sinceboth the cartridge and flat sheet are made from the same material.


(b) Flow to permeate outlet

Car

trid

ge

Ple

at v

alle

ys

Ple

at t

ips

ZY

feed inletFlow from

X

Figure 2: (a) (From [5], reproduced with permission.) Schematic showing the external housing and pleated filtercartridge within it. (b) Idealization of the pleated filter cartridge geometry (the X-direction from Fig.3(b) below isthe inward radial direction, while the Y -direction is arc length around the outer cylinder boundary, measured in thedirection indicated).

(a)

S.G

iglia

etal

./Jo

urna

lofM

embr

ane

Scie

nce

365

(201

0) 3

47–3

5535

1

Fig.

7.D

efine

dpe

rfor

man

cera

nge

for

smal

lsc

ale

devi

cere

duce

ssc

ale-

upun

cer-

tain

ty.

isde

fined

asth

esc

alin

gfa

ctor

unce

rtai

nty

rati

o(U

sf)

acco

rdin

gto

the

follo

win

gfo

rmul

a:

Usf

=F h

/Sl

F l/S

h=

F h F l

S h S l(2

)

whe

reF h

isth

efu

llsc

ale

devi

cehi

ghen

dpo

tent

ial

perf

orm

ance

,F l

isth

efu

llsc

ale

devi

celo

wen

dpo

tent

ialp

erfo

rman

ce,S

his

the

scal

ing

devi

cehi

ghen

dpo

tent

ialp

erfo

rman

ce,a

ndS l

isth

esc

alin

gde

vice

low

end

pote

ntia

lper

form

ance

.

4.2.

Redu

cing

mem

bran

eva

riab

ility

unce

rtai

nty

Bysp

ecif

ying

only

ana

rrow

rang

eof

the

dist

ribu

tion

for

scal

ing

devi

ces,

the

unce

rtai

nty

insc

alin

gfr

omsm

all

scal

eto

larg

esc

ale

devi

ces

ism

inim

ized

(lar

gesc

ale

devi

ces

can

cont

ain

any

qual

i-fie

dm

embr

ane,

soth

esy

stem

mus

tbe

size

dto

acco

mm

odat

eth

efu

llra

nge

ofpo

tent

ialm

embr

ane

perf

orm

ance

espe

cial

lyif

filte

rsar

ere

plac

edfo

reac

hba

tch

orar

eot

herw

ise

peri

odic

ally

repl

aced

).Fo

rex

ampl

e,if

only

the

mid

dle

thir

dof

the

dist

ribu

tion

rang

eis

sele

cted

fors

mal

lsca

lede

vice

s,as

illus

trat

edin

Fig.

7,th

enth

epe

r-fo

rman

ceof

the

smal

lsca

lede

vice

will

rang

efr

om0.

9to

1.1.

Sinc

eth

ela

rge

scal

ede

vice

sw

illra

nge

from

0.7

to1.

3,th

esc

alin

gsa

fety

fact

orw

illbe

(in

acco

rdan

cew

ith

Eq.(

1)),

(1.3

/0.9

)/(0

.7/1

.1)=

2.3,

whe

reS h

beco

mes

the

scal

ing

devi

cehi

ghen

dpo

tent

ial

perf

or-

man

cew

ithi

nth

esu

bset

ofth

edi

stri

buti

on,a

ndS l

beco

mes

the

scal

ing

devi

celo

wen

dpo

tent

ialp

erfo

rman

cew

ithi

nth

esu

bset

ofth

edi

stri

buti

on.I

nth

isex

ampl

e,th

ism

etho

dre

sult

sin

abou

ta

0

20406080

100

120

140

140

120

100

8060

4020

0Pr

edic

ted

Flow

Rat

e (lp

m)


SHR

P-G

SHR

P-A

SHC

-GSH

C-A

SHF-

ASH

F-G

SHR

-ASH

R-G

Fig.

8.W

ater

flow

rate

scal

ing

pred

icti

ons,

excl

udin

gho

usin

gpr

essu

relo

sses

and

plea

ting

effe

cts.

0246810

020

4060

8010

012

014

016

0

Flow

Rat

e (L

PM)

Housing ΔP (kPa)

0123

040

8012

016

0Fl

ow R

ate

(LPM

)


Fig.

9.N

on-m

embr

ane

pres

sure

loss

esas

soci

ated

wit

hho

usin

gan

dde

vice

con-

stru

ctio

n.In

sets

how

sex

pect

edlin

earr

elat

ions

hip

betw

een

squa

rero

otof

pres

sure

drop

and

flow

rate

.

35%

savi

ngs

insc

ale-

upsi

zing

requ

irem

ents

com

pare

dto

conv

en-

tion

alra

ndom

mem

bran

ese

lect

ion

used

fors

calin

gde

vice

s.It

isno

tne

cess

ary

that

the

mem

bran

ein

the

scal

ing

devi

ceor

igin

ate

from

the

cent

erpo

rtio

nof

the

mem

bran

epo

pula

tion

.Any

defin

edpo

r-ti

onof

the

mem

bran

epo

pula

tion

will

redu

cesc

alin

gun

cert

aint

yin

acco

rdan

cew

ith

Eq.(

1).

Fig.

10.

Plea

ted

stru

ctur

es.

(b)

Symmetry line Y=H

Symmetry line Y=−H

Membrane

Membrane

Pleat valleyNo flux

OutflowNo flux

Inflow

Pleat tipX

Y

Figure 3: (a) Section of the pleated geometry, which is repeated periodically (adapted from [8]). The Z-axis infigure 2(b) is perpendicular to the page and out of it. Green/blue correspond to backer layers exterior/interior tothe annulus; gray represents the membrane filter (in reality much thinner than the backer layers). (b) Idealizedmembrane geometry to be considered in our model. Symmetry lines (dashed) are located at Y = ±H, and the pleatoccupies 0 ≤ X ≤ L.

3

A.I. Brown et al. / Journal of Membrane Science 341 (2009) 76–83 79







Average membrane resistances (RM) for the flat sheet discsand the 10′′ cartridge shown in Fig. 4 were calculated [21] as1.60 × 1010 m−1 and 3.43 × 1010 m−1, respectively. In principle themembrane resistance due to the porosity of the membrane and theresistance to flow that the pores create [22] should be identical sinceboth the cartridge and flat sheet are made from the same material.



Pleated filter: Modeling assumptions

Membrane

No fluxPleat tip

y=HInflow

y=−H Outflow

x


Pleat valleyNo flux

y

L

Periodic geometry – consider single pleat.

Neglect curvature & axial variation and simplify to 2Drectangular geometry.

Membrane thickness D, support layer thickness H, pleatlength L: D � H � L.

Neglect flow through pleat tips and valleys.

Pores cylindrical, initial radius A0, and traverse membrane.


Pleated filter: Modeling

Membrane

No fluxPleat tip

y=HInflow

y=−H Outflow

x


Pleat valleyNo flux

y

L

Y=−D/2

Y

−H

H

=0PX

=0

P =0

PX

=0

L X

Y

PY

0P=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


Flux through membrane pores

Support permeability K constant in time (no fouling) but mayvary 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 inseries and write

Qb,pore =1

Rb(P|Y =D/2 − P|Y =−D/2), Rb =

8µD

πA40

((A0

A

)4

+ ρb

).


Blocking & membrane permeability

Assume a bimodal distribution of particle sizes in feed: verysmall particles that are adsorbed within pores and shrinkthem; and large particles that can block pores from above.

Let N(X ,T ) be number density of unblocked pores, withN0 = 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 quantitiesby

Km =πA4

0

8

(N

(A0/A)4+

N0 − N

(A0/A)4 + ρb)

)

Close model by specifying evolution of N and A.

Permeability evolution

For large particles assume cumulative size distributionfunction G (S) (no. of particles per unit volume with radius< S). Then[

Probability per unit time thatpore 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 atypical large-particle size).

Adsorption: Pore radius A shrinks in time. Assume simplestpossible law: uniform adsorption within pores (will considermore complicated laws later):

∂A

∂T= −E , A|T=0 = A0.


Asymptotic solution to model

Introduce scalings

P = P0p, (X ,Y ) = L(x , εy) (ε = H/L), T = TBt,

K = Kavk, Km = Km0km, A(T ) = A0a(t)

(TB = 8µD/(πP0G∞A40) is pore-blocking timescale).

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

0 (x), p−0 (x).

Anticipate solution as asymptotic expansion in ε2.

Flux continuity across membrane requires

p+y |y=δ/2 = p−y |y=−δ/2 = ε2Γ

km

k[p+|y=δ/2 − p−|y=−δ/2],

whereΓ =

Km0L2

KavHD=

Km0

ε2δKav(δ = D/H)

measures relative importance of resistance of membrane andsupport material.


Asymptotics: ε2 � 1; δ � 1

Seek solution as asymptotic expansion in ε2.

Darcy model gives 2nd order coupled BVPs for p±0 (need togo to O(ε2) for solvability condition).

∂

∂x

(k(x)

∂p+0

∂x

)= Γkm(x)(p+

0 − p−0 ),

∂

∂x

(k(x)

∂p−0∂x

)= −Γkm(x)(p+

0 − p−0 ),

p+0 |x=0 = 1,

∂p+0

∂x

∣∣∣∣x=1

= 0,

∂p−0∂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.


Iterative solution scheme

At t = 0 assign km(x , 0) = km0 = 1 and support permeabilityk(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

πA50P0G∞

� 1.

4 Return to 1.

Iterative solution scheme

Simulate model for relevant parameter values, investigatingcases of uniform support permeability, and decreasing supportpermeability gradients k ′(x) < 0.A.I. Brown et al. / Journal of Membrane Science 341 (2009) 76–83 79







Average membrane resistances (RM) for the flat sheet discsand the 10′′ cartridge shown in Fig. 4 were calculated [21] as1.60 × 1010 m−1 and 3.43 × 1010 m−1, respectively. In principle themembrane resistance due to the porosity of the membrane and theresistance to flow that the pores create[22] should be identical sinceboth the cartridge and flat sheet are made from the same material.


(b) Flow to permeate outlet

Car

trid

ge

Ple

at v

alle

ys

Ple

at t

ips

ZY

feed inletFlow from

X

Figure 2: (a) (From [5], reproduced with permission.) Schematic showing the external housing and pleated filtercartridge within it. (b) Idealization of the pleated filter cartridge geometry (the X-direction from Fig.3(b) below isthe inward radial direction, while the Y -direction is arc length around the outer cylinder boundary, measured in thedirection indicated).

(a)

S.G

iglia

etal

./Jo

urna

lofM

embr

ane

Scie

nce

365

(201

0) 3

47–3

5535

1

Fig.

7.D

efine

dpe

rfor

man

cera

nge

for

smal

lsc

ale

devi

cere

duce

ssc

ale-

upun

cer-

tain

ty.

isde

fined

asth

esc

alin

gfa

ctor

unce

rtai

nty

rati

o(U

sf)

acco

rdin

gto

the

follo

win

gfo

rmul

a:

Usf

=F h

/Sl

F l/S

h=

F h F l

S h S l(2

)

whe

reF h

isth

efu

llsc

ale

devi

cehi

ghen

dpo

tent

ial

perf

orm

ance

,F l

isth

efu

llsc

ale

devi

celo

wen

dpo

tent

ialp

erfo

rman

ce,S

his

the

scal

ing

devi

cehi

ghen

dpo

tent

ialp

erfo

rman

ce,a

ndS l

isth

esc

alin

gde

vice

low

end

pote

ntia

lper

form

ance

.

4.2.

Redu

cing

mem

bran

eva

riab

ility

unce

rtai

nty

Bysp

ecif

ying

only

ana

rrow

rang

eof

the

dist

ribu

tion

for

scal

ing

devi

ces,

the

unce

rtai

nty

insc

alin

gfr

omsm

all

scal

eto

larg

esc

ale

devi

ces

ism

inim

ized

(lar

gesc

ale

devi

ces

can

cont

ain

any

qual

i-fie

dm

embr

ane,

soth

esy

stem

mus

tbe

size

dto

acco

mm

odat

eth

efu

llra

nge

ofpo

tent

ialm

embr

ane

perf

orm

ance

espe

cial

lyif

filte

rsar

ere

plac

edfo

reac

hba

tch

orar

eot

herw

ise

peri

odic

ally

repl

aced

).Fo

rex

ampl

e,if

only

the

mid

dle

thir

dof

the

dist

ribu

tion

rang

eis

sele

cted

fors

mal

lsca

lede

vice

s,as

illus

trat

edin

Fig.

7,th

enth

epe

r-fo

rman

ceof

the

smal

lsca

lede

vice

will

rang

efr

om0.

9to

1.1.

Sinc

eth

ela

rge

scal

ede

vice

sw

illra

nge

from

0.7

to1.

3,th

esc

alin

gsa

fety

fact

orw

illbe

(in

acco

rdan

cew

ith

Eq.(

1)),

(1.3

/0.9

)/(0

.7/1

.1)=

2.3,

whe

reS h

beco

mes

the

scal

ing

devi

cehi

ghen

dpo

tent

ial

perf

or-

man

cew

ithi

nth

esu

bset

ofth

edi

stri

buti

on,a

ndS l

beco

mes

the

scal

ing

devi

celo

wen

dpo

tent

ialp

erfo

rman

cew

ithi

nth

esu

bset

ofth

edi

stri

buti

on.I

nth

isex

ampl

e,th

ism

etho

dre

sult

sin

abou

ta

0

20406080

100

120

140

140

120

100

8060

4020

0Pr

edic

ted

Flow

Rat

e (lp

m)


SHR

P-G

SHR

P-A

SHC

-GSH

C-A

SHF-

ASH

F-G

SHR

-ASH

R-G

Fig.

8.W

ater

flow

rate

scal

ing

pred

icti

ons,

excl

udin

gho

usin

gpr

essu

relo

sses

and

plea

ting

effe

cts.

0246810

020

4060

8010

012

014

016

0

Flow

Rat

e (L

PM)

Housing ΔP (kPa)

0123

040

8012

016

0Fl

ow R

ate

(LPM

)


Fig.

9.N

on-m

embr

ane

pres

sure

loss

esas

soci

ated

wit

hho

usin

gan

dde

vice

con-

stru

ctio

n.In

sets

how

sex

pect

edlin

earr

elat

ions

hip

betw

een

squa

rero

otof

pres

sure

drop

and

flow

rate

.

35%

savi

ngs

insc

ale-

upsi

zing

requ

irem

ents

com

pare

dto

conv

en-

tion

alra

ndom

mem

bran

ese

lect

ion

used

fors

calin

gde

vice

s.It

isno

tne

cess

ary

that

the

mem

bran

ein

the

scal

ing

devi

ceor

igin

ate

from

the

cent

erpo

rtio

nof

the

mem

bran

epo

pula

tion

.Any

defin

edpo

r-ti

onof

the

mem

bran

epo

pula

tion

will

redu

cesc

alin

gun

cert

aint

yin

acco

rdan

cew

ith

Eq.(

1).

Fig.

10.

Plea

ted

stru

ctur

es.

(b)

Symmetry line Y=H

Symmetry line Y=−H

Membrane

Membrane

Pleat valleyNo flux

OutflowNo flux

Inflow

Pleat tipX

Y

Figure 3: (a) Section of the pleated geometry, which is repeated periodically (adapted from [8]). The Z-axis infigure 2(b) is perpendicular to the page and out of it. Green/blue correspond to backer layers exterior/interior tothe annulus; gray represents the membrane filter (in reality much thinner than the backer layers). (b) Idealizedmembrane geometry to be considered in our model. Symmetry lines (dashed) are located at Y = ±H, and the pleatoccupies 0 ≤ X ≤ L.

3

A.I.

Brow

net

al./

Jour

nalo

fMem

bran

eScienc

e34

1 (2

009)

76–

8377

Fig.

1.(a

)Ex

plo

ded

view

ofa

1′′

cartridge

(withouthousing),i

llustra

tingth

ew

ayin

whichth

eple

ated

mem

bra

ne

isin

stal

led

insideth

eca

rtridge

;in

this

case

aPP

Dof0.6

5is

show

n.d

Cre

pre

sents

the

dia

met

erofth

eca

rtridge

and

L Cth

ele

ngt

h.(

b)

Illu

stra

tion

ofth

ediffere

ntple

atden

sities

and

mem

bra

ne

configu

rations

use

din

this

work

andth

eto

talm

embra

ne

surfac

ear

eaav

aila

ble

per

1′′ca

rtridge

.

tore

duce

the

over

allpre

ssure

dro

pby

optim

isin

gth

eple

atco

unt

per

unit

lengt

h.Ithas

bee

nre

ported

that

ple

athei

ghtal

soim

pac

tsupon

over

allpre

ssure

dro

p[9

].Th

em

ost

rece

nt

studie

s[5

,6,14]

hav

epro

duce

dm

odel

sutilising

com

putational

fluid

dyn

amic

sto

explore

thein

fluen

ceth

atple

atdes

ignhas

onairfilter

per

form

ance

.Th

ese

include

com

pre

ssio

nofth

em

ediu

m,p

leat

def

orm

atio

nan

dple

atcr

owdin

gat

hig

hPP

Dva

lues

.W

hen

sim

ulations

wer

eper

-fo

rmed

inco

rpora

ting

med

ium

com

pre

ssio

nan

dar

ealo

ss,th

ere

was

good

agre

emen

tw

ith

exper

imen

taldat

a.Ex

amin

atio

nofth

edat

apre

sente

din

thes

est

udie

ssu

gges

ted

that

ofth

etw

ofa

ctors

inco

rpora

ted

into

them

odel,a

realo

ssap

pea

redto

hav

eth

egr

eate

stim

pac

tonper

form

ance

[5,6

].Air

filter

sar

esim

ilar

inbas

icdes

ignto

those

use

din

the

bio

phar

mac

eutica

lin

dustry,b

utas

not

edab

ove

are

gener

ally

optim

ised

soas

tore

duce

the

pre

ssure

dro

pac

ross

the

ple

at.Th

eca

rtridge

suse

din

bio

phar

mac

eutica

lm

anufa

cture

are

optim

ised

tom

axim

iseth

efiltra

tion

area

andto

han

dle

hig

her

visc

osity

fluid

s.W

hilst

findin

gsge

ner

ated

forairfilter

sm

ayw

ell

bea

ronth

ech

arac

terist

icsoft

heirbio

phar

mac

eutica

lcounte

rpar

tsth

ese

key

differe

nce

sm

ust

be

born

ein

min

d.

For0.4

5 m

PVDF

mem

bra

ne

filter

cartridge

suse

dw

ith

liqu

idfe

eds,

ple

atin

ghas

bee

nas

soci

ated

with

adro

pin

cartridge

per

-fo

rman

cew

hen

com

par

edw

ith

flat

shee

tm

embra

nes

[15].Se

vera

lre

sear

cher

s[1

6,17]hav

eco

nsider

edth

eef

fect

ofth

eper

mea

bility

ofth

edra

inag

em

ater

ialupon

cartridge

per

form

ance

.Gola

nan

dPa

rekh

[16]dev

eloped

per

mea

bility

model

sto

acco

untfo

rth

edif-

fere

ntpro

cess

feed

sth

ata

cartridge

may

be

requ

ired

tofilter.A

dra

inag

eper

mea

bility

term

toac

count

forth

eco

mpre

ssib

ility

of

thedra

inag

em

ater

ial,w

asin

troduce

d.T

his

varied

acro

ssth

ele

ngt

h

Fig.

2.

(a)Ph

otogr

aph

ofa

cross

-sec

tion

of10

′′UEA

Vm

embra

new

ith

anUltip

leat

®

ple

atco

nfigu

ration

and

(b)p

hot

ogr

aph

ofa

cross

-sec

tion

ofa

10′′EA

Vca

rtridge

with

aFa

nple

atco

nfigu

ration.

oft

heple

at,b

utn

oex

per

imen

tald

ataw

aspre

sente

dfo

rm

odel

vali-

dat

ion.G

igliaan

dYa

vors

ky[1

7]b

uiltuponth

isw

ork

and

com

par

edth

eper

form

ance

ofa

range

ofs

terile

cartridge

sco

nsist

ing

ofd

iffer-

entpore

size

san

dm

embra

nem

ater

ials.A

differe

nce

inm

embra

ne

per

mea

bilityw

asobse

rved

bet

wee

nflat

shee

tdis

csan

d10

′′ple

ated

mem

bra

ne

cartridge

sas

has

bee

nre

ported

pre

viousl

y[1

5,18].

This

studypre

sents

asy

stem

atic

exper

imen

talinve

stig

atio

nin

toth

eef

fect

that

mem

bra

ne

ple

atin

ghas

upon

the

clea

nw

ater

flux

ofa

larg

esc

ale

cartridge

(mem

bra

ne

area

∼1m

2).By

studyi

ngth

ech

ange

sin

clea

nw

ater

flux

when

tran

sitionin

gfrom

aflat

shee

tdis

cofm

embra

neto

alarg

esc

ale

cartridge

weai

mto

des

crib

ean

dto

quan

tify

the

effe

ctth

atple

atdes

ign

has

upon

cartridge

per

for-

man

ce.D

ilute

yeas

tsusp

ensionsar

eal

souse

das

apro

beto

visu

alise

and

quan

tify

the

acce

ssib

ility

ofm

icro

n-s

ized

par

ticles

,su

chas

pro

tein

aggr

egat

es,into

them

embra

neple

ats.

2.Exp

erim

ental

2.1.

Fabricatio

nan

dde

sign

ofpl

eate

dm

embr

aneca

rtridg

es

Inord

erto

inve

stig

ateth

ein

fluen

ceofp

leat

des

ign

and

configu

-ra

tionase

ries

of2

.54

cm(1

′′ )se

ctio

nsfrom

a25.4

cm(1

0′′ )

cartridge

(Pal

lEuro

peLt

d.,Po

rtsm

outh

,UK)w

ere

spec

ially

fabrica

ted.T

hes

eca

rtridge

sw

ere

des

igned

topro

vide

ara

nge

ofple

atge

om

etries

bet

wee

nth

atof10

′′ca

rtridge

san

dflat

shee

tdis

cs.F

ully

pac

ked

1′′


Typical parameter values12 P. Sanaei et al.

Parameter Description Typical Value

L Length of the pleat 1.3 cmH Support layer thickness 1 mmD Membrane thickness 300 µmA0 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 membraneand feed solution)

G∞ Total particle concentration Depends on applicationN0 Number of pores per unit area 7×1010 m−2 (very variable)P0 Pressure drop Depends on applicationKav Average support layer permeability 10−11 m2 (very variable)Km0 Clean membrane permeability 5×10−13 m2 (very variable)

Table 1. Approximate dimensional parameter values (Kumar 2014)

in terms of which the pressures in the support layers are given by (2.21), (2.32). Notethat we have four boundary conditions (2.46), (2.47) for the second-order equation (2.45),which ensures that the unknown functions c1(t) and c2(t) are fixed also. The membranepermeability km(x, t) varies quasistatically in (2.45) due to the fouling; it satisfies (2.40)

km(x, t) = a(t)4[n(x, t) +

(1− n(x, t))(1 + ρba(t)4)

], where a(t) = 1− βt. (2.48)

The number density of unblocked pores, n(x, t), varies according to (2.41),

∂n(x, t)∂t

= −n(x, t)a(t)4e−ba(t)

(2p+

0 (x, t) + c1(t)∫ x

0

dx

k(x)+ c2(t)

), n(x, 0) = 1. (2.49)

The solution scheme for this system is straightforward: At time t = 0 assign km(x, 0) =km0 = 1. Then: (i) solve the boundary value problem (2.45), (2.46), (2.47) for p+

0 (x, t);(ii) use this solution, and the current membrane permeability km(x, t) and pore radiusa(t) as given by (2.48) to solve (2.49) for n(x, t); (iii) update km(x, t) and a(t) via (2.48)according to the new n(x, t); and (iv) use the updated km(x, t) and return to step 1;repeat.

3. ResultsThe model contains a number of parameters, which are summarized in Tables 1

(dimensional parameters) and 2 (dimensionless parameters) along with typical values,where known. Considerable variation in the exact values is possible as indicated in thetable, but exhaustive investigation of the effects of each parameter is impractical, hencefor most of our simulations we fix their values as discussed below.

The relative measure of the resistance of the packing material to that of the membrane,Γ , could certainly vary quite widely from one system to another depending on the detailedstructure of the filter membrane and the support layers. Our analysis assumes Γ = O(1),which appears to be in line with data for real pleated filters (Kumar 2014). Based onthe values given in Tables 1 and 2 we take Γ = 10 throughout most of our simulations


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

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 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x

k m

.01tf

.02tf

.05tf

.08tf

.13tf

.18tf

.25tftf

With uniform support permeability k(x) = 1 permeabilityremains symmetric about pleat centerline.

Fouling occurs preferentially at the pleat ends.

As total blocking occurs km → 0 and permeability gradientsalong pleat ultimately smooth out.

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

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x

k m

.01tf

.02tf

.05tf

.08tf

.13tf

.18tf

.25tftf

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x

k m

.01tf

.02tf

.05tf

.08tf

.13tf

.18tf

.25tftf

Compare only profiles with same average support permeability(two examples shown).

Membrane permeability now develops asymmetry. Fluid hasan initially easy path through upper support layer, so lesspasses through leftmost part of membrane.

Fouling occurs preferentially at the far pleat end.


Results: Membrane performance (Γ = 10)

Performance characterized by flux-throughput curves.

Compare our pleated filter model with closest equivalentnon-pleated filter:

0 1 2 3 4 5 6 70

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Throughput

Tota

l Flu

x

Flat

k1

k2

k3

Flat filter significantly outperforms pleated filters.

Among pleated filters, uniform support permeability “best”.


Parametric dependence: ρb

0 0.5 1 1.5 2 2.5 30

0.02

0.04

0.06

0.08

0.1

0.12

0.14

Throughput

Tota

l Flu

x

ρb=0.25

ρb=1

ρb=2

ρb=5

ρb=10

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


Parametric dependence: b

0 0.5 1 1.5 2 2.5 30

0.02

0.04

0.06

0.08

0.1

0.12

0.14

Throughput

Tota

l Flu

x

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 largeparticle in our bimodal distribution. (When b � 1 most largeparticles are sieved.)


Dependence on Γ = Km0/(ε2δKav)

Γ is key system parameter characterizing where primarysystem resistance arises.

In limit Γ→ 0 anticipate pleated model to approach “flatfilter” solution. Simulations bear this out.

0 1 2 3 4 5 6 70

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Throughput

Tota

l Flu

x

Flat

Pleated, Γ=0.1

Pleated, Γ=1

Pleated, Γ=10

Pleated, Γ=100

Conclusions (i)

Simple model of pleated filter, which we believe captures keyfeatures.

Important factors for performance are permeability ratioΓ = Km0H/(L2DKav), and support permeability profile k(x).

Performance converges to that of the equivalent flat filter as Γdecreases (use more permeable support material, or decreasepleat thickness H, or increase pleat length L).

However, increasing L means either using bigger cylinders(bad), or decreasing inner cylinder radius, which effectivelyintroduces negative permeability gradients k ′(x) < 0,negatively impacting performance – tradeoff.

More precise predictions require both accurate data, and moresophisticated modeling – work in progress.

Improved membrane description (Pall, MPI 2014)

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!"#" $%&'()'%*+ ,-.,(%/,+'0









U = −K(X, T )

µ

∂P

∂X,

∂

∂X

(K(X, T )

∂P

∂X

)= 0, 0 ≤ X ≤ D, (1) eq:darcy


P (0, T ) = P0, P (D, T ) = 0. (2) pressBC







U = − πA4

8µ(2W )2

∂P

∂X= −φKp

µ

∂P

∂X, (4) darcy

3

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!"#" $%&'()'%*+ ,-.,(%/,+'0









U = −K(X, T )

µ

∂P

∂X,

∂

∂X

(K(X, T )

∂P

∂X

)= 0, 0 ≤ X ≤ D, (1) eq:darcy


P (0, T ) = P0, P (D, T ) = 0. (2) pressBC







U = − πA4

8µ(2W )2

∂P

∂X= −φKp

µ

∂P

∂X, (4) darcy

3

Most filtration/fouling literature assumes identical cylindricalpores, 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 havepermeability depth gradients, being typically more permeableon the upstream side.

Industry interested in improved mathematical descriptions ofinternal membrane structure and particle deposition, whichcan better predict membrane fouling and lifetime, and guidemembrane design.


Model membrane geometry

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

C(X,T)

2W

2WX

D

A(X,T)


Flow within slender pore (no blocking)

Incompressible Darcy flow U = (U(X ,T ), 0, 0) withinmembrane, 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); Λcaptures physics of attraction between particles and pore wall.


Membrane fouling (adsorption)

As before we consider two fouling modes: adsorption &blocking (cake formation can be bolted on later). Foradsorption:

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 previousslide.


Membrane fouling (blocking)

To include blocking of pores by large particles also, followpleated filter modeling, tracking number of unblocked poresper 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 = −πA40

8µ

∂P

∂X

(N

(A0/A)4+

N0 − N

(A0/A)4 + ρb

);

∂U

∂X= 0

(permeability K (X ,T ) implicit in this equation).


Parameter values

To investigate how pore shape and permeability gradientsaffect filtration & fouling, simulate filtration through pores ofvarying profiles (fix initial membrane porosity/permeability).

Typical parameter values, where known:

β̃ =1

QporeG∞, λ̃ =

πΛW 3D

Qpore

, (40) a_c_blocking_specified flux

with modified Darcy pressure p within the membrane given by

p =

∫ 1

x

dx′

a4( 1−n1+ρba4 + n)

. (41)

In particular, this last expression allows the pressure p(0, t) at the membrane inlet (the dimensionlesspressure drop in this constant flux case) to be evaluated.

3 Resultssec:simulation

In this section we present some sample simulations of the models presented in §§fouling12.1 andsec:modified_darcy2.2, show-

ing how results depend on the pore features and parameters, and how results change depending onwhether purely adsorptive fouling, or a combination of adsorptive fouling and sieving, is considered.Our models contain several dimensionless parameters and functional inputs: the constant parame-ters λ̂ and λ̃, which capture the physics of the attraction between particles and the pore wall; theratio ρb of the additional resistance due to pore-blocking to the original resistance of the unblockedpore; the ratio b of pore size to characteristic particle size; the dimensionless pore shrinkage rates βand β̃. An exhaustive investigation of the effects of each of these parameters is impractical. Theirvalues depend on physical dimensional parameters that must be measured for the particular systemunder investigation, and we lack such detailed experimental data; hence we have to make our bestguess as to the most appropriate values to use in our simulations. These parameters are summarizedin Tables

t:parameters11 (dimensional parameters) and

t:parameters22 (dimensionless parameters) along with typical values,

where known.

Parameter Description Typical Value2W 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 µmA0 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 applicationN0 Number of pores per unit area 7×1010 m−2 (very variable)P0 Pressure drop Depends on applicationQpore Flux through a single pore Depends on applicationC0 Intial particle concentration Depends on application

Table 1: Approximate dimensional parameter values [14]t:parameters1params

Given the number of parameters, most of them will be fixed throughout our simulations. Thevalues of the dimensionless attraction coefficients between pore wall and particles, λ̂ and λ̃, areunknown, and could certainly vary widely from one system to another depending on the detailedstructure of the filter membrane. In the absence of firm data on their value we take λ̂ = λ̃ = 1

9


Parameter values

Where parameters are unknown, experimental data may beavailable to help estimate them, e.g., can estimateparticle-wall attraction coefficient Λ by fitting model solutionsfor C (X ,T ) to fluorescence data:

identical to that for the un-fouled membrane; the 50%decline in filtrate flux due to IgG fouling had no effect oneither the initial LRV or the rate of LRV decline during thesubsequent phage challenge. These data also demonstrate theneed to perform virus spiking studies over the full processthroughput, since the initial LRV after prefouling is unlikelyto be representative of the true process LRV.

The results in Figure 2 demonstrate that LRV for theDV20 membrane is strongly dependent on throughput of thevirus challenge, in contrast to the strong dependence on fluxdecline seen by Bolton et al.3 The different behaviorobserved in these studies is likely due to differences in themorphology of the virus filters; the DV20 membrane is rela-tively homogeneous while the Viresolve NFP membranesexamined by Bolton et al.3 are highly asymmetric with avery open structure (with micron-size pores) at the inletregion of the filter. It is also possible that the results areaffected by differences in the proteins; Bolton et al.3 prefil-tered their IgG through the Viresolve Prefilter media whilethe IgG used in this work was prefiltered only through a 0.2-mm sterilizing grade filter. SEC analysis of the IgG used inthis work showed no measurable aggregates >40 nm in size,suggesting that the protein fouling was not due to cake for-mation. Note that Khan et al.7 observed an increase in LRVwith increasing throughput due to protein cake formation,which is exactly opposite to the behavior seen in this study.

Confocal microscopy

Figure 3 shows a fluorescent image of a cross-sectionthrough a single layer of two DV20 membranes after filtra-

tion of 11 L/m2 (top panel) and 44 L/m2 (bottom panel) of asuspension containing approximately 108 pfu/mL of the fluo-rescently labeled PP7 bacteriophage. These cross-sectionalviews were constructed from 54 in-focus images of the x–yplanes inside the membrane. These images were stackedalong the z-axis and then sliced through an arbitrary x–zplane, approximately 1 mm thick, using the Olympus Fluo-ViewTM viewer software. The fluorescently labeled PP7 bac-teriophage are easily visible throughout the upper region ofthe DV20 membrane, extending approximately one-quarterof the way through the approximately 40 mm thick mem-brane. The depth of penetration was very similar at the twoloadings, with the image at 44 L/m2 showing a much greaterfluorescent intensity. Additional details on the confocalmicroscopy are provided by Bakhshayeshi et al.19

Independent experiments were performed to verify that themajority of filtered bacteriophage actually enter the DV20membrane. 100 mL of a solution containing a mixture of 2.83 107 pfu/mL PP7 and 2.5 3 104 pfu/mL PR772 bacterio-phages were filtered through a DV20 membrane. The filtratewas collected in four samples, each containing approximately24 mL, with approximately 4 mL of retentate left in the hous-ing above the filter. No PR772 phage were observed in the fil-trate samples and less than 0.01% of the PP7 phage passedthrough the membrane. The residual retentate was collectedby draining the housing, with any remaining phage collectedby circulating 200 mL of acetate buffer through the filterheadspace upstream of the membrane at a flow rate of approx-imately 40 mL/min. Approximately 20 6 15% of the PP7phage were recovered in the residual retentate plus bufferrinse, with the remainder being retained within the filter (con-sistent with the confocal image in Figure 3). In contrast,75 6 25% of the large PR772 bacteriophage were recoveredin the residual retentate plus the buffer rinse. The large errorbars on the calculated values of the percent recovery reflectthe inherent uncertainties in the plaque assay.

Internal polarization model

The confocal images in Figure 3 suggest that the retainedbacteriophage accumulate within the upper region of themembrane, which we refer to as the “reservoir zone” withinthe filter. This accumulation of retained virus would lead toa reduction in LRV as the “rejection zone” of the membraneis challenged by a continually increasing concentration ofvirus from the reservoir zone. As with classical membraneprocesses, the polarization occurs when the virus is selec-tively retained by the membrane so that the concentrationincreases near the surface of the rejection layer. In contrastto standard concentration polarization theory, rejection ininternal polarization occurs within the membrane after con-vective transfer of viruses through the reservoir zone. Thethickness of this internal concentration polarization layer isthus determined by the physical dimensions of the reservoirzone as opposed to the balance between convection and dif-fusion in a concentration polarization boundary layer.

The slightly asymmetric structure of the Ultipor VF GradeDV20 membrane, with a more open pore size in the reservoirzone and a tighter pore size near the filter exit, has beenobserved previously. In contrast to scanning electron micros-copy cross-sections showing relative homogeneity, dextranretention tests performed with the membrane oriented with theshiny-side up yielded contrasting results to tests with theshiny-side down.20 Further demonstration of this asymmetry

Figure 3. Confocal scanning image of the cross-section of aDV20 membrane after filtration of 11 L/m2 (toppanel) and 44 L/m2 (bottom panel) of a suspensioncontaining approximately 108 pfu/mL fluorescentlylabeled PP7 bacteriophage.

4 Biotechnol. Prog., 2014, Vol. 00, No. 00

From Jackson et al., Biotechnol. Prog. 2014.

Note however that this is not an example of an efficientfiltration scenario!


Schematic geometry

Membrane


Example results: Initially uniform pore profile

Plot normalized pore radius a(x , t) = 1W A(X ,T ), and particle

concentration c(x , t) = 1C0

C (X ,T ), x = X/D, at varioustimesteps t = T/TB .

Though porosity & permeability are initially uniform, gradientsquickly develop.

Pore closes first at upstream side.

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x

Ra

diu

s o

f p

ore

an

d p

art

icle

co

nce

ntr

atio

n

a(x,0)a(x,0.4t

f)

a(x,0.7tf)

a(x,tf=9.05)

c(x,0)c(x,0.4t

f)

c(x,0.7tf)

c(x,tf)

(a)0 0.2 0.4 0.6 0.8 1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x

Ra

diu

s o

f p

ore

an

d p

art

icle

co

nce

ntr

atio

n

a(x,0)a(x,0.4t

f)

a(x,0.7tf)

a(x,tf=8.2)

c(x,0)c(x,0.4t

f)

c(x,0.7tf)

c(x,tf)

(b)

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x

Ra

diu

s o

f p

ore

an

d p

art

icle

co

nce

ntr

atio

n

a(x,0)a(x,0.4t

f)

a(x,0.7tf)

a(x,tf=9.8)

c(x,0)c(x,0.4t

f)

c(x,0.7tf)

c(x,tf)

(c)0 0.2 0.4 0.6 0.8 1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x

Ra

diu

s o

f p

ore

an

d p

art

icle

co

nce

ntr

atio

n

a(x,0)a(x,0.4t

f)

a(x,0.7tf)

a(x,tf=9.7)

c(x,0)c(x,0.4t

f)

c(x,0.7tf)

c(x,tf)

(d)

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x

Ra

diu

s o

f p

ore

an

d p

art

icle

co

nce

ntr

atio

n

a(x,0)a(x,0.4t

f)

a(x,0.7tf)

a(x,tf=8.5)

c(x,0)c(x,0.4t

f)

c(x,0.7tf)

c(x,tf)

(e)0 0.5 1 1.5 2 2.5

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Throughput

Flu

x

a1

a2

a3

a4

a5

a1, b=0.5

a2, b=0.5

a3, b=0.5

a4, b=0.5

a5, b=0.5

(f)

Figure 3: Filtration simulations at constant pressure drop: The pore radius and particle concentration at severaldifferent times up to the final blocking time (tf , indicated in the legends) for different initial pore radius profiles(a-e). Figure (a): a1(x, 0) = 0.904, (b): a2(x, 0) = 0.16x + 0.82, (c): a3(x, 0) = 0.98 − 0.16x, (d): a4(x, 0) =0.87 + .39(x − 0.5)2, (e): a5(x, 0) = 0.93 − 0.33(x − 0.5)2. (f) shows total flux vs throughput for those initial poreradius profiles for homogenous (28) and exponential (29) distributions of large particles, with λ = 2, β = 0.1, ρb = 2and b = 0.5.

11

λ = 8ΛµD2/(P0W ) = 2


Results: Comparing selected (normalized) pore profiles

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x

Ra

diu

s o

f p

ore

an

d p

art

icle

co

nce

ntr

atio

n

a(x,0)a(x,0.4t

f)

a(x,0.7tf)

a(x,tf=9.05)

c(x,0)c(x,0.4t

f)

c(x,0.7tf)

c(x,tf)

(a)0 0.2 0.4 0.6 0.8 1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x

Ra

diu

s o

f p

ore

an

d p

art

icle

co

nce

ntr

atio

n

a(x,0)a(x,0.4t

f)

a(x,0.7tf)

a(x,tf=8.2)

c(x,0)c(x,0.4t

f)

c(x,0.7tf)

c(x,tf)

(b)

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x

Ra

diu

s o

f p

ore

an

d p

art

icle

co

nce

ntr

atio

n

a(x,0)a(x,0.4t

f)

a(x,0.7tf)

a(x,tf=9.8)

c(x,0)c(x,0.4t

f)

c(x,0.7tf)

c(x,tf)

(c)0 0.2 0.4 0.6 0.8 1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x

Ra

diu

s o

f p

ore

an

d p

art

icle

co

nce

ntr

atio

n

a(x,0)a(x,0.4t

f)

a(x,0.7tf)

a(x,tf=9.7)

c(x,0)c(x,0.4t

f)

c(x,0.7tf)

c(x,tf)

(d)

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x

Ra

diu

s o

f p

ore

an

d p

art

icle

co

nce

ntr

atio

n

a(x,0)a(x,0.4t

f)

a(x,0.7tf)

a(x,tf=8.5)

c(x,0)c(x,0.4t

f)

c(x,0.7tf)

c(x,tf)

(e)0 0.5 1 1.5 2 2.5

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Throughput

Flu

x

a1

a2

a3

a4

a5

a1, b=0.5

a2, b=0.5

a3, b=0.5

a4, b=0.5

a5, b=0.5

(f)


11

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x

Ra

diu

s o

f p

ore

an

d p

art

icle

co

nce

ntr

atio

n

a(x,0)a(x,0.4t

f)

a(x,0.7tf)

a(x,tf=9.05)

c(x,0)c(x,0.4t

f)

c(x,0.7tf)

c(x,tf)

(a)0 0.2 0.4 0.6 0.8 1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x

Ra

diu

s o

f p

ore

an

d p

art

icle

co

nce

ntr

atio

n

a(x,0)a(x,0.4t

f)

a(x,0.7tf)

a(x,tf=8.2)

c(x,0)c(x,0.4t

f)

c(x,0.7tf)

c(x,tf)

(b)

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x

Ra

diu

s o

f p

ore

an

d p

art

icle

co

nce

ntr

atio

n

a(x,0)a(x,0.4t

f)

a(x,0.7tf)

a(x,tf=9.8)

c(x,0)c(x,0.4t

f)

c(x,0.7tf)

c(x,tf)

(c)0 0.2 0.4 0.6 0.8 1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x

Ra

diu

s o

f p

ore

an

d p

art

icle

co

nce

ntr

atio

n

a(x,0)a(x,0.4t

f)

a(x,0.7tf)

a(x,tf=9.7)

c(x,0)c(x,0.4t

f)

c(x,0.7tf)

c(x,tf)

(d)

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x

Ra

diu

s o

f p

ore

an

d p

art

icle

co

nce

ntr

atio

n

a(x,0)a(x,0.4t

f)

a(x,0.7tf)

a(x,tf=8.5)

c(x,0)c(x,0.4t

f)

c(x,0.7tf)

c(x,tf)

(e)0 0.5 1 1.5 2 2.5

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Throughput

Flu

x

a1

a2

a3

a4

a5

a1, b=0.5

a2, b=0.5

a3, b=0.5

a4, b=0.5

a5, b=0.5

(f)


11

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x

Ra

diu

s o

f p

ore

an

d p

art

icle

co

nce

ntr

atio

n

a(x,0)a(x,0.4t

f)

a(x,0.7tf)

a(x,tf=9.05)

c(x,0)c(x,0.4t

f)

c(x,0.7tf)

c(x,tf)

(a)0 0.2 0.4 0.6 0.8 1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x

Ra

diu

s o

f p

ore

an

d p

art

icle

co

nce

ntr

atio

n

a(x,0)a(x,0.4t

f)

a(x,0.7tf)

a(x,tf=8.2)

c(x,0)c(x,0.4t

f)

c(x,0.7tf)

c(x,tf)

(b)

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x

Ra

diu

s o

f p

ore

an

d p

art

icle

co

nce

ntr

atio

n

a(x,0)a(x,0.4t

f)

a(x,0.7tf)

a(x,tf=9.8)

c(x,0)c(x,0.4t

f)

c(x,0.7tf)

c(x,tf)

(c)0 0.2 0.4 0.6 0.8 1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x

Ra

diu

s o

f p

ore

an

d p

art

icle

co

nce

ntr

atio

n

a(x,0)a(x,0.4t

f)

a(x,0.7tf)

a(x,tf=9.7)

c(x,0)c(x,0.4t

f)

c(x,0.7tf)

c(x,tf)

(d)

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x

Ra

diu

s o

f p

ore

an

d p

art

icle

co

nce

ntr

atio

n

a(x,0)a(x,0.4t

f)

a(x,0.7tf)

a(x,tf=8.5)

c(x,0)c(x,0.4t

f)

c(x,0.7tf)

c(x,tf)

(e)0 0.5 1 1.5 2 2.5

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Throughput

Flu

x

a1

a2

a3

a4

a5

a1, b=0.5

a2, b=0.5

a3, b=0.5

a4, b=0.5

a5, b=0.5

(f)


11

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x

Ra

diu

s o

f p

ore

an

d p

art

icle

co

nce

ntr

atio

n

a(x,0)a(x,0.4t

f)

a(x,0.7tf)

a(x,tf=9.05)

c(x,0)c(x,0.4t

f)

c(x,0.7tf)

c(x,tf)

(a)0 0.2 0.4 0.6 0.8 1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x

Ra

diu

s o

f p

ore

an

d p

art

icle

co

nce

ntr

atio

n

a(x,0)a(x,0.4t

f)

a(x,0.7tf)

a(x,tf=8.2)

c(x,0)c(x,0.4t

f)

c(x,0.7tf)

c(x,tf)

(b)

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x

Ra

diu

s o

f p

ore

an

d p

art

icle

co

nce

ntr

atio

n

a(x,0)a(x,0.4t

f)

a(x,0.7tf)

a(x,tf=9.8)

c(x,0)c(x,0.4t

f)

c(x,0.7tf)

c(x,tf)

(c)0 0.2 0.4 0.6 0.8 1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x

Ra

diu

s o

f p

ore

an

d p

art

icle

co

nce

ntr

atio

n

a(x,0)a(x,0.4t

f)

a(x,0.7tf)

a(x,tf=9.7)

c(x,0)c(x,0.4t

f)

c(x,0.7tf)

c(x,tf)

(d)

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x

Ra

diu

s o

f p

ore

an

d p

art

icle

co

nce

ntr

atio

n

a(x,0)a(x,0.4t

f)

a(x,0.7tf)

a(x,tf=8.5)

c(x,0)c(x,0.4t

f)

c(x,0.7tf)

c(x,tf)

(e)0 0.5 1 1.5 2 2.5

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Throughput

Flu

x

a1

a2

a3

a4

a5

a1, b=0.5

a2, b=0.5

a3, b=0.5

a4, b=0.5

a5, b=0.5

(f)


11


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

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

porosity πA20

∫ 10 a(x , 0)2dx/(2W )2D (results with same initial

membrane permeability are almost identical).

In all cases shown pore closure occurs first at upstreammembrane surface (flux then goes to zero).

Can, by careful choice of initial pore profile, or by anunreasonably small choice of λ, obtain pore closure at internalpoints, or even the downstream end.

The pore with linear decreasing initial profile stays openlongest (longest filter lifetime).

In addition to simply sustaining a nonzero flux, however, wantto know which pore profiles give maximal total throughputover the filter lifetime.


Results: Flux vs throughput

Compare 5 different pore profiles (a1-a4 on previous slide, plusone other: uniform, linear increasing, linear decreasing,concave, convex).

Linear decreasing profile clearly “best”, among thoseconsidered.

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

x

Radiu

s o

f pore

time=0.2t

f

.5tf

.8tf

tf=9.05

(a)0 0.2 0.4 0.6 0.8 1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x

Radiu

s o

f pore

time=0.2t

f

.5tf

.8tf

tf=8.25

(b)

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x

Radiu

s o

f pore

time=0.2t

f

.5tf

.8tf

tf=9.9

(c)0 0.2 0.4 0.6 0.8 1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x

Radiu

s o

f pore

time=0.2t

f

.5tf

.8tf

tf=9.75

(d)

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x

Radiu

s o

f pore

time=0.2t

f

.5tf

.8tf

tf=8.55

(e)0 0.5 1 1.5 2

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Throughput

Flu

x

a1

a2

a3

a4

a5

(f)

Figure 4: Constant pressure drop case, Darcy model with blocking by large particles: The pore radius at severaldifferent times at different final blocking times (tf , indicated in the legends) for different initial pore radius profiles(a-e), Figure (a): a1(x, 0) = .9, (b): a2(x, 0) = 0.16x+0.82, (c): a3(x, 0) = 0.98−0.16x, (d): a4(x, 0) = 0.86+ .39(x−0.5)2, (e): a5(x, 0) = 0.92− 0.33(x− 0.5)2, and (f) total flux vs throughput for those initial pore radius profiles withλ̂ = 1, β = 0.1, ρb = 2 and b = 0.5.radius_revised

13


Results: Particle concentration within pores

0 0.5 1 1.5 20

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Throughput

Pa

rtic

le c

on

ce

ntr

atio

n a

t p

ore

ou

tle

t

a1

a2

a3

a4

a5

(a)0 0.2 0.4 0.6 0.8 1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

time

Pa

rtic

le c

on

ce

ntr

atio

n a

t p

ore

ou

tle

t

!=0.1

!=0.5

!=1

!=2

(b)

Figure 7: (a) Particle concentration vs throughput at the filter downstream (x = 1), with λ̂ = 1, β = 0.1, ρb = 2and b = 0.5; and (b) Particle concentration graph for several different values of λ̂ (a measure of the attraction betweenwall and particle), for the uniform initial pore profile a(x, 0) = 1 with β = 0.1, ρb = 2 and b = 0.5.pc-lambda-rho

4 Conclusionssec:conclusion

We have presented a model that can describe the key features of membrane morphology on sep-aration efficiency and fouling of a membrane filter. Our model accounts for Darcy flow throughthe membrane, and for fouling by two distinct mechanisms: adsorption and pore-blocking. Whileessentially predictive, our model contains several parameters that may be difficult to measure fora given system (most notably, the relative importance of blocking to adsorption, ρb, the ratio ofinitial pore size to characteristic particle size b, and the dimensionless attraction coefficient betweenthe membrane pore wall and particles, λ̂). In practice such parameters could be inferred by fittingto a reliable dataset; but even so these parameters will vary from one membrane-filtrate systemto another, since they depend on membrane structure, and the chemical interactions between thefiltrate particles and the membrane material.

In the absence of firm data on model parameters we have chosen what we believe to be plausibleparameter values (summarized at the start of §sec:simulation3) for most of our simulations. The focus in this paperis on development of a model that can be used to quantify the effects of membrane morphology onseparation efficiency by the performance curve of a membrane filter with known characteristics undergiven operating conditions and by the particle concentration curve. We present some preliminaryresults that bear out the expected particle concentration discrepancy, but we do not, in this paper,investigate exhaustively how this discrepancy depends on model parameters.

Our model can account, in the simplest possible way, for variations in membrane pore profiles.The pore profile variation in real membranes is undoubtedly highly complex: we restrict attentionto simple axisymmetric pore profiles characterized by depth-dependent initial radius a(x, 0), whichspan the entire membrane depth, and we investigate how filtration performance varies as these initialpore profiles change. Our results indicate firstly that such variations in initial pore profile can leadto different fouling patterns within the membrane. More importantly, if the initial pore radius at thetop of membrane is small (a2 in (

profile42)), it can give rise to a marked decrease in filter performance as

quantified by the total amount of filtrate processed under the same operating conditions, as shownby Figure

radius_revised4(f). This figure shows that, in addition to the pronounced difference in performance

between uniform and increasing initial pore profile, the case where the initial pore profile is given bya(x, 0) = a3(x) (see equation (

profile42)) gives significantly higher total throughput when compared with

the cases where the initial pore profile is uniform along the membrane pore (a(x, 0) = a1(x)). The

16

Dimensionless parameter λ = 8ΛµD2/(P0W ) measuresstrength of attraction between small particles & pore wallsand governs deposition.

Sample simulations show how the normalized concentration,c(1, t) = C (D,T )/C0, at pore outlet, varies both with poreprofile, and with λ (λ = 1 in plot (a); a(x , 0) = a1(x),uniform pore, in plot (b)).


Results: Particle concentration at pore outlet

0 0.5 1 1.5 20

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Throughput

Pa

rtic

le c

on

ce

ntr

atio

n a

t p

ore

ou

tle

t

a1

a2

a3

a4

a5

(a)0 0.2 0.4 0.6 0.8 1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

time

Pa

rtic

le c

on

ce

ntr

atio

n a

t p

ore

ou

tle

t

!=0.1

!=0.5

!=1

!=2

(b)

Figure 7: (a) Particle concentration vs throughput at the filter downstream (x = 1), with λ̂ = 1, β = 0.1, ρb = 2and b = 0.5; and (b) Particle concentration graph for several different values of λ̂ (a measure of the attraction betweenwall and particle), for the uniform initial pore profile a(x, 0) = 1 with β = 0.1, ρb = 2 and b = 0.5.pc-lambda-rho

4 Conclusionssec:conclusion

We have presented a model that can describe the key features of membrane morphology on sep-aration efficiency and fouling of a membrane filter. Our model accounts for Darcy flow throughthe membrane, and for fouling by two distinct mechanisms: adsorption and pore-blocking. Whileessentially predictive, our model contains several parameters that may be difficult to measure fora given system (most notably, the relative importance of blocking to adsorption, ρb, the ratio ofinitial pore size to characteristic particle size b, and the dimensionless attraction coefficient betweenthe membrane pore wall and particles, λ̂). In practice such parameters could be inferred by fittingto a reliable dataset; but even so these parameters will vary from one membrane-filtrate systemto another, since they depend on membrane structure, and the chemical interactions between thefiltrate particles and the membrane material.

In the absence of firm data on model parameters we have chosen what we believe to be plausibleparameter values (summarized at the start of §sec:simulation3) for most of our simulations. The focus in this paperis on development of a model that can be used to quantify the effects of membrane morphology onseparation efficiency by the performance curve of a membrane filter with known characteristics undergiven operating conditions and by the particle concentration curve. We present some preliminaryresults that bear out the expected particle concentration discrepancy, but we do not, in this paper,investigate exhaustively how this discrepancy depends on model parameters.

Our model can account, in the simplest possible way, for variations in membrane pore profiles.The pore profile variation in real membranes is undoubtedly highly complex: we restrict attentionto simple axisymmetric pore profiles characterized by depth-dependent initial radius a(x, 0), whichspan the entire membrane depth, and we investigate how filtration performance varies as these initialpore profiles change. Our results indicate firstly that such variations in initial pore profile can leadto different fouling patterns within the membrane. More importantly, if the initial pore radius at thetop of membrane is small (a2 in (

profile42)), it can give rise to a marked decrease in filter performance as

quantified by the total amount of filtrate processed under the same operating conditions, as shownby Figure

radius_revised4(f). This figure shows that, in addition to the pronounced difference in performance

between uniform and increasing initial pore profile, the case where the initial pore profile is given bya(x, 0) = a3(x) (see equation (

profile42)) gives significantly higher total throughput when compared with

the cases where the initial pore profile is uniform along the membrane pore (a(x, 0) = a1(x)). The

16

Fixed λ, (a): There is a clear distinction between performanceof different pore profiles.

Again we see that the membrane with linear decreasing poreprofile (a3(x)) gives the greatest throughput for any givenparticle 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)).


Results: Optimum permeability profile

Given that results depend rather strongly on pore profile, agood question to ask is: can we determine the optimumprofile?

Difficult question to address in general (work in progress!) butwithin selected function classes can answer.

Here compare linear pore profiles, of equal averaged porosityor permeability (same values of

∫ 10 a(x , 0)2dx or∫ 1

0 a(x , 0)4dx), and see which gives greatest total throughputover filter lifetime.



!1 !0.5 0 0.5 10

0.5

1

1.5

2

2.5

3

Pore Gradient

Th

rou

gh

pu

t

!=.1

!=.2

!=.5

!=1

!=2

(a)!1 !0.5 0 0.5 10

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Pore Gradient

Th

rou

gh

pu

t

k=.1

k=.2

k=.3

k=.45

k=.7

(b)

Figure 6: (a) Throughput∫ t

0q(t′)dt′ versus slope of linear initial pore profile with fixed permeability k =

∫ 1

0(a0x

′ +b0)4dx′ = .2, for several different values of λ̂, with ρb = 2, b = 0.5 and β changes proportionally to λ̂; (b) Throughput∫ t

0q(t′)dt′ versus slope of linear initial pore profile a0, for several different values of membrane permeability k with

λ̂ = 1, β = 0.1, ρb = 2 and b = 0.5.pore-gradient

3.2 Constant flux simulations

We now present simulations for the case where the total flux through the system, rather than thepressure drop across it, is specified. Figure

radius-constant-flux8 shows results for both Darcy models, with and without

the pore blocking by sieving of large particles (§sec:flux2.1.2, §sec:flux_modified2.2.2) for the same initial pore profiles givenin (

profile42). Unlike in the constant pressure simulations of §sec:press2.1.1 the pore profile evolution is now

indistinguishable for the two models, hence figuresradius-constant-flux8(a)-(e) are the same for the two models. Figure

radius-constant-flux8(f) shows the inverse pressure drop versus throughput for each of those pore profiles, for the twomodels: here, as before, the results for the two blocking models differ.

The results differ quite significantly from those for the constant pressure case. In contrast tothose simulations, the pore radius evolution is now much more uniform along the pore length. Thereis still a tendency for pore closure to happen first at the upstream end of the pores (this happensin four of the five cases considered), but this is no longer inevitable. The pore of linear decreasingradius (Fig.

radius-constant-flux8(c)) evolves to a state of almost uniform radius, and closes very nearly uniformly

(though closure does appear to happen marginally sooner at the upstream end x = 0). Theconcave-up parabolic pore profile actually closes up first at an interior point (Fig.

radius-constant-flux8(d)). Since the

total flux through the system is held constant the flux-throughput graph gives no characterizationof the system in these simulations, hence we instead plot how the pressure drop rises over time asblocking occurs in order to maintain the specified flow rate (Fig.

radius-constant-flux8(e)). Mathematically pressure

goes to infinity as time passes but this is not practical. In reality pressure is increased until specificvalue (which is practical and may depend on the system endurance) then it will be fixed and thesystem runs on specified pressure drop as discussed in §Constant pressure drop simulations

3.1. The two models give rise to qualitativelysimilar behavior; as anticipated, the main difference is that the model with additional blockage bylarge particles is associated with higher pressures throughout, due to the larger total resistance.As with the constant pressure simulations, the best overall performance (in terms of efficiency) isprovided by the pores of monotone decreasing radius (profile a3(x) in (

profile42)), and the worst by pores

of monotone increasing radius profile (profile a2(x)).

15

!1 !0.5 0 0.5 10

0.2

0.4

0.6

0.8

1

1.2

1.4

Pore gradient

Th

rou

gh

pu

t

!=1, "=0.05

!=1.5, "=0.075

!=2, "=0.1

!=4, "=0.2

!=1, "=0.0707

!=1.5, "=0.0866

!=4, "=0.1414

(a)!1 !0.5 0 0.5 10

0.5

1

1.5

2

2.5

3

Pore Gradient

Th

rou

gh

pu

t

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)

Figure 6: (a) Total throughput∫ t

0q(t′)dt′ versus slope of initial pore profile a0, keeping dimensionless initial net

permeability, k0 =∫ 1

0(a0x

′ + b0)4dx′ = 0.2, fixed, for several different values of λ, with cumulative particle sizedistribution function g(s) given by (28) and ρb = 2. For the thin curves we set β ∝ λ (corresponding to varyingΛ) and for the thick curves β ∝

√λ (corresponding to varying membrane thickness D). Red dots and blue squares

denote maximum throughput for each value of λ. (b) Total throughput versus initial slope a0, for several differentvalues of dimensionless initial permeability k0 with g(s) given by (28), λ = 2, β = 0.1 and ρb = 2. The red dots aremaximum throughput for each given initial permeability k0.

large fraction of the period-box (within which it must be entirely confined), and hence the range ofvalues of a0 will be limited in such cases.

Figures 6(a) and (b) illustrate our results, plotting throughput versus pore gradient for severaldifferent scenarios. In figure 6(a), the dimensionless initial permeability is fixed at k0 = 0.2 (a valuesmall enough that a wide range of pore gradients are available), and throughput is plotted as afunction of pore gradient for several different values of the deposition coefficient λ. Recalling thediscussion at the end of §3.1 above, we cannot change λ in isolation; here we change β proportionallyto λ (with β = 0.1 when λ = 2), modeling changes in the dimensional particle-membrane attractioncoefficient Λ, and we change β proportionally to

√λ, which means we are changing the membrane

thickness D. In Figure 6(b) total throughput is again plotted versus slope of the initial pore profile,for several different values of the membrane permeability k0. As noted above, only a limited rangeof pore gradients are realizable at larger permeabilities.

In all cases considered, the optimum (as measured by maximal total throughput) is achievedat a negative value of the pore profile gradient. Note that these results say nothing about theproportion of small particles captured in each filtration scenario (though they assume capture bysieving of all large particles).

4 Conclusions

We have presented a model that can describe the key effects of membrane morphology on separa-tion efficiency and fouling of a membrane filter. Our model accounts for Darcy flow through themembrane, and for fouling by two distinct mechanisms: adsorption of small particles within pores,and pore-blocking (sieving) by large particles. While essentially predictive, our model containsseveral parameters that may be difficult to measure for a given system (most notably, the relativeincrease in pore resistance due to a blocking event, ρb; the dimensionless attraction coefficient be-tween the membrane pore wall and particles, λ; and (in the case where a non-uniform distribution

15

For fixed net permeability, how does the gradient of a linearpore affect net throughput over filter lifetime?

Not all pore gradients may be allowable for a fixedpermeability. Plot (a) shows results for a permeability valuefor which a wide range of pore gradients is possible.

Observe that as λ ↓, the pore gradient is less significant (lessdeposition, and hence less filtration, is occurring!).



!1 !0.5 0 0.5 10

0.5

1

1.5

2

2.5

3

Pore Gradient

Th

rou

gh

pu

t

!=.1

!=.2

!=.5

!=1

!=2

(a)!1 !0.5 0 0.5 10

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Pore Gradient

Th

rou

gh

pu

t

k=.1

k=.2

k=.3

k=.45

k=.7

(b)

Figure 6: (a) Throughput∫ t

0q(t′)dt′ versus slope of linear initial pore profile with fixed permeability k =

∫ 1

0(a0x

′ +b0)4dx′ = .2, for several different values of λ̂, with ρb = 2, b = 0.5 and β changes proportionally to λ̂; (b) Throughput∫ t

0q(t′)dt′ versus slope of linear initial pore profile a0, for several different values of membrane permeability k with

λ̂ = 1, β = 0.1, ρb = 2 and b = 0.5.pore-gradient

3.2 Constant flux simulations

We now present simulations for the case where the total flux through the system, rather than thepressure drop across it, is specified. Figure

radius-constant-flux8 shows results for both Darcy models, with and without

the pore blocking by sieving of large particles (§sec:flux2.1.2, §sec:flux_modified2.2.2) for the same initial pore profiles givenin (

profile42). Unlike in the constant pressure simulations of §sec:press2.1.1 the pore profile evolution is now

indistinguishable for the two models, hence figuresradius-constant-flux8(a)-(e) are the same for the two models. Figure

radius-constant-flux8(f) shows the inverse pressure drop versus throughput for each of those pore profiles, for the twomodels: here, as before, the results for the two blocking models differ.

The results differ quite significantly from those for the constant pressure case. In contrast tothose simulations, the pore radius evolution is now much more uniform along the pore length. Thereis still a tendency for pore closure to happen first at the upstream end of the pores (this happensin four of the five cases considered), but this is no longer inevitable. The pore of linear decreasingradius (Fig.

radius-constant-flux8(c)) evolves to a state of almost uniform radius, and closes very nearly uniformly

(though closure does appear to happen marginally sooner at the upstream end x = 0). Theconcave-up parabolic pore profile actually closes up first at an interior point (Fig.

radius-constant-flux8(d)). Since the

total flux through the system is held constant the flux-throughput graph gives no characterizationof the system in these simulations, hence we instead plot how the pressure drop rises over time asblocking occurs in order to maintain the specified flow rate (Fig.

radius-constant-flux8(e)). Mathematically pressure

goes to infinity as time passes but this is not practical. In reality pressure is increased until specificvalue (which is practical and may depend on the system endurance) then it will be fixed and thesystem runs on specified pressure drop as discussed in §Constant pressure drop simulations

3.1. The two models give rise to qualitativelysimilar behavior; as anticipated, the main difference is that the model with additional blockage bylarge particles is associated with higher pressures throughout, due to the larger total resistance.As with the constant pressure simulations, the best overall performance (in terms of efficiency) isprovided by the pores of monotone decreasing radius (profile a3(x) in (

profile42)), and the worst by pores

of monotone increasing radius profile (profile a2(x)).

15

!1 !0.5 0 0.5 10

0.2

0.4

0.6

0.8

1

1.2

1.4

Pore gradient

Thro

ughput

!=1, "=0.05

!=1.5, "=0.075

!=2, "=0.1

!=4, "=0.2

!=1, "=0.0707

!=1.5, "=0.0866

!=4, "=0.1414

(a)!1 !0.5 0 0.5 10

0.5

1

1.5

2

2.5

3

Pore Gradient

Thro

ughput

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)

Figure 6: (a) Total throughput∫ t

0q(t′)dt′ versus slope of initial pore profile a0, keeping dimensionless initial net

permeability, k0 =∫ 1

0(a0x

′ + b0)4dx′ = 0.2, fixed, for several different values of λ, with cumulative particle sizedistribution function g(s) given by (28) and ρb = 2. For the thin curves we set β ∝ λ (corresponding to varyingΛ) and for the thick curves β ∝

√λ (corresponding to varying membrane thickness D). Red dots and blue squares

denote maximum throughput for each value of λ. (b) Total throughput versus initial slope a0, for several differentvalues of dimensionless initial permeability k0 with g(s) given by (28), λ = 2, β = 0.1 and ρb = 2. The red dots aremaximum throughput for each given initial permeability k0.

large fraction of the period-box (within which it must be entirely confined), and hence the range ofvalues of a0 will be limited in such cases.

Figures 6(a) and (b) illustrate our results, plotting throughput versus pore gradient for severaldifferent scenarios. In figure 6(a), the dimensionless initial permeability is fixed at k0 = 0.2 (a valuesmall enough that a wide range of pore gradients are available), and throughput is plotted as afunction of pore gradient for several different values of the deposition coefficient λ. Recalling thediscussion at the end of §3.1 above, we cannot change λ in isolation; here we change β proportionallyto λ (with β = 0.1 when λ = 2), modeling changes in the dimensional particle-membrane attractioncoefficient Λ, and we change β proportionally to

√λ, which means we are changing the membrane

thickness D. In Figure 6(b) total throughput is again plotted versus slope of the initial pore profile,for several different values of the membrane permeability k0. As noted above, only a limited rangeof pore gradients are realizable at larger permeabilities.

In all cases considered, the optimum (as measured by maximal total throughput) is achievedat a negative value of the pore profile gradient. Note that these results say nothing about theproportion of small particles captured in each filtration scenario (though they assume capture bysieving of all large particles).

4 Conclusions

We have presented a model that can describe the key effects of membrane morphology on separa-tion efficiency and fouling of a membrane filter. Our model accounts for Darcy flow through themembrane, and for fouling by two distinct mechanisms: adsorption of small particles within pores,and pore-blocking (sieving) by large particles. While essentially predictive, our model containsseveral parameters that may be difficult to measure for a given system (most notably, the relativeincrease in pore resistance due to a blocking event, ρb; the dimensionless attraction coefficient be-tween the membrane pore wall and particles, λ; and (in the case where a non-uniform distribution

15

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 structureassumed here, but optimal pore gradient translates into apermeability gradient, which has meaning for any membrane.


Conclusions (ii)

Allowing (axisymmetric) variations in pore radius provides asimple way to model gradations in membrane structure.

Simulation results bear out empirical findings that negativepermeability gradients are preferable.

Within a given class of pore shapes, and given appropriatedata to fix model parameters, can address issue of which poreshape maximizes total throughput over filter lifetime (or otherperformance-related questions).

Generalizing the formulation of this optimization is ongoingwork.


Outlook

Membrane filtration is a complex process, but simplemathematical models can provide significant insight.

Considered three specific modeling directions here, but manyextensions are possible and should be explored.

Interaction force (electrostatics) may vary as depositionoccurs – may need to allow deposition parameter Λ to varydepending on concentration of particles already deposited.

In slow filtration (or with very small particles) diffusion maybe important within pores – more complicated model forparticle transport (careful averaging needed).

A particular area for future study is pore branching – modelshere all consider simple isolated pores that span membranefrom upstream to downstream. Both branching andrecombining of pores may occur in real membranes, and maybe important for filtration dynamics.


Acknowledgements

NSF DMS-1261596NSF DMS-1211713

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

models for membrane filtration

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