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chemical engineering research and design 8 6 ( 2 0 0 8 ) 1281–1293

Contents lists available at ScienceDirect

Chemical Engineering Research and Design

journa l homepage: www.e lsev ier .com/ locate /cherd

nalysing flux decline in dead-end filtration

lexia Grenier, Martine Meireles ∗, Pierre Aimar, Philippe Carvin1

niversité de Toulouse, Laboratoire de Génie Chimique, UMR CNRS 5503, Université Paul Sabatier,18 route de Narbonne, 31062 Toulouse Cedex 9, France

a b s t r a c t

Can we rely on the analysis of flux decline to evaluate the risks of a filter media to be clogged during filtration of a

given particle suspension? This important issue can be dealt with a macroscopic approach described in this paper.

We seek to identify and quantify the successive prevailing mechanisms which occur during a filtration run, directly

and solely from experimental flux data. This is achieved from the collection of experimental data (filtrate volume V

vs. time t) and the use of the differential equation (d2t/dV2) = k(dt/dV)n. A methodology is then proposed to define and

validate experimental procedures with the purpose of quantifying occurring fouling mechanism. For the purpose of

illustrating its valuable impact for a bench marking procedure, the methodology has been applied on a model system

composed of a bentonite suspension and a series of microfiltration membranes of different structures.

© 2008 The Institution of Chemical Engineers. Published by Elsevier B.V. All rights reserved.

Keywords: Microfiltration; Flux decline; Blocking; Fouling mechanisms; Filter media

Each of these mechanisms has been individually invoked

. Introduction

icrofiltration is a separation process by which some con-tituents of a suspension are separated from the liquid by aembrane mainly acting as a sieve. Ideally the membrane

llows the passage of the fluid through its pores while retain-ng all suspended solid particles originally present in the fluid.he ideal picture stands if solid particles are all larger than

he pores of the membrane and the pore structure consists inapillaries. However in almost all practical cases a fairly wideange of effective particle sizes exists in the feed as well as aandom distribution of pores exists in the membrane. In usualases, when filtering suspensions containing more than a fewercents of solids, blocking of particles inside or on the top ofhe membrane occurs, leading to a reduction in filtration flux.here are four mechanistic models that are generally used toescribe fouling (Grace, 1956; Hermia, 1966).

Complete blocking assumes a seal of pore entrances and theprevention of any flow through them. As pore entrances aresealed, the area open to flow is reduced.Intermediate blocking also assumes a seal of pore entrances

by a fraction of particles and a deposition of the rest on thetop of them.

∗ Corresponding author.E-mail address: meireles@chimie.ups-tlse.fr (M. Meireles).Received 7 September 2007; Accepted 10 June 2008

1 RHODIA Recherche, Centre de Recherche de Lyon, 85 rue des Frères263-8762/$ – see front matter © 2008 The Institution of Chemical Engioi:10.1016/j.cherd.2008.06.005

• Cake filtration is a mechanism by which particles accu-mulate at the surface in a permeable cake of increasingthickness that adds a hydraulic resistance to filtration.

• Finally standard blocking assumes an accumulation insidethe membrane on the pore walls. As the pores areconstricted, the membrane permeability is reduced. Themathematical expressions for these laws are summarizedin Table 1. These expressions were derived by Hermans andBredée (1936), on the assumption of separate mechanisms.All these different laws stem from a common differentialequation. For a constant pressure filtration they have beenreformulated in a common frame of power-law relation-ships (Hermia, 1982):

d2t

dV2= k

(dtdV

)n(1)

Table 1 presents the physical basis for each fouling mecha-nism and the corresponding n-value according to Eq. (1). Theblocking filtration laws have also been extensively used undertheir linear form detailed in Table 4.

Perret, B.P. 62, 69192 Saint Fons Cedex, France.

alone or in combinations, to explain experimental obser-vations. Most of the time the model which best fits the

neers. Published by Elsevier B.V. All rights reserved.

1282 chemical engineering research and design 8 6 ( 2 0 0 8 ) 1281–1293

Nomenclature

A0 total membrane surface area (m2)AB/mB blocked surface area of the membrane per unit

of blocking particle mass (m2 kg−1)C feed suspension concentration (kg m−3)Cv volume of the particles which uniformly

deposit along the pore wall per unit of permeatevolume (dimensionless)

dp mean particle diameter (m)dpore pore diameter (m)J permeate flux (m s−1)k multiplicative constant in Eq. (1) (s1−n m3n−6)mC deposit mass per unit of membrane surface

(kg m−2)n exponent in Eq. (1) (dimensionless)�P transmembrane pressure (Pa)Q permeate flow rate (m3 s−1)QB,0 permeate flow rate at the start point of pore

blocking (m3 s−1)QI,0 permeate flow rate at the start point of internal

fouling (m3 s−1)Q0 initial permeate flow rate (m3 s−1)R hydraulic resistance in Darcy’s law (m−1)RC cake hydraulic resistance (m−1)Re Reynolds number (dimensionless)Rm membrane hydraulic resistance (m−1)Rm,0 clean membrane hydraulic resistance (m−1)Rm,B fouled membrane hydraulic resistance by pore

blocking (m−1)Rm,B,f final fouled membrane hydraulic resistance by

pore blocking (m−1)Rm,I fouled membrane hydraulic resistance by inter-

nal fouling (m−1)V total volume permeated through the mem-

brane at time t (m3)VB,0 permeate volume value at the start point of

pore blocking (m3)VB,f permeate volume value at the end of pore

blocking (m3)VC,0 permeate volume value at the start point of

cake filtration (m3)v̄ fluid velocity at the pore entrance (m s−1)t time (s)

Greek letters˛ specific resistance of the cake (m kg−1)ˇB surface coverage ratio for pore blocking (dimen-

sionless)ˇB,f final surface coverage ratio at the end of the

pore blocking regime (dimensionless)εC cake porosity (dimensionless)εm membrane porosity (dimensionless)�B blocked surface area by unit of time and surface

(m−1)�C cake volumic specific resistance (m−2)�I volume of particle accumulated per unit of per-

meate and unit of void volume (m−3)� permeate dynamic viscosity (Pa s)� permeate density (kg m−3)�p particle density (kg m−3)�w shear stress ate the pore entrance (Pa)

p/pore blocked pores quantity by one blocking particle(dimensionless)

experimental data is claimed to capture the fouling mecha-nism.

For filtration of BSA proteins suspensions, standard block-ing, intermediate blocking or complete blocking models werefound to fit the experimental data (Hlavacek and Bouchet,1993) with the intermediate model providing the best fit. Acombination of standard or complete model and subsequentlycake model was also found to fit the data well for the same sys-tem (Tracey and Davis, 1994). In some other conditions, (Bowenet al., 1995), it was observed that fouling of microfiltrationmembranes by BSA did not follow any of the individual modelsand is more likely described by a combination of several mod-els. These results show that the predominant mechanismsmay vary with operating conditions.

Recently a model has been proposed based on a com-bination of blocking laws in a two-stage mechanism (Hoand Zydney, 2000). Fouling initially occurred through com-plete blocking and subsequently through cake formation. Thedeposited assemblages were assumed to be permeable so thatthe flow through the blocked areas also participate to cakeformation. The authors found this model provided good datafits for the fouling of microporous track-etched membranesby five proteins. The previous model was further expandedby using a method to combine the four individual block-ing laws (Bolton et al., 2006). The combined models (cakefiltration—complete blocking model, cake filtration—standardblocking model) were assessed by testing solutions of bovineserum albumin and human IgG under constant pressure andconstant flow modes. The combined cake-complete blockingmodel provides good fits of both data sets.

Combined best fitted models allow a better determinationof the membrane capacity, defined as the amount of fluid permembrane are which can be processed until the flux declinesto a preset fraction of the initial flux or until the caking startsleading to a several-fold increase in the filtration resistance.

Combination of mechanisms has been also been reportedfor filtration of particles suspensions. For example, the anal-ysis of experimental flux decline during filtration of sizedistributed carbonyl iron suspensions through woven filtermedia shows that fouling is initially controlled by standardblocking mechanism prolonged by a cake filtration period(Grace, 1956). Owing to the impact of porous structure clog-ging on the lifetime of the filter media, the characterizationof the mechanism occurring in the initial period may largelyserve to choose the appropriate filter media. An importantdifficulty lies in the determination at lab scale of this criti-cal initial period. Dilution which extends its duration can be away but the question may be raised if the same mechanismsoccur as for higher solid concentrations.

This works intends to complete previous studies by set-ting a methodology to analyse experimental flux declinefor dead-end filtration runs and identify successive prevail-ing mechanisms. As developed further on, we shall simplyassume three types of blocking mechanisms (cake, completeblocking and standard fouling) The methodology can be imple-mented even in the absence of precise information on the

“suspension” or on the filter (for instance, particle size distribu-tion, mean particle diameter, pore size distribution, nominal

chemical engineering research and design 8 6 ( 2 0 0 8 ) 1281–1293 1283

Table 1 – Fouling mechanisms and their corresponding physical basis

Fouling mechanism n Corresponding linear form Physical concept

Cake filtration 0 dtdV = 1

Q = f (V) Formation of a surface deposit

Intermediate blocking 1 dtdV = 1

Q = f (t) Pore blocking + surface deposit

Standard blocking 1.5(

dVdt

)1/2 = Q1/2 = f (V) Pore constriction

Complete blocking 2 dV = Q = f (V) Pore blocking

psam

tddisitr

Ttbwt

2

Ts(fls

•

•

•

2

Tb

J

w

dt

ore diameter, zeta potential, etc.) and is applicable to variousystems (in this paper, we report the results obtained withbentonite suspension filtered through four microfiltrationembranes of different pore size).In our approach, we made the choice of not to describe the

ransitions between two apparent fouling regimes when twoifferent mechanisms may compete, a point of view whichiffers from other work (Ho and Zydney, 2000), but to approx-

mate the filtration curve into several successive apparentteps during which only one fouling mechanism is predom-nant. We also made the choice not to dilute the feeds buto develop an appropriate data analysis to capture the initialegime of blocking.

In this paper, the main steps of the approach are detailed.hen we see how one could select a filter medium adapted to

he filtration of a given suspension and we present how thelocking coefficients deduced with our methodology evolvehen parameters such as feed suspension concentration,

ransmembrane pressure or mean pore diameter change.

. Modelling flux decline

he approach proposed is based on the flux decline analy-is. Three types of fouling mechanisms have been consideredcake, complete blocking and standard blocking) leading to aux decline. Each mechanism is denoted by a specific sub-cript as:

(B) for complete blocking defined as particle capture at the‘suspension–filter’ interface;(C) for cake formation defined as particle capture at the topsurface of the filter (to form a cake or a deposit);(I) for standard blocking defined as particle capture insidethe filter.

.1. A series description for the different mechanisms

he evolution of the permeate flux, J, with filtration time haseen modelled by a classical Darcy law (Darcy, 1856):

= �P(2)

�R

here R is a hydraulic resistance.

Accordingly, the permeate flux is expressed by

J = �P

�(Rm + RC)(3)

where Rm and RC correspond to the hydraulic resistances ofthe filter (fouled or not) and the hydraulic resistance of thecake, respectively. As in an unstirred system, we use the clas-sical assumption that the deposit is covering the entire areaof the filter.

2.2. Complete blocking (B)

Concerning the complete blocking mechanism, we use a sur-face coverage ratio, ˇB (Ho and Zydney, 2000), to characterizethe surface fraction open to the flow. This ratio may evolveduring a certain time and then remains constant when com-plete blocking is not prevailing anymore. The surface coverageratio ˇB increases with the cumulative permeate volume, V, ata rate characterized by the blocking parameter �B

dˇB

dt= �B × J (4)

Eq. (4) can also be reformulated as

ˇB = �B × V

A0(5)

�B is the blocked surface area by unit of time and surface ofthe membrane (see Appendix A).

When a surface blocking prevails, the filter hydraulic resis-tance, Rm,B, is directly a function of ˇB according to

Rm,B = Rm,0

1 − ˇB(6)

also reformulated as

Rm,B = Rm,0

1 − �B(V/A0)(7)

Rm,0 is the clean membrane hydraulic resistance. It can becalculated from permeability tests on the clean (non-fouled)membrane before any filtration experiment.

2.3. Cake formation (C)

The cake hydraulic resistance, RC, increases with the cumula-tive permeate volume V, at a rate characterized by �C for cake

and d

1284 chemical engineering research

formation (C) according to the following equation:

RC = �C

A0V (8)

�C is the volumic specific resistance of the cake (see AppendixA).

2.4. Internal fouling (I)

The filter resistance, Rm,I, increases with the cumulative per-meate volume V, at a rate characterized by �I for internalfouling (I), according to the standard blocking law:

Rm,I = Rm,0

(1 − �IV)2(9)

�I is the volume of particle accumulated in the media per unitof permeate volume and per unit of void volume.

3. Materials

3.1. Suspension

Bentonite (clay) particles are platelets; the larger dimension ofthese platelets of about 1 �m is about 100 times the thickness(showing a large shape factor). Their basal face is positivelycharged, whereas the particle edge is negatively charged.

A first suspension was prepared by dispersing 30 g ofclay powder in 1 L of deionised water. Four settlings (each4-h long) lead to a stock suspension of 19 g L−1consistingof micron-sized particles. This suspension exhibits a broadparticle size distribution around 0.5 �m (see Fig. 1) deter-mined by dynamic light scattering experiments (Mastersizer2000, Malvern Instruments). Suspensions in the concentra-tion range 10−3–10 g L−1 are prepared by dilution of the stocksolution with deionised water.

3.2. Filter media

Four microfiltration membranes have been tested with dif-ferent nominal pore sizes as provided by manufacturers (0.2,0.8, 5 and 8 �m). The 0.2 and 0.8 �m membranes are made

Fig. 1 – Particle size distribution of a bentonite suspensionafter four settlings.

esign 8 6 ( 2 0 0 8 ) 1281–1293

of cellulose acetate (Schleicher and Schuell). The 5 and 8 �mmembranes are made of mixed cellulose esters (Millipore).According to information provided by manufacturers, nominalpore sizes are measured by liquid displacement methods.

3.3. Experimental set-up

The filtration system is composed of an air-pressurized feedtank connected to a 50-mL filtration cell (Millipore). A freshmembrane is mounted up in the filtration cell and thereservoir is first filled with deionised water. The system is pres-surized until a steady state flux is obtained. The cell is thenemptied and refilled with the suspension to be filtered. Fil-tration experiments are conducted at different pressures anddifferent concentrations of suspension without any stirring ofthe suspension. The permeate is collected in a beaker and thecumulative permeate mass is measured via an electronic bal-ance connected to a computer for on-line data acquisition. Atthe end of the filtration run, the used microfiltration mem-brane is discarded and the filtration cell is emptied. For agiven set of operating conditions (feed concentration, trans-membrane pressure), the permeate versus volume data arecollected from a series of five runs.

4. Methodology background

The first objective is to identify which fouling mechanisms areprevailing and when, during a filtration experiment. Even if wedo know that fouling phenomena can simultaneously occur,we assume that a filtration run can be split into several succes-sive steps in which only one fouling mechanism is prevailing.Flux decline analysis requires a rigorous procedure so as todiscriminate between the different fouling mechanisms froma differentiation of experimental data such as in Eq. (1).

From a filtration data set, V(t), it is necessary to developan optimized data post-processing method in order to deriveparameters from the (d2t/dV2) = f(dt/dV) equations. In orderto attenuate the noise in experimental data, a smoothingmethod is used: the moving average filtering method. It is a lowpass filter that averages five neighbouring data. This methodproved to be simple and efficient. Lastly, the finite differencesmethod (Constantinides, 1987) is used to numerically deriveV(t) into (d2t/dV2) and (dt/dV).

4.1. Prevailing fouling mechanisms

In Table 1, it has been noticed that a specific value of n charac-terizes each given fouling mechanism. In the cases for whichthe value of n found experimentally is different from the onesgiven in Table 1, we do not have any physical fouling modelthat corresponds to the flux decline: there may be a combina-tion of several identified or not identified mechanisms.

Let us now focus on the exponent n. The value of n enablesthe identification of the controlling fouling mechanisms,whereas the determination of k permits the quantification ofthe identified fouling mechanism (see Table 2).

According to Fig. 2 and Table 3, when k is derived by fittingEq. (1) to experimental data, it is greatly influenced by the timestep �t used in the differentiating method.

Accordingly, it seems more appropriate to use Eq. (1) for

mechanism identification purposes only (evaluation of n) butnot for quantification (estimation of k). k will therefore bedetermined by using the linear form of the models (Table 1).

chemical engineering research and design 8 6 ( 2 0 0 8 ) 1281–1293 1285

Table 2 – Relationships between k and the specific andsystem parameters

Fouling mechanisms k n

Cake filtration ��CA0�P

0

Standard blocking 2�IQ1/20 1.5

Complete blocking Q0A0�B 2

Fig. 2 – Influence of the time step �t on the k-valuedetermination by using the representation for theexperiment using a 0.2 �m membrane (C = 10−2 g L−1 and�

ocaoo

4

Omt

4s

At

•

Fig. 3 – Example of the determination of the slope n in therepresentation for the fouling mechanisms identification

−2 −1

fouling behaviour is dominated by the complete pore block-ing phenomenon (Step II) (n = 2); in this regime we calculate

P = 0.3 bar).

Besides, the use of Eq. (1) also allows the determinationf a range of specific data for which a given mechanism isontrolling the flux decline process. The blocking parametersre then evaluated on the data range corresponding to a periodf time where the blocking law prevails, which is a key pointf this methodology.

.2. Determination of blocking parameters

nce the data range has been determined for each foulingechanism, the parameters can be evaluated from the plot of

he linear form of the considered fouling mechanism (Table 4).

.3. Approximation of the filtration in severaluccessive fouling regimes

s for an example, for the ‘bentonite–.5 �m membrane’ sys-em (Fig. 3), this method allows to identify two fouling regimes

A very low fouling (VLF) regime, in our description the flowrate is indeed assumed to remain constant. We use theaverage value over the time period (Step I). This durationis actually an approximation since the permeate flow rateslightly decreases during this period (Fig. 4). But accordingto the V(t) into (d2t/dV2) = f(dt/dV) representation, the slope

n is greater than 2 for this period and it is therefore notpossible to refer to a well-identified fouling mechanism. We

Table 3 – k and n values for two time steps (see Fig. 3)

Time step, �t (s) k n

10 1.4 × 10−2 2.020 3.7 × 10−2 1.9

(experimental data for 5 �m membrame, C = 10 g L and�P = 0.3 bar).

came to suppose that this period may correspond to thedeposit of the few first particles having some pore bridgingeffect; since the mean pore diameter is larger than the meanparticle diameter r;

• surface blocking or pore blocking (B) regime for which the

Fig. 4 – Split of the curves into successive mechanisms: (A)very low fouling mechanism, (B) blocking, (C) cake filtration.

1286 chemical engineering research and design 8 6 ( 2 0 0 8 ) 1281–1293

Table 4 – Fouling mechanisms and their corresponding linearized forms

Fouling mechanism Linear form Equation

Cake formationdtdV

= f (V)dtdV

= 1QB,0(1 − ˇB,f)

+ ��C

A02�P

(V − VC,0) (10)

Surface blockingdVdt

= Q = f (V)Q

QB,0= 1 − ˇB = 1 − �B

A0(V − VB,0) (11)(

dV)1/2

V)(

Q)1/2

Internal foulingdt

= Q1/2 = f (

the value of �B by fitting experimental data with Eq. (11)(Fig. 4);

• cake filtration (C) regime for which the fouling behaviour isdominated by the build-up of a deposit at the top surface ofthe filter media (Step III) (n = 0) in this regime we calculatethe value of �C and ˇB,f by fitting experimental data with Eq.(10) (Fig. 4).

When the mean pore diameter is close to the mean particlediameter, we often observe the last two steps (Steps II and III).When the mean pore diameter is much smaller than the meanparticle diameter, only cake filtration (C) appears to controlflux decline from the very beginning (Step III).

4.4. Methodology approximations

Splitting the experimental filtration curve into several suc-cessive mechanisms might be a rough approximation ascommented before. To make an assessment of the approach,we seek to rebuild the filtration curve from the equationspresented in the previous section, and from the parametersderived from experimental data. This step is not a compulsorystep of the methodology but a test ensuring that approxima-tions do not introduce significant bias in the quantification ofblocking parameters.

For the ‘VLF’ period, the constant flow rate is taken as theaverage between the initial flow rate (Q0) when the membraneis clean (flow rate measured before the filtration run) and theflow rate (QB,0), the y-intercept of the plot of Q versus (V − VB,0)for the pore blocking regime (Fig. 4).

In order to determine the transition between the poreblocking regime and the cake formation regime, two straightlines (n = 2 and n = 0 (plateau)) can be drawn (Fig. 3). If weextrapolate these two lines, we consider their intersectiondetermines a transition between the two regimes.

We then compare the data calculated by approximationin successive fouling regimes with the experimental data.We show the comparison for some given operating condi-tions in Fig. 5: a suspension concentration of 10−2 g L−1 anda transmembrane pressure of 0.3 bar, for the four membranescovering a large spectrum of pore sizes (from 0.2 to 8 �m).

Fig. 5 shows the plot of experimental and calculated curvesfor the four membranes.

The methodology gives quite good results for 0.2, 5 and8 �m membranes, as the calculated curves are very close tothe experimental points. The difference between experimen-tal curve and calculated curve for 0.8 �m membrane is closeto 15% which is still acceptable.

4.5. Methodology robustness

According to the model, ˇB,f can be estimated as describedabove by fitting experimental data with a linear form for theblocking law in the cake filtration regime But it can also be

QI,0= 1 − �IV (12)

evaluated from �B as calculated from the experimental flowrate decline in the surface blocking regime and from the dif-ference VB,f − VB,0, according to the expression:

ˇB,f = �B

A0(VB,f − VB,0) (13)

VB,0 and VB,f respectively correspond to the first and the lastexperimental points of the defined range for which pore block-ing regime is dominant (n = 2).

In Fig. 6, we compared the values of ˇB,f determined fromthe first procedure (exp.) and from the second one (calc.). Thevalues obtained with both procedures are very close to eachother. We can conclude a good agreement between parame-ters determined in the surface blocking dominant regime withthose determined on the experimental data range for whichthe cake filtration regime is dominant. From this result, wecan deduce that our methodology is able to describe the foul-ing phenomena occurring during a filtration run even thoughthe transition periods have been neglected.

In the second part of the paper we shall describe how block-ing parameters (ˇB,f, �B, �c) correlate to filter type and operatingparameters, such as transmembrane pressure or feed concen-tration.

5. Results and analysis

5.1. Choice of a filter medium

When having to filter a given suspension, the industrial issueis often to get or maintain the highest flow rate at a fixed trans-membrane pressure. Whenever filtration is used to collect andconcentrate valuable solids, cake formation can be consid-ered as a production regime. The fouling of the filter medium,occurring before or during production is detrimental to theprocess efficiency. The selection of the filter medium shouldtherefore primarily be based on these filter-fouling issues only.

In this work, we run three series of filtration experimentswith a bentonite suspension through different microfiltrationmembranes. Blocking of the filter surface then cake formationonto the filter media were the two main successive foulingregimes which were identified by the methodology.

One can consider that cake formation does not contributeto filter fouling. However, the cake filtration generates a fluxresistance, whose effect is to decrease the filtration processproductivity. Its quantification is then worth to be known.

The (�c) parameter is an intrinsic characteristic of thefiltration cake as we observe the same variations with suspen-sion concentration and pressure, whatever the microfiltrationmembrane). Therefore, it does not help at selecting the bestfilter media.

The (�B) parameter characterizes the rate of surface block-

ing by a particle, whereas the surface parameter, ˇB, quantifiesthe surface coverage, and ˇB,f quantifies the coverage at theend of the blocking regime.

chemical engineering research and design 8 6 ( 2 0 0 8 ) 1281–1293 1287

F mpae .8 �

(tist

pfic

er

••

•

f0t

f

membranes are very sensitive to fouling by surface blockingmechanism as their fraction of surface open to flow is below25% at the end of the blocking regime. The knowledge of the

ig. 5 – Plot of cumulative permeate volume V vs. time t—coxperiments (C = 10−2 g L−1 and �P = 0.3 bar). [(�) 0.2 �m; (�) 0

These parameters are linked by the relation ˇB,f =�B/A0)(VB,f − VB,0) (Eq. (13)) but they do not show the samerends towards a variation (increase or decrease) of an operat-ng or system parameter. The study of the pore diameter effecthows that using a filter, whose mean pore diameter is closeo the mean particle diameter is obviously to be avoided.

The knowledge of the particle size distribution of a sus-ension to be filtered is often useful to avoid choosing certainlter types which could lead to severe media blocking, but thisriteria cannot remain the only basis.

Our goal is to choose a filter whose flux resistance at thend of the blocking regime (before starting the cake filtrationegime), RB,f = (Rm,0/1 −ˇB,f), has the smallest value.

Therefore, the characteristics of the “best” filter can be:

a low clean filter media resistance Rm,0;a low surface coverage ratio ˇB,f at the end of the blockingregime;or the best compromise between both parameters.

In Fig. 7, we plotted (VB,f − VB,0) versus �B, for a series of dataor which the pore diameter effect has been studied (dpore = 0.2,.8, 5, 8 �m). Operating conditions were kept the same for all

hese experiments: C = 10−2 g L−1, �P = 0.3 bar.

Only the 0.2 �m membrane displays a ‘high’ filtering sur-ace open to the flow, close to 50%, before the beginning of the

rison between experimental and calculated curves for fourm; (�) 5 �m; (�) 8 �m].

regime dominated by the “cake formation”. The three other

Fig. 6 – Effect of the transmembrane pressure �P on thefinal surface coverage ratio, ˇB,f for two differentmembranes (0.2 and 8 �m).

1288 chemical engineering research and design 8 6 ( 2 0 0 8 ) 1281–1293

Fig. 7 – Symbols are the values of VB,f–VB,0 vs. �B, for aseries of data (dpore = 0.2, 0.8, 5, 8 �m). Operating conditionswere kept the same for all these experiments:

−2 −1

Fig. 9 – Effect of feed suspension concentration C on thespecific parameter, �C for cake formation at constant

C = 10 g L , �P = 0.3 bar. Lines are the calculated data ofVB,f–VB,0 vs. �B, for different values of ˇB,f.

�B is therefore not sufficient to choose between different filtermedia as a low �B parameter does not necessarily indicate alow fouling. Indeed, Rm,B,f, the fouled filter media resistanceat the end of the blocking regime, is a key parameter in thefiltration process. Fig. 8 shows two curves:

• the clean filter media resistance Rm,0 versus dpore (©);• the fouled filter media resistance Rm,B,f versus dpore (�).

In Fig. 8, the data which correspond to the 5 and 8 �m mem-branes are not considered in the following discussion as wealready know that a membrane whose mean pore diameteris way greater than the particle size distribution should beavoided.

Usually, the choice of a filter media is usually made on thesole basis of the knowledge of the clean filter media resistance,

Rm,0, value. In Fig. 8, we can note that the 0.2 �m membranehas a clean filter media resistance which is higher than the oneof the 0.8 �m membrane whereas, at the end of the blocking

Fig. 8 – Evolution of a clean and fouled filter mediaresistance (respectively, Rm,0 and Rm,B,f) with its initialmean pore diameter, dpore.

pressure 0.3 bar.

regime period, the former membrane displays a flux resistancewhich is lower than the one of the latter. We can then deducethat the 0.2 �m membrane is the best compromise and is ableto maintain a good productivity during the cake formation.

In short, the choice of a filter media, based on the method-ology developed in this paper, remains on the combination ofthree key parameters:

• the clean filter media resistance Rm,0;• the �B parameter;• the filtrate volume during the blocking phase, (VB,f − VB,0).

5.2. Blocking parameters variations when changingoperating parameters

In this section, we focus on parameters variations for the sys-tem ‘bentonite suspension–cellulose MF membrane’ with thetwo main mechanisms: complete blocking and cake formation.

5.2.1. Effect of feed concentrationWe conducted filtration experiments for a wide range of feedconcentrations, from 10−3 to 10 g L−1. The transmembranepressure was set to 0.3 bar.

Fig. 9 shows the increase in �C the volumic specific resis-tance of the cake with bentonite concentration for the fourmembrane pore diameters. A correlation can be determined:�C = 7.2 × 1014 × C1.1, whatever the studied membrane, whichindicates that the value thus obtained characterizes the cakestructure, and id independent on the filter characteristics.

As shown in Fig. 10, �B also increases with the suspensionconcentration but in a nonlinear way. This response could beexplained by the presence of some interactions between theparticles and the filter media surface (when the concentrationof particles is doubled, the �B value is not necessarily doubled)and particles steric hindrance at the pores entrance. Besides,two zones can be distinguished according to the ratio of dpore

(pore diameter) to dp (mean particle diameter): the systemsfor which dpore ≤ dp and those for which dpore > dp.

We defined p/pore as the ratio of pore to particle pro-

jected surfaces (assuming spherical particles). Physically; itrepresents the number of blocked pores quantity per blocking

chemical engineering research and design 8 6 ( 2 0 0 8 ) 1281–1293 1289

Fig. 10 – Effect of feed suspension concentration C on thespecific parameter, �B for cake formation at constantpressure 0.3 bar.

Fig. 11 – Plot of �B vs. the product of feed concentration, Ctimes the number of blocked pores per unit of blockingp

p

sta

5TpbCf

Fig. 12 – Effect of transmembrane pressure on cakeformation: comparison between two membranes (0.2 and

is not influenced by the mean pore diameter of the filtermedia.

Fig. 13 – Effect of transmembrane pressure on pore

article, p/pore (C × p/pore) at constant pressure 0.3 bar.

article when pore size is smaller than particle size.

p/pore =(

dp

dpore

)2

when dpore > dp, whereas p/pore

=(dpore

dp

)2

when dpore ≤ dp

When �B is plotted versus C × p/pore (Fig. 11), data follow aame trend for concentrations below 1 g L−1. But for concen-rations above 1 g L−1, interactions between particles may playmore significant role.

.2.2. Effect of transmembrane pressurehe effect of transmembrane pressure on cake filtrationarameter is investigated for the range 0.03–1.2 bar. Two mem-ranes are considered: 0.2 �m and C = 0.1 g L−1; 8 �m and

= 1 g L−1. Since (�C) is proportional to C1.1, the sets of data

or the two pore diameters (0.2 and 8 �m) can be normalised

8 �m) by considering the plot of �C/C1.1 vs. �P.

by dividing (�C) by C1.1 (see Section 5.2.1). Data then fall onthe same line (Fig. 12) which confirms that cake filtration

blocking for two membranes (0.2 and 8 �m) by consideringthe plot of �B vs. �P.

1290 chemical engineering research and design 8 6 ( 2 0 0 8 ) 1281–1293

Table 5 – Wall shear stress values calculated using Eq. (i-16) for two membranes (0.2 and 8 �m) at the beginning of theblocking regime

dpore (�m) �P (Pa) QB,0 (m3 s−1) v̄ (m s−1) Re �w (Pa)

0.2 120,000 5.45 × 10−7 5.08 × 10−4 1.02 × 10−4 2030,000 3.67 × 10−7 3.42 × 10−4 6.84 × 10−5 14

3,000 4.51 × 10−8 4.20 × 10−5 8.40 × 10−6 1.7

8 120,000 3.01 × 10−6 2.81 × 10−3 2.25 × 10−2 2.860,000 2.48 × 10−6 2.31 × 10−3 1.85 × 10−2 2.330,000 1.83 × 10−6 1.70 × 10−3 1.36 × 10−2 1.7

9,000 8.02 × 10−7 7.48 × 10−4 5.98 × 10−3 0.758.83 × 10−5 7.07 × 10−4 0.09

Fig. 15 – Effect of the mean pore diameter of the filter mediaon the specific parameter, �B, for pore blocking

3,000 9.47 × 10−8

As shown in Fig. 13, (�B) decreases when the transmem-brane pressure increases for both membranes (0.2 and 8 �m).In Fig. 13, for the 8 �m membrane, the value of (�B) is high(1500 m−1) for very low transmembrane pressure, whereasit is much lower for the higher pressures. This shows thatbentonite particles have a strong tendency to stick to themembrane pores at low filtration rate. The trend is different forthe 0.2 �m membrane. As the mean particle diameter is herefour times greater than the mean pore diameter, the filtrationrate has a weak effect, compared to the 8 �m membrane.

From these data, we see that when increasing the trans-membrane pressure, a decrease of the �B value has beenobserved. The shear stress, �w, at the pore entrance should beconsidered in order to explain these observed variations. Fora cylindrical pore, the shear stress, �w, at the pore entrance isa function of the pore diameter, dpore, the fluid flow rate at thepore mouth, v̄, and the fluid viscosity, �.

�w = 8�v̄dpore

(14)

According to Eq. (14), we calculated the shear stresses (seeTable 5) corresponding to filtration experiments.

In Fig. 14, we have plotted �B versus the wall shear stress,�w. At low wall shear stress, �B has a high value, and con-versely. This result is in good accordance with the fact thatthe effect of a high wall shear stress at the pore entrance isto lift particles captured at the membrane surface, in other

words, to decrease the total amount of particles actually stickto in the filter media.

Fig. 14 – Effect of the wall shear stress, �w on pore blockingmechanism for two different membranes (0.2 and 8 �m).

(C = 10−2 g L−1, �P = 0.3 bar).

Also, a threshold value for the wall shear stress (�w ∼= 2 Pa)appears in Fig. 14, above which �B tends to stabilize when thetransmembrane pressure increases. The data for the 0.2 �mare all above the threshold value, which explains why the poreblocking mechanism is not as much affected by an increase oftransmembrane pressure as it is for the 8 �m membrane Asthe average size of bentonite particle is around 0.5 �m; it canbe expected that hydrodynamics will only play a role whenparticle size are smaller than pores. Accounting for the shearstress in the pore sheds some light, and some details on thephysics of the particle capture efficiency in this system.

5.2.3. Effect of pore diameterFour microfiltration membranes have been tested (dpore = 0.2,0.8, 5 and 8 �m). Transmembrane pressure was set at 0.3 barand the feed suspension concentration, C, at 10−2 g L−1.

If we now consider the effect of the mean pore diameteron �B, Fig. 15 shows a maximum for the 0.8 �m membrane forwhich dpore ≈ dp. This result confirms the fact that a maximumof fouling by pore blocking can be reached when using a filtermedia with a mean pore diameter close to the mean diameterof the particles of the suspension to be filtered.

We qualitatively compare Fig. 15 to the particle size dis-tribution of the bentonite suspension presented in Fig. 1.According to Fig. 1, the quantity CB, the concentration of allthe particles which block pores of diameter dpore, is lower fordpore = 0.2 �m than for dpore = 0.8 �m. Fig. 15 could be inter-preted as the variation of CB with the mean pore diameterdpore: the concentration of the potential particles which plug

pores of diameter dpore varies when using different mem-branes for a same given suspension.

desi

6

Aitcgmhof

sfTc

af

oadipasmtatpb

mvst

eoipaspae

A

Wp

A

A

Aad

b

chemical engineering research and

. Conclusions

phenomenological approach, which includes a qualitativedentification of the main fouling mechanisms prior to a quan-itative determination of parameters (blocking parameters)haracterizing the “suspension–filter” system, allows a ratherood description of the flux decline and underlying foulingechanisms during filtration of a particle suspension. We

ave considered here the filtration of a bentonite suspensionn several microfiltration membranes over a wide range ofeed concentrations and transmembrane pressures.

This work is based on splitting the fouling analysis into twoteps: one is dedicated to the identification of the prevailingouling mechanisms, and the fraction of process they control.his one much preferably will use Eq. (1), and the concept ofapture efficiency.

The second step uses the integrated form of Eq. (1) and isimed at finding the values of the model parameters, for eachouling mechanism.

Even if several fouling phenomena may simultaneouslyccur, the assumption of successive prevailing fouling mech-nisms used here enables a good description of flux declineuring a filtration run. The good agreement between exper-

mental and calculated data over wide ranges of operatingarameters is a favourable indication of the robustness to thispproach, even though the level of description of the filter-uspension system was kept as low as possible all along whichakes this approach quite a valuable one for complex sys-

ems such as industrial suspension. As a matter of fact, timend volume filtered are the only basic data used. Correlationo other information regarding the filter (pore size) or the sus-ension (concentration, particles size distribution) is possible,ut not necessary.

In the systems tested here, we identified two prevailingechanisms namely surface blocking and cake filtration. A

aluable further step would be to extend this work to otherystems implying pore blocking as well as internal fouling byesting various ‘suspension–filter’ pairs.

An original finding was that surface blocking may consid-rably reduce the filtering surface open to flow, as up to 80%f the filter surface could get blinded at the end of the block-

ng regime. Phenomenological correlations between blockingarameters for internal, surface or cake blocking mechanismsnd operating parameters such as feed concentration or pres-ure give first insights into physical aspects which rule thearticle trapping in different locations of the filter. Such anpproach should also be useful to develop more physical mod-ls for blocking phenomena.

cknowledgements

e would like to thank Rhodia Recherche for its financial sup-ort and its technical expertise.

ppendix A

.1. Complete blocking (B)

surface coverage ratio, ˇB, has been defined in order to char-cterize the membrane surface which is no more open to flow

ue to complete pore blocking.

The parameter ˇB may evolve with time, with ˇB = 0 at theeginning of a filtration run as the membrane surface is clean.

gn 8 6 ( 2 0 0 8 ) 1281–1293 1291

The extreme case for ˇB is ˇB = 1 for which the whole mem-brane surface is totally blocked and the permeate flux J isnull.

Complete pore blocking mechanism is directly illustratedthrough this parameter as its effect is to reduce the membranesurface open to flow.

A = A0(1 − ˇB) (i-1)

in terms of flow rate, Eq. (i-1) becomes:

Q = Q0(1 − ˇB) (i-2)

The latter equation (Eq. (i-2)) is valid for a blocking surfaceregime.

Let us now consider CB is the mass of particles whichblock the membrane surface per unit of permeate volume andAB/mB the ratio of the blocked membrane surface per unit ofmass of blocking particles. This ratio is assumed to be kept con-stant and an intrinsic characteristic of the studied system (blockingparticles–filter media pores).

CB × J × AB/mB is the blocked surface per unit of time andper unit of membrane surface, this quantity represents alsothe surface coverage ratio per unit of time, (dˇB/dt)

dˇB

dt= CB × J×AB/mB (i-3)

We herein define the quantity, �B as the blocked surfacearea by unit of time and surface of the membrane where�B = {CB × AB/mB}.

dˇB

dt= �B × J,

A.2. Cake formation (C)

In classical filtration theory, the hydraulic resistance of thecake filtration is linked to the deposit mass by means of thespecific cake resistance, ˛:

dRC

dt= ˛

dmC

dt(i-4)

When all particles deposit onto the membrane surface toform a cake a mass balance on the cake yields

dmC

dt= A0CCJ, (i-5)

where mC is the deposit mass per unit of membrane surface,CC is the mass of particles which participate to the build-upof the filtration cake per unit of permeate volume.

By combining Eqs. (i-4) and (i-5), we have the followingrelation:

dRC

dt= ˛A0CCJ (i-6)

We herein define the quantity, �C as the volumic specific resis-tance of the cake where �C = {CC˛}.

The permeate flux J, is also equals to:

J = 1A0

dVdt

(i-7)

and d

1292 chemical engineering research

By combining Eqs. (i-6) and (i-7), then by integrating from(V = 0) to V, we have a relation between RC and V

RC = �CV

A0(i-8)

If particles accumulating in the cake at the surface are specifi-cally deposited by a sieving mechanism, �C = 1. There are somecases for which �C could be different from 1, for instance sed-imentation (�C > 1) and back-diffusion (�C < 1).

Note: a relation is able to link CC to C (C is the feed suspen-sion concentration):

CC = C�p(1 − εC)�p(1 − εC) − C

(i-9)

For very diluted suspensions (C � �p(1 − εC)), CC can be assimi-lated to C. But for concentrated suspensions, Eq. (i-5) is neededto properly calculate CC.

A.3. Internal fouling (I)

Internal fouling results in an accumulation of particles intothe filter media, and therefore an increase of the filter mediaresistance, named in this case Rm,I. Internal fouling mecha-nism can be described by the standard blocking law (Bolton etal., 2006).

The standard blocking law’s assumptions are a Poiseuilleflow in a network of uniform cylindrical pores, and parti-cles should uniformly deposit on the pore walls resulting ina decrease of the inner diameter of the pores.

According to the standard blocking filtration law, the porevolume variation is supposed to be proportional to the perme-ate volume variation, due to the particles retention along thepore wall:

Npore(−2�ada)L = Cv dV (i-10)

where Npore is the number of the cylindrical pores in the mem-brane; a, the pore radius of one cylindrical pore; L, the porelength; Cv, the volume of the particles which uniformly depositalong the pore wall per unit of permeate volume.

By integrating Eq. (i-11), from a = a0; V = 0 to a; V, a relationbetween a and V is obtained

Npore�(a02 − a2)L = CvV (i-11)

According to Poiseuille’s equation, the permeate flow ratethrough the clean membrane can be expressed:

Q0 = Npore

(�

8

a40

L

)�P

�(i-12)

Eq. (i-13) shows proportionality between Q and a4. We canhence deduce these following ratios:

Q

Q0=

(a

a0

)4⇒

(Q

Q0

)1/2

=(a

a0

)2(i-13)

Eq. (i-12) can be reformulated:

1 −(a

a0

)2= CvV

NporeL�a20

(i-14)

esign 8 6 ( 2 0 0 8 ) 1281–1293

By combining Eqs. (i-13) and (i-14), a linear relation betweenQ1/2 and V is obtained:

(Q

Q0

)1/2

= 1 − CvV

NporeL�a20

(i-15)

We tried to generalize Eq. (i-16) to a filter media which doesnot necessarily have a network of uniform cylindrical pores.

If no assumption is made on the filter media structure, thevoid volume of a clean filter media is A0Lmεm, where Lm andεm are, respectively, the thickness and the porosity of the filtermedia.

By analogy, we have two relations:

Cv ⇔ CI

�pand NporeL�a

20 ⇔ A0Lmεm

where CI is the mass of particles which deposit along the porewall per unit of permeate volume.

Hence, from Eq. (i-16), we have a generalized relationbetween Q1/2 and V

(Q

Q0

)1/2

= 1 − (CI/�p)VA0Lmεm

(i-16)

According to Darcy’s law, the permeate flux, J, is inversely pro-portional to the filter media resistance, Rm.

Eq. (i-17) can be rewritten and we obtain a relation betweenRm,I and V

(Rm,0

Rm,I

)1/2= 1 − (CI/�p)V

A0Lmεm(i-17)

And gives the following relations for Rm,I the hydraulic resis-tance of the filter media internally fouled

Rm,I = Rm,0

(1 − �IV)2, (i-18)

where �I = {(CI/�p)/(A0Lmεm)} is the volume of particle accu-mulated in the filter media per unit of permeate volume andunit of void volume.

Appendix B. Supplementary data

Supplementary data associated with this article can be found,in the online version, at doi:10.1016/j.cherd.2008.06.005.

References

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