Developments of Blocking Filtration Model in Membrane Filtration †

Blocking filtration laws consist of four different filtration mechanisms: complete blocking, standard blocking, intermediate blocking, and cake filtration. Blocking filtration laws for describing both the pore blocking and cake formation have been extensively employed over the past several decades to evaluate the increase in filtration resistance with the progress of filtration in the field of classical particulate filtration. In recent years, blocking filtration laws become widely used also in membrane filtration such as microfiltration and ultrafiltration of colloids. This paper gives an overview of the developments of blocking filtration laws and equations under constant pressure and constant rate conditions reported for the filtrate flow of Newtonian and non-Newtonian fluids. The fouling index evaluating the degree of membrane fouling was examined on the basis of the blocking filtration equations. The blocking filtration laws were reexamined to extend the range of their application. Moreover, various combined models developed based on the blocking filtration laws were introduced for describing more rigorously the complicated filtration behaviors controlled by more than one mechanism which occurs successively or simultaneously.


Introduction
Membrane filtration processes such as microfiltration and ultrafiltration of dilute colloids play an increasingly important and indispensable role in widely diversified fields ranging from industry to drinking water production, treatment of domestic and industrial effluents, and production of water suitable for reuse.While membrane filtration is a key process which has the widespread application, it is generally recognized that one of the major drawbacks to more widespread use of membrane filtration is a significant increase in the filtration resistance known as membrane fouling, resulting in a dramatic flux decline over time under constant pressure conditions or a remarkable pressure rise over time under constant rate conditions.The membrane fouling is affected by several factors, e.g., pore blocking and/or pore constriction (Hermans andBredée, 1935, 1936;Grace, 1956;Shirato et al., 1979;Hermia, 1982;Iritani et al, 1992Iritani et al, , 2009Iritani et al, , 2013)), cake formation (Reihanian et al., 1983;Chudacek and Fane, 1984;Iritani et al., 1991aIritani et al., , 2014a, b;, b;Nakakura et al., 1997;Mohammadi et al., 2005;Thekkedath et al., 2007;Sarkar, 2013;Salinas-Rodriguez et al., 2015), solute adsorption (Fane et al., 1983;Iritani et al., 1994), and concentration polarization (Kimura and Sourirajan, 1967;Vilker et al., 1981).Initially, foulants smaller than the pore size of membrane deposit or adsorb onto the pore walls, thereby leading to the pore constriction.This induces a significant reduction in the cross-sectional area available to the filtrate flow.In contrast, larger foulants deposit or adsorb onto the pore entrances, resulting in a marked increase in the filtrate flow resistance.In either case, the pore constriction and pore plugging are followed by the formation of filter cake accumulating on the membrane surface, thus severely increasing the filtration resistance.Therefore, it is essential to elucidate the underlying mechanism controlling the membrane fouling such as the pore constriction, pore plugging, and cake formation during the course of membrane filtration.
So far, a number of models have been proposed to describe the fouling of filter medium during the classical liquid filtration.The theory of cake filtration in which the filter cake forms on the surface of filter medium was initially established by Ruth (1935Ruth ( , 1946) ) and then extended to deal with the case of the compressible filter cake, as referred to as the modern filtration theory (Grace, 1953;Tiller, 1953;Okamura and Shirato, 1955;Tiller and Shirato, 1964;Shirato et al., 1969).
The classical blocking filtration laws describe three types of physical mechanisms controlling the blockage of membrane pores, in addition to the cake filtration model.The blocking filtration laws were originally presented by Hermans andBredée (1935, 1936) and later systematized by Grace (1956), Shirato et al. (1979), andHermia (1982).The model consists of four different filtration mechanisms: complete blocking, standard blocking, intermediate blocking, and cake formation.Among them, both complete and intermediate blocking laws describe the pore plugging due to foulants reaching the top surfaces of pores.In contrast, the standard blocking law deals with the pore constriction caused by the deposition of foulants onto the pore wall.Interestingly enough, these four filtration mechanisms reduce to a common differential equation with different values of power index.While blocking filtration laws are summarized in quite simple mechanisms, still present today they provide a powerful tool to reasonably evaluate the increasing behavior of filtration resistance in liquid filtration of relatively dilute suspension.Nowadays, blocking filtration laws have become widely used in the analysis of the flux decline behaviors observed not only in classical liquid filtration but also in membrane filtration such as microfiltration and ultrafiltration.Therefore, it is considered that there is a significant value to have an overview of the developments of blocking filtration laws and the related mechanisms.
This review paper initially describes the classical blocking filtration laws derived under constant pressure conditions.Then, the blocking filtration laws are extended to be applied to constant rate (flux) filtration and are generalized through the inclusion of filtrate (permeate) flow of non-Newtonian fluids.A common characteristic filtration form derived from the blocking filtration laws is revisited by considering the membrane pore fouling represented by Kozeny-Carman equation describing the flow through the granular bed.The paper explains that the blocking filtration laws are made available for evaluating the degree of membrane fouling, e.g., the maximum filtrate volume and the fouling index such as SDI and MFI.Finally, the combined models stemming from the blocking filtration laws are described to reasonably evaluate two fouling mechanisms occurring sequentially or simultaneously, which are frequently observed in the actual processes of membrane filtration of colloids.Specifically, much emphasis is placed on the combination of membrane pore blockage and cake formation on the membrane surface.These combined model can approximate highly complicated membrane fouling behaviors often encountered in membrane filtration.

Blocking filtration law 2.1 Mechanism of blocking filtration
Blocking filtration laws are applied to four different fouling patterns for describing the deposit of particles on filter media and membranes, as schematically illustrated in Fig. 1.The complete, intermediate, and standard blocking laws describe the blocking of membrane pores, while the cake filtration law is applied to the description of the growth of filter cake comprised of particles accumulating on the membrane.For simplicity, it is postulated that the membrane consists of parallel pores with constant diameter and length.Both the complete and intermediate blocking laws are applicable in case that the particle diameter is larger than the pore size.Thus, each particle reaching the membrane due to convection is inevitably trapped on the membrane surface in either case.However, pore blocking behaviors are substantially different from each other.In the complete blocking law, it is assumed that each particle blocks an open pore completely, as shown in Fig. 1(a).When more appropriate, it is assumed that the probability that a particle blocks an open pore is constant during the course of filtration, considering the possibility that a particle deposits on the membrane surface other than pores.Therefore, the number of blocked pores is directly proportional to the filtrate volume v per unit effective membrane area.The variation of the number of open pores during the course of filtration is given by   the filtration rate J is directly proportional to the number of open pores, it can be written as where t is the filtration time, k c is the proportional constant, and p is the applied filtration pressure.Differentiating Eq. ( 1) under the constant pressure condition (p = p 0 = const.),one obtains where K b (= k c p 0 x) is the blocking constant for complete blocking law.However, in practice, the probability that a particle blocks an open pore varies with v during the course of filtration.As the number of open pores decreases due to the progress of filtration, particles newly reaching the membrane may deposit onto the particles that have already blocked the open pores, as shown in Fig. 1(c).In the intermediate blocking law, it is assumed that the rate of pore blocking is proportional to the number of open pores, and thus dN′/ dv may be written as where N′ is the number of open pores per unit effective membrane area at the filtrate volume v per unit effective membrane area, and K i is the blocking constant for intermediate blocking law.Integrating Eq. ( 3), the number of open pores becomes . Consequently, the filtration rate is given by Differentiating Eq. ( 4) under the constant pressure condition, one obtains (Hermia, 1982) Although the intermediate blocking law had been considered to be empirical over the years (Grace, 1956), Hermia (1982) originally verified the theoretical background of the intermediate blocking law by considering the decrease in the probability blocking membrane pores with the progress of filtration.Around the same time, Hsu and Fan (1984) proposed the intermediate blocking equations applicable to constant rate filtration on the basis of the stochastic model (pure birth model) in order to use them in the analysis of sand filtration behaviors.Iritani et al. (1991b) derived the intermediate blocking equations for constant pressure filtration based on the stochastic model by considering the probabilistic event with the progress of the filtrate volume v per unit membrane area instead of the filtration time t because the number of particles reaching the membrane is proportional to v in the case of constant pressure filtration.Fan et al. (1985a, b) modified the intermediate blocking equations based on the birth-death model.In the model, scouring of particles blocking pores was also taken into consideration as the death process.
In the standard blocking law shown in Fig. 1(b), the particle diameter is considerably smaller than the pore size.Therefore, solid-liquid separation proceeds by the deposition of particles on the pore wall and the pore gradually constricts with the progress of filtration.For simplicity, it is postulated that the pore volume decreases proportionally to the filtrate volume v per unit membrane area.Consequently, the filtration rate under the constant pressure condition gradually decreases with decreasing pore size.On the assumption that the pore radius decreases by dr by obtaining an infinitesimal amount of filtrate volume dv per unit membrane area, the mass balance produces (Grace, 1956) where L is the membrane thickness, c is the volume of particles trapped per unit filtrate volume v per unit membrane area, and ε p is the packing porosity of the particle layer formed on the pore wall.Integrating Eq. ( 6) from r = r 0 at v = 0 to r = r at v = v, one obtains where K s is the blocking constant for standard blocking law and defined by In general, the relation between the average flow rate u and pore radius r can be given by the Hagen-Poiseuille law applicable to the laminar flow in a capillary, and it is written as where μ is the viscosity of the filtrate.Thus, the filtration rate through the membrane comprised of capillaries can be described by at the start of filtration; at any time of filtration, where the subscript "0" indicates the initial value.Substituting Eqs. ( 10) and (11) into Eq.( 7) under the constant pressure condition (p = p 0 = const.),one obtains Differentiation of Eq. ( 12) yields The filtration data of most dilute suspensions is described by the standard blocking law.
In contrast to the pore blocking, in cake filtration shown in Fig. 1(d), the filter cake consisted of the particles deposited on the membrane surface gradually grows as filtration proceeds.The filter cake is viewed as a kind of granular bed which produces the increase in the thickness with the progress of filtration, thereby resulting in the increase in the additional resistance to flow.According to the Ruth theory, the filtration rate J is given by (Ruth, 1935(Ruth, , 1946) where α av is the average specific cake resistance, ρ is the density of filtrate, s is the mass fraction of solids in colloids, m is the ratio of the mass of wet to the mass of dry cake, and v m is the fictitious filtrate volume per unit membrane area required to obtain the cake with the flow resistance equivalent to that of the membrane.Since both α av and m are considered to be constant throughout the period of filtration under the constant pressure condition, Eq. ( 14) becomes where K v (= 2p(1 -ms)/(μα av ρs)) is the Ruth coefficient in constant pressure cake filtration and constant (Ruth, 1935(Ruth, , 1946)).Although the filter cake generally exhibits compressible behavior in which α av varies with the applied filtration pressure, it should be noted that Eq. ( 15) is applicable to not only the incompressible cake but also compressible cake (Tiller and Cooper, 1960;Tiller and Shirato, 1964).Differentiating Eq. ( 15) with respect to the time t, one gets where K c (= 2/K v ) is the blocking constant for cake filtration law.Interestingly enough, Eqs.(2), ( 5), (13), and (16) derived for four different filtration mechanisms reduce to the following common differential equation (Hermans andBredée, 1935, 1936;Grace, 1956).
where k and n are constants.The value of the blocking index n depends on the mode of filtration mechanisms, and indicates 2.0 for complete blocking, 1.5 for standard blocking, 1.0 for intermediate blocking, and 0 for cake filtration.The constant k is the resistance coefficient depending on the properties of suspension, membrane, and the operating conditions in filtration such as the applied filtration pressure.It is found from Eq. ( 17) that the changing rate in the filtration resistance is proportional to the filtration resistance raised to a power n.When the blocking filtration laws are represented by using the filtration rate J (= dv/dt), Eq. ( 17) reduces to (Sun et al., 2003) A double logarithmic plot of d 2 t/dv 2 vs. dt/dv is depicted based on the flux decline behaviors in order to understand the membrane fouling mechanism of constant pressure filtration with the aid of Eq. ( 17).The fouling mode can be easily determined from the slope of a linear regression fitting to the plot.However, since d 2 t/dv 2 is the second order differential of the filtration time t with respect to the filtrate volume v per unit membrane area, the value of n determined from the direct use of Eq. ( 17) is likely to be influenced by the noise in the experimental data measured as the cumulative filtrate volume v per unit membrane area vs. the time t.Nevertheless, this plot provides important information on the fouling mechanism of membranes, as mentioned later (Bowen et al., 1995;Iritani et al., 1995Iritani et al., , 2014c)).
Table 1 systematically edits mathematical equations derived for each blocking filtration law under the constant pressure condition and represents v as functions of t and the filtration rate J as functions of t or v (Hermans andBredée, 1935, 1936;Grace, 1956).These expressions can be used as a means for identifying the membrane fouling mechanism, just like the double logarithmic plot of d 2 t/dv 2 vs. dt/dv.The fouling pattern of membranes can be judged from the plots shown in Fig. 2 based on the linear expressions.For example, if the membrane fouling is controlled by the complete blocking filtration mechanism, the plot of J vs. v and the semi-logarithmic plot of J vs. t should show straight lines.Thus, the predominant blocking filtration law describing membrane fouling pattern can be determined from the graphical expression best fitted based on a linear regression analysis.
Integration of Eq. ( 18) with respect to the filtration time t produces the relation between the filtration rate J vs. the time t (Herrero et al., 1997;Ho and Zydney, 1999).It should be noted that the standard blocking (n = 1.5), intermediate blocking (n = 1.0), and cake filtration (n = 0) modes are represented as a common equation given by where ) and n j (= 1/(2 -n)) are constants.In contrast, in the case of the complete blocking law (n = 2), integrating Eq. ( 18), one obtains Conventionally, the experimental data of the flux decline during the blocking filtration period have been analyzed by only one of above mentioned blocking filtration laws (Granger et al., 1985;Hodgson et al., 1993;Blanpain et al., 1993;Ruohomäki and Nyström, 2000;Gironès et al., 2006;Lee et al., 2008;de Lara and Benavente, 2009;Nandi et al., 2010;Li et al., 2012;Lim and Mohammad, 2012;Pan et al., 2012;Masoudnia et al., 2013, Palencia et al., 2014).Tettamanti (1982) proposed five blocking filtration laws by adding the adhesive filtration law in which the value of k in Eq. ( 17) is represented as zero.When a stepwise procedure is employed, the blocking filtration equation is derived in the form (Heertjes, 1957) where a and S c are constants.In the case of complete blocking, a = 1.0 and S c = 1.0.For partial blocking, a < 1.0 and S c > 1.0.As the slurry concentration increases, a decreases and S c increases, and both a and S c depend on the form of pore and of the particle.In recent years, several problems are pointed out in applying the blocking filtra- Table 1 Blocking filtration equations for constant pressure filtration tion laws to the analysis of the experimental data of the flux decline during blocking filtration (Tien and Ramarao, 2011;Tien et al., 2014).

Blocking filtration equations for constant rate filtration
Although the membrane blocking research has been focused mostly on constant pressure filtration, which is easily tested on the laboratory scale, the membrane blocking in constant rate filtration is also crucially important from the viewpoint of the industrial level.While the flux decline behaviors are examined in constant pressure filtration, the pressure rising behaviors resulting from the increase in the filtration resistance are investigated in constant rate filtration in which the filtration rate is kept constant (Blankert et al., 2006;Liu and Kim, 2008;Sun et al., 2008;Mahdi and Holdich, 2013;Raspati et al., 2013).
The blocking filtration equations for constant rate filtration can be obtained in accordance with a procedure similar to that used in constant pressure filtration.For example, in complete blocking law, Eq. ( 1) is rewritten as Differentiating Eq. ( 34) with respect to v under constant rate condition (J = J 0 = const.),one gets where K br (= k c x/J 0 ) is the blocking constant for complete blocking law in constant rate filtration.Eventually, also in the case of constant rate filtration, four blocking filtration laws are represented by a common differential equation in the form (Grace, 1956) where k′ and n′ are constants.A power index n′ is the blocking index which defines the filtration mechanism, and indicates the same value as constant pressure filtration for each blocking filtration law.Table 2 lists the equations derived for constant rate filtration (Grace, 1956).Graphical indications are illustrated in Fig. 3 and are employed in order to judge the blocking filtration law.
The filtrate volume v per unit membrane area is directly proportional to the filtration time t in constant rate filtration, i.e., v = J 0 t.Consequently, it is also possible to identify the dominant blocking filtration law by plotting the data against t in place of v shown in Fig. 3.
The value of n′ is 0 for cake filtration in which an incompressible filter cake forms during filtration.However, when the filter cake exhibits a compressible behavior, the use of Eq. ( 36) requires a considerable attention.Equation ( 14) is rewritten as The average specific cake resistance α av in Eq. ( 41) is related to the applied filtration pressure p by (Sperry, 1921;Murase et al., 1987;Iritani et al., 2002;Zhou et al., 2015) where α 1 and n 1 are constants, and n 1 is especially termed the compressibility coefficient of the filter cake.The higher the n 1 -value, the more compressibility the cake is.Substituting Eq. ( 42) into Eq.( 41), one obtains (Tiller, 1955;Shirato et al., 1968)   Differentiation of Eq. ( 43) with respect to v under the constant rate conditions leads to where ) is a constant.Therefore, the value of n′ in Eq. ( 36) is not 0 for constant rate cake filtration in which a compressible filter cake forms during filtration and increases as the compressible behaviors become more pronounced.If the compressibility coefficient n 1 is 1.0, as seen in highly compressible filter cake, the power index n′ in the differential equation for cake filtration shows the same value as that derived for the intermediate blocking law, thereby leading to serious confusion.
Since v is directly proportional to the time t in constant rate filtration, Eq. ( 36) is rewritten as where k 2 is a constant.It should be noted that the characteristic differential equation can be represented also in the form (Hlavacek and Bouchet, 1993) where 1 k and 1 n are constants.Equation ( 46) is similar to Eq. ( 17) in the form, by considering that p in Eq. ( 46) corresponds to v in Eq. ( 17).Alternatively, the characteristic form is represented by (Chellam and Xu, 2006;Chellam and Cogan, 2011) where 2 k and 2 n are constants.Also in this case, the use of Eq. ( 47) requires a considerable attention in applying it to filtration forming compressible filter cakes.Chellam and Xu (2006) employed the expression of the average specific cake resistance α av increasing with pressure p in the analysis of constant rate filtration behaviors, as described by where α 0 and α 2 are constants, and α 0 is the average specific cake resistance at null stress.Eq. ( 48) has been applied to microbial suspensions, which exhibit highly compressible behaviors under relatively high pressure conditions.In this case, the value of exponent 2 n in Eq.
(47) becomes 1.5, and thus is the same as the complete blocking law.
Blocking filtration equations applicable to variable pressure and variable rate filtration were proposed by Suarez and Veza (2000).For instance, in the complete blocking law, the relation among the filtration rate J, applied filtration pressure p, and filtrate volume v per unit membrane area can be obtained on the basis of Eq. ( 1) and thus is represented by The model adequately described the blocking filtration behaviors of effluent water from a municipal wastewater treatment plant.

Blocking filtration equations for filtrate of non-Newtonian fluids
When the filtrate flow presents with non-Newtonian behaviors, the analysis for blocking filtration law becomes even more complex than the filtrate flow of Newtonian fluids.In power-law non-Newtonian fluids, which are the simplest case, the rheological equation representing the relation between the shear stress τ and shear rate  is written as where K is the fluid consistency index, and N is the fluid behavior index representing the intensity of non-Newtonian behaviors.When the fluid is characterized by Newtonian fluid, the value of N is equal to 1.0 and departs from 1.0 as non-Newtonian behaviors are more pronounced.The values of N less than 1.0 characterize pseudo-plastic or shear thinning fluids and melts.On the basis of Eq. ( 50), the relation between the average flow rate u and the tube radius r for power-law non-Newtonian fluids is given by (Kozicki et al., 1966) When one puts N as 1.0 and K as the Newtonian viscosity μ, Eq. ( 51) reduces to the Hagen-Poiseuille equation applicable to Newtonian flow represented as Eq. ( 9).While the number of open pores varies with the prog-ress of filtration in both complete blocking and intermediate blocking laws, the radius of open pores remains constant since the start of filtration.Therefore, the forms of the blocking filtration equations for complete blocking and intermediate blocking laws are independent of the flow behaviors in pores, irrespective of the difference between Newtonian and non-Newtonian fluids.However, since the pore radius gradually decreases with the progress of filtration in the standard blocking law, the difference between Newtonian and non-Newtonian fluid is quite obvious, as inferred by Eq. ( 51).The filtration rates at the start and any time of filtration are, respectively, represented as Substituting Eqs. ( 52) and ( 53) into Eq.( 7) under the constant pressure condition (p = p 0 = const.),one gets Differentiating Eq. ( 54), one obtains (Shirato et al., 1979) The filtration rate in cake filtration is described by (Kozicki et al., 1968(Kozicki et al., , 1972;;Shirato et al., 1977Shirato et al., , 1980;;Murase et al., 1989) where γ av is the average specific cake resistance for powerlaw non-Newtonian flow.Differentiating Eq. ( 56) with respect to v under the constant pressure condition (p = p 0 = cont.)where γ av and m are treated as constant throughout the course of filtration, one obtains Consequently, four blocking filtration laws for powerlaw non-Newtonian filtration under the constant pressure condition can be represented by a common differential equation with two constants k N and n N , as described by (Shirato et al., 1980;Hermia, 1982;Rushton, 1986 In the case of constant rate filtration, the characteristic form is represented as where N k and N n are constants.The blocking filtration equations are listed in Table 3 for constant pressure and constant rate filtration processes for power-law non-Newtonian flow.In the case of constant pressure filtration, the complete and intermediate blocking filtration equations for the filtrate flow of non-Newtonian fluids are the same as those for the flow of Newtonian fluids and thus It should be stressed that the equations for Newtonian fluids is the special case of those for power-law non-Newtonian fluids.Fig. 4 shows the experimental results of clarification filtration of dilute suspensions prepared by suspending diatomaceous earth in aqueous sodium polyacrylate exhibiting the behavior of power-law non-Newtonian fluid (Iritani et al., 1991b).The plots of J 2N/(3N+1) vs. v yield linear relationships in accordance with Eq. ( 54) for the standard blocking law since the diameter d p of the suspended solids is much smaller than the pore size d m .As the solids mass fraction s in suspension decreases, the slope of straight line decreases and thus the filtrate volume obtained during clarification filtration increases.

Significance of blocking index n in blocking filtration laws
The unified characteristic form of blocking filtration laws is derived from four different filtration mechanisms.The blocking index n in Eq. ( 17) results in values of 2.0, 1.5, 1.0, and 0 for complete blocking, standard blocking, intermediate blocking, and cake filtration laws, respectively.However, in practice, the experimental data frequently exhibit the value other than these.Moreover, even though the pore size is much larger than the particle size, the blocking index n occasionally exhibits the value of 2.0.Therefore, from this point of view it is of significance to throw a new look at the underlying model of the characteristic form of blocking filtration laws.
In the blocking filtration model, for simplicity, it is assumed that the membrane consists of parallel cylindrical pores with constant diameter and length.However, in practice, the porous structure of most membranes is of a complex geometry with irregular pore morphology.The Kozeny-Carman equation can describe the flow through such a porous medium and is written as where ε is the porosity, S is the specific surface area, and k 0 is the Kozeny constant.Iritani et al. (2007a) derived the characteristic form of blocking filtration laws on the basis of the Kozeny-Carman equation ( 70), by considering the variations of the porosity and the specific surface area of the membrane caused by the particle deposition within the porous membrane during filtration, as schematically illustrated in Fig. 5.The porosity ε in Eq. ( 70) decreases by particle deposition on the pore walls with the progress of filtration and is represented by where ε 0 is the initial porosity of clean membrane, and K p is a constant.The specific surface area S of the membrane in Eq. ( 70) also varies with the progress of filtration and it is assumed that S is described as where S 0 is the initial specific surface area of the clean membrane, and β is a constant which depends on the mode of the morphology of the deposit assemblages and defined as  where D s is the representative diameter of pores on a wetted perimeter basis, D is the representative diameter of pores on a flow cross-sectional area basis, and the subscript 0 indicates the clean membrane.The characteristic form represented by Eq. ( 17) can be obtained by substituting Eqs. ( 71) and (72) into Eq.( 70) and then by differentiating the reciprocal filtration rate (dt/dv) with respect to v under constant pressure condition (p = p 0 = const.).It should be stressed that Eq. ( 17) derived in this way is no longer limited by the specific values of n.In this sense, it is concluded that the derivation of Eq. ( 17) based on the Kozeny-Carman equation is more universal than the classical blocking filtration laws.In a similar way, Eq. ( 36) applicable to constant rate filtration can be derived based on the Kozeny-Carman equation (Iritani et al., 2011).Some researches were conducted on pore fouling behaviors by employing the Kozeny-Carman equation (Broeckmann et al., 2006;Zhong et al., 2011;Wu et al., 2012).Cheng et al. (2011) derived the characteristic form described by Eq. ( 17) based on not the Kozeny-Carman equation but the Hagen-Poiseuille equation.
In the derivation of Eq. ( 17), it is implicitly assumed that the particles are in complete retention, or that the amount of particles deposited within the pores of the membrane increases linearly with v.However, in practice, there often exists some solid leakage through the membrane (Iritani et al., 1994;Rodgers et al., 1995;Hwang et al., 2006;Hwang and Sz, 2010;Polyakov, 2008;Polyakov and Zydney, 2013).As a result, the sieving coefficient of solids varies during the course of filtration.Therefore, the characteristic form represented by Eq. ( 17) can be generalized by accounting for the variation with time of the amount of the particle deposition within the pores of the membrane.By employing the mass σ of particles deposited on the pore wall per unit membrane area, referred to as the specific deposit (Maroudas and Eisenklam, 1965;Ives and Pienvichitr, 1965;Tien and Payatakes, 1979;Choo and Tien, 1995), the variations of ε and S with the progress of filtration can be, respectively, represented by where K m is a constant.In a similar way to the derivation of Eq. ( 17), on the basis of Eq. ( 70), (74), and (75), the modified form of Eq. ( 17) can be derived as (Iritani et al., 2010)   Therefore, when the relation between σ and v is experimentally obtained, the left-hand side value of Eq. ( 76) may be calculated from the stepwise difference quotient using the experimental data of dt/dv vs. v with the aid of the relation between σ and v. Consequently, the values of n and k in Eq. ( 76) can be easily obtained from the double logarithmic plots of d(dt/dv)/dσ vs. dt/dv.Fig. 6 compares the logarithmic plots of d 2 t/dv 2 vs. dt/dv with those of d(dt/dv)/dσ vs. dt/dv for the experimental results in membrane filtration of dilute suspensions of monodisperse polystyrene latex (PSL) under the constant pressure condition using diatomaceous ceramic membranes which are semi-permeable to the PSL (Iritani et al., 2010).The plots for each run show a convex curve with the exception of the case when the initial porosity ε 0 of the membrane is 0.7.The slope of curve gradually decreases with the increase in dt/dv in accordance with the progress of filtration.In contrast, the plots of d(dt/dv)/dσ vs. dt/dv show a linear relation with the same slope of 2.35 for each initial porosity except for the last part of filtration.Thus, Eq. ( 76) provides a much better description of blocking filtration behaviors obtained for semi-permeable membranes than Eq. ( 17).
On the basis of the Kozeny-Carman equation, Bolton et al. (2005) proposed a fiber-coating model in which the filter medium becomes plugged as solids coat the surface of cylindrical fibers that constitutes the filter medium.In the model, the fibers become thicker with the progress of filtration, as shown in Fig. 7.As a result, the effective radius of fibers increases with time, and correspondingly the porosity decreases, reducing the filter permeability.Therefore, with the aid of the Kozeny-Carman equation, the relation between the filtration rate J and v becomes where K f is the fiber coating constant and equal to the inverse of the solution volume filtered until the filter void volume is completely filled with solids.

Evaluation of degree of membrane fouling
It is essential to evaluate the degree of membrane fouling during filtration on the basis of the theoretical background in the design of new filter equipment and optimization of commercial filtration operations.The maximum filtrate volume v max per unit membrane area is defined as the value of v obtained by the time when the filtration rate drops to zero.If the flux decline behavior is controlled by the standard blocking law, on the basis of Eq. ( 24), v max can be given as (Badmington et al., 1995;van Reis and Zydney, 2007) Combining Eq. ( 78) with Eq. (24) yields Consequently, one can evaluate what percentage of v max has been already obtained when the filtration rate ratio (J/J 0 ) is known.On the basis of Eqs. ( 10) and ( 11), the thickness Δr of the layer deposited on the pore wall is given by (Zeman, 1983;Bowen and Gan, 1991;Blanpain-Avet et al., 1999;Persson et al., 2003) 0.25 Similarly to the derivation of Eq. ( 78), v max for complete blocking law is written as Although infinite time is required until the filtration rate becomes zero for intermediate blocking and cake filtration laws, it is possible to calculate the filtrate volume per unit membrane area at an arbitrary ratio (J/J 0 ) of the flux decline.For instance, when the filtration rate J decreases to y percent of the initial filtration rate J 0 , influenced by the intermediate blocking law, the filtrate volume v y per unit membrane area can be obtained based on Eq. ( 27) and is represented as i ln 100 For the cake filtration law, on the basis of Eq. ( 30), v y can be written as Fouling index (FI) such as the silt density index (SDI) has been employed to predict and evaluate the fouling potential of the feed water in membrane filtration.The SDI is the most widely applied method for many decades (Nagel, 1987;Yiantsios and Karabelas, 2003;Alhadidi et al., 2011aAlhadidi et al., , 2011b;;Koo et al., 2012).According to the ASTM standard (2007), the filtration test is performed under the constant pressure condition of 207 kPa (30 psi) using a microfiltration membrane with the pore size of 0.45 μm.Both the time t 1 required to collect the first 500 ml and the time t 2 required to collect the second 500 ml after 15 min are measured to obtain SDI.Based on the time ratio (t 1 /t 2 ), SDI is calculated as SDI values less than 1.0 and 4-5 are preferable for hollow fiber and spiral wound reverse osmosis membranes, respectively.In spite of the widespread application of SDI, it has been long thought that SDI lacks a theoretical basis and that it makes no distinction between different filtration mechanisms.In response to this criticism, White (1996) first clarified the theoretical background based on the cake filtration model.In this case, SDI is written by More recently, the relation between SDI and blocking filtration laws has been examined in detail for four types of blocking filtration laws consisted of complete blocking, intermediate blocking, standard blocking, and cake filtration (Matsumoto et al., 2009;Alhadidi et al., 2011a;Wei et al., 2012).
The modified fouling index (MFI) proposed by Schippers et al. (1981) was also developed to measure the fouling potential of feed water in membrane filtration.While the feed water is filtered under the constant pressure condition through a 0.45 μm microfiltration membrane in dead-end mode, as is the case in SDI, the filtrate volume is recorded every 30 seconds over the filtration period in the MFI measurement.On the basis of cake filtration model, integrating Eq. ( 15) under the initial condition that v = 0 at t = 0, one obtains Consequently, MFI is the reciprocal of the Ruth coefficient K v in constant pressure filtration appeared in Eq. ( 15).Thus, MFI is calculated from the slope obtained by the linear approximation to the plot of the reciprocal average filtration rate (t/v) vs. the filtrate volume v per unit membrane area (Keskinler et al., 2004;Srisukphun et al., 2009).This means that MFI is defined on the assumption that the separation mechanism is controlled by cake filtration.Since MFI includes the solid concentration s in feed water (Park et al., 2006), it is more convenient than the average specific filtration resistance α av to evaluate the fouling potential of feed water in which the solid concentration is unknown.In order to evaluate the fouling potential of smaller colloidal particles or macromolecules, MFI was developed by using ultrafiltration membranes and nanofiltration membranes (referred to as MFI-UF (Boerlage et al., 2002(Boerlage et al., , 2003(Boerlage et al., , 2004) ) and MFI-NF (Khirani et al., 2006), respectively).Jin et al. (2015) proposed the cake fouling index (CFI) in which the true fouling cake layer resistance can be accurately evaluated by eliminating the effect of pore blocking.

Developments of consecutive combined model
Classical blocking filtration laws comprise of three pore blocking mechanisms and a cake formation mechanism.In the generality of cases, the membrane fouling proceeds in two steps: the initial membrane fouling caused by pore blockage and/or pore constriction followed by the long-term fouling arising from the filter cake gradually accumulating on the membrane surface (Kim et al., 1993;Tracey and Davis, 1994;Madaeni and Fane, 1996;Huang and Morrissey, 1998;Blanpain-Avet et al., 1999;Altman et al., 1999;Lim and Bai, 2003;Purkait et al., 2004Purkait et al., , 2005;;Wang and Tarabara, 2008;Juang et al., 2010;Mohd Amin et al., 2010;Ozdemir et al., 2012).The initial pore blocking frequently causes the irreversible fouling of membranes, resulting in the decrease in the efficiency of membrane cleaning.Once a sufficient fraction of the pores becomes clogged depending on the retentiveness of the membrane, an external cake begins to form on the fouled membrane.
In Fig. 8, the logarithmic plots of d 2 t/dv 2 as a function of dt/dv are shown as the characteristic filtration curves for filtration of pond water in which the turbidity and concentration of suspended solids are 19.4NTU and 18.0 mg/l, respectively (Iritani et al., 2007a).In the first stage of filtration (i.e., small dt/dv), the plots show a unique linear relationship, irrespective of the applied filtration pressure p.As filtration proceeds, the reciprocal filtration rate (dt/dv) increases and thus its derivative (d 2 t/dv 2 ) increases.Once the value of d 2 t/dv 2 reaches the limiting value, which depends on the filtration pressure, the second stage begins and the value of d 2 t/dv 2 remains constant, as given by Eq. ( 16).Therefore, the transition point from the initial pore blocking to the following cake filtration during a filtration run can be determined from the change of the slope of the straight line in the double logarithmic plot of d 2 t/dv 2 vs. dt/dv according to Eq. ( 17).It should be noted that the pore fouling is frequently represented by the pore constriction described by the standard blocking law followed by the pore plugging described by the complete or intermediate blocking law (Herrero et al., 1997;Griffiths et al., 2014).
The pore blockage and cake formation may be treated as two resistances in series.According to the resistancein-series model based on Darcy's law, the filtration rate J is related to the filtration resistances in series as (Iritani et al., 2007b)   where R is the overall filtration resistance, and R c is the filter cake resistance.It should be noted that R m indicates not the resistance of clean membrane but the resistance of fouled membrane.Bowen et al. (1995) and Iritani et al. (1995) found that the value of n in Eq. ( 17) gradually varied with the course of filtration in constant pressure dead-end microfiltration of bovine serum albumin (BSA) solution.Later, Hwang et al. (2007) reported a similar result in constant pressure dead-end microfiltration of particulate suspension.In their study, complete blocking (n = 2) initially occurred, then gradually changed to standard blocking (n = 1.5), and finally cake filtration (n = 0) started (Hwang and Chiu, 2008).Therefore, the values of n successively decreased with the progress of filtration.
It should be stressed that several researchers (Bowen et al., 1995;Iritani et al., 1995;Costa et al., 2006;Kim et al., 2007, Yukseler et al., 2007) reported the negative values of n in the later stages of filtration.Fig. 9 shows the logarithmic plots of d 2 t/dv 2 vs. dt/dv as the characteristic form of blocking filtration described by Eq. ( 17) for constant pressure microfiltration of BSA solution (Iritani et al., 1995).The curve shows a convex shape.The slope of the curve decreases with the increase in dt/dv due to the progress of filtration.Eventually, the slope of the curve has negative values after the slope reaches zero.
Strictly speaking, the blocking filtration laws can be applied only to unstirred dead-end filtration.It is impossible to apply the blocking filtration laws to crossflow filtration where the filter cake growth is restricted by external crossflow of the feed suspension.However, at the earlier stage of crossflow filtration where the solids deposited inside the pore structure, the blocking filtration laws are frequently employed to describe the progressive pore clogging.For instance, Murase and Ohn (1996) adopted the intermediate blocking law to describe the membrane fouling behavior in the initial of crossflow microfiltration of polymethyl methacrylate (PMMA) suspension.
The blocking filtration law has been frequently employed in the analysis of the flux decline in crossflow filtration (Jonsson et al., 1996;Prádanos et al., 1996;Keskinler et al., 2004).In this case, the flux decline behaviors should be analyzed by introducing the term of a steady-state flux controlled by crossflow (Field et al., 1995;de Bruijn et al., 2005).Field et al. (1995) modified the blocking filtration equation ( 18) by accounting for the back-transport effect arising from crossflow as where J lim is the critical filtration rate under steady-state condition.Equation ( 88) has been widely employed in the analysis of the flux decline behaviors in crossflow membrane filtration of a variety of colloids (Todisco et al., 1996;Blanpain and Lalande, 1997;de Barros et al., 2003;Rai et al., 2006;Cassano et al., 2007;Mondal and De, 2009;Vera et al., 2009;Ma et al., 2010;Chang et al., 2011;Daniel et al., 2011;Field and Wu, 2011;Huang et al., 2014).Electric field-assisted membrane filtration in which the particle deposition was restricted due to an external electric field was also investigated on the basis of blocking filtration laws as well as crossflow membrane filtration (Sarkar and De, 2012).

Developments of concurrent combined model
While membrane fouling generally proceeds in two steps consisted of pore blocking followed by cake formation, as mentioned above, pore blocking and cake formation may be frequently occurring simultaneously during the filtration process (Takahashi et al., 1991;Matsumoto et al., 1992;Katsoufidou et al., 2005;Fernández et al., 2011;Li et al., 2011;Nakamura et al., 2012).Bolton et al. (2006a) combined two blocking filtration laws occurring concurrently among four blocking filtration laws.On the basis of the Darcy's law, the volumetric flow rate Q through the membrane is related to the overall filtration resistance R and the effective filtration area A in the form Since the apparent filtration area remains constant during the course of filtration, the filtration rate J (= dv/dt) is directly proportional to the volumetric flow rate Q in Eq. ( 89) as where the subscript "0" indicates the value at the start of filtration.Substituting Eq. ( 89) into Eq.( 90), one obtains Both complete and intermediate blocking mechanisms contribute the decrease in the effective filtration area A.
In contrast, both standard blocking and cake filtration mechanisms increase the filtration resistance R. On the basis of Eqs. ( 21) and ( 27), the variations of the effective filtration area A in Eq. ( 91) for complete and intermediate blocking mechanisms are, respectively, represented as where b K and i K are the blocking constants for complete and intermediate blocking laws, respectively, and V is the filtrate volume.On the basis of Eq. ( 24) and ( 30), the variations of the filtration resistance R in Eq. ( 91) for standard blocking and cake filtration models are, respectively, given in where s K and c K are the blocking constants for standard blocking and cake filtration laws, respectively.Consequently, for instance, the governing equation for the com-bined process in which both intermediate blocking and cake formation occur simultaneously is derived using Eqs.( 91), (93), and (95) and is given by Just around the same time, Duclos-Orsello et al. ( 2006) also proposed a very similar combined model for describing the membrane fouling.Rezaei et al. (2011) employed the combined model developed by Bolton et al. (2006a) to analyze the fouling mechanism in crossflow microfiltration of whey.Affandy et al. (2013) well described fouling behaviors in sterile microfiltration of large plasmids DNA with the use of the standard-intermediate model.Bolton et al. (2006b) combined the adsorption model with the classical blocking filtration model by a method similar to that mentioned above.In the adsorption model, it is assumed that foulant adsorption occurs at the pore walls with zeroth-order kinetics, thereby reducing the pore size and thus increasing the filtration resistance.As a result, the increase in the resistance R with filtration time t is written as where K a is the adsorption blocking constant.Consequently, for instant, in a combined intermediate blocking-adsorption model, substituting Eqs. ( 93) and (97) into Eq.( 91), one gets Integration of Eq. ( 98) lead to the relation between V and t in the form: Giglia and Straeffer (2012) applied the combined intermediate blocking-adsorption model to the evaluation of filtration performance of microfiltration membranes operated in series.The combined cake-adsorption model can be described by adding the filtration resistances due to the adsorption and cake formation as In the initial stage of cake filtration, pore blocking of a membrane often occurs.As a result, the membrane resistance gradually increases with time and approaches to the saturated value, as indicated by Notebaert et al. (1975).On the assumption that the membrane resistance approaches a finite value due to the progress of filtration, the following equation was presented for describing the variation of the clogged membrane resistance R m during the course of filtration for cake filtration of liquefied coal (Tiller et al., 1981;Leu and Tiller, 1983).
where R m0 and R m∞ are the initial and infinite membrane resistances, respectively, η is the blocking rate constant, and w is the net solid mass in the filter cake per unit membrane area.Thereafter, Lee (1997) where λ is a constant.Interestingly, Eq. ( 102) reduces to Eq. ( 101) found experimentally by Tiller et al. (1981) when the membrane clogging is quite-serious, i.e., at the limit λ → 0.
By imposing the condition that some of pores of the membrane still remain open finally in the intermediate blocking law, Iritani et al. (2005) described the clogged membrane resistance R m as It should be noted that the variation of R m with w can be represented by only one fitting parameter η in Eq. ( 103) while two fitting parameters η and λ are required in the analysis of pore clogging using Eq. ( 102).The cake resistance R c is related to w as Thus, the increasing behaviors of filtration resistance can be described by adding the cake resistance R c represented by Eq. ( 104) to clogged membrane resistance R m represented by Eq. ( 103) on the basis of Eq. ( 87) describing the resistance-in-series model.In this case, all the particles retained by the membrane contribute as the cake resistance R c and a part of them also serve as the clogged membrane resistance R m by clogging the membrane pores in accordance with the intermediate blocking law considering the limiting value of the clogging resistance on the basis of the concurrent combined model, as schematically shown in Fig. 10.Fig. 11 shows the characteristic curves of blocking filtration laws, which is plotted in the form of d 2 t/dv 2 vs. dt/dv, for different feed concentrations in constant pressure microfiltration of monodisperse PSL with a particle diameter d p of 0.522 μm filtered under the applied pressure p of 196 kPa using the track-etched polycarbonate membrane with a nominal pore size d m of 0.2 μm (Iritani et al., 2015).Each plot results in the distinct negative slope in the initial period of filtration since the filtration behaviors are influenced both by the pore blocking of membrane and by cake formation, as reported by several researchers (Bowen et al., 1995;Iritani et al., 1995;Ho and Zydney, 2000;Hwang et al., 2007;Yukseler et al., 2007).However, as filtration proceeds, cake filtration has a dominant influence on the filtration behaviors and thus the slope of the plot becomes equal to zero.Substituting Eqs. ( 103) and (104) into Eq.( 87) and using the relation that w = ρsv on the assumption that suspension is very dilute, one obtains where F is a function defined by  Subsequently, differentiating Eq. ( 105) with respect to v under constant pressure condition, one gets Therefore, the curve of d 2 t/dv 2 vs. dt/dv can be evaluated from Eq. ( 107) with the aid of Eq. ( 105).The solid curves in Fig. 11 are the calculations and it is of significance to note that the calculations roughly reflect the trends of experimental data, indicating the negative slope in the initial period followed by the straight line with the slope of zero.Hwang et al. (2006) evaluated the protein capture into the interstices of filter cake in crossflow microfiltration of particle/protein binary mixtures on the basis of the deepbed filtration mechanism.The apparent protein rejection R obs is evaluated from where C p and C b are protein concentrations in the filtrate and in the bulk feed suspension, respectively, γ is a screening parameter which represents the protein fraction rejected by the filter cake per unit cake thickness, and L c is the thickness of the filter cake.Ho and Zydney (2000) presented a unique model describing pore blocking and cake formation occurring simultaneously in microfiltration processes.As schematically shown in Fig. 12, the filter cake only forms over the regions of the membrane which have already been blocked by the initial deposit in the membrane pores.This means that the pore blockage did not lead to complete loss of flow through the pore.As a result, according to Darcy's law, the flux J blocked through the already blocked pores can be written as The cake resistance R c increases with time during the course of filtration as where f ′ is the fraction of solutes which contribute to the growth of filter cake.Integrating Eq. ( 110) with the aid of Eq. ( 109), one obtains where R c0 is the flow resistance of the first cake layer.The filtration rate J can be given as the sum of the flow rate through open and blocked pores.Thus, with the use of Eq. ( 111), the variation of J with t can be approximately written as where α is the pore blocking constant in the complete blocking law.The first term in Eq. ( 112) represents a simple exponential decline in the flux through the open pores controlled by the classical complete blocking law.As filtration proceeds, the deposit continued to grow due to some fluid flow through pores partially blocked by the complete blocking mechanism, as illustrated by Fig. 12, and thus the filtration rate becomes governed by the classical cake filtration law represented by the second term in Eq. ( 112).The model has been extensively used to analyze the fouling behaviors in membrane filtration of a number of colloids under various operating conditions (Yuan et al., 2002;Ho and Zydney, 2002;Palacio et al., 2002Palacio et al., , 2003;;Taniguchi et al., 2003;Ye et al., 2005;Chandler and Zydney, 2006;Cogan and Chellam, 2009;Byun et al., 2011).

Conclusions and prospective view
The present article overviewed the blocking filtration laws comprised of the complete blocking, standard blocking, intermediate blocking, and cake filtration, which could describe the increase in the filtration resistance during the course of filtration in membrane filtration of colloids.The equations derived based on the blocking filtration laws were reported to describe the filtrate flow of Newtonian and non-Newtonian fluids through membranes for both constant pressure and constant rate filtration processes.The blocking filtration laws are quite useful due to the simplicity in use of the model to identify the prevailing fouling mechanism from the experimental data of the flux decline in constant pressure filtration or pressure rise in constant rate filtration.In order to evaluate more com-Fig.12 Schematic view for illustrating the mechanism of membrane fouling comprised of both pore blocking and cake formation occurring simultaneously in the model presented by Ho and Zydney (2000).
plicate fouling behaviors in membrane filtration, several combined models have been developed based on the blocking filtration laws and well described the fouling phenomena in which more than one filtration mechanism occurred successively or simultaneously.While blocking filtration laws and their combinations largely contributed to the optimal choice of the membrane and membrane-cleaning strategy in industrial use, it is essential to develop more sophisticated models which can describe more accurately the complicated behaviors of membrane fouling actually encountered in industrial membrane filtration.In particular, there is a pressing need for developing the models which are applicable not only the simple model colloids but also to the actual colloids containing a wide variety of ingredients, as frequently encountered in water treatment.
In any case, the elucidation of mechanism predominating the membrane fouling in membrane filtration is ever lasting problems crying out for solutions.We believe that this article provides a valuable insight into the further developments of models which can reasonably describe the fouling behaviors in membrane filtration.K br blocking constant in Eq. ( 35) for complete blocking law in constant rate filtation (kg -1 s 2 ) K c blocking constant in Eq. ( 16) for cake filtration law (s/ m 2 ) c K blocking constant in Eq. ( 95) for cake filtration law (s/ m 4 ) K f fiber coating constant in Eq. ( 77)

Fig. 1
Fig. 1 Schematic view of four fouling patterns in blocking filtration laws: (a) complete blocking law, (b) standard blocking law, (c) intermediate blocking law, and (d) cake filtration law.

Fig. 2
Fig. 2 Graphical identification of blocking filtration laws for constant pressure filtration: (a) complete blocking law, (b) standard blocking law, (c) intermediate blocking law, and (d) cake filtration law.

Fig. 3
Fig. 3 Graphical identification of blocking filtration laws for constant rate filtration: (a) complete blocking law, (b) standard blocking law, (c) intermediate blocking law, and (d) cake filtration law.

Fig. 4
Fig. 4 Flux decline behaviors in clarification filtration of dilute suspensions prepared by suspending diatomaceous earth in aqueous sodium polyacrylate exhibiting behavior of power-law non-Newtonian fluid.

Fig. 5
Fig. 5 Schematic view for illustrating the mechanism of membrane fouling in the model presented by Iritani et al. (2007a).

Fig. 6
Fig. 6 Characteristic filtration curves for blocking filtration of dilute suspensions of PSL under constant pressure condition using diatomaceous ceramic membranes which are semi-permeable to PSL: (a) logarithmic plots of d 2 t/dv 2 vs. dt/dv and (b) logarithmic plots of d(dt/dv)/dσ vs. dt/dv.

Fig. 7
Fig. 7 Schematic view for illustrating the mechanism of membrane fouling in the fiber-coating model presented by Bolton et al. (2005): (a) clean fibers and (b) fouled fibers.

Fig. 8
Fig. 8 Characteristic filtration curves for constant pressure filtration of pond water.

Fig.
Fig. Characteristic filtration curve for constant pressure microfiltration of BSA solution.

Fig. 10
Fig. 10 Schematic view for illustrating the mechanism of membrane fouling comprised of both pore blocking and cake formation occurring simultaneously in the model presented by Iritani et al. (2005).

Fig. 11
Fig. 11 Effect of solid mass fraction in suspension on characteristic filtration curves in constant pressure microfiltration.
of suspended solids (m) F function defined by Eq. (106) f ′ fraction of solutes which contribute to growth of filter cake J filtration rate (m/s) J 0 initial filtration rate (m/s) J blocked flux through already blocked pores (m/s) J lim critical filtration rate under steady-state condition in crossflow filtration (m/s) K fluid consistency index for power-law non-Newtonian fluids (kg m -1 s N-2 ) Eq.(17) (m n-2 s 1-n ) k′ resistant coefficient in Eq. (36) for constant rate filtration (kg 1-n′ m n′-2 s 2n′-2 ) of filter cake (m) m ratio of mass of wet to mass of dry cake MFI modified fouling index defined by Eq. (86) (s/m 2 ) N fluid behavior index for power-law non-Newtonian fluids N′ number of open pores per unit effective membrane area at filtrate volume v per unit effective membrane area (per unit membrane area required to obtain cake with flow resistance equivalent to that of membrane (m) v max maximum filtrate volume per unit membrane area (m) v y filtrate volume per unit membrane area obtained until filtration rate decreases to y percent of initial filtration rate (m) w net solid mass in filter cake per unit membrane area (kg/ m 2 ) x number of particles blocking pores per unit filtrate volume (m -3 ) y percentage of initial value of filtration rate α pore blocking constant in Eq. (112) for complete blocking law (m 2 /kg) α 0 average specific cake resistance at null stress in Eq. specific cake resistance for power-law non-Newtonian flow (m 2-N /kg) Δr thickness of layer deposited on pore wall (m) i.e., mass of particles deposited on pore wall per unit membrane area (kg/m 2 ) τ shear stress (Pa)

Table 2
Blocking filtration equations for constant rate filtration

Table 3
Blocking filtration equations for power-law non-Newtonian fluids-solids mixtures: (a) constant pressure filtration and (b) constant rate filtration they are omitted from the table.