2016 Volume 33 Pages 179-202
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.
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 and Bredée, 1935, 1936; Grace, 1956; Shirato et al., 1979; Hermia, 1982; Iritani et al, 1992, 2009, 2013), cake formation (Reihanian et al., 1983; Chudacek and Fane, 1984; Iritani et al., 1991a, 2014a, 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 (1935, 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 and Bredée (1935, 1936) and later systematized by Grace (1956), Shirato et al. (1979), and Hermia (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 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, (N′0 − xv) where N′0 is the total number of open pores per unit effective membrane area at start of filtration, x is the number of particles blocking pores per unit filtrate volume. Since the filtration rate J is directly proportional to the number of open pores, it can be written as
(1) |
(2) |
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.
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
(3) |
(4) |
(5) |
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)
(6) |
(7) |
(8) |
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
(9) |
(10) |
(11) |
(12) |
(13) |
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, 1946)
(14) |
(15) |
(16) |
Interestingly enough, Eqs. (2), (5), (13), and (16) derived for four different filtration mechanisms reduce to the following common differential equation (Hermans and Bredée, 1935, 1936; Grace, 1956).
(17) |
(18) |
A double logarithmic plot of d2t/dv2 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 d2t/dv2 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., 1995, 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 and Bredée, 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 d2t/dv2 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.
Function | (a) Complete blocking | (b) Standard blocking | (c) Intermediate blocking | (d) Cake filtration | ||||||||
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n = 2.0 | n = 1.5 | n = 1.0 | n = 0 | ||||||||
v =f(t) |
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| ||||||||
J =f(t) |
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| ||||||||
J =f(v) |
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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.
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
(31) |
(32) |
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)
(33) |
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
(34) |
(35) |
(36) |
Function | (a) Complete blocking | (b) Standard blocking | (c) Intermediate blocking | (d) Cake filtration | ||||||||
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n′ = 2.0 | n′ = 1.5 | n′ = 1.0 | n′ = 0 | ||||||||
p =f(v) |
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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.
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
(41) |
(42) |
(43) |
(44) |
Since v is directly proportional to the time t in constant rate filtration, Eq. (36) is rewritten as
(45) |
(46) |
(47) |
(48) |
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
(49) |
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 γ̇ written as
(50) |
(51) |
While the number of open pores varies with the progress 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
(52) |
(53) |
Substituting Eqs. (52) and (53) into Eq. (7) under the constant pressure condition (p = p0 = const.), one gets
(54) |
Differentiating Eq. (54), one obtains (Shirato et al., 1979)
(55) |
(56) |
(57) |
Consequently, four blocking filtration laws for power-law non-Newtonian filtration under the constant pressure condition can be represented by a common differential equation with two constants kN and nN, as described by (Shirato et al., 1980; Hermia, 1982; Rushton, 1986)
(58) |
(59) |
(a) | ||||||
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Function | Standard blocking | Cake filtration | ||||
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nN = 1−N | ||||
v =f(t) |
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J =f(t) |
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J =f(v) |
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(b) | ||||||||||||
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Function | Complete blocking | Standard blocking | Intermediate blocking | Cake filtration | ||||||||
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n′N = 1 | n′N = 0 | ||||||||
p =f(v) |
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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 J2N/(3N+1) vs. v yield linear relationships in accordance with Eq. (54) for the standard blocking law since the diameter dp of the suspended solids is much smaller than the pore size dm. 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.
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.
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
(70) |
(71) |
(72) |
(73) |
Schematic view for illustrating the mechanism of membrane fouling in the model presented by Iritani et al. (2007a).
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
(74) |
(75) |
(76) |
Fig. 6 compares the logarithmic plots of d2t/dv2 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 ε0 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).
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 d2t/dv2 vs. dt/dv and (b) logarithmic plots of d(dt/dv)/dσ vs. dt/dv.
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
(77) |
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.
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 vmax 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), vmax can be given as (Badmington et al., 1995; van Reis and Zydney, 2007)
(78) |
(79) |
(80) |
(81) |
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/J0) of the flux decline. For instance, when the filtration rate J decreases to y percent of the initial filtration rate J0, influenced by the intermediate blocking law, the filtrate volume vy per unit membrane area can be obtained based on Eq. (27) and is represented as
(82) |
(83) |
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., 2011a, 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 t1 required to collect the first 500 ml and the time t2 required to collect the second 500 ml after 15 min are measured to obtain SDI. Based on the time ratio (t1/t2), SDI is calculated as
(84) |
(85) |
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
(86) |
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., 2004, 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 d2t/dv2 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 19.4 NTU 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 (d2t/dv2) increases. Once the value of d2t/dv2 reaches the limiting value, which depends on the filtration pressure, the second stage begins and the value of d2t/dv2 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 d2t/dv2 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).
Characteristic filtration curves for constant pressure filtration of pond water.
The pore blockage and cake formation may be treated as two resistances in series. According to the resistance-in-series model based on Darcy’s law, the filtration rate J is related to the filtration resistances in series as (Iritani et al., 2007b)
(87) |
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 d2t/dv2 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.
Characteristic filtration curve for constant pressure microfiltration of BSA solution.
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 stage 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
(88) |
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
(89) |
(90) |
(91) |
(92) |
(93) |
(94) |
(95) |
(96) |
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
(97) |
(98) |
(99) |
(100) |
(101) |
(102) |
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 Rm as
(103) |
(104) |
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 shows the characteristic curves of blocking filtration laws, which is plotted in the form of d2t/dv2 vs. dt/dv, for different feed concentrations in constant pressure microfiltration of monodisperse PSL with a particle diameter dp of 0.522 μm filtered under the applied pressure p of 196 kPa using the track-etched polycarbonate membrane with a nominal pore size dm 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
(105) |
(106) |
(107) |
Effect of solid mass fraction in suspension on characteristic filtration curves in constant pressure microfiltration.
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 deep-bed filtration mechanism. The apparent protein rejection Robs is evaluated from
(108) |
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 Jblocked through the already blocked pores can be written as
(109) |
(110) |
(111) |
(112) |
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).
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 complicate 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 the 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.
This work has been partially supported by Grant-in-Aid for Scientific Research from The Ministry of Education, Culture, Sports, Science and Technology, Japan. The authors wish to acknowledge with sincere gratitude the financial support leading to the publication of this article.
A
effective filtration area (m2)
A0initial effective filtration area (m2)
aconstant in Eq. (33)
Cbprotein concentration in bulk feed suspension (kg/m3)
Cpprotein concentration in filtrate (kg/m3)
cvolume of particles trapped per unit filtrate volume v per unit membrane area (m)
Drepresentative diameter of pores on flow cross-sectional area basis (m)
Dsrepresentative diameter of pores on wetted perimeter basis (m)
dmpore size (m)
dpdiameter of suspended solids (m)
Ffunction defined by Eq. (106)
f′fraction of solutes which contribute to growth of filter cake
Jfiltration rate (m/s)
J0initial filtration rate (m/s)
Jblockedflux through already blocked pores (m/s)
Jlimcritical filtration rate under steady-state condition in crossflow filtration (m/s)
Kfluid consistency index for power-law non-Newtonian fluids (kg m−1 sN−2)
Kaadsorption blocking constant in Eq. (97) (s−1)
Kbblocking constant in Eq. (2) for complete blocking law (s−1)
K′bblocking constant in Eq. (92) for complete blocking law (m−2 s−1)
Kbrblocking constant in Eq. (35) for complete blocking law in constant rate filtation (kg−1 s2)
Kcblocking constant in Eq. (16) for cake filtration law (s/m2)
K′cblocking constant in Eq. (95) for cake filtration law (s/m4)
Kffiber coating constant in Eq. (77)
Kiblocking constant in Eq. (3) for intermediate blocking law (m−1)
K′iblocking constant in Eq. (93) for intermediate blocking law (m−3)
Kmconstant in Eq. (74) (m2/kg)
Kpconstant in Eq. (71) (m−1)
Ksblocking constant in Eq. (7) for standard blocking law (m−1)
K′sblocking constant in Eq. (94) for standard blocking law (m−3)
KvRuth coefficient in constant pressure cake filtration (m2/s)
kresistant coefficient in Eq.(17) (mn−2 s1−n)
k′resistant coefficient in Eq. (36) for constant rate filtration (kg1−n′ mn′− 2 s2n′− 2)
k0Kozeny constant
k1constant in Eq. (44) (kg1−n1 mn1−2s2n1−2)
k′1constant in Eq. (46) (kgn′1−2 m2−2n′1s5−5n′1)
k2constant in Eq. (45) (kg1−n′ mn′− 1 s2n′− 3)
k′2constant in Eq. (47) (kg1−n′2 m2n′2−3s2n′2−2)
kcproportional constant in Eq. (1) (kg− 1 m4 s)
kjconstant in Eq. (31) (s− 1)
kNconstant in Eq. (58) (mnN−2s1−nN)
k′Nconstant in Eq. (59) (kg1−n′N mn′N−2s2n′N−2)
Lmembrane thickness (m)
Lcthickness of filter cake (m)
mratio of mass of wet to mass of dry cake
MFImodified fouling index defined by Eq. (86) (s/m2)
Nfluid 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 (m− 2)
N′0total number of open pores per unit effective membrane area at start of filtration (m− 2)
nblocking index in Eq. (17)
n′blocking index in Eq. (36) for constant rate filtration
n1compressibility coefficient in Eq. (42)
n′1constant in Eq. (46)
n′2constant in Eq. (47)
njconstant in Eq. (31)
nNconstant in Eq. (58)
n′Nconstant in Eq. (59)
papplied filtration pressure (Pa)
p0initial applied filtration pressure (Pa)
Qvolumetric flow rate (m3/s)
Q0initial volumetric flow rate (m3/s)
Roverall filtration resistance (m− 1)
R0initial overall filtration resistance (m− 1)
Rcfilter cake resistance (m− 1)
Rc0flow resistance of the first cake layer (m− 1)
Rmclogged membrane resistance (m− 1)
Rm0initial membrane resistance (m− 1)
Rm∞infinite membrane resistance (m− 1)
rpore radius (m)
r0initial pore radius (m)
Sspecific surface area of membrane (m− 1)
S0initial specific surface area of clean membrane (m− 1)
Scconstant in Eq. (33)
smass fraction of solids in colloids
SDIsilt density index defined by Eq. (84)
tfiltration time (s)
t1time required to collect the first 500 ml of filtrate volume (s)
t2time required to collect the second 500 ml of filtrate volume after 15 min (s)
uaverage flow rate (m/s)
Vfiltrate volume (m3)
vfiltrate volume per unit effective membrane area (m)
vmfictitious filtrate volume per unit membrane area required to obtain cake with flow resistance equivalent to that of membrane (m)
vmaxmaximum filtrate volume per unit membrane area (m)
vyfiltrate volume per unit membrane area obtained until filtration rate decreases to y percent of initial filtration rate (m)
wnet solid mass in filter cake per unit membrane area (kg/m2)
xnumber of particles blocking pores per unit filtrate volume (m− 3)
ypercentage of initial value of filtration rate
αpore blocking constant in Eq. (112) for complete blocking law (m2/kg)
α0average specific cake resistance at null stress in Eq. (48) (m/kg)
α1constant in Eq. (42) (kg−1−n1m1+n1s2n1)
α2constant in Eq. (48) (kg− 2 m2 s2)
αavaverage specific cake resistance (m/kg)
βconstant which depends on mode of morphology of deposit assemblages
γscreening parameter in Eq. (108) (m− 1)
γ˙shear rate (s− 1)
γavaverage specific cake resistance for power-law non-Newtonian flow (m2−N/kg)
Δrthickness of layer deposited on pore wall (m)
εporosity of membrane
ε0initial porosity of clean membrane
εppacking porosity of particle layer formed on pore wall
ηblocking rate constant (m2/kg)
λconstant in Eq. (102)
μviscosity of filtrate (Pa s)
ρdensity of filtrate (kg/m3)
σspecific deposit, i.e., mass of particles deposited on pore wall per unit membrane area (kg/m2)
τshear stress (Pa)
Eiji Iritani
Eiji Iritani is currently Professor of Department of Chemical Engineering at Nagoya University. He received his Docter degree of Engineering from Nagoya University in 1981. His research interests are in the area of cake filtration, membrane filtration, deliquoring due to expression, sedimentation, centrifugation, and water treatments based on solid-liquid separation technology.
Nobuyuki Katagiri
Nobuyuki Katagiri is currently Assistant Professor of Department of Chemical Engineering at Nagoya University. He received his Doctor degree of Agriculture from Gifu University in 1998. His research interests are in the area of filtration and expression of biologically-derived materials, and waste water treatments.