Cake formation and particle rejection in microfiltration of binary mixtures of particles with two different sizes

Separation and Purification Technology 123 (2014) 214–220

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Cake formation and particle rejection in microfiltration of binary mixtures of particles with two different sizes Eiji Iritani ⇑, Nobuyuki Katagiri, Yoshihito Ishikawa, Da-Qi Cao Department of Chemical Engineering, Nagoya University, Furo-cho, Chikusa-ku, Nagoya 464-8603, Japan

a r t i c l e

i n f o

Article history: Received 17 October 2013 Received in revised form 19 December 2013 Accepted 20 December 2013 Available online 4 January 2014 Keywords: Binary mixtures Cake filtration Filtration rate Particle rejection Pore blocking

a b s t r a c t The filtration behaviors were examined in microfiltration of binary mixtures of particles with two different sizes. Dilute suspension of monodisperse polystyrene latexes with particle diameters of 0.522 and 0.091 lm was filtered using the microfiltration membranes with the nominal pore size of 0.3 lm, making them essentially impermeable to larger particles but permeable to smaller particles. The filter cake comprised of larger particles alone initially formed because smaller particles permeate through the membrane. However, the flux decline became gradually marked since smaller particles were trapped into the pores of the filter cake of larger particles. Eventually, smaller particles were fully rejected, and thereafter the binary cake of both larger and smaller particles grew. This filtration behavior was reflected by both data of flux decline and particle rejection. The logistic equation was employed to describe the variation of the rejection of smaller particles with the progress of filtration. The flux decline behaviors were well described using the logistic equation on the basis of the resistance-in-series model that the total cake resistance was represented by adding the increased cake resistance caused by the capture of smaller particles to the cake resistance of larger particles alone. Ó 2014 Elsevier B.V. All rights reserved.

1. Introduction Recently, clarifying membrane filtration of dilute suspension containing fine particles has been developed specifically to meet the requirement of such water purification processes as drinking water treatment, effluent polishing, and ultrapure water production [1–3]. One of the critical issues in the developments of efficient processes of clarifying membrane filtration is a significant flux decline, resulting from the membrane fouling such as pore blocking and cake formation. A number of mathematical expressions have been developed over the past few decades to describe the membrane fouling due to pore blocking and cake formation during membrane filtration [4,5]. The classical blocking filtration law [6,7] has been exclusively used in the analysis of clogging behaviors of membranes and cake formation during membrane filtration. The importance of cake formation in filtration is recognized by numerous authors, and the compressible cake filtration model [8] has been extensively employed in the analysis of cake formation during particulate membrane filtration. When cake formation and pore blocking may be occurring simultaneously, the fouling mechanism becomes more complex.

⇑ Corresponding author. Tel.: +81 52 789 3374; fax: +81 52 789 4531. E-mail address: [email protected] (E. Iritani). 1383-5866/$ - see front matter Ó 2014 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.seppur.2013.12.033

In this case, the total resistance increases with the progress of filtration due to a combination of cake formation and pore blocking according to the resistance-in-series model [9]. Matsumoto et al. [10] developed the combined model describing pore blocking during cake growth. For instance, when the particles with different sizes and types are present in suspension filtered, the situation is complicated [11–18]. The existence of the filter cake comprised of larger particles rejected by the membrane plays a crucial role in the capture of smaller particles in filtration of binary particulate mixtures which differ in size. The cake layer frequently removes a significant portion of fine particles before they reach the membrane surface [19]. Such behavior is highly important also from the viewpoint of secondary dynamic membranes in membrane filtration [20–22]. Hwang et al. [23] analyzed a rise in the apparent protein rejection in crossflow microfiltration of particle/protein mixtures on the basis of the deep-bed filtration model. The focus of the present study is to delineate the complicated behaviors of cake formation and pore blocking occurring during dead-end microfiltration under constant pressure conditions using dilute suspension of binary mixtures comprised of the model particles with two different sizes. The flux decline behaviors with the progress of filtration are analyzed on the basis of the resistance-inseries model by using the model equation describing the variation of the rejection of smaller particles.

E. Iritani et al. / Separation and Purification Technology 123 (2014) 214–220

215

Nomenclature c dm dp k m n p R⁄ Rm Rt r s u1

solid mass in filter cake divided by cumulative filtrate volume in case where all particles are rejected (kg/m3) nominal pore size of membrane (m) particle diameter (m) resistance coefficient in Eq. (4) (mn2 s1n) ratio of mass of wet to mass of dry cake (–) blocking index in Eq. (4) (–) applied filtration pressure (Pa) apparent rejection of particles (–) membrane resistance including blocking resistance of membrane (m1) total filtration resistance to filtrate flow (m1) empirical constant in Eq. (6) (m1) mass fraction of particles in suspension (–) filtration rate (m/s)

v v0,k

cumulative filtrate volume per unit membrane area (m3/m2) cumulative filtrate volume per unit membrane area when R2 is 0.5 (m3/m2)

Greek letters aav average specific cake resistance (m/kg) h filtration time (s) l viscosity of filtrate (Pa s) q density of filtrate (kg/m3) Subscripts 1 larger particles 2 smaller particles

2. Materials and methods 2.1. Materials The particles used in the experiments were monodisperse polystyrene latexes (PSL) with particle diameters of 0.522 and 0.091 lm and true density of 1.05 g/cm3 (Dow Chemical Japan Ltd., Japan). It should be stressed that the effect of particle sedimentation on filtration behaviors examined in this work is negligible small, judging from the diameters and density of particles [24]. The thick suspension provided from the manufacturer was diluted with ultrapure, deionized water produced by purifying tap water through ultrapure water systems equipped with both Elix-UV20 and Milli-Q Advantage for laboratory use (Millipore Corp., USA). The mixed particulate suspension was prepared by mixing each single particulate suspension. The weight fraction of particles in the mixed suspension was maintained at 1.0  104 for larger particles and ranged from 1.0  105 to 4.0  104 for smaller particles, and was set to very low values in order to simulate clarifying membrane filtration, which is encountered in advanced treatment in drinking water treatment, effluent polishing, and ultrapure water production.

Fig. 1. Schematic layout of experimental setup for dead-end microfiltration.

2.2. Experimental apparatus and technique Fig. 1 illustrates a schematic layout of the experimental setup used to carry out dead-end microfiltration experiments, in conjunction with the appearance of filtration. An unstirred batch filtration cell with an effective membrane area of 3.14 cm2 was utilized in this research in order to promote fouling in filtration operation. Microfiltration experiments were performed using suspension of binary mixtures under conditions of a constant pressure varying over the range 49–294 kPa by adjusting the applied filtration pressure automatically by a computer-driven electronic pressure regulator by applying compressed nitrogen gas, as shown in the figure. The membrane employed is mixed cellulose ester microfiltration membranes (Advantec Toyo Corp., Japan) with a nominal pore size of 0.3 lm, and the SEM image of the membrane is shown in Fig. 2. As shown later, the membranes are essentially impermeable to larger particles but permeable to smaller particles since two types of constituent particles larger and smaller than the pore size of membranes were chosen as the binary mixture. The filtrate was collected in a reservoir placed on an electronic balance connected to a personal computer to collect and record mass vs. time data every 5 s. The weights were converted to volumes using density correlations. The values of the filtration rate

Fig. 2. SEM image of membrane used.

at various volumes were computed by numerical differentiation of the volume vs. time data. The reservoir was replaced over measured time intervals in order to measure the temporal variation of

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particle rejection. Both absorbance values at the wavelengths of 286 and 340 nm in the filtrate were measured. Each concentration of both larger and smaller particles in the filtrate was determined from the comparison between both absorbance values through use of the fact that the relation between concentration and absorbance at a given wavelength is different between larger and smaller particles. For comparison, constant pressure microfiltration experiments were conducted using a single-component suspension comprised of each of both larger and smaller particles, and the variations with time of the filtration rate and each rejection of both larger and smaller particles were measured. 3. Results and discussion 3.1. Flux decline behaviors of mixed suspension Typical data of constant pressure filtration experiments conducted using binary particulate mixtures are plotted in Fig. 3 as the form of the reciprocal filtration rate (dh/dv) against the cumulative filtrate volume v per unit membrane area, where h is the filtration time. In the figure, p is the applied filtration pressure, dm is the nominal pore size of the membrane, dp is the particle diameter, s is the mass fraction of solids in suspension, and the subscripts 1 and 2 stand for the first (larger) and the second (smaller) particles components, respectively. The results for each single component are also included in the figure. It should be noted that the term (dh/dv) is a measure of the total filtration resistance Rt to the filtrate flow and is expressed by the Darcy’s law in the form:

dh lRt ¼ dv p

ð1Þ

where l is the viscosity of the filtrate. In the single-component system of particles of 0.522 lm larger than the pore size (0.3 lm) of the membrane, the plot shows a linear relationship throughout the course of filtration because the filter cake readily forms on the membrane surface from the beginning of filtration. Thus, the variation characteristic of the filtration resistance is described by the Ruth filtration rate equation represented by [25,26]

1 dh l ¼ ¼ ðaav cv þ Rm Þ u1 dv p

ð2Þ

300 PSL p = 98 kPa dm = 0.3 μm dp1 = 0.522 μ m s1 = 1.0x10-4 dp2 = 0.091 μ m s2 = 1.0x10-4

d θ /dv (s/cm)

200

Single (dp1) Single (dp2) Mix Eq. (9)

100

0 0

50

100

150

v (cm) Fig. 3. Typical flux decline behaviors for binary particulate mixtures and each single component.

where u1 is the filtration rate, aav is the average specific resistance of the filter cake, and Rm is the membrane resistance including blocking resistance of the membrane. The term c is given by the solid mass in the filter cake divided by the cumulative filtrate volume in the case where all the particles are rejected and is written as [27]

c¼

qs  qs 1  ms

ð3Þ

where m represents the average ratio of the mass of wet to the mass of dry cake. As shown in the above equation, c is approximated by qs for dilute suspension [27]. In contrast, in the single-component system of particles of 0.091 lm much smaller than the pore size (0.3 lm) of the membrane, the increase in the filtration resistance resulting in the flux decline is invisible up to v of about 110 cm since most particles permeate through the membrane. After a while, the pore of the membrane gradually becomes clogged with particles and the filter cake forms on the surface of the clogged membrane. Finally, filtration proceeds in accord with the mechanism of cake filtration described by Eq. (2). In the case of binary particulate mixtures, the flux decline behavior is initially very similar to that of the single-component system of larger particles and dh/dv increases linearly with v. However, dh/dv increases more rapidly with v with the progress of filtration because smaller particles become trapped into the pores of the filter cake consisting of larger particles. Therefore, the filter cake contains not only larger particles but also smaller particles. Eventually, the plots show a straight line with much steeper slope than that in the initial period of filtration and thus the specific resistance of the binary cake are calculated from the slope of the straight line. The characteristic filtration form for the blocking filtration laws including cake filtration law can be expressed as [6,7,28,29] 2

d h dv

2

¼k



dh dv

n ð4Þ

or



 3n dv ¼ k dh dh2

d

2

v

ð5Þ

where k is the resistance coefficient, and n is the blocking index that characterizes the mode of the fouling model involved, with n = 2 for complete blocking, n = 3/2 for standard blocking, n = 1 for intermediate blocking, and n = 0 for cake filtration. Therefore, a double logarithmic plot of d2h/dv2 as a function of dh/dv is frequently utilized as the characteristic filtration curve based on Eq. (4), in order to provide a clear indication of the dominant fouling mechanism [30–37], and the flux decline data indicated in Fig. 3 are illustrated in Fig. 4 in the form of a double logarithmic plot of d2h/dv2 vs. dh/dv. The value of d2h/dv2 remains almost constant throughout the course of filtration in accordance with the cake filtration law in filtration of the single-component suspension of larger particles. The plots for single-component suspension of smaller particles are approximately divided into a two-step process: the initial fouling due to pore blocking represented as the curve in which d2h/dv2 increases with the increase in dh/dv followed by the long-term fouling arising from cake accumulation on the membrane surface represented as horizontal line [38,39]. In the case of binary particulate mixtures, initially the plots lie on the horizontal line and are consistent with those for singlecomponent suspension of larger particles because the filter cake comprised of larger particles forms on the membrane surface. Subsequently, d2h/dv2 increases with increasing dh/dv since smaller particles becomes trapped in the filter cake formed. The value of

217

10

2

10

1

10

0

10

-1

10

It is assumed that the capture of smaller particles is accelerated by the increase in the number of smaller particles trapped in the pores of the filter cake: the rate of increase in the rejection,  dR2 =dv , is directly proportional to R2 , where the subscript ‘2’ indicates the value for smaller particles. In contrast, as the upper limit  of the rejection is set to unity, it is assumed that dR2 =dv is directly   proportional to ð1  R2 Þ. Hence, dR2 =dv is related to R2 by 

-2

10

-3

10

-4

dR2 ¼ rR2 ð1  R2 Þ dv

PSL p = 98 kPa dm = 0.3 μm dp1 = 0.522 μm s1 = 1. 1 0x10-4 dp2 = 0.091 μm s2 = 1.0x10-4

10

0

10

1

10

2

10

3

dθ/dv (s/cm) Fig. 4. Typical plots based on characteristic filtration form for blocking filtration law.

d2h/dv2 is eventually reached the limiting value and thereafter it remains constant. At this stage of filtration, it is inferred that smaller particles are fully rejected and that the mixed cake of both larger and smaller particles grows.

3.2. Particle rejection behaviors of mixed suspension In filtration of mixed suspension, the increase in the filtration resistance strikingly depends on the solid mass of particles deposited, particularly smaller particles. Each apparent rejection R⁄ of both larger and smaller particles in filtration of binary particulate mixtures illustrated in Fig. 3 is plotted in Fig. 5 against the filtrate volume v per unit membrane area. The rejection of larger particles is nearly equal to unity because the pore size of the membrane is smaller than the diameter of larger particles. However, the rejection of smaller particles is not quite as simple as that of larger particles. Smaller particles permeate through the membrane in the incipient period of filtration. However, once the value of v is over about 15 cm, the rejection R⁄ significantly increases and approaches the value of unity. This means that smaller particles become trapped in the pores of the filter cake consisted of larger particles. Initially, the rate of increase in the rejection R⁄ increases with v and then it decreases with v as it approaches the value of unity.

R2 ¼

1 1 þ expðrðv  v 0 ÞÞ

(a) 1.0

r = 0.50 cm-11 r = 0.25 cm-1 r = 0.15 0 15 cm-1

0 .6

0.4

Eq. (7) v0 = 35.0 cm

0.2

0

0

0

10

20

30

40

50

60

30

40

50

60

70

(b)

v0 = 20.0 cm 27 5 cm v0 = 27.5 v0 = 35.0 35 0 cm

0 .6

0.4

Mix (dp11) Mix ((dp2) Eq. (7)

0.2

20

Eq. (7) Eq r = 0.25 0 25 cm-1

0.8

0.4

10

v (cm)

R*2 (−))

R* (−))

r = 0.75 0 75 cm-1

0.8

0.8

0

r = 1.00 cm-11

1.0

PSL p = 98 kP kPa dm = 0.3 0 3 μm dp1 = 0.522 0 522 μm s1 = 1.0x10 1 0x10-4 dp2 = 0.091 μ μm s2 = 1.0x10-4

ð7Þ

where v0 is the filtrate volume per unit membrane area when R2 is 0.5. Eq. (7) is the same formula as the logistic function, which is widely used in describing the population growth behavior represented by the S-shaped curve (sigmoid curve). Fig. 6(a) illustrates particle rejection curves represented by Eq. (7) with the different values of the parameter r. As r increases, R2 rises more rapidly. As indicated in Fig. 6(b), the smaller v0 is, sooner the particle rejection R2 increases. Consequently, it may be possible to fit the experimental data of particle rejection with Eq. (7) by using appropriate values of r and v0. It can be seen that the plots

1.0

06 0.

ð6Þ

where r is the empirical constant associated with the rate of increase in the rejection. Integrating Eq. (6) with respect to v under the boundary condition that R2 is 0.5 when v = v0, the relation between R2 and v is represented as

Single (dp1) Single (dp22) Mix Eqs. (9), (10)

R*2 (−))

d2θ/d dv 2 (s/cm2)

E. Iritani et al. / Separation and Purification Technology 123 (2014) 214–220

v0 = 42.5 42 5 cm

0.2

v0 = 50.0 cm 70

v (cm) Fig. 5. Typical particle rejection behaviors for binary particulate mixtures.

0

0

10

20

30

40

50

60

70

v (cm) Fig. 6. Particle rejection curves based on Eq. (7): (a) effect of r, (b) effect of

v0 .

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E. Iritani et al. / Separation and Purification Technology 123 (2014) 214–220

1 .0

3.3. Description of flux decline behaviors based on particle rejection data

0 .8

The amount of trapped particles evaluated from Eq. (7) representing the particle rejection mentioned above increases the resistance to the filtrate flow in filtration [40,41]. It is assumed that the total cake resistance in filtration of binary particulate mixtures is represented by adding the increased cake resistance caused by the capture of smaller particles to the resistance of the filter cake comprised of larger particles alone. The cake solid mass divided by filtrate volume is constant for larger particles, whereas it varies with time for smaller particles because of the variation of observed rejection for smaller particles. Therefore, on the basis of the resistance-in-series model for convenience, the filtration rate u1 is described by

0.6

Z v 0

0 .2

0 0

v þ Rm Þ

    dh l c2 aav2 1 þ expðrðv  v 0 ÞÞ þ Rm ¼ ln c1 aav1 v þ dv p 1 þ expðr v 0 Þ r

8

2

dv

2

 c2 aav2 ¼ c1 aav1 þ p 1 þ expðrðv  v 0 ÞÞ

l

4

3

5x10

-4

4

2

10

(b)

PSL p = 98 kPa dm = 0.3 μ m dp1 = 0.522 μm s1 = 1.0x10-4 dp2 = 0.091 μ m

6

2

8 6 4

ð9Þ 2

Differentiating Eq. (9) with respect to v, one obtains

d h

2

3

ð8Þ

where the subscripts ‘1’ and ‘2’ indicate the values for larger and smaller particles, respectively. It must be stressed that the average specific cake resistance aav2 is the characteristic value in which the effect of the interaction between larger and smaller particles is taken into consideration. Substituting Eq. (7) describing the rejection of smaller particles into Eq. (8), the relation between dh/dv and v is rewritten as

1

s2 (−) 10

R2 d

(a)

PSL p = 98 kPa m dm = 0.3 μm dp1 = 0.522 μm s1 = 1.0x10-4 dp2 = 0.091 μm

0 .4

v0 (cm)

1 dh l ¼ ¼ ðc1 aav1 v þ c2 aav2 u1 dv p

r (cm-11)

of the rejection of smaller particles shown in Fig. 5 are well approximated by the solid curve described by Eq. (7).

10



10

ð10Þ

The solid curves in Figs. 3 and 4 are the calculations based on Eqs. (9) and (10) and are in relatively good agreement with the experimental data throughout the course of filtration of binary particulate mixtures. It is, therefore, concluded that the analysis presented in this article provides an appropriate description of fouling behaviors in dead-end microfiltration of binary mixtures comprised of particles with two different sizes.

-6

10

-5

10

-4

10

-3

s2 (−) Fig. 7. Effect of concentration of smaller particles on parameters in Eq. (7): (a) dependence of r, (b) dependence of v0.

10

d2θ /dv2 (s/cm)

3.4. Effect of concentration of smaller particles It is considered essential to reveal the effect of concentration of smaller particles on the filtration behaviors because the concentration largely affects the rejection behaviors. Thus, filtration experiments of binary mixtures were conducted with varying concentration of smaller particles, and the influences on the filtration properties such as the flux decline and particle rejection behaviors were investigated. Fig. 7 illustrates the effect of the concentration s2 of smaller particles on r and v0 in Eq. (7) obtained from the rejection data for smaller particles. As the concentration s2 increases, r significantly increases and v0 dramatically decreases, indicating a more fulminant capture of smaller particles. The relation between r and s2 is roughly approximated by a straight line from the origin, as shown in Fig. 7(a). The logarithmic plot of v0 vs. s2 is represented by a linear relationship within the range of conditions tested, as shown in Fig. 7(b). The effect of s2 on the characteristic filtration curve based on Eq. (4) is illustrated in Fig. 8. Although the behaviors are similar irrespective of concentration s2 in the initial stage of filtration, the increase in d2h/dv2 gradually becomes more pronounced with increasing concentration s2. However, it is currently difficult to

1

3

10

2

10

1

10

0

10

-1

10

PSL p = 98 kPa dm = 0.3 μm dp1 = 0.522 μm s1 = 1.0x10-4 dp2 = 0.091 μm

s2 = 1.0x10-4 s2 = 2.0x10-4 s2 = 4.0x10-4 Eqs. (9), (10) 0

10

1

10

2

10

3

10

4

dθ /dv (s/cm) Fig. 8. Effect of concentration of smaller particles on characteristic filtration form for blocking filtration law.

accurately evaluate the change of n in Eq. (4) with increasing dh/ dv because of variability in the experimental data. Eventually, d2h/dv2 is kept at an almost constant value depending on the concentration s2 in accordance with the cake filtration theory

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because the mixed cake comprised of larger and smaller particles grows. This constant value of d2h/dv2 tends to increase with increasing concentration s2. This is because the term c2 in Eq. (10) is proportional to s2, and means that the amount of smaller particles in the growing filter cake increases with increasing s2. The solid curves in Fig. 8 represent the calculations based on Eqs. (9) and (10), being in reasonable agreement with the experimental data.

150 dp1 = 0.522 μm p = 49 kPa s1 = 1.0x10-4 p = 98 dp2 = 0.091 μ m p = 196 s2 = 1.0x10-4 p = 294 Eqs. (9), (11)

v (cm)

100

3.5. Effect of applied filtration pressure

50

It is well known that the applied filtration pressure exerts a profound impact on the particle rejection and resultant flux decline behaviors in filtration of single-component suspension [31,40,42]. It is, therefore, expected that the filtration pressure plays an important role in controlling filtration behaviors also in filtration of binary mixtures. In Fig. 9, r and v0 in Eq. (7) derived from the rejection behaviors of smaller particles are plotted as functions of the applied filtration pressure p. It is found from the figure that r linearly decreases with the pressure p and that v0 linearly increases with p through the origin. This means that lower pressure leads to the marked increase in the rejection of smaller particles. This is because lower shear rate brought about by lower pressure pro-

0.4

100 PSL dm = 0.3 μm dp1 = 0.522 μm s1 = 1.0x10-4 dp2 = 0.091 μm s2 = 1.0x10-4

80

60

v0 (cm)

r (cm-1)

0.3

0.2 40 0.1 20

0 300

0 0

100

200

p (kPa)

PSL dm = 0.3 μ m

0

0

0.2

0.4

0.6

0.8

1.0 x10 4

θ (s) Fig. 11. Effect of applied filtration pressure on relation between filtrate volume per unit membrane area and filtration time.

motes the trapping of smaller particles in the filter cake of larger particles. The characteristic filtration curve for blocking filtration law is illustrated in Fig. 10 for several values of the applied filtration pressure p. It is found that the plots of d2h/dv2 vs. dh/dv shift upwards with decreasing pressure. The solid curves are the calculations based on Eqs. (9) and (10) and roughly describe the experimental data. The influence of the pressure p on the relation between v and h is shown in Fig. 11. The increase in the pressure p brings about the increase in the filtrate volume collected within the same filtration time especially in the latter stage of filtration. Larger filtrate volume is obtained under higher filtration pressure. This is because smaller particles are poorly trapped in the filter cake under higher pressure, in addition to the increase in the driving force of the filtrate flow through the incompressible cake comprised of PSL particles with increasing filtration pressure [10]. The solid curves represent the calculations obtained from the numerical integral of Eq. (9) in accordance with [43,44]

h¼

Z v  dh dv dv 0

ð11Þ

Fig. 9. Effect of applied filtration pressure on parameters in Eq. (7).

It is seen that the calculation roughly describes binary mixture filtration data.

d2θ /dv2 (s/cm2)

10

3

10

2

10

1

10

0

10

-1

10

4. Conclusions

PSL dm = 0.3 μm dp1 = 0.522 μm s1 = 1.0x10-4 dp2 = 0.091 μm s2 = 1.0x10-4

p = 49 kPa p = 98 p = 196 p = 294 Eqs. (9), (10) 0

10

1

10

2

10

3

10

4

dθ /dv (s/cm) Fig. 10. Effect of applied filtration pressure on characteristic filtration form for blocking filtration law.

The filtration behaviors such as the filtration rate and particle rejection were examined in dead-end microfiltration of binary mixtures comprised of two types of particles of monodisperse PSL larger and smaller than the pore size of the membrane. The flux decline behaviors were well described based on the resistance-inseries model by using the logistic equation in the description of the variation of the rejection of smaller particles with the progress of filtration. The calculations based on the model were in relatively good agreement with the experimental data plotted as the reciprocal filtration rate vs. the filtrate volume per unit membrane area and the plots based on the characteristic filtration form for the blocking filtration law. Moreover, it was revealed that the concentration of smaller particles and the applied filtration pressure strikingly affected the rejection behaviors of smaller particles, thereby influencing the flux decline behaviors. It is expected that the analytical method developed here may serve as a basis for the analysis of clarifying membrane filtration behavior of very dilute

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