A study on dynamic separation of silica slurry using a rotating membrane filter: 2. Modelling of cake formation

Journal of Membrane Science 157 (1999) 177±187

A study on dynamic separation of silica slurry using a rotating membrane ®lter: 2. Modelling of cake formation Chang Kyun Choia,*, Jin Yong Parkb, Won Cheol Parka, Jae Jin Kimc a

Department of Chemical Engineering, Seoul National University, San 56-1 Shinlim-dong, Kwanak-ku, Seoul 151-742, South Korea b Department of Environmental Science, Hallym University, Chunchon 200-702, South Korea c Membrane Laboratory, Korea Institute of Science and Technology, Seoul 130-650, South Korea Received 9 March 1998; received in revised form 21 July 1998; accepted 25 November 1998

Abstract In micro®ltration the transport, deposition and removal of particles control cake formation on a ®lter. In this connection a new empirical model on cake formation, based on the wall shear stress, was tested here in comparison with experiments of ®ne silica slurry under Taylor-vortex ¯ow. This model on a constant pressure operation expresses the deposition process for particles as two ®rst-order steps in series of mass transfer and adhesion, and their removal process as a linear relation to the wall shear stress. The correlation resulting from ®tting to experimental data represented the present cake resistances reasonably well. This study complements the work of Park et al. [J.Y. Park, C.K. Choi, J.J. Kim, A study on dynamic separation of silica slurry using a rotating membrane ®lter: 1. Experiments and ®ltrate ¯uxes, J. Membr. Sci. 97 (1994) 263] and it will also be helpful in analyzing fouling in heat exchangers. # 1999 Published by Elsevier Science B.V. All rights reserved. Keywords: Micro®ltration; Rotating membrane ®lter; Cake formation; Couette±Taylor ¯ow; Wall shear stress

1. Introduction In the micro®ltration system fouling is common. During its operation the ®lter surface gradually becomes covered with a solid fouling deposit, i.e. cake, and a cake layer formed creates a barrier to the ®ltrate ¯ux. Cake formation can be reduced by high shear stresses at the ®lter surface. For its reduction the micro®ltration method to use Taylor vortices generated in the annular gap between two coaxial cylinders has been tested intensively [1±6]. *Corresponding author. Tel.: +82-2-880-7407; fax: +82-2-8887295; e-mail: [email protected]

Park et al. [1] reported experimental results on micro®ltration of ®ne silica particles suspended in pure water under the Couette±Taylor ¯ow under gravity forces. The dynamic behavior of ®ltrate ¯uxes in their work can be summarized: the ®ltration ¯ux decreases sharply for a short time and with time it approaches a higher value with increasing rotating speed of the inner ®lter. Therefore, they divided the experimental data into two groups, transient and pseudosteady state, and analyzed the system as a function of the Taylor number and the ratio of the annular gap to the radius of the inner ®lter. In the present work, their work is extended by conducting complementary experiments and also analyzing char-

0376-7388/99/$ ± see front matter # 1999 Published by Elsevier Science B.V. All rights reserved. PII: S0376-7388(98)00377-9

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C.K. Choi et al. / Journal of Membrane Science 157 (1999) 177±187

acteristics of cake formation on the basis of the wall shear stress. Before cake forms, there is usually an induction period owing to plugging processes in the ®lter itself. Fortunately, this period was rather short in the present system. Therefore here the mechanism of cake formation is concentrated. Particles that become deposit will be transported from the bulk to the cake surface after the ®rst layer is formed on the ®lter surface. Then adhesion or sticking will be followed on it. Therefore the process will be at least a two-stage one. In this connection it has been well known that the wall shear stress plays an important role on deposition and removal of particles [7,8]. Therefore, an empirical model for predicting the cake resistance on a constant pressure operation is proposed here as a function of the wall shear stress in connection with fouling in heat exchangers. The resulting correlation ®tted to the experimental data is compared reasonably well with the whole experimental data of the present micro®ltration system under the Couette±Taylor ¯ow. The present results will be helpful in analyzing dynamic systems such as cross¯ow micro®ltration and heat exchangers experiencing transient fouling.

where C is the cake density, xC the cake thickness, and the proportionality coef®cient including the effect of the particle size and porosity. Here RC denotes the cake resistance having the unit of Pa s/m and it is the measure of operation. In the present system, RC is obtained from the following equation: Jm ˆ


P ; RM ‡ R C

(3)

where Jm denotes the average ®ltrate ¯ux, RM the membrane resistance including the ®lter plugging and viscosity effect, and
P the actual pressure difference across the ®lter. In the present dynamic separation using a rotating ®lter as is shown in Fig. 1, the driving force of gravity is reduced owing to centrifugal forces and Taylor vortices [1]. But the Couette±Taylor ¯ow has been studied intensively and the wall shear stress  !i is well known. Atsumi et al. [7] suggested the following correlation: !i ˆ f ri2 !2i =2

for 0:03  d=ri  1:0;

(4a)

2. Theory 2.1. Dynamic filtration overview In most of dynamic ®ltration systems fouling is a transient process; the ®lter surface starts clean and ends up fouled. The ®rst layer of cake can give extra roughness to the ®lter surface. In this case cake forms rapidly initially but with time the rate of cake formation becomes slow. Finally, a pseudosteady state deposit is reached. Since cake formation is a transient process, its rate is commonly expressed as a balance between deposition and removal processes: dMC _ Cd ÿ M _ Cr ; ˆM dt

(1)

where MC represents the mass of the cake layer per _ Cd the deposition rate and unit ®lter surface area, M _ M Cr the removal rate. The total deposit mass MC is given by MC ˆ C xC ˆ RC ;

(2)

Fig. 1. Schematic of the present system under the Couette±Taylor flow.

C.K. Choi et al. / Journal of Membrane Science 157 (1999) 177±187

in Eq. (1) may be expressed as

with f ˆ

2

4…1 ‡ d=ri † …2 ‡ d=ri †Re

f ˆ 0:80…d=ri †

179

0:35

for 20  Re  ReC ; Re

ÿ0:53

(4b) 4

for ReC  Re  10 ; (4c)

where f denotes the friction factor,  the slurry density, ri the radius of the inner ®lter and !i its rotating angular velocity. The Reynolds number Re is based on the angular velocity !iri and the gap d. ReC is the critical Reynolds number to mark the onset of Taylor vortices and ReC is given by ReC ˆ 41:2…d=ri †ÿ0:5 ‡ 27:2…d=ri †0:5 ‡ 2:8…d=ri †1:5 : (5) Furthermore, the present micro®ltration system under Taylor vortices may be considered as continuous well-stirred ®lters in series. The driving force
P is almost independent of RM and RC. The characteristics of RM and RC themselves may be analyzed by using Darcy's law and the Kozeny±Carman equation. But here the primary concern is to know how the wall shear stress  !i affects RC. Since  !i is given by Eq. (4a), the present Couette±Taylor ¯ow system will become a good one to show usefulness of theoretical models. Mechanisms of the cake resistance in above systems seem to be analogous to those of the fouling resistance in heat exchangers. Fouling of heat transfer surfaces is well illustrated by Krause [8]. For its analysis, several models have been suggested [8±10]. Usually the cake formation process in micro®ltration is fast in comparison to the heat exchanger fouling because in the former case the ®ltrate ¯ux enhances the cake growth and the particle size becomes larger. By transfusing the latter into the former a modeling approach is conducted here. 2.2. Modeling approach Cake formation usually results from a combination of mass transfer and adhesion steps. To form cake on the ®lter surface the particle must be transported from the slurry bulk to the surface and then undergo an adhesion process. Starting from the initial cake formation at the ®lter surface, the deposition rate of cake

^b ÿ C ^ i † ‡ Jm …C ^b ÿ C ^ i †; _ Cd ˆ km …C M

(6)

where km denotes the particle transport rate analogous ^ b the bulk to the usual mass transfer coef®cient, C ^ concentration of particles in slurry, and C i that at the ^ has the unit of kg-particle/m3. cake surface. Here C This equation has been derived under the assumption that the deposition rate would be the sum of the diffusive ¯ux based on the ®lm theory and the convective one by through¯ow, i.e. Jm. On particle deposits at a non-transpirating wall, Metzner and Friend [11] and Cleaver and Yates [12] showed the following theoretical relation of km for ®ne particles suspended in ¯uid: km ˆ u ;

(7)

where  is a dimensionless empirical constant. The friction velocity u* is given by r !i  : (8) u ˆ  For the present system in the Couette±Taylor ¯ow the wall shear stress  !i can be obtained from Eq. (4a). It is known that Eq. (7) can be applied to the fully developed turbulent boundary layers of momentum and concentration [13]. In general, the particle transport is very much complicated and some models are summarized in the work of Melo and Pinheiro [14]. The particle transport is followed by the surface adhesion and in the present micro®ltration system it is assumed that the adhesion rate of particles will be the same as the deposition rate. If the adhesion is ®rst order, the following relation is possible: ^ i; _ Cd ˆ ka C M

(9)

where ka is the adhesion rate constant. By combination of Eqs. (6) and (9) to eliminate the unknown surface ^ i the deposition rate can be written as a concentration C ^ b: function of C   1 1 ^b _ Cd ˆ C ‡ M : (10) Jm ‡ km ka Vasak et al. [13] assumed that ka is directly proportional to the particle residence time in the vicinity of the interface having a timescale of =…u2 †, so that ka ˆ ‰ u2 Šÿ1 ;

(11)

180

C.K. Choi et al. / Journal of Membrane Science 157 (1999) 177±187 1=2

where is an empirical coef®cient. Since u / !i from Eq. (8), it is known that the wall shear stress  !i _ Cd is the most important parameter in estimating M (see Eqs. (7) and (11)). Removal of particles from the cake surface occurs due to the force exerted by the ¯uid, adversely to their deposition. The removal rate has been known to be proportional to the cake thickness xC and also to the wall shear stress t!i [8,10]: _ Cr ˆ kr xC !i ; M

(12)

where kr is the removal coef®cient. Now, Eqs. (2), (3)±(12) are substituted into Eq. (1). Then the process of deposition and removal of particles on a constant pressure operation is represented in terms of  !i as follows:  ÿ1 dRC ^ 1 1 ˆ Cb ‡ 0:5 dt K3 =!i K1 …
P=…RM ‡ RC †† ‡ K2 !i ÿ K4 !i RC

(13)

or 

d…1=Jm † ^ 1 1 ˆ Cb ‡ 0:5 dt k3 =!i k1 Jm ‡ k2 !i   1 1 ÿ k4 !i ÿ ; Jm Jm0

K1ˆK4ˆ0 is the same as that of Vasak et al. [13] for the initial deposition process of particles involving no reaction in a heat exchanger. In Eqs. (13) and (14), the  !i-dependence of each term needs a further study and they are not the rigorous ones. But they involve the essential feature of dynamic micro®ltration using Taylor vortices or cross¯ow and also that of fouling in heat exchangers. Now, let us compare Eq. (13) with the following model of Kern and Seaton [10]: RC ˆ RCS ‰1 ÿ exp…ÿKt†Š;

where RCS denotes the asymptotic value at the pseudosteady state and K is an empirical constant ®tted to the experimental data. The value 1/K is a kind of time constant to represent the resistance to removal. This simple model can be obtained by assuming that the decreasing rate of the number of particles deposited at the cake surface is proportional to their number at that time like the ®rst-order reaction system [15]. The above equation is transformed by its differentiation: dRC ˆ K…RCS ÿ RC † ˆ KRCS exp…ÿKt†: dt

ÿ1

(14)

where Ki and ki (iˆ1±4) are the empirical coef®cients and Jm0 the initial ®ltrate ¯ux at that time when the ®rst layer of cake has been formed. Using Eq. (3), Eq. (13) can be converted to Eq. (14) and vice versa. Dead-end ®ltration corresponds to the case of Kiˆkiˆ0 (iˆ2±4) and with K1ˆ0 the trend of Eq. (13) is almost the same as that of fouling in heat exchangers. In heat exchangers particles are transferred from the suspension bulk at the concentration ^ b to the wall surface without permeation (Jmˆ0, i.e. C K1ˆ0) and adhere to it. In the present system the strength of turbulent Taylor-vortex ¯ow controls the boundary layer ¯ow formed near the surface (see Fig. 1), while in heat exchangers that of the bulk ¯ow does. Both ¯ows have a common boundary-¯ow layer and its characteristics are connected to each other through the wall shear stress  !i. In this sense there exists an analogy between a rotating ®lter and a heat exchanger in the hydrodynamically fully developed state. Therefore, the meaning of Eq. (13) with

(15)

(16)

At the initial stage of ®ltration, the above equation can be written as dRC ˆ Kd RCS dt

for t ! 0

(17)

by letting KˆKd. By comparing the terms corresponding to the removal rate in Eqs. (13) and (16) and letting KˆKr the following relation is obtained: Kr ˆ K4 !i ;

(18)

where Kd and Kr represent the overall deposition and removal coef®cients, respectively. The K-value in Eq. (15) becomes Kd or Kr, depending on the mechanism. Melo and Pinheiro [14] used Kd while Krause [8] recommended Kr in place of K in Eq. (16). Therefore, this simple model means that the initial deposition rate is KdRCS (from Eqs. (17) and (2)) and the value of RCS is the measure of the deposition rate. From Eq. (13) it is known that Kd is a function of  !i. With time the removal rate will become signi®cant and Eq. (16) with KˆKr looks proper. This means that Eq. (15) may represent the limiting cases of the initial and the pseudosteady state. The above postulate will be tested in comparison with our experimental results.

C.K. Choi et al. / Journal of Membrane Science 157 (1999) 177±187

Of course, Eq. (15) is not applicable to the case of dead-end ®ltration because its RCS!1.

Table 1 Experimental results by the present rotating filter with the residence time of 10.4 min for CFˆ1.0 wt% !i (rad/s)

3. Experimental 3.1. Experiments By using ®ne silica particles (average sizeˆ4.0 mm) suspended in water micro®ltration experiments were conducted under the Couette±Taylor ¯ow. Detail of the material and experimental procedure is given in the work of Park et al. [1]. The inner rotating ®lter contained MF-cellulose ester membrane having an average pore size of 1.2 mm. The inner ®lter of outside radius riˆ2.6 cm rotated with a ®xed angular velocity !iˆ0±62.8 rad/s and the outer cylinder was motionless. The annular gap d between the outer cylinder and the inner ®lter ranged from 0.45 to 1.70 cm and the feed concentration of silica in water was kept constant at CFˆ1.0 wt%. TheaxialReynoldsnumber(ˆ2uzd/) based on the gap d and the axial velocity uz ranged from 14.6 to 36.5 and in this range its effect on ®ltration was negligible. The slurry level inside the annular gap (h) was 67 cm and the resulting hydrostatic pressure was the main driving force in ®ltration. The residence time was 10.4 min and experiments were conducted here with d/riˆ0.33. 3.2. Data processing Experimental data for d/riˆ0.65, 0.54 and 0.17 were already reported by Park et al. [1]. The data of the pseudosteady state ¯ux Jms for d/riˆ0.33 and 0.65 are given in Table 1. This value was obtained by averaging the data after the ®ltration time tˆ180 min. P* denotes the average pressure reduction caused by Taylor vortices, which was obtained with pure water. The average pressure difference
P was approximated by the relation of
Pˆgh/2ÿP*, where g denotes the gravity acceleration. With increasing rotating speed !i,
P decreased but Jms increased. The direct measurement of the initial ®ltrate ¯ux Jm0 was very dif®cult in the present experiments. Its value listed in Table 2 was obtained here by extrapolating from the experimental data of Jm vs. t, based on Eq. (14). Therefore, its values are a little different from that of Park et al. [1]. Jm0 was converted to the

181

0.0 10.5 20.9 31.4 41.8 52.4 62.8

P* (Pa) (pure water)

Jms105 (m/s) d/riˆ0.33 a

0.0 18.1 71.6 161.7 286.6 450.3 646.8

0.22 0.31a 0.42 0.54 0.55 0.56 0.57

d/riˆ0.65 0.21a 0.26a 0.40 0.54 0.67 0.74 0.95

180 min data after the startup of experiments. Transient.

a

membrane resistance RM by using Eq. (3). For analysis these four sets of experimental results were used. ^ F since Jm was ^b  C Now, it was assumed that C small. With the experimental results of !iˆ0 the K1value in Eq. (13) was obtained from the relation: R2C =2 ‡ RC RM ˆ K1 t ^b C

for !i ˆ 0:

(19)

This equation was derived from Eq. (13) by letting  !iˆ0 and integrating the resulting equation with respect to time. This corresponds to the case of dead-end ®ltration for incompressible particles under complete blocking [16]. In order to obtain the value of Ki (iˆ2±4) in Eq. (13) the data points of RC vs. t were ®tted by a polynomial of a maximum degree of ®ve. Based on these polynomials, the values of dRC/dt were estimated and then the Ki-values were obtained directly from Eq. (13) by using a nonlinear regression method. This computation was conducted using computer package programs of Mathmatica and Matlab. 4. Results and discussion In the present micro®ltration system under the Couette±Taylor ¯ow the  !i-value ranged from 0 to 6 Pa by Eq. (4a). First, the experimental Jm-value was converted to RC by using Eq. (3). Eq. (19) was well ®tted to the RC-values obtained from the experimental Jm-data of !iˆ0, which corresponds to the case of dead-end ®ltration. With the K1-value ®xed, Eq. (13) ®tted to the experimental RC-values for each experimental run represents the present system well except

182

C.K. Choi et al. / Journal of Membrane Science 157 (1999) 177±187

Table 2 Initial filtration fluxes Jm0 and membrane resistances RM from present experimental results d/ri

!i (rad/s)

Jm0105 (m/s)

RM10ÿ8 (Pa s/m)

tR

0.65

0.0 10.5 20.9 31.4 41.8 52.4 62.8

1.59 1.04 1.02 1.16 1.39 1.33 1.52

1.75 2.37 2.36 2.00 1.59 1.53 1.21

± 0.10 0.28 0.51 0.78 1.08 1.41

0.54

0.0 10.5 20.9 31.4 41.8 52.4 62.8

1.82 1.70 1.32 1.37 1.40 1.36 1.41

1.36 1.45 1.83 1.69 1.57 1.49 1.30

± 0.11 0.29 0.53 0.80 1.12 1.46

0.33

0.0 10.5 20.9 31.4 41.8 52.4 62.8

1.58 1.45 1.07 1.13 0.92 0.84 0.76

1.57 1.70 2.25 2.05 2.39 2.42 2.41

± 0.12 0.32 0.58 0.88 1.22 1.60

0.17

0.0 10.5 20.9 31.4 41.8 52.4 62.8

1.58 1.32 1.42 1.48 1.45 1.42 1.32

1.57 1.87 1.70 1.57 1.51 1.43 1.39

± 0.13 0.36 0.65 0.99 1.38 1.80

those at low ®ltration times of small t, as shown in Fig. 2. At the early stage of ®ltration the agreement between the calculated and the measured RC-values are not so good because the K1-value is ®xed for each d/ri. The resulting K1-values for each d/ri are listed in Table 3(a), which were 2.303109 sÿ1 for d/riˆ0.65, 1.466109 sÿ1 for d/riˆ0.54, 1.984109 sÿ1 for d/riˆ 0.33, and 1.629109 sÿ1 for d/riˆ0.17. Their average value was found to be 1.845109 sÿ1. The other average Ki-values for each set are also listed in Table 3(a). Since the Ki-values ®tted to each set of experimental results are almost constant for the pre^ F 10 kg/m3, their all-averaged values sent system of C listed in Table 3(b) were used in Eq. (13). In the present experiments the order of magnitude having the unit of m/s2 was found to be that of K1JmO(104), 0:5 O(102) and K3/ !iO(104) with  !iˆO(1 Pa) K2 !i

and JmˆO(10ÿ5 m/s). This means that in the present system, the deposition rate is controlled by the ®ltrate ¯ux itself and adhesion. The calculated RC-values from the resulting correlation is compared with the experimental ones in Fig. 3. Except for the three initial data of the ®ltration the average relative errors with increasing d/ri are 10.0%, 8.9%, 9.5% and 12.3%, respectively. With d/riˆ0.65 and 0.54, the calculated RC-values deviate more from the experimental ones with decreasing !i and increasing RC (also, see Fig. 2) but with decreasing RC their relative errors become smaller than those of d/riˆ0.33 and 0.17. This might be caused by both experimental dif®culties encountered and the different mechanisms. In an overall view, it is known that the present single, empirical correlation represents the whole experimental data to a certain degree.

C.K. Choi et al. / Journal of Membrane Science 157 (1999) 177±187

183

where mˆ0.5±1.0 and nˆ1/3±1/2. Sh denotes the Sherwood number and Sc the Schmidt number. The 0:5 term K2 !i is equivalent to that of mˆ0.735 (see Eqs. (4a), (4c) and (6)). The term K3/ !i in Eq. (13) concerns the adhesion resistance. In connection with the particle transport it has been known that the dimensionless particle relaxation time de®ned by tR ˆ

Fig. 2. Comparison of experimental and calculated cake resistance RC using Eq. (13) and Table 3(a) for d/riˆ0.65.

0:5 The term K2 !i in Eq. (13) is analogous to that of the conventional mass transfer coef®cient derived from correlation of

Sh / Rem Scn ;

(20)

Fig. 3. Comparison of cake resistance RC between its values calculated from Eq. (13) using Table 3(b) and present experimental ones.

…P ÿ L †dp2 u2 18

(21)

plays an important role [17]. Here P represents the particle density, L the liquid density, dp the particle size, and  the kinematic viscosity. The characteristic length …P ÿ L †dp2 u2 =…18† represents the stopping distance over which the particle velocity u* is reduced to zero on account of viscous drag in the Stokes region. tR is a measure of particle size and velocity. For tR<0.1, diffusion is assumed to be the dominant mechanism without the adhesion resistance. With increasing tR, the particle transport mechanism becomes different. Davis [18] suggested that for tR>0.22 interception and impact effects becomes signi®cant and for large, fast particles (tR>33) only one jump is necessary for the particles to impact on to the surface. Since 0 2:25 PaŠ by C assuming that for tR>0.2 particles would transport along the ¯owlines with interception by roughness elements. They supposed that the particle transport would be represented by the term K3/ !i since in the present system tR>0.22 for !i20.9 rad/s (see Table 2). In their correlation K4 depended strongly on d/ri but in the present analysis K4 is almost constant, as shown in Table 3. Therefore, their interpretation on the term K3/ !i may not be applicable to the present system. In the present tR-range it may be proper to consider the term K3/ !i as the measure of adhesion at the cake surface (see Table 2 and Eqs. (10) and (11)). It is _ Cr / tR . The two interesting that ka/1/tR and M mechanisms of particle adhesion and removal at the cake surface are closely related to tR or  !i through u*. Their dependencies on  !i for !iˆ10.5 rad/s may be somewhat different from those shown in Eq. (13) since the corresponding tR-values range from 0.10

184

C.K. Choi et al. / Journal of Membrane Science 157 (1999) 177±187

Table 3 The values of empirical coefficients in Eq. (13) d/ri

!i (rad/s)

K110ÿ9 (sÿ1)

(a) Averaged values for each experimental set 0.65 0.0 2.303 10.5 20.9 31.4 41.8 52.4 62.8 Average

K210ÿ2 (m2/3/kg1/2 s)

K310ÿ5 (kg/s4)

K4104 (m s/kg)

± 1.412 1.398 1.330 1.169 1.275 1.123 1.284

± 1.055 1.127 1.081 0.896 1.036 0.873 1.012

± 1.485 2.004 2.329 2.866 2.514 2.672 2.312

0.54

0.0 10.5 20.9 31.4 41.8 52.4 62.8 Average

1.466

± 1.323 1.586 1.372 1.352 1.206 1.022 1.310

± 1.185 1.417 1.054 1.026 0.894 0.905 1.080

± 2.424 1.671 2.100 2.360 2.690 2.824 2.345

0.33

0.0 10.5 20.9 31.4 41.8 52.4 62.8 Average

1.984

± 1.452 1.536 1.223 1.349 1.350 1.450 1.393

± 0.881 1.174 0.925 1.045 1.193 1.261 1.080

± 2.472 1.925 2.322 2.152 2.907 2.085 2.176

0.17

0.0 10.5 20.9 31.4 41.8 52.4 62.8 Average

1.629

± 1.441 1.460 1.286 1.306 1.258 1.289 1.340

± 1.146 1.340 1.063 1.088 1.061 1.027 1.121

± 1.506 1.882 2.026 2.213 2.259 2.011 1.983

1.845

1.332

1.073

2.204

(b) All-averaged values

to 0.13. In aerosol studies diffusion has been assumed to be the dominant mechanism for tR<0.1±0.2 [12,13,17,18]. Over this characteristic tR-value the complicated particle-transport mechanism becomes different depending on the tR-value. This may be the reason why the present model does not ®t experimental results well at low rotation velocities for the higher d/ri ratios. But it is interesting that the present model using Table 3(b) ®ts very well the experimental data points of d/riˆ0.17 and 0.33 for large RC, as

shown in Fig. 3. Therefore, it may be stated that the Ki-values in Table 3(b) should be used for ®ne silica slurry of dpˆ4 mm under the condition of tR<2. With this the empirical correlation (13) or (14) may be used in predicting the operation performance of other module geometries in micro®ltration, dead-end (K2ˆK3ˆK4ˆ0) or cross¯ow. In the case of cross¯ow, its characteristics are almost the same as the present ones when the ¯ow is fully developed and turbulent. Therefore, in a channel ¯ow, one set of experiments

C.K. Choi et al. / Journal of Membrane Science 157 (1999) 177±187

Fig. 4. Pseudosteady state cake resistance RCS using Eqs. (13) and (22).

for a given Reynolds number will produce the Ki-values through  !i. In order to evaluate the operation performance in micro®ltration the prediction of the pseudosteady state resistance RCS is critical. In the present system RCS was obtained by letting dRC/dtˆ0 in Eq. (13) and using the Ki values in Table 3(b). The resulting overall correlation for  !i<6 Pa was derived with the relative error of 9.3% as ÿ0:938 : RCS ˆ 4:32  108 !i

(22)

Fig. 4 shows that its comparison with experimental values looks good for  !i>1 Pa. For  !i<1 Pa the predicted RCS-values are higher than the experimental ones. This may be caused by different mechanisms of particle transport in view of tR, as is aforementioned. Murase and Ohn [20] reported that in cross¯ow micro0:91 . ®ltration the ®ltrate ¯ux in steady state Jms / !i ÿ0:91 This means that RCS / !i and therefore Eq. (22) is comparable with theirs. Since Eq. (15) is very simple, it will be interesting to test its applicability by using Eq. (22). By comparison of Eq. (17) with Eq. (13) at tˆ0, the following relationship is obtained:  ÿ1 ^b C 1 1 ‡ : (23) Kd ˆ 0:5 RCS K1 Jmo ‡ K2 !i K3 =!i

185

Fig. 5. The deposition coefficient Kd and the removal coefficient Kr as a function of the wall shear stress  !i.

For  !i!1, Eq. (23) reduces to ^ b K3 =…RCS !i †: Kd ˆ C

(24)

^ F 10 kg/m3, ^b  C Since in the present system C 5 4 from Table 3(b), and K3ˆ1.07310 kg/s RCS !i4.32108 from Eqs. (22) and (24), Kd has the upper bound of 2.4810ÿ3 sÿ1. From Table 3(b) and Eq. (18), it is known that Krˆ2.2010ÿ4 !i. This Kr-relation and the Kd-values obtained from Eq. (23), Tables 2 and 3(a) and actual RCS for each experimental run are plotted as a function of  !i in Fig. 5. Also, correlation (23) with Jm0ˆ1.8210ÿ5 m/s (the maximum Jm0-value in Table 2), RCS from Eq. (22) and Ki-values from Table 3(b) is shown in this ®gure. It is known that Kd is larger than Kr in the present experimental range and Kd is almost equal to Kr near  !i6.8 Pa. This  !i-value increases gradually with an increase in Jm0 and in the range of  !i larger than this value it may be stated that the relation of KdˆKr is possible. For the test of Eq. (15) the case of d/riˆ0.65 with KˆKd or Kr is compared in Fig. 6 with correlation (13) using the Ki-values of each experimental run. In this ®gure, the data of !iˆ0 and 10.5 rad/s were not plotted because the pseudosteady state was not observed experimentally under these rotating speeds. It is known that with an increase in !i, i.e.  !i, the two

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C.K. Choi et al. / Journal of Membrane Science 157 (1999) 177±187

experimental data of silica slurry under the Couette± Taylor ¯ow represented reasonably well the present system of  !i<6 Pa and tR<2. 6. Nomenclature CF ^b C ^i C ^F C d

Fig. 6. Prediction of cake resistance RC using Eq. (15) for d/riˆ0.65.

results agree well but except for the case of !iˆ20.9 rad/s the correlation with KˆKr is better than that with KˆKd. The latter one supports the interpretation of Krause [8] on the model of Kern and Seaton [10] in fouling of heat exchangers. In comparison with Fig. 2, it is known that Eq. (15) with KˆKr can be used effectively for large  !i, while for small  !i, Kd becomes more important rather than Kr. With increasing  !i the distinction between the results of KˆKd and KˆKr will disappear since Kr becomes equal to Kd. Cake formation in micro®ltration is similar to fouling in heat exchangers with K1ˆ0, i.e. Jmˆ0, as illustrated above. In both cases, Eq. (13) can be used by considering common features of the coef®cients K2, K3 and K4. These coef®cients will depend on the physical properties such as the viscosity, particle size and particle concentration. In this connection experiments are being conducted in our laboratory. 5. Conclusion In the present study a new model on a constant pressure operation was introduced to predict the cake resistance in micro®ltration. This model expresses the deposition and removal rates of particles as a function of the wall shear stress. The correlation ®tted to the

dp f g h Jm Jm0 Jms ka km kr K K1 K2 K3 K4 Kd Kr MC _ Cd M _ Cr M P P*
P ri RC RCS RM Re

feed concentration (wt%) bulk concentration of particles in slurry (kgparticle/m3) concentration of particles at cake surface (kg-particle/m3) concentration of particles in feed slurry (kgparticle/m3) annular gap size between two coaxial cylinders (m) particle diameter (m) friction factor (dimensionless) gravitational constant (m/s2) fluid depth in dynamic filter (m) average filtrate flux (m/s) initial filtrate flux (m/s) filtrate flux in pseudosteady state (m/s) adhesion rate constant of particles (m/s) transport rate constant of particles (m/s) removal rate constant of particles (m/s) constant in Eq. (15) (sÿ1) empirical coefficient in Eq. (13) (sÿ1) empirical coefficient in Eq. (13) (m3/2/kg1/2 s) empirical coefficient in Eq. (13) (kg/s4) empirical coefficient in Eq. (13) (m s/kg) overall deposition coefficient (sÿ1) overall removal coefficient (sÿ1) mass of cake layer per unit surface area of filter (kg/m2) particle deposition rate (kg/m2 s) particle removal rate (kg/m2 s) pressure (Pa) pressure drop by inner filter rotation (Pa) pressure difference across filter (Pa) outer radius of inner filter (m) cake resistance,
P/JmÿRM (Pa s/m) cake resistance at pseudosteady state (Pa s/m) membrane resistance defined in Eq. (3) (Pa s/m) Reynolds number, !irid/ (dimensionless)

C.K. Choi et al. / Journal of Membrane Science 157 (1999) 177±187

ReC Rez Sc Sh tR u* uz xC   C L P

   !i !i

critical Reynolds number (dimensionless) axial Reynolds number, 2uzd/ (dimensionless) Schmidt number (dimensionless) Sherwood number (dimensionless) dimensionless particle relaxation time defined by Eq. (21) friction velocity defined by Eq. (8) (m/s) axial velocity (m) cake thickness (m) dimensionless empirical constant in Eq. (7) empirical coefficient in Eq. (11) (dimensionless) slurry density (kg/m3) cake density (kg/m3) liquid density (kg/m3) particle density (kg/m3) proportionality coefficient in Eq. (2) (s) viscosity (kg/m s) kinematic viscosity (m2/s) wall shear stress at rotating filter surface (Pa) angular velocity of inner filter (rad/s)

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