A new method for the prediction of pore size distribution and MWCO of ultrafiltration membranes

Journal of Membrane Science 279 (2006) 558–569

A new method for the prediction of pore size distribution and MWCO of ultrafiltration membranes Jizhong Ren ∗ , Zhansheng Li, Fook-Sin Wong Institute of Environmental Science and Engineering, Nanyang Technological University, 18 Nanyang Drive, Singapore 637723, Singapore Received 25 June 2005; received in revised form 3 December 2005; accepted 24 December 2005 Available online 3 February 2006

Abstract This paper describes the transport process, rejection curve and molecular weight cut-off (MWCO) of ultrafiltration membranes by using a log-normal distribution, Poiseuille flow and steric interaction between solute molecules and pores. The rejection coefficient of membranes was expressed with an analytical function of D* /a (the ratio of geometric mean diameter to solute diameter) and σ (geometric standard deviation). For ultrafiltration membranes with different MWCO, the pore size distribution can be divided into three zones by σ: Zone I, long tail-effect of big pores (σ > 1.55); Zone II, linearization of pore size distribution parameters (D* , σ) (1.15 ≤ σ ≤ 1.55); Zone III, sieving effect (1 ≤ σ < 1.15). The relationship between D* /DMWCO (DMWCO is the diameter of solute molecule at R = 0.9) and σ was described by the Extreme model: D∗ DMWCO

= A e(−e

(−z) −z+1)

,

z=

σ−1 w

(σ ∈ [1, ∞]),

A = 1.296,

w = 0.299

For any rejection coefficient, the relationship between D* /a and σ can also be described using the Extreme model. A method to predict the pore size distribution and MWCO of ultrafiltration membranes by two points was first presented and verified by hollow fiber membranes spun at different shear rates. The shear rate in the spinning of hollow fiber membranes dominated the pore size distribution parameters (D* , σ), which played an important role in membrane performance. With an increase in shear rate, the geometric standard deviation (σ) was strongly suppressed, which resulted in hollow fiber membranes with low MWCO. © 2006 Elsevier B.V. All rights reserved. Keywords: Log-normal distribution; Rejection coefficient; Ultrafiltration; Theory

1. Introduction Membrane processes have been applied in various types of industries such as the separation, concentration and purification in food technology, biotechnology and petrochemical processes, as well as water and wastewater treatment. The pore size and its distribution are very important parameters for membrane quality and membrane transport mechanisms. The pore size distribution dominates the separation characteristics of asymmetric membranes, which can be used to predict the molecular weight cut-off (MWCO) of porous membranes and the rejection for different solute molecules or particles. This allows an appropriate membrane process to be chosen to achieve specific separation and purification goals. The suppression of pore size

∗

Corresponding author. Fax: +65 67921291. E-mail address: [email protected] (J. Ren).

0376-7388/$ – see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.memsci.2005.12.052

distribution is very important in advancing commercial ultrafiltration membranes. The pore size distribution of ultrafiltration membranes can be determined by different techniques such as bubble point method, liquid displacement, thermoporometry and visual observation (scanning electron microscopy (SEM), transmission electron microscopy (TEM) or atomic force microscopy (AFM)), which have been reviewed by Nakao [1] and Zhao et al. [2]. Usually, these methods mentioned above cannot characterize the pore size distribution and the sieving characteristics of ultrafiltration membranes accurately [1,3,4]. The direct measurement of solute rejection is a relatively accurate method to characterize the separation characteristics of ultrafiltration membranes [5–8]. This method measures the rejection characteristics of standard solutes with known molecular weight such as dextran or polyethylene glycol (PEG). In order to characterize the pore size distribution, there are a number of studies to examine the relationship between solute size and solute rejection with different distribution functions

J. Ren et al. / Journal of Membrane Science 279 (2006) 558–569

and transport theories [8–13,15–25]. Michaels [8] introduced a log-normal linear dependency of the rejection profile on the probability coordinates and obtained a simple estimation of the membrane mean radius and a standard deviation in both biological and synthesis membranes. Wendt et al. [9,10] assumed a monomodal homoporous, bimodal pore size distribution to evaluate the sieving coefficients, and suggested that the dependence of sieving coefficient on molecular size was relatively independent of the membrane pore size distribution. Kassotis et al. [11] developed a mathematical model to relate the flux and solute rejection to the pore size distribution function and the total number of pores based upon the assumption that solute rejection was the result of purely geometric considerations. Leypoldt [12] investigated the mathematical problem of using the rejection profile to calculate the pore size distribution, and concluded that the exact membrane pore size distribution could not be determined uniquely from a single probability correlation function. Derjani-Bayeh and Rodgers [13] evaluated the effect of different standard probabilistic models (gamma, log-normal, normal, Weibel and Rayleigh) on area average flux and area average membrane sieving coefficient, and concluded that an uncertainty in the choice of distribution in describing the membrane morphology could lead to a propagated uncertainty in predicting the overall membrane performance. Now, the log-normal distribution [14] has been widely used to describe the actual pore size distribution in a wide range of porous membranes since it was introduced by several authors to analyse the sieving data of different ultrafiltration membranes [8,15]. Belfort et al. [16] also demonstrated that the pore size distribution evaluated from the particle fouling data with two district particle sizes was consistent with a log-normal density function. Zydney et al. [17] have reviewed the log-normal distributions with different forms. The appropriate equations required to transform between the different functional forms and their statistical means and variances were also provided and evaluated properly. Mochizuki and Zydney [18] studied the effect of the standard deviation of the log-normal and normal distribution on the asymptotic sieving coefficient. Aimar et al. [19,20] obtained the log-normal pore size distribution parameters of clean and BSA-fouled polysulfone IRIS membranes based upon the normalization of the sieving curves with solute rejection experimentally measured. The solute rejection was simplified to a pore steric mechanism and the geometric standard deviation ranged from 1.35 to 1.6. By ignoring the dependence of solute separation on the steric and hydrodynamic interaction between solute and pore size, Singh et al. [4] and Mosqueda-Jimenez et al. [21] obtained the log-normal distribution parameters of various ultrafiltration membranes with the geometric standard deviation ranging from approximately 1.7–3.5. The total number of pores and surface porosity were obtained using the Hagen-Poiseuille equation, which was not taken into account in evaluating the pore size distribution parameters. Due to this significant drawback, the MWCO of ultrafiltration membranes could not be predicted properly, and the pore size distribution parameters could not be obtained correctly. From the above analysis, these methods can be used to evaluate the pore size distribution of various mem-

559

branes, but some limitations are also exhibited with the different methods. In this paper, the transport process, rejection coefficient and MWCO of ultrafiltration membranes were studied theoretically using a log-normal distribution, Poiseuille flow and steric interaction between solute molecules and pores. The analytical solution for the rejection of ultrafiltration membranes was expressed as a function of D* /a (geometric mean diameter/solute diameter) and σ (geometric standard deviation). The relationship between D* /DMWCO (the diameter of solute molecular at R = 0.9) and σ was also discussed with the Extreme model. A new mathematical model was developed to predict the pore size distribution and MWCO of ultrafiltration membranes. The model was evaluated with hollow fiber membranes spun at different shear rates. In addition, the influence of shear rate on the pore size distribution parameters of hollow fiber membranes was also investigated. 2. Experimental In order to predict the pore size distribution and MWCO of ultrafiltration membranes, BTDA-TDI/MDI co-polyimide (P84, CAS: #58698-66-1) hollow fiber membranes spun at different shear rates were used in this paper. The spinning process, analysis method and the performance of these membranes were discussed in detail in a previous paper [26]. Seven dextrans ((C6 H10 O5 )n , CAS: #9004-54-0, Mr : 1500–200,000 Da, from Fluka and Sigma) were used to characterize the rejection coefficient of the hollow fiber membranes. The test solution was approximately 1500 ppm of mixed dextran with different molecular weight in salt-free Milli-Q water. The rejection of different dextrans was tested by a standard laboratory procedure. The hollow fibers were cut into length of 30 cm, and small test modules were made with 8–10 fibers in a glass tube with a diameter of 1 cm. The effective length of the glass module was 25 cm. The ultrafiltration apparatus is shown in Fig. 1. MasterFlex® L/STM peristaltic pump (Model 7554-95, pump head: easy-load® Model 7518-02, Cole-Parmer Instru-

Fig. 1. Ultrafiltration apparatus: (1) feed tank; (2) MasterFlex® L/STM peristaltic pump; (3) pressure gauges; (4) glass membrane module; (5) pressure control valve; (6) flow meter.

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coefficient of hollow fiber membranes is calculated according to C f − Cp (1) R(a) = Cf where Cf and Cp are the concentrations of dextran molecule with diameter a in the feed and permeate solution, which can be represented by the GPC spectrum intensity (I) of dextran molecular. So the rejection coefficient of hollow fiber membranes for different dextran molecules can be calculated. The MWCO of asymmetric hollow fiber membranes is defined as the molecular weight of a solute that has an R value of 0.9. The correlation ˚ and the molecular weight (MW, in between the radius (r, in A) g/mol) of dextran is as follows [19,27]: r = 0.33(MW)0.46

(2)

The solute diameter a (a = 0.2r, in nm) is used in the following paragraphs. In the testing process, the influence of concentration polarization on the rejection of hollow fiber membranes should be eliminated greatly. Fig. 2b shows the influence of the circulating feed flux on the rejection of a out-skinned cellulose acetate hollow fiber membrane of our lab for 11,000 and 100,000 Da dextran moleculars. It is clear that the concentration polarization can be eliminated greatly for out-skinned cellulose acetate hollow fiber membranes when the circulating feed flux is greater than 0.6 L/min. So in this experiment, the feed flux was controlled at 1.1 L/min (feed velocity in empty glass tube: 0.23 m/s; Reynolds number: ∼2500) and the operation pressure was controlled at 0.5 × 105 Pa (0.5 bar), and the concentration polarization can be eliminated greatly. 3. Theoretical 3.1. Theoretical background Fig. 2. (a) The molecular weight distribution of dextran molecules in feed solution. (b) The influence of circulating feed flux on the rejection of cellulose acetate hollow fiber membranes (() 11,000 Da dextran; (䊉) 100,000 Da dextran; pressure: 0.5 × 105 Pa (0.5 bar); pure water flux: 50 × 10−5 L/m2 h Pa (50 L/m2 h bar)).

ment Company, USA) was used to circulate the feed solution, which can provide the constant flux at different feed pressure. As the BTDA-TDI/MDI co-polyimide hollow fiber membranes were outer-skinned, the feed solution was pumped through the shell side of the glass membrane modules. Because only 8–10 fibers were put into the glass tube, the hollow fiber membranes vibrated inside the glass module when the feed solution was pumped to the shell side of the membrane module from the side inlet. The dextran solution in the permeate side was collected in the lumen of hollow fiber membranes. The molecular weight distributions of dextran in the feed and permeate were analyzed by gel permeation chromatography (GPC). Fig. 2a shows the molecular weight distribution of dextran molecules in the feed solution. At each dextran molecule with diameter a, the rejection

In this method, the concentration polarization at the membrane surface and the interference in pore transport between solute molecules of different sizes are not considered. That is to say, this theoretical model can only be applied to analysis the pore size distribution of the asymmetric membranes without concentration polarization. The transport properties for membrane process are assumed to base on the hydrodynamic and hindered transport theory. The flux of a pore with diameter d can be described with the Hagen-Poiseuille equation [28]: j(d) =

πd 4 P 128ηδ

(3)

where η is the viscosity of feed solution through pores, δ the equivalent length of the pore and d is the pore diameter. The total flux is the cumulation of the fluxes in all pores per unit area.
∞ J =N j(d)f (d) dd (4) 0

where N is the total number of pores per unit area and f(d) is the pore size distribution function. The concentration (Cp ) of

J. Ren et al. / Journal of Membrane Science 279 (2006) 558–569

a solute molecule with diameter a in the permeate solution is expressed by Eq. (5): ∞ Cp =

0  j(d)C(d)f (d) dd ∞ 0 j(d)f (d) dd

∞ =

4 0  d C(d)f (d) dd ∞ 4 0 d f (d) dd

(5)

where C(d) is the concentration of a solute molecule with diameter a inside a pore. For a solute with diameter a, the retention coefficient R(d) in a pore of diameter d is defined as: R(d) = 1 −

C(d) Cf

(6)

In this paper, the interaction between solute molecules of different sizes in pore transport are not considered, the retention coefficient R(d) can be described by [19,28]: 

2

R(d) = (1 − (1 − λ)2 ) , R(d) = 1,

λ≤1 λ>1

(7)

where λ = a/d. Substituting for C(d) in Eq. (5) ∞

d 4 (1 − R(d))f (d) dd ∞ 4 0 d f (d) dd   ∞ 4 0  d R(d)f (d) dd = Cf 1 − ∞ 4 0 d f (d) dd

Cp =

Cf

Cp = R(a) = 1 − Cf

(8)

j(d)f (d) d 4 f (d) ff (d) =  ∞ = µ4 0 j(d)f (d) dd  4 d 1 √ = D∗ ln(σ)d 2π    1 ln(d/D∗ ) 2 2 × exp − − 8(ln σ) 2 ln(σ)

(12)

(13)

4 0  d R(d)f (d) dd . ∞ 4 0 d f (d) dd

(9)

are the geometric mean diameter and the geometric standard deviation, respectively. The nth raw moment of the pore size distribution function f(d) is d n f (d) dd

0

   d n−1 1 ln(d/D∗ ) 2 √ = exp − dd 2 ln(σ) ln(σ) 2π 0 

1 = (D∗ )n exp n2 (ln σ)2 2

∞

Dm =



df (d) dd =

0

Da = =

where D* , σ

∞

The mean pore diameter (Dm ), the average pore diameter based on pore area (Da ) and the average pore diameter based on pore flow (Df ) are given by Eqs. (14)–(16), respectively.

∞

The most common form of the two-parameter log-normal distribution function can be written as follows:    1 ln(d/D∗ ) 2 1 √ (10) exp − f (d) = 2 ln(σ) ln(σ)d 2π




((π/4)d 2 )f (d) fa (d) =  ∞ 2 0 ((π/4)d )f (d) dd  2 d 2 f (d) 1 d √ = = µ2 D∗ ln(σ)d 2π    1 ln(d/D∗ ) 2 2 × exp − − 2(ln σ) 2 ln(σ)




3.2. The pore size distribution and rejection coefficient




The pore area distribution function fa (d) and pore flux distribution function ff (d) are given by Eqs. (12) and (13), respectively.

0

A relationship for the overall rejection R(a) of a solute with diameter a can be written as follows [19]:

µn =

561

=




∞

(ln σ)2 = D exp 2

128ηδ jm = π P

4

 4

(14)

d 2 f (d) dd

0


4

∞

(15)

d 4 f (d) dd

0

µ4 = D∗ exp[2(ln σ)2 ]

(16)

where Am and jm are the average pore area and the average pore flux, respectively. By substitution in Eq. (9), the rejection coefficient for a solute with diameter a can be given as Eq. (17) ∞

∞

R(a) = (11)



∗

 µ2 = D∗ exp[(ln σ)2 ]

Df =

4 Am = π

µ1

4 0  d R(d)f (d) dd ∞ 4 0 d f (d) dd

∞ =

1 = exp[−8(ln σ)2 ] (D∗ )4

0


0

∞

d 4 R(d)f (d) dd µ4 d 4 R(d)f (d) dd

(17)

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J. Ren et al. / Journal of Membrane Science 279 (2006) 558–569

By integration of Eq. (17), R(a) can be given by 
a  
a
∞ 1 d3 1 2 4 2 2 4 2 √ exp[−8(ln σ) ] d f (d) dd + (1 − (1 − a/d) ) d f (d) dd = exp[−8(ln σ) ] (D∗ )4 (D∗ )4 0 a 0 ln(σ) 2π       
∞ 3 2 1 ln(d/D∗ ) 2 d (1 − (1 − a/d)2 ) 1 ln(d/D∗ ) 2 √ dd + × exp − dd exp − 2 ln(σ) 2 ln(σ) ln(σ) 2π a 

 

  ∗  a 2 1 1 1 D 2 √ = 1− 1 + erf √ +2 exp[−6(ln σ) ] 1 + erf ln + 4(ln σ)2 2 D∗ a 2 ln σ 2 ln σ   ∗  



   ∗  a 3 D 1 15 D 2 2 2 × ln −2 1 + erf √ exp − (ln σ) + 2(ln σ) ln + (ln σ) D∗ a 2 a 2 ln σ 

 ∗  1  a 4 1 D 2 + exp[−8(ln σ) ] 1 + erf √ ln (18) 2 D∗ a 2 ln σ

R(a) =

4. Result and discussion

The above equation is defined as R(D∗ , σ, a) = q



D∗ ,σ a

 (19)

where D* /a is a dimensionless number, which represents the ratio of geometric mean diameter of ultrafiltration membrane (D* ) to the diameter of the rejected solute molecule (a). In Eq. (19), the rejection coefficient of ultrafiltration membranes was dominated by two independent variables, D* /a and σ. The relationship between DMWCO (the diameter of solute molecule at R = 0.9) and the pore size distribution parameters (D* , σ) can be described as:  q



D∗ DMWCO

,σ

= 0.9

(20)

where DMWCO = 0.066 (MWCO)0.46 (DMWCO : nm; MWCO: Da), the above equation can also be written as: D∗ DMWCO

= h(σ).

(21)

4.1. The relationship of pore size distribution parameters (D* , σ) at different MWCO The distribution parameters (mean diameter (Dm ), average diameter based on pore area (Da ), average diameter based on pore flux (Df ), most probable in pore size distribution (dp ), most probable in pore area distribution (da ), most probable in flux pore distribution (df ) and DMWCO ) of ultrafiltration membranes can be expressed as a function of the geometric mean diameter (D* ) and the geometric standard deviation (σ), which are shown in Table 1. The relationship between these parameters is as follows: dp < D∗ < Dm < da = Da < Df < df ,

σ>1

DMWCO can vary from the smallest to the largest among these parameters when the geometric standard deviation (σ) tends towards one or great value. Moreover, DMWCO can also be equal to any of the distribution parameters mentioned above at a specific geometric standard deviation (σ). According to Eq. (18), Fig. 3A and B shows the influence of pore size distribution parameters (D* , σ) on the rejection

Fig. 3. The influence of pore size distribution parameters (D* , σ) on the rejection curves for dextran molecules at 10,000 and 100,000 Da. ((A) MW = 10,000 Da, a = 4.57 nm; (B) MW = 100,000 Da, a = 13.17 nm; D* : nm).

J. Ren et al. / Journal of Membrane Science 279 (2006) 558–569 Table 1 The distribution parameters and their relations to geometric mean diameter (D* ) and geometric standard deviation (σ) Mean diameter, Dm Average diameter by pore area, Da Average diameter by pore flux, Df Most probable in pore size distribution, dp Most probable in pore area distribution, da Most probable in pore flux distribution, df Diameter of solute molecular at R = 0.9

D* exp[(ln σ)2 /2] D* exp[(ln σ)2 ] D* exp[2(ln σ)2 ] D* exp[−(ln σ)2 ] D* exp[(ln σ)2 ] D* exp[3(ln σ)2 ] DMWCO

curves for dextran molecules of 10,000 Da (a = 4.57 nm) and 100,000 Da (a = 13.17 nm). As D* and σ increase the rejection coefficient of ultrafiltration membranes clearly decreases. It is also clear that the contour curves at the same rejection in Fig. 3A and B are different. For ultrafiltration membranes with different MWCO (10,000, 30,000, 50,000, 100,000 Da), the contour curves (R = 0.9; a = DMWCO ) are shown in Fig. 4. These contour curves reflect the inherent relationship of the pore size distribution parameters (D* , σ) of the ultrafiltration membranes at different MWCO. From this figure, the pore size distribution can be divided into three zones according to the geometric standard deviation (σ): Zone I: Long tail-effect of big pores (σ > 1.55). Zone II: Linearization of pore size distribution parameters (D* , σ) (1.15 ≤ σ ≤ 1.55). Zone III: Sieving effect (1 ≤ σ < 1.15).

parameters (D* , σ) will result in a great change of MWCO. At the same MWCO, as the geometric mean diameter (D* ) decreases, the geometric standard deviation (σ) increases greatly, which results in a strong long tail of big pores. Meanwhile, in Zone I, the DMWCO of ultrafiltration membranes is usually more than the geometric mean diameter (D* ). For an ultrafiltration membrane with 50,000 MWCO (D50,000 = 9.57 nm), the geometric mean diameter (D* ) of the membrane in Zone I is lower than 5 nm. Obviously, because of the long tail-effect of big pores, the diameter of the solute molecule rejected (at R = 0.9) is 9.57 nm. 4.1.2. Zone II: linearization of pore size distribution parameters (D* , σ) (1.15 ≤ σ ≤ 1.55) When 1.15 ≤ σ ≤ 1.55, the geometric standard deviation (σ) is almost a linear function of the geometric mean diameter (D* ) for the ultrafiltration membranes with the same MWCO. From the Zone II in Fig. 4, the distribution of the contour curves (R = 0.9) for different MWCO are relatively loose. The MWCO of ultrafiltration membranes is not heavily influenced by a slight change of pore size distribution parameters (D* , σ). Extrapolating the lines for different MWCO cuts the Y-axis at the same σ 0 of 1.8054. Using the linear regression, the relationship between the geometric standard deviation (σ) and the geometric mean diameter (D* ) at different MWCO can be described as: σ = 1.8054 − 0.5608 1.15 ≤ σ ≤ 1.55,

4.1.1. Zone I: long tail-effect of big pores (σ > 1.55) When σ > 1.55 the pore size distribution is wide and the tail dragged is very serious. The big pores strongly influence the MWCO of ultrafiltration membranes. From Fig. 4, the distribution of the contour curves (R = 0.9) at different MWCO in Zone I is relatively tight. A slight variance of pore size distribution

Fig. 4. The inherent relationship of pore size distribution parameters (D* , σ) for ultrafiltration membranes with different MWCO at the same rejection coefficient (R = 0.9).

563

D∗ DMWCO

,

r2 = 0.999

(22)

For a quality polymeric ultrafiltration membrane, its pore size distribution parameters (D* , σ) should be in Zone II. At the same MWCO, ultrafiltration membranes with a pore size distribution in Zone II should have a relatively higher flux than those with a pore size distribution in Zone I due to their bigger pore size. Meanwhile, the geometric mean diameter (D* ) can be higher or lower than the DMWCO of ultrafiltration membranes depending on the geometric standard deviation (σ). For example, the geometric mean diameter (D* ) of a membrane with 50,000 MWCO (D50,000 = 9.57 nm) in Zone II changes from almost 5 to 12 nm. In Zone II, the long tail-effect and sieving effect influence the performance of ultrafiltration membranes simultaneously. 4.1.3. Zone III: sieving effect (1 ≤ σ < 1.15) When 1 ≤ σ < 1.15 the pore size distribution of ultrafiltration membranes is very narrow, and the MWCO of ultrafiltration membranes is mainly controlled by the sieving effect. The DMWCO of ultrafiltration membranes is lower than the geometric mean diameter (D* ) because of the steric interactions in the transport process of solute molecule through the pores. The ultrafiltration membranes with a pore size distribution in this zone usually have very good performance and rejection. However, it is difficult to fabricate polymeric ultrafiltration membranes with the pore size distribution in Zone III by the phase inversion processes.

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J. Ren et al. / Journal of Membrane Science 279 (2006) 558–569

Fig. 5. The influence of pore size distribution parameters (D* , σ) on the MWCO of ultrafiltration membranes.

4.2. The relationship of pore size distribution parameters (D* , σ) and solute diameter (a) at different rejection coefficients The MWCO of ultrafiltration membranes is a very important parameter for the evaluation of membrane performance. The influence of pore size distribution parameters (D* , σ) on the MWCO of ultrafiltration membranes is clearly demonstrated in Fig. 5. With an increase in D* or σ, the MWCO of ultrafiltration membranes increases greatly. From Eq. (21), it is obvious that the dimensionless number (D* /DMWCO ) can also be described as a function of the geometric standard deviation (σ). Hence, the curved surface in Fig. 5 calculated according to Eq. (18) can be transferred to the curved line in Fig. 6. The three zones defined in Section 4.1 can still be observed clearly. Moreover, Eq. (22) obtained by linear regression in Section 4.1 is still in accordance

Fig. 7. The relationship between the geometric standard deviation σ and (D* /a) at different rejection coefficients (() calculated by equation by Eq. (18) with R = 0.2, 0.5, 0.7, 0.95, respectively; (—) regression by the Extreme model).

with the linear relationship of pore size distribution parameters (D* , σ) (1.15 ≤ σ ≤ 1.55), as shown in Fig. 6. In order to describe the relationship between D* /DMWCO and σ in the full range (σ ∈ [1, ∞]), a new function, the Extreme model, is first introduced in this paper. The parameters of this model were obtained by the regression and listed in Eq. (23). D∗ DMWCO

(−z) −z+1)

= A e(−e

A = 1.296,

w = 0.299,

σ−1 w

(σ ∈ [1, ∞]),

r 2 = 1.0

(23)

Obviously, the dimensionless number (D* /DMWCO ) can be described by Extreme Model successfully. The DMWCO of ultrafiltration membranes changes at different geometric standard deviation (σ), which can be equal to any of the pore size distribution parameters listed in Table 1 at a specific geometric standard deviation (σ). Usually it is also necessary to study the relationship between the pore size distribution parameters (D* , σ) and the solute diameter (a) at any other rejection coefficient R. Analogically, the relationship between D* /a and σ at different rejection coefficients were calculated according to Eq. (18) and shown in Fig. 7, which was also regressed with the Extreme model (see Eq. (24)). The regression parameters in the Extreme model are listed in Table 2. D∗ (−z) = A e(−e −z+1) , a

Fig. 6. The relationship between the geometric standard deviation σ and (D* /DMWCO ).

z=

,

z=

σ−1 w

(σ ∈ [1, ∞])

(24)

This equation involves some important information of the ultrafiltration membrane such as pore size distribution parameters (D* , σ) and the diameter of solute separated (a) at specific rejection coefficients, which is a simple form of Eq. (18). From Table 2, the regression parameters (A, w) in the Extreme model changed with the rejection coefficient (R). In order to obtain the

J. Ren et al. / Journal of Membrane Science 279 (2006) 558–569 Table 2 The regression parameters (A, w) in the Extreme model at different rejection coefficients R

A

w

Related coefficient

0.15 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.95

4.604 3.901 3.061 2.546 2.199 1.915 1.687 1.487 1.296 1.189

0.400 0.395 0.385 0.375 0.366 0.354 0.341 0.324 0.299 0.277

1 1 0.9999 0.9998 0.9998 0.9998 0.9998 0.9999 1 0.9998

D∗ a

= A e(−e

(−z) −z+1)

, z=

σ−1 w

model parameters (A1 , w1 ) and (A2 , w2 ) can be calculated using Eqs. (25) and (26). The pore size distribution parameters (D* , σ) can be expressed as follows: Curve 1 : D∗ = g1 (σ) = a1 A1 e(−e

(−z) −z+1)

Curve 2 : D∗ = g2 (σ) = a2 A2 e(−e

(−z) −z+1)

(25)

w = 0.4030 + 0.0425R − 0.5596R2 + 0.9161R3 r2 = 1.

(26)

4.3. The theoretical prediction of pore size distribution and MWCO by two points and experimental verification by hollow fiber membranes spun at different shear rates The rejection coefficient of ultrafiltration membranes for solutes with different molecular weights can usually be obtained experimentally. From Eq. (24), the geometric mean diameter (D* ) can be expressed as a function of the solution diameter (a) and geometric standard deviation (σ): D∗ = g(σ) = aA e(−e

(−z) −z+1)

,

z=

σ−1 w

σ−1 w1

,

z=

σ−1 w2

The pore size distribution parameters (D* , σ) can be obtained at g1 (σ) = g2 (σ). Finally, the DMWCO of ultrafiltration membranes can be obtained by Eq. (22) (σ ∈ [1.15, 1.55]) or Eq. (23) (σ ∈ [1, ∞]). In spinning process of hollow fiber membranes, the shear rate of dope solution insider the spinneret strongly influenced the performance and morphology of hollow fiber membranes [26,29–32]. In this paper, the results of BTDA-TDI/MDI copolyimide hollow fiber membranes spun at different shear rates were used [26]. For the hollow fiber membranes spun at different shear rates, the rejection coefficients of two selected solutes [(a1 , R1 ), (a2 , R2 )] and their corresponding parameters [(A1 , w1 ), (A2 , w2 )] in the Extreme model are listed in Table 3. Fig. 8A and B demonstrates the prediction of the pore size distribution parameters (D* , σ) at g1 (σ) = g2 (σ) for hollow fiber membranes spun at 1640 s−1 (Batch I-2) and 8200 s−1 (Batch I-5), respectively. The predicted pore size distribution parameters (D* , σ), predicted and experimental MWCO are also listed in Table 3. As can be seen, the MWCO predicted is in consistent with the experimental results for different hollow fiber membranes. Fig. 9 shows the experimental and predicted rejection curves based on two points of hollow fiber membranes spun at 1640, 3280, 4920 and 8200 s−1 , respectively. Obviously, the experimental rejection curves fit the theoretical prediction very well for a wide range of molecular weights. This proves that the prediction of pore size distribution and MWCO by two points is feasible for ultrafiltration membranes. The other pore size distribution parameters of hollow fiber membranes spun at different shear rates are listed in Table 4. As seen in Table 4, the shear rate in the spinning process dominates the pore size distribution parameters (D* , σ). With an increase in shear rate, the geometric standard deviation (σ) is strongly suppressed, but the geometric mean diameter (D* ) of the hol-

A = 7.613 − 27.251R + 54.228R2 − 52.299R3

− 0.5467R4 ,

z=

(29)

(σ ∈ [1, ∞]).

r2 = 1

,

(28)

parameters (A, w) of Extreme model at different rejection coefficient easily, polynomial regression is used and the regression parameters (A, w) can be expressed as a function of rejection coefficient (R)

+ 18.883R4 ,

565

(σ ∈ [1, ∞]) (27)

The Extreme model parameters (A, w) in Eq. (27) can be obtained from Eqs. (25) and (26). Using two solutes with different diameter a1 and a2 , two different rejection coefficients R1 and R2 can be obtained experimentally. And two sets of the Extreme

Table 3 The prediction of the pore size distribution parameters (D* , σ) and MWCO by two points for hollow fiber membranes spun at different shear rates Shear rate (s−1 )

Batch I-2 Batch I-3 Batch I-4 Batch I-5

1640 3280 4920 8200

First point

Second point

Solute diameter, a1 (nm)

Rejection, R1

6.773 5.675 4.242 4.777

0.558 0.549 0.552 0.742

Extreme model parameters A1

w1

2.036 2.058 2.051 1.608

0.359 0.360 0.359 0.335

Solute diameter, a2 (nm)

Rejection, R2

16.764 11.623 8.750 6.773

0.950 0.911 0.954 0.940

Pore size distribution parameters Extreme model parameters A2

w2

1.205 1.257 1.202 1.217

0.278 0.293 0.277 0.283

MWCO (Da)

σ

D* (nm)

Prediction Experiment

1.570 1.466 1.343 1.237

6.245 6.617 6.198 6.287

111426 67861 30382 19451

110000 69000 26000 19000

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J. Ren et al. / Journal of Membrane Science 279 (2006) 558–569

Table 4 The distribution parameters of hollow fiber membranes spun at different shear rates

Batch I-2 Batch I-3 Batch I-4 Batch I-5 a

Shear rate (s−1 )

Geometric mean diameter, D* (nm)

Geometric Mean standard diameter, deviation, Dm (nm) σ

Average diameter Average diameter by pore area, Da by pore flux, Df (nm) (nm)

Most probable in pore size distribution, dp (nm)

Most probable in pore area distribution, da (nm)

Most probable in pore flux distribution, df (nm)

Molecular weight cut-offa , DMWCO

1640 3280 4920 8200

6.245 6.617 6.198 6.287

1.570 1.466 1.343 1.237

7.654 7.660 6.761 6.578

5.095 5.716 5.682 6.009

7.654 7.660 6.761 6.578

11.498 10.264 8.046 7.201

13.759 11.102 7.086 6.134

6.914 7.119 6.473 6.431

9.381 8.867 7.376 6.882

Test by GPC.

low fiber membranes is almost unaffected. It can be concluded that with an increase in shear rate, the decrease in the MWCO of hollow fiber membranes is mainly due to the suppression of the geometric standard deviation (σ), and not the geometric mean diameter (D* ). Figs. 10 and 11 show the distribution functions f(d), fa (d), ff (d) and distribution parameters for two hollow

fiber membranes spun at 1640 and 8200 s−1 . At a low shear rate (Fig. 10, 1640 s−1 ), the distribution functions become wider and also shift towards the right in the sequence of f(d), fa (d), ff (d). Due to the high geometric standard deviation (σ = 1.570), the DMWCO is greater than any other distribution parameters. At a high shear rate (Fig. 11, 8200 s−1 ), the distribution functions

Fig. 9. The comparison between the experimental [26] and predicted rejection curves for hollow fiber membranes spun at different shear rates ((
) 1640 s−1 ; () 3280 s−1 ; () 4920 s−1 ; (䊉) 8200 s−1 ; (—) calculated).

Fig. 8. The prediction of the pore size distribution parameters (D* , σ) by two different molecular rejection coefficients for hollow fiber membranes spun at (A) 1640 s−1 (Batch I-2) and (B) 8200 s−1 (Batch I-5).

Fig. 10. The distribution functions f(d), fa (d), ff (d) and some distribution parameters for hollow fiber membranes spun at 1640 s−1 .

J. Ren et al. / Journal of Membrane Science 279 (2006) 558–569

Fig. 11. The distribution functions f(d), fa (d), ff (d) and some distribution parameters for hollow fiber membranes spun at 8200 s−1 .

become very sharp and the other distribution parameters are also very tightly distributed. The DMWCO is less than the mean diameter (Dm ) and geometric mean diameter (D* ), which results in good performance of hollow fiber membranes. The influence of shear rate on the pore size distribution f(d) and flux distribution ff (d) is shown in Figs. 12 and 13, respectively. In Fig. 12 the increase in shear rate suppresses the formation of big pores and decreases the long tail of pore size distribution by big pores. Usually, the long tail of pore size distribution by big pores strongly influences the MWCO of hollow fiber membranes. At a low shear rate (1640 s−1 ), the long tail of f(d) by big pores dominates the flux distribution (see Fig. 13), which results in poor performance of the ultrafiltration membranes (Table 3, Fig. 10; MWCO = 110,000 Da). With an increase in shear rate (especially 8200 s−1 ), the pore size distribution function f(d) for big pores declines greatly and the flux distribution also becomes tighter, resulting in good performance (Table 3, Fig. 11; MWCO = 19,000 Da). Compared to the pore size distribution function f(d) (see Fig. 12), the shear

Fig. 12. The influence of shear rate on the pore size distribution of hollow fiber membranes.

567

Fig. 13. The influence of shear rate on the flux distribution of hollow fiber membranes.

rate has a large influence on the flux distribution function ff (d) (see Fig. 13). The increase in shear rate not only sharpens the peak of the flux distribution function, but also shifts it to the left, resulting in a lower MWCO of hollow fiber membranes. The influence of shear rate on the pore size distribution parameters (D* , σ) is shown in Fig. 14. At a low shear rate, the pore size distribution parameters (D* , σ) approach the Zone I [long tail-effect of big pores (σ > 1.55)], which leads to poor performance for ultrafiltration membranes. With an increase in shear rate, the pore size distribution parameters (D* , σ) mainly lie in the Zone II [linearization of pore size distribution parameters (D* , σ) (1.15 ≤ σ ≤ 1.55)], and tend towards Zone III [sieving effect (1 ≤ σ < 1.15)]. The pore size distribution parameters (D* , σ) in this zone are suitable for good polymeric ultrafiltration membranes. So it can be concluded that the shear rate in the spinning of hollow fiber membranes plays an important role in their performance.

Fig. 14. The pore size distribution parameters (D* , σ) for hollow fiber membranes spun at different shear rate ((
) 1640 s−1 ; () 3280 s−1 ; () 4920 s−1 ; (䊉) 8200 s−1 ).

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J. Ren et al. / Journal of Membrane Science 279 (2006) 558–569

5. Conclusion The log-normal distribution, Poiseuille flow and steric interaction between solute molecules and pores are used to study the transport process, rejection curves and MWCO of ultrafiltration membranes, and some conclusions can be made as follows: 1. The pore size distribution of ultrafiltration membranes can be divided into three zones mainly due to the geometric standard deviation (σ): Zone I, long tail-effect of big pores (σ > 1.55); Zone II, linearization of pore size distribution parameters (D* , σ) (1.15 ≤ σ ≤ 1.55); Zone III, sieving effect (1 ≤ σ < 1.15). 2. The relationship between D* /DMWCO and σ can be described with the Extreme model: D∗ DMWCO z=

(−z) −z+1)

= A e(−e

σ−1 w

(σ ∈ [1, ∞]),

, A = 1.296,

w = 0.299

For different rejection coefficients, (D* /a, σ) can also be described with the Extreme model successfully. 3. The theoretical prediction of the pore size distribution and MWCO of ultrafiltration membranes by two points is first developed and verified with hollow fiber membranes spun at different shear rates. 4. The shear rate in the spinning process dominates the pore size distribution parameters (D* , σ). With an increase in shear rate, the geometric standard deviation (σ) is strongly suppressed, resulting in a low MWCO of hollow fiber membranes. The shear rate in the spinning of hollow fiber membranes plays an important role in their performances.

Nomenclature a solute diameter A, A1 , A2 parameters for the Extreme model Am average pore area C(d) solute concentration in a pore with diameter d Cf solute concentration in feed solution solute concentration in permeate solution Cp d pore diameter da most probable in area distribution df most probable in flux distribution dp most probable in pore size distribution Da average diameter based on pore area Df average diameter based on pore flux Dm mean pore diameter D* geometric mean diameter DMWCO the solute diameter at R = 0.9 f(d) pore size distribution function fa (d) area distribution function ff (d) flux distribution function

g1 (σ), g2 (σ) function of σ h(σ) function of σ j(d) the flux through the membrane open pores with diameter of d jm average pore flux J total flux per unit area MW molecular weight (g/mol) MWCO molecular weight cut-off P pressure difference between feed and permeate side q(D* /a, σ) a function of D* /a and σ r solute radius r2 correlation coefficient R, R1 , R2 rejection coefficient R(a), R(D* , σ, a) rejection coefficient of a solute with diameter a for a membrane R(d) rejection coefficient of a solute in a pore with diameter a w, w1 , w2 parameters for the Extreme model z a variable of the Extreme model Greek letters δ the equivalent length of a pore η the viscosity of the feed solution through a pore λ the ratio of solute diameter to pore diameter (=a/d) µn the nth moment σ the geometric standard deviation σ0 the intercept in Y-axis of Eq. (22)

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