Modeling and Simulation of Stirred Dead End Ultrafiltration Process

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Ind. Eng. Chem. Res. 2009, 48, 4428–4439

SEPARATIONS Modeling and Simulation of Stirred Dead End Ultrafiltration Process Using the Aspen Engineering Suite Sabuj Das, Prabirkumar Saha,* and G. Pugazhenthi Department of Chemical Engineering, Indian Institute of Technology, Guwahati, Assam 781 039, India

Simulation of pressure-driven membrane process has been carried out considering two different types of solute (silica and dextran) and membrane (partially permeable and totally retentive) in a stirred cell using the Aspen Engineering Suite. One solute, silica, exerts negligible osmotic pressure but the other, dextran, offers adequate osmotic pressure. Silica is very susceptible to form a gel layer while dextran has no such tendency. Two types of membrane are considered for this work. One membrane completely separates solute from the solvent (totally retentive membrane) and the other does it partially (partially permeable membrane). The second type of membrane is regarded as highly compact granulated particle layer which has a different range of porosity. So the diffusivity within the membrane layer is referred to as hindered back diffusion. Osmotic pressure model and gel polarization model have been considered for batch and continuous mode operation. Silica suspension has been simulated with totally retentive membrane and dextran solution is simulated with partially retentive membrane in batch and continuous mode, respectively. The effect of applied pressure, stirring speed, and initial feed concentration on permeate flux, membrane surface concentration, permeate concentration, and true and observed rejection have been studied. 1. Introduction Pressure-driven membrane processes are being increasingly incorporated in the process industries for the separation of valuable chemical and biological compounds from mixtures. Of these processes, reverse osmosis (RO) and ultrafiltration (UF) have become increasingly important during the past decades due to their mild and energetically favorable separation and concentration technique, which have been commercialized in pharmaceutical, chemical, and food industries. Among the main problems hampering its widespread use, concentration polarization as well as fouling may be mentioned, because both phenomena give rise to flux decline. The flux decrease could also be due to an increase in osmotic pressure, formation of gel layer, solute adsorption on membrane, and pore plugging. In order to minimize the concentration polarization, much attention has been paid to rather mechanical aspects, such as cross flow, back pulsing, and promotion of turbulence, e.g., by gas purging.1,2 Moreover, any attempt to increase the flux by increasing pressure causes severe increase in concentration polarization, thus giving little increase in flux compared to increase in pressure. Eventually, the flux does not increase beyond a certain point (called limiting flux) even if the pressure is increased further. Types of solute and membrane have great impact for gel formation and concentration polarization. Flux decline rate also depends on the flow direction and operating condition. In the dead end filtration, the feed flow is directed perpendicular to the membrane, whereas in the crossflow filtration, feed flow is directed parallel or tangential to the membrane. The former one is most often used for small volume laboratory applications and the later one is used for large volume process applications. Dead end filtration may be carried with stirred and unstirred cells. Nicolas et al.3 showed a comparison study of rejection in a stirred and an unstirred batch cell for * To whom correspondence should be addressed. E-mail: p.saha@ iitg.ac.in. Phone/Fax: +91.361.2582257.

small noncharged molecules in an aqueous solution using reverse osmosis and ultrafiltration membranes. The results revealed that in a stirred cell with high stirring velocity, the concentration near the membrane was close to the bulk concentration, resulting in improved rejection compared to the unstirred batch cell results. Several studies have reported the experimental results as well as modeling of limiting flux for the separation of macro solute and colloids by UF that provided strong support to the gel polarization model.4-6 Karode7 simulated the unsteady-state permeate flux response for three types of solutes, such as (i) solutes which exert an osmotic pressure but do not form a gel; (ii) solutes which do not exert an osmotic pressure but form a gel, and (iii) solutes which exert an osmotic pressure and also form a gel. He showed that the concentration at the surface of a membrane was a function of operating parameters and it was possible to predict the membrane wall concentration as a function of time. In addition, a step change in transmembrane pressure showed that a solute, which exerts osmotic pressure but does not form a gel layer, exhibited a gradual flux decline over time. Experimental verification of this work was presented by Zaidi and Kumar.8-10 They experimentally studied unsteady-state flux response to a step change in transmembrane pressure for dextran8 and polyethylene glycol (PEG) solution9 and reported that polarization resistance depended on applied pressures and initial bulk feed concentrations, whereas the thickness of polarization layer affected by only feed concentration. They have also investigated the characteristics of a deposited layer formed in dead end UF of silica after reaching steady-state flux.10 In addition, the effects of operating conditions such as applied pressure and bulk concentration on specific resistance, thickness, porosity, and silica concentration of gel were reported. Zaidi and Kumar11 have also experimentally investigated the influence of ethanol concentration on flux in ultrafiltration of PEG and dextran with a solvent resistance polymeric membrane. They observed that

10.1021/ie801293d CCC: $40.75  2009 American Chemical Society Published on Web 04/09/2009

Ind. Eng. Chem. Res., Vol. 48, No. 9, 2009 4429

both steady-state flux and gel formation were significantly influenced in the UF of both PEG and dextran from blended solvent (ethanol-water). The unsteady-state flux behavior for silica and dextran in an ultrafiltration cell has been studied experimentally by van Oers et al.12 During the filtration of dextran, only a polarization layer was built up and the osmotic pressure model provided a good description of the flux for the experiments with dextran. For silica suspension also, a gel layer formation occurred and was well predicted by the gel polarization model. An unsteady-state model was developed to predict the feed surface and permeate concentration for ultrafiltration of macromolecules using the diffusion coefficient of the solution through the membrane.13 This model was proficient in predicting the feed surface concentration once the operating conditions were fixed and did not make an implicit assumption of constant feed surface concentration. It was further shown that, for a certain range of system parameters, a negative rejection (based on the feed concentration) was possible at steady state. Nakao et al.14 reported that no gel layer was found during the ultrafiltration of dextran even with a very high solute concentration of 400 kg/m3. The above reason was also further confirmed by Choe et al.15 They obtained the permeate flux of dextran for totally retentive membranes in a stirred UF cell as a function of time for sudden variations of transmembrane pressures and concluded that only the polarization layer was formed. The osmotic pressure model was well fitted with the dextran flux data. They also studied the flux decline of dextran in UF and concluded that the change in the driving force could be explained in terms of osmotic pressure at the membrane surface. Bhattacharjee and Bhattacharya16,17 have developed a model to predict the flux using a generalized formulation which takes into account the unsteady-state behavior during the initial stages of continuous stirred ultrafiltration of PEG. They17 showed the effect of operating variables on limiting flux phenomena and the membrane parameters, i.e., reflection coefficient and solute permeability, were determined to characterize the membrane. A correlation was developed to relate polarized layer resistance with concentration polarization, osmotic pressure to applied pressure ratio and Reynolds number. Bhattacharjee and Datta18 used the experimental data reported in the literature16,17 to integrate the governing partial differential equation and calculate the back transport coefficient by a leastsquares fit. The model predicted that the flux decline during ultrafiltration of PEG in a continuous stirred cell using a cellulose acetate membrane occurred mainly due to the resistance offered by solute molecules during their back transport to the bulk. The effects of osmotic pressure and cake/gel formation were assumed to be negligible in this study. A threeparameter model based on unsteady-state mass transfer was developed for the prediction of flux and rejection of solute during ultrafiltration in an unstirred batch cell.19 The main feature of this model was that it allowed variation of solute diffusivity with concentration in the boundary layer whose effect was more pronounced in the case of unstirred batch cell. Bhattacharjee et al.20 utilized Kedem-Katchalsky (KK) and Spiegler-Kedem (SK) models for the parameter estimation like solute permeability and reflection coefficient for ultrafiltration of black liquor by using an asymmetric membrane in a stirred batch cell, which was modified to work on a continuous mode. The variations of parameters with other process variable like bulk concentration, pressure difference, and stirrer speed were also established. Shukla and Kumar21-24 have modeled the separation of aqueous solution of FeCl3, AlCl3, and chromic

Figure 1. Schematic representation of a concentration polarization layer over gel layer and mass balance over the concentration polarization layer.

acid by zeolite-clay composite membrane using a twodimensional space charge model. They reported a new scheme for the solution of space-charge model which reduces the computational time considerably. An unsteady-state mass transfer model was developed taking the effect of reversible pore plugging25 by the diffusing solute molecules into account. The osmotic pressure model has been used to predict the membrane surface concentration. The rate of available fractional area of the membrane that blocked at any time due to pore plugging phenomenon was assumed to be the function of dimensionless membrane surface concentration existing at that time. In this article, we have modeled the effect of various operating parameters on the performance of membrane for separation of dextran and silica suspension using a process engineering software, viz., Aspen Engineering Suite. The advantage of this software is manifold. First, this has a dedicated user interface for both steady-state and dynamic process simulation. Unlike high-level programming languages such as C++ or Fortran where the user needs to write rigorous codes, Aspen software has inbuilt object modules that can handle codes for ODE/PDE solver, optimization, real-time graphical output, etc. Second, the software has a rich library of commonly used unit operations such as distillation column, reactors, heat exchanger, etc., which are menu-driven and can easily be integrated with various process operations. And third and most importantly, Aspen has a huge database of physical as well as thermodynamic properties of various chemical compounds which could be called in while simulation progresses as and when required. In other words, even a minute change of physicochemical property of a target chemical can be predicted and used in the simulation when process condition such as temperature, pressure, or concentration changes. This is contrary to the poor assumption that is usually considered by other custom-made simulation code that takes constant density/viscosity etc. while performing real-time simulation. Two different types of membrane are considered for batch and continuous mode of operation in a stirred dead end UF cell. In this analysis, variations of permeate flux, membrane surface concentration, permeate concentration, observed and true rejection with the transmembrane pressure, stirrer speed, and initial feed concentration have been studied.

4430 Ind. Eng. Chem. Res., Vol. 48, No. 9, 2009

2. Theory Both the models discussed above, viz., gel polarization model and osmotic pressure model, incorporate the phenomenon of concentration polarization (see Figure 1).7 Based on the film theory, the formation of a polarization layer can be described with the following partial differential equation (PDE), assuming flux J(t), and diffusivity D, as independent of concentration c of the macromolecular solutes ∂c ∂2c ∂c ) -J(t) + D 2 ∂t ∂x ∂x

(1)

Within the membrane (δpol < x < δpol + δmem), the concentration is governed by13 ∂c ∂2c ) Dm 2 ∂t ∂x

(2)

The gel layer and membrane resistance may be considered as two resistances in series, and the permeate flux is then described by Darcy’s law: J)

∆P - ∆π 1 dVP ) Am dt µ(Rm + Rg)

(3)

where ∆π ) π(cm) - π(cp)

(4)

The gel-polarization model assumes that the concentration at the membrane surface cannot exceed the gel concentration Cg.12 Initially, polarization layer forms when the membrane surface concentration reaches the gel concentration. The gel layer thickness increases with time and reduces its hydraulic permeability. Figure 1 shows the schematic representation of concentration polarization layer over the gel layer. For totally retentive membrane (where cp ) 0), the rate of growth of the gel layer thickness can be calculated by a mass balance as follows7

( )

∂δg D ∂c )J∂t Cg ∂x |x)δpol

(5)

The hydraulic resistance of the gel layer of particles can be calculated using the Kozeny-Carmen equation. Rg ) 180

(1 - n)2 δg dp2n3

(6)

3. Model In this work, two different types of solute separation are considered such as (i) solute which has negligible osmotic pressure in the solution but has a great tendency to form gel layer (silica suspension) and (ii) solute which has observable osmotic pressure and no tendency to form gel layer (dextran) using two different categories of membrane, viz., (i) totally retentive membrane where solutes cannot pass through the membrane and (ii) partially retentive membrane where solute may pass with solvent through the membrane. The following two cases have been considered here for the performance study using stirred dead end UF setup and is shown in Figure 2. Case I: Separation performance of dextran solution using partially retentive membrane in the continuous mode of operation (Figure 2a). Case II: Separation performance of silica suspension using totally retentive membrane in the batch mode operation (Figure 2b).

Figure 2. Schematic representation of continuous (a) and batch (b) setup.

Both the cases, membranes are considered as black box, i.e., membrane parameters like permeability, reflection coefficient, and charge density are not judged. In partially retentive membrane, diffusivity within the membrane layer is referred to as hindered back diffusion. To simplify the modeling, the membrane layer is represented as granulated layer characterized by equivalent diffusivity. The same assumption was used by Agashichev26 and Boudreau27 for calculating effective diffusion coefficient within the granulated cake layer. The diffusivity of solute within the membrane can be calculated as ε D (7) 1 - ln ε2 In this simulation study, we have considered a stirred dead end UF cell with a membrane area of Am and initial concentration and volume of the feed are cf0 and Vf0, respectively. If the mass arrested into the polarization layer is neglected, then for a totally retentive membrane (no permeation of solute through membrane), the change of bulk volume (Vf) and concentration (cf) with time in the UF cell for batch mode operation (Figure 2b) can be written as7 Dm )

dVf ) -JAm dt cf dcf ) JAm dt Vf

(8) (9)

The mass transfer coefficient for a stirred batch cell is calculated by the following empirical equation:16,18,20

( Dr )( Dν ) ( ωrν )

k ) 0.0443

0.33

2 0.8

(10)


It is also assumed that the time taken for formation of concentration polarization layer is negligible. Polarization layer thickness is calculated by film theory. Many researchers have reported that there is no significant influence of molecular weight of dextran on osmotic pressure.8,9,12 So in this report, osmotic pressure of dextran is calculated using the virial expansion π ) A1c + A2c2 + A3c3

(11)

The initial and boundary conditions for solving eqs 1 and 2 for both the cases are given below. Case I: t ) 0;

c ) cf0

0 e x e δpol ;

t > 0;

x ) 0;

c ) cf ;

and x ) δpol ;

( ∂x∂c ) ∂c D ( ) ∂x

Jc ) D

x)δpol

m

+ x)δpol

Within the membrane, the PDE eq 2 is solved with the following boundary condition:13 δpol < x < δpol + δmem ;

t ) 0;

t > 0;

x ) δpol ;

c)0 ∂c ∂c Jc ) D - Dm ∂x x)δpol ∂x

( )

x ) δpol + δmem ;

( ) ∂c Jc ) -D ( ) ∂x

+ x)δpol

m

x)δpol+mem

Case II: t ) 0; t > 0;

0 e x e δpol ; x ) 0;

c ) cf0

c ) cf(t);

and x ) δpol ;

∂c ∂x

( )

Jc ) D

x)δpol

4. Simulation All the model equations discussed so far are either ordinary differential equations (ODE) or PDE and most of them are nonlinear. Simultaneous solutions of these equations are computationally extensive and can only be done through appropriate numerical techniques. The entire task is gigantic and computationally extensive. Aspen Engineering Suite has been used in this work to overcome the above difficulties. The suite also provides an extensive database for physical and thermodynamic property library, viz., DIPPR, that can be directly linked in while performing the simulation. It has readily available GUI, various types PDE, ODE, nonlinear or linear solvers, or optimization tools. Aspen Engineering Suite is a chemical process simulation software that provides a complete library of models for a number of commonly used unit operations in the chemical processes such as distillation, absorption, extraction, heat exchanger, reactor, etc. Process industries that use Aspen Engineering Suite include petroleum, petrochemical, gas processing, polymer, mineral processing, and so on. Unfortunately, the membrane separation units have not so far been included in the Aspen Engineering Suite as a ready model. In the present work, Aspen Custom Modeler (ACM), version 20.0, is used for the numerical solution of the model equations. All the membrane model equations have been coded in Aspen Custom Modeler (ACM) and for getting property data (which are pressure, temperature, and concentration dependent) are called within the code by using Aspen Property file. For the case I (Figure 2a), both the concentration polarization layer and membrane are divided into 100 parts, while in the case II (Figure 2b), the concentration polarization layer is divided into 100 parts. These PDE are solved with backward finite

Table 1. Properties of Membrane, Solute, and Operating Conditions of the Dead-End UF Cell7 Membrane Parameter membrane radius membrane area membrane porosity thickness of the membrane hydraulic resistance

r ) 0.06 m Am ) πr2 ) 0.0137 m2 ε ) 0-35% ∂mem ) 10-6 m Rm ) 1013 m-1

Cell Parameter transmembrane pressure ∆P ) 1-12 bar operating temperature T ) 298.15 K stirring speed ω ) 4.8-10000 rpm Solute and Solvent Parameter 1. dextran molecular wt water solution initial concentration initial volume virial coefficients 2. silica viscosity water solution initial concentration initial volume particle diameter particle density solution density diffusivity molecular wt gel porosity gel concentration

73500 Da (av dia )11.4 nm) cf0 ) 28 kg/ m3 Vf0 ) 0.02 m3 A1 ) 37.5 A2 ) 0.752 A3 ) 76.4 × 10-4 µ ) 0.001 Pa · s cf0 ) 2.5-120 kg/ m3 Vf0 ) 0.02 m3 dp ) 13 × 10-4 m from Aspen property file from Aspen property file from Aspen property file from Aspen property file n ) 0.37 (assuming spherical particle) Cg ) 230 kg/m3

difference method and the nonlinear equations are solved using mixed Newton method. Implicit Euler method has been used as integrator. For modeling, a flat sheet with circular shape membrane is considered for both the cases and its properties are given in Table 1. It is assumed that there is no dead zone in the cell and the stirrer radius (assumed equal to membrane radius) is sufficient to maintain the uniform mass transfer rate throughout the membrane area. Table 1 gives the summary of all the parameter value and the solution (dextran and silica) properties used in this model.12 5. Results and Discussion 5.1. Separation Performance of Dextran Solution Using Partially Retentive Membrane in the Continuous Mode of Operation. It is generally accepted that concentration polarization of solute at the membrane surface can lead to solute adsorption, precipitation, and gel layer formation and this results in flux decline. This increases operating costs due to the need for frequent cleaning and possibly reduces membrane lifetime. In this article, we have investigated to understand this phenomenon so that its effect can be minimized. Figure 3 shows the steady-state permeate flux of dextran solution with variation of stirrer speed for three different transmembrane pressures (100, 200, and 300 kPa) with feed concentration of 28 kg/m3. It is clear from the figure that the flux increases with raise of stirrer speed, and after a certain speed, there is no further increase and it becomes steady. This means that the increasing stirrer speed results in the brushing away of solute deposited on the membrane surface and membrane surface acquires in the vicinity to the bulk concentration. However, higher transmembrane pressure (300 kPa) requires higher stirrer speed to reach steady-state permeate flux compared to the lower pressure (100 kPa). This


Figure 3. Effect of stirring speed on steady-state permeate flux of dextran solution for different transmembrane pressure (at feed concentration ) 28 kg/m3; membrane porosity ) 10%; membrane thickness ) 1 µm).

is due to the increase of concentration at the membrane surface as the driving pressure increases and consequently the osmotic pressure also increases. In a stirred cell, the growth of concentration polarization layer is usually less due to turbulence and, at higher speed, the layer resistance drops. The polarization layer is gradually removed by increasing the stirrer speed and beyond certain extent of turbulence; the resistance offered by this additional layer is negligible. It can be seen from eq 3 that the osmotic pressure is only for the bulk concentration which turns into fixed (as continuous mode) and the flux becomes unchanging. As expected, an increase of pressure stands for increase of flux which is also confirmed by the above reason. van Oers et al.12 have proved experimentally that the ultrafiltration of dextran was found to satisfy the case of only osmotic pressure limited flux, even if flux at higher concentration was found to be pressure independent. In our case also, there is osmotic pressure limited flux established due to no gel formation. Figures 4 and 5 demonstrate the steady-state decrease of concentration at the surface of membrane (Cm) and permeate concentration (Cp), respectively, with variation of stirrer speed. From these figures, it is understandable that with the augment of stirrer speed, the membrane surface is obtaining toward the bulk concentration. It is expected that an increase of applied pressure raises the concentration at the surface of membrane. The membrane used in this analysis retained the solute partially. This high concentration at the membrane surface increases the true rejection (see Figure 6) and the low diffusion coefficient of dextran (4.6 × 10-11 m2 s-1) reduces the rate of back transport at higher applied pressure. It is confirmed from the results that more stirring is required for closely obtaining the bulk concentration at the membrane surface. The permeate concentration also depends on the permeate flux and mass accumulated on the membrane surface and hence increases with applied pressure for a fixed stirrer speed (see Figure 5). This implies that increasing pressure forces the solute to pass more through the membrane pores. Bhattacharjee and Datta18 have also reported that the membrane surface concentration depends on stirring speed.

Figure 4. Effect of stirring speed on steady-state membrane surface concentration of dextran solution, Cm, for different transmembrane pressures (at feed concentration ) 28 kg/m3; membrane porosity ) 10%; membrane thickness ) 1 µm).

Figure 5. Effect of stirring speed on steady-state permeate concentration of dextran solution, Cp, for different transmembrane pressures (at feed concentration ) 28 kg/m3; membrane porosity ) 10%; membrane thickness ) 1 µm).

Figure 6 explains the observed and true rejection of dextran with variation of stirrer speed for different applied transmembrane pressures (100, 200, and 300 kPa). As the permeate concentration diminishes with the higher stirrer speed, the observed rejection increases for all the pressures studied (Figure 6). For a particular stirrer speed, permeate concentration increases with the applied pressure, resulting in the reduction of observed rejection (Ro). It can be seen that, for lower stirrer speed, negative observed rejection is obtained, which is because the adsorbed solute particles within the membrane pores try to come out at higher pressure and cause the permeate concentration to become more than the feed concentration. Karode13 has also observed the same and explained the possibility of this condition by simulation.


Figure 6. Effect of stirring speed on steady state observed and true rejection of dextran solution for different transmembrane pressures (at feed concentration ) 28 kg/m3; membrane porosity ) 10%; membrane thickness ) 1 µm).

Figure 7. Effect of stirring speed on steady-state permeate concentration of dextran solution for different membrane porosities (at feed concentration ) 28 kg/m3; transmembrane pressure ) 100 kPa; membrane thickness ) 1 µm).

The same was experimentally proved by Balakrishnan et al.28 The observed rejection is found to be lower than the true rejection for all the ranges of stirrer speed and transmembrane pressure studied. This indicates that the concentration polarization effects are significant and leads to considerable loss of efficiency. This clearly shows that Ro is strongly dependent on the operating conditions while Rt represents the true rejection of the membrane. The porosity of the membrane has also great impact on permeate concentration. It can be seen from Figure 7 that the permeate concentration increases with porosity of membrane which is due to the more permeation of dextran through the pores. For a particular porosity, as the stirrer speed increases, membrane surface concentration decreases and so the permeate concentration also decreases.

Figure 8. Unsteady-state permeate flux of dextran solution with time for different stirrer speeds (at feed concentration ) 28 kg/m3; transmembrane pressure ) 200 kPa; membrane porosity ) 10%; membrane thickness ) 1 µm).

Figure 8 depicts the unsteady-state permeate flux variation of dextran solution with time for different stirrer speed with initial feed concentration of 28 kg/m3 and transmembrane pressure of 200 kPa. As expected, there is an initial sharp drop in the permeate flux from the initial value for short filtration time. Zaidi and Kumar8 have experimentally found the similar type of flux decline trend for the dextran water solution using unstirred batch cell. Chudacek and Fane29 have also used stirred batch cell with stirrer speed of 1000 rpm for the separation of dextran solution and obtained the same type of results what we reported in Figure 8. Many researchers suggested that the permeate flux versus time curve can be divided into two domains.30-32 Domain 1 is the initial decline of flux that may be attributed to the membrane fouling. This domain is characterized by an operationally irreversible deposition of material in the membrane pores and the sorption on the membrane surface of the macromolecules initially present in the bulk solution. Domain 2, the remaining flux versus time curve, may be associated with the concentration polarization and gel layer formation phenomena. This domain is characterized by the reversible deposition of macromolecules on the membrane surface. Mehta33 has also developed a model and reported that the rate of flux decline with time is a linear function of the difference between the flux at any time during the time period and the flux at the end of the time period investigated. The resulting independent solutions were combined to provide the global solution applicable for the entire flux vs time curve. However, one can minimize the flux decline due to adsorption of solute layers at the membrane surface by changing the membrane material or chemically modifying it. The adsorption at the membrane surface occurs due to the strong interaction between the solute and membrane material. The result of Cherkasov et al.34 study showed that hydrophobic membranes attracted a thicker irreversible adsorption layer than hydrophilic membranes. Figures 9 and 10 demonstrate the unsteady-state effect of membrane surface concentration and permeate concentration with time for different stirrer speed with constant transmembrane pressure of 200 kPa, respectively. Many studies


Figure 9. Unsteady-state effect of membrane surface concentration (Cm) of dextran solution with time for different stirrer speeds (at feed concentration ) 28 kg/m3; transmembrane pressure ) 200 kPa; membrane porosity ) 10%; membrane thickness ) 1 µm).

Figure 10. Unsteady-state effect of permeate concentration (Cp) of dextran solution with time for different stirrer speeds (at feed concentration ) 28 kg/m3; transmembrane pressure ) 200 kPa; membrane porosity ) 10%; membrane thickness ) 1 µm).

reported experimentally that, at the start of separation, membrane surface concentration is equal to bulk concentration. So, osmotic pressure is contributed only by bulk concentration of the solution. As the time progress, membrane surface acquires more mass from the bulk due to the convection. Some acquired mass on the membrane surface is returned to the bulk by diffusion due to the higher concentration than the bulk. After a certain time, membrane surface reaches the equilibrium concentration (see Figure 9), which is more than bulk concentration and flux becomes steady for that transmembrane pressure and stirrer speed. It is observed from the Figure 9 that flux attains steady state

Figure 11. Unsteady-state effect of permeate flux of dextran solution with time for different transmembrane pressures (at feed concentration ) 28 kg/m3; stirrer speed ) 57.3 rpm; membrane porosity ) 10%; membrane thickness ) 1 µm).

quickly with increase of stirrer speed. This is because of the augmentation of the back diffusion rate resulting in less concentration at the membrane surface and so flux reaches steady state quickly. It elucidates that higher stirrer speed makes the flux variation small for a particular pressure that is due to less membrane surface concentration and the permeate concentration decreases with increasing stirrer speed. It can be clearly seen from the Figures 9 and 10 that the time required to reach steady-state permeate concentration (Cp) and membrane surface concentration (Cm) is not same. In all the cases, membrane surface concentration requires less time to obtain steady state than the permeate concentration. Figures 11-13 explain the unsteady-state effects of permeate flux, membrane surface concentration, and permeate concentration of dextran for different transmembrane pressure (100, 200, and 300 kPa), respectively, at 57.3 rpm stirrer speed. For a fixed stirrer speed, increasing applied pressure enhances the flux. Initially, the change of flux with time (dJ/ dt) is different for different pressures and after a certain time, it becomes the same and the gradient becomes zero after 10 s. From Figure 11, one can notice that at 1 s, dJ/dt for the pressures of 100, 200, and 300 kPa are almost equal. This gradient becomes equal for the pressure of 200 and 300 kPa at 0.1 s, suggesting that at higher pressure there would be no gradient difference. The experimental results reported in ref 8 also support the trend presented in Figure 11, though the time scales do not match well because experiments were performed by using unstirred batch cells with low dextran concentration. The flux decline due to increase of the concentration of membrane surface is also time dependent. However, the doubling of applied pressure does not lead to the doubling of permeate flux, because the increase of applied pressure partially was compensated by an increase of osmotic pressure due to increase of membrane surface concentration. It is observed from Figures 12 and 13 that the membrane surface and permeate concentration acquire higher concentration for higher pressure as is common in filtration practice. Zaidi and Kumar8 experimentally proved that the rejection


Figure 12. Unsteady-state effect of membrane surface concentration of dextran solution with time for different transmembrane pressures (at feed concentration ) 28 kg/m3; stirrer speed ) 57.3 rpm; membrane porosity ) 10%; membrane thickness ) 1 µm).

Figure 13. Unsteady-state effect of permeate concentration of dextran solution with time for different transmembrane pressures (at feed concentration ) 28 kg/m3; stirrer speed ) 57.3 rpm; membrane porosity ) 10%; membrane thickness ) 1 µm).

of dextran was more for lower pressure compared to that of higher applied pressure. This implies that the permeate concentration is low at low applied pressure while higher pressure gives high permeate concentration. Our simulation results reported in Figures 12 and 13 also support this experimental phenomenon and show the dynamics of membrane surface concentration and permeate concentration. 5.2. Separation Performance of Silica Suspension Using Totally Retentive Membrane in the Batch Mode Operation. In this case, we have used a totally retentive membrane to study the separation performance of silica in the batch

Figure 14. Unsteady-state effect of membrane surface concentration of silica suspension with time for different transmembrane pressures for stirred batch mode (initial feed concentration cf0 ) 2.5 kg/m3; stirrer speed ) 9.55 rpm).

Figure 15. Unsteady-state effect of permeate flux of silica suspension with time for different transmembrane pressures for stirred batch mode (initial feed concentration cf0 ) 2.5 kg/m3; stirrer speed ) 9.55 rpm).

mode of operation using stirred UF cell (see Figure 2b). The mass of silica in the upstream side is same as the initial mass for all the process time, as there is no passage of silica through the membrane and only water is passing through membrane. It is well-known that the separation of silica suspension by membrane forms a gel layer on the membrane surface. The osmotic pressure effect is negligible and hence flux is governed by a gel polarization model. Figures 14 and 15 demonstrate the plot of unsteady-state membrane surface concentration (Cm) and permeate flux with time for silica suspension with different transmembrane pressures (200-900 kPa). The results indicate that there is no reduction of applied


pressure because of negligible osmotic pressure of silica. There is no flux reduction with the time until the membrane surface obtains the gel concentration (shown in Figure 15) and this time is termed as characteristic time.10 Higher pressures result in less characteristic time than the lower pressure. For this simulation, we have taken the gel concentration value of silica solute as 230 kg/m3 from the published literature12 for different operating pressures, though the thermodynamic relation of gel concentration depends on both temperature and pressure35,36 and their experimental work showed almost constant (230 kg/m3)12 for varying pressure ranges from 100-200 kPa. The gel layer initiates to form over the membrane surface and increases with the time of operation. This layer provides an additional resistance that results in a flux decline. For higher pressures, the membrane surface acquires the gel concentration much faster than for the lower pressure for a fixed initial feed concentration, which can be seen from Figures 14 and 15. For lower pressure, no gel layer appears within 2000 s. For an example, at 400 kPa transmembrane pressure, membrane surface reaches gel concentration at 1135 s while for 500 kPa pressure it takes 170 s. Before gel formation, there is small flux decline that is due to the increase of viscosity. For all the pressure ranges studied, the increase of membrane surface concentration has two slopes (see Figure 14). The steepest slope is for initial unsteady effect due to the formation of concentration polarization. The second slope is due to the increase of upstream side concentration. So membrane surface concentration increases until it reaches the gel concentration. This high concentration at the membrane surface is due to high rejection of silica which is found to be nearly 100%. The concentration at the membrane surface increases with the transmembrane pressure for a fixed initial feed concentration. This is mainly due to the low rate of back transport of silica (because of low diffusion coefficient of silica). This implies that gel formation is rapid and instantaneous at the beginning of separation, particularly at higher applied pressure. From Figure 15, it is unambiguous that the permeate flux follows the same path after a certain time for 700, 800, and 900 kPa transmembrane pressure; i.e., the flux becomes independent of pressure which is known as the limiting flux. Figure 16 depicts the unsteady-state effect of initial feed concentration of silica suspension on the membrane surface concentration with time. It demonstrates that pressure of 200 kPa is not sufficient for the initial feed concentration of 120 kg/m3 to acquire the gel concentration within 2000 s at the membrane surface. Figure 17 shows the comparison of permeate flux versus time obtained from simulation by using Aspen with experimental data reported by Zaidi and Kumar10 (initial silica feed concentration of 28 kg/m3, pressure of 270 and 405 kPa). It can be seen that experimental results match well with the simulation done in Aspen. So it is strongly advocated that Aspen Engineering Suite is justified as a modeling tool for NF/UF/MF. The effect of gel layer thickness of silica suspension with time for different transmembrane pressures with initial feed concentration of 30 kg/m3 is shown in Figure 18. At higher pressure, gel layer thickness is more which results in more resistance. With the initial feed concentration of 30 kg/m3 and stirrer speed of 9.55 rpm, the applied pressure of 200 kPa is not enough to form gel within 400 s. It is expected that with increasing pressure the gel layer formation is faster. To study the influence of stirrer speed on flux, concentration of the membrane surface (Cm), and gel layer thickness, we have varied

Figure 16. Unsteady-state effect of membrane surface concentration of silica suspension with time for different initial feed concentrations (transmembrane pressure ) 200 kPa; stirrer speed ) 9.55 rpm).

Figure 17. Comparison of unsteady-state permeate flux of silica suspension versus time simulated by using Aspen with experimental data reported by Zaidi and Kumar10 (initial silica feed concentration of 28 kg/m3, pressure of 270 and 405 kPa).

the stirrer speed of the UF cell from 4.8 to 47.7 rpm by keeping the initial feed concentration and transmembrane pressure of 30 kg/m3 and 1200 kPa, respectively, and the results are shown in Figures 19-21. As can be seen from Figure 19, at the beginning, the permeate flux is a time invariant for all the stirrer speed studied. After a few seconds, a sudden drop in the flux is observed due to the buildup of polarization at the membrane surface. This leads to an increase in the concentration at the membrane surface and the same can be seen in Figure 20. As the stirrer speed increases, the time required to reach the gel layer formation increases which is due to the removal of silica particle from surface of membrane. Figure 21 represents the deposited gel layer thickness as function of time for an initial feed concentration of 30 kg/m3 and transmembrane pressure of 1200 kPa. At a particular time, the rate of solute deposition is higher for lower stirrer speed that leads to build up of a thick


Figure 18. Unsteady-state effect of gel layer thickness of silica suspension with time for different transmembrane pressures (initial feed concentration cf0 ) 30 kg/m3; stirrer speed ) 9.55 rpm).

Figure 20. Unsteady-state effect of membrane surface concentration of silica suspension with time for different stirrer speeds in batch mode operation (initial feed concentration cf0 ) 30 kg/m3; transmembrane pressure ) 1200 kPa).

Figure 19. Unsteady-state permeate flux of silica suspension with time for different stirrer speeds in batch mode operation (initial feed concentration cf0 ) 30 kg/m3; transmembrane pressure ) 1200 kPa).

Figure 21. Unsteady-state effect of gel layer thickness of silica suspension with time for different stirrer speeds in batch mode operation (initial feed concentration cf0 ) 30 kg/m3; transmembrane pressure ) 1200 kPa).

layer and is also confirmed from Figure 19. The reason for this behavior may be due to the increase in the bulk concentration at the upstream side of membrane that provides additional resistance. 6. Conclusion We have successfully modeled the separation performance of dextran and silica suspension in a dead-end stirred UF using Aspen Engineering Suite. For dextran solution, it is observed that the permeate flux increases with the stirrer speed up to a certain range beyond which the flux remains constant. Higher stirrer speed with a constant transmembrane pressure offers more permeate flux and observed rejection.

The flux decline rate is larger for higher applied pressure with constant stirrer speed. As the membrane porosity increases, the observed rejection and permeate concentration decrease. For silica suspension, an increase of stirrer speed reduces the flux and increases the time required to reach the gel layer formation at the membrane surface. At the beginning, permeate flux is time invariant for all the stirrer speeds studied; after a few seconds, a sudden drop in the flux is observed due to build up of polarization at the membrane surface. For a fixed transmembrane pressure and stirrer speed, fouling is more for higher initial feed concentration. It is also concluded from the above study that higher stirrer speed and less initial feed concentration are favorable for pressure-


driven membrane filtration process by providing less flux decline. Simulation results of UF of silica suspension using Aspen Engineering Suite was compared with experimental data reported in the literature and it matches quite well. It justifies that the Aspen Engineering Suite can also be used as a modeling tool for NF/UF/MF. Acknowledgment The authors gratefully acknowledge the financial support from CSIR project grant (ref no: 22(0435)/07/EMR-II) from which the Aspen Engineering Suite was procured.

NOMENCLATURE Am A1-3 c cp cm cf0 D Dm dp J k M MW MWCO n NF r Ro Rt Rg Rm RO UF t Vf0 Vf Vp x ∆P ∆π ∆t ∆x υ µ Fp Fg π ε ω υ

membrane cross-sectional area (m2) virial coefficient solute concentration (kg m-3) solute concentration of the permeate (kg m-3) solute concentration at the membrane surface (kg m-3) initial feed solute concentration (kg m-3) diffusivity of solute (m2 s-1) effective diffusivity of the solute through the membrane (m2 s-1) solute particle diameter (m) permeate flux (m3 m-2 s-1) mass transfer coefficient (m/s) molecular weight molecular weight (Da) molecular weight cutoff porosity of cake or gel layer nanofiltration radius of the membrane (m) observed rejection true or intrinsic rejection membrane hydraulic resistance gel resistance reverse osmosis ultrafiltration time (s) initial feed volume (m3) feed volume (m3) permeate volume (m3) distance from the membrane (m) applied pressure difference between permeate and retenate side (Pa) osmotic pressure difference between permeate and retenate side (Pa) differential time (s) differential thickness (m) stoichiometric coefficient of solute viscosity of the solution (P) solute density (kg m-3) gel density (kg m-3) osmotic pressure (Pa) porosity of membrane stirring speed (rad/s) kinematic viscosity (m2 s-1)

υg ∂mem ∂pol

thickness of the gel layer thickness of membrane (m) thickness of concentration polarization layer (m)

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ReceiVed for reView August 25, 2008 ReVised manuscript receiVed January 14, 2009 Accepted March 16, 2009 IE801293D

Modeling and Simulation of Stirred Dead End Ultrafiltration Process

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