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Modeling of suspension fouling in nanofiltration
Desalination 346 (2014) 80–90
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Desalination journal homepage: www.elsevier.com/locate/desal
Modeling of suspension fouling in nanofiltration F. Faridirad a, Z. Zourmand a, N. Kasiri a,⁎, M. Kazemi Moghaddam b, T. Mohammadi b a b
Computer Aided Process Engineering (CAPE) Lab, School of Chemical Engineering, Iran University of Science and Technology (IUST), Narmak, Tehran 1684613114, Iran Research Centre for Membrane Separation Processes, School of Chemical Engineering, Iran University of Science and Technology (IUST), Narmak, Tehran 1684613114, Iran
H I G H L I G H T S • • • • •
Dissolution and sedimentation of solute particles during NF were considered. Considering the fouling of solutes during NF of suspensions Derive a new mathematical model which calculates permeate flux Verifying the new model with experimental data Comparing the performance of the new model with other existing models
a r t i c l e
i n f o
Article history: Received 23 March 2014 Received in revised form 6 May 2014 Accepted 7 May 2014 Available online 2 June 2014 Keywords: Membrane process Mathematical model Nanofiltration Dissolution Cross flow
a b s t r a c t Prediction of membrane process performance using experimental and mathematical models facilitates optimization of membrane processes and their operating conditions. On the other hand investigation of nanofiltration process performance over time and estimation of the time for recovery are very essential in the membrane industry. In this work a mathematical model considering membrane resistance changes and also dissolution of deposited particles in nanofiltration feed, for a cross flow nanofiltration system was developed. The validation of this model with experimental data demonstrated a good agreement. At a constant concentration of 0.2 g/L error of the present models varied from 3% (at the pressure of 1 bar) up to about 3.8% (at the pressure of 2.9 bar) while at a constant pressure of 2 bar error varied from 2.45% (at the concentration of 0.1 g/L) up to about 4.2% (at the concentration of 0.4 g/L). A comparison between this model and three other previous models proved a better performance for the present model with an average error of 3.25%. © 2014 Elsevier B.V. All rights reserved.
1. Introduction Nanofiltration is a membrane process for liquid separation which has been extended recently [1]. Reverse osmosis is replaced by nanofiltration in many applications, due to the fact that NF requires less energy comparatively. Going by its specifications, nanofiltration lays between the ultrafiltration membrane and RO. Membrane filtration processes, such as nanofiltration (NF) and reverse osmosis, play an important role in the production of high quality reclaimed water when small organic compounds, (e.g., pesticides, endocrine disruptors and pharmaceutically active compounds) are to be removed from polluted water [2]. The important application of NF is for water treatment, waste water treatment and desalination. NF is a pressure driven separation process which can separate particles based on their size and electrostatic interactions between them. For the uncharged molecules, the dominant mechanism for separation is molecular size while for the separation of ions with similar size, electrostatic forces ⁎ Corresponding author. E-mail address: [email protected] (N. Kasiri).
http://dx.doi.org/10.1016/j.desal.2014.05.014 0011-9164/© 2014 Elsevier B.V. All rights reserved.
play a more important role [3]. NF membranes have pores of around 1 nm [4], therefore they got higher water flux at lower pressures compared to RO. The main problem in using NF technology is capacity reduction of industrial units caused by membrane fouling, particularly during inorganic separation [5,6], leading to membrane life reduction [7]. Inorganic fouling induced by concentration polarization and scale formation is one of the major limitations of NF as in other membrane processes used in water treatment [6,8]. This necessitates fouling control in membrane processing [9]. Suspension fouling is of particular interest in the membrane field. Suspensions are defined as fine particles whose characteristic size is roughly in the range of 1 nm–1 μm [10]. In these processes flux decline is dependent on many factors such as membrane characteristics & module geometry, feed condition, solute type and also operating conditions [11]. Concentration polarization refers to the reversible accumulation of solute within a thin boundary layer adjacent to the membrane surface. Membrane fouling can be irreversible with solute adsorption on or in the membrane pore walls, leading to complete or partial pore blocking [12]. Formation of cake layer over the membrane surface can determine rejection properties of the system since the deposited layer will act as a “secondary”
F. Faridirad et al. / Desalination 346 (2014) 80–90
membrane prior to the “real” membrane or support material [13]. Fouling importance has initiated many efforts for modeling flux decline profile during NF [14,15]. To improve fouling models many dynamic models have been derived [4,11,16]. Theoretical models are not capable of predicting the filtration flux precisely, due to the simplifying assumptions required to enable their analytical solution [17]. Various flux decline mechanisms are studied which are generally governed by cake or Gel layer [18–20], resistance in series and pore blocking [21,22]. Ballet et al. studied experimentally the effect of feed pressure, ionic strength, concentration, and pH on the retention of phosphate anions [23]. Li et al. investigated the effects of pressure, flow rate on flux, and retention in seawater desalination process [24]. Darvishmanesh et al. developed a new semi-empirical model based on the traditional solution diffusion with imperfection model for solvent resistant NF [25]. This model demonstrated a good prediction for the flux of solvent through the membrane. Fadaei et al. developed a mass transfer model to predict ion transport through the NF membrane to account for the concentration polarization phenomenon and its influence on ion separation [26]. They used CFD techniques and successfully predicted the local concentration of ions; permeate flux, and rejection of ions in a rectangular cross flow. In the field of CFD many other works have been done such as modeling the impacts of feed spacer geometry on NF processes by Guillen and Hoek [27] and CFD modeling of porous membranes by Pak et.al [28]. Field et al. introduced the concept of critical flux for microfiltration, stating that there is a permeate flux below which fouling is not observed. It was a very powerful optimization tool for this kind of separation operation. It was possible then to identify a critical flux for UF and NF membranes [29]. The purpose of this research is to introduce a new fouling model based on cake formation to describe the flux decline caused by inorganic solutes. Fouling based models are appropriate for predicting the filtration and clean up time duration. In model development the effect of many parameters such as operating conditions, cake filtration characteristics, membrane permeability, cross flow velocity, specific cake resistance and other experimental parameters on flux reduction has been investigated. The developed model was then validated against experimental data obtained from water–oil solution NF. Finally model performance was compared with three other models developed by Lihan Huang [30] and Wu [31], as well as the cake or Gel Layer Filtration model [32]. In these models the fact of dissolution of solutes, transmission of particles through the membrane, membrane resistance changes, and also the effect of operating parameters such as cross flow velocity and temperature were neglected. Considering the solute dissolution, membrane resistance changes and also the effect of operating conditions lead to a model with better performance and better agreement with experimental data.
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accumulation on the membrane leading to reduced cake formation and fouling [33]. Membrane fouling is related to cross flow nanofiltration velocity with larger flow velocities causing more turbulence hence less fouling [34]. One of the theories for explaining membrane fouling is cake filtration model which was developed by Hoek et al. to evaluate reverse osmosis and nanofiltration processes [35,36]. According to this model flux can be explained as: J ðt Þ ¼
Δp : Rm þ Rcp þ Rc ðt Þ
ð1Þ
In which J(t) is solvent (permeate) flux which is a function of effective pressure applied on the membrane surface (Δp), hydraulic membrane resistance (Rm) and cake layer resistance (Rc) and also the resistance related to concentration polarization layer (Rcp). Although concentration polarization layer effect is significant at the very early stages of the separation process, this gradually fades away when pore blockage takes over and due to which concentration polarization has been neglected in the model. Eq. (1) is therefore simplified to: J ðt Þ ¼
Δp : Rm þ Rc ðt Þ
ð2Þ
Cake layer resistance is calculated through Eq. (3) using specific cake resistance (α). Rc ¼ αM d
ð3Þ
In Eq. (3), Md is accumulated mass per surface unit. The constant α is related to membrane porosity through Carman–Kozeny's equation [36]: α¼
45μ ð1−ε Þ2 : ρa2p ε3
ð4Þ
Here μ is water (solvent) viscosity, ε is cake porosity, ρ is cake density and ap is particle radius. If temperature is fixed, viscosity would remain constant. Despite this, Arrhenius equation can be used to take into consideration the effect of temperature on permeate flux. E μ ¼ μ 0 exp − T R
ð5Þ
Pressure difference is calculated through Eq. (6) by using pure water permeate. ð6Þ
Δp ¼ J 0 Rm 2. Model development
In this model membrane fouling is modeled as a dynamic process. The model is based on the following assumptions:
In the NF process feed enters from one side of the cell, having passed over the membrane surface, leaves the cell from the other side with permeate passing through the membrane and exiting from a second exit of the chamber. Cross flow arrangement reduces solute particle
1- Due to the small membrane length, a uniform cake layer of a uniform resistance will be formed through the membrane surface.
Table 1 Estimated parameters of the developed model, obtained from GA, using 70% of experimental data.
c = 0.2 g/L
P = 2 bar
Pressure
ρ
α
b
a
M1
M2
M3
P P P P
2300 2400 2450 2500
1011 1012 1.4 × 1012 1013
0.00027
3.42 × 109
0.03 0.07 0.1 0.1
0.00004 0.00002 0.00001 0.000001
1000 1100 1180 1200
= = = =
1 bar 2 bar 2.4 bar 2.9 bar
Concentration
ρ
α
b
a
M1
M2
M3
c c c c
2350 2400 2400 2450
1011 1.4 × 1011 1.6 × 1011 1.8 × 1011
0.000087
3.34 × 109
1000 1070 1100 1130
0.000006 0.00003 0.00003 0.00002
0.06 0.06 0.05 0.4
= = = =
0.1 0.2 0.3 0.4
g/L g/L g/L g/L
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F. Faridirad et al. / Desalination 346 (2014) 80–90
Fig. 1. Variation of present model parameters with pressure.
2- Solute particle sedimentation on the membrane surface is proportional to the volumetric filtrate rate passing through it [16]. 3- Sedimentation and dissolution happen simultaneously. 4- Fouling will occur when solute concentration is equal to the saturation concentration. Before this, neither sedimentation nor dissolution will take place. 5- Particles can pass through the pores and therefore there are some solute particles in the permeate side. By passing the particles through the membrane pores, some particles sediment within them, and therefore the diameter of pores reduces which causes membrane resistance to be changed. Thus membrane resistance
changes over time. As time passes and cake formation takes place leading to cake ΔP build up passing flux diminishes leading to cesar of further particle sedimentation within membrane pores. 6- Cake layer resistance is time dependent. As time passes membrane resistance to flow increases while rate of its increase over time or the increasing gradient is reduced. As this behavior could mathematically be represented by a logarithmic function, in the present work this is proposed to be presented by the following function. Rm ðt Þ ¼ Rm ð0Þ ða ln ðb t ÞÞ
Fig. 2. Variation of present model parameters with concentration.
ð7Þ
F. Faridirad et al. / Desalination 346 (2014) 80–90
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In Eq. (7), a and b are constants that could be determined using experimental data. According to the previous assumptions solute sedimentation rate is described as [19]: Rdep ¼ M1 Q p :
ð8Þ
And solute dissolution rate is calculated through Eq. (9) [37]. Rdis ¼ M2 Q c
ð9Þ
Eq. (10) is used to describe particles passing through the membrane.
Fig. 3. Porosity changes of the cake layer with pressure in present model.
passed ¼
2
ðM 3 Þ
ð10Þ
Rm ðt Þ
In Eqs. (8), (9) and (10), M1, M2 and M3 are deposition, dissolution and transmission constants respectively. Total accumulation of solute : (M ) can be calculated through Eq. (11). ðM Þ M˙ ¼ M1 Q p −M 2 Q c − 3
2
Rm ðt Þ
¼ ρA
dl dt
ð11Þ
In Eq. (11), ρ is cake density, A is effective surface of the dl is the cake layer growth rate. Permeate flux and membrane and dt rate, are determined through Eqs. (12) and (13) respectively. J ðt Þ ¼
J 0 Rm ðt Þ Rm ðt Þ þ ρ lðt Þ α
Q p ðt Þ ¼ Fig. 4. Porosity changes of the cake layer with concentration in present model.
Q0
^ 1þk
lðt Þ ða ln ðb t ÞÞ
ð12Þ
ð13Þ
^ ¼ Q 0 ρα is the mass transfer Here l(t) is cake filtration thickness and k ΔP resistance constant.
Fig. 5. Model and experimental data (flux vs. time) at different pressures.
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Fig. 6. Error variation of developed model versus pressure.
A mass balance for flows gives Eq. (14). 1
0
ρA
B dl B ¼ M1 B @ dt 1þ
Q0 ^ ðt Þ kl a ln ðb t Þ
0
C B C B C−M2 BQ − A @
1þ
Q0 ^ ðt Þ kl
Fig. 8. Error variation of developed model versus concentration.
1
of particle concentration and β is a constant which is determined experimentally [30].
C ðM 3 Þ2 C C− A Rm;0 ða ln ðb t ÞÞ
J0 ¼
a ln ðb t Þ
ð14Þ Therefore the amount of cake layer thickness could be determined through solving Eq. (14) and the flux quantity will be calculated afterwards.
J¼
−3V 0 þ 4V 1 −V 2 2AΔt
1 dðV−V c Þ ΔPαβ 1 ¼ : A dðt−t c Þ μ ðV−V c Þα1−1
ð15Þ
ð16Þ
3.2. Gel Layer Filtration model [32,39] 3. Previous models 3.1. Lihan Huang model In this model it is assumed that fouling is a dynamic process which starts with pore blocking and then continues with cake formation on the membrane surface [38]. The initial permeate flux is calculated through Eq. (15). In Eq. (15) V0, V1 and V2 are the first infiltrated volumes and Δt is the time step between these measurements. In this model Eq. (16) is used to calculate flux. In Eq. (16) α is a function
In this model it is assumed that the size of particles is the same hence the porosity can be assumed constant. The transient flux is described by Eq. (17) [40]. In Eq. (17) solvent flux is a function of TMP, Δp(=ΔP − Δπm), hydraulic resistance of the membrane, Rm(=ΔP/μ × Jw), and cake layer resistance. At constant pressure the flux is calculated only through multiplying specific cake resistance (rc) and the mass of transient layer per surface unit (Mc), as it is shown in Eq. (18). Specific cake resistance is described through Carman– Kozeny's equation which is shown in Eq. (19). μ0 is solvent viscosity, ε is
Fig. 7. A comparison between developed model and experimental data flux at different concentrations.
F. Faridirad et al. / Desalination 346 (2014) 80–90
Fig. 9. The effect of time passage on cake thickness (a) and flux (b) with V
85
m s
as parameter.
Fig. 10. The effect of time passage on cake thickness (a) and flux (b) with P (bar) as parameter.
cake layer porosity, dp is particle diameter and ρc is particle density. Δp J ðt Þ ¼ ð17Þ Rm þ Rc ðt Þ " # 45μ 0 ð1−εÞ Mc ðt Þ ð18Þ Rc ðt Þ ¼ r c M c ðt Þ ¼ ρc d2p ε3 " # 45μ 0 ð1−εÞ rc ¼ ð19Þ ρc d2p ε3
This model has two constants which are determined through experimental data.
dJ ¼ −kP expð−k f t Þ J dt
ð20Þ
4. Results and discussions 4.1. Parameter estimation
3.3. Wu model According to this model at the beginning of the process, flux decline is proportional to flux according to Eq. (20) [40].
Experimental data for water–oil feed which has been obtained from the experimental work carried out by T. Mohammadi et al. [41] have been carried out at 4 different pressures and concentrations assuming
Fig. 11. The effect of time passage on cake thickness (a) and flux (b) with M1 as parameter.
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Fig. 12. The variation of cake thickness (a) and flux (b) versus time with M2 as parameter.
a constant temperature of 30 °C. The membrane made of polysulfone was produced by DOW of Denmark. Of course the membranes which were used in this reference were ultrafiltration membranes, but due to their pore size that is close enough to the range of nanofiltration membrane size, using the data obtained from this paper does not make any significant problem in our work. The membrane surface was 28.64 cm2 and the cross flow velocity was 1 cm/s. The concentrated flow which goes out from the cell and the permeated flow from the membrane are returned to the feed tank to make the feed concentration
constant. The permeate fluxes were measured with an accuracy of 0.001 g. Flow system was laminar according to the following calculations: Re ¼
ρuDh μ
rh ¼ 4
Dh ¼ 4r h
0:0030 0:0335 ¼ 0:0055 2 ð0:0030 þ 0:0335Þ
Fig. 13. The variation of cake thickness (a) and flux (b) versus time with M3 as parameter.
Fig. 14. The variation of cake thickness (a) and flux (b) versus time with α as parameter.
F. Faridirad et al. / Desalination 346 (2014) 80–90
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Fig. 15. A comparison of different model flux predictions at 4 different pressures (present model and previous models).
u ¼ 0:01
m s
→ Re ¼ 220:
In the first part, experiments have been done at 1, 2, 2.4, and 2.9 bar with a constant concentration of 0.2 g/L. In the second part tests have been carried out at 2 bar and at different concentrations of 0.1, 0.2, 0.3 and 0.4 g/L. Oil density was set equal to 890 kg/m3. The parameters of the model have been calculated from experimental data using the Genetic Algorithm method [42,43]. In the GA optimization method 70% of the data are used for fitting the best values to these parameters, the other 30% for goodness of fit evaluation and validation, thus providing the best fit given the data circumstances. The results have been tabulated in Table 1. Figs. 1 and 2, show the parameters of the model versus pressure and concentration respectively. From Fig. 1 it is clear that by increasing the amount of pressure M1, M3 and α increase while M2 decreases. It is reasonable as higher pressures cause more particles to permeate which means larger M1 and M3, on the other hand this causes the cake
layer to be more resistant and therefore specific resistance to increase. But this increase leads to less particle dissolution and thus decreases the amount of M2. Concentration has had the same effect on model parameters as well. By increasing the concentration a larger amount of solute finds a chance for passing through membrane pores and therefore both sedimentation (M1) and transmission (M3) constants will be increased as shown in Fig. 2. Also a larger amount of solute causes more resistant cake layer leading to a larger value for the specific cake resistance (α). The increase of specific cake resistance as a result of the concentration increase is demonstrated in Fig. 2 which also shows that increasing the concentration decreases the dissolution constant as more sedimentation takes place. Porosity changes of the cake layer with pressure and concentration are shown in Figs. 3 and 4. In reality by increasing the pressure the driving force towards the membrane increases thus applying larger force on the cake which causes it to become more compact reducing the free volume within it. This phenomenon leads to less porosity which is obvious from Fig. 3. Concentration has the same influence
Fig. 16. A comparison of error at different pressures for different model flux predictions (present model and previous models).
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F. Faridirad et al. / Desalination 346 (2014) 80–90
Fig. 17. A comparison between different models at 4 different concentrations for flux prediction (present model and previous models).
on porosity. By increasing the amount of solute, free space of the cake layer will be decreased leading to less porosity as shown in Fig. 4.
4.2. Model validation and verification Permeate flux variations with pressure as observed experimentally and predicted by the model and their relative error are demonstrated in Figs. 5 and 6 respectively. This demonstrates good compatibility. As is clear from Fig. 6 the model error increases slightly as the pressure increases. Error changes from 3% at P = 1 bar to about 3.8% at P = 2.9 bar which is not significant. A comparison between experimental data and model results has been done at different concentrations, which is shown in Fig. 7 again demonstrating a good agreement. Error variation at different concentrations is shown in Fig. 8. As shown in Fig. 8, by increasing the concentration, error increases from 2.45%, at a concentration of 0.1 g/L to 4.2% at a concentration of 0.4 g/L.
4.3. Assessment of model parameters on membrane performance The effect of different parameters on flux decline according to the time is investigated in this section. The influence of parameters such as pressure, specific cake resistance, cross flow velocity, model constants and temperature on flux and fouling will be evaluated. 4.3.1. Cross flow velocity More cross flow velocity causes more flow tension and therefore reduces concentration polarization layer [44]. It also leads to more turbulency in flow which prevents solute sedimentation and membrane fouling [5]. Cross flow velocity can be changed by feed flow rate. Fig. 9 shows the effect of cross flow velocity on flux profile and cake thickness. In Fig. 9 cross flow velocity changes from 0.08 m/s to 0.6 m/s while other parameters remain constant. As can be seen, flux reduces over time (Fig. 9b) and at higher rates initially, while cake thickness increases (Fig. 9a) with both eventually becoming constant. The process of course tends to a steady state status. By increasing the cross flow velocity, as it
Fig. 18. A comparison of error at different concentrations for different model predictions of flux (present model and previous models).
F. Faridirad et al. / Desalination 346 (2014) 80–90
is clear from Fig. 9, the final flux increases, while cake layer thickness decreases. It can also be seen from Fig. 9 that all the curves start from the same point. As can be seen the flux changes from 1:22 10−5 ms , −5 m when the cross flow velocity is 0.08 m/s, to 2:03 10 s when the velocity is 0.6 m/s. This fact can be explained as flow turbulency will increase by increasing the cross flow velocity, which leads to less solute sedimentation on the membrane surface, smaller cake layer thickness and therefore membrane fouling, which allows more solvent to permeate through the membrane and will increase the flux. 4.3.2. Transmembrane pressure Transmembrane pressure, TMP, is a pressure difference between two sides of the membrane which has a direct effect on permeated flux. Fig. 10 shows the effect of pressure increase on NF, when it is changed from 20.7 kPa to 62.1 kPa. As the permeated flux is related to transmembrane pressure, the amount of solvent which passes through the membrane will be increased by increasing the pressure. This fact is demonstrated in Fig. 10 where both permeated flux and cake layer thickness increase over time. As it can be seen in Fig. 10b eventually all systems yield to the same amount of final flux. 4.3.3. Sedimentation parameter (M1) Here the amount of solute which has been accumulated on the membrane surface is equal to the difference between the amount which has been deposited and the amount which has been dissolved in the feed solution, therefore the process is controlled by two parameters M1 and M2 and it is important to evaluate the effect of these two parameters on the flux profile. Fig. 11 shows the effect of sedimentation parameter on solvent flux (picture a) and cake thickness (picture b). As it can be seen from Fig. 11, by increasing the amount of M1, permeate flux decreases as the cake layer thickness increases. On the other hand larger sedimentation coefficient causes more sedimentation which causes less solvent to permeate through the membrane. It can be explained from Fig. 11 that the value of M 1 does not affect the final flux but it has a great influence on the slope of flux diagram and increases the rate of flux reduction. 4.3.4. Dissolution parameter (M2) It was mentioned before that M2 is the dissolution parameter of the solute which has been deposited on the membrane surface. The effect of this parameter on the flux of permeated solvent (picture b) and cake layer thickness (picture a) is shown in Fig. 12. Here M2 varies from a near zero value of 0.000001 to 0.0002. It can be deduced from Fig. 12 that by increasing M2 the amount of dissolution solutes in feed solution increases and therefore the amount of sedimentation on membrane surface or cake layer thickness decreases. Sedimentation and also fouling reduction, cause more solvent to permeate. This is apparently clear from Fig. 12 (picture b). 4.3.5. Transmission parameter (M3) This parameter denotes particle transmission. The effect of this term is shown in Fig. 13. It is clear that by increasing the amount of transmission parameter, more solution permeates through the membrane therefore permeate flux will be increased (picture a) while cake layer thickness (picture b) will be decreased. As it is obvious all the diagrams in both pictures a and b start from the same point but terminate at different values of flux and cake thickness. 4.3.6. Specific cake resistance (α) This parameter expresses the amount of cake layer resistance which is formed on the membrane surface. Therefore it is obvious that by increasing the amount of this parameter its effect on solvent flux is more, allowing a lesser amount of solvent to pass through the membrane leading to a decrease in the amount of cake layer thickness. In Fig. 14 the effect of specific cake resistance on permeated flux (picture b) and cake layer thickness (picture a) is shown. Here specific
89
cake resistance was changed from 50 1011 1 h to 12 1013 1 h . As it is shown in Fig. 14 by increasing α the amount of permeated flux is decreased leading to a shortening of the transient period. By increasing the amount of α, cake layer resistance increases leading to less solvent permeation. Another point is that by increasing α permeated flux reaches to its constant amount in a shorter time. Also from Fig. 14 it is clear that specific cake resistant does not affect the initial and final flux and in the long run all diagrams become close together and their final flux is similar. The cake layer thickness reduction is shown in Fig. 14 (picture a). Other than the influence on flux, by increasing the amount of specific cake resistance the slope of the diagrams is increased which causes the system to take longer to become a steady state. 4.4. The comparison of present model against previous models The developed model has an acceptable performance with an average error of 3.2%. This model can predict the behavior of oil–water systems in an appropriate way which is due to the realistic set of assumptions the model is based on for the oil–water system. A comparison between different models at various pressures has been carried out with the results shown in Fig. 15. According to the assumptions of the previous models and the presented model it is apparently clear that this model performs better in flux prediction. Lihan Huang model had an acceptable performance in predicting experimental data related to oil–water mixtures. This relatively good performance is due to the acceptable assumptions that fouling is a dynamic process starting by pore blockage and continuing by cake formation. Pore blocking is a natural phenomenon which occurs at the beginning of oil–water separation, because of the small molecules capable of passing through the membrane pores. This little deviation increases by increasing the pressure. This can be justified by increasing the pressure cake layer resistance increases resulting to reduced flux while at the same time driving force increases leading to permeation of more water. In reality the second feature was dominant hence the model under predicts the flux. This difference can also be due to neglecting the dissolution effect of the particles which causes the cake layer resistance to be reduced. This difference increases by increasing the driving force. Gel Layer Filtration and Wu models do not consider membrane fouling and pore blocking which causes these models to evaluate a lower resistance against permeate flow leading to the over prediction of permeate flux compared to the experimental data. The amount of error and its changes versus pressure for different models is shown in Fig. 16. It is clear that the developed model has the smallest error compared to the others. Increased pressure only increases the error slightly. After this model, Lihan Huang showed the best performance compared to other previously developed models. A comparison among models and experimental data has been done at different concentrations, as shown in Fig. 17. As in the previous section, it can be seen from Fig. 18 that the error related to the present model is the least compared to other models. By increasing the concentration a little raise in error was detected. 5. Conclusions Modeling is an appropriate tool for predicting processes behavior in nanofiltration of suspensions. Here a mathematical model was extended for predicting and describing the behavior of a nanofiltration system and fouling. In this work available suitable models were analyzed and compared to the experimental data and also with the presently developed model. The comparison between model results and experimental data showed that the model assumptions play an important role in the final relations and therefore prediction of experimental data. As mentioned before, due to the conditions of the evaluated system and the fact that membrane resistance changes are accompanied with cake layer formation during filtration, the assumption of current model is close enough to the behavior of oil–water system enabling the model
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F. Faridirad et al. / Desalination 346 (2014) 80–90
to predict the experimental data better than the three other models evaluated here. The average error of developed model is approximately 3.2% compared to the average errors of Lihan Huang, Wu and Gel Layer Filtration models which are 6%, 8.2% and 9.6% respectively. This increases only slightly as pressure and concentration increase demonstrating the acceptable performance of the model at higher pressures and concentrations of up to 2.9 bar and 0.4 g/L respectively. Symbols J(t) J0 Δp Rm Rcp Rc(t) μ μ0 α Md Rm (0) a b t Rdep Rdis Q0 Qc Qp M1 M2 M3 l ρ ρc A : M ^ k v0 α ε dp
permeation flux (m/s) initial permeation flux (m/s) pressure difference between two sides of the membrane (Pa) membrane resistance (kg/s·m2) polarization layer resistance (kg/s·m2) cake layer resistance (kg/s·m2) solution viscosity (Pa·s) solvent viscosity (Pa·s) specific cake resistance (1/s) accumulated mass surface unit (kg/m2) distinct membrane resistance (kg/m2) membrane resistance parameter (1/s) membrane resistance parameter (s) nanofiltration time (s) deposition rate of solute (kg/s) dissolution of solute (kg/s) feed intel rate (m3/s) cake formation rate (m3/s) permeate rate (m3/s) deposition constant (kg/m3) dissolution constant (kg/m3) passed constant (kg/m·s) cake layer thickness (m) solution density (kg/m3) cake layer density (kg/m3) membrane surface area (m2) mass accumulation on the membrane surface (kg/s) mass transfer resistance constant (1/s·m3) initial permeate volume (m3) Lihan Huang constant (1) cake layer porosity (1) particle diameter (m)
References [1] W.R. Bowen, J.S. Welfoot, Modelling the performance of membrane nanofiltration— critical assessment and model development, Chem. Eng. Sci. 57 (2002) 1121–1137. [2] E.-E. Chang, Y.-C. Chang, C.-H. Liang, C.-P. Huang, P.-C. Chiang, Identifying the rejection mechanism for nanofiltration membranes fouled by humic acid and calcium ions exemplified by acetaminophen, sulfamethoxazole, and triclosan, J. Hazard. Mater. 221 (2012) 19–27. [3] A.E. Childress, M. Elimelech, Relating nanofiltration membrane performance to membrane charge (electrokinetic) characteristics, Environ. Sci. Technol. 34 (2000) 3710–3716. [4] B. Tansel, W. Bao, I. Tansel, Characterization of fouling kinetics in ultrafiltration systems by resistances in series model, Desalination 129 (2000) 7–14. [5] S. Bhattacharjee, G.M. Johnston, A model of membrane fouling by salt precipitation from multicomponent ionic mixtures in crossflow nanofiltration, Environ. Eng. Sci. 19 (2002) 399–412. [6] C.-J. Lin, P. Rao, S. Shirazi, Effect of operating parameters on permeate flux decline caused by cake formation — a model study, Desalination 171 (2005) 95–105. [7] A. Seidel, M. Elimelech, Coupling between chemical and physical interactions in natural organic matter (NOM) fouling of nanofiltration membranes: implications for fouling control, J. Membr. Sci. 203 (2002) 245–255. [8] S. Mattaraj, C. Jarusutthirak, C. Charoensuk, R. Jiraratananon, A combined pore blockage, osmotic pressure, and cake filtration model for crossflow nanofiltration of natural organic matter and inorganic salts, Desalination 274 (2011) 182–191. [9] Y.A. Le Gouellec, M. Elimelech, Calcium sulfate (gypsum) scaling in nanofiltration of agricultural drainage water, J. Membr. Sci. 205 (2002) 279–291.
[10] C.Y. Tang, T. Chong, A.G. Fane, Colloidal interactions and fouling of NF and RO membranes: a review, Adv. Colloid Interf. Sci. 164 (2011) 126–143. [11] H.F. Shaalan, Development of fouling control strategies pertinent to nanofiltration membranes, Desalination 153 (2003) 125–131. [12] B. Sarkar, S. De, A combined complete pore blocking and cake filtration model for steady‐state electric field‐assisted ultrafiltration, AICHE J. 58 (2012) 1435–1446. [13] M.E. Ersahin, H. Ozgun, R.K. Dereli, I. Ozturk, K. Roest, J.B. van Lier, A review on dynamic membrane filtration: materials, applications and future perspectives, Bioresour. Technol. 122 (2012) 196–206. [14] E. Fountoukidis, Z. Maroulis, D. Marinos-Kouris, Crystallization of calcium sulfate on reverse osmosis membranes, Desalination 79 (1990) 47–63. [15] L. Song, M. Elimelech, Theory of concentration polarization in crossflow filtration, J. Chem. Soc. Faraday Trans. 91 (1995) 3389–3398. [16] H.R. Rabie, P. Côté, N. Adams, A method for assessing membrane fouling in pilotand full-scale systems, Desalination 141 (2001) 237–243. [17] S. Ghandehari, M.M. Montazer-Rahmati, M. Asghari, A comparison between semitheoretical and empirical modeling of cross-flow microfiltration using ANN, Desalination 277 (2011) 348–355. [18] P. Rai, G. Majumdar, S. DasGupta, S. De, Modeling of permeate flux of synthetic fruit juice and mosambi juice (Citrus sinensis (L.) Osbeck) in stirred continuous ultrafiltration, LWT-Food Sci. Technol. 40 (2007) 1765–1773. [19] B. Sarkar, S. Pal, T.B. Ghosh, S. De, S. DasGupta, A study of electric field enhanced ultrafiltration of synthetic fruit juice and optical quantification of gel deposition, J. Membr. Sci. 311 (2008) 112–120. [20] S. De, P. Bhattacharya, Modeling of ultrafiltration process for a two-component aqueous solution of low and high (gel-forming) molecular weight solutes, J. Membr. Sci. 136 (1997) 57–69. [21] S. Nataraj, R. Schomäcker, M. Kraume, I. Mishra, A. Drews, Analyses of polysaccharide fouling mechanisms during crossflow membrane filtration, J. Membr. Sci. 308 (2008) 152–161. [22] Ö. ÇEtinkaya, V. GÖKmen, Assessment of an exponential model for ultrafiltration of apple juice, J. Food Process Eng. 29 (2006) 508–518. [23] G.T. Ballet, A. Hafiane, M. Dhahbi, Influence of operating conditions on the retention of phosphate in water by nanofiltration, J. Membr. Sci. 290 (2007) 164–172. [24] X.-m. LI, D. WANG, T. CHAI, B.-w. SU, C.-j. GAO, Study on seawater softening by nanofiltration membrane, J. Chem. Eng. Chin. Univ. 4 (2009) 008. [25] S. Darvishmanesh, A. Buekenhoudt, J. Degrève, B. Van der Bruggen, General model for prediction of solvent permeation through organic and inorganic solvent resistant nanofiltration membranes, J. Membr. Sci. 334 (2009) 43–49. [26] F. Fadaei, S. Shirazian, S.N. Ashrafizadeh, Mass transfer modeling of ion transport through nanoporous media, Desalination 281 (2011) 325–333. [27] G. Guillen, E. Hoek, Modeling the impacts of feed spacer geometry on reverse osmosis and nanofiltration processes, Chem. Eng. J. 149 (2009) 221–231. [28] A. Pak, T. Mohammadi, S. Hosseinalipour, V. Allahdini, CFD modeling of porous membranes, Desalination 222 (2008) 482–488. [29] M. Iaquinta, M. Stoller, C. Merli, Optimization of a nanofiltration membrane process for tomato industry wastewater effluent treatment, Desalination 245 (2009) 314–320. [30] L. Huang, M.T. Morrissey, Fouling of membranes during microfiltration of surimi wash water: roles of pore blocking and surface cake formation, J. Membr. Sci. 144 (1998) 113–123. [31] D. Wu, J. Howell, N. Turner, New method for modelling the time-dependence of permeation flux in ultrafiltration, Food Bioprod. Process. 69 (1991) 77–82. [32] P. Hermans, H. Bredée, Principles of the mathematical treatment of constantpressure filtration, J. Soc. Chem. Ind. 55 (1936) 1–4. [33] W.R. Bowen, A. Mongruel, P.M. Williams, Prediction of the rate of cross-flow membrane ultrafiltration: a colloidal interaction approach, Chem. Eng. Sci. 51 (1996) 4321–4333. [34] R. Sheikholeslami, Fouling mitigation in membrane processes: report on a workshop held January 26–29, 1999, Technion—Israel Institute of Technology, Haifa, Israel, Desalination 123 (1999) 45–53. [35] R.H. Davis, Modeling of fouling of crossflow microfiltration membranes, Sep. Purif. methods 21 (1992) 75–126. [36] E.M. Hoek, A.S. Kim, M. Elimelech, Influence of crossflow membrane filter geometry and shear rate on colloidal fouling in reverse osmosis and nanofiltration separations, Environ. Eng. Sci. 19 (2002) 357–372. [37] R. Bian, K. Yamamoto, Y. Watanabe, The effect of shear rate on controlling the concentration polarization and membrane fouling, Desalination 131 (2000) 225–236. [38] R.S. Faibish, M. Elimelech, Y. Cohen, Effect of interparticle electrostatic double layer interactions on permeate flux decline in crossflow membrane filtration of colloidal suspensions: an experimental investigation, J. Colloid Interface Sci. 204 (1998) 77–86. [39] V. Gonsalves, A critical investigation on the viscose filtration process, Recueil des Travaux Chimiques des Pays-Bas 69 (1950) 873–903. [40] T. Arnot, R. Field, A. Koltuniewicz, Cross-flow and dead-end microfiltration of oilywater emulsions: part II. Mechanisms and modelling of flux decline, J. Membr. Sci. 169 (2000) 1–15. [41] T. Mohammadi, A. Kohpeyma, M. Sadrzadeh, Mathematical modeling of flux decline in ultrafiltration, Desalination 184 (2005) 367–375. [42] R. Poli, J. Koza, Genetic Programming, Springer, 2014. [43] D.E. Goldberg, Genetic Algorithms in Search, Optimization, and Machine Learning, Addison-Wesley Reading Menlo Park, 1989. [44] M.A. Razavi, A. Mortazavi, M. Mousavi, Dynamic modelling of milk ultrafiltration by artificial neural network, J. Membr. Sci. 220 (2003) 47–58.
Contents lists available at ScienceDirect
Desalination journal homepage: www.elsevier.com/locate/desal
Modeling of suspension fouling in nanofiltration F. Faridirad a, Z. Zourmand a, N. Kasiri a,⁎, M. Kazemi Moghaddam b, T. Mohammadi b a b
Computer Aided Process Engineering (CAPE) Lab, School of Chemical Engineering, Iran University of Science and Technology (IUST), Narmak, Tehran 1684613114, Iran Research Centre for Membrane Separation Processes, School of Chemical Engineering, Iran University of Science and Technology (IUST), Narmak, Tehran 1684613114, Iran
H I G H L I G H T S • • • • •
Dissolution and sedimentation of solute particles during NF were considered. Considering the fouling of solutes during NF of suspensions Derive a new mathematical model which calculates permeate flux Verifying the new model with experimental data Comparing the performance of the new model with other existing models
a r t i c l e
i n f o
Article history: Received 23 March 2014 Received in revised form 6 May 2014 Accepted 7 May 2014 Available online 2 June 2014 Keywords: Membrane process Mathematical model Nanofiltration Dissolution Cross flow
a b s t r a c t Prediction of membrane process performance using experimental and mathematical models facilitates optimization of membrane processes and their operating conditions. On the other hand investigation of nanofiltration process performance over time and estimation of the time for recovery are very essential in the membrane industry. In this work a mathematical model considering membrane resistance changes and also dissolution of deposited particles in nanofiltration feed, for a cross flow nanofiltration system was developed. The validation of this model with experimental data demonstrated a good agreement. At a constant concentration of 0.2 g/L error of the present models varied from 3% (at the pressure of 1 bar) up to about 3.8% (at the pressure of 2.9 bar) while at a constant pressure of 2 bar error varied from 2.45% (at the concentration of 0.1 g/L) up to about 4.2% (at the concentration of 0.4 g/L). A comparison between this model and three other previous models proved a better performance for the present model with an average error of 3.25%. © 2014 Elsevier B.V. All rights reserved.
1. Introduction Nanofiltration is a membrane process for liquid separation which has been extended recently [1]. Reverse osmosis is replaced by nanofiltration in many applications, due to the fact that NF requires less energy comparatively. Going by its specifications, nanofiltration lays between the ultrafiltration membrane and RO. Membrane filtration processes, such as nanofiltration (NF) and reverse osmosis, play an important role in the production of high quality reclaimed water when small organic compounds, (e.g., pesticides, endocrine disruptors and pharmaceutically active compounds) are to be removed from polluted water [2]. The important application of NF is for water treatment, waste water treatment and desalination. NF is a pressure driven separation process which can separate particles based on their size and electrostatic interactions between them. For the uncharged molecules, the dominant mechanism for separation is molecular size while for the separation of ions with similar size, electrostatic forces ⁎ Corresponding author. E-mail address: [email protected] (N. Kasiri).
http://dx.doi.org/10.1016/j.desal.2014.05.014 0011-9164/© 2014 Elsevier B.V. All rights reserved.
play a more important role [3]. NF membranes have pores of around 1 nm [4], therefore they got higher water flux at lower pressures compared to RO. The main problem in using NF technology is capacity reduction of industrial units caused by membrane fouling, particularly during inorganic separation [5,6], leading to membrane life reduction [7]. Inorganic fouling induced by concentration polarization and scale formation is one of the major limitations of NF as in other membrane processes used in water treatment [6,8]. This necessitates fouling control in membrane processing [9]. Suspension fouling is of particular interest in the membrane field. Suspensions are defined as fine particles whose characteristic size is roughly in the range of 1 nm–1 μm [10]. In these processes flux decline is dependent on many factors such as membrane characteristics & module geometry, feed condition, solute type and also operating conditions [11]. Concentration polarization refers to the reversible accumulation of solute within a thin boundary layer adjacent to the membrane surface. Membrane fouling can be irreversible with solute adsorption on or in the membrane pore walls, leading to complete or partial pore blocking [12]. Formation of cake layer over the membrane surface can determine rejection properties of the system since the deposited layer will act as a “secondary”
F. Faridirad et al. / Desalination 346 (2014) 80–90
membrane prior to the “real” membrane or support material [13]. Fouling importance has initiated many efforts for modeling flux decline profile during NF [14,15]. To improve fouling models many dynamic models have been derived [4,11,16]. Theoretical models are not capable of predicting the filtration flux precisely, due to the simplifying assumptions required to enable their analytical solution [17]. Various flux decline mechanisms are studied which are generally governed by cake or Gel layer [18–20], resistance in series and pore blocking [21,22]. Ballet et al. studied experimentally the effect of feed pressure, ionic strength, concentration, and pH on the retention of phosphate anions [23]. Li et al. investigated the effects of pressure, flow rate on flux, and retention in seawater desalination process [24]. Darvishmanesh et al. developed a new semi-empirical model based on the traditional solution diffusion with imperfection model for solvent resistant NF [25]. This model demonstrated a good prediction for the flux of solvent through the membrane. Fadaei et al. developed a mass transfer model to predict ion transport through the NF membrane to account for the concentration polarization phenomenon and its influence on ion separation [26]. They used CFD techniques and successfully predicted the local concentration of ions; permeate flux, and rejection of ions in a rectangular cross flow. In the field of CFD many other works have been done such as modeling the impacts of feed spacer geometry on NF processes by Guillen and Hoek [27] and CFD modeling of porous membranes by Pak et.al [28]. Field et al. introduced the concept of critical flux for microfiltration, stating that there is a permeate flux below which fouling is not observed. It was a very powerful optimization tool for this kind of separation operation. It was possible then to identify a critical flux for UF and NF membranes [29]. The purpose of this research is to introduce a new fouling model based on cake formation to describe the flux decline caused by inorganic solutes. Fouling based models are appropriate for predicting the filtration and clean up time duration. In model development the effect of many parameters such as operating conditions, cake filtration characteristics, membrane permeability, cross flow velocity, specific cake resistance and other experimental parameters on flux reduction has been investigated. The developed model was then validated against experimental data obtained from water–oil solution NF. Finally model performance was compared with three other models developed by Lihan Huang [30] and Wu [31], as well as the cake or Gel Layer Filtration model [32]. In these models the fact of dissolution of solutes, transmission of particles through the membrane, membrane resistance changes, and also the effect of operating parameters such as cross flow velocity and temperature were neglected. Considering the solute dissolution, membrane resistance changes and also the effect of operating conditions lead to a model with better performance and better agreement with experimental data.
81
accumulation on the membrane leading to reduced cake formation and fouling [33]. Membrane fouling is related to cross flow nanofiltration velocity with larger flow velocities causing more turbulence hence less fouling [34]. One of the theories for explaining membrane fouling is cake filtration model which was developed by Hoek et al. to evaluate reverse osmosis and nanofiltration processes [35,36]. According to this model flux can be explained as: J ðt Þ ¼
Δp : Rm þ Rcp þ Rc ðt Þ
ð1Þ
In which J(t) is solvent (permeate) flux which is a function of effective pressure applied on the membrane surface (Δp), hydraulic membrane resistance (Rm) and cake layer resistance (Rc) and also the resistance related to concentration polarization layer (Rcp). Although concentration polarization layer effect is significant at the very early stages of the separation process, this gradually fades away when pore blockage takes over and due to which concentration polarization has been neglected in the model. Eq. (1) is therefore simplified to: J ðt Þ ¼
Δp : Rm þ Rc ðt Þ
ð2Þ
Cake layer resistance is calculated through Eq. (3) using specific cake resistance (α). Rc ¼ αM d
ð3Þ
In Eq. (3), Md is accumulated mass per surface unit. The constant α is related to membrane porosity through Carman–Kozeny's equation [36]: α¼
45μ ð1−ε Þ2 : ρa2p ε3
ð4Þ
Here μ is water (solvent) viscosity, ε is cake porosity, ρ is cake density and ap is particle radius. If temperature is fixed, viscosity would remain constant. Despite this, Arrhenius equation can be used to take into consideration the effect of temperature on permeate flux. E μ ¼ μ 0 exp − T R
ð5Þ
Pressure difference is calculated through Eq. (6) by using pure water permeate. ð6Þ
Δp ¼ J 0 Rm 2. Model development
In this model membrane fouling is modeled as a dynamic process. The model is based on the following assumptions:
In the NF process feed enters from one side of the cell, having passed over the membrane surface, leaves the cell from the other side with permeate passing through the membrane and exiting from a second exit of the chamber. Cross flow arrangement reduces solute particle
1- Due to the small membrane length, a uniform cake layer of a uniform resistance will be formed through the membrane surface.
Table 1 Estimated parameters of the developed model, obtained from GA, using 70% of experimental data.
c = 0.2 g/L
P = 2 bar
Pressure
ρ
α
b
a
M1
M2
M3
P P P P
2300 2400 2450 2500
1011 1012 1.4 × 1012 1013
0.00027
3.42 × 109
0.03 0.07 0.1 0.1
0.00004 0.00002 0.00001 0.000001
1000 1100 1180 1200
= = = =
1 bar 2 bar 2.4 bar 2.9 bar
Concentration
ρ
α
b
a
M1
M2
M3
c c c c
2350 2400 2400 2450
1011 1.4 × 1011 1.6 × 1011 1.8 × 1011
0.000087
3.34 × 109
1000 1070 1100 1130
0.000006 0.00003 0.00003 0.00002
0.06 0.06 0.05 0.4
= = = =
0.1 0.2 0.3 0.4
g/L g/L g/L g/L
82
F. Faridirad et al. / Desalination 346 (2014) 80–90
Fig. 1. Variation of present model parameters with pressure.
2- Solute particle sedimentation on the membrane surface is proportional to the volumetric filtrate rate passing through it [16]. 3- Sedimentation and dissolution happen simultaneously. 4- Fouling will occur when solute concentration is equal to the saturation concentration. Before this, neither sedimentation nor dissolution will take place. 5- Particles can pass through the pores and therefore there are some solute particles in the permeate side. By passing the particles through the membrane pores, some particles sediment within them, and therefore the diameter of pores reduces which causes membrane resistance to be changed. Thus membrane resistance
changes over time. As time passes and cake formation takes place leading to cake ΔP build up passing flux diminishes leading to cesar of further particle sedimentation within membrane pores. 6- Cake layer resistance is time dependent. As time passes membrane resistance to flow increases while rate of its increase over time or the increasing gradient is reduced. As this behavior could mathematically be represented by a logarithmic function, in the present work this is proposed to be presented by the following function. Rm ðt Þ ¼ Rm ð0Þ ða ln ðb t ÞÞ
Fig. 2. Variation of present model parameters with concentration.
ð7Þ
F. Faridirad et al. / Desalination 346 (2014) 80–90
83
In Eq. (7), a and b are constants that could be determined using experimental data. According to the previous assumptions solute sedimentation rate is described as [19]: Rdep ¼ M1 Q p :
ð8Þ
And solute dissolution rate is calculated through Eq. (9) [37]. Rdis ¼ M2 Q c
ð9Þ
Eq. (10) is used to describe particles passing through the membrane.
Fig. 3. Porosity changes of the cake layer with pressure in present model.
passed ¼
2
ðM 3 Þ
ð10Þ
Rm ðt Þ
In Eqs. (8), (9) and (10), M1, M2 and M3 are deposition, dissolution and transmission constants respectively. Total accumulation of solute : (M ) can be calculated through Eq. (11). ðM Þ M˙ ¼ M1 Q p −M 2 Q c − 3
2
Rm ðt Þ
¼ ρA
dl dt
ð11Þ
In Eq. (11), ρ is cake density, A is effective surface of the dl is the cake layer growth rate. Permeate flux and membrane and dt rate, are determined through Eqs. (12) and (13) respectively. J ðt Þ ¼
J 0 Rm ðt Þ Rm ðt Þ þ ρ lðt Þ α
Q p ðt Þ ¼ Fig. 4. Porosity changes of the cake layer with concentration in present model.
Q0
^ 1þk
lðt Þ ða ln ðb t ÞÞ
ð12Þ
ð13Þ
^ ¼ Q 0 ρα is the mass transfer Here l(t) is cake filtration thickness and k ΔP resistance constant.
Fig. 5. Model and experimental data (flux vs. time) at different pressures.
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Fig. 6. Error variation of developed model versus pressure.
A mass balance for flows gives Eq. (14). 1
0
ρA
B dl B ¼ M1 B @ dt 1þ
Q0 ^ ðt Þ kl a ln ðb t Þ
0
C B C B C−M2 BQ − A @
1þ
Q0 ^ ðt Þ kl
Fig. 8. Error variation of developed model versus concentration.
1
of particle concentration and β is a constant which is determined experimentally [30].
C ðM 3 Þ2 C C− A Rm;0 ða ln ðb t ÞÞ
J0 ¼
a ln ðb t Þ
ð14Þ Therefore the amount of cake layer thickness could be determined through solving Eq. (14) and the flux quantity will be calculated afterwards.
J¼
−3V 0 þ 4V 1 −V 2 2AΔt
1 dðV−V c Þ ΔPαβ 1 ¼ : A dðt−t c Þ μ ðV−V c Þα1−1
ð15Þ
ð16Þ
3.2. Gel Layer Filtration model [32,39] 3. Previous models 3.1. Lihan Huang model In this model it is assumed that fouling is a dynamic process which starts with pore blocking and then continues with cake formation on the membrane surface [38]. The initial permeate flux is calculated through Eq. (15). In Eq. (15) V0, V1 and V2 are the first infiltrated volumes and Δt is the time step between these measurements. In this model Eq. (16) is used to calculate flux. In Eq. (16) α is a function
In this model it is assumed that the size of particles is the same hence the porosity can be assumed constant. The transient flux is described by Eq. (17) [40]. In Eq. (17) solvent flux is a function of TMP, Δp(=ΔP − Δπm), hydraulic resistance of the membrane, Rm(=ΔP/μ × Jw), and cake layer resistance. At constant pressure the flux is calculated only through multiplying specific cake resistance (rc) and the mass of transient layer per surface unit (Mc), as it is shown in Eq. (18). Specific cake resistance is described through Carman– Kozeny's equation which is shown in Eq. (19). μ0 is solvent viscosity, ε is
Fig. 7. A comparison between developed model and experimental data flux at different concentrations.
F. Faridirad et al. / Desalination 346 (2014) 80–90
Fig. 9. The effect of time passage on cake thickness (a) and flux (b) with V
85
m s
as parameter.
Fig. 10. The effect of time passage on cake thickness (a) and flux (b) with P (bar) as parameter.
cake layer porosity, dp is particle diameter and ρc is particle density. Δp J ðt Þ ¼ ð17Þ Rm þ Rc ðt Þ " # 45μ 0 ð1−εÞ Mc ðt Þ ð18Þ Rc ðt Þ ¼ r c M c ðt Þ ¼ ρc d2p ε3 " # 45μ 0 ð1−εÞ rc ¼ ð19Þ ρc d2p ε3
This model has two constants which are determined through experimental data.
dJ ¼ −kP expð−k f t Þ J dt
ð20Þ
4. Results and discussions 4.1. Parameter estimation
3.3. Wu model According to this model at the beginning of the process, flux decline is proportional to flux according to Eq. (20) [40].
Experimental data for water–oil feed which has been obtained from the experimental work carried out by T. Mohammadi et al. [41] have been carried out at 4 different pressures and concentrations assuming
Fig. 11. The effect of time passage on cake thickness (a) and flux (b) with M1 as parameter.
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F. Faridirad et al. / Desalination 346 (2014) 80–90
Fig. 12. The variation of cake thickness (a) and flux (b) versus time with M2 as parameter.
a constant temperature of 30 °C. The membrane made of polysulfone was produced by DOW of Denmark. Of course the membranes which were used in this reference were ultrafiltration membranes, but due to their pore size that is close enough to the range of nanofiltration membrane size, using the data obtained from this paper does not make any significant problem in our work. The membrane surface was 28.64 cm2 and the cross flow velocity was 1 cm/s. The concentrated flow which goes out from the cell and the permeated flow from the membrane are returned to the feed tank to make the feed concentration
constant. The permeate fluxes were measured with an accuracy of 0.001 g. Flow system was laminar according to the following calculations: Re ¼
ρuDh μ
rh ¼ 4
Dh ¼ 4r h
0:0030 0:0335 ¼ 0:0055 2 ð0:0030 þ 0:0335Þ
Fig. 13. The variation of cake thickness (a) and flux (b) versus time with M3 as parameter.
Fig. 14. The variation of cake thickness (a) and flux (b) versus time with α as parameter.
F. Faridirad et al. / Desalination 346 (2014) 80–90
87
Fig. 15. A comparison of different model flux predictions at 4 different pressures (present model and previous models).
u ¼ 0:01
m s
→ Re ¼ 220:
In the first part, experiments have been done at 1, 2, 2.4, and 2.9 bar with a constant concentration of 0.2 g/L. In the second part tests have been carried out at 2 bar and at different concentrations of 0.1, 0.2, 0.3 and 0.4 g/L. Oil density was set equal to 890 kg/m3. The parameters of the model have been calculated from experimental data using the Genetic Algorithm method [42,43]. In the GA optimization method 70% of the data are used for fitting the best values to these parameters, the other 30% for goodness of fit evaluation and validation, thus providing the best fit given the data circumstances. The results have been tabulated in Table 1. Figs. 1 and 2, show the parameters of the model versus pressure and concentration respectively. From Fig. 1 it is clear that by increasing the amount of pressure M1, M3 and α increase while M2 decreases. It is reasonable as higher pressures cause more particles to permeate which means larger M1 and M3, on the other hand this causes the cake
layer to be more resistant and therefore specific resistance to increase. But this increase leads to less particle dissolution and thus decreases the amount of M2. Concentration has had the same effect on model parameters as well. By increasing the concentration a larger amount of solute finds a chance for passing through membrane pores and therefore both sedimentation (M1) and transmission (M3) constants will be increased as shown in Fig. 2. Also a larger amount of solute causes more resistant cake layer leading to a larger value for the specific cake resistance (α). The increase of specific cake resistance as a result of the concentration increase is demonstrated in Fig. 2 which also shows that increasing the concentration decreases the dissolution constant as more sedimentation takes place. Porosity changes of the cake layer with pressure and concentration are shown in Figs. 3 and 4. In reality by increasing the pressure the driving force towards the membrane increases thus applying larger force on the cake which causes it to become more compact reducing the free volume within it. This phenomenon leads to less porosity which is obvious from Fig. 3. Concentration has the same influence
Fig. 16. A comparison of error at different pressures for different model flux predictions (present model and previous models).
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F. Faridirad et al. / Desalination 346 (2014) 80–90
Fig. 17. A comparison between different models at 4 different concentrations for flux prediction (present model and previous models).
on porosity. By increasing the amount of solute, free space of the cake layer will be decreased leading to less porosity as shown in Fig. 4.
4.2. Model validation and verification Permeate flux variations with pressure as observed experimentally and predicted by the model and their relative error are demonstrated in Figs. 5 and 6 respectively. This demonstrates good compatibility. As is clear from Fig. 6 the model error increases slightly as the pressure increases. Error changes from 3% at P = 1 bar to about 3.8% at P = 2.9 bar which is not significant. A comparison between experimental data and model results has been done at different concentrations, which is shown in Fig. 7 again demonstrating a good agreement. Error variation at different concentrations is shown in Fig. 8. As shown in Fig. 8, by increasing the concentration, error increases from 2.45%, at a concentration of 0.1 g/L to 4.2% at a concentration of 0.4 g/L.
4.3. Assessment of model parameters on membrane performance The effect of different parameters on flux decline according to the time is investigated in this section. The influence of parameters such as pressure, specific cake resistance, cross flow velocity, model constants and temperature on flux and fouling will be evaluated. 4.3.1. Cross flow velocity More cross flow velocity causes more flow tension and therefore reduces concentration polarization layer [44]. It also leads to more turbulency in flow which prevents solute sedimentation and membrane fouling [5]. Cross flow velocity can be changed by feed flow rate. Fig. 9 shows the effect of cross flow velocity on flux profile and cake thickness. In Fig. 9 cross flow velocity changes from 0.08 m/s to 0.6 m/s while other parameters remain constant. As can be seen, flux reduces over time (Fig. 9b) and at higher rates initially, while cake thickness increases (Fig. 9a) with both eventually becoming constant. The process of course tends to a steady state status. By increasing the cross flow velocity, as it
Fig. 18. A comparison of error at different concentrations for different model predictions of flux (present model and previous models).
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is clear from Fig. 9, the final flux increases, while cake layer thickness decreases. It can also be seen from Fig. 9 that all the curves start from the same point. As can be seen the flux changes from 1:22 10−5 ms , −5 m when the cross flow velocity is 0.08 m/s, to 2:03 10 s when the velocity is 0.6 m/s. This fact can be explained as flow turbulency will increase by increasing the cross flow velocity, which leads to less solute sedimentation on the membrane surface, smaller cake layer thickness and therefore membrane fouling, which allows more solvent to permeate through the membrane and will increase the flux. 4.3.2. Transmembrane pressure Transmembrane pressure, TMP, is a pressure difference between two sides of the membrane which has a direct effect on permeated flux. Fig. 10 shows the effect of pressure increase on NF, when it is changed from 20.7 kPa to 62.1 kPa. As the permeated flux is related to transmembrane pressure, the amount of solvent which passes through the membrane will be increased by increasing the pressure. This fact is demonstrated in Fig. 10 where both permeated flux and cake layer thickness increase over time. As it can be seen in Fig. 10b eventually all systems yield to the same amount of final flux. 4.3.3. Sedimentation parameter (M1) Here the amount of solute which has been accumulated on the membrane surface is equal to the difference between the amount which has been deposited and the amount which has been dissolved in the feed solution, therefore the process is controlled by two parameters M1 and M2 and it is important to evaluate the effect of these two parameters on the flux profile. Fig. 11 shows the effect of sedimentation parameter on solvent flux (picture a) and cake thickness (picture b). As it can be seen from Fig. 11, by increasing the amount of M1, permeate flux decreases as the cake layer thickness increases. On the other hand larger sedimentation coefficient causes more sedimentation which causes less solvent to permeate through the membrane. It can be explained from Fig. 11 that the value of M 1 does not affect the final flux but it has a great influence on the slope of flux diagram and increases the rate of flux reduction. 4.3.4. Dissolution parameter (M2) It was mentioned before that M2 is the dissolution parameter of the solute which has been deposited on the membrane surface. The effect of this parameter on the flux of permeated solvent (picture b) and cake layer thickness (picture a) is shown in Fig. 12. Here M2 varies from a near zero value of 0.000001 to 0.0002. It can be deduced from Fig. 12 that by increasing M2 the amount of dissolution solutes in feed solution increases and therefore the amount of sedimentation on membrane surface or cake layer thickness decreases. Sedimentation and also fouling reduction, cause more solvent to permeate. This is apparently clear from Fig. 12 (picture b). 4.3.5. Transmission parameter (M3) This parameter denotes particle transmission. The effect of this term is shown in Fig. 13. It is clear that by increasing the amount of transmission parameter, more solution permeates through the membrane therefore permeate flux will be increased (picture a) while cake layer thickness (picture b) will be decreased. As it is obvious all the diagrams in both pictures a and b start from the same point but terminate at different values of flux and cake thickness. 4.3.6. Specific cake resistance (α) This parameter expresses the amount of cake layer resistance which is formed on the membrane surface. Therefore it is obvious that by increasing the amount of this parameter its effect on solvent flux is more, allowing a lesser amount of solvent to pass through the membrane leading to a decrease in the amount of cake layer thickness. In Fig. 14 the effect of specific cake resistance on permeated flux (picture b) and cake layer thickness (picture a) is shown. Here specific
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cake resistance was changed from 50 1011 1 h to 12 1013 1 h . As it is shown in Fig. 14 by increasing α the amount of permeated flux is decreased leading to a shortening of the transient period. By increasing the amount of α, cake layer resistance increases leading to less solvent permeation. Another point is that by increasing α permeated flux reaches to its constant amount in a shorter time. Also from Fig. 14 it is clear that specific cake resistant does not affect the initial and final flux and in the long run all diagrams become close together and their final flux is similar. The cake layer thickness reduction is shown in Fig. 14 (picture a). Other than the influence on flux, by increasing the amount of specific cake resistance the slope of the diagrams is increased which causes the system to take longer to become a steady state. 4.4. The comparison of present model against previous models The developed model has an acceptable performance with an average error of 3.2%. This model can predict the behavior of oil–water systems in an appropriate way which is due to the realistic set of assumptions the model is based on for the oil–water system. A comparison between different models at various pressures has been carried out with the results shown in Fig. 15. According to the assumptions of the previous models and the presented model it is apparently clear that this model performs better in flux prediction. Lihan Huang model had an acceptable performance in predicting experimental data related to oil–water mixtures. This relatively good performance is due to the acceptable assumptions that fouling is a dynamic process starting by pore blockage and continuing by cake formation. Pore blocking is a natural phenomenon which occurs at the beginning of oil–water separation, because of the small molecules capable of passing through the membrane pores. This little deviation increases by increasing the pressure. This can be justified by increasing the pressure cake layer resistance increases resulting to reduced flux while at the same time driving force increases leading to permeation of more water. In reality the second feature was dominant hence the model under predicts the flux. This difference can also be due to neglecting the dissolution effect of the particles which causes the cake layer resistance to be reduced. This difference increases by increasing the driving force. Gel Layer Filtration and Wu models do not consider membrane fouling and pore blocking which causes these models to evaluate a lower resistance against permeate flow leading to the over prediction of permeate flux compared to the experimental data. The amount of error and its changes versus pressure for different models is shown in Fig. 16. It is clear that the developed model has the smallest error compared to the others. Increased pressure only increases the error slightly. After this model, Lihan Huang showed the best performance compared to other previously developed models. A comparison among models and experimental data has been done at different concentrations, as shown in Fig. 17. As in the previous section, it can be seen from Fig. 18 that the error related to the present model is the least compared to other models. By increasing the concentration a little raise in error was detected. 5. Conclusions Modeling is an appropriate tool for predicting processes behavior in nanofiltration of suspensions. Here a mathematical model was extended for predicting and describing the behavior of a nanofiltration system and fouling. In this work available suitable models were analyzed and compared to the experimental data and also with the presently developed model. The comparison between model results and experimental data showed that the model assumptions play an important role in the final relations and therefore prediction of experimental data. As mentioned before, due to the conditions of the evaluated system and the fact that membrane resistance changes are accompanied with cake layer formation during filtration, the assumption of current model is close enough to the behavior of oil–water system enabling the model
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to predict the experimental data better than the three other models evaluated here. The average error of developed model is approximately 3.2% compared to the average errors of Lihan Huang, Wu and Gel Layer Filtration models which are 6%, 8.2% and 9.6% respectively. This increases only slightly as pressure and concentration increase demonstrating the acceptable performance of the model at higher pressures and concentrations of up to 2.9 bar and 0.4 g/L respectively. Symbols J(t) J0 Δp Rm Rcp Rc(t) μ μ0 α Md Rm (0) a b t Rdep Rdis Q0 Qc Qp M1 M2 M3 l ρ ρc A : M ^ k v0 α ε dp
permeation flux (m/s) initial permeation flux (m/s) pressure difference between two sides of the membrane (Pa) membrane resistance (kg/s·m2) polarization layer resistance (kg/s·m2) cake layer resistance (kg/s·m2) solution viscosity (Pa·s) solvent viscosity (Pa·s) specific cake resistance (1/s) accumulated mass surface unit (kg/m2) distinct membrane resistance (kg/m2) membrane resistance parameter (1/s) membrane resistance parameter (s) nanofiltration time (s) deposition rate of solute (kg/s) dissolution of solute (kg/s) feed intel rate (m3/s) cake formation rate (m3/s) permeate rate (m3/s) deposition constant (kg/m3) dissolution constant (kg/m3) passed constant (kg/m·s) cake layer thickness (m) solution density (kg/m3) cake layer density (kg/m3) membrane surface area (m2) mass accumulation on the membrane surface (kg/s) mass transfer resistance constant (1/s·m3) initial permeate volume (m3) Lihan Huang constant (1) cake layer porosity (1) particle diameter (m)
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