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Theoretical investigation of cross flow ultrafiltration by mixed matrix membrane: A case study on fluoride removal
Desalination 365 (2015) 347–354
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Desalination journal homepage: www.elsevier.com/locate/desal
Theoretical investigation of cross flow ultrafiltration by mixed matrix membrane: A case study on fluoride removal Sourav Mondal, Somak Chatterjee, Sirshendu De ⁎ Department of Chemical Engineering, Indian Institute of Technology Kharagpur, Kharagpur 721302, India
H I G H L I G H T S
G R A P H I C A L
A B S T R A C T
• Modeling performance of mixed matrix membrane in cross flow ultrafiltration • The model includes concentration polarization coupled with adsorption. • Ultrafiltration of fluoride using alumina doped mixed matrix membrane • Experimental validation and sensitivity analysis of the model parameters
The present analysis captures the physics of simultaneous concentration polarization and adsorption during ultrafiltration in mixed matrix membrane.
a r t i c l e
i n f o
Article history: Received 7 January 2015 Received in revised form 11 March 2015 Accepted 12 March 2015 Available online 19 March 2015 Keywords: Mixed matrix membrane Ultrafiltration Adsorption Cross flow Transport
a b s t r a c t Selective membrane filtration with high throughput can be achieved using mixed matrix membrane (MMM). The application of MMM in integrated membrane processing requires a continuous mode of operation. Therefore, understanding the mechanism of cross flow ultrafiltration of MMM is important from the design and operational point of view. Theoretical analysis based on first principles presented in this study takes into account the simultaneous occurrence of adsorption in the matrix and spatially developing concentration polarization layer over the membrane surface. The change in the filtration regime from adsorption dominated to diffusion governed can be identified. The developed model is validated with cross flow ultrafiltration experiments of fluoride contaminated solution using activated alumina MMM, for different operating conditions. The impact of the adsorption isotherm constants on the system performance is also evaluated. © 2015 Elsevier B.V. All rights reserved.
1. Introduction The main advantage of mixed matrix membrane (MMM) is the separation of the targeted solute with enhanced selectivity and high ⁎ Corresponding author. E-mail address: [email protected] (S. De).
http://dx.doi.org/10.1016/j.desal.2015.03.017 0011-9164/© 2015 Elsevier B.V. All rights reserved.
throughput. Typically, MMMs are prepared by doping inorganic filler into the base matrix, to impart the desired characteristics [1–3]. Several researchers have reported the use of MMM for the separation and purification of gas mixture [4–7]. The primary removal mechanism is the adsorption of the targeted species into the membrane matrix. However, in the case of filtration of liquid stream, the species diffusion and convection along with adsorption occur simultaneously [8,9]. In membrane
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filtration, the solutes are screened by physical sieving and the balance between diffusion and convection is the prevailing mechanism for transport species across the membrane. On the other hand, physical adsorption occurs in the matrix of the MMM, resulting in reduced membrane surface concentration compared to that without adsorption. Adsorption study on the membrane was first reported by Matthiasson, who found that adsorption had a significant influence on ultrafiltration [10]. The first mathematical analysis was developed by Doshi (1986) considering the simultaneous occurrence of adsorption and concentration polarization [11]. He derived a steady state model for two limiting cases of adsorption or diffusion dominating. In the case of adsorption limited process, the solute concentration at the membrane surface is negligible due to its desorption from the membrane to the permeate side. In the case of diffusion limited process, he indicated that ultrafiltration was dominated by the difference in osmotic pressure. In the modeling of protein ultrafiltration, Gekas et al., combined both adsorption and concentration polarization in a single model [12]. The theory was based on the generalized integral approach for concentration boundary layer, with a sink term for adsorption in the boundary condition at the membrane solution interface. Numerical computation for the system of equations was carried out with the finite discretization technique. The discrepancies in the modeling approach by Gekas et al., were corrected and an improved version of the model was reported by Bevia et al. [13]. They suggested that the adsorption dynamics could not be solved explicitly as the system of equations were coupled with the boundary layer equation, which was not considered by Gekas et al. They also clarified the convention of co-ordinate system selected in calculating the diffusive term. The quantification of the permeate concentration was not included in their model. Extending the analysis of coupled adsorption in MMM, Mondal et al., presented a comprehensive analysis incorporating the modifications suggested by Bevia et al., to predict the system performance for ultrafiltration in dead end configuration [8]. They developed a transport model for ultrafiltration which could predict both permeate flux and permeate concentration in batch filtration considering the effects of volume reduction with time and adsorption isotherms. The present study aims to develop a theoretical understanding of performance of MMM during cross flow ultrafiltration. Being continuous in operation, the cross flow mode of operation is important from the practical standpoint. The effect of forced convection on concentration polarization in the presence of adsorption is highlighted in this study. The transient state nature of the system is analyzed considering the adsorption dynamics which is an order of magnitude higher than the time dependent concentration term in the species convection– diffusion equation. The developing concentration boundary layer is solved using the integral method of solution. Further, the model predictions are validated for the ultrafiltration of the fluoride solution using activated alumina-cellulose acetate phthalate membrane. The model sensitivity analysis is also reported for the model parameters and constants of adsorption isotherm. 2. Theoretical development The transport mechanism of concentration polarization together with adsorption in the mass transfer boundary layer (Fig. 1) for a rectangular geometry, is described by two-dimensional convective-diffusive species transport equation. In this work, a more generalized approach of developing concentration polarization layer is considered and therefore, boundary layer thickness is not constant and varies with the axial location. ∂c ∂c ∂c ∂2 c þu þv ¼D 2 ∂t ∂x ∂y ∂y
ð1Þ
The assumptions involved in theoretical consideration are: (i) the transient term in Eq. (1) is significantly less than the order of magnitude
of the remaining terms, especially beyond 100 s of filtration [8]. Therefore, one can take recourse to quasi steady-state analysis for the determination of the concentration boundary layer profile compared to adsorption dynamics; (ii) The thickness of the concentration boundary layer is small, so that the transverse velocity is approximated to the permeation velocity at the wall [14]; (iii) laminar flow in the channel; (iv) solute diffusivity and solution viscosity are constant; (v) rheology of the fluid is Newtonian; (vi) uniform rate of adsorption in the membrane matrix and the flow channel does not contain any spacers. According to assumption (ii) the transverse velocity is expressed as, v ¼ −vw :
ð2Þ
The x-component velocity, expressed by the Poiseuille velocity for a rectangular channel is given as, 2
u¼
3 y y u 2 − 2 2 p h h
! ð3Þ
where, up is the average axial velocity and h is the channel half-height. The relevant boundary conditions for Eq. (1) are: at x ¼ 0; c ¼ c0
ð4aÞ
∂c ∂q þ ρm ð1−ε m Þt m at y ¼ 0; vw cm −cp ¼ D ∂y y¼0 ∂t y¼0
ð4bÞ
at y ¼ δðxÞ; c ¼ c0
ð4cÞ
where, cm is the solute concentration at the membrane surface (y = 0). It may be noted here, that in mass-transfer analysis, the diffusive flux D ∂c , accounting for concentration polarization is always away from ∂yy¼0 the membrane (in the form of back diffusion) as the solute concentration over the membrane surface is higher than the bulk concentration. However, this may not be true in the case of adsorption occurring on the membrane surface as in the case of MMM. The solutes polarized over membrane surface get adsorbed by the membrane and in the process, the surface concentration decreases. Hence, more solutes are diffused from the bulk toward the membrane surface, due to the positive concentration gradient [13]. The permeate flux of the membrane at any point of time is given by [15], vw ¼
ΔP−Δπ μ ðRm þ Rad Þ
ð5Þ
where, Rad is the adsorption resistance and Δπ is the osmotic pressure difference across the membrane feed and permeate side. For low molecular weight solutes and salts, the osmotic pressure is quantified using the Vant-Hoff's relation for ideal solution [16], R T þ − g π ¼ γ þ jγ j c Mw
ð6Þ
where, Mw is the molecular weight of the salt. Thus, the osmotic pressure difference (Δπ) is represented as, R T þ − g cm −cp : Δπ ¼ πjm −πjp ¼ γ þ jγ j Mw
ð7Þ
The kinetics of adsorption process is considered a first order rate process, represented by the Lagergren model as [17], h i q ¼ qe 1− exp −kp t
ð8Þ
S. Mondal et al. / Desalination 365 (2015) 347–354
349
2h
h
Membrane Permeate flux
Fig. 1. Schematic of the adsorption phenomena in cross flow mode of filtration.
where, qe is the equilibrium adsorption amount, determined by the adsorption isotherm equation. Most membrane–solute adsorption processes are described by the Langmuir isotherm [2,3] as, qe ¼
Acm : 1 þ Bcm
ð9Þ
The adsorption resistance (Rad) in the membrane separation process is proportional to the amount of solute adsorbed, independent of whether the adsorbed solute is in equilibrium with the solution or not [13]. Thus, adsorption resistance is defined as, n
Rad ¼ kad q :
ð10Þ
The concentration across the membrane feed and permeate side can be correlated using a real retention factor, specific for a particular membrane-solute system [8], Rr ¼ 1−
cp : cm
ð11Þ
Using Eqs. (2) and (3), the species transport equation (Eq. (1)) can be simplified as, y 1 y2 ∂c ∂c ∂2 c −vw ¼D 2: 3up − h 2 h ∂x ∂y ∂y
ð12Þ
The above equation can be non-dimensionalized as, 3 d ReSc e 16 L
y −
y 2
c ¼1
ð14aÞ
at; y ¼ δ ;
c ¼1
and at; y ¼ δ x ;
ð14bÞ ∂c ¼ 0: ∂y
ð14cÞ
Now, an integral method of solution to Eq. (13) is presented [14]. A quadratic concentration profile in the boundary layer is assumed,
2
c ¼ a0 þ a1 y þ a2 y :
ð15Þ
The constants a0, a1 and a2 are determined using the boundary conditions presented by Eqs. (14b) and (14c) along with the condition that at y* = 0, c* = cm⁎. Thus, the concentration profile is expressed as,
ð16Þ
2
∂c ∂c ∂ c , and 2 are evaluated and substituted in ∂x ∂y ∂y Eq. (13). Multiplying both sides of the resultant equation by dy* and integrating across the boundary layer thickness from 0 to δ*, the following equation is resulted, The derivatives
!
at; x ¼ 0;
! 2 y y : c ¼ cm −2 cm −1 − δ 2δ2
2.1. Solution of the model equations
2
ρup de μ v d and Pew ¼ w e . Consequently, the non; Sc ¼ ρD μ D dimensional boundary conditions needed to solve the above equation are, where, Re ¼
2
∂c Pe ∂c ∂ c − w ¼ 2 4 ∂y ∂x ∂y
ð13Þ
1 δ2 δ − 2ðcm −1Þ 5
!
dcm 3δ dδ − 1− ¼ dx 10 dx
8−Pew δ : ð17Þ d δ2 ReSc e L 8
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The solvent flux-adsorption equation (Eq. (4b)) at the membrane– fluid interface is non-dimensionalized as, Pew cm Rr δ þ 8 cm −1 ¼
de dq : δ dt c0 D
ð18Þ
Also, the permeate flux equation in Eq. (5) can be nondimensionalized as, 1− 0
Pew ¼ Pew
1þ
Δπ ΔP .
Rad
(iv) The above set of computational steps (i–vii) is repeated again for a next time interval, t + Δt.
:
ð19Þ
It must be mentioned here that the present problem involves differential solution of two space dimensions and there is no differential equation that is solved for the time domain. The normal initial condition of Eq. (17) is at x* = 0, δ* = 0. A close look at Eq. (17) reveals that the solution becomes indeterminate by this initial condition as δ* appears in the denominator. To circumvent this problem, an asymptotic solution of δ* is derived at x* → 0 [14]. δ x→0 ¼
Rm
2.2. Numerical solution strategy The following sequence of mathematical steps is involved in the solution of Eqs. (17)–(19) at a particular time instant, t: dq (i) The term in Eq. (18) is obtained by substituting the expression dt of qe from Eq. (9) into Eq. (8) and subsequently differentiating dq ¼ f cm ; t . Eq. (8). Thus, dt (ii) The expression of Δπ in Eq. (19) is evaluated from Eq. (7). Note that, substituting the expression of Rr from Eq. (11), Eq. (7) is simplified to, R Tc þ − g o Rc : Δπ ¼ γ þ jγ j Mw r m
ð20Þ
(iii) Similarly, the term Rad in Eq. (19) is computed from Eq. (10), substituting the values of q from Eq. (8) and qe from Eq. (9). Therefore, the quantity Rad = f(cm⁎, t). (iv) Next, Eqs. (18) and (19) are differentiated with respect to x*, to get the variation of the system variables Pew, cm⁎ and δ*, 2
Rad
!1 =
Rm
ð21Þ
de 1 dq δ Pew Rr δ þ 8− dt cm ð1 þ Bc0 cm Þ dx c0 D dPew þ c m Rr δ
dx de dq dδ −Pew cm Rr ¼ : c0 D dt dx
dcm
ð22Þ
3
ð23Þ
Using the value of δ* at x* → 0, the values of cm⁎ and Pew at x* → 0 are calculated using Eqs. (18) and (19) for each time instant t. It may be noted that the solution of the above set of DAEs' would result in the profiles of Pew, cm⁎ and δ* as a function of x⁎ at different time instants, t. Pew, cm⁎ and δ* are length averaged using Simpson's one-third rule for every time instant. The entire length of the x co-ordinate is graded at 1000 equi-spaced grids, while the time domain is divided equally in logarithmic mode at 100 intervals. It has been found that increasing the number of grids by one order would only enhance the accuracy by less than 1.5% at the cost of increased computational time by sixfold. 2.3. Determination of unknown model parameters The model parameters of this system are the adsorption resistance co-efficient kad and n, and the real retention factor Rr. The values of these parameters are optimized by minimizing the sum of the relative error of the experimental and theoretical prediction, using an optimization routine of interior point algorithm following a trust region method [19]. The objective function for error minimization is given as,
3
.
0 6 7 dcm dPew Pew Δπ n Rm ¼ −6 . ΔP cm þ Pew . c ð1 þ Bc c Þ7 4 5 dx dx Rad Rad 0 m m 1þ 1þ Rm
128x 3ReScde L
S¼
Nexp Nd j;i j;i X X vw; exp −vw i¼1 j¼1
!
j;i vw; exp
þ
N exp N d X X cp;j;iexp −cpj;i i¼1 j¼1
cp;j;iexp
! :
ð24Þ
The absolute tolerance for the objective goal attainment was given as 0.01. The number of optimization iterations required was more than 12, depending on the experimental conditions. The numerical computation is carried out using Intel Xenon (E5606) quad core 2.13 GHz with 24 GB device memory. The computational time required for one simulation of one operating conditions (corresponding to physical time of 2 h) is 13.7 min. 3. Experimental section
On substitution of
dPew from Eq. (21) into Eq. (22), one can get dx
dδ dcm ¼ f cm ; δ ; Pew , which is substituted into Eq. (17) to obtain dx dx the governing ODE for solution of δ⁎(x*).
(i) The above set of time dependent differential-algebraic equation (DAE) system (Eq. (17)–(19)) is solved numerically using MATLAB, invoking the state independent mass matrix [18]. (ii) The differential equation is solved using the 4th order Runge– Kutta method, while the algebraic equations (Eqs. (18) and (19)) are solved using the non-linear Newton–Raphson technique at every x interval. The convergence tolerance used for Newton–Raphson is 0.001. (iii) Next, the profile of cm⁎, δ* and Pew is integrated using Simpson's 3/8th rule, in the space domain to obtain the length averaged quantities, which can be compared experimentally.
The MMM was prepared using cellulose acetate phthalate (CAP) as the base polymer and activated alumina as an additive. The casting solution was prepared by doping 25 wt.% of activated alumina and 20 wt.% CAP in preheated Di-methyl formamide (DMF) at 45 °C. Uniform stirring was given to the CAP-DMF solution with intermittent heating for 4 h to ensure proper dissolution of CAP for 4 h. Next, activated alumina was added at a particular concentration and the mixture was sonicated for 6 h. The casting solution was drawn manually with a speed of 20 mm/s using a casting knife set at a fixed gap of 200 μm. The membrane was placed in a water bath at 27 °C for 16 h to complete the phase inversion. The membrane permeability (Lp), porosity (εm), density (ρm) and thickness were found to be (1.6 ± 0.03) × 10− 11 m/Pa·s, 0.52 ± 0.07, 700 ± 11 kg/m3 and 0.9 mm, respectively. Details of the physical characteristics of the membrane such as molecular weight cutoff, contact angle, surface roughness and pores size, are presented elsewhere [3].
S. Mondal et al. / Desalination 365 (2015) 347–354
The MMM was used for the ultrafiltration of the fluoride solution (feed concentration, 4 mg/l) in cross flow mode of filtration at different operating transmembrane pressures and cross flow rates. The dimension of the rectangular channel was 14.5 cm in length, 5.5 cm in width and 1.5 mm thickness. The available membrane area was 0.08 m2. Details of the cross flow ultrafiltration experimental setup are available [20]. The fluoride concentration in the permeate was measured using an ion-selective electrode (ISE). The batch adsorption experiments were conducted at a fixed temperature of 27 °C for different fluoride concentrations. The adsorption results were found to be in good agreement with the Langmuir isotherm. The isotherm constants were found to be A = 0.26 ± 0.02 m3/kg and B = 130 ± 1.5 m3/kg, corresponding to Eq. (9). The adsorption kinetic experiment was conducted, and the values of q were noted at different time instants till equilibrium was reached. The kinetic result obeyed the standard Lagergren 1st order kinetic model (Eq. (8)). The kinetic constant kp was found to be 0.58 ± 0.02 h− 1. 4. Results and discussion The optimum values of the physical constants (kad, n and Rr), obtained from the minimization of the experimental and simulated profiles of fluoride concentration using activated-alumina doped MMM are 4 × 1013 m−2, 0.15 and 0.72 respectively. The diffusivity of the fluoride molecule used in the simulation is 1.4 × 10−9 m2/s [21]. The profile of the length averaged membrane surface concentration for different operating conditions is presented in Fig. 2. The figure shows that, at the beginning of the filtration, adsorption is dominant resulting to cm⁎ b 1. This suggests that the fluoride concentration in the boundary layer is strongly adsorbed by the membrane, and the concentration gradient in this case is from the bulk to the membrane surface, unlike the case of cm⁎ N 1. This figure provides an idea about the transition time (cm⁎ ~ 1), when the governing transport mechanism changes from adsorption to concentration polarization. Comparing curves 1 and 3, it can be understood that on increasing the transmembrane pressure (TMP), cm⁎ increases sharply and is dominated by concentration polarization. This suggests that the filtration is dominated by adsorption for longer a duration (curve 1) at lower TMP. An interesting observation is found by comparing curves 1 and 2. On increasing the cross flow rate, the adsorption effect is minimized due to reduction in contact time, thereby cm⁎ increases, in the case of adsorption dominated filtration. However, when the filtration mechanism shifts to the concentration polarization regime, high cross flow rate reduces the membrane polarization and hence cm⁎ decreases. This phenomenon
351
is well established by curve 2. Thus, in order to maximize rejection of the targeted solute, one should operate at a lower flow rate, if the system is adsorption controlling and vice-versa in the case of concentration polarization dominated. As presented in Fig. 3, the adsorption resistance increases with time of filtration and becomes constant at steady state. It can be observed that fluoride removal using MMM, the magnitude of Rad/Rm b 1 and reaches a maximum of 0.18 till 3 h. The adsorption resistance for this system does not vary significantly with cross flow rate (refer to curves 1 and 2). The resistance increases marginally with TMP. A comparison of the model prediction with experimental results for different cross flow rates is presented in Fig. 4. The experimental results show that for different cross flow velocities, there is hardly any variation of permeate flux profile (Fig. 5a). The Reynolds numbers corresponding to 20, 60 and 100 L/h are 202, 606 and 1010, respectively, which suggests the flow to be in the laminar regime. The theoretical prediction also follows the same. It can be understood from Fig. 2 that the adsorption resistance is not affected by flow rate, and the same is reflected in this figure. Permeate concentration (refer Fig. 4b) increases with cross flow rate and time of operation. The permeate concentration profile is a direct consequence of the cm profile. With increase in flow rate, the contact time of solute with membrane decreases and correspondingly cm increases in the adsorption dominated filtration regime, as shown in Fig. 2. The dominant regime of filtration switches from adsorption to concentration polarization around 2 h (as shown by theoretical curves). In the case of much higher flow rate (100 L/h), the experimental result deviated from the theoretical prediction. This may be due to the very small contact time and adsorption kinetics may not follow the Lagergren model as used for the theoretical calculations. Faster kinetic mechanisms represented by second rate equations and doubleexponential models may be more applicable [22]. Also, the real retention of the membrane (Rr) is considered constant, which can be dependent on the rate of adsorption. This particular aspect can be taken up as an improvement on the existing model. Moreover, fluoride being highly electronegative and charged, there exists the possibility of solutemembrane electrostatic interactions due to which the experimental values are over predicted compared to the theoretical results. Similarly, differences in the predicted and experimental results of the permeate concentration values are also observed for modeling the filtration of steel industrial effluent using cellulose-acetate phthalate-activated carbon MMM [23]. The effect of TMP on the permeate flux and permeate concentration is illustrated in Fig. 5. As expected, the permeate flux decline is nominal as the magnitude of adsorption resistance (Rad) is maximum up to 20%
0.20
1.4
0.18
1.2
0.16
3
Rad / Rm
Cm / C0
1.6
3
1.0
2
0.8 TMP (kPa) 1: 276 2: 276 3: 690
1
0.6 0.4 0
30
60
90
120
150
cross flow rate (L/h) 20 100 20 180
Time (min) Fig. 2. Profile of the average membrane surface concentration at different operating conditions (Feed concentration of fluoride is 4 mg/l).
1 0.14 2 TMP (kPa) 1: 276 2: 276 3: 690
0.12 0.10 0
30
60
90 120 Time (min)
Cross flow rate (L/h) 20 100 20 150
180
Fig. 3. Profile of the relative adsorption resistance with different operating conditions (Feed concentration of fluoride is 4 mg/l).
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S. Mondal et al. / Desalination 365 (2015) 347–354
27
27
18
error bar ± 10%
Permeate flux (L/m2.h)
15 12 9
symbols: experiment solid line: theory
error bar ± 10% 24
Permeate flux (L/m2.h)
21
Cross flow rate (L/h) 100 60 20
(a)
24
21
TMP = 414 kPa symbols: experiment solid line: theory
18 15
TMP = 276 kPa
12
(a)
9
TMP = 138 kPa
6
6 0
30
60
90
120
150
0
180
30
60
1.4
(b)
2.4
1.2 1.0 error bar ± 10% 0.8
Cross flow rate (L/h) 100 60 20
0.6 0.4 0
30
60
90
120
150
180
Time (min)
Permeate concentration (mg/l)
Permeate concentration (mg/l)
1.8 1.6
120
150
180
Time (min)
Time (min) symbols: experiment solid line: theory
90
2.1
symbols: experiment solid line: theory
1.8
error bar ± 10%
1.5
(b)
1.2 0.9 TMP (kPa) 414 276 138
0.6 0.3 0
30
60
90
120
150
180
Time (min)
Fig. 4. Comparison of the experimental and theoretical results for different cross flow rates (a) permeate flux and (b) permeate concentration. Feed concentration of fluoride is 4 mg/l and TMP = 276 kPa.
Fig. 5. Comparison of the experimental and theoretical results for different operating TMPs (a) permeate flux and (b) permeate concentration. Feed concentration of fluoride is 4 mg/l and cross flow rate is 20 L/h.
of Rm (refer Fig. 3) and Rm is independent of TMP. Since, permeate flux is directly proportional to the TMP of the system, Fig. 5(a) shows the expected trend. For example, increasing the TMP from 138 kPa to 276 kPa, the permeate flux almost doubles from 7 L/m2 ·h to 13 L/m2·h. It must be noted here, that the theoretical prediction is almost within ±2% of the experimental data. In the case of permeate concentration, the theoretical results follow closely the experimental data in the adsorption controlled filtration regime, beyond that the permeate concentration at 414 kPa increases and deviates from the theoretical prediction. This is due to the fact that adsorption is affected by the system pressure [24], which becomes significant at higher TMP and is not included in the present model. Such pressure dependent adsorption model can be incorporated to make the model predictions more accurate at higher TMP. Fig. 5(b) also illustrates that permeate concentration increases with TMP due to higher transmembrane flux and lower solute contact time with membrane. Therefore, in order to achieve higher removal efficiency of fluoride using MMM, one must operate at lower TMP. There is a point of optimization for the TMP depending on the level of feed concentration, as increasing TMP increases both system throughput as well as permeate concentration, at the cost of higher energy consumption. The specific permeate flux in this case, is 0.05 L/m2 ·h·kPa. It may be noted that in the same range of cross flow rate, the specific permeate flux for arsenic contaminated groundwater was reported as 0.9 L/m2·h·kPa, using activated laterite-
polyacrylonitrile MMM [25]. Similarly, 0.21 L/m2·h·kPa was obtained at 20 L/h cross flow rate using alumina nanoparticle-polyacrylonitrile MMM for the removal of nitrate in synthetic solution [26]. This value was 0.14 L/m 2·h·kPa for cellulose-acetate phthalateactivated carbon MMM for the removal of phenol from steel industry effluent [23]. The effect of the model parameters, Rr, kad and n on the system performance is depicted in Fig. 6. It is clearly observed that real retention Rr has no effect on the permeate flux of the system (refer to curves 1 and 2 in Fig. 6a). On the other hand, parameters affecting the adsorption resistance influence the permeate flux considerably. Increasing kad increases the adsorption resistance thereby decreasing the permeate flux significantly (refer to curves 2 and 3). Increasing the power co-efficient n (in Eq. (10)), increases the adsorption resistance. However, in the adsorption dominated filtration regime, the relative membrane surface concentration is less than 1, thereby increasing n reduces the adsorption resistance. As a result, the permeate flux increases with n, as clearly observed by comparing curves 3 and 4. The effect of model parameters on permeate concentration is shown in Fig. 6b. As expected, the magnitude of Rr shows strong influence on the permeate concentration (curves 1 and 2). As Rr decreases, the overall rejection capacity of the membrane reduces leading to high permeate concentration. The effects of the parameters (affecting adsorption resistance) on permeate concentration are related to its impact on permeate flux. The higher
S. Mondal et al. / Desalination 365 (2015) 347–354
16
4
15.4
14
(a)
kad(m-1)
Rr
13
1: 2: 3: 4:
12 11
1013
0.6 0.8 0.8 0.8
n 0.1 0.1 0.1 1.0
1013 1014 1014
Permeate flux (L/m2.h)
2 Permeate flux (L/m2.h)
(a)
1
15
353
1: 2: 3: 4:
15.2
B(m3/kg) 50 150 150 150
kp(h-1) 0.2 0.2 0.2 1
15.0 2 14.8
1
14.6
3
4
3
10
A(m3/kg) 0.1 0.1 0.5 0.5
14.4
0
30
60
90
120
150
0
180
30
60
1: 2: 3: 4:
1013 1013 1014 1014
0.6 0.8 0.8 0.8
n
Permeate concentration (mg/l)
Permeate concentration (mg/l)
kad(m-1)
Rr
1
0.1 0.1 0.1 1.0
1.0 4
0.8
2 3
0.6 0
180
1
(b)
1.4
0.4
150
2
1.4
1.8
1.2
120
Time (min)
Time (min)
1.6
90
30
60
90
120
150
180
1.2 1.0
3
(b)
0.8 0.6 4
0.4 0.2 0
30
1: 2: 3: 4: 60
A(m3/kg) 0.1 0.1 0.5 0.5
B(m3/kg) 50 150 150 150
90 120 Time (min)
150
kp(h-1) 0.2 0.2 0.2 1 180
Time (min) Fig. 6. Effects of the model parameters on the profiles of the (a) permeate flux and (b) permeate concentration, TMP = 276 kPa; cross flow rate = 20 L/h and fluoride concentration 4 mg/l.
the adsorption resistance, the lower is the permeate flux, leading to low permeate concentration due to extended contact time between the adsorbate and the matrix. As a result, on increasing kad (curves 2 and 3) and decreasing n (curves 3 and 4), the permeate flux decreases and it is responsible for lower permeate concentration, due to higher contact time. The effect of the adsorption isotherm parameters (A and B) and kinetic constant (kp) on the permeate flux is illustrated in Fig. 7(a). The overall range of variation in permeate flux is small for different values of constants. The adsorption efficiency increases with the magnitude of B. As adsorption increases, the membrane surface concentration decreases, leading to reduced adsorption resistance. Thus, permeate flux increases with B, which is evident from curves 1 and 2. In the case of A, the phenomenon is just reverse, leading to a decrease in permeate flux (curves 2 and 3). This fact is also supported by the profiles of permeate concentration as explained in Fig. 7(b). The permeate concentration increases with membrane surface concentration. The dynamics of adsorption and the contact time during transport of the adsorbate are greatly affected by the magnitude of the kinetic constant kp. The adsorption dynamics is faster with kp. Contact time increases with kp and therefore, permeate concentration at the initial stage of filtration (curves 3 and 4 of Fig. 7b) is reduced. As the value of kp increases, adsorption is more prominent in the beginning of the filtration leading
Fig. 7. Effects of the adsorption isotherm and kinetic constants on the profiles of the (a) permeate flux and (b) permeate concentration, TMP = 276 kPa; cross flow rate = 20 L/h and fluoride concentration 4 mg/l.
to high values of q. However, the effects of kp on permeate flux (Fig. 7b) are not significant. 5. Conclusion A theoretical model for cross flow filtration in MMM in a rectangular channel has been developed and validated with the experimental data of fluoride removal by CAP-alumina membrane. The following are the conclusions of this work: (i) The developed model was able to identify the regime at filtration, i.e., adsorption dominated or concentration polarization filtration. (ii) Lower TMP and higher cross flow rate favor adsorption dominated filtration. (iii) About 2 h was required to shift the filtration mechanism from adsorption to concentration polarization. (iv) The effect of cross flow on permeate flux and permeate concentration was marginal and that of TMP was significant. (v) Real retention had marginal effects on permeate flux but its effect on permeate concentration was significant. (vi) Adsorption resistance parameter kad had more significant effect of permeate flux compared to permeate concentration.
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Similar effects were observed for variation of n on permeate flux and concentration. (vii) Effects of adsorption isotherm parameters A, B and kinetic parameter kp had significant effects on permeate concentration compared to permeate flux. (viii) Incorporation of the diffusion and adsorption mechanism inside the membrane pore and effects of concentration gradient in the membrane matrix, for more realistic description of the adsorption phenomenon, is a scope of improvement to the present analysis.
Nomenclature A, B Adsorption isotherm constants, m3/kg a0, a1, a2 Co-efficient of the concentration boundary layer profile in Eq. (15) c Concentration of the transported species, kg/m3 c* Non-dimensional concentration equivalent to c/c0 c0 Inlet concentration, kg/m3 cm Membrane surface concentration, kg/m3 cm* Non-dimensional membrane surface concentration cp Permeate concentration, kg/m3 D Diffusivity of the transported solute, m2/s de Equivalent channel diameter equivalent to 4 h, m h Channel half height, m kad Adsorption resistance constant in Eq. (10), m−1 kp Adsorption kinetic constant in Lagergren model, s−1 L Length of the cross flow membrane module, m Mw Molecular weight of the salt containing the transported species, g/mol n Adsorption resistance power co-efficient in Eq. (10), m−1 Nd Total number of data points in each experiment Nexp Total number of experimental data sets Pew Wall Peclet number equivalent to vwDde 0 Pew Wall Peclet number at x* → 0 q Solute adsorption capacity, mg/g qe Equilibrium solute adsorption capacity, mg/g Rad Membrane adsorption resistance, m−1 Re Reynolds number Rg Universal gas constant, J/mol·K Rm Membrane hydraulic resistance, m−1 Rr Real retention of the membrane S Variable defining the sum of the relative error between experiment and theoretical predictions Sc Schmidt number T Temperature, K t Time of filtration, s tm Thickness of the membrane, m u Axial velocity, m/s up Cross-sectional average axial velocity, m/s v Transverse velocity, m/s vw Permeate flux, m3/m2·s x* Non-dimensional axial co-ordinate equivalent to x/h y y-direction co-ordinate, m y* Non-dimensional y co-ordinate equivalent to y/h Greek symbols γ+ Valence of the cation in the salt γ− Valence of the anion in the salt ΔP Transmembrane pressure drop, Pa Δπ Osmotic pressure difference between the feed and permeate side, Pa δ Thickness of the concentration boundary layer, m
δ* εm μ π ρm
Non-dimensional thickness of the concentration boundary layer Porosity of the membrane matrix Viscosity of the solution, Pa·s Osmotic pressure, Pa Membrane density, kg/m3
References [1] P.S. Goh, A.F. Ismail, S.M. Sanip, B.C. Ng, M. Aziz, Recent advances of inorganic fillers in mixed matrix membrane for gas separation, Sep. Purif. Technol. 81 (2011) 243–264. [2] R. Mukherjee, S. De, Adsorptive removal of phenolic compounds using cellulose acetate phthalate–alumina nanoparticle mixed matrix membrane, J. Hazard. Mater. 265 (2014) 8–19. [3] S. Chatterjee, S. De, Adsorptive removal of fluoride by activated alumina doped cellulose acetate phthalate (CAP) mixed matrix membrane, Sep. Purif. Technol. 125 (2014) 223–238. [4] H.B.T. Jeazet, C. Staudt, C. Janiak, A method for increasing permeability in O2/N2 separation with mixed-matrix membranes made of water-stable MIL-101 and polysulfone, Chem. Commun. 48 (2012) 2140–2142. [5] B. Zornoza, C. Tellez, J. Coronas, J. Gascon, F. Kapteijn, Metal organic framework based mixed matrix membranes: an increasingly important field of research with a large application potential, Microporous Mesoporous Mater. 166 (2013) 67–78. [6] M.A. Aroon, A.F. Ismail, T. Matsuura, M.M.M. Rahmati, Performance studies of mixed matrix membranes for gas separation: a review, Sep. Purif. Technol. 75 (2010) 229–242. [7] A. Galve, D. Sieffert, C. Staudt, M. Ferrando, C. Güell, C. Téllez, J. Coronas, Combination of ordered mesoporous silica MCM-41 and layered titanosilicate JDF-L1 fillers for 6FDA-based copolyimide mixed matrix membranes, J. Membr. Sci. 431 (2013) 163–170. [8] S. Mondal, R. Mukherjee, S. Chatterjee, S. De, Adsorption-concentration polarization model for ultrafiltration in mixed matrix membrane, AIChE J. 60 (2014) 2354–2364. [9] R. Mukherjee, S. De, Adsorptive removal of phenolic compounds using cellulose acetate phthalate–alumina nanoparticle mixed matrix membrane, J. Membr. Sci. 265 (2014) 8–19. [10] E. Matthiasson, The role of macromolecular adsorption in fouling of ultrafiltration membranes, J. Membr. Sci. 16 (1983) 23–36. [11] M.R. Doshi, Limiting flux in the ultrafiltration of macromolecular solutions, in: S. Sourirajan, T. Matsuura (Eds.),Reverse Osmosis and Ultrafiltration, vol. 281, ACS Symposium Series, Washington, DC, 1985, pp. 209–223. [12] V. Gekas, P. Aimar, J.P. Lafaille, V. Sanchez, A simulation study of the adsorptionconcentration polarization interplay in protein ultrafiltration, Chem. Eng. Sci. 48 (1993) 2753–2765. [13] F.R. Bevia, V.G. Yagues, J.F. Sempere, M.J.F. Torres, An improved model with timedependent adsorption for simulating protein ultrafiltration, Chem. Eng. Sci. 52 (1997) 2343–2352. [14] S. De, S. Bhattacharjee, A. Sharma, P.K. Bhattacharya, Generalized integral and similarity solutions of the concentration profiles for osmotic pressure controlled ultrafiltration, J. Membr. Sci. 130 (1997) 99–121. [15] S. Whitaker, Flow in porous media I: a theoretical derivation of Darcy's Law, Transp. Porous Media 1 (1986) 3–25. [16] S. Mondal, S.B. Mlouka, M. Dhahbi, S. De, A physico-chemical model for polyelectrolyte enhanced ultrafiltration, J. Membr. Sci. 376 (2011) 142–152. [17] S. Lagergren, About the theory of so-called adsorption of soluble substances, Kungl. Svensk. Vet. Akad. Handl. 24 (1898) 1–39. [18] L.F. Shampine, M.W. Reichelt, J.A. Kierzenka, Solving Index-1 DAEs in MATLAB and Simulink, SIAM Rev. 41 (1999) 538–552. [19] R.H. Byrd, J.C. Gilbert, J. Nocedal, A trust region method based on interior point techniques for nonlinear programming, Math. Program. 89 (2000) 149–185. [20] S. Mondal, Chhaya, S. De, Modeling of cross flow ultrafiltration of Stevia extract in a rectangular cell, J. Food Eng. 112 (2012) 326–337. [21] R.L. Ramos, J.R. Utrilla, N.A.M. Castillo, M.S. Polo, Kinetic modeling of fluoride adsorption from aqueous solution onto bone char, Chem. Eng. J. 159 (2010) 458–467. [22] H. Qiu, L. Lv, B.-C. Pan, Q.-J. Zhang, W.-M. Zhang, Q.-X. Zhang, Critical review in adsorption kinetic models, J. Zhejiang Univ. Sci. A 10 (2009) 716–724. [23] S. Mondal, R. Mukherjee, S. De, Process modeling for the removal of phenolic compounds from industrial wastewater using a mixed-matrix membrane, Ind. Eng. Chem. Res. 54 (2015) 514–521. [24] S.S. Dukhin, G. Kretzchmar, B. Miller, Dynamics of Adsorption at Liquid Interfaces, Elsevier Science, Netherlands, 1995. [25] S. Chatterjee, S. De, Adsorptive removal of arsenic from groundwater using a novel high flux polyacrylonitrile (PAN)–laterite mixed matrix ultrafiltration membrane, Environ. Sci. Water Res. Technol. (2015), http://dx.doi.org/10.1039/c4ew00075g. [26] R. Mukherjee, S. De, Adsorptive removal of nitrate from aqueous solution by polyacrylonitrile–alumina nanoparticle mixed matrix hollow-fiber membrane, J. Membr. Sci. 466 (2014) 281–292.
Contents lists available at ScienceDirect
Desalination journal homepage: www.elsevier.com/locate/desal
Theoretical investigation of cross flow ultrafiltration by mixed matrix membrane: A case study on fluoride removal Sourav Mondal, Somak Chatterjee, Sirshendu De ⁎ Department of Chemical Engineering, Indian Institute of Technology Kharagpur, Kharagpur 721302, India
H I G H L I G H T S
G R A P H I C A L
A B S T R A C T
• Modeling performance of mixed matrix membrane in cross flow ultrafiltration • The model includes concentration polarization coupled with adsorption. • Ultrafiltration of fluoride using alumina doped mixed matrix membrane • Experimental validation and sensitivity analysis of the model parameters
The present analysis captures the physics of simultaneous concentration polarization and adsorption during ultrafiltration in mixed matrix membrane.
a r t i c l e
i n f o
Article history: Received 7 January 2015 Received in revised form 11 March 2015 Accepted 12 March 2015 Available online 19 March 2015 Keywords: Mixed matrix membrane Ultrafiltration Adsorption Cross flow Transport
a b s t r a c t Selective membrane filtration with high throughput can be achieved using mixed matrix membrane (MMM). The application of MMM in integrated membrane processing requires a continuous mode of operation. Therefore, understanding the mechanism of cross flow ultrafiltration of MMM is important from the design and operational point of view. Theoretical analysis based on first principles presented in this study takes into account the simultaneous occurrence of adsorption in the matrix and spatially developing concentration polarization layer over the membrane surface. The change in the filtration regime from adsorption dominated to diffusion governed can be identified. The developed model is validated with cross flow ultrafiltration experiments of fluoride contaminated solution using activated alumina MMM, for different operating conditions. The impact of the adsorption isotherm constants on the system performance is also evaluated. © 2015 Elsevier B.V. All rights reserved.
1. Introduction The main advantage of mixed matrix membrane (MMM) is the separation of the targeted solute with enhanced selectivity and high ⁎ Corresponding author. E-mail address: [email protected] (S. De).
http://dx.doi.org/10.1016/j.desal.2015.03.017 0011-9164/© 2015 Elsevier B.V. All rights reserved.
throughput. Typically, MMMs are prepared by doping inorganic filler into the base matrix, to impart the desired characteristics [1–3]. Several researchers have reported the use of MMM for the separation and purification of gas mixture [4–7]. The primary removal mechanism is the adsorption of the targeted species into the membrane matrix. However, in the case of filtration of liquid stream, the species diffusion and convection along with adsorption occur simultaneously [8,9]. In membrane
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filtration, the solutes are screened by physical sieving and the balance between diffusion and convection is the prevailing mechanism for transport species across the membrane. On the other hand, physical adsorption occurs in the matrix of the MMM, resulting in reduced membrane surface concentration compared to that without adsorption. Adsorption study on the membrane was first reported by Matthiasson, who found that adsorption had a significant influence on ultrafiltration [10]. The first mathematical analysis was developed by Doshi (1986) considering the simultaneous occurrence of adsorption and concentration polarization [11]. He derived a steady state model for two limiting cases of adsorption or diffusion dominating. In the case of adsorption limited process, the solute concentration at the membrane surface is negligible due to its desorption from the membrane to the permeate side. In the case of diffusion limited process, he indicated that ultrafiltration was dominated by the difference in osmotic pressure. In the modeling of protein ultrafiltration, Gekas et al., combined both adsorption and concentration polarization in a single model [12]. The theory was based on the generalized integral approach for concentration boundary layer, with a sink term for adsorption in the boundary condition at the membrane solution interface. Numerical computation for the system of equations was carried out with the finite discretization technique. The discrepancies in the modeling approach by Gekas et al., were corrected and an improved version of the model was reported by Bevia et al. [13]. They suggested that the adsorption dynamics could not be solved explicitly as the system of equations were coupled with the boundary layer equation, which was not considered by Gekas et al. They also clarified the convention of co-ordinate system selected in calculating the diffusive term. The quantification of the permeate concentration was not included in their model. Extending the analysis of coupled adsorption in MMM, Mondal et al., presented a comprehensive analysis incorporating the modifications suggested by Bevia et al., to predict the system performance for ultrafiltration in dead end configuration [8]. They developed a transport model for ultrafiltration which could predict both permeate flux and permeate concentration in batch filtration considering the effects of volume reduction with time and adsorption isotherms. The present study aims to develop a theoretical understanding of performance of MMM during cross flow ultrafiltration. Being continuous in operation, the cross flow mode of operation is important from the practical standpoint. The effect of forced convection on concentration polarization in the presence of adsorption is highlighted in this study. The transient state nature of the system is analyzed considering the adsorption dynamics which is an order of magnitude higher than the time dependent concentration term in the species convection– diffusion equation. The developing concentration boundary layer is solved using the integral method of solution. Further, the model predictions are validated for the ultrafiltration of the fluoride solution using activated alumina-cellulose acetate phthalate membrane. The model sensitivity analysis is also reported for the model parameters and constants of adsorption isotherm. 2. Theoretical development The transport mechanism of concentration polarization together with adsorption in the mass transfer boundary layer (Fig. 1) for a rectangular geometry, is described by two-dimensional convective-diffusive species transport equation. In this work, a more generalized approach of developing concentration polarization layer is considered and therefore, boundary layer thickness is not constant and varies with the axial location. ∂c ∂c ∂c ∂2 c þu þv ¼D 2 ∂t ∂x ∂y ∂y
ð1Þ
The assumptions involved in theoretical consideration are: (i) the transient term in Eq. (1) is significantly less than the order of magnitude
of the remaining terms, especially beyond 100 s of filtration [8]. Therefore, one can take recourse to quasi steady-state analysis for the determination of the concentration boundary layer profile compared to adsorption dynamics; (ii) The thickness of the concentration boundary layer is small, so that the transverse velocity is approximated to the permeation velocity at the wall [14]; (iii) laminar flow in the channel; (iv) solute diffusivity and solution viscosity are constant; (v) rheology of the fluid is Newtonian; (vi) uniform rate of adsorption in the membrane matrix and the flow channel does not contain any spacers. According to assumption (ii) the transverse velocity is expressed as, v ¼ −vw :
ð2Þ
The x-component velocity, expressed by the Poiseuille velocity for a rectangular channel is given as, 2
u¼
3 y y u 2 − 2 2 p h h
! ð3Þ
where, up is the average axial velocity and h is the channel half-height. The relevant boundary conditions for Eq. (1) are: at x ¼ 0; c ¼ c0
ð4aÞ
∂c ∂q þ ρm ð1−ε m Þt m at y ¼ 0; vw cm −cp ¼ D ∂y y¼0 ∂t y¼0
ð4bÞ
at y ¼ δðxÞ; c ¼ c0
ð4cÞ
where, cm is the solute concentration at the membrane surface (y = 0). It may be noted here, that in mass-transfer analysis, the diffusive flux D ∂c , accounting for concentration polarization is always away from ∂yy¼0 the membrane (in the form of back diffusion) as the solute concentration over the membrane surface is higher than the bulk concentration. However, this may not be true in the case of adsorption occurring on the membrane surface as in the case of MMM. The solutes polarized over membrane surface get adsorbed by the membrane and in the process, the surface concentration decreases. Hence, more solutes are diffused from the bulk toward the membrane surface, due to the positive concentration gradient [13]. The permeate flux of the membrane at any point of time is given by [15], vw ¼
ΔP−Δπ μ ðRm þ Rad Þ
ð5Þ
where, Rad is the adsorption resistance and Δπ is the osmotic pressure difference across the membrane feed and permeate side. For low molecular weight solutes and salts, the osmotic pressure is quantified using the Vant-Hoff's relation for ideal solution [16], R T þ − g π ¼ γ þ jγ j c Mw
ð6Þ
where, Mw is the molecular weight of the salt. Thus, the osmotic pressure difference (Δπ) is represented as, R T þ − g cm −cp : Δπ ¼ πjm −πjp ¼ γ þ jγ j Mw
ð7Þ
The kinetics of adsorption process is considered a first order rate process, represented by the Lagergren model as [17], h i q ¼ qe 1− exp −kp t
ð8Þ
S. Mondal et al. / Desalination 365 (2015) 347–354
349
2h
h
Membrane Permeate flux
Fig. 1. Schematic of the adsorption phenomena in cross flow mode of filtration.
where, qe is the equilibrium adsorption amount, determined by the adsorption isotherm equation. Most membrane–solute adsorption processes are described by the Langmuir isotherm [2,3] as, qe ¼
Acm : 1 þ Bcm
ð9Þ
The adsorption resistance (Rad) in the membrane separation process is proportional to the amount of solute adsorbed, independent of whether the adsorbed solute is in equilibrium with the solution or not [13]. Thus, adsorption resistance is defined as, n
Rad ¼ kad q :
ð10Þ
The concentration across the membrane feed and permeate side can be correlated using a real retention factor, specific for a particular membrane-solute system [8], Rr ¼ 1−
cp : cm
ð11Þ
Using Eqs. (2) and (3), the species transport equation (Eq. (1)) can be simplified as, y 1 y2 ∂c ∂c ∂2 c −vw ¼D 2: 3up − h 2 h ∂x ∂y ∂y
ð12Þ
The above equation can be non-dimensionalized as, 3 d ReSc e 16 L
y −
y 2
c ¼1
ð14aÞ
at; y ¼ δ ;
c ¼1
and at; y ¼ δ x ;
ð14bÞ ∂c ¼ 0: ∂y
ð14cÞ
Now, an integral method of solution to Eq. (13) is presented [14]. A quadratic concentration profile in the boundary layer is assumed,
2
c ¼ a0 þ a1 y þ a2 y :
ð15Þ
The constants a0, a1 and a2 are determined using the boundary conditions presented by Eqs. (14b) and (14c) along with the condition that at y* = 0, c* = cm⁎. Thus, the concentration profile is expressed as,
ð16Þ
2
∂c ∂c ∂ c , and 2 are evaluated and substituted in ∂x ∂y ∂y Eq. (13). Multiplying both sides of the resultant equation by dy* and integrating across the boundary layer thickness from 0 to δ*, the following equation is resulted, The derivatives
!
at; x ¼ 0;
! 2 y y : c ¼ cm −2 cm −1 − δ 2δ2
2.1. Solution of the model equations
2
ρup de μ v d and Pew ¼ w e . Consequently, the non; Sc ¼ ρD μ D dimensional boundary conditions needed to solve the above equation are, where, Re ¼
2
∂c Pe ∂c ∂ c − w ¼ 2 4 ∂y ∂x ∂y
ð13Þ
1 δ2 δ − 2ðcm −1Þ 5
!
dcm 3δ dδ − 1− ¼ dx 10 dx
8−Pew δ : ð17Þ d δ2 ReSc e L 8
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The solvent flux-adsorption equation (Eq. (4b)) at the membrane– fluid interface is non-dimensionalized as, Pew cm Rr δ þ 8 cm −1 ¼
de dq : δ dt c0 D
ð18Þ
Also, the permeate flux equation in Eq. (5) can be nondimensionalized as, 1− 0
Pew ¼ Pew
1þ
Δπ ΔP .
Rad
(iv) The above set of computational steps (i–vii) is repeated again for a next time interval, t + Δt.
:
ð19Þ
It must be mentioned here that the present problem involves differential solution of two space dimensions and there is no differential equation that is solved for the time domain. The normal initial condition of Eq. (17) is at x* = 0, δ* = 0. A close look at Eq. (17) reveals that the solution becomes indeterminate by this initial condition as δ* appears in the denominator. To circumvent this problem, an asymptotic solution of δ* is derived at x* → 0 [14]. δ x→0 ¼
Rm
2.2. Numerical solution strategy The following sequence of mathematical steps is involved in the solution of Eqs. (17)–(19) at a particular time instant, t: dq (i) The term in Eq. (18) is obtained by substituting the expression dt of qe from Eq. (9) into Eq. (8) and subsequently differentiating dq ¼ f cm ; t . Eq. (8). Thus, dt (ii) The expression of Δπ in Eq. (19) is evaluated from Eq. (7). Note that, substituting the expression of Rr from Eq. (11), Eq. (7) is simplified to, R Tc þ − g o Rc : Δπ ¼ γ þ jγ j Mw r m
ð20Þ
(iii) Similarly, the term Rad in Eq. (19) is computed from Eq. (10), substituting the values of q from Eq. (8) and qe from Eq. (9). Therefore, the quantity Rad = f(cm⁎, t). (iv) Next, Eqs. (18) and (19) are differentiated with respect to x*, to get the variation of the system variables Pew, cm⁎ and δ*, 2
Rad
!1 =
Rm
ð21Þ
de 1 dq δ Pew Rr δ þ 8− dt cm ð1 þ Bc0 cm Þ dx c0 D dPew þ c m Rr δ
dx de dq dδ −Pew cm Rr ¼ : c0 D dt dx
dcm
ð22Þ
3
ð23Þ
Using the value of δ* at x* → 0, the values of cm⁎ and Pew at x* → 0 are calculated using Eqs. (18) and (19) for each time instant t. It may be noted that the solution of the above set of DAEs' would result in the profiles of Pew, cm⁎ and δ* as a function of x⁎ at different time instants, t. Pew, cm⁎ and δ* are length averaged using Simpson's one-third rule for every time instant. The entire length of the x co-ordinate is graded at 1000 equi-spaced grids, while the time domain is divided equally in logarithmic mode at 100 intervals. It has been found that increasing the number of grids by one order would only enhance the accuracy by less than 1.5% at the cost of increased computational time by sixfold. 2.3. Determination of unknown model parameters The model parameters of this system are the adsorption resistance co-efficient kad and n, and the real retention factor Rr. The values of these parameters are optimized by minimizing the sum of the relative error of the experimental and theoretical prediction, using an optimization routine of interior point algorithm following a trust region method [19]. The objective function for error minimization is given as,
3
.
0 6 7 dcm dPew Pew Δπ n Rm ¼ −6 . ΔP cm þ Pew . c ð1 þ Bc c Þ7 4 5 dx dx Rad Rad 0 m m 1þ 1þ Rm
128x 3ReScde L
S¼
Nexp Nd j;i j;i X X vw; exp −vw i¼1 j¼1
!
j;i vw; exp
þ
N exp N d X X cp;j;iexp −cpj;i i¼1 j¼1
cp;j;iexp
! :
ð24Þ
The absolute tolerance for the objective goal attainment was given as 0.01. The number of optimization iterations required was more than 12, depending on the experimental conditions. The numerical computation is carried out using Intel Xenon (E5606) quad core 2.13 GHz with 24 GB device memory. The computational time required for one simulation of one operating conditions (corresponding to physical time of 2 h) is 13.7 min. 3. Experimental section
On substitution of
dPew from Eq. (21) into Eq. (22), one can get dx
dδ dcm ¼ f cm ; δ ; Pew , which is substituted into Eq. (17) to obtain dx dx the governing ODE for solution of δ⁎(x*).
(i) The above set of time dependent differential-algebraic equation (DAE) system (Eq. (17)–(19)) is solved numerically using MATLAB, invoking the state independent mass matrix [18]. (ii) The differential equation is solved using the 4th order Runge– Kutta method, while the algebraic equations (Eqs. (18) and (19)) are solved using the non-linear Newton–Raphson technique at every x interval. The convergence tolerance used for Newton–Raphson is 0.001. (iii) Next, the profile of cm⁎, δ* and Pew is integrated using Simpson's 3/8th rule, in the space domain to obtain the length averaged quantities, which can be compared experimentally.
The MMM was prepared using cellulose acetate phthalate (CAP) as the base polymer and activated alumina as an additive. The casting solution was prepared by doping 25 wt.% of activated alumina and 20 wt.% CAP in preheated Di-methyl formamide (DMF) at 45 °C. Uniform stirring was given to the CAP-DMF solution with intermittent heating for 4 h to ensure proper dissolution of CAP for 4 h. Next, activated alumina was added at a particular concentration and the mixture was sonicated for 6 h. The casting solution was drawn manually with a speed of 20 mm/s using a casting knife set at a fixed gap of 200 μm. The membrane was placed in a water bath at 27 °C for 16 h to complete the phase inversion. The membrane permeability (Lp), porosity (εm), density (ρm) and thickness were found to be (1.6 ± 0.03) × 10− 11 m/Pa·s, 0.52 ± 0.07, 700 ± 11 kg/m3 and 0.9 mm, respectively. Details of the physical characteristics of the membrane such as molecular weight cutoff, contact angle, surface roughness and pores size, are presented elsewhere [3].
S. Mondal et al. / Desalination 365 (2015) 347–354
The MMM was used for the ultrafiltration of the fluoride solution (feed concentration, 4 mg/l) in cross flow mode of filtration at different operating transmembrane pressures and cross flow rates. The dimension of the rectangular channel was 14.5 cm in length, 5.5 cm in width and 1.5 mm thickness. The available membrane area was 0.08 m2. Details of the cross flow ultrafiltration experimental setup are available [20]. The fluoride concentration in the permeate was measured using an ion-selective electrode (ISE). The batch adsorption experiments were conducted at a fixed temperature of 27 °C for different fluoride concentrations. The adsorption results were found to be in good agreement with the Langmuir isotherm. The isotherm constants were found to be A = 0.26 ± 0.02 m3/kg and B = 130 ± 1.5 m3/kg, corresponding to Eq. (9). The adsorption kinetic experiment was conducted, and the values of q were noted at different time instants till equilibrium was reached. The kinetic result obeyed the standard Lagergren 1st order kinetic model (Eq. (8)). The kinetic constant kp was found to be 0.58 ± 0.02 h− 1. 4. Results and discussion The optimum values of the physical constants (kad, n and Rr), obtained from the minimization of the experimental and simulated profiles of fluoride concentration using activated-alumina doped MMM are 4 × 1013 m−2, 0.15 and 0.72 respectively. The diffusivity of the fluoride molecule used in the simulation is 1.4 × 10−9 m2/s [21]. The profile of the length averaged membrane surface concentration for different operating conditions is presented in Fig. 2. The figure shows that, at the beginning of the filtration, adsorption is dominant resulting to cm⁎ b 1. This suggests that the fluoride concentration in the boundary layer is strongly adsorbed by the membrane, and the concentration gradient in this case is from the bulk to the membrane surface, unlike the case of cm⁎ N 1. This figure provides an idea about the transition time (cm⁎ ~ 1), when the governing transport mechanism changes from adsorption to concentration polarization. Comparing curves 1 and 3, it can be understood that on increasing the transmembrane pressure (TMP), cm⁎ increases sharply and is dominated by concentration polarization. This suggests that the filtration is dominated by adsorption for longer a duration (curve 1) at lower TMP. An interesting observation is found by comparing curves 1 and 2. On increasing the cross flow rate, the adsorption effect is minimized due to reduction in contact time, thereby cm⁎ increases, in the case of adsorption dominated filtration. However, when the filtration mechanism shifts to the concentration polarization regime, high cross flow rate reduces the membrane polarization and hence cm⁎ decreases. This phenomenon
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is well established by curve 2. Thus, in order to maximize rejection of the targeted solute, one should operate at a lower flow rate, if the system is adsorption controlling and vice-versa in the case of concentration polarization dominated. As presented in Fig. 3, the adsorption resistance increases with time of filtration and becomes constant at steady state. It can be observed that fluoride removal using MMM, the magnitude of Rad/Rm b 1 and reaches a maximum of 0.18 till 3 h. The adsorption resistance for this system does not vary significantly with cross flow rate (refer to curves 1 and 2). The resistance increases marginally with TMP. A comparison of the model prediction with experimental results for different cross flow rates is presented in Fig. 4. The experimental results show that for different cross flow velocities, there is hardly any variation of permeate flux profile (Fig. 5a). The Reynolds numbers corresponding to 20, 60 and 100 L/h are 202, 606 and 1010, respectively, which suggests the flow to be in the laminar regime. The theoretical prediction also follows the same. It can be understood from Fig. 2 that the adsorption resistance is not affected by flow rate, and the same is reflected in this figure. Permeate concentration (refer Fig. 4b) increases with cross flow rate and time of operation. The permeate concentration profile is a direct consequence of the cm profile. With increase in flow rate, the contact time of solute with membrane decreases and correspondingly cm increases in the adsorption dominated filtration regime, as shown in Fig. 2. The dominant regime of filtration switches from adsorption to concentration polarization around 2 h (as shown by theoretical curves). In the case of much higher flow rate (100 L/h), the experimental result deviated from the theoretical prediction. This may be due to the very small contact time and adsorption kinetics may not follow the Lagergren model as used for the theoretical calculations. Faster kinetic mechanisms represented by second rate equations and doubleexponential models may be more applicable [22]. Also, the real retention of the membrane (Rr) is considered constant, which can be dependent on the rate of adsorption. This particular aspect can be taken up as an improvement on the existing model. Moreover, fluoride being highly electronegative and charged, there exists the possibility of solutemembrane electrostatic interactions due to which the experimental values are over predicted compared to the theoretical results. Similarly, differences in the predicted and experimental results of the permeate concentration values are also observed for modeling the filtration of steel industrial effluent using cellulose-acetate phthalate-activated carbon MMM [23]. The effect of TMP on the permeate flux and permeate concentration is illustrated in Fig. 5. As expected, the permeate flux decline is nominal as the magnitude of adsorption resistance (Rad) is maximum up to 20%
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Fig. 4. Comparison of the experimental and theoretical results for different cross flow rates (a) permeate flux and (b) permeate concentration. Feed concentration of fluoride is 4 mg/l and TMP = 276 kPa.
Fig. 5. Comparison of the experimental and theoretical results for different operating TMPs (a) permeate flux and (b) permeate concentration. Feed concentration of fluoride is 4 mg/l and cross flow rate is 20 L/h.
of Rm (refer Fig. 3) and Rm is independent of TMP. Since, permeate flux is directly proportional to the TMP of the system, Fig. 5(a) shows the expected trend. For example, increasing the TMP from 138 kPa to 276 kPa, the permeate flux almost doubles from 7 L/m2 ·h to 13 L/m2·h. It must be noted here, that the theoretical prediction is almost within ±2% of the experimental data. In the case of permeate concentration, the theoretical results follow closely the experimental data in the adsorption controlled filtration regime, beyond that the permeate concentration at 414 kPa increases and deviates from the theoretical prediction. This is due to the fact that adsorption is affected by the system pressure [24], which becomes significant at higher TMP and is not included in the present model. Such pressure dependent adsorption model can be incorporated to make the model predictions more accurate at higher TMP. Fig. 5(b) also illustrates that permeate concentration increases with TMP due to higher transmembrane flux and lower solute contact time with membrane. Therefore, in order to achieve higher removal efficiency of fluoride using MMM, one must operate at lower TMP. There is a point of optimization for the TMP depending on the level of feed concentration, as increasing TMP increases both system throughput as well as permeate concentration, at the cost of higher energy consumption. The specific permeate flux in this case, is 0.05 L/m2 ·h·kPa. It may be noted that in the same range of cross flow rate, the specific permeate flux for arsenic contaminated groundwater was reported as 0.9 L/m2·h·kPa, using activated laterite-
polyacrylonitrile MMM [25]. Similarly, 0.21 L/m2·h·kPa was obtained at 20 L/h cross flow rate using alumina nanoparticle-polyacrylonitrile MMM for the removal of nitrate in synthetic solution [26]. This value was 0.14 L/m 2·h·kPa for cellulose-acetate phthalateactivated carbon MMM for the removal of phenol from steel industry effluent [23]. The effect of the model parameters, Rr, kad and n on the system performance is depicted in Fig. 6. It is clearly observed that real retention Rr has no effect on the permeate flux of the system (refer to curves 1 and 2 in Fig. 6a). On the other hand, parameters affecting the adsorption resistance influence the permeate flux considerably. Increasing kad increases the adsorption resistance thereby decreasing the permeate flux significantly (refer to curves 2 and 3). Increasing the power co-efficient n (in Eq. (10)), increases the adsorption resistance. However, in the adsorption dominated filtration regime, the relative membrane surface concentration is less than 1, thereby increasing n reduces the adsorption resistance. As a result, the permeate flux increases with n, as clearly observed by comparing curves 3 and 4. The effect of model parameters on permeate concentration is shown in Fig. 6b. As expected, the magnitude of Rr shows strong influence on the permeate concentration (curves 1 and 2). As Rr decreases, the overall rejection capacity of the membrane reduces leading to high permeate concentration. The effects of the parameters (affecting adsorption resistance) on permeate concentration are related to its impact on permeate flux. The higher
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the adsorption resistance, the lower is the permeate flux, leading to low permeate concentration due to extended contact time between the adsorbate and the matrix. As a result, on increasing kad (curves 2 and 3) and decreasing n (curves 3 and 4), the permeate flux decreases and it is responsible for lower permeate concentration, due to higher contact time. The effect of the adsorption isotherm parameters (A and B) and kinetic constant (kp) on the permeate flux is illustrated in Fig. 7(a). The overall range of variation in permeate flux is small for different values of constants. The adsorption efficiency increases with the magnitude of B. As adsorption increases, the membrane surface concentration decreases, leading to reduced adsorption resistance. Thus, permeate flux increases with B, which is evident from curves 1 and 2. In the case of A, the phenomenon is just reverse, leading to a decrease in permeate flux (curves 2 and 3). This fact is also supported by the profiles of permeate concentration as explained in Fig. 7(b). The permeate concentration increases with membrane surface concentration. The dynamics of adsorption and the contact time during transport of the adsorbate are greatly affected by the magnitude of the kinetic constant kp. The adsorption dynamics is faster with kp. Contact time increases with kp and therefore, permeate concentration at the initial stage of filtration (curves 3 and 4 of Fig. 7b) is reduced. As the value of kp increases, adsorption is more prominent in the beginning of the filtration leading
Fig. 7. Effects of the adsorption isotherm and kinetic constants on the profiles of the (a) permeate flux and (b) permeate concentration, TMP = 276 kPa; cross flow rate = 20 L/h and fluoride concentration 4 mg/l.
to high values of q. However, the effects of kp on permeate flux (Fig. 7b) are not significant. 5. Conclusion A theoretical model for cross flow filtration in MMM in a rectangular channel has been developed and validated with the experimental data of fluoride removal by CAP-alumina membrane. The following are the conclusions of this work: (i) The developed model was able to identify the regime at filtration, i.e., adsorption dominated or concentration polarization filtration. (ii) Lower TMP and higher cross flow rate favor adsorption dominated filtration. (iii) About 2 h was required to shift the filtration mechanism from adsorption to concentration polarization. (iv) The effect of cross flow on permeate flux and permeate concentration was marginal and that of TMP was significant. (v) Real retention had marginal effects on permeate flux but its effect on permeate concentration was significant. (vi) Adsorption resistance parameter kad had more significant effect of permeate flux compared to permeate concentration.
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Similar effects were observed for variation of n on permeate flux and concentration. (vii) Effects of adsorption isotherm parameters A, B and kinetic parameter kp had significant effects on permeate concentration compared to permeate flux. (viii) Incorporation of the diffusion and adsorption mechanism inside the membrane pore and effects of concentration gradient in the membrane matrix, for more realistic description of the adsorption phenomenon, is a scope of improvement to the present analysis.
Nomenclature A, B Adsorption isotherm constants, m3/kg a0, a1, a2 Co-efficient of the concentration boundary layer profile in Eq. (15) c Concentration of the transported species, kg/m3 c* Non-dimensional concentration equivalent to c/c0 c0 Inlet concentration, kg/m3 cm Membrane surface concentration, kg/m3 cm* Non-dimensional membrane surface concentration cp Permeate concentration, kg/m3 D Diffusivity of the transported solute, m2/s de Equivalent channel diameter equivalent to 4 h, m h Channel half height, m kad Adsorption resistance constant in Eq. (10), m−1 kp Adsorption kinetic constant in Lagergren model, s−1 L Length of the cross flow membrane module, m Mw Molecular weight of the salt containing the transported species, g/mol n Adsorption resistance power co-efficient in Eq. (10), m−1 Nd Total number of data points in each experiment Nexp Total number of experimental data sets Pew Wall Peclet number equivalent to vwDde 0 Pew Wall Peclet number at x* → 0 q Solute adsorption capacity, mg/g qe Equilibrium solute adsorption capacity, mg/g Rad Membrane adsorption resistance, m−1 Re Reynolds number Rg Universal gas constant, J/mol·K Rm Membrane hydraulic resistance, m−1 Rr Real retention of the membrane S Variable defining the sum of the relative error between experiment and theoretical predictions Sc Schmidt number T Temperature, K t Time of filtration, s tm Thickness of the membrane, m u Axial velocity, m/s up Cross-sectional average axial velocity, m/s v Transverse velocity, m/s vw Permeate flux, m3/m2·s x* Non-dimensional axial co-ordinate equivalent to x/h y y-direction co-ordinate, m y* Non-dimensional y co-ordinate equivalent to y/h Greek symbols γ+ Valence of the cation in the salt γ− Valence of the anion in the salt ΔP Transmembrane pressure drop, Pa Δπ Osmotic pressure difference between the feed and permeate side, Pa δ Thickness of the concentration boundary layer, m
δ* εm μ π ρm
Non-dimensional thickness of the concentration boundary layer Porosity of the membrane matrix Viscosity of the solution, Pa·s Osmotic pressure, Pa Membrane density, kg/m3
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