Process Modeling for the Removal of Phenolic Compounds from

Dec 17, 2014 - Process modeling of the performance of the mixed-matrix ultrafiltration ... couple of decades.1−3 Typically, the membrane process inc...
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Process Modeling for the Removal of Phenolic Compounds from Industrial Wastewater Using a Mixed-Matrix Membrane Sourav Mondal, Raka Mukherjee, and Sirshendu De* Department of Chemical Engineering, Indian Institute of Technology Kharagpur, Kharagpur 721302, India S Supporting Information *

ABSTRACT: Application of a mixed-matrix membrane in the treatment of industrial wastewater provides an added advantage of coupling adsorption with membrane filtration. This is particularly useful in achieving high selectivity with enhanced throughput. Continuous cross-flow experiments were conducted with effluent from a secondary treatment plant of the steel industry containing a high concentration of phenol. The transport phenomenon involves the concentration polarization over the gel layer, growth of the gel-layer dynamics, and adsorption in the membrane matrix. Because the real-life effluent is a complex mixture containing several components, knowledge of the physical properties is challenging. Process modeling of the performance of the mixed-matrix ultrafiltration membrane of industrial effluent in a cross-flow configuration is attempted in this study. The mathematical analysis is based on the transport mechanism of the low-molecular-weight solute through a gel of large-molecularweight solutes and consequent adsorption in the membrane matrix. The model results are compared with the actual experimental data. The effects of the adsorption isotherm and sensitivity of the model parameters on the system performance are also evaluated.

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INTRODUCTION The treatment of wastewater by membrane separation processes has been an active area of research for the past couple of decades.1−3 Typically, the membrane process includes microfiltration, ultrafiltration (UF), or nanofiltration or a combination of them, depending upon the process requirement.4−6 The membrane selectivity is improved using lower-pore-size membranes, at the cost of reduced throughput. The use of a hybrid process particularly adsorption coupled with membrane separation has gained considerable interest because of its merit of achieving high throughput with selectivity.7,8 However, such hybrid processes are associated with several problems: (i) regeneration of the adsorbent from the reject stream is difficult; (ii) in the case of low-cost adsorbent, disposal of the spent adsorbent becomes a problem; (iii) fouling of the membrane becomes significant in the long run; (iv) competition among the targeted solute with other solutes reduces the rate of adsorption. The use of a mixed-matrix membrane (MMM) is an active area of research because it combines both adsorption and membrane separation in one step and one can achieve high selectivity with enhanced throughput.9,10 Inorganic filler in the polymeric matrix imparts the selective separation of a targeted solute.11 Therefore, the filtration mechanism of MMM involves preferential solute adsorption together with solute convection and diffusion.12 Understanding the underlying transport mechanism is important in the context of the design, operation, and performance prediction of such systems. The first mathematical attempt in describing such a coupled phenomenon was reported by Doshi.13 He calculated the limiting cases of an adsorption- or a diffusion-dominated situation. His analysis showed that, in the case of an adsorption-limited process, the membrane surface concentration is insignificant because of desorption of the adsorbed solute from the © 2014 American Chemical Society

membrane to the permeate stream. On the other hand, the osmotic pressure difference controls filtration in the diffusionlimited case. Its subsequent development including the simultaneous occurrence of adsorption and concentration polarization was modeled by Gekas et al.14 The model was based on the generalized concentration boundary layer equation including an additional source term in the membrane−liquid interface to account for adsorption. However, this study had certain limitations, which were pointed out by Ruiz-Bevia et al., proposing a revised model to quantify the process.15 The major assumption by Gekas et al. was that the adsorption dynamics remained unaffected by the solute adsorbed, leading to explicit solution adsorption resistance, which was incorrect. Also, there was an apparent inconsistency in the sign convention of the diffusive term in the membrane− fluid interface boundary condition. The work by Ruiz-Bevia et al. corrected these, and the developed model provided insight into the complex mechanism of the coupled adsorption− concentration polarization.15 However, this study did not give any information about the permeate concentration because it is related to complete retention of the solute and is valid only for a dead-end configuration. The study of Ruiz-Bevia et al. dealt with the separation of macromolecules, like protein from an aqueous solution having high osmotic pressure, and solute adsorption was mostly on the membrane−solution interface. Mondal et al.16 developed a comprehensive model following the derivations by Ruiz-Bevia et al. to predict the permeate flux and permeate concentration of smaller-sized solutes, like phenol, fluoride, etc., during UF by MMM in an unstirred Received: Revised: Accepted: Published: 514

November 4, 2014 December 14, 2014 December 17, 2014 December 17, 2014 DOI: 10.1021/ie504358j Ind. Eng. Chem. Res. 2015, 54, 514−521

Article

Industrial & Engineering Chemistry Research batch cell. The model takes into account the developing masstransfer boundary layer, adsorption dynamics influencing the adsorption resistance and solute adsorption rate, and increasing feed concentration with time. However, the present analysis is more focused on a relatively more complex phenomenon of UF of a real-life industrial effluent using MMM in a continuous cross-flow configuration. The effluent contains several components, and therefore the formation of a gel-type layer over the membrane surface occurs along with adsorption and concentration polarization. The pore size of the UF membrane varies from 2 to 10 nm, while the size of the low-molecularweight solute (phenolics) is in the range of 0.5−0.7 nm and the particle size distribution of the effluent stream varies from 0.5 to 100 μm (refer to Figure S1 in the Supporting Information). Therefore, the occurrence of gel formation is quite likely. In the present study, a two-component model considering the development of a gel layer by a high-molecular-weight solute and transport of a low-molecular-weight solute through the gel is coupled with the adsorption dynamics in the membrane matrix. The interplay between the adsorption resistance, gel resistance, and concentration boundary layer is described by the mathematical equations. The dynamics of the adsorption resistance is fast compared to that in the gel layer, and the effect of adsorption is more dominant on the first few hours of filtration. The plot in the graphical abstract represents this phenomenon. It clearly explains the relative influence of the two governing mechanisms (gel formation and adsorption) on the overall flux decline profile. The unknown physical properties of the real-life stream make the problem challenging. The model is applied to the UF of effluent from a secondary treatment step of a steel industry using a powdered activated carbon (PAC)−cellulose acetate phthalate (CAP) MMM. The effects of the operating conditions on the filtration performance and understanding the transport phenomena are quantified. The impact of the model parameters and adsorption isotherm on the system behavior is also studied. In summary, the novelty of this work addresses the following aspects: (i) transport modeling of a continuous membrane process with three simultaneous effectsconcentration polarization, gel-layer formation, and adsorption; (ii) performance prediction of MMM-based membrane filtration of a real-life industrial effluent; (iii) effect of the adsorption isotherm on the system performance, so that model analysis can be generalized for any such system; (iv) determination of the unknown physical parameters for a real-life effluent, which is a mixture of several components.

The mass-transfer coefficients for the gel-forming and smallersized solute are k1 and k2, respectively, and they are evaluated using the Sherwood number relation. 1/3 ⎛ d ⎞ Sh = 1.86⎜Re × Sc e ⎟ ⎝ L⎠

In the case of a synthetic solution (gel-forming solutes are absent) containing only the adsorbed solute, it has been experimentally observed that flux decline occurs and adsorption is solely responsible for it.16 Once the solute gets adsorbed, the permeability gets affected, which is reflected through an additional resistance due to adsorption. Considering a species balance of the smaller-sized solute in the membrane, the adsorption dynamics can be expressed as15,16 vw(c 2m − c 2p) = ρm (1 − εm)tm

(5)

where qe is the equilibrium adsorption amount, determined by the adsorption isotherm equation. Most membrane−solute adsorption processes are described by the Langmuir isotherm as

qe =

Ac 2m 1 + Bc 2m

(6)

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.15 Thus, the adsorption resistance is defined as R ad = kadc 2m n

(7)

The gel-layer resistance (Rg) can be quantified according to the conventional cake filtration theory20 as R g = α(1 − εg)ρg L = βL

(8)

where β = α(1 − εg)ρg is a constant for a particular solute. The osmotic pressure due to the gel-forming solute is negligible compared to the smaller-sized solute21 and is adsorbed by the membrane. The osmotic pressure difference due to the lowmolecular-weight solute is determined by the van’t Hoff relation for the solution containing a smaller-sized solute:22 Δπ = π (c 2m) − π (c 2p) =

RT (c 2m − c 2p) M2

(9)

Thus, the filtration flux is calculated considering the resistances due to gel formation, adsorption, and osmotic pressure as

(1)

vw =

The concentration of the low-molecular-weight, smaller-sized solute across the MMM is expressed as18

Δπ

(

)

vw0 1 − ΔP ΔP − Δπ = Rg R μ(R m + R g + R ad) 1 + R + Rad m

⎛vL⎞ 1 exp⎜⎜ w ⎟⎟[c 2p(1 − γg) + (c 2b − c 2p) γg ⎝ εgD2 ⎠ exp(vw /k 2)]

(4)

q = qe[1 − exp(−k pt )]

THEORY Considering a material balance of the gel-forming solute over the membrane surface results in the dynamics of gel formation,17

c 2m − c 2p =

dq dt

Now, the kinetics of the adsorption process is assumed to be first-order and represented by the Lagergren model as19

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c1b − c1g exp( −vw /k1) dL ρg (1 − εg) = vw dt 1 − exp( −vw /k1)

(3)

m

(10)

In the above equation, Rm, Rg, and Rad are the hydraulic resistance of the membrane, gel-layer resistance, and adsorption resistance, respectively. The system variables for the above equations (eqs 1−10) are L, vw, c2m, and c2p. The initial conditions of eq 1 are at t = 0 and L = 0.

(2) 515

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The batch adsorption experiments were conducted at a fixed temperature of 27 ± 1 °C for different phenol concentrations. The adsorption results were found to be in good agreement with the Langmuir isotherm. The constants were found to be A = 0.486 ± 0.03 m3/kg and B = 7.29 ± 0.15 m3/kg corresponding to eq 6. The adsorption kinetic experiment was conducted at different time intervals until equilibration was reached in a glass beaker. The kinetic results obeyed the standard Lagergren first-order kinetic model (eq 5). The kinetic constant kp was found to be 0.872 ± 0.02 h−1.

The above set of equations (eqs 1, 2, and 4) constitute a system of differential algebraic equations (DAEs), that is, solved numerically using the state-independent matrix in MATLAB.23 Thus, the profiles of the permeate flux (vw), permeate concentration (c2p), adsorption resistance (Rad), gel-layer thickness (L), and gel-layer resistance (Rg) are computed for different operating conditions. The unknown parameters of the system of equations are D1, ρg, εg, c1g, β, γg, kad, and n. Among these eight parameters, D1, ρg, εg, and c1g are the physical properties of the system particularly for this feed composition. The remaining four parameters β, γg, kad, and n are the model constants specific to a solute−membrane pair. The unknown parameters are evaluated by simultaneously optimizing the experimental profiles of the permeate flux and concentration for all of the different operating conditions (S).17,24 The objective function (S) for error minimization is given as Nexp Nd

S=

⎛ v j,i

∑ ∑ ⎜⎜ i=1 j=1

⎝

j,i ⎞ w,exp − vw ⎟⎟ j,i vw,exp ⎠

+

Nexp Nd

⎛ c j,i

i=1 j=1

⎝

∑ ∑ ⎜⎜

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RESULTS AND DISCUSSION The developed model is applied to predict the system performance of cross-flow UF of a steel-plant effluent. The gel-forming solutes constitutes a complex mixture of the total nonvolatile compounds and other solid components (concentration: c1 = 2.5 kg/m3). The low-molecular-weight solute is represented by phenol and its derivatives (concentration: c2 = 0.023 kg/m3), which are selectively adsorbed by the MMM. The diffusivity of the phenolic compound (D2) is 8.9 × 10−10 m2/s as obtained from the literature.27 The values of the optimization parameters for the case of cross-flow UF of steel effluent using PAC MMM are presented in Table 1.

j,i ⎞ 2p,exp − c 2p ⎟⎟ j,i c 2p,exp ⎠

(11)

S is minimized using an optimization routine of the interior point algorithm following a trust region method.25

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Table 1. Values of the Optimization Parameters

METHODS

variable Physical Property Constants D1, m2/s ρg, kg/m3 εg c1g/c0 Model Constants γg kad, m−1 n β, m−2

The MMM was prepared using CAP as the base polymer and 25% (w/v) PAC as an additive. Dimethylformamide was used to prepare the casting solution. The casting solution was drawn manually with a speed of 20 mm/s using a doctor’s blade with a gap of 200 μm over nonwoven polyester fabric, acting as a support to the membrane. The membranes were put in a water bath at 27 °C for 16 h to complete phase inversion. The membrane permeability is calculated by measuring the pure water flux at different transmembrane pressures (TMPs). The slope of the straight line representing the pure water flux with respect to the pressure gives the membrane permeability. The membrane density is measured by calculating the dry weight of known dimensions of a membrane sample. The membrane porosity is measured by calculating the difference of the dry and wet weights of a known volume of the membrane sample. The permeability of the membrane was found to be (7.8 ± 0.3) × 10−11 m/Pa·s. The density of the prepared membrane (ρm) was 1550 ± 23 kg/m3, and the porosity (εm) was 0.55 ± 0.03. The membrane experiments were conducted in a cross-flow filtration setup. A flow schematic diagram (indicating the flow lines) of the experimental setup is presented in the graphical abstract. The available membrane area was 0.08 m2. Effluent from the secondary treatment step of the steel industry was collected from Tata Steel, Jamshedpur, Jharkhand, India. The effluent was fed to this system after pretreatment by centrifugation to reduce the total solid load on the membrane. The total phenol content, which was the adsorbate for the present analysis, was found to be 23 mg/L, and the total solid content was 2.5 g/L, which was considered to be the gelforming material. The total solid was measured gravimetrically, and the total phenol was measured using an APHA method.26 Each experiment was conducted in triplicate for identical operating conditions; the standard deviation of each experimental data point was calculated and found to be within ±10% of the mean data. The experimental results are therefore reported as mean ±10%, in the subsequent figures.

value (3.5 ± 0.2) × 10−10 2800 ± 21 0.5 ± 0.02 55 ± 3.1 150 ± 11.4 (4.2 ± 0.2) × 1013 0.2 ± 0.003 (4.8 ± 0.09) × 1015

The simulation results are compared with the experimental data for operation at different TMPs in Figure 1. The results show that the model prediction is within ±10% of the experimental flux, as shown in Figure 1a. One can understand from the mathematical analysis that the filtration mechanism is governed by both gel formation and adsorption in the membrane matrix. With an increase in the pressure, the permeate flux increases as expected. In the case of gel controlling filtration, the permeate flux is invariant with the TMP if the filtration is conducted in a true pressureindependent region.28 However, in the presence of adsorption, attainment of such an operating zone may be difficult. The steady state for the system is attained within 1 h of operation. Within this time (1 h), the permeate flux declines sharply (about 50%) because of buildup of the gel layer. The increase in the steady-state permeate flux is linearly proportional to the TMP. For example, when the pressure is increased from 414 to 552 kPa, the permeate flux increases from 56.7 to 73.8 L/m2·h (30%). In the case of the permeate concentration, the model prediction follows the experimental profiles for all TMPs (refer to Figure 1b). It can be observed from this figure that the permeate concentration decreases with the filtration time. The permeate flux is high at the start of filtration; therefore, the contact time between the adsorbate and adsorbent is less, leading to a higher concentration of phenol in the permeate. 516

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Figure 1. Comparison of the experimental and simulated results of the (a) permeate flux and (b) permeate concentration for different TMPs.

Figure 2. Comparison of the experimental and simulated results of the (a) permeate flux and (b) permeate concentration for different cross-flow rates.

observed from this figure that model predictions are close to the experimental results after the permeate concentration approaches steady-state values. However, there is a deviation between the experimental and simulated profiles for the permeate concentration (both Figures 1b and 2b) for the initial 90 min of filtration. There are several reasons to this observation. The change in the adsorption reaction order in the initial duration could be one of the reasons for the deviation, which is not accounted for in the model. The adsorption isotherm (Langmuir) constants are experimentally calculated using a synthetic phenol solution having different concentrations. However, in the presence of other pollutants in the stream, the magnitude of the constants can be affected by it, which is not measured or reported in this study. This could be one of the causes for the deviation in the model predictions. Also, the feed is a complex mixture (effluent) containing several solutes, which are also adsorbed together with phenol in the membrane matrix. However, in modeling, it is assumed that only phenol gets preferentially adsorbed and only the adsorption resistance is affected by it, which does not happen in reality. The effect of multicomponent adsorption on the phenol concentration, and its subsequent impact on the adsorption resistance, is possible, which is beyond the scope of the present analysis. Variation of the gel thickness with the operating conditions is illustrated in Figure 3. As observed from the figure, the gel layer grows with the filtration time. The gel-layer thickness increases with the TMP and decreases with the cross-flow rate. With a

For example, the permeate concentration drops from 4.3 to 1.9 mg/L. In the case of higher TMP, the permeate concentration is also higher because of the short contact time. Hence, the permeate concentration is associated with lower permeate flux. Therefore, there exists a trade-off between the permeate flux and concentration, as the permeate concentration at the cost of reduced throughput, which needs to be optimized considering the discharge regulations and required output. The effect of the cross-flow rate on the system performance is depicted in Figure 2. Analysis of the concentration boundary layer suggests that the permeate flux increases with the crossflow rate because of enhanced mass transport at higher forced convection. The steady-state permeate flux increases from 56.7 to 81.7 L/m2·h as the cross-flow rate is increased from 20 to 60 L/h. However, because the model takes into account the adsorption resistance together with the gel resistance, the permeate flux is affected by the competing phenomena and, hence, the model prediction of the permeate flux does not follow Re1/3 variation as in the case of pure mass-transfercontrolled filtration without adsorption.29 The permeate concentration also increases with the cross-flow rate, as observed from Figure 2b. This is due to the fact that, with an increase in the flow rate, the permeate flux increases, thereby decreasing the contact time between the solute and adsorbent, similar to the case of variation of the permeate concentration with the TMP. In this situation also, the permeate concentration is high at the start of the filtration when the permeate flux is high and decreases subsequently. It can be 517

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demonstrate that the breakthrough profile does not occur within 8 h of filtration. The adsorption resistance, on the other hand, increases sharply during the start of the experiment and becomes steady beyond 2 h. The adsorption phenomenon is influenced by the TMP and cross-flow rate (refer to Figure 4). There are several factors that influence the adsorption resistance. The adsorption resistance sharply increases in the initial 90 min, as a result of a change in the permeate concentration. Also, as observed from Figure 4b, it is clear that the adsorption resistance is dominating the overall filtration process and in the beginning it is the most influential. So, it is understood that the more realistic modeling of the adsorption resistance in the beginning of the filtration (during which it has a significant impact) would help in close model predictions of the actual phenomenon. The adsorption resistance increases with pressure, as can be observed from curves 1 and 2. When the pressure is increased from 414 to 690 kPa, the relative magnitude of the adsorption resistance increases by 25%. However, the pressure does not have any effect on the time taken to reach steady state; that is, both curves 1 and 2 attain steady state at around 2 h, as shown in Figure 4a. The effect of forced convection can be understood by comparing curves 2 and 3. When the cross-flow rate is increased, the adsorption resistance decreases. This is because the gel-layer formation is reduced because of enhanced convection, thereby increasing the permeate flux and subsequently reducing solute adsorption. The relative effect of the adsorption resistance to gel resistance is presented in Figure 4b. Because the magnitudes of Rg/Rad and Rad/Rm are less than 1, the overall ratio of Rg/Rm is always less than 1, which suggests that gel formation is not dominating for the present operating conditions, but it exists. The adsorption resistance remains constant beyond a critical time (about 90 min). The rise in the relative ratio of Rg/Rad beyond this time suggests that Rg increases continuously (during the entire 8 h of filtration time) when operated at higher TMP (curve 1). Beyond the critical time, when Rad becomes constant, the profile of Rg/Rad is dictated by Rg. The general trend of Rg/Rm follows a line similar to the growth of the gel-layer dynamics, as referred to in Figure 4. In summary, the cross-flow rate influences the contact time of the adsorbate to the adsorbent. The contact time decreases with the flow rate; therefore, the adsorption decreases. Second, the membrane surface concentration (which subsequently governs adsorption via eq 7) itself decreases with the cross-flow rate. This also accounts for the reduction in adsorption. Consequently, the permeate concen-

Figure 3. Growth of the gel-layer thickness for different operating conditions.

rise in the pressure, the solvent flux increases toward the membrane, leading to enhanced deposition of gel-forming particles. For example, in a comparizon of curves 1 and 2, upon an increase of ΔP from 414 to 690 kPa, the gel thickness doubles from 130 to 260 μm after 3 h of operation and becomes 500 μm after 6 h of operation. The growth of the gel layer is arrested by forced convection imparted by higher crossflow rate. It is observed from curve 3 that the growth of the gel layer becomes steady after 1 h, unlike the case with 20 L/h. This is confirmed by observing the thickness after 2 and 6 h of operation, which increases by only 20 μm. Also, in a comparison of curves 2 and 3, the gel thickness is significant in the case of lower flow rate. The gel grows from 50 to 250 μm (5 times) when the flow rate is decreased from 60 to 20 L/h (33%) at the end of 6 h. The gel resistance follows a similar trend, in line with the gel-layer dynamics. Adsorption is inherently a transient process. However, in the present analysis, adsorption is coupled with the concentration polarization, which is basically dictated by gel formation. The smaller-sized solute gets screened to some extent by this dynamic gel layer (represented by the parameter γg) before reaching the membrane surface, thereby subduing their adsorption by the membrane matrix. This phenomenon results in a relatively steady profile of the permeate concentration of smaller-sized solutes, as described by Figures 1b and 2b. On the other hand, in the long run, the adsorption capacity of the membrane matrix gets saturated and the permeate concentration expresses a breakthrough profile. The figures clearly

Figure 4. Variation of the (a) ratio of adsorption to membrane hydraulic resistance and (b) ratio of the gel to adsorption resistance with time for different operating conditions. 518

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Figure 5. Effects of the various model parameters on the profiles of (a) permeate flux and (b) concentration of phenol in the permeate.

Figure 6. Effects of the adsorption isotherm and kinetic constants on the profile of the (a) permeate flux and (b) concentration of phenol in permeate.

on the permeate flux (compare curves 2 and 3). The effect of γg on the permeate flux and concentration is observed by comparing curves 2 and 3. Higher γg (curve 3) indicates enhanced retention of the lower-molecular-weight solute (phenol) in the gel layer, leading to a lower permeate concentration (curve 3 in Figure 5b). Curves 2 and 3 in Figure 5a indicate that the effect of γg on the permeate flux is insignificant. This may be due to the gel-layer resistance remaining unaffected by screened phenol molecules. The adsorption resistance coefficient kad has a strong influence on the system performance, as observed by a comparison of curves 3 and 4. When the adsorption resistance (high kad) is increased, the permeate flux decreases sharply and attains steady state quickly. Also, the decrease in the permeate concentration is gradual (curve 3 in Figure 5b) for lower values of kad. This is due to the fact that the permeate concentration decreases with the permeate flux, as explained earlier. In the case of the permeate concentration, it drops an order of magnitude (at 2 h of filtration) when kad is increased from 1013 to 1014 m−1. Increasing the value of n also increases the adsorption, but its effect is prominent only after significant adsorption has taken place. For this reason, the flux decline of curve 5 is the steepest. The permeate concentration shows a similar trend. With low values of n, the effect of kad is prominent because a change in the amount of the adsorbed solute has a small impact on the overall adsorption resistance, which is not the case for high values of n (Rad = kadc2mn). The effect of the Langmuir adsorption isotherm constants (A and B) and the kinetic constant kp has a considerable influence

tration increases while the adsorption resistance decreases. In the case of the TMP, the permeate flux increases with the TMP, leading to a reduced contact time between the adsorbate and adsorbent (matrix) and a rise in the permeate concentration. However, with an increase in the permeate flux, the concentration polarization also increases, leading to higher adsorption resistance (the concentration polarization increases cm, and from eq 7, Rad increases with cm). These effects become dominant at higher TMP, leading to an increase in the adsorption resistance. The effect of the model parameters (β, γg, kad, and n) on the system performance is illustrated in Figure 5. The effect of the specific gel resistance parameter β on the permeate flux and concentration can be observed by comparing curves 1 and 2. An increase in β leads to an increase in the gel resistance, Rg, thereby reducing the permeate flux. It can also be observed that the onset of steady state is delayed at higher β (curve 2). At higher β, the gel-layer resistance is sufficiently high and the shear imparted by cross-flow of a retentate is not adequate to achieve steady state within 8 h. Trends of profiles of the permeate concentration with β are observed from Figure 5b. The permeate concentration decreases with β. As shown in Figure 5a, higher β results in lower permeate flux, thereby increasing the contact time of the transported solute with the MMM, leading to lower solute concentration in the permeate. Because the permeate flux decreases as a result of an increase in the gel resistance, solute adsorption is favored. This leads to a decrease in the permeate concentration. The change in magnitude of the partition coefficient γg has a marginal effect 519

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on the system dynamics, as represented in Figure 6. The solute adsorption capacity increases with A and decreases with B. Increasing magnitude of the isotherm constant A decreases the permeate flux and concentration significantly (compare curves 1 and 2). For example, the permeate flux reduces by 15% as the magnitude of A increases 10-fold. As the solute adsorption capacity increases, the permeate concentration is decreased, leading to a rise in the adsorption resistance, which is responsible for higher flux decline. On the other hand, the permeate flux and concentration increase with B. Similar observations were also noted for the case of the batch cell configuration.16 This suggests that the effects of the adsorption isotherm constants are independent of the membrane configuration. The effect of the adsorption kinetic constant on the system performance is also significant. Physically, the magnitude of kp quantifies the rate of adsorption. The higher the value of kp, the faster is the adsorption, leading to quick saturation of the membrane adsorption capacity. Upon comparison of curves 1 and 4, it can be observed that the flux decline and decline in the permeate concentration become gradual after 2 h of filtration (curve 2) for kp = 1 h−1. Curve 4 also suggests that the adsorption rate by the membrane decreases considerably after 2 h and thereby the permeate concentration decreases marginally for the next 4 h, which is unlike the case for curve 2 (kp = 0.1 h−1). Hence, higher kp enhances the rate of adsorption in the beginning of filtration, but lower levels of the permeate concentration can be achieved with low kp for a long duration of filtration.

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Article

ASSOCIATED CONTENT

S Supporting Information *

Size distribution of the particles in the steel-plant effluent. This material is available free of charge via the Internet at http:// pubs.acs.org.

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AUTHOR INFORMATION

Corresponding Author

*Tel: + 91-3222-283926. Fax: +91-3222-255303. E-mail: sde@ che.iitkgp.ernet.in. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work is partially supported by a grant from the Department of Science and Technology (DST), New Delhi, Government of India, under the scheme DST/TMC/2K11/ 339, dated 23-05-2012, and by the Board of Research in Nuclear Sciences (BRNS), Department of Atomic Energy, Mumbai, Government of India, under the scheme 2012/21/03BRNS, dated 25-07-2012. Any opinions, findings and conclusions expressed in this paper are those of the authors and do not necessarily reflect the views of DST or BRNS.

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A B c1b c1g c2b

CONCLUSION

c2m

The transport-based model of UF of an industrial effluent using a carbon-based MMM has been reported in this work. Other than the sieving mechanism, adsorption plays a significant role in the retention behavior of the membrane for effluent treatment. The important findings of this study can be summarized as follows: 1. The transport-based model is developed to include the effects of concentration polarization, gel formation, and adsorption simultaneously. The proposed model is able to predict the experimental results quantitatively, as shown in Figures 1 and 2. 2. The model results are useful in describing the internal physical parameters of the system such the gel-layer thickness and adsorption resistance. It is observed that the adsorption dynamics is much faster than the gel-layer development. The adsorption and gel-layer resistances increase with the TMP and decrease with the cross-flow rate. 3. The sensitivity analysis of the model parameters shows that γg has the least influence on the permeate flux, but it has a considerable impact on the permeate concentration. 4. Both the adsorption isotherm (Langmuir) model and the kinetic (Lagergren) model constants have a significant influence on the permeate flux and concentration. 5. Model analysis is useful in determining the physical properties. The optimum values of the physical properties of the system are found to be D1 = 3.5 × 10−10 m2/s, ρg = 2800 kg/m3, εg = 0.5. and c1g/c0 = 55. 6. Model analysis is useful in the design and scaling up of MMM-based filtration systems.

c2p D1 D2 de k1 k2 kad kp L M2 n q qe R Rad Re Rg Rm S Sc Sh T t tm 520

NOMENCLATURE Langmuir adsorption coefficient in eq 6, m3/kg Langmuir adsorption coefficient in eq 6, m3/kg bulk concentration of the gel-forming solute, kg/m3 gel-layer concentration, kg/m3 bulk concentration of the low-molecular-weight solute, kg/m3 membrane surface concentration of the low-molecularweight solute, kg/m3 permeate concentration of the low-molecular-weight solute, kg/m3 diffusivity of the gel-forming solute, m2 /s diffusivity of the low-molecular-weight solute, m2/s equivalent channel diameter, m mass-transfer coefficient corresponding to the gel-layer solute, m/s mass-transfer coefficient of the low-molecular-weight solute, m/s adsorption resistance coefficient in eq 7, m−2 adsorption kinetic constant corresponding to the Lagergren model in eq 5, s−1 thickness of the gel layer, m molecular weight of the low-molecular-weight solute, kg/ kmol adsorption resistance power coefficient in eq 7 solute adsorbed by MMM at time t equilibrium solute adsorption universal gas law constant adsorption resistance, m−1 Reynolds number gel-layer resistance, m−1 membrane hydraulic resistance, m−1 objective function (S) for error minimization in eq 11 Schmidt number Sherwood number temperature, K time of filtration, s thickness of the membrane matrix, m DOI: 10.1021/ie504358j Ind. Eng. Chem. Res. 2015, 54, 514−521

Article

Industrial & Engineering Chemistry Research vw permeate flux predicted from the model, m3/m2·s vw,exp experimental permeate flux data, m3/m2·s v0w pure water flux at a constant TMP, m3/m2·s

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Greek Symbols

α β γg

specific gel-layer resistance coefficient in eq 8, m/kg gel-layer resistance parameter in eq 8, m−2 partition coefficient defining selective retention of the lowmolecular-weight solute in the gel layer15 ΔP transmembrane pressure drop (TMP), Pa Δπ osmotic pressure difference between the feed and permeate side, Pa εm porosity of the MMM εg gel-layer porosity μ viscosity of the permeate solution, Pa·s π osmotic pressure, Pa ρm membrane density, kg/m3 ρg gel-layer density, kg/m3

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DOI: 10.1021/ie504358j Ind. Eng. Chem. Res. 2015, 54, 514−521