A mass transfer model for the prediction of permeate concentration during ultrafiltration of methyl violet dye solution

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1 Indian Journal of Chemical Technology Vol. 11, March 2005, pp A mass transfer model for the prediction of permeate concentration during ultrafiltration of methyl violet dye solution S Chatterjee & D K Acharjee * Department of Chemical Engineering, Indian Institute of Technology, Kharagpur , India Received 14 May 2004; revised received 24 December 2004; accepted 20 January 2005 A mass transfer model is proposed for prediction of permeate concentration and percent rejection at different inlet pressures and feed dye concentrations in membrane ultrafiltration. A differential equation in cylindrical co-ordinates considering the spiral wound membrane equivalent to a cylindrical module is derived which coupled with mass balance and permeate flux equations are solved to yield the theoretical permeate concentration. As the mass transfer coefficient used from literature gave high values for predicted permeate concentration, an empirical correlation for mass transfer coefficient was obtained based on experimental data on the dissolution of solid in aqueous solution from the wall of a hollow cylindrical tube where the mass transfer occurred in the boundary layer. The experimental permeate concentrations agreed reasonably well with those predicted from the model developed. Keywords: Ultrafiltration, permeate concentration, spiral wound module, mass transfer model. IPC Code: C02F1/00; C02F1/46 A wide range of dissolved contaminants present in wastewater from textile, tanneries and metallurgical industries etc. can be removed using the technique of ultrafiltration (UF). This membrane separation process uses the pressure difference as the driving force for the concentration of macromolecular solutions. In UF, the solutes which are rejected by the membrane accumulate on the membrane surface leading to concentration polarization A number of models were used to explain the concentration polarization such as resistance in series model, gel polarisation model and osmotic pressure model. However, it was found 11 that all classical models mentioned above cannot fully explain the experimental data on UF. This may be due to failure of the classical models to incorporate the effect of membrane module configurations such as tubular, spiral wound, hollowfibre, etc. The spiral wound and hollow fibre membrane modules are very effective for the desalination and waste water treatment operations. The spiral wound membranes are relatively better suited for the waste water treatment 12. In the earlier work 12 because of the *For correspondence ( dka@che.iitkgp.ernet.in; Fax: ) complex geometry of the spiral wound membrane, a transient model was reported for a spiral wound membrane considering that it was unwound and the flow within the module to be the flow in a channel with two porous walls neglecting the spiraling effects. The transient model based on differential equations in cartesian co-ordinates predicted concentration polarisation. In the present work, a spiral wound membrane module has been used for ultrafiltration of aqueous dye solution. For this, a new approach has been made to consider the spiral wound membrane module to be equivalent to cylindrical module and differential equation in cylindrical co-ordinates derived to analyze and predict permeate concentration. The mass transfer coefficent, K C, required for estimation of permeate flux, is usually obtained from Leveque or Dittus-Boelter equations 13. A critical review of the correlations for K C for membrane separation processes is available 14. These correlations are not universally applicable for UF where there is significant variation in transmembrane pressure, membrane module configuration, flow regimes (Reynolds numbers) and Schmidt number. Initial estimation of theoretical permeate concentration using K C values from literature yielded large deviations when compared with the

2 206 INDIAN J. CHEM. TECHNOL., MARCH 2005 experimental data. Therefore, K C values were determined experimentally measuring the rates of dissolution of benzoic acid from the wall of cylindrical tube in aqueous solutions and the mass transfer occurred in the boundary layer. Measurement of K C for dissolution of benzoic acid has been reported by others Theoretical model development Because of the complex velocity profile of liquid in axial and radial direction in spiral wound (SW) module, previous workers 12 neglecting the spiraling effects derived model equations for SW membrane in rectangular coordinates considering it unwound, and the flow within the module approximated the flow in channel. The present work (neglecting the spiraling effect) considers the SW module to be equivalent to a cylindrical module. Based on this, one dimensional model in cylindrical coordinates has been derived to predict theoretical permeate concentration. The concentration polarization phenomena over a membrane is shown in Fig. 1. At steady state, there is no accumulation over the differential element of thickness Δr and only convective and back diffusive fluxes are present. The solute mass balance over Δr in the boundary layer at a distance r = R + δ, yields Eq (1). λ(dc/dr) + 1/r [r(d 2 c/dr 2 ) + dc/dr] = 0 Boundary conditions: c (R) c m = 0 (6) (7) c(r + δ) c b = 0 (8) The second order differential equation is converted into two first order equations. dc/dr = w = F 1 (r, c, w) d 2 c/dr 2 = (λ + 1/r) w = F 2(r, c, w) x = r 0 y = c 0 z = w 0 (9) (10) J (dc/dr) + D[(1/r).(d/dr). {r.(dc/dr) ] = 0 The boundary conditions are At r = R, c = c m At r = R + δ, c = c b (1) (2) (3) (a) Analytical solution The analytical solution of Eq. (1) gives c = A (1/ r) e --λ r dr + B Eq.(4) gives the series solution as (4) c = A [ln r + Σ (-1) i {(λ r) i / i(i!)}] +B (5) i= 1 where λ = (J/D), and A and B are integration constants. (b) Numerical solution by Runge-Kutta method The algorithm for numerical solution of Eq. (1) by Runge-Kutta is given below. (c) Numerical solution by Finite Difference (with Wegstein Convergence) method h = x n x n-1 y n y n-1 y n = h

CHATTERJEE & ACHARJEE: A MASS TRANSFER MODEL FOR PREDICTION OF PERMEATE CONCENTRATION 207 y n y n-1 y n 2 y n-1 + y n-2 y n = = h h 2

(1), in the difference form 1 1 y n = (y n-1 { 2 [λ + ) h} y n-2 )/(1 + {λ + } h) x x Boundary conditions are c initial = c m, c final = c b

1 Mass balance over the boundary layer At r = r 0, c = c P On differentiation, Eq.

(11), one can get, V P (R + δ) c b + D A e -λ (R + δ) = r 0 V P c p (14) For ultrafiltration the permeate flux, V P, can be obtained from Eq.

violet dye in aqueous solution, the following equation was obtained, (d) Mass balance equation Equation (5) has two unknowns A and B which

A mass balance where the flux of solute due to convection plus diffusion in the boundary layer is equated to the flux of solute in the

3 CHATTERJEE & ACHARJEE: A MASS TRANSFER MODEL FOR PREDICTION OF PERMEATE CONCENTRATION 207 y n y n-1 y n 2 y n-1 + y n-2 y n = = h h 2 Converting the second order differential equation, Eq. (1), in the difference form 1 1 y n = (y n-1 { 2 [λ + ) h} y n-2 )/(1 + {λ + } h) x x Boundary conditions are c initial = c m, c final = c b x ε [R, R + δ ] Algorithm for numerical solution by Finite Difference Fig. 1 Mass balance over the boundary layer At r = r 0, c = c P On differentiation, Eq.(4) gives (12) (dc/dr) = A [(e -λ r )/r ] (13) By substituting the boundary conditions of Eq. (3) and Eq. (13) in Eq. (11), one can get, V P (R + δ) c b + D A e -λ (R + δ) = r 0 V P c p (14) For ultrafiltration the permeate flux, V P, can be obtained from Eq. (15) V P = L P (Δ P Δ π) The osmotic pressure difference is (15) Δ π = π m π P (16) From experimental data on osmotic pressure for the methyl violet dye in aqueous solution, the following equation was obtained, (d) Mass balance equation Equation (5) has two unknowns A and B which cannot be evaluated using the boundary conditions of Eqs 2 & 3. A mass balance where the flux of solute due to convection plus diffusion in the boundary layer is equated to the flux of solute in the permeate yields: 2 π rl V P c + 2 π r L D (dc/dr) =2 π r 0 L V P c P (11) π = c c c 3 (17) In Eq. (17), the osmotic pressure π is in atm. Eq. (17) was formulated on the approach given by others 18,19. The experimental values of osmotic pressures were also compared with those calculated using Van t Hoff equation, π = crt and and the variation was within five percent. The osmotic pressure for the membrane and the permeate can be expressed as

4 208 INDIAN J. CHEM. TECHNOL., MARCH 2005 At π = π m, c = c m (18) At π = π P, c = c P (19) The simultaneous solution of Eqs (5), (14) and (15) will provide the values of three unknowns, A, c P and c m respectively. (e) Mass transfer coefficient, K C The boundary layer thickness, δ is determined from Eq. (20) δ = (D/K C ) (20) The mass transfer coefficient, K C, values used from literature gave large deviations between the theoretical c p and the experimental c p obtained in the present work. Therefore, K C values were determined experimentally from the rate of dissolution of benzoic acid from the inner wall of a cylindrical tube into the flowing water and aqueous solution of viscous liquids. The following correlation was obtained for K C. Sh = (R e ) 0.5 (S c ) (21) The diffusivity of the dye solution was computed from Wilke-Chang equation 20. The R e varied from to , R e is based on equivalent diameter, d e = m, of SW membrane, the d e is calculated from π/4(d 2 1 d 2 2 ) = π/4(d 2 e ), where d 1 = cm, d 2 = cm. Similar approach for determining equivalent height considering SW module unwound (open-up condition) was reported 21 earlier. Experimental Procedure Determination of L P The solvent permeability, L P, was estimated from pure water permeation experiments using the SW membrane at different inlet pressures. The value of L P was m 2 s/kg. Experimental set-up The details of experimental set-up are available elsewhere 22. The set-up essentially consists of three tanks for feed, permeate and reject (effluent), centrifugal pump for feed, pressure gauges and a spiral wound membrane module. The specifications of the module are: A cm ( ) spiral wound cellulose acetate UF membrane, MWCO: Dalton, membrane area per unit volume: 800 m 2 /m 3. The module was supplied by M/s Permionics, Vadodara, India. The feed under steady state to the UF membrane module was a solution of the dye methyl violet (C 24 H 28 ClN 3 ). The inlet solution pressure to the membrane was varied from 202 to 908 K Pa. The aqueous feed dye concentration was in the range of 0.1 to 0.8 % (w/w). The dye solution was analysed measuring the absorbance by UV Spectrophotometer, Shimadzu Model No: UV-2100, wave-length range: nm. The calibration data as percent dye concentration versus absorbance are given in Table 1. Results and Discussion Effect of feed flow rate on permeate concentration Figure 2 shows that for a feed of 1.0 kg/m 3 dye cocentration (0.1% w/w), the permeate concentration decreases from 0.2 to kg/m 3 as the feed flow rate increases from to , m 3 /s. Similar observations were made for other dye Table 1 Calibration data for UV-spectrophotometer % Dye concentration (w/w) Absorbance Fig. 2 Effect of feed flowrate on permeate concentration

5 CHATTERJEE & ACHARJEE: A MASS TRANSFER MODEL FOR PREDICTION OF PERMEATE CONCENTRATION 209 Fig. 3 Effect of transmemnbrane pressure drop on permeate concentration concentrations such as 0.8%. The MWCO for the membrane module of the present work is Dalton. The molecular weight of methyl violet dye is 393. The solute dye rejection by this membrane is expected to be 100 percent. At low feed flow rate some solute dye permeates through the membrane. With increase in feed rate solute dye molecules buildup on the membrane surface which becomes a hindrance to the permeation of solute dye and back diffusion of solute into the bulk liquid within the boundary layer occurs (Fig. 1) and hence the solute dye concentration in the permeate decreases. Similar observations have been reported for SW membrane 21. Effect of transmembrane pressure drop on permeate concentration Figure 3 shows that for a feed dye concentration of 1.0 kg/m 3, as the transmembrane pressure drop is increased from 0.5 to 9.0 kg/cm 2, the permeate concentration decreases from 0.20 to kg/m 3. The rejection of dye molecules is more effective at higher values of transmembrane pressure drop. As the transmembrane pressure drop increases, permeation of more water occurs thereby decreasing the solute dye concentration in the permeate. Figure 3 shows the maximum transmembrane pressure drop as 4.5 Kg/cm 2 (inlet feed pressure is 5.5 Kg/cm 2 and permeate pressure is 1.0 Kg/cm 2 ). Fig. 3 if extrapolated for a transmembrane pressure drop of 7.0 Kg/cm 2 (inlet feed pressure of 8.0 Kg/cm 2 ), can predict permeate concentrations for various feed dye concentrations. At inlet pressure of 8.0 Kg/cm 2, permeate concentration was measured for 0.3% feed dye concentration which compared well with the predicted concentration from Fig. 3, but this has not Fig. 4 Effect of feed flowrate on percent rejection been shown in Fig. 3 as permeate concentration could not be measured for 0.1 and 0.2% feed dye concentration because of leakage in the set-up at 8.0 Kg/cm 2 inlet pressure. Effect of feed flow rate on percent rejection Figure 4 shows that for feed dye concentration 1.0 kg/m 3, the percent rejection, R J, of the dye increases from 80 to 98.4 % in the range of feed flow rate from to m 3 /s. The rejection R J depends on the ratio of c P /c F. At a constant feed dye concentration, the c P decreases with increase in feed rate and this causes an increase in R J. Such observation has been reported 23 earlier. The high rejection of the dye exhibits very good performance of the SW membrane. Evaluation of K C, Sh and Sc Using the data on benzoic acid dissolution in water and viscous solution, the mass transfer coefficient, K C and Sherwood number, Sh, were evaluated. K C varied from to m/s and Sh from 7302 to 17412, respectively. The Schmidt number was in the range of 2698 to Evaluation of the performance of the membrane The performance of the membrane in terms of the permeate flux, percent rejection and concentration of the dye in the permeate was evaluated. Table 2 shows the values of various parameters for evaluation of the performance of the membrane. The Eqs (5), (14) and (15) with their boundary conditions were solved using a computer program for determining the three unknowns A, c m and c p. The experimental and theoretical values of c p are shown in Fig. 5 for dye concentration of 1.0 kg/m 3. It may be noted from this

6 210 INDIAN J. CHEM. TECHNOL., MARCH 2005 Table 2 Values of various parameters for evaluation of membrane performance Parameters Range R e Sh Sc K C m/s D m 2 /s μ d e L P V P r 0 R Kg/m s m m 2 s /Kg m/s m m Fig. 5 Comparison between experimental and theoretical permeate concentration predicted by analytical solution figure that c p given by analytical solution deviated from experimental values by less than 10 percent whereas, c p by Runge-Kutta and finite difference method had shown a maximum deviation of about 17 percent. Similar deviations in c p were observed for other dye concentrations 22,24 as well. Predicted dye concentration at membrane surface, c m Figure 6 shows the variation in the predicted values of c m for various inlet feed dye concentrations. For a particular feed dye concentration, 1 kg/m 3, c m increases with increase in feed flow rate and for a particular feed flow rate, c m increases with increase in feed dye concentration. This is because of high solute (dye) rejection at the membrane surface. The concentration polarization, β, may be defined 17,19 as β = c m /c b. Using the data of c m from Fig. 6, β was calculated. In the range of feed flow rate m 3 /s, for c b = 1.0 Kg/m 3, β = , for c b = 2.0 Kg/m 3, β = , and for c b = 3.0 Kg/m 3, β = respectively. For ultrafiltration of whey protein in tubular membrane values of β were reported 19 in the range of (gel formation) and β increased with bulk solute concentration as observed in the present work. Conclusion In this study the spiral wound membrane module was considered to be equivalent to a cylindrical module. Accordingly, model equations were derived Fig. 6 Effect of feed flowrate on theoretical membrane surface concentration in cylindrical coordinates to predict permeate concentraion, c p and percent rejection of the solute dye. Eq. (1) was solved using analytical as well as numerical methods (Runge Kutta and finite difference). The percent deviation of the predicted c p was less for analytical method compared to the two other methods. In the range of experimental dye concentrations and feed flow rates studied, the derived model predicted the performance of the spiral wound membrane for UF quite satisfactorily. As the mass transfer coefficient, K C, used from literature gave high theoretical c p, experiments were performed using a cylindrical module whose inside was coated with benzoic acid which dissolved in flowing water and viscous solutions.

7 CHATTERJEE & ACHARJEE: A MASS TRANSFER MODEL FOR PREDICTION OF PERMEATE CONCENTRATION 211 Nomenclature A = constant in Eq. (4) B = constant in Eq. (4) c = solute dye conentration (kg/m 3 ) c b = bulk(feed) solute concentration (kg/m 3 ) c m = membrane solute concentration (kg/m 3 ) c p = permeate solute concentration (kg/m 3 ) D = diffusivity of solute dye in water (m 2 /s) d 1 = diameter of SW module (m) d 2 =diameter of permeate tube (m) d e = equivalent diameter of membrane (m) J = volumetric permeate flux (m 3 /m 2.s) K C = mass transfer coefficient (m/s) l = length of membrane (m) L P = solvent permeability (m 3 /kg.s) MWCO = molecular cut off (abbr) ΔP = transmembrane pressure drop (Pa) r o = radius of permeate tube(m) R = membrane radius (m), universal gas constant (m 3 atm/kgmole.k) Re = Reynolds number (d e V f ρ f /μ) R J = percent rejection (1 c p /c f ) 100 Sc = Schmidt number (μ f /ρ f D) Sh = Sherwood number (K C d e /D) T = Absolute temperature (K) V f = axial feed velocity based on d e (m/s) V P = permeate flux (m/s) Greek symbols δ = thickness of concentration boundary layer(m) π = osmotic pressure (Pa) μ = viscosity of solution (Pa.s) ρ = density of solution (kg/m 3 ) Subscripts m =membrane p = permeate References 1 Denisov G A, J Membr Sci, 91 (1994) Bhattacharjee C & Dutta S, J Membr Sci, 125 (1997) Yeh H M & Cheng T W, J Membr Sci, 154 (1999) Minnikanti V S, Das Gupta S & De S, J Membr Sci, 157 (1999) Karode K, J Membr Sci, 171 (2000) Foley G & Jose G, J Membr Sci, 175 (2000) Zaki S M, Meriam S N & Beicha A, J Membr Sci, 189 (2001) Cheng T, Wen H & Cheng Y, J Membr Sci, 209 (2002) Marchese J, Ponce M, Ochoa N A, Pradanos P L, Palacio N & Hernandez, J Membr Sci, 211 (2003) Yeh H M, Wu H P & Dong J F, J Membr Sci, 213 (2003) Danes F E, Boriou B & Poyen B S, J Membr Sci, 50 (1990) Madireddi K, Babcock R B, Levine B, Kim J H & Stenstrom M K, J Membr Sci, 157 (1999) Porter M C, Ind Eng Chem Prod Res Dev, 11 (1972) Van den Berg G B, Racz I G & Smolders C A, J Membr Sci, 47 (1989) Harriot P, AIChE J, 8 (1962) F E Marble & Adamson C, Jet Propulsion, 24 (1954) Geankoplis C J,Transport Process and Unit Operations (Prentice Hall of India), Leung W & Probstein R F, Ind Eng Fundam, 18 (1979) Jonsson G, Desalination, 51 (1984) Wilke C R & Chang P, AIChE J, 1 (1955) Avlonitis S, Hanbury W T & Boudinar M B, Desalination, 89 (1973) Patel S, Modeling and simulation of spiral wound membrane for ultrafiltration of dye solution, M. Tech. Thesis, Indian Institute of Technology, Kharagpur, Tsapiuk E A, J Membr Sci, 116 (1996) Jha P K, Prediction of permeate concentration for ultrafiltration of dye solution using spiral wound membrane module, M. Tech. Thesis, Indian Institute of Technology, Kharagpur, 2004.

Estimate the extent of concentration polarization in crossflow filtration Select filtration unit operations to meet product requirements, consistent

Membrane Separation Process Objectives Estimate the extent of concentration polarization in crossflow filtration Select filtration unit operations to meet product requirements, consistent with product