MODIFIED SERIES RESISTANCE MODEL - DETERMINATION OF MEAN CONCENTRATION BY INTEGRAL TRANSFORMATION

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1 MODIFIED SERIES RESISTANCE MODEL - DETERMINATION OF MEAN CONCENTRATION BY INTEGRAL TRANSFORMATION A. L. Venezuela a, R. F. Cantão a, R. S. Ongaratto b, and R. N. Haneda c a Universidade Federal de São Carlos Departaento de Física, Quíica e Mateática Rodovia João Lee dos Santos, K Bairro do Itinga CEP: , Sorocaba, SP, Brasil alvenez@ ufscar.br b Instituto Nacional da Propriedade Industrial Divisão de Patentes VI Rua Mayrink Veiga, 9, Bairro Centro CEP: 29-9, Rio de Janeiro, RJ, Brasil c Centro Paulista de Estudos e Agronegócio Rua 7 de Setebro 2378, Bairro Centro CEP 356-8, São Carlos, SP, Brasil ABSTRACT There are several atheatical odels that describe pereate flow in ebrane separation processes. Aong these, the series resistance odel plays a proinent role. It takes into account ebrane strength, concentration polarization, polarized layer and fouling to describe the pereate flow over tie. In this work, an analysis of the odified series resistance odel was perfored, in which the resistance by polarization of the concentration is defined as being directly proportional to the transebrane pressure. The proportionality constant is given by the product of a specific coefficient of resistance which is deterined by eans of experiental data the thickness of the boundary layer of concentration and the ean concentration. Due to the inability to obtain experientally the value of the average concentration within the boundary layer of concentration, its siulation is carried out fro the conservation equation of the cheical species. Thus, the objective of the present work was to solve the equation of the conservation of cheical species using GITT (Generalized Integral Transfor Technique) and apply the odified series resistance odel to describe the pereate flow of a solution of dextran through a pereable tube under lainar flow. GITT provided satisfactory results for the ean concentration, verified by coparison with the pereate flow obtained by the series resistance odel with experiental results reported in the literature. Received: April 4, 28 Revised: May, 28 Accepted: May 3, 28 Keywords: hybrid ethods, polarization resistance, cross flow NOMENCLATURE Diensionless c concentration, -3 kg c inlet concentration, -3 kg c average concentration in concentration boundary layer, -3 kg D diffusivity, 2 s - J experiental pereate flux, s - J siulated pereate flux (GITT), s - J initial pereate flux, s - J Y siulated pereate flux (Yeh, 23), s - k ass transfer coefficient, s - L length of the tube, r po pore radius, R p radius of the tube, R resistance of concentration polarization, - R f fouling resistence, - u inlet ean axial velocity, s - r radial coordinate, x axial coordinate, C concentration (c/ c ) U axial velocity (u/ u ) V radial velocity (v/ u ) R radial coordinate (r/ R ) X axial coordinate (x/ R ) Re Reynolds nuber, u2r Sc Schidt nuber, D Pe Péclet nuber, Re Sc Re w Reynolds nuber in the pereable wall, J 2R Greek sybols thickness of concentration boundary layer, ebrane thickness, porosity dynaic viscosity, - kg s - density, -3 kg 94 Engenharia Térica (Theral Engineering), Vol. 7 No. June 28 p

2 p transebrane pressure, - kg s -2 (=Pa) tortuosity Subscripts inlet condition f fouling li liiting flux average experiental and ebrane p polarization po pore Y yeh INTRODUCTION In the ebrane separation processes (MSP), especially for ultrafiltration and icrofiltration, there is an initial drop in pereate flow over tie which is caused by phenoena that liit the transport of the solvent such as concentration polarization, polarized layer and fouling. The resistance in series odel is a theoretical odel that has been used to describe the phenoena of polarization and fouling and the pereate flux over tie (Cheryan, 998). The Generalized Integral Transfor Technique (GITT) is a hybrid ethod (nuerical and analytical) which is very attractive for the treatent of ultidiensional probles. Allied to this, the use of analytical filters, the choice of the ost representative basis of the original proble and the use of sybolic anipulation platfors like (Cotta, 993), contribute to the iproveent of this technique. Developent of hybrid techniques, which are able to assist in the understanding of the phenoena responsible for the pereate flow drop have been conducted (Venezuela et al., 29; Kuar et al., 2). The objective of this work was to solve by hybrid ethods (GITT), the conservation equation of cheical species in cylindrical coordinates to deterine theoretical odels that will assist the analysis of polarization phenoena responsible for pereate flow decreasing during the icrofiltration process. MATHEMATICAL MODEL The developent of the ethodology for deterining the pereate flow follows the steps below. Modified resistance-in-series odel The pereate flow, J, varying with a pressure is (Paris, 22): J p p R R c p f For odeling purposes, a cylindrical ebrane with fixed length L, radius R and resistance R was considered (Yeh, 23). We assue that through it flows a dextran solution with velocity u, carrying a solute with initial concentration c, diffusivity D, viscosity, and transebrane pressure p. Deterination of J li, k and The thickness of the concentration boundary layer, δ, depends on the ass transport coefficient, k, which in turn depends on the pereate flow liit, J. We have developed a ethodology to obtain li J li, which consists of a nonlinear expression with three unknown paraeters, deterined by nonlinear regression. Using the gel-polarization odel along with the aforeentioned experiental data, we can obtain the ass transfer coefficient k using a siple linear regression. Fro the stagnant fil theory we evaluate the thickness of concentration boundary layer as δ = D/k (Zean and Zydney, 996). The diffusivity, D, according to Yeh et al. (23). Obtaining c, via GITT The deterination of the ean concentration in the adjacencies of the inner wall of the pereable tube is given by: c cc, where C is the ean diensionless concentration defined by: C R L 2 LR R C X, R dr dx The diensionless concentration profile, C C( X, R), for R and X L R, is deterined, by GITT (Venezuela et al., 29), fro the following boundary value proble in the diensionless for: C C C V U R R X Pe R R R C X, () C, R (2) C R R (3) (4) J X D To deterine the C profile, we ust have the following: (a) The radial, V V( R), and axial, U U( X, R), velocity profiles, which are obtained fro the work Engenharia Térica (Theral Engineering), Vol. 7 No. June 28 p

3 of Yuan et al. (956). (b) The forulation of the boundary layer thickness of local concentration, X, is obtained fro the work of (Venezuela et al., 29). (c) The initial pereate flow, J, is deterined using the Kade-Katchalsky analysis (Zean and Zydney, 996): J 2 po r 8 p where,, r po, and are given in the work of Yeh et al. (23). Deterination of and In order to calculate the specific resistance,, and the fouling resistance, R f, we use: where R f p J c p R Rf J is the experientally easured pereate flow. The slope, c, and the intercept, R Rf, are deterined by standard linear regression. The value of ebrane resistance is given in the work of Yeh et al. (23). RESULTS AND DISCUSSION As entioned in the text, the deterination of the ean concentration in the vicinity of the pereable wall was perfored by GITT, according to Venezuela et al. (29). Thus, the eigenvalues and the noralization integrals were analyzed analytically. The integral coefficients were evaluated nuerically. Finally, the resulting syste of ordinary differential equations truncated together with the initial conditions was solved nuerically. The coputational ipleentation of GITT was carried out on the coputer algebra syste SAGe ( accessed in 2/5/28). In this section we present the nuber of ters (order) for the truncation of the transfored ordinary differentiated syste. Siulation results and physical proble data were copared through the behavior of the concentration profile with varying Schidt and Reynolds nubers. In addition, we copared the results for the pereate flux with experiental data fro the literature. Table presents results for the concentration profile, C, as a function of the variable X, setting R =.98, for several values of N. The values of C were considered with tree decial places. For this analysis, we can take any value of R in the vicinity of the pereable wall that we will have the sae truncation orders. To verify the order of truncation, we fixed a line, for exaple, in X = 22, and we find that for N = 5 we have C =.9, for N = we have C =.78 and for N 5 the values C =.72 are repeated. Table. Convergence results for concentration C varying X in R =.98, c. kg, u.5 s 5 - p.25 Pa., R =2.5-4 e X N=5 N= N=5 N= Figure show the results of concentration, C, near to the surface tube (R =.98) as a function of the diensionless axial position, X. The values correspond to Schidt, Sc, are. 4,.8 4 and.6 4 ; a concentration increase tendency is observed next to the wall as a function of an increase of the Schidt nuber. Sc and diffusivity D are inversely proportional, that is, the increase of Sc ay be associated to the decreasing of the diffusion coefficient. The results obtained fro the integral transfor are in agreeent with this type of physical behavior. Figure. Schidt s nuber effect over the concentration profile depending on the diensionless axial position, X, near the pereable wall (R=.98) for N=5, Re=7 and Re w =.4. Figure 2 illustrates when the Reynolds nuber decreases, there is an increase in solute concentration near the surface of the tube. In this case, it is expected that, with decreasing Reynolds nuber, the C Engenharia Térica (Theral Engineering), Vol. 7 No. June 28 p

4 ass transfer echanis decreased, causing an increase in the rate liitation and the concentration of solute near the porous surface. In this case, the behavior of the concentration profile is in line with the expected physical patterns. Figure 2. Reynolds s nuber effect over the concentration profile depending on the diensionless axial position, X, near the pereable wall (R=.98) for N=5, Sc=.6 4 and Re w =.4. Pereate flux results obtained with GITT, J, were copared with experiental data, J, and siulated data, J Y, as reported by Yeh et al. (23). Table 2-a and 2-b shows that the relative differences D J J J and D2 JY J J are adequate. In Table 2-a the relative differences, D and D 2, show that the pereate flow results J and J Y have good approxiation with the experiental pereate flow results, J. However, in Table 2-b, D and D 2 illustrate that the pereate flow results J have a better approxiation than J Y, when copared with J. Therefore, for c =. kg -3, we have that the experiental pereate flux, J, is best represented by J and, for c =. kg -3, J and J Y are good representatives of J. Table 2. (a) c =. kg -3 and (b) c =. kg -3. Coparison aongst experiental. J. and siulated data fro Yeh et al. (23), J Y, and the results obtained with the odified resistance in series odel, J, depending on pressure, Δp. For each case relative 6 differences are also shown. J, J, J ( s ). p 5 u Pa ( s - ) (a) Y 3 kg Rel. Diff. c. J J Y J D D 2 (%) (%) (b) p 3 c.kg Rel. Diff. 5 u Pa ( s - J ) J J Y D D 2 (%) (%) CONCLUSIONS Polarized concentration resistance was deterined with the help of coputational siulations of the ean concentration of solute within the boundary layer, via GITT. Next, pereate flux was calculated using the odified resistance-inseries odel. Relative errors are satisfactory. when copared with the literature. Thus, GITT can be regarded as an efficient tool to evaluate the pereate flow associated with icrofiltration processes. REFERENCES Cheryan, M., 998, Ultrafiltration and Engenharia Térica (Theral Engineering), Vol. 7 No. June 28 p

5 Microfiltration Handbook, CRC Press, Boca Raton, Florida, 2nd edition. Cotta, R. M., 993, Integral Transfors in Coputational Heat and Fluid Flow, CRC Press, Boca Raton, Florida. Kuar, V., Pandey, R. N., and Upadhyay, S. N., 2, Exact Solution of Convective Mass Transfer Model for Calciu Response of Endotheliu, Matheatical Probles in Engineering, Vol. 5, pp Özisik, M. N., 98, Heat Conduction, John Wiley, New York. Paris, J., Guichardon, P., and Charbit, F., 22, Transport Phenoena in Ultrafiltration: a New Two- Diensional Model Copared with Classical Model, Journal Mebrane Science, Vol. 27, pp Venezuela, A. L., Guerrero, J. P., and Fontes, S. R., 29, Hybrid Modeling of Convective Lainar Flow in a Pereable Tube Associated with the Cross- Flow Process, Counications in Nonlinear Science and Nuerical Siulation, Vol. 4, pp Zean, L. J., and Zydney, A. L., 996, Microfiltration and Ultrafiltration Principles and Applications, Marcel Dekker, New York. Yeh, H. M., Wu, H. P., and Dong, J. F., 23, Effects of Design and Operating Paraeters on the Declination of Pereate Flux for Mebrane Ultrafiltration Along Hollow-Fiber Modules, Journal of Mebrane Science, Vol. 23, pp Yuan, S. W., Funkelstein, A. B., and Brooklyn, N. Y., 956, Lainar Pipe Flow with Injection and Suction through a Porous Wall, Transactions of the ASME, Vol. 78, pp Engenharia Térica (Theral Engineering), Vol. 7 No. June 28 p