A NEW INTEGRAL EQUATION APPROACH FOR

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MASS TRANSFER THROUGH PERMEABLE WALLS: A NEW INTEGRAL EQUATION APPROACH FOR CYLINDRICAL TUBES WITH LAMINAR FLOW a

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RICARDO J. GRAU , ALBERTO E. CASSANO & HORACIO A. IRAZOQUI

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INTEC§ , Guemes 3450, Santa Fe, 3000, Argentina Published online: 27 Apr 2007.

To cite this article: RICARDO J. GRAU , ALBERTO E. CASSANO & HORACIO A. IRAZOQUI (1988) MASS TRANSFER THROUGH PERMEABLE WALLS: A NEW INTEGRAL EQUATION APPROACH FOR CYLINDRICAL TUBES WITH LAMINAR FLOW, Chemical Engineering Communications, 64:1, 47-65, DOI: 10.1080/00986448808940227 To link to this article: http://dx.doi.org/10.1080/00986448808940227

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Chem. Eng. Comm. 1988, Vol. 64, pp. 47-65 Reprints available directly from the publisher. Photocopying permitted by license only. © 1988 Gordon and Breach Science Publishers S.A. Printed in the United States of America

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MASS TRANSFER THROUGH PERMEABLE WALLS: A NEW INTEGRAL EQUATION APPROACH FOR CYLINDRICAL TUBES WITH LAMINAR FLOW RICARDO J. GRAUt, ALBERTO E. CASSANO:j: and HORACIO A. IRAZOQUI:j: INTEC§, Guemes 3450, 3000, Santa Fe, Argentina (Received June 1, 1987) Mass transfer through cylindrical semipermeable walls is analyzed. The solution is obtained in terms of integral equations. Despite the existence of a non-homogeneous boundary condition on the semipermeable wall, the solution thus obtained is particularly advantageous since the associated eigenvalue problem is independent of the Sherwood number. This parameter takes into account the main conductances at the tube wall. The approach is applied to the case of mass transfer from the interior of a capillary tube with semipermeable walls to an external fluid. The flow in the tube is laminar, and the external flow is assumed turbulent. The mathematical methodology employed provides a framework to develop numerical schemes of fast and sure convergence. KEYWORDS Mass transfer Semi-permeable walls Cylindrical tube Laminar flow Integral Equations solution

I.

INTRODUCTION

Heat and mass transfer processes occurring through permeable walls of cylindrical shape containing a fluid that flows in laminar regime, have been frequently analyzed and their complete review is beyond the scope of this work. We may instead make reference to the variety and extension of their applications which range from separation processes in multi-component mixtures studied from both the theoretical and experimental viewpoints (Schell, 1977), to reverse osmosis and ultrafiltration processes to obtain proteins from milk or drinkable water from hard water (Sourivajan , 1970; Lightfoot, 1974; etc.), or to important biomedical uses such as mass transfer processes for artificial kidneys (Grimsrud and Babb, 1966; Cooney et al., 1974a; Cooney et al., 1974b, etc.). t Research Assistant from CONICET. ~

Member of CONICETs Scientific and Technological Research Staff and Professor at U.N.L.

§ Instituto de Desarrollo Tecnol6gico para la Industria Quimica. Universidad Nacional del Litoral

(U.N.L.) and Consejo Nacional de tnvestigaciones Cientificas y Tecnicas (CONICET). Correspondence related to this paper should be addressed to A.E. Cassano. 47

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RJ. GRAU, A.E. CASSANO AND H.A. IRAZOQUI

The process is important and of old data since its treatment goes back to the classical formulation of the well-known Graetz problem, already described in 1883 for the first time. Concerning this problem and its variations, numerous papers have been written among which we can cite Schenk and Dumore (1953), Siegel et al. (1958), Singh (1958), Brown (1960), Sideman et al. (1965), Sparrow and Chen (1969), Jones (1971), Michelsen and Villadsen (1974), Lee and Aris (1977), Papoutsakis (1979) and Papoutsakis et al. (1980), just to mention some of the best known fundamental contributions from the engineering viewpoint. An even longer list could be drawn up with those papers dealing exclusively with the mathematical aspect of the problem. More recently, Arce et al. (1987) analyzed the case of a tubular reactor of circular cross section in laminar flow regime, taking into account the presence of homogeneous reactions inside the tube. The problem was tackled for the case in which the kinetics may be arbitrary. The solution was obtained in terms of integral equations by constructing the appropriate Green functions. These formal solutions have the advantage of being intermediate results which already reveal the main features of the final solution. This work also showed that the proposed approach provides a solution scheme that allows the use of numerical techniques that are fast and stable and that, at the same time, ensure convergency. A valuable sequel to that work is to extend its mathematical approach to mass transfer processes through the permeable walls of a tube. The problem has been previously solved with the aid of power series expansions, direct numerical solution of the partial differential equations and, more recently, by using confluent hypergeometric functions. However, when the boundary conditions are non-homogeneous (which is the case of permeable walls), the eigenvalues of the associated transcendental equation are obtained as a function of Sh w . The wall Sherwood number accounts for the transport conductance at the external side of the tube and the conductance of the membrane (see for example, Cooney et al., 1974a). The methodology proposed in this work to treat the boundary condition, will render a solution whose eigenvalues are independent of Sh w . This is particularly interesting since with this approach the eigenvalue problem can be calculated regardless of the physical parameters of the system. In contrast to previous results (Cooney et al. 1974a), the solution thus obtained can be used for any wall Sherwood number. After depicting the general problem and the solution, the use of the resulting equations is illustrated by applying them to the case of a hemodialyzer of the capillary-tube type, excluding the existence of a coupled ultrafiltration phenomenon. It should be noticed that this example has been selected as one among the numerous applications of mass transfer through permeable walls. II.

MODEL

Let us consider the transfer of a solute between two fluids separated by a semi-permeable tubular membrane of cylindrical cross section. The model is

MASS TRANSFER THROUGH PERMEABLE WALLS

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established by making the following assumptions:

* For the internal fluid: a) Isothermal regime; b) Steady state; c) Newtonian fluid; d) Constant physical properties; e) Parabolic and totally developed velocity profile; f) Convection in the axial direction only; g) Transport by diffusion in the axial direction negligible with respect to the convective flux; h) Fickian diffusion in the radial direction; i) No mass sinks or sources in the bulk of the fluid. * For the membrane: a) Diffusional mass transport through the membrane; b) Mass transfer coefficient independent of the axial position. * For the external fluid: a) Turbulent flow regime; b) Uniform solute concentration. Under these hypotheses, the solute concentration distribution in the flowing solution mal' be represented by:

x = x(r, z)} O