Mass transfer in the vicinity of a separation membrane the applicability of the stagnant film theory

Transcription

1 Journal of Membrane Science 202 (2002) Mass transfer in the vicinity of a separation membrane the applicability of the stagnant film theory João M. Miranda, João B.L.M. Campos Centro de Estudos de Fenómenos de Transporte, Departamento de Engenharia Química, Faculdade de Engenharia da Universidade do Porto, Rua Dr. Roberto Frias, Porto, Portugal Received 10 April 2001; received in revised form 19 November 2001; accepted 23 November 2001 Abstract In membrane separation cells, the permeate velocity is usually predicted applying the stagnant film equation with mass transfer data from impermeable systems, Sh I. In this paper, the applicability of the stagnant film equation is discussed. Laminar momentum and solute transport equations are numerically solved in a permeable parallel plate cell and in a conical cell with a laminar jet impinging vertically to the membrane surface. Membrane surface concentration and permeate velocity predictions are applied to stagnant film equation to obtain Sh I. Sh I data are compared with Sherwood data from impermeable systems with uniform mass production at the wall, and with Sherwood data from impermeable systems with uniform concentration at the wall. This analysis is done for a wide range of the dimensionless numbers characterizing the membrane separation process: Schmidt, Reynolds (or Peclet for the parallel plate cell), Π v and Π π0 numbers. The stagnant film equation is a quite accurate equation to predict permeate velocity, and the type of impermeable data that must be used depends on the dimensionless groups. The conclusions are identical for both cells Elsevier Science B.V. All rights reserved. Keywords: Mass transfer; Membrane separation processes; Stagnant film theory 1. Introduction The pressure-driven fluid flow through a separation membrane convectively transports solute to the membrane surface which, during the start of the separation process (non-steady-state), is much larger than the solute that turns back to the bulk by diffusion. The solute accumulates at the membrane surface until equilibrium between diffusive and convective solute fluxes is reached. The solute concentration changes from a maximum at the membrane surface to a min- Corresponding author. Tel.: ; fax: address: jmc@fe.up.pt (J.B.L.M. Campos). imum in the bulk, outside the mass boundary layer. The membrane surface with rejected solute is said to be polarized. Concentration polarization increases the difficulty of the design of membrane separation systems. In polarized systems, the solute transport equation must be solved to predict the solute distribution at the membrane surface: ( C V x X + V C 2 ) z Z = D C X C Z 2 (1) No exact analytical solution exists for Eq. (1) with the respective boundary conditions, and several approximate solutions have been developed. Among them, the simple stagnant film model originally developed /02/$ see front matter 2002 Elsevier Science B.V. All rights reserved. PII: S (01)

2 138 J.M. Miranda, J.B.L.M. Campos / Journal of Membrane Science 202 (2002) Nomenclature A 1 virial coefficient c normalized solute concentration c I normalized solute concentration in the permeable system (C P /C 0 ) c P normalized solute concentration in the impermeable system ( (V 0 C I /ṁ)(1/pe)) c s normalized solute concentration at the membrane surface C solute concentration (kg/m 3 ) C 0 solute concentration in the feed (kg/m 3 ) C b solute concentration in the bulk (kg/m 3 ) C p solute concentration in the filtrate flux (kg/m 3 ) C s solute concentration at the membrane surface (kg/m 3 ) D j inlet jet diameter in the conical cell (m) D solute molecular diffusivity (m 2 /s) H distance between parallel plates (m) j normalized permeate velocity j x normalized perturbation induced in vx I by the suction. j z normalized perturbation induced in vz I by the suction. J permeate velocity (m/s) k mass transfer coefficient (m/s) l m normalized membrane length L f final impermeable length (m) L i initial impermeable length (m) L m membrane length (m) ṁ mass flux produced at the impermeable wall (kg m 2 s 1 ) R m membrane flow resistance (Pa s m 1 ) v x normalized tangential velocity component v z normalized normal velocity component V 0 mean inlet velocity (m/s) V x tangential velocity component (m/s) V z normal velocity component (m/s) x normalized tangential coordinate X tangential coordinate (m) z normalized normal coordinate Z normal coordinate (m) Non-dimemensional numbers Pe Peclet number (HV 0 /D) or(d j V 0 /D) Re Sc Sh Reynolds number (ρhv 0 /µ) or(ρd j V 0 /µ) Schmidt number (µ/ρd) Sherwood number (kh/d) or(kd j /D) Π v (( P m π 0 )/R m V 0 ) Π π0 ( π 0 / P m ) Π π0 (π 0 / P m ) Greek symbols P m static pressure difference across the membrane at the cell axis (Pa) π(x) osmotic pressure difference between the liquid at the membrane surface and the filtrate flux (Pa) π 0 osmotic pressure difference between the liquid over the membrane with a concentration C 0 and the filtrate flux (Pa) δ film thickness (m) δ normalized film thickness φ (ln c P /jpe) µ dynamic viscosity of the feed solution (Pa s) π(c) osmotic pressure (Pa) π 0 osmotic pressure in the the liquid over the membrane with concentration C 0 (Pa) θ ln c ρ density of the feed solution (kg/m 3 ) Superscripts I impermeable system P permeable system by Brian [1] has been widely used. The mass transfer is assumed to occur in a stagnant liquid film of thickness δ. According to the model assumptions, the solute transport equation, Eq. (1), is simplified and after integration across the stagnant film gives: J = D ( ) ( ) δ ln Cs C p Cs C p = kln (2) C b C p C b C p

3 J.M. Miranda, J.B.L.M. Campos / Journal of Membrane Science 202 (2002) where J is the permeate velocity, C s, C b and C p respectively, the solute concentration at the membrane surface, the solute concentration in the bulk, and the solute concentration in the filtrate solution. The ratio of the solute diffusivity to the stagnant film thickness is set equal to the solute mass transfer coefficient, k. The application of the stagnant film equation has the problem of quantification of k, since δ is a model parameter. Mass transfer coefficients from impermeable systems were adopted and the usual form of the stagnant film equation is: ( ) J = k I Cs C p ln (3) C b C p where the superscript I refers to impermeable systems. From now on, this equation is mentioned in the text as the film equation. Two types of empirical correlations are frequently used to obtain k I : correlations from mass transfer data in impermeable soluble systems (uniform surface concentration) and; correlations from momentum or heat transfer data in impermeable systems, coupled with theoretical mass/heat/momentum analogies (uniform temperature or uniform heat flux). The use of the film equation with mass transfer coefficient data from impermeable systems has been theme of discussion in the literature [2 4]. Zydney [4] provided a fundamental understanding of the mathematical basis for the stagnant film formalism as well as a quantitative description of the limitations of the model for a parallel plate cell. To clarify the limitations of Eq. (3), it is necessary to have a complete description of the concentration and velocity profiles in both permeable and impermeable systems. To reach this objective, the governing flow and mass transport equations and the respective boundary conditions must be numerically solved. The numerical procedure is not simple since, at the membrane surface, the solute concentration depends on the permeate velocity, and so, flow and mass transport equations must be solved simultaneously. Shen and Probstein [5], Leung and Probstein [6], and Probstein et al. [2] solved the governing equations by a boundary layer integral method. The basis of this numerical method is to assume the shapes of the velocity and concentration profiles across the diffusion layer. After integration, the partial differential governing equations are transformed in ordinary differential equations easy to solve. The accuracy of this numerical method depends on how the adopted profiles describe the reality. The shapes of these profiles are not easy to assume in high-polarized cells and even more when the cell has a complex geometry. Probstein et al. [2] applied the integral method to solve the laminar governing equations in a parallel plate ultrafiltration cell. One of the objectives was to compare the film equation with the numerical solution when gelation takes place at the membrane surface, i.e. when the concentration along the membrane is uniform and equal to the gelling concentration. They evaluated k I from Leveque solution in an impermeable system with uniform concentration at the surface. They find a good agreement between numerical solution and film equation, in particular when the value of the gelling concentration is low. Shen and Probstein [5], and Leung and Probstein [6] also studied a parallel plate ultrafiltration cell in laminar regime with a gel at the surface. They took the film equation with k I from Leveque solution and compared the predictions with experimental and numerical data. The deviations were justified by the way the physical properties were quantified, in particular, the molecular diffusivity. In summary, the numerical studies published about the applicability of the film equation were all performed in a parallel plate cell with uniform concentration along the membrane. These studies applied approximate numerical methods to predict concentration and velocity fields in the cell. In the present study, the governing equations are solved simultaneously, applying a numerical finite-difference scheme. Two cells with different geometries are studied: a parallel plate cell, with the fluid flowing along the membrane surface and a conical cell with a jet impinging vertically to the membrane and spreading radially. The flow in the parallel plate cell is a well-behaved flow, while in the conical cell the jet flow is complex and has an important feature: in the boundary layer, the normal velocity component does not arise only from suction effects. The applicability of the film equation is analyzed for a large range of operation conditions. This general analysis is possible after a parametric study based on dimensional analysis. In Section 2, the hydrodynamic and mass transfer assumptions of the film model are going to be analyzed and discussed in a parallel plate cell.

4 140 J.M. Miranda, J.B.L.M. Campos / Journal of Membrane Science 202 (2002) Description of the film models 2.1. Introduction The permeable parallel plate cell is schematically represented in Fig. 1a. The width is very large, the distance between plates is H and the membrane length is L m. A liquid solution (solvent/solute) is fed at a mean velocity V 0, and the solute concentration at the cell inlet is C 0. This cell is going to be designated along the text by the letter or superscript P. An impermeable cell with the same geometry is represented in Fig. 1b. Instead of the membrane, there is an impermeable wall producing mass at a uniform rate ṁ. A pure solvent is fed into the cell and the hydrodynamic conditions at the inlet are identical to those of the cell P. This cell is going to be designated along the text by the letter or superscript I. When the permeate velocity is very small (zero in the limit), the flow fields in P and in I are identical, becoming different for increasing permeate velocities. Anyway, the permeate flow rate is always a very small fraction of the feed flow rate and the suction (permeability) effects on the flow are only felt in a thin layer over the membrane. The velocity fields in both systems (I and P) can be related in the following way: The tangential velocity component in a point (x, z) inside P, vx P, can be expressed by: vx P = vi x + j x (4) where vx I is the tangential velocity component in point (x, z) inside I and j x is the tangential perturbation induced in the flow by the presence of the membrane (suction effect). The tangential velocity components are normalized by the mean inlet velocity V 0. The normal velocity component in a point (x, z) inside P, vz P, can be expressed by: vz P = vi z + j z (5) where vz I is the normal velocity component in point (x, z) inside I and j z is the axial perturbation induced in the flow by the presence of the membrane (suction effect). The normal velocity components are also normalized by the mean inlet velocity V Impermeable cell model (I) The model adopted to describe the solute transport in the vicinity of the impermeable wall producing mass at a uniform rate, ṁ, is schematically represented in Fig. 2a. The model assumptions are described as follow. Fig. 1. Representation of: (a) permeable parallel plate device; (b) impermeable parallel plate device with uniform mass production at the wall.

5 J.M. Miranda, J.B.L.M. Campos / Journal of Membrane Science 202 (2002) Normalizing the coordinates and the film thickness by the distance between plates, H, and the solute concentration by, Pe ṁ V 0 ( c I = V 0 ṁpe CI ) the solute transport equation and the boundary conditions become: 2 c I 2 z = 0 (9) c I = 1 z at z = 0 (10) Fig. 2. Stagnant film theory applied to: (a) impermeable system with uniform mass production at the wall; (b) permeable system Hydrodynamic assumptions The fluid in the layer just above the wall is stagnant. Outside this layer, the fluid is well mixed and flows along the tangential direction with a mean velocity V Mass transport assumptions The mass produced at the surface is transported into the bulk by molecular diffusion through the stagnant film. The variation of solute concentration along the tangential direction X is small, 2 C I / X 2 = 0, and the diffusive transport along this direction is negligible, 2 C I / X 2 = 0. In the bulk, after solute dilution, the concentration remains very low and is taken as negligible, (Cb I = 0). The model is a totally hydrodynamic model, since the mass transport assumptions are imposed by the flow conditions. The hydrodynamic assumption, Vx I = 0, and the solute transport assumption, 2 C I / X 2 = 0, induce simplifications on the mass transport equation, Eq. (1), which becomes: 2 C I Z 2 = 0 (6) According to the model assumptions, the mass boundary conditions are: D CI = ṁ at Z = 0 (7) Z C I = 0 at Z = δ I (8) c I = 0 at z = [δ ] I (11) The concentration profile, c I, is linear, Eqs. (9) and (10), and the normalized film thickness, [δ ] I, is equal to the symmetric value of the normalized solute concentration at the surface, c I s : c I z = 1 ci s [δ ] I = 1 [δ ] I = c I s (12) 2.3. Permeable cell model (P) The model adopted to describe the solute transport in the vicinity of the membrane is schematically represented in Fig. 2b. The model assumptions are described as follow Hydrodynamic assumptions The fluid layer just over the membrane is quasi-stagnant. The permeate flux crosses this layer, flowing in the normal direction with uniform velocity, V z = J. Outside this layer, the fluid is well mixed and flows along the tangential direction with a mean velocity V Mass transport assumptions The solute transported by the permeate flux is retained at the membrane surface and returns to the bulk by molecular diffusion along Z. The variation of solute concentration along the tangential direction X is small, C P / X = 0, and the diffusive transport along this direction is negligible, 2 C P / X 2 = 0. The bulk is well mixed and after solute dilution, the concentration remains equal to the inlet concentration (C P b = CP 0 ).

6 142 J.M. Miranda, J.B.L.M. Campos / Journal of Membrane Science 202 (2002) The hydrodynamic assumptions, Vx P = 0 and Vz P = constant = J, and the mass transfer assumption, 2 C P / X 2 = 0, induce simplifications on the mass transport equation. Normalizing the velocity component by the inlet velocity, V 0, the solute concentration by the feed solute concentration, C0 P, and the coordinates by the distance between plates, H, the solute transport equation becomes: 1 2 c P Pe z 2 = j cp z (13) Assuming total solute retention at the membrane surface (cb P = 0), the normalized boundary conditions are: 1 c P Pe z = jcp at z = 0 (14) c P = cb P = 1 at z = [δ ] P (15) Applying the following variable transformation, φ = (ln c P /jpe), to the mass transport equation and respective boundary conditions, leads to: 2 φ 2 z = 0 (16) φ = 1 z at z = 0 (17) φ = 0 at z = [δ ] P (18) The φ profile is linear, Eqs. (16) and (17), and the normalized film thickness, [δ ] P, is equal to the symmetric of the value of φ at the membrane surface, φ s : φ z = 1 φ s [δ ] P = 1 [δ ] P = φ s (19) 2.4. Association between models I and P In models I and P, the solute transport equations, Eqs. (9) and (16), are equivalent. The boundary conditions, Eqs. (10), (11), (17) and (18), are also equivalent if the thickness of the stagnant films is the same, δ P = δ I. According to Eq. (12), the Sherwood number for model I is given by: Sh I = ki H D = 1 [δ ] I = 1 cs I (20) and, assuming identical fields: Sh I = 1 cs I = 1 Sh I = jpe φ s ln cs P (21) The stagnant film equation of Brian [1] is obtained solving Eq. (21) in order to the permeate velocity: J = ShI V 0 Pe ln CP s C0 P (22) According to the theory developed, Eq. (22) is accurate if the thickness of the films is similar. This is impossible to verify and so a strategy was outlined to discuss the applicability of the film equation. For established operation conditions, laminar momentum and solute transport equations in a permeable system were solved by a numerical procedure. Permeate velocity and membrane surface concentration predictions were used in Eq. (22) to calculate values of Sh I. These values are compared with Sherwood values numerically obtained in an identical impermeable system with uniform mass production at the wall. From this comparison, the strengths and limitations of the film equation are going to be understood. 3. Numerical work 3.1. Introduction An improved finite-difference scheme was used to solve the momentum and solute transport equations. This improved numerical scheme is described in detailed in [7] for a parallel plate cell. Therefore, only the basic principles are going to be briefly described. In solid liquid contact systems, the mass boundary layer is very thin and to achieve accurate numerical solutions, a very dense grid near the membrane must be used. Inside the mass boundary layer, the solute concentration changes intensely, in particular for high permeate fluxes; the first and second derivatives in order to the normal direction are high. The shape of the concentration profiles is close to exponential and so a variable transformation, θ = ln c, is used to attenuate the derivatives. Applying this transformation, the shape of the new θ profiles is almost linear and accurate solutions are obtained with large grid spacing.

7 J.M. Miranda, J.B.L.M. Campos / Journal of Membrane Science 202 (2002) Governing equations The flow equations were applied in their stream function vorticity formulation [7]. The dimensionless solute transport equation is written as: c v x x + v c z z = 1 ( 2 ) c Pe x c z 2 (23) with the solute concentration normalized by the feed solute concentration, C 0, for the permeable cell, and by ṁ/v 0 for the impermeable cell with uniform mass flux at the wall. All the equations applied were written assuming incompressible flow and constant physical properties Dimensional analysis The flow inside the cell, the solute distribution and the permeate flux depend on several variables. These variables were combined in dimensionless groups in a previous work [7], and so, only the physical meaning of the most important groups are going to be briefly referred Convective diffusive groups The flow equations introduce the Reynolds number based on the parallel plate distance, H: Re = ρv 0H (24) µ The solute transport equation introduces the Peclet number: Pe = Sc Re = V 0H (25) D where Sc is the Schmidt number Permeate flow groups The permeate velocity depends on the osmotic pressure difference between the liquid at the membrane surface and at the filtrate flux, π. The osmotic pressure is related to the solute concentration and this relationship is well represented by a linear equation: π(c) = A 1 C (26) where the virial coefficient A 1 depends on the solute properties. The importance of the osmotic pressure difference in the permeate velocity is measured by the following dimensionless number: Π π0 = π 0 (27) P m where π 0 is the osmotic pressure difference between the liquid over the membrane with concentration C 0, and the permeate, and P m is the static pressure difference across the membrane. If the solute is completely rejected, the dimensionless group is re-written as: Π π0 = π 0 (28) P m The permeate velocity also depends on the membrane resistance to the permeate flux, R m. The ratio ( P m π 0 )/R m represents the permeate velocity when the solute concentration at the membrane surface is constant and equal to the feed concentration, C 0. The dimensionless group Π v = P m π 0 (29) R m V 0 represents the ratio between the permeate velocity at the conditions defined above and the average velocity at the cell inlet Boundary conditions Fig. 3 shows the numerical domain and the respective boundaries for cells I and P. The flow is symmetric and so, only half of the cell is represented. Most of the boundary conditions and respective equations are described in [7], and so, they are going to be briefly referred Boundary I cell inlet At the cell inlet, the laminar flow is fully developed. The solute concentration is C 0 for cell P and C 0 = 0 for cell I. Fig. 3. Numerical domain and respective boundaries.

8 144 J.M. Miranda, J.B.L.M. Campos / Journal of Membrane Science 202 (2002) Boundary II at the middle plane of the rectangular channel The middle plane is a plane of symmetry and so there are no momentum or mass fluxes crossing this boundary Boundary III cell exit At the cell exit, a fully developed laminar profile and a fully developed concentration profile is assumed Boundary IV permeable and impermeable walls Permeable wall. At the membrane surface, the velocity component along x is zero (non-slip condition) and the normalized permeate velocity is given by the solution diffusion model originally developed by Lonsdale et al. [8]: j(x) = P m(x) π(x) (30) V 0 R m The introduction of the dimensionless groups previously defined gives: Π v j(x) = (1 c s (x)π π0 ) (31) 1 Π π0 The solute retained at the membrane surface is assumed to be transported by diffusion into the bulk: J(x)C s (x) = D dc s(x) (32) dz or using the dimensionless groups: j(x) = 1 1 Pe c s (x) dc s (x) dz (33) Impermeable wall. The velocity is zero along the impermeable wall, due to the non-slip condition. The solute produced at the wall is transported by diffusion into the flowing fluid: ṁ = D dc s(x) (34) dz or in a dimensionless form (c s = C s V 0 /ṁ): 1 dc s (x) = 1 (35) Pe dz Boundary V and VI initial and final impermeable walls The velocity is zero along the impermeable sections, due to the non-slip condition. There are no mass fluxes crossing the impermeable walls Numerical grid The orthogonal grid used was denser at the beginning and at the end of the numerical domain and in the layer over the membrane, where concentration and velocity gradients have their highest values. The grid was uniformly spaced in those zones Numerical procedure The numerical procedure employed to study the permeable cell is described in detail in [7]. The flow and solute (transformed by θ = ln c) equations were discretized by a finite-difference technique. The θ equation was transformed before discretization. After discretization, an iterative numerical method was used to solve the discretized equations. The convergence of the iterative process was studied by two complementary ways: analyzing the evolution of the θ values in the layer adjacent to the membrane, and analyzing the sum of the normalized total residues of the discretized θ equation and respective boundary conditions. The accuracy of the numerical method was tested obtaining solutions on successively refined grids. 4. Results and discussion As previously referred, a strategy was outlined to understand the strengths and limitations of the film equation. Momentum and solute transport equations in permeable systems were numerically solved, and permeate velocity and membrane surface concentration predictions were applied to Eq. (22) to calculate Sh I. The values of Sh I are compared with Sherwood values numerically obtained in an impermeable system with uniform mass production at the wall and also with Sherwood values numerically obtained in an impermeable system with uniform concentration at the wall.

9 4.1. Parallel plate cell J.M. Miranda, J.B.L.M. Campos / Journal of Membrane Science 202 (2002) Uniform permeate velocity The permeate velocity is uniform along the membrane when the osmotic pressure difference between the liquid at the membrane surface and the permeate, π, attains a nearly constant value along the membrane (i.e. when the surface concentration is constant) or when the static pressure difference across the membrane, P m, is much higher than π. This is a limit situation and according to Eqs. (28) and (31), when P m π 0 then Π π0 0 and Π v becomes equal to the absolute normalized permeate velocity, j. In the following figures, the curves I represent the Sherwood values for an impermeable system with uniform mass production at the wall and the curves II represent the Sherwood values for an impermeable system with uniform solute concentration at the wall. Fig. 4 shows Sh I data along the membrane for Pe = ; curves A to D are for different values of Π v at Π π0 = 0. Curves A D and curve I are coincident and they are deviated from curve II. For this Peclet number, Sh I values are independent of Π v. The film equation is exact if the k I data are from impermeable systems with uniform mass production at the wall (I). The relative deviation between curves I and II increases slightly along the membrane. Fig. 5 shows Sh I data along the membrane for a higher value of Peclet number, Pe = Curves A and B are coincident with curve I, but curves C and D are over curve I. For this Peclet number, Sherwood values are no longer independent of Π v number. The Fig. 4. Sh I along the membrane in the parallel plate cell for different values of Π v at Pe = 10 5 and Π π0 = 0. Fig. 5. Sh I along the membrane in the parallel plate cell for different values of Π v at Pe = 10 6 and Π π0 = 0. film equation gives more accurate results, if the k I data are from impermeable systems with uniform mass production at the wall (I). This accuracy decreases with increasing Π v number, particularly for values of Π v higher than 10 5.ForΠ v = (D), at the end of the membrane (x = 12), the relative deviation to curve I is about 28% while the deviation to curve II is about 50% Non-uniform permeate velocity When the permeate velocity is non-uniform, the osmotic pressure difference between the liquid at the membrane surface and the permeate, π, is not negligible when compared with the static pressure difference across the membrane, P m. The local pressure driving force is now given by P m (x) π(x). The value of Π π0 is in the range between 0 and 1, and Π v represents the upper limit of the absolute normalized permeate velocity, j(x) (Eq. (31) with c s (x) = 1 whatever is Π π0 ). Fig. 6 shows Sh I data along the membrane for Pe = and Π v = ; curves A D are for different values of Π π0. For increasing values of Π π0, Sh I values decrease and tend to those of curve II. For values of Π π0 greater than approximately 0.7, the film equation gives more accurate results if the k I data are from impermeable systems with uniform concentration at the wall (II). This is not a surprise, since for high values of Π π0, the membrane is strongly polarized, the surface concentration tends to be uniform and also tends to the bulk concentration (zero mass flux at the membrane surface).

10 146 J.M. Miranda, J.B.L.M. Campos / Journal of Membrane Science 202 (2002) Conical cell with an impinging jet Fig. 6. Sh I along the membrane in the parallel plate cell for different values of Π π0 at Pe = 10 6 and Π v = Several conclusions can be stated from the analysis of the parallel plate cell: whatever is the type of the k I data applied, the accuracy of the film equation decreases with increasing Peclet and Π v numbers; for very high values of Π π0 (Π π0 > 0.7), the film equation should be applied with k I data from an impermeable system with uniform concentration at the wall; for moderate and low values of Π π0 (Π π0 < 0.7), the film equation should be applied with k I data from an impermeable system with uniform mass flux at the wall; for the operation conditions studied, the maximum relative deviation observed between the accurate value of k I and the value of k I from an impermeable system with uniform mass flux at the wall is of the order of 28% (Pe = , Π v = and Π π0 = 0). The parallel plate device is very suitable to develop the theory, however, the flow is rather simple: well developed from the inlet until the exit of the cell, without normal velocity component. The stagnant film equation should be tested in a more complex flow. The device chosen to perform this test was a conical cell with a laminar jet impinging vertically to the membrane surface (Fig. 7). The conical wall extends from the nozzle to a short distance above the impingement membrane. The laminar jet flow in an impermeable conical cell was studied by the authors [9], as well as the mass transfer coefficients from a soluble plate into a liquid jet [10]. The concentration and velocity profiles in a permeable conical cell were also studied [11]. New concentration polarization indexes were defined, but no mass transfer data were presented due to the lack of information about the applicability of the film equation. The models developed, the numerical methods employed and the dimensional analysis done for the parallel plate cell can be applied to the conical cell with some adaptations to the new geometry. In order to analyze the results, some important differences must be kept in mind: the characteristic dimension is not the distance between plates but the inlet jet diameter (D j ); the flow is not fully developed and so the principle of similarity of the flow solution with respect to Reynolds number is not observed. This is the reason why the Sherwood number is no longer dependent on the Peclet number but on Reynolds and Schmidt numbers individually [10]. Fig. 7. Conical cell: streamlines for an impermeable cell at Re = 860.

11 J.M. Miranda, J.B.L.M. Campos / Journal of Membrane Science 202 (2002) A brief description of the flow is needed to better understand the mass transfer data. The streamlines in an impermeable cell are represented in Fig. 7 for Re = 860, and three regions are observed: an impingement region, a wall region and an expansion region. At the impingement region, the plate imposes a shift on the fluid direction, that occurs very close to the surface. At the wall region, the fluid flows in a thin channel confined downside by the plate and upside by fluid in recirculation. Along this channel, the radial velocity component is high inside the mass boundary layer. At the expansion region, the fluid expands to the whole area of the cell and from there on, the fluid flows confined by the conical wall and by the impingement plate. Between the wall zone and the expansion zone, there is, over the plate, a second recirculation zone. The dimensions of this recirculation zone increase with increasing Reynolds number Low Schmidt number Fig. 8 shows Sh I data along the membrane for two values of Reynolds number (A and B) at Sc = 900, Π v = 10 4 and Π π0 = 0 (uniform permeate velocity). The shape of the curves is according to the flow pattern in the cell. Curves A and B are coincident with curves I and are deviated from curves II. Along the membrane, the relative deviation is higher in the wall zone (1 <r<2), where the fluid decelerates after the impingement. Fig. 9 shows Sh I data along the membrane for different values of Π v (A C) at Sc = 900, Re = 860 and Π π0 = 0 (uniform permeate velocity). Curves A C and curve I are coincident and once more deviated from curve II. Fig. 9. Sh I along the membrane in the conical cell for different values of Π v at Sc = 900, Re = 860 and Π π0 = 0. Fig. 10 shows Sh I data along the membrane for different values of Π π0 (A C) at Sc = 900, Re = 860 and Π v = For increasing values of Π π0, Sh I values decrease and move from curve I (Π π0 = 0) to curve II (Π π0 = 1). The conclusions for low Schmidt numbers are identical to those in the parallel plate cell for low Peclet numbers: the accuracy of the stagnant film equation does not depend on Reynolds and Π v numbers; for low and moderate values of Π π0 (Π π0 < 0.8), the film equation should be applied with k I data from impermeable systems with uniform mass production at the wall and the film equation is exact for Π π0 = 0, whatever are the Reynolds and Π v numbers; for very high values of Π π0 (Π π0 > 0.8), the film equation should be applied with k I data from an Fig. 8. Sh I along the membrane in the conical cell for two values of Re at Re = 900, Π v = 10 4 and Π π0 = 0. Fig. 10. Sh I along the membrane in the conical cell for different values of Π π0 at Sc = 900, Re = 860 and Π v = 10 4.

12 148 J.M. Miranda, J.B.L.M. Campos / Journal of Membrane Science 202 (2002) Fig. 11. Sh I along the membrane in the conical cell for two values of Re at Sc = 28600, Π v = 10 4 and Π π0 = 0. impermeable system with uniform concentration at the wall; for the operation conditions studied, the maximum relative deviation between the accurate value of k I and the value of k I from an impermeable system with uniform mass flux at the wall is of the order of 25% (Pe = , and Π π0 = 1). This maximum deviation is located at the wall zone High Schmidt number Fig. 11 shows Sh I data along the membrane for two values of Reynolds number (A and B) at Sc = 28600, Π v = 10 4 and Π π0 = 0 (uniform permeate velocity). Curves A and B are deviated from curves I (I A and I B ) and the deviation increases with increasing Reynolds number. Fig. 12 shows Sh I data along the membrane for different values of Π v (A and C) at Sc = 28600, Re = 300 and Π π0 = 0 (uniform permeate velocity). Fig. 13. Sh I along the membrane in the conical cell for different values of Π π0 at Sc = 28600, Re = 300 and Π v = Curve A is coincident with curve I but curve C has higher values, which means that the deviation to curve I increases with increasing Π v number. Fig. 13 shows Sh I data along the membrane for different values of Π π0 (A C) at Sc = 28600, Re = 300 and Π v = Once more, for increasing values of Π π0, Sh I values decrease and for Π π0 = 1, they are coincident with those of curve II. The conclusions for high Schmidt numbers are again similar to those in the parallel plate cell for high Peclet numbers: whatever is the type of k I data applied, the accuracy of the film equation decreases with increasing Reynolds, Schmidt and Π v numbers; for high values of Π π0 (Π π0 > 0.5), the film equation should be applied with k I data from an impermeable system with uniform concentration at the wall; for low values of Π π0 (Π π0 < 0.5), the film equation should be applied with k I data from an impermeable system with uniform mass flux at the wall; for the operation conditions studied, the maximum relative deviation between the accurate value of k I and the value of k I from an impermeable system with uniform mass production at the wall is of the order of 28% (Sc = 28600, Re = 300, Π v = 10 4 and Π π0 = 1) and this maximum deviation is located at the wall zone Mass transfer data in permeable systems Fig. 12. Sh I along the membrane in the conical cell for different values of Π v at Sc = 28600, Re = 300 and Π π0 = 0. As was proved, the stagnant film equation with mass transfer data from impermeable systems can be applied with relative success to membrane separation

13 J.M. Miranda, J.B.L.M. Campos / Journal of Membrane Science 202 (2002) significantly with increasing Π v number and the relative deviation between Sh I and Sh P data can reach 300% (in both cells for the highest value of Π v number studied). This behavior, stresses the idea that the film equation is suitable to obtain accurate permeate velocity predictions in spite of the unrealistic hydrodynamic and mass transfer assumptions of the film model. Fig. 14. Sh I, Sh P along the membrane in a parallel plate cell for different values of Π v at Pe = 10 6 and Π π0 = 0. Curves III A (Π v = 10 6 ) and III D (Π v = ) were obtained applying mass transfer predictions to Eq. (36). processes design. However, it is interesting to compare Sh I values with the real Sherwood values in permeable systems: Sh P = c/ z z=0 (36) c s 1 This comparison is made in Figs. 14 and 15, respectively for a parallel plate cell and for a conical cell (curves III were obtained applying numerical concentration predictions to Eq. (36)). The operation conditions chosen were the most unfavorable (between those studied) and they can be seen as limit conditions; for the parallel plate cell, high Peclet number and Π π0 = 0, and for the conical cell, high Schmidt number and Π π0 = 0. The values of Sh P increase 5. Conclusions The film equation is established combining two stagnant film models, one for a permeable system and another for an impermeable system. Both models have severe mass transport and flow assumptions and the real behavior is far from being described in a precise way. In spite of this, the film equation gives accurate permeate velocity data, either in a permeable parallel plate cell, or in a conical cell with a jet impinging vertically to the membrane. This agreement was much more unexpected in the conical cell, where the flow is much more complex: the normal component velocity does not arise only from suction effects, but also from the impingement of the jet into the membrane surface; the radial velocity component reaches high values inside the mass boundary layer and so the fluid is far from being stagnant and; just over the membrane there is a small recirculation zone. The following conclusions about the applicability of the film equation are valid for both cells: for low and moderate values of Π π0, the film equation should be applied with mass transfer data from impermeable systems with uniform mass production at the wall; for high values of Π π0, the film equation should be applied with mass transfer data from impermeable systems with uniform concentration at the wall; whatever is the value of Π π0 and the type of Sh I data, the accuracy of the film equation decreases with increasing Schmidt, Reynolds (or Peclet for the parallel plate device) and Π v numbers. Fig. 15. Sh I, Sh P along the membrane in a conical cell for different values of Π v at Sc = 28600, Re = 300 and Π π0 = 0. Curves III A (Π v = 10 6 ) and III D (Π v = ) were obtained applying mass transfer predictions to Eq. (36). Acknowledgements The authors acknowledge the financial support given by JNICT, PBIC/C/CEN/1337/92 and by F.C.T.,

14 150 J.M. Miranda, J.B.L.M. Campos / Journal of Membrane Science 202 (2002) PRAXIS XXI/BD/3280/94 and PRAXIS/C/EQU/ 12141/98. References [1] P. Brian, Mass transport in reverse osmosis, in: U. Merten (Ed.), Desalination by Reverse Osmosis, Cambridge, MA: MIT Press, [2] R.F. Probstein, J.S. Shen, W.F. Leung, Ultrafiltration of macromolecular solutions at high polarization in laminar channel flow, Desalination 24 (1978) [3] D.R. Trettin, M.R. Doshi, Ultrafiltration in an unstirred batch cell, Ind. Eng. Chem. Fund. 19 (1980) [4] A.L. Zydney, Stagnant film model for concentration polarization in membrane systems, J. Membr. Sci. 130 (1997) [5] J.S. Shen, R.F. Probstein, On the prediction of limiting flux in laminar ultrafiltration of macromolecular solutions, Ind. Eng. Chem. Fund. 16 (1977) [6] W.F. Leung, R.F. Probstein, Low polarization in laminar ultrafiltration of macromolecular solutions, Ind. Eng. Chem. Fund. 18 (1977) [7] J.M. Miranda, J.B.L.M. Campos, An improved numerical scheme to study mass transfer over a separation membrane, J. Membr. Sci. 188 (2001) [8] H. Lonsdale, U. Merten, R. Riley, Transport properties of cellulose acetate osmotic membranes, J. Appl. Polym. Sci. 9 (1965) [9] J.M. Miranda, J.B.L.M. Campos, Impinging jets confined by a conical wall: laminar flow predictions, AIChE J. 45 (1999) [10] J.M. Miranda, J.B.L.M. Campos, Impinging jets confined by a conical wall: high Schmidt mass transfer predictions in laminar flow, Int. J. Heat Mass Transfer 44 (2001) [11] J.M. Miranda, J.B.L.M. Campos, Concentration polarization in a membrane placed under an impinging jet confined by a conical wall a numerical approach, J. Membr. Sci. 182 (2001)