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Analysis of mass and momentum transfer in an annular electrodialysis cell in pulsed flow
Chemical Engineering Science 54 (1999) 1667—1675
Analysis of mass and momentum transfer in an annular electrodialysis cell in pulsed flow V. Pe´rez-Herranz*, J.L. Guin o´n, J. Garcı´ a-Anto´n Departamento de Ingenierı& a Quı& mica y Nuclear, E.T.S.I. Industriales, Universidad Polite& cnica de Valencia, P.O. Box. 22012, 46071 Valencia, Spain Received 19 March 1998; received in revised form 18 November 1998; accepted 17 December 1998
Abstract An analysis of mass and momentum transfer in an annular pulsating electrodialysis cell has been made. The mass and momentum balance equations are solved numerically for the case of reversing and non-reversing oscillatory flow. The numerical mass transfer is compared with the experimental results obtained from the determination of the limiting current density in an electrodialysis cell. From the numerical calculations and the experimental results, an enhancement of mass transfer due to superimposed pulsation can take place in the case of reversing flow. Increases in frequency and amplitude of pulsation improve the mass transfer coefficient, the effect of frequency being greater. 1999 Elsevier Science Ltd. All rights reserved. Keywords: Annular pulsed flow; Electrodialysis; Mass transfer
1. Introduction Annular flow has been used in heat transfer and mass transfer applications as electrochemical metal removal from dilute solutions. These heat or mass transfer devices can be designed using particular flows in which high transfer coefficients are encountered. One way to increase heat or mass transfer coefficients is to use a pulsating flow in the annular geometry, that could be applied to metal removal from dilute solutions using electrochemical reactors, or to chromate ions removal coming from the rinsing baths of the electroplating industry using electrodialysis. In these applications, pulsation has the advantage of increasing mass transfer due to the increase of the local velocity maintaining high residence times. The problem of unsteady or periodic turbulent and laminar flow through a tube has been studied experimentally as well as computationally by several research workers. Mao and Hanratty (1985, 1986) found that at high enough frequencies, fluid inertia is dominant over most of the flow field, and at high Reynolds numbers (Re'15 000), the imposed oscillation has no effect on the time-mean velocity gradient at the wall. Tu and
* Corresponding author. Tel.: 00 96 387 7630; fax: 00 96 387 7639.
Ramaprian (1983), and Ramaprian and Tu (1983), studying a fully developed periodic turbulent pipe flow, found that the distortion in the shape of the velocity profiles due to pulsation is large enough to cause inflexion points at some phase positions. This amplitude overshoot can be predicted by a quasi-steady model. Laminar pulsating flows in a tube were studied by Selx (1930), Kusama (1952) and Uchida (1956). Khodadadi (1991) obtained an analytic solution to the fully developed oscillatory fluid flow through a porous medium. Finally, Sudo et al. (1992) studied the fully developed laminar pulsating flow in a curved tube. In all cases a velocity overshoot near the wall has been observed. Pulsation has been used to increase the mass or heat transfer by convection between a solid wall and a liquid. The increase in turbulence by flow pulsation can only play a relevant role in mass and heat transfer when natural turbulence is not significant. So, it is in laminar or transition flow when the use of such disturbances can increase mass or heat transfer. Many works dealing with liquid—solid mass transfer in the presence of oscillating flow have been concerned with oscillatory flow around an isolated sphere (Gibert and Angelino, 1973, 1974; Guin o´n et al., 1995). The mass transfer in a particle bed with oscillating flow has been studied by Krasuk and Smith (1964), Ratel et al. (1988) and Cordonet et al. (1989). Krasuk and Smith, (1963) studied the mass
0009-2509/99/$ — see front matter 1999 Elsevier Science Ltd. All rights reserved. PII: S 0 0 0 9 - 2 5 0 9 ( 9 8 ) 0 0 5 3 7 - 5
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transfer in a pulsed column. Sobey in 1980 and 1982 has analysed numerically the oscillatory flow in a sinusoidal wavy channel for laminar flow at Reynolds number between 5 and 600. He studied the effect of the parameters about values of interest in the design of the high-efficiency membrane oxygenator of Bellhouse et al. (1973). In a second paper, in 1980, Stephanof et al., present the experimental observations of the flow corresponding to the calculations made by Sobey in the previous work. In 1983, Sobey extended the calculations to a wide range of Reynolds and Strouhal numbers. He defines three regions depending on the value of the Strouhal number, whose value can be classified as small, intermediate or large. Finally, in 1983, the same author presented a numerical analysis and experimental observations of oscillatory flow for different channel geometry at intermediate Strouhal numbers. The mass transfer in pulsating flow and the flow structure for different conditions have been studied by Nishimura et al. in several papers. In 1986, Nishimura et al. studied the relation between the flow structure and mass transfer in a wavy channel in laminar and turbulent flow. They found that pulsatile flow increases mass transfer due to the renewal of the concentration boundary layer. The measurements of local wall shear stress and mass transfer rate were carried out by the electrochemical method. In 1987 they developed a numerical method for the analysis of pulsatile flow at moderate Reynolds number. They compare their results with experimental flow visualisations that shows good agreement with the numerical analysis. In other work, in 1987, they studied experimentally the flow behaviour and the mass transfer rates in unsteady flow for a wide range of the Reynolds number from 8;10\ to 10. They found that the mass transfer rate strongly depends on the Reynolds number but the effect of the Strouhal number is insignificant. In 1989, Nishimura et al. studied numerically and experimentally an oscillatory flow in symmetric wavy-walled channels for Reynolds numbers between 5 and 200 and Strouhal numbers between 0.06 and 0.3 using a flow visualization technique. They found that the flow structures depend on the Womerseley number. Finally Mackley and Ni (1991) and Harrison and Mackley (1992) studied the mass transfer in a pulsatile flow bioreactor. On the other hand, mass or heat transfer in unsteady flows has been used to determine wall shear stress using small electrodes (Mao and Hanratty, 1991), or hot films (Kaiping, 1983). These authors, have developed mathematical models to calculate mass or heat transfer in pulsating flow solving the mass or heat transfer equations, respectively. In the case of annular ducts, mass and momentum transfer has been studied in the absence of pulsation. In this way, Ould-Rouis et al. in 1989 studied the laminar steady flow in an annular electrochemical cell. Cheng and Hsu, in 1986, studied the heat transfer and velocity profiles in a fully developed forced convective flow through
an annular packed-sphere bed. Legentilhomme and Legrand, in 1993, studied the velocity profiles in a laminar swirling flow induced by a tangential inlet in a cylindrical annulus. Finally Pe´rez-Herranz et al. in 1997 obtained the velocity profiles of a laminar pulsating flow in an annulus. These authors found a velocity overshoot near the walls of the inner and outer cylinders of the annulus. This paper presents the effect of pulsation frequency, pulsation amplitude and fluid velocity on the wall shear stress of the inner cylinder of an annular duct composed by an anionic membrane in laminar flow. The mass transfer across the anionic membrane is calculated numerically for different pulsation parameters (frequency, amplitude and fluid velocity), solving the mass balance equation. The results obtained from the mass balance solution are compared with the experimental results determined from the limiting current density. Although other works have been made about mass and momentum transfer in pulsatile flows, different hydrodynamic conditions or different configurations have been studied. This study is confined to laminar flow and low Reynolds numbers (30(Re(75), in order to maintain high residence time in the electrodialysis cell.
2. Theoretical analysis Non-steady flow of an incompressible fluid with axial symmetry through an annulus has been considered. The problem has been solved taking cylindrical coordinates, whose z-axis is identified with the centre line of the inner cylinder. The geometric configuration of the problem is shown in Fig. 1. In Section 1 the shear stress on the inner wall of the annulus is calculated from the solution of the periodic motions superposed on the steady Poiseuille
Fig. 1. Schematic diagram of the system configuration.
V. Pe´ rez-Herranz et al./Chemical Engineering Science 54 (1999) 1667—1675
flow which causes mean transmission of fluid in one direction. In Section 2, the mass transfer across the inner cylinder of the annulus is calculated numerically from the solution of the mass balance assuming that the thickness of the diffusion boundary layer can be neglected with respect to the thickness of the momentum boundary layer. 2.1. Momentum transfer analysis When fluid enters an annular duct, a hydrodynamic boundary layer begins to develop on the wall of the annulus. As the fluid progresses downstream, the boundary layer on the inner cylinder thickens until it reaches the boundary layer of the outer cylinder. At this point, the velocity profile is quasi-established. The diffusion boundary layer develops in a similar manner to the pure diffusion concentration profile. The imposition of a sinusoidal oscillation on the mean flow through an annular duct causes a periodically varying pressure field. If the amplitude is small enough, a linear response is obtained from which dp ! "k #k cos (ut) dz
(1)
*u 1 *p 1 *u *u X"! X# X . #l *t o *z r *r *r
(2)
The solution of Eqs. (2) and (1) was obtained by PerezHerranz et al. in 1997, and consists of a steady component u and a transient component u which is a periodic function of time, and are given by
D "D !D . J is the zeroth-order Bessel function of the first kind, and K is the zeroth-order modified Bessel function of the first kind. The velocity profiles in oscillatory flow are affected by two dimensionless groups, the dimensionless velocity, a "au/v, and the Womersley number, a "D (u/l . As shown in a previous work, a maximum in the velocity profile is obtained in the proximity of the wall due to pulsation. The contribution of frequency to the shape of the velocity profiles is greater than the contribution of amplitude. 2.1.1. Time-average shear stress The effect of liquid pulsation on mass transfer across the anionic membrane is essentially due to changes in the velocity gradient close to the membrane wall. So, before solving the mass transfer balance, the effect of pulsation parameters, frequency, amplitude, and velocity, on the membrane wall shear stress is studied. Using Eqs. (3) and (4), the local instantaneous shear stress, q (t) exerted on the wall by a single laminar pulsating flowing fluid element is obtained from q (t)"!k
du dr
rPR
(5)
and the time-average shear stress in a pulsation cycle
with z being the distance in the flow direction. The pressure is constant across the section of the annulus and becomes only function of z and t, and then, the pressure gradient becomes only a function of t. The time-mean pressure gradient is designated by k , and the amplitude of the oscillations by k . These oscillations give rise to oscillations in the shear stress at the wall. The simplest model for the flow assumes that the induced oscillations behave in the same way as for a laminar pulsating flow, i.e.
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1 R!r R u " k !(R) ln , 2k K 2 r
(3)
1 2 qN " q(t) dt (6) ¹ can be evaluated by a Simpson rule integration of the instantaneous wall shear stress. The study will be made considering the modulus sign of the instantaneous shear stress, taking into account the possibility of flow reversal for part of a pulsation cycle, and that the effect of the instantaneous velocity on mass transfer is independent of flow direction. Fig. 2a—c, shows in logarithmic coordinates a study of the influence of pulsation parameters, v, a, and u, on time-average shear stress, calculated from expression (6) in the inner cylinder of the annulus. In these figures, a line delimits the value a "1, which indicates if reversing flow can take place. Fig. 2a shows the effect of the steady fluid velocity, v, on the time-average shear stress, for different amplitude and frequency values. When there is flow reversal, a '1,
[K (a R*i )!K (a R*i )] J (a r*i )#[J (a R*i )!J (a R*i )] K (a r*i ) u "!iva 1# e S , J (a R*i ) K (a R*i )!J (a R*i ) K (a R*i )
where u and, u are the steady and the oscillatory components of velocity, respectively. R*, R* and r* are the inner radius, outer radius and radial coordinate, dimensionalized with respect to the equivalent diameter,
(4)
the steady component of the shear stress, can be neglected with respect to the oscillatory component, and the time-average shear stress, qN , is independent of the steady fluid velocity. However, for a (1, the time-average
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Fig. 2. a. Effect of the steady velocity, v, on the time-averaged shear stress: (———) a"8 mm, u"3n/2; (— — —) a"2 mm, u"3n/2; (— — — —) a"8 mm, u"n/2; (— — —) a"2 mm, u"n/2. b. Effect of pulsation amplitude, a on the time-averaged shear stress. v"0.001 m/s, (———) u"2n; (— — —) u"3n/2; (— — — —) u"n; (— — —) u"n/2. c. Effect of pulsation frequency, u on the time-averaged shear stress. v"0.001 m/s, (———) a"0.008 m; (— — —) a"0.004 m; (— — — —) a"0.002 m.
shear stress tends towards the shear stress in steady flow, with a slope of one, as it could be expected. In Fig. 2b, the variation of qN is presented with respect to the pulsation amplitude for a given steady fluid velocity and different values of frequency. As it can be seen in this figure, when flow reversal takes place, a '1, qN varies linearly with amplitude in logarithmic coordinates, having a slope of one, while for values of a (1, amplitude does not exert any influence on the time-average shear stress. Finally, from Fig. 2c, it can be concluded that for a (1, when there is no flow reversal, the time-average shear stress is independent of frequency, while for a '1,
qN varies linearly with frequency. In this case the slope is 1.5. Then, in the case of non-reversing flow, a (1, pulsa tion, has no effect on the time-average shear stress, qN , which varies linearly with velocity (Fig. 2a). However, in the case of reversing flow, qN increases with amplitude and frequency, and the effect of the fluid velocity on the time-average shear stress can be neglected. In order to increase the velocity gradient at the wall it is better to operate at high frequencies than at high amplitudes as observed from the slope of the curves obtained in Fig. 2b and c for the variation of the time-averaged shear stress with amplitude and frequency.
V. Pe´ rez-Herranz et al./Chemical Engineering Science 54 (1999) 1667—1675
The Pe´clet number, Pe, is defined as
2.2. Mass transfer analysis In an electrodialysis cell, the operational current density must be kept below the limiting current density to avoid water splitting. This phenomenon takes place in the diluting compartment of the electrodialysis cell, since it is in this compartment where mass transfer is taking place. The limiting current density can be related to the Sherwood number by the following expression: i D Sh" , (7) nFDC where Sh is the Sherwood number, i , is the limiting current density, n is the electrical charge of the ions transferred across the membrane, D is the diffusion coefficient and C the bulk concentration. The limiting current density increases as the flow rate increases. Garcı´ a-Anto´n et al. demonstrated in 1997 that the use of a pulsating flow overimposed to a steady flow increases the limiting current density. Since the Schmidt number for liquids is large, the concentration boundary layer is very thin compared with the hydrodynamic boundary layer. Then, the concentration field within the boundary layer can be described by a two-dimensional mass balance equation:
*C *C *C *C *C #u #u "D # X *z W *y *t *z *y
(8)
being z the coordinate in the streamwise direction, and y the coordinate in the direction normal to the membrane. At high Schmidt numbers, within the concentration boundary layer, the velocity profile may be considered as a linear function of the distance from the wall, with a slope equal to the velocity gradient at the wall: u (t)"S (t) ) y , (9) X where S (t) is the instantaneous velocity gradient on the wall. If the Pe´clet number is high enough, diffusion in the flow direction can be neglected in comparison with the convection term. This last assumption is valid except near the edges of the anionic membrane, however, Kaiping, in 1983, and Mao and Hanratty in 1991, have shown that if the inertia term, (*C/*t), is important, diffusion in the y coordinate dominates over diffusion in the flow direction. Then, Eq. (8) can be written in a dimensionless form as *C* *C* *C* #S* (t*) " , *t* *z* *y*
(10)
where the following dimensionless variables have been used: C z y C*" , z*" , y*" Pe , C ¸ ¸ t u¸ S (t) t*" Pe , u*" Pe\ , S* (t*)" . ¸/D D S
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Pe"S ¸/D (11) and the reference velocity gradient at the membrane wall, S , will be the corresponding velocity gradient at the steady velocity, v. The previous differential equation is solved using the following boundary conditions: C* (z*, 0, t* )"0 for 0)z*)1, *C* *y* *C* *y*
"0 for z*(0 and z*'1,
y *"0
"0 for all z* ,
y *"R
C*"1 for z*P!R and z*PR. The first of these boundary conditions means that the concentration on the membrane wall is zero because of the assumption that the electrodialysis cell is operated under limiting current conditions. The solution of the momentum equation leads to the value of the velocity gradient at the membrane wall, and then the differential equation (10) can be solved numerically to obtain the mass flux across the membrane. The overall mass flux is described by the modified Sherwood number, Sh*, that is calculated from the integration of the concentration gradient at the surface along the membrane using the following equation:
*C* dz* . (12) *y* y *"0 For steady flow, Le´veˆque’s solution gives for the modified Sherwood number
Sh*"Sh Pe "
Sh*"0.8075 .
(13)
For a given condition of fluid flow, amplitude, frequency and mean velocity, the time variation of the velocity gradient, S (t) can be obtained from the velocity profiles given by Eqs. (3) and (4). Then, the differential equation (10) is solved numerically using the control volume method by Patankar (1980). By solving the system of finite difference equations derived from the application of this method using the same calculation procedure of Mao and Hanratty (1991), the concentration field is obtained, and then the modified Sherwood number, Sh*, is calculated from Eq. (12). In order to test the calculation procedure, Eq. (10) has been solved assuming a sinusoidal variation of shear rate using the same conditions as Kaiping (1983) and Mao and Hanratty (1991). So, a sinusoidal variation of the velocity gradient can be considered in a dimensionless form: S* (t)"1#S
cos (u*t*) ,
(14)
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Fig. 3. Effect of the velocity gradient on the modified Sherwood number. Case of non-reversing flow. S*"1#0.9 cos(u*t*). (———) quasisteady solution; (— — —) u*"0.1; (— — — —) u*"0.5; (— — — ) ) u*"1; (— — — —) u*"2.
S being the amplitude of the dimensionless velocity
gradient. The mass transfer for different shear functions is studied in Figs. 3 and 4 in terms of the modified Sherwood number, comparing the results obtained with the quasisteady mass transfer for non-reversing and reversing flow respectively. In the case of non-reversing flow, Fig. 3, the mass transfer reaches a maximum and a minimum value in a pulsation period, and the increase in frequency causes a decrease in the amplitude of the mass transfer. The minima of the mass flux change strongly with frequency, while the maximum is the same for all frequencies. Moreover a phase lag that increases with frequency appears with respect to the quasi-steady solution. However, in the case of reversing flow, Fig. 4, when the amplitude of the shear rate is greater than one, the velocity gradient can reach negative values, but the mass flux is always positive, and the modified Sherwood number reaches two maxima and two minima in a pulsation period. The increase in the pulsation frequency has a greater effect on the mass transfer than in the case of non-reversing flow. The maximum as well as the minimum values of mass transfer are attenuated with the increase in frequency, moving away from the quasisteady solution. A phase lag that increases with frequency is also observed.
3. Experimental procedure Details regarding the experimental arrangement and the experimental techniques used may be found in Garcia-Anto´n et al. (1997) and Pe´rez-Herranz et al. (1997).
Fig. 4. Effect of the velocity gradient on the modified Sherwood number. Case of reversing flow. S*"1#2.0 cos (u*t*). (———) quasisteady solution; ( — — — ) u*"0.2; ( — — — — )u*"1.0; (— — —) u*"5.
The annular electrodialysis cell used in this work is applied to study the mass transfer across an anionic membrane of chromate ions coming from a bath whose concentration C was 9.62;10\ M CrO and 0.05 M H SO . The solution used was the same in the two sections, in order to minimize diffusion problems through the membrane owing to the concentration gradients on the two sides of the membrane. The cathode was a 7 cm diameter and 20 cm high AISI 316 stainless steel cylindrical grid. The anode was a 20 cm high Pb—Sn alloy bar. The membrane used in this work was the ARA 17-10 membrane produced by Morgane . This membrane is specially designed for the recovery of acids by electrodialysis. The major independent variables investigated were fluid velocity, v, pulsation frequency, u, and pulsation amplitude, a, which were varied in the limits: 1.33;10\)v)6.63;10\ m/s, 0)u)2n s\, 2;10\)a)8;10\ m.
4. Results The mass transfer across the anionic membrane is studied in terms of the limiting current density, i . In the case of pulsating flow conditions, average values of the limiting current density, iM , in a pulsation period have been considered. Then the experimental modified Sherwood number, Sh*, averaged in time is calculated from iM D Sh*" Pe , nFDC
(14)
V. Pe´ rez-Herranz et al./Chemical Engineering Science 54 (1999) 1667—1675
where C is the bulk concentration, and iM , the time average limiting current density. The limiting current density has been determined by the Cowan and Brown method (Cowan and Brown, 1959), which consists of the plotting of M /iM versus 1/iM ,
M being the time-average voltage drop across the membrane. The limiting current density corresponds to a minimum in this plot, and as it was found by Garcı´ a-Anto´n et al. in 1997, the time-average limiting current density increases with amplitude and frequency, but the effect of frequency on the time-average limiting current density is greater than the effect of amplitude, as can be predicted by the theoretical variation of the wall shear stress with the pulsation parameters discussed previously. Figs. 5—7 compare the experimental results obtained for the mass transfer across the membrane with the predicted mass transfer calculated from the solution of Eq. (10) using the velocity gradient at the membrane wall calculated from Eqs. (3) and (4) for the same values of pulsation frequency, u, pulsation amplitude, a, and fluid velocity, v, investigated in the experimental work. The results are compared in terms of the modified Sherwood number. Figs. 5 and 6 show the effect of frequency on the enhancement of mass transfer for different values of velocity and amplitude, respectively. The experimental results are compared with those obtained from the numerical solution of the mass balance. These figures show that in the range of experimental conditions, corresponding to reversing flow, a great increase in mass transfer, expressed in terms of Sh*/Sh* is observed when the frequency increases. For a given frequency, the enhancement of mass transfer increases with increasing amplitude and with decreasing velocity.
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Fig. 6. Effect of pulsating flow on the modified Sherwood number. v"3.98;10\ m/s, (䊏) a"2;10\ m, (䉬) a"8;10\ m.
Fig. 7. Comparison of calculated modified Sherwood number with respect to the experimental results: (———) calculated; (䊏) Experimental.
Fig. 5. Effect of pulsating flow on the modified Sherwood number. a"8;10\ m, (䊏) v"1.326;10\ m/s, (䉬) v"3.98;10\ m/s, (䉱) v"6.63;10\ m/s.
In Fig. 7 the experimental data of Sh*/Sh* are compared with the data calculated from the mass balance for different pulsation conditions. As it can be seen, pulsation contributes to increase mass transfer with respect to the steady flow. However, the results obtained from the solution of the mass balance are slightly smaller than the experimental results. This difference can be explained from the fact that the theoretical development neglects diffusion in the flow direction, which can be a significant effect at the two ends of the membrane. Another cause of error could be the possibility that the flow field was not sufficiently developed in the region of the test section, despite the calming section near the membrane. These
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two situations will be more relevant in the case of reversing flow. Finally, the exact nature of the mass transport mechanism can be another possible source of error. In the classical theory of concentration polarization, it is assumed that mass transport from the bulk solution to the membrane takes place by diffusion across the diffusion boundary layer whose thickness depends only on the hydrodynamical conditions near the membrane. However, an additional mass transport by natural convection can take place, apart from diffusion. This natural convection is due to the voltage-dependent concentrations and electric field gradients within the boundary layer. Then, apart from the hydrodynamical conditions, the thickness of the boundary layer decreases as voltage increases (Rubinstein and Shtilman, 1978). This effect will increase with the instantaneous velocity as the voltage drop corresponding to the limiting current density increases with velocity.
Marı´ a Asuncio´n Jaime for her help in translating this paper into English.
5. Conclusions
K
Time-averaged shear stress is evaluated numerically for the case of pulsating annular flow. The results indicate that shear stress is affected by two dimensionless groups, the dimensionless velocity, a , and the Womer sley number, a , frequency being the parameter with a greater influence on the shear stress. The mass transfer across an anionic membrane in an annular pulsating flow is compared with the numerical results obtained from the mass balance for different values of the pulsation parameters, frequency, u, amplitude, a and fluid velocity, v. In general the experimental enhancement of mass transfer due to pulsation is higher than the numerical results, probably due to the assumptions made in the application of the boundary layer theory. Also, end effects on the mass balance, that have not been considered, can contribute to the differences between experimental mass transfer and the numerical calculations. From the results obtained, it is concluded that laminar oscillatory flow can be used to increase mass transfer in electrodialysis. Oscillatory flow has the advantage of increasing mass transfer due to the increase of the local velocity gradient at the membrane surface maintaining high residence times. This situation is specially interesting in the removal of metals from dilute solutions, such as chromate ions coming from the rinsing baths of the electroplating industries.
¸ n p r r* R R R* R* S S* S S
Acknowledgements We are indebted to the Generalitat Valenciana for a Postgraduate grant for V. Pe´rez-Herranz and the D.G.I.C.Y.T. for the support of this work. We thank
Notation a C C C* D D D D F i iM J k k
t t* ¹ u u u W u X v y y* z z*
pulsation amplitude, m concentration, mol/m bulk concentration, mol/m dimensionless concentration diffusion coefficient, m/s diameter of the inner cylinder (membrene wall), m diameter of the outer cylinder (cathode), m equivalent diameter, m Faraday constant, 96500 C/eq limiting current density, A/m time-average limiting current density, A/m zeroth-order Bessel function of the first kind pressure gradient for steady flow, kg/m s maximum pressure gradient due to pulsation, kg/m s zeroth-order modified Bessel function of the first kind length of the membrane in the flow direction, m valence of chromate ions pressure, kg/m s radial distance, m dimensionless radial distance radius of the inner cylinder (membrane), m radius of the outer cylinder (cathode), m dimensionless radius of the inner cylinder dimensionless radius of the outer cylinder velocity gradient at the wall, s\ dimensionless velocity gradient at the wall velocity gradient at the wall in steady flow, s\ amplitude of the dimensionless velocity gradient in pulsating flow time, s dimensionless time pulsation period, s steady component of the fluid velocity, m/s oscillatory component of the fluid velocity, m/s velocity normal to the wall in the mass balance, m/s velocity in z-direction in pulsed flow, m/s bulk mean velocity for steady flow, m/s axial distance normal to the membrane wall in the mass balance, m dimensionless axial distance normal to the membrane wall in the mass balance axial distance along the annulus, m dimensionless axial distance
Dimensionless groups Pe Re
Pe´clet number"S ¸/D Reynolds number"vD /l
V. Pe´ rez-Herranz et al./Chemical Engineering Science 54 (1999) 1667—1675
Sh Sh* a a
Sherwood number"i D /n F D C modified Sherwood number"Sh Pe dimensionless velocity"a u/v Womersley number"D (u/l)
Greek letters
M k l o q qN u
time-average voltage drop, V fluid viscosity, kg/m s kinematic viscosity, m/s density, kg/m wall shear stress, N/m time-average wall shear stress, N/m pulsation frequency, s\
References Bellhouse, B.J., Bellhouse, F.H., Curl, C.M., MacMillan, T.I., Gunning, A.J., Spratt, E.H., MacMurray, S.B., & Nelems, J.M. (1973). A high efficiency oxigenator and pulsatile pumping system, its application to animal trials. Am. Soc. Artif. Internal Organs, 19, 72—79. Cheng, P., & Hsu, C.T. (1986). Fully-developed, forced convective flow through an annular packed-sphere bed with wall effects. Int. J. Heat Mass ¹ransfer, 29, 1843—1853. Cordonet, J.S., Riba, J.P., & Angelino, H. (1989). Mass transfer in a particle bed with oscillating flow. Chem. Engng. Sci., 44, 2107—2111. Cowan, D.A., & Brown, J.H. (1959). Effect of turbulence on limiting current in electrodialysis cells. Ind. Engng Chem., 51, 1445—1448. Garcı´ a-Anto´n, J., Pe´rez-Herranz, V., & Guin o´n, J.L. (1997). Mass transfer in an annular electrodialysis cell in pulsating flow. J. Appl. Electrochem., 27, 469—476. Gibert, H., & Angelino, H. (1973). Influence de la pulsation sur les transferts de matie`re entre une sphe`re et un liquide. Can. J. Chem. Engng, 51, 319—325. Gibert, H., & Angelino, H. (1974). Transferts de matie`re entre une sphe`re soumise a des vibrations et un liquide en mouvement. Int. J. Heat Mass ¹ransfer, 17, 625—632. Guin o´n, J.L., Pe´rez-Herranz, V., Garcı´ a-Anto´n, J., & Lacoste, G. (1995). Enhancement of mass transfer at a espherical electrode in pulsating flow. J. Appl. Electrochem., 25, 267—272. Harrison, S.T.L., & Mackley, M.R. (1992). A pulsatile flow bioreactor. Chem. Engng Sci., 47, 490—493. Kaiping, P. (1983). Unsteady forced convective heat transfer from a hot film in non-reversing and reversing shear flow. Int. J. Heat Mass ¹ransfer, 26, 545—557. Khodadadi, J.M. (1991). Oscillatory fluid flow through a porous medium chanel bounded by two impermeable parallel plates. J. Fluid Engng, 113, 509—511. Krasuk, J.H., & Smith, J.M. (1963). Mass transfer in a pulsed column. Chem. Engng Sci., 18, 591—598. Krasuk, J.H., & Smith, J.M. (1964). Mass transfer in a packed, pulsed column. A.I.Ch.E. J., 10, 759—763. Kusama, H. (1952). Study of pulsating flow (Pulsating flow in a circular pipe). Soc. Mech. Engrs ¹rans., 18, 27—31. Legentilhomme, P., & Legrand, J. (1993). Mode´lisation nume´rique du transfert de matie`re dans un e´coulement annulaire faiblement tourbillonaire non-entretenu. Can. J. Chem. Engng, 71, 299—311.
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Mackley, M.R., & Ni, X. (1991). Mixing and dispersion in a buffled tube for steady laminar and pulsatile flow. Chem. Engng Sci., 46, 3139—3151. Mao, Z.X., & Hanratty, T.J. (1985). The use of scalar transport probes to measure wall shear stress in a flow with imposed oscillations. Exp. Fluids, 2, 129—135. Mao, Z.X., & Hanratty, T.J. (1986). Studies of the wall shear stress in a turbulent pulsating pipe flow. J. Fluid Mech., 170, 545—564. Mao, Z.X., & Hanratty, T.J. (1991). Analysis of wall shear stress probes in large amplitude unsteady flows. Int. J. Heat Mass ¹ransfer, 34, 281—290. Nishimura, T., Arakawa, S., Murakami, S., & Kawamura, Y. (1989). Oscillatory viscous flow in symmetric wavy-walled channels. Chem. Engng Sci, 44, 2137—2148. Nishimura, T., Kajimoto, Y., Tarumoto, A., & Kawamura, Y. (1986). Flow structure and mass transfer for a wavy channel in transitional flow regime. J. Chem. Engng Jpn., 19, 449—455. Nishimura, T., Tarumoto, A., & Kawamura, Y. (1987). Flow and mass transfer characteristics in wavy channels for oscillatory flow. Int. J. Heat Mass ¹ransfer, 30, 1007—1015. Nishimura, T., Yoshino, T., & Kawamura, Y. (1987). Numerical flow analysis of pulsatile flow in a channel with symmetric wavy walls at moderate Reynolds numbers. J. Chem. Engng Jpn., 20, 479—485. Oud-Rouis, M., Nouar, C., Salem, A., Legrand, J., Legentilhomme, P., & Aouabed, H. (1989). Hydrodynamics and mass transfer of the laminar flow in the entrance region of an annular electrochemical reactor. Int. Chem. Engng Symp. Ser., 112, 17—28. Patankar, S.V. (1980). Numerical heat transfer and fluid flow. New York: McGraw-Hill. Pe´rez-Herranz, V., Garcı´ a-Anto´n, J., & Guin o´n, J.L. (1997). Velocity profiles and limiting current in an annular electrodialysis cell in pulsed flow. Chem. Engng Sci., 52, 843—851. Ramaprian, B.R., & Tu, S.W. (1983). Fully developed periodic turbulent pipe flow. Part 2. The detailed structure of the flow. J. Fluid Mech., 137, 59—81. Ratel, A., Duverneuil, P., & Lacoste, G. (1988). Influence de la pulsation liquide sur le comportement d’un re´acteur e´lectrochimique constitue´ par un lit de particules conductrices. I. Transfert de matie`re. J. Appl. Electrochem., 18, 394—400. Rubinstein, I., & Shtilman, L. (1978). Voltage against current curves of cation exchange membranes. J. Chem. Soc. Faraday ¹rans., 2, 231—246. Sexl, T. (1930). U®ber den von E.G. Richardson entdeckten annulareffekt. Z. Phys., 61, 349—362. Sobey, I.J. (1980). On flow patterns through furrowed channels. Part 1. Calculated flow patterns. J. Fluid Mech., 96, 1—26. Sobey, I.J. (1982). Oscillatory flows at intermediate Strouhal number in assymetric channels. J. Fluid Mech., 125, 359—373. Sobey, I.J. (1983). The occurrence of separation in oscillatory flow. J. Fluid Mech., 134, 247—257. Stephanoff, K.D., Sobey, I.J., & Bellhouse, B.J. (1980). On flow patterns through furrowed channels. Part 2. Observed flow patterns. J. Fluid Mech., 96, 27—32. Sudo, K., Sumida, M., & Yamane, R. (1992). Secondary motion of fully developed oscillatory flow in a curved pipe. J. Fluid Mech., 237, 189—208. Tu, S.W., & Ramaprian, B.R. (1983). Fully developed periodic turbulent pipe flow. Part 1. Main experimental results and comparison with predictions. J. Fluid Mech., 137, 31—58. Uchida, S. (1956). The pulsating viscous flow superimposed on the steady laminar motion of incompressible fluid in a circular pipe. Zamp., 7, 403—421.
Analysis of mass and momentum transfer in an annular electrodialysis cell in pulsed flow V. Pe´rez-Herranz*, J.L. Guin o´n, J. Garcı´ a-Anto´n Departamento de Ingenierı& a Quı& mica y Nuclear, E.T.S.I. Industriales, Universidad Polite& cnica de Valencia, P.O. Box. 22012, 46071 Valencia, Spain Received 19 March 1998; received in revised form 18 November 1998; accepted 17 December 1998
Abstract An analysis of mass and momentum transfer in an annular pulsating electrodialysis cell has been made. The mass and momentum balance equations are solved numerically for the case of reversing and non-reversing oscillatory flow. The numerical mass transfer is compared with the experimental results obtained from the determination of the limiting current density in an electrodialysis cell. From the numerical calculations and the experimental results, an enhancement of mass transfer due to superimposed pulsation can take place in the case of reversing flow. Increases in frequency and amplitude of pulsation improve the mass transfer coefficient, the effect of frequency being greater. 1999 Elsevier Science Ltd. All rights reserved. Keywords: Annular pulsed flow; Electrodialysis; Mass transfer
1. Introduction Annular flow has been used in heat transfer and mass transfer applications as electrochemical metal removal from dilute solutions. These heat or mass transfer devices can be designed using particular flows in which high transfer coefficients are encountered. One way to increase heat or mass transfer coefficients is to use a pulsating flow in the annular geometry, that could be applied to metal removal from dilute solutions using electrochemical reactors, or to chromate ions removal coming from the rinsing baths of the electroplating industry using electrodialysis. In these applications, pulsation has the advantage of increasing mass transfer due to the increase of the local velocity maintaining high residence times. The problem of unsteady or periodic turbulent and laminar flow through a tube has been studied experimentally as well as computationally by several research workers. Mao and Hanratty (1985, 1986) found that at high enough frequencies, fluid inertia is dominant over most of the flow field, and at high Reynolds numbers (Re'15 000), the imposed oscillation has no effect on the time-mean velocity gradient at the wall. Tu and
* Corresponding author. Tel.: 00 96 387 7630; fax: 00 96 387 7639.
Ramaprian (1983), and Ramaprian and Tu (1983), studying a fully developed periodic turbulent pipe flow, found that the distortion in the shape of the velocity profiles due to pulsation is large enough to cause inflexion points at some phase positions. This amplitude overshoot can be predicted by a quasi-steady model. Laminar pulsating flows in a tube were studied by Selx (1930), Kusama (1952) and Uchida (1956). Khodadadi (1991) obtained an analytic solution to the fully developed oscillatory fluid flow through a porous medium. Finally, Sudo et al. (1992) studied the fully developed laminar pulsating flow in a curved tube. In all cases a velocity overshoot near the wall has been observed. Pulsation has been used to increase the mass or heat transfer by convection between a solid wall and a liquid. The increase in turbulence by flow pulsation can only play a relevant role in mass and heat transfer when natural turbulence is not significant. So, it is in laminar or transition flow when the use of such disturbances can increase mass or heat transfer. Many works dealing with liquid—solid mass transfer in the presence of oscillating flow have been concerned with oscillatory flow around an isolated sphere (Gibert and Angelino, 1973, 1974; Guin o´n et al., 1995). The mass transfer in a particle bed with oscillating flow has been studied by Krasuk and Smith (1964), Ratel et al. (1988) and Cordonet et al. (1989). Krasuk and Smith, (1963) studied the mass
0009-2509/99/$ — see front matter 1999 Elsevier Science Ltd. All rights reserved. PII: S 0 0 0 9 - 2 5 0 9 ( 9 8 ) 0 0 5 3 7 - 5
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transfer in a pulsed column. Sobey in 1980 and 1982 has analysed numerically the oscillatory flow in a sinusoidal wavy channel for laminar flow at Reynolds number between 5 and 600. He studied the effect of the parameters about values of interest in the design of the high-efficiency membrane oxygenator of Bellhouse et al. (1973). In a second paper, in 1980, Stephanof et al., present the experimental observations of the flow corresponding to the calculations made by Sobey in the previous work. In 1983, Sobey extended the calculations to a wide range of Reynolds and Strouhal numbers. He defines three regions depending on the value of the Strouhal number, whose value can be classified as small, intermediate or large. Finally, in 1983, the same author presented a numerical analysis and experimental observations of oscillatory flow for different channel geometry at intermediate Strouhal numbers. The mass transfer in pulsating flow and the flow structure for different conditions have been studied by Nishimura et al. in several papers. In 1986, Nishimura et al. studied the relation between the flow structure and mass transfer in a wavy channel in laminar and turbulent flow. They found that pulsatile flow increases mass transfer due to the renewal of the concentration boundary layer. The measurements of local wall shear stress and mass transfer rate were carried out by the electrochemical method. In 1987 they developed a numerical method for the analysis of pulsatile flow at moderate Reynolds number. They compare their results with experimental flow visualisations that shows good agreement with the numerical analysis. In other work, in 1987, they studied experimentally the flow behaviour and the mass transfer rates in unsteady flow for a wide range of the Reynolds number from 8;10\ to 10. They found that the mass transfer rate strongly depends on the Reynolds number but the effect of the Strouhal number is insignificant. In 1989, Nishimura et al. studied numerically and experimentally an oscillatory flow in symmetric wavy-walled channels for Reynolds numbers between 5 and 200 and Strouhal numbers between 0.06 and 0.3 using a flow visualization technique. They found that the flow structures depend on the Womerseley number. Finally Mackley and Ni (1991) and Harrison and Mackley (1992) studied the mass transfer in a pulsatile flow bioreactor. On the other hand, mass or heat transfer in unsteady flows has been used to determine wall shear stress using small electrodes (Mao and Hanratty, 1991), or hot films (Kaiping, 1983). These authors, have developed mathematical models to calculate mass or heat transfer in pulsating flow solving the mass or heat transfer equations, respectively. In the case of annular ducts, mass and momentum transfer has been studied in the absence of pulsation. In this way, Ould-Rouis et al. in 1989 studied the laminar steady flow in an annular electrochemical cell. Cheng and Hsu, in 1986, studied the heat transfer and velocity profiles in a fully developed forced convective flow through
an annular packed-sphere bed. Legentilhomme and Legrand, in 1993, studied the velocity profiles in a laminar swirling flow induced by a tangential inlet in a cylindrical annulus. Finally Pe´rez-Herranz et al. in 1997 obtained the velocity profiles of a laminar pulsating flow in an annulus. These authors found a velocity overshoot near the walls of the inner and outer cylinders of the annulus. This paper presents the effect of pulsation frequency, pulsation amplitude and fluid velocity on the wall shear stress of the inner cylinder of an annular duct composed by an anionic membrane in laminar flow. The mass transfer across the anionic membrane is calculated numerically for different pulsation parameters (frequency, amplitude and fluid velocity), solving the mass balance equation. The results obtained from the mass balance solution are compared with the experimental results determined from the limiting current density. Although other works have been made about mass and momentum transfer in pulsatile flows, different hydrodynamic conditions or different configurations have been studied. This study is confined to laminar flow and low Reynolds numbers (30(Re(75), in order to maintain high residence time in the electrodialysis cell.
2. Theoretical analysis Non-steady flow of an incompressible fluid with axial symmetry through an annulus has been considered. The problem has been solved taking cylindrical coordinates, whose z-axis is identified with the centre line of the inner cylinder. The geometric configuration of the problem is shown in Fig. 1. In Section 1 the shear stress on the inner wall of the annulus is calculated from the solution of the periodic motions superposed on the steady Poiseuille
Fig. 1. Schematic diagram of the system configuration.
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flow which causes mean transmission of fluid in one direction. In Section 2, the mass transfer across the inner cylinder of the annulus is calculated numerically from the solution of the mass balance assuming that the thickness of the diffusion boundary layer can be neglected with respect to the thickness of the momentum boundary layer. 2.1. Momentum transfer analysis When fluid enters an annular duct, a hydrodynamic boundary layer begins to develop on the wall of the annulus. As the fluid progresses downstream, the boundary layer on the inner cylinder thickens until it reaches the boundary layer of the outer cylinder. At this point, the velocity profile is quasi-established. The diffusion boundary layer develops in a similar manner to the pure diffusion concentration profile. The imposition of a sinusoidal oscillation on the mean flow through an annular duct causes a periodically varying pressure field. If the amplitude is small enough, a linear response is obtained from which dp ! "k #k cos (ut) dz
(1)
*u 1 *p 1 *u *u X"! X# X . #l *t o *z r *r *r
(2)
The solution of Eqs. (2) and (1) was obtained by PerezHerranz et al. in 1997, and consists of a steady component u and a transient component u which is a periodic function of time, and are given by
D "D !D . J is the zeroth-order Bessel function of the first kind, and K is the zeroth-order modified Bessel function of the first kind. The velocity profiles in oscillatory flow are affected by two dimensionless groups, the dimensionless velocity, a "au/v, and the Womersley number, a "D (u/l . As shown in a previous work, a maximum in the velocity profile is obtained in the proximity of the wall due to pulsation. The contribution of frequency to the shape of the velocity profiles is greater than the contribution of amplitude. 2.1.1. Time-average shear stress The effect of liquid pulsation on mass transfer across the anionic membrane is essentially due to changes in the velocity gradient close to the membrane wall. So, before solving the mass transfer balance, the effect of pulsation parameters, frequency, amplitude, and velocity, on the membrane wall shear stress is studied. Using Eqs. (3) and (4), the local instantaneous shear stress, q (t) exerted on the wall by a single laminar pulsating flowing fluid element is obtained from q (t)"!k
du dr
rPR
(5)
and the time-average shear stress in a pulsation cycle
with z being the distance in the flow direction. The pressure is constant across the section of the annulus and becomes only function of z and t, and then, the pressure gradient becomes only a function of t. The time-mean pressure gradient is designated by k , and the amplitude of the oscillations by k . These oscillations give rise to oscillations in the shear stress at the wall. The simplest model for the flow assumes that the induced oscillations behave in the same way as for a laminar pulsating flow, i.e.
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1 R!r R u " k !(R) ln , 2k K 2 r
(3)
1 2 qN " q(t) dt (6) ¹ can be evaluated by a Simpson rule integration of the instantaneous wall shear stress. The study will be made considering the modulus sign of the instantaneous shear stress, taking into account the possibility of flow reversal for part of a pulsation cycle, and that the effect of the instantaneous velocity on mass transfer is independent of flow direction. Fig. 2a—c, shows in logarithmic coordinates a study of the influence of pulsation parameters, v, a, and u, on time-average shear stress, calculated from expression (6) in the inner cylinder of the annulus. In these figures, a line delimits the value a "1, which indicates if reversing flow can take place. Fig. 2a shows the effect of the steady fluid velocity, v, on the time-average shear stress, for different amplitude and frequency values. When there is flow reversal, a '1,
[K (a R*i )!K (a R*i )] J (a r*i )#[J (a R*i )!J (a R*i )] K (a r*i ) u "!iva 1# e S , J (a R*i ) K (a R*i )!J (a R*i ) K (a R*i )
where u and, u are the steady and the oscillatory components of velocity, respectively. R*, R* and r* are the inner radius, outer radius and radial coordinate, dimensionalized with respect to the equivalent diameter,
(4)
the steady component of the shear stress, can be neglected with respect to the oscillatory component, and the time-average shear stress, qN , is independent of the steady fluid velocity. However, for a (1, the time-average
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Fig. 2. a. Effect of the steady velocity, v, on the time-averaged shear stress: (———) a"8 mm, u"3n/2; (— — —) a"2 mm, u"3n/2; (— — — —) a"8 mm, u"n/2; (— — —) a"2 mm, u"n/2. b. Effect of pulsation amplitude, a on the time-averaged shear stress. v"0.001 m/s, (———) u"2n; (— — —) u"3n/2; (— — — —) u"n; (— — —) u"n/2. c. Effect of pulsation frequency, u on the time-averaged shear stress. v"0.001 m/s, (———) a"0.008 m; (— — —) a"0.004 m; (— — — —) a"0.002 m.
shear stress tends towards the shear stress in steady flow, with a slope of one, as it could be expected. In Fig. 2b, the variation of qN is presented with respect to the pulsation amplitude for a given steady fluid velocity and different values of frequency. As it can be seen in this figure, when flow reversal takes place, a '1, qN varies linearly with amplitude in logarithmic coordinates, having a slope of one, while for values of a (1, amplitude does not exert any influence on the time-average shear stress. Finally, from Fig. 2c, it can be concluded that for a (1, when there is no flow reversal, the time-average shear stress is independent of frequency, while for a '1,
qN varies linearly with frequency. In this case the slope is 1.5. Then, in the case of non-reversing flow, a (1, pulsa tion, has no effect on the time-average shear stress, qN , which varies linearly with velocity (Fig. 2a). However, in the case of reversing flow, qN increases with amplitude and frequency, and the effect of the fluid velocity on the time-average shear stress can be neglected. In order to increase the velocity gradient at the wall it is better to operate at high frequencies than at high amplitudes as observed from the slope of the curves obtained in Fig. 2b and c for the variation of the time-averaged shear stress with amplitude and frequency.
V. Pe´ rez-Herranz et al./Chemical Engineering Science 54 (1999) 1667—1675
The Pe´clet number, Pe, is defined as
2.2. Mass transfer analysis In an electrodialysis cell, the operational current density must be kept below the limiting current density to avoid water splitting. This phenomenon takes place in the diluting compartment of the electrodialysis cell, since it is in this compartment where mass transfer is taking place. The limiting current density can be related to the Sherwood number by the following expression: i D Sh" , (7) nFDC where Sh is the Sherwood number, i , is the limiting current density, n is the electrical charge of the ions transferred across the membrane, D is the diffusion coefficient and C the bulk concentration. The limiting current density increases as the flow rate increases. Garcı´ a-Anto´n et al. demonstrated in 1997 that the use of a pulsating flow overimposed to a steady flow increases the limiting current density. Since the Schmidt number for liquids is large, the concentration boundary layer is very thin compared with the hydrodynamic boundary layer. Then, the concentration field within the boundary layer can be described by a two-dimensional mass balance equation:
*C *C *C *C *C #u #u "D # X *z W *y *t *z *y
(8)
being z the coordinate in the streamwise direction, and y the coordinate in the direction normal to the membrane. At high Schmidt numbers, within the concentration boundary layer, the velocity profile may be considered as a linear function of the distance from the wall, with a slope equal to the velocity gradient at the wall: u (t)"S (t) ) y , (9) X where S (t) is the instantaneous velocity gradient on the wall. If the Pe´clet number is high enough, diffusion in the flow direction can be neglected in comparison with the convection term. This last assumption is valid except near the edges of the anionic membrane, however, Kaiping, in 1983, and Mao and Hanratty in 1991, have shown that if the inertia term, (*C/*t), is important, diffusion in the y coordinate dominates over diffusion in the flow direction. Then, Eq. (8) can be written in a dimensionless form as *C* *C* *C* #S* (t*) " , *t* *z* *y*
(10)
where the following dimensionless variables have been used: C z y C*" , z*" , y*" Pe , C ¸ ¸ t u¸ S (t) t*" Pe , u*" Pe\ , S* (t*)" . ¸/D D S
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Pe"S ¸/D (11) and the reference velocity gradient at the membrane wall, S , will be the corresponding velocity gradient at the steady velocity, v. The previous differential equation is solved using the following boundary conditions: C* (z*, 0, t* )"0 for 0)z*)1, *C* *y* *C* *y*
"0 for z*(0 and z*'1,
y *"0
"0 for all z* ,
y *"R
C*"1 for z*P!R and z*PR. The first of these boundary conditions means that the concentration on the membrane wall is zero because of the assumption that the electrodialysis cell is operated under limiting current conditions. The solution of the momentum equation leads to the value of the velocity gradient at the membrane wall, and then the differential equation (10) can be solved numerically to obtain the mass flux across the membrane. The overall mass flux is described by the modified Sherwood number, Sh*, that is calculated from the integration of the concentration gradient at the surface along the membrane using the following equation:
*C* dz* . (12) *y* y *"0 For steady flow, Le´veˆque’s solution gives for the modified Sherwood number
Sh*"Sh Pe "
Sh*"0.8075 .
(13)
For a given condition of fluid flow, amplitude, frequency and mean velocity, the time variation of the velocity gradient, S (t) can be obtained from the velocity profiles given by Eqs. (3) and (4). Then, the differential equation (10) is solved numerically using the control volume method by Patankar (1980). By solving the system of finite difference equations derived from the application of this method using the same calculation procedure of Mao and Hanratty (1991), the concentration field is obtained, and then the modified Sherwood number, Sh*, is calculated from Eq. (12). In order to test the calculation procedure, Eq. (10) has been solved assuming a sinusoidal variation of shear rate using the same conditions as Kaiping (1983) and Mao and Hanratty (1991). So, a sinusoidal variation of the velocity gradient can be considered in a dimensionless form: S* (t)"1#S
cos (u*t*) ,
(14)
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Fig. 3. Effect of the velocity gradient on the modified Sherwood number. Case of non-reversing flow. S*"1#0.9 cos(u*t*). (———) quasisteady solution; (— — —) u*"0.1; (— — — —) u*"0.5; (— — — ) ) u*"1; (— — — —) u*"2.
S being the amplitude of the dimensionless velocity
gradient. The mass transfer for different shear functions is studied in Figs. 3 and 4 in terms of the modified Sherwood number, comparing the results obtained with the quasisteady mass transfer for non-reversing and reversing flow respectively. In the case of non-reversing flow, Fig. 3, the mass transfer reaches a maximum and a minimum value in a pulsation period, and the increase in frequency causes a decrease in the amplitude of the mass transfer. The minima of the mass flux change strongly with frequency, while the maximum is the same for all frequencies. Moreover a phase lag that increases with frequency appears with respect to the quasi-steady solution. However, in the case of reversing flow, Fig. 4, when the amplitude of the shear rate is greater than one, the velocity gradient can reach negative values, but the mass flux is always positive, and the modified Sherwood number reaches two maxima and two minima in a pulsation period. The increase in the pulsation frequency has a greater effect on the mass transfer than in the case of non-reversing flow. The maximum as well as the minimum values of mass transfer are attenuated with the increase in frequency, moving away from the quasisteady solution. A phase lag that increases with frequency is also observed.
3. Experimental procedure Details regarding the experimental arrangement and the experimental techniques used may be found in Garcia-Anto´n et al. (1997) and Pe´rez-Herranz et al. (1997).
Fig. 4. Effect of the velocity gradient on the modified Sherwood number. Case of reversing flow. S*"1#2.0 cos (u*t*). (———) quasisteady solution; ( — — — ) u*"0.2; ( — — — — )u*"1.0; (— — —) u*"5.
The annular electrodialysis cell used in this work is applied to study the mass transfer across an anionic membrane of chromate ions coming from a bath whose concentration C was 9.62;10\ M CrO and 0.05 M H SO . The solution used was the same in the two sections, in order to minimize diffusion problems through the membrane owing to the concentration gradients on the two sides of the membrane. The cathode was a 7 cm diameter and 20 cm high AISI 316 stainless steel cylindrical grid. The anode was a 20 cm high Pb—Sn alloy bar. The membrane used in this work was the ARA 17-10 membrane produced by Morgane . This membrane is specially designed for the recovery of acids by electrodialysis. The major independent variables investigated were fluid velocity, v, pulsation frequency, u, and pulsation amplitude, a, which were varied in the limits: 1.33;10\)v)6.63;10\ m/s, 0)u)2n s\, 2;10\)a)8;10\ m.
4. Results The mass transfer across the anionic membrane is studied in terms of the limiting current density, i . In the case of pulsating flow conditions, average values of the limiting current density, iM , in a pulsation period have been considered. Then the experimental modified Sherwood number, Sh*, averaged in time is calculated from iM D Sh*" Pe , nFDC
(14)
V. Pe´ rez-Herranz et al./Chemical Engineering Science 54 (1999) 1667—1675
where C is the bulk concentration, and iM , the time average limiting current density. The limiting current density has been determined by the Cowan and Brown method (Cowan and Brown, 1959), which consists of the plotting of M /iM versus 1/iM ,
M being the time-average voltage drop across the membrane. The limiting current density corresponds to a minimum in this plot, and as it was found by Garcı´ a-Anto´n et al. in 1997, the time-average limiting current density increases with amplitude and frequency, but the effect of frequency on the time-average limiting current density is greater than the effect of amplitude, as can be predicted by the theoretical variation of the wall shear stress with the pulsation parameters discussed previously. Figs. 5—7 compare the experimental results obtained for the mass transfer across the membrane with the predicted mass transfer calculated from the solution of Eq. (10) using the velocity gradient at the membrane wall calculated from Eqs. (3) and (4) for the same values of pulsation frequency, u, pulsation amplitude, a, and fluid velocity, v, investigated in the experimental work. The results are compared in terms of the modified Sherwood number. Figs. 5 and 6 show the effect of frequency on the enhancement of mass transfer for different values of velocity and amplitude, respectively. The experimental results are compared with those obtained from the numerical solution of the mass balance. These figures show that in the range of experimental conditions, corresponding to reversing flow, a great increase in mass transfer, expressed in terms of Sh*/Sh* is observed when the frequency increases. For a given frequency, the enhancement of mass transfer increases with increasing amplitude and with decreasing velocity.
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Fig. 6. Effect of pulsating flow on the modified Sherwood number. v"3.98;10\ m/s, (䊏) a"2;10\ m, (䉬) a"8;10\ m.
Fig. 7. Comparison of calculated modified Sherwood number with respect to the experimental results: (———) calculated; (䊏) Experimental.
Fig. 5. Effect of pulsating flow on the modified Sherwood number. a"8;10\ m, (䊏) v"1.326;10\ m/s, (䉬) v"3.98;10\ m/s, (䉱) v"6.63;10\ m/s.
In Fig. 7 the experimental data of Sh*/Sh* are compared with the data calculated from the mass balance for different pulsation conditions. As it can be seen, pulsation contributes to increase mass transfer with respect to the steady flow. However, the results obtained from the solution of the mass balance are slightly smaller than the experimental results. This difference can be explained from the fact that the theoretical development neglects diffusion in the flow direction, which can be a significant effect at the two ends of the membrane. Another cause of error could be the possibility that the flow field was not sufficiently developed in the region of the test section, despite the calming section near the membrane. These
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two situations will be more relevant in the case of reversing flow. Finally, the exact nature of the mass transport mechanism can be another possible source of error. In the classical theory of concentration polarization, it is assumed that mass transport from the bulk solution to the membrane takes place by diffusion across the diffusion boundary layer whose thickness depends only on the hydrodynamical conditions near the membrane. However, an additional mass transport by natural convection can take place, apart from diffusion. This natural convection is due to the voltage-dependent concentrations and electric field gradients within the boundary layer. Then, apart from the hydrodynamical conditions, the thickness of the boundary layer decreases as voltage increases (Rubinstein and Shtilman, 1978). This effect will increase with the instantaneous velocity as the voltage drop corresponding to the limiting current density increases with velocity.
Marı´ a Asuncio´n Jaime for her help in translating this paper into English.
5. Conclusions
K
Time-averaged shear stress is evaluated numerically for the case of pulsating annular flow. The results indicate that shear stress is affected by two dimensionless groups, the dimensionless velocity, a , and the Womer sley number, a , frequency being the parameter with a greater influence on the shear stress. The mass transfer across an anionic membrane in an annular pulsating flow is compared with the numerical results obtained from the mass balance for different values of the pulsation parameters, frequency, u, amplitude, a and fluid velocity, v. In general the experimental enhancement of mass transfer due to pulsation is higher than the numerical results, probably due to the assumptions made in the application of the boundary layer theory. Also, end effects on the mass balance, that have not been considered, can contribute to the differences between experimental mass transfer and the numerical calculations. From the results obtained, it is concluded that laminar oscillatory flow can be used to increase mass transfer in electrodialysis. Oscillatory flow has the advantage of increasing mass transfer due to the increase of the local velocity gradient at the membrane surface maintaining high residence times. This situation is specially interesting in the removal of metals from dilute solutions, such as chromate ions coming from the rinsing baths of the electroplating industries.
¸ n p r r* R R R* R* S S* S S
Acknowledgements We are indebted to the Generalitat Valenciana for a Postgraduate grant for V. Pe´rez-Herranz and the D.G.I.C.Y.T. for the support of this work. We thank
Notation a C C C* D D D D F i iM J k k
t t* ¹ u u u W u X v y y* z z*
pulsation amplitude, m concentration, mol/m bulk concentration, mol/m dimensionless concentration diffusion coefficient, m/s diameter of the inner cylinder (membrene wall), m diameter of the outer cylinder (cathode), m equivalent diameter, m Faraday constant, 96500 C/eq limiting current density, A/m time-average limiting current density, A/m zeroth-order Bessel function of the first kind pressure gradient for steady flow, kg/m s maximum pressure gradient due to pulsation, kg/m s zeroth-order modified Bessel function of the first kind length of the membrane in the flow direction, m valence of chromate ions pressure, kg/m s radial distance, m dimensionless radial distance radius of the inner cylinder (membrane), m radius of the outer cylinder (cathode), m dimensionless radius of the inner cylinder dimensionless radius of the outer cylinder velocity gradient at the wall, s\ dimensionless velocity gradient at the wall velocity gradient at the wall in steady flow, s\ amplitude of the dimensionless velocity gradient in pulsating flow time, s dimensionless time pulsation period, s steady component of the fluid velocity, m/s oscillatory component of the fluid velocity, m/s velocity normal to the wall in the mass balance, m/s velocity in z-direction in pulsed flow, m/s bulk mean velocity for steady flow, m/s axial distance normal to the membrane wall in the mass balance, m dimensionless axial distance normal to the membrane wall in the mass balance axial distance along the annulus, m dimensionless axial distance
Dimensionless groups Pe Re
Pe´clet number"S ¸/D Reynolds number"vD /l
V. Pe´ rez-Herranz et al./Chemical Engineering Science 54 (1999) 1667—1675
Sh Sh* a a
Sherwood number"i D /n F D C modified Sherwood number"Sh Pe dimensionless velocity"a u/v Womersley number"D (u/l)
Greek letters
M k l o q qN u
time-average voltage drop, V fluid viscosity, kg/m s kinematic viscosity, m/s density, kg/m wall shear stress, N/m time-average wall shear stress, N/m pulsation frequency, s\
References Bellhouse, B.J., Bellhouse, F.H., Curl, C.M., MacMillan, T.I., Gunning, A.J., Spratt, E.H., MacMurray, S.B., & Nelems, J.M. (1973). A high efficiency oxigenator and pulsatile pumping system, its application to animal trials. Am. Soc. Artif. Internal Organs, 19, 72—79. Cheng, P., & Hsu, C.T. (1986). Fully-developed, forced convective flow through an annular packed-sphere bed with wall effects. Int. J. Heat Mass ¹ransfer, 29, 1843—1853. Cordonet, J.S., Riba, J.P., & Angelino, H. (1989). Mass transfer in a particle bed with oscillating flow. Chem. Engng. Sci., 44, 2107—2111. Cowan, D.A., & Brown, J.H. (1959). Effect of turbulence on limiting current in electrodialysis cells. Ind. Engng Chem., 51, 1445—1448. Garcı´ a-Anto´n, J., Pe´rez-Herranz, V., & Guin o´n, J.L. (1997). Mass transfer in an annular electrodialysis cell in pulsating flow. J. Appl. Electrochem., 27, 469—476. Gibert, H., & Angelino, H. (1973). Influence de la pulsation sur les transferts de matie`re entre une sphe`re et un liquide. Can. J. Chem. Engng, 51, 319—325. Gibert, H., & Angelino, H. (1974). Transferts de matie`re entre une sphe`re soumise a des vibrations et un liquide en mouvement. Int. J. Heat Mass ¹ransfer, 17, 625—632. Guin o´n, J.L., Pe´rez-Herranz, V., Garcı´ a-Anto´n, J., & Lacoste, G. (1995). Enhancement of mass transfer at a espherical electrode in pulsating flow. J. Appl. Electrochem., 25, 267—272. Harrison, S.T.L., & Mackley, M.R. (1992). A pulsatile flow bioreactor. Chem. Engng Sci., 47, 490—493. Kaiping, P. (1983). Unsteady forced convective heat transfer from a hot film in non-reversing and reversing shear flow. Int. J. Heat Mass ¹ransfer, 26, 545—557. Khodadadi, J.M. (1991). Oscillatory fluid flow through a porous medium chanel bounded by two impermeable parallel plates. J. Fluid Engng, 113, 509—511. Krasuk, J.H., & Smith, J.M. (1963). Mass transfer in a pulsed column. Chem. Engng Sci., 18, 591—598. Krasuk, J.H., & Smith, J.M. (1964). Mass transfer in a packed, pulsed column. A.I.Ch.E. J., 10, 759—763. Kusama, H. (1952). Study of pulsating flow (Pulsating flow in a circular pipe). Soc. Mech. Engrs ¹rans., 18, 27—31. Legentilhomme, P., & Legrand, J. (1993). Mode´lisation nume´rique du transfert de matie`re dans un e´coulement annulaire faiblement tourbillonaire non-entretenu. Can. J. Chem. Engng, 71, 299—311.
1675
Mackley, M.R., & Ni, X. (1991). Mixing and dispersion in a buffled tube for steady laminar and pulsatile flow. Chem. Engng Sci., 46, 3139—3151. Mao, Z.X., & Hanratty, T.J. (1985). The use of scalar transport probes to measure wall shear stress in a flow with imposed oscillations. Exp. Fluids, 2, 129—135. Mao, Z.X., & Hanratty, T.J. (1986). Studies of the wall shear stress in a turbulent pulsating pipe flow. J. Fluid Mech., 170, 545—564. Mao, Z.X., & Hanratty, T.J. (1991). Analysis of wall shear stress probes in large amplitude unsteady flows. Int. J. Heat Mass ¹ransfer, 34, 281—290. Nishimura, T., Arakawa, S., Murakami, S., & Kawamura, Y. (1989). Oscillatory viscous flow in symmetric wavy-walled channels. Chem. Engng Sci, 44, 2137—2148. Nishimura, T., Kajimoto, Y., Tarumoto, A., & Kawamura, Y. (1986). Flow structure and mass transfer for a wavy channel in transitional flow regime. J. Chem. Engng Jpn., 19, 449—455. Nishimura, T., Tarumoto, A., & Kawamura, Y. (1987). Flow and mass transfer characteristics in wavy channels for oscillatory flow. Int. J. Heat Mass ¹ransfer, 30, 1007—1015. Nishimura, T., Yoshino, T., & Kawamura, Y. (1987). Numerical flow analysis of pulsatile flow in a channel with symmetric wavy walls at moderate Reynolds numbers. J. Chem. Engng Jpn., 20, 479—485. Oud-Rouis, M., Nouar, C., Salem, A., Legrand, J., Legentilhomme, P., & Aouabed, H. (1989). Hydrodynamics and mass transfer of the laminar flow in the entrance region of an annular electrochemical reactor. Int. Chem. Engng Symp. Ser., 112, 17—28. Patankar, S.V. (1980). Numerical heat transfer and fluid flow. New York: McGraw-Hill. Pe´rez-Herranz, V., Garcı´ a-Anto´n, J., & Guin o´n, J.L. (1997). Velocity profiles and limiting current in an annular electrodialysis cell in pulsed flow. Chem. Engng Sci., 52, 843—851. Ramaprian, B.R., & Tu, S.W. (1983). Fully developed periodic turbulent pipe flow. Part 2. The detailed structure of the flow. J. Fluid Mech., 137, 59—81. Ratel, A., Duverneuil, P., & Lacoste, G. (1988). Influence de la pulsation liquide sur le comportement d’un re´acteur e´lectrochimique constitue´ par un lit de particules conductrices. I. Transfert de matie`re. J. Appl. Electrochem., 18, 394—400. Rubinstein, I., & Shtilman, L. (1978). Voltage against current curves of cation exchange membranes. J. Chem. Soc. Faraday ¹rans., 2, 231—246. Sexl, T. (1930). U®ber den von E.G. Richardson entdeckten annulareffekt. Z. Phys., 61, 349—362. Sobey, I.J. (1980). On flow patterns through furrowed channels. Part 1. Calculated flow patterns. J. Fluid Mech., 96, 1—26. Sobey, I.J. (1982). Oscillatory flows at intermediate Strouhal number in assymetric channels. J. Fluid Mech., 125, 359—373. Sobey, I.J. (1983). The occurrence of separation in oscillatory flow. J. Fluid Mech., 134, 247—257. Stephanoff, K.D., Sobey, I.J., & Bellhouse, B.J. (1980). On flow patterns through furrowed channels. Part 2. Observed flow patterns. J. Fluid Mech., 96, 27—32. Sudo, K., Sumida, M., & Yamane, R. (1992). Secondary motion of fully developed oscillatory flow in a curved pipe. J. Fluid Mech., 237, 189—208. Tu, S.W., & Ramaprian, B.R. (1983). Fully developed periodic turbulent pipe flow. Part 1. Main experimental results and comparison with predictions. J. Fluid Mech., 137, 31—58. Uchida, S. (1956). The pulsating viscous flow superimposed on the steady laminar motion of incompressible fluid in a circular pipe. Zamp., 7, 403—421.