Diffusion coefficients of ferricyanide ions in polymeric solutions comparison of different experimental methods

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1 Electrochimica Acta 45 (2000) Diffusion coefficients of ferricyanide ions in polymeric solutions comparison of different experimental methods J. Legrand a, *, E. Dumont a,b,1, J. Comiti a,2, F. Fayolle b,3 a Laboratoire de Génie des Procédés, UPRES EA 1152, Uni ersité de Nantes, IUT de Saint-Nazaire, CRTT, B.P. 406, Saint-Nazaire Cedex, France b Laboratoire de Génie des Procédés Alimentaires, ENITIAA, B.P , Nantes Cedex 03, France Received 22 June 1999; received in revised form 9 September 1999 Abstract The molecular diffusion of ferricyanide ions in polymeric Newtonian and non-newtonian solution has been studied at 25 C with different experimental methods: rotating-disc flow, Couette-flow and unsteady diffusion. It is shown that the Levich equation established for the determination of diffusion coefficients with the rotating-disc electrode method cannot be applied for Reynolds number values less than 30 for Newtonian and power-law fluids, when the momentum boundary layer thickness is of the same order of magnitude than the disc radius. The unsteady diffusion method seems to be the most suitable technique to determine the diffusion coefficient in highly or viscous non-newtonian electrolytes. For the studied polymer solution, it is shown that the decrease of diffusion coefficient is much slower that the increase in dynamic viscosity. Then, the classical StokesEinstein equation, D /T=cst, is not valid for electrolyte solution with high viscosity Elsevier Science Ltd. All rights reserved. Keywords: Couette-flow; Diffusion coefficient; Polymeric electrolyte; Rotating-disc; Unsteady diffusion 1. Introduction * Corresponding author. Tel.: ; fax: addresses: jack.legrand@lgp.univ-nantes.fr (J. Legrand), eric.dumont@lgp.univ-nantes.fr (E. Dumont), jacques.comiti@lgp.univ-nantes.fr (J. Comiti), fayolle@enitiaa-nantes.fr (F. Fayolle) Fax: Fax: Fax: The determination of molecular diffusion coefficient in electrolyte solutions is generally obtained by the measurement of steady-state diffusion mass flux in known hydrodynamic conditions. The most popular device is the rotating-disc electrode. The diffusion convection equation was solved by Levich [1] in order to give a relationship between the diffusional limiting current and the angular velocity of the disc. The Couetteflow between two concentric cylinders is sometimes used [2] to simultaneously measure the diffusion coefficient and the viscosity. Transient methods have been developed also. Robertson et al. [3] compared the DC method and the electrohydrodynamic (EHD) method by using the rotating-disc electrode. The interest of the EHD method is that the values of the electrode surface area and of the viscosity are not needed to determine to the Schmidt number. The objective of this work is the determination of molecular diffusion coefficient of ferricyanide ions in highly viscous and non-newtonian electrolyte solutions. These solutions have been selected as model fluids for the determination of transport phenomena in food industry with electrochemical method. The flow characteristics of scraped heat exchanger is of special interest for us [4], in particular to relate the heat transfer /00/$ - see front matter 2000 Elsevier Science Ltd. All rights reserved. PII: S (99)

2 1792 J. Legrand et al. / Electrochimica Acta 45 (2000) performance to flow regimes. Arvia et al. [5,6] used the rotating-disc electrode for the study of diffusion coefficients of ferro- and ferricyanide ions in potassium hydroxide solutions [5] and in potassium chloride solutions with carboxymethylcellulose (CMC) [6]. They have verified that the expression D /T was constant and temperature-independent. Bourne et al. [7] tabulated viscosities and diffusivities of ferri- and ferrocyanide solutions in aqueous sodium hydroxide as function of temperature and electrolyte composition. Agbangla et al. [2] measured the diffusion of ferricyanide ions in viscoelastic electrolyte solutions composed of different concentrations of Polyox WSR 301 in a Couette-flow cell. They obtained a constant value for the expression DK 1.54 /T, where K is the consistency coefficient of the solution. Deviations from diffusion classic law have been analysed by Mashelkar and Dutta [8], who have shown that slip effects dominate transport phenomena in non-homogenous flows of polymer solutions, referred as structured fluids by [8]. Otherwise, Deslouis and Tribollet [9] have shown that for concentrated solutions of Polyox WSR 301, the rotating-discflow is characterised by a centripetal flow for low rotational speeds. The flow becomes centrifugal for rotational speeds greater than the one corresponding to the flow-inversion point. The Weissenberg effect, i.e. centripetal flow around the rotating-disc is caused by the presence of normal stress difference. This result has to be taken into account for the determination of molecular diffusion coefficient in complex solutions. Wein and Assaf [10] have proposed a semi-empirical modification of the Levich equation for low Reynolds numbers in the rotating-disc electrode for viscoelastic liquids with non-zero normal stress differences. A Russian book [11] gives an overview of the use of electrochemical methods for the study of polymer solutions flows. In the present work, we have measured diffusion coefficients of ferricyanide ions in highly viscous solutions of polyethylene glycol, in power-law CMC solutions and in viscoelastic solutions of guar. The measurements have been performed with a rotating disc electrode, with a Couette-flow cell and by using an unsteady diffusion method based on the Fick s second law. The results are discussed and compared in order to show the limitations of these methods. 2. Characteristics of the electrolytes Three polymer solutions have been studied: one Newtonian fluid and two shear-thinning fluids. The viscosity measurements were performed at 25 C with a Couette rheometer TA instrument AR 1000 with imposed torque. Emkarox HV 45, from ICI, is a mixture of polypropylene glycol and polyethylene glycol. The dynamic viscosity of the pure products is equal to 3.7 Pa s at 25 C. The electrolyte solution is composed of a mixture of potassium ferri- and ferrocyanide and potassium sulfate as supporting electrolyte. The viscosities of the different aqueous solutions are given in Table 1. We have also added a low viscosity solution done with polyethylene glycol (PEG). The Newtonian behaviour of these solutions has been checked. Aqueous solutions of carboxymethylcellulose (CMC; high viscosity from Sigma) have a pseudoplastic behaviour. The non-newtonian characteristics depend on the CMC concentration. CMC powder is added progressively in agitated cold water. After dissolution of CMC sodium salt, potassium sulphate and potassium ferro- and ferricyanide are added in the solution. For the highest CMC concentrations (1.0 and 1.1%), Weissenberg effects appear on the agitator axis. The rheological behaviour is modelled by a power-law according to the shear rate domain, 580 s 1, corresponding to the operating conditions of Couette-flow method. The physical properties of CMC solution are given in Table 2. The preparation of guar gum solutions is similar to the one of CMC solutions, except that the guar gum dissolution is operated at 40 C. A power-law equation is proposed to model the non-newtonian behaviour of Table 1 Physical properties of Emkarox HV45 solutions at 25 C Solution Water (w/w%) K 3 [Fe(CN) 6 ] K 4 [Fe(CN) 6 ] K 2 SO 4 (kg m 3 ) (Pa s) (mol m 3 ) (mol m 3 ) (mol m 3 ) HV HV HV HV HV HV HV PEG

3 J. Legrand et al. / Electrochimica Acta 45 (2000) Table 2 Physical properties of CMC and guar gum solutions at 25 C Solution (kg m 3 ) K 3 [Fe(CN) 6 ] (mol m 3 ) K 4 [Fe(CN) 6 ] (mol m 3 ) K 2 SO 4 (mol m 3 ) n () K (Pa s n ) Shear rate domain (s 1 )580 CMC 0.5% CMC 0.7% CMC 0.9% CMC 1.0% CMC 1.1% Shear rate domain (s 1 )220 Guar 0.7% Guar 0.8% Guar 1.0% Guar 1.2% guar gum solutions for going from 2 to 20 s 1. The rheological data are summarised in Table Experimental methods 3.1. The rotating-disc electrode method Two rotating disc devices were used: the first (EDI from Tacussel) consisted of a 2-mm diameter disc electrode embedded in an 11-mm insulated tip, which is driven by a motor allowing a rpm range of rotation speed; the second (from AE, Asservissement Electronique, Bagnolet, France) consisted of a 5-mm diameter electrode in a 8.3 mm insulated tip with a rpm motor. These two devices allow analysing the influence of the geometric factors on the mass transfer on the rotating disc. For Newtonian fluids, the mass transfer at the rotating disc is expressed [1] by: Sh D =0.62 Re D 1/2 Sc 1/3 (1) Eq. (1) was established by neglecting the edge effect of the disc, which is true if: h 3.6 / R D (2) 1 I d = N(n) (n+1) 1 n 1 efac b K R 3(1+n) D D 2/3 1+n (3) where N(n) is a function of the fluid index, n, [12]: 1+n 2n1/3 N(n)= 12a (4) 7+5n where a is a numerical coefficient depending on the flow index and tabulated by Mitschka and Ulbrecht [14]. The limiting current, I d, is proportional to 1/1+n, which allows to determine the flow index, n, from the experimental I curves and to compare with the values of n obtained by viscosimetry. The diffusion coefficient is obtained from the slope of the experimental curves: I d = NN 1/1+n Couette-flow method The Couette-flow cell consists of two concentric cylinders (Fig. 1), the radius of inner and outer cylinders are respectively equal to 20 and 32.5 mm. The length of the cell is equal to 380 mm. The mass flux is measured on seven microelectrodes of 0.4 mm in diameter embedded in the outer cylinder (Fig. 1). The wall velocity gradient, S w, on the outer cylinder is obtained where h is the momentum boundary layer thickness and R D the disc radius. For the diffusional limiting current condition at the rotating electrode, the diffusion coefficient, D, is obtained from the experimental curves, I d = N, with the help of Eq. (1). For non-newtonian power-law ( =K n ) fluids, the mass flux at a rotating disc was obtained theoretically by Greif and Patterson [12] who have improved the initial determination of Hansford and Litt [13] by taking into account the convective transport in the radial direction. It can be written in terms of diffusional limiting current as: Fig. 1. The Couette-flow cell.

4 1794 J. Legrand et al. / Electrochimica Acta 45 (2000) from the Couette-flow velocity profile for a power-law fluid by: R 1 2/n S w = 2 (5) 2/n 2/n n (R 2 R 1 ) The relationship between the diffusion limiting current and the wall velocity gradient for circular microelectrode embedded in an inert wall is given [15] by: I d =0.862 e FAC b D 2 S w 1/3 (6) d where d is the diameter of the microelectrode. The experimental curves, I d = co 1/3, allow the determination of the diffusion coefficient. Eq. (5) is only valid for laminar flow without Taylor vortices, which appear for a critical Taylor number, equal to 41.3 [16]. This critical value of Ta g can be used also for non-newtonian fluids with respect to the definition of generalised Taylor number, Ta g, given in the nomenclature Unsteady diffusion method The principle of the method is based on the measurement after a voltage step of a transient diffusion limiting current on an electrode placed in electrolyte at rest. Under these conditions, Fick s second law can express the concentration profile of electroactive ions. The diffusional limiting current is obtained by solving unsteady diffusion equation in the following form: I d = e FAC b D (7) t The voltage-step transient method was used by Sobolik et al. [17] for the calibration of electrodiffusional wall shear-stress probes. The experimental procedure starts from the zero-current state. At t=0 a constant voltage, corresponding to the diffusional plateau, is applied and the time evolution of the diffusional limiting current is recorded. The experimental device is constituted by a disc platinum electrode of 2 mm in diameter and stainless steel counter-electrode. The data acquisition system (Fastlab card) allows sampling frequency of 1 khz. Eq. (7) can be written as I d = UD t 1/2, then the diffusion coefficient is obtained from the experimental value of UD. 4. Results and discussion 4.1. Rotating-disc electrode The determination of the diffusion coefficient is made with the 2-mm diameter rotating-disc electrode. For high viscosity Newtonian solutions, Eq. (1) is verified except for the highest viscous solutions (Fig. 2). When the water concentration is less than 30%, the exponent of is greater than 0.6, instead of 0.5 corresponding to the Levich equation. This value of 0.6 appears also for the polymer concentration equal to 65 and 60%, but only for the lowest values of the rotation speed of the disc (Fig. 2). For these two concentrations, the exponent becomes equal to 0.5 for the highest values of. This result is explained by the edge effect of the disc. The increase in viscosity leads to an increase of the momentum boundary layer thickness (Eq. (2)), which becomes of the same order of magnitude than the disc radius. Eq. (1) is not valid in this case and the diffusion coefficient cannot be calculated. The limiting value of h /R D will be analysed in the Section 4.4 in order to use non-dimensional numbers (Eq. (1)), in which D is needed, for a general significance of the obtained result. For HV45 solutions verifying the Levich equation, the diffusion coefficients of ferricyanide ions are given in Table 3. The reproducibility of results was within 5%. For CMC solutions, the same phenomenon is observed (Fig. 3). The straight lines, corresponding to the slope calculated with the flow index determined by viscometer, show that only low concentration CMC solutions verify Eq. (3). The increase in apparent viscosity of the power-law fluids leads to a thick boundary layer with respect to the disc radius. The diffusion coefficients calculated for 0.5 and 0.7% CMC solutions are given in Table 3. The diffusional limiting current obtained at the rotating disc electrode with guar gum solutions is given on Fig. 4. The curves are characterised by a flow-inversion point, which appears at increasing values of the rotation speed when the guar gum concentration increases. The same result was obtained by Deslouis and Tribollet [9] with polyoxyethylene solutions. These authors have visualised with argon bubbles the streamlines near the rotating disc. For rotating velocity values below the flow-inversion point, the gas bubbles follow a helicoïdal trajectory with the same axis than the disc one, then, near the liquid disc interface, they are submitted to a centripetal motion in the direction of the disc centre and finally they move away from the disc along the disc axis. Deslouis et al. [18] have shown also that the rotational speed value corresponding to the flow-inversion point is inversely proportional to the polyoxyethylene concentration. The flow modification, with respect to the flow structure described by Levich [1], is explained by the sharp variation of the apparent viscosity with the velocity gradient near the disc. Thus, the diffusion coefficient cannot be determined with rotating disc with finite value of the external radius for guar gum solutions.

5 J. Legrand et al. / Electrochimica Acta 45 (2000) Fig. 2. Experimental curves I d vs. for Newtonian fluids with a 2-mm diameter rotating-disc electrode (EWPA, electrolyte without polymer added) Couette-flow cell The experimental curves giving I d as a function of 1/3 for HV45 solutions are shown in Fig. 5. The presence of Taylor vortices is shown by a sharp increase of the slope of I 1/3 curves. The same conclusion can be drawn for CMC solutions (Fig. 6). It is necessary to emphasize that for non-newtonian fluids the determination of diffusion coefficient in the Couette-flow cell depends on the value of the flow behaviour index. The wall velocity gradient (Eq. (5)), which depends on the flow index n, is estimated to determine the convenient rheological law =K n for the flow conditions in the measurement cell. This procedure can lead to an increase of the uncertainties of the determination of D if non-newtonian fluids are described by several powerlaws according to the range of velocity gradient. Some differences between theoretical equation and experimental data obtained for high concentrated guar gum solutions are shown on Fig. 7. When the proportionality between I and 1/3 is not respected (Fig. 7), the diffusion coefficient is not calculated. These differences could be due to non-explained viscoelastic effects that are also responsible of time-perturbations of the diffusional limiting current for high concentrations of guar gum ( 1%). The diffusion coefficients of ferricyanide ions, which can be calculated, are given in Table 3. The accuracy is 5%, calculated from the standard deviation of the experimental data obtained with seven microelectrodes.

6 1796 J. Legrand et al. / Electrochimica Acta 45 (2000) Unsteady diffusion Examples of time evolution curves of diffusional limiting current, after a potential step, for the three types of electrolyte solutions are given in Figs It can be shown that Eq. (7) is verified for the different solutions and then, this equation can be used for the determination of the diffusion coefficient (Table 3). This method can be applied whatever the rheological behaviour of polymeric solutions used. The key point is to have a well-defined diffusional plateau for a wide range of overpotential in order to be independent of potential step values. Large electrodes, with diameter of at least 1 mm, are preferable than microelectrodes to Table 3 Comparison of the ferricyanide ion diffusion coefficient determined with the different methods at 25 C Solutions Rotating-disc electrode Couette-flow method, Unsteady diffusion method, D /T method, D (m 2 s 1 ) D (m 2 s 1 ) D (m 2 s 1 ) (m 2 Pa K 1 ) Electrolyte without 7.43 polymer added 7.50 HV HV HV HV HV HV HV PEG CMC 0.5% CMC 0.7% CMC 0.9% CMC 1.0% CMC 1.1% Guar 0.7% Guar 0.8% 6.74 Guar 1.0% 6.98 Guar 1.2% 6.98 Fig. 3. I d vs. for CMC solutions with a 2-mm diameter rotating-disc electrode.

7 J. Legrand et al. / Electrochimica Acta 45 (2000) Fig. 4. Experimental curves I d vs. for guar gum solution with a 2-mm diameter rotating-disc electrode. Fig. 5. Evolution of the diffusional limiting current with rotating speed for HV45 solutions in Couette-flow at 25 C. obtain a well-defined I D t 1/2 curve. In these conditions, measurements of D are highly reproducible with an accuracy of 5% Discussion Table 3 summarises the values of the diffusion coefficient obtained with the three experimental methods. The first comment is that only the unsteady-diffusion method can allow the determination of D for all the electrolyte solutions. It is not possible to use the Couette-flow method for low viscosity solution due to the existence of Taylor vortices in the annular gap. The rotating-disc method cannot be used for viscoelastic fluids and for high viscous fluids because of the edge effects. For the HV45 solutions, there is a good agreement between the three methods; except for the HV45 50% polymer solution for Couette-flow method. This result is explained by the fact that, for dilute polymer concentration (20 60% w/w polymer content), it is necessary to have a strong agitation to obtain an homogeneous solution. It is not achieved in the Couette-flow

8 1798 J. Legrand et al. / Electrochimica Acta 45 (2000) cell and there is a segregation between the heavy phase, composed of salted water, and light phase, composed of concentrated polymer solution. The results obtained with Newtonian solutions are shown on Fig. 11 with the data of Rabehanta [19]. The agreement is satisfactory. Fig. 11 shows also the wide range of viscosity investigated. For comprised between 0.01 and 1.2 Pa s, the diffusion coefficient in nearly proportional to 0.5 (Fig. 11). It seems that for high viscosity solutions the classical law, D /T=cst, cannot be applied, and the values of D /T are much higher (Table 3) than the one obtained for low viscosity electrolytes [5]: D T m 2 Pa K 1 (8) However, the StokesEinstein law, D /T=cst, allows to take into account the temperature changes [5 7]. The differences between the three methods are obvious for high concentrated solutions of guar gum. It is shown (Fig. 7) that for concentrations of guar gum greater than 0.8% there are some discrepancies between the experimental curves and the theoretical current given by Eq. (6), with S w defined by Eq. (5). For these reasons, we have considered that the best accuracy of the values of diffusion coefficient is given by the unsteady-diffusion method. They will be considered as reference values. This method allows to obtain a very good fitting between experimental data and theoretical equation. An another interest of the unsteady diffusion method is its flexibility, because the determination of D is independent of, then of hydrodynamical and rheological experimental conditions. Béreiziat [20] has also shown with a Couette-flow method that D is non-dependent on for CMC solutions. It is shown (Table 3) that the diffusion coefficient is quasi-similar for CMC solution and for electrolyte without polymer added. Arvia et al. [6] have obtained the same result with rotating-disc electrode and have explained this experimental observation by the fact that the ferricyanide ions could diffuse for low CMC concentration, in the aqueous solution through the macromolecules. From molecular viewpoint, CMC molecules may be considered as a quasi-static frame. Then, the diffusion media is the electrolyte without polymer, corresponding to a viscosity similar to the water one. Deslouis and Tribollet [21] have obtained a similar result with an Ostwald fluid. They have shown that the molecular diffusion coefficient of ferricyanide ions is the same in polyethylene oxide solutions and in Newtonian fluids. In semi-dilute solution the polymer is organised as a pseudo network whose knots are the entanglement points [21]. The absence of variation of diffusion coefficient could be explained by small size of the ion involved with respect to the correlation length of the entangled polymer chains. The diffusion coefficient is also independent on guar gum concentration (Table 3), but the value is slightly lower than the one obtained for the reference electrolyte Critical Reynolds number for the rotating-disc electrode It has been seen that the Levich equation cannot be applied for highly viscous solution. By using the diffu- Fig. 6. Variation of I d with for CMC solutions in Couette-flow at 25 C.

9 J. Legrand et al. / Electrochimica Acta 45 (2000) Fig. 7. Evolution of the diffusional limiting current with rotating speed for guar gum solutions in Couette-flow at 25 C. sion coefficient determined by the unsteady-diffusion method, we have represented (Fig. 12) all the experimental data obtained with the rotating disc electrode in terms of Sh D /(Re D 1/2 Sc 1/3 ) in function of Reynolds number, Re D, in order to visualise a critical Reynolds number, corresponding to the minimum value for which Eq. (1) is valid for the determination of diffusion coefficient. It can be seen on Fig. 12 that the differences between experimental data and Eq. (1) can be considered as significant for Re D =30. In Fig. 12 we have reported also experimental data obtained with the same electrode embedded in a cylindrical tip of 25 mm in diameter and with a rotating electrode of 5 mm diameter embedded in 8.3 mm diameter insulator tip in order to investigate the influence of the geometric parameter on the critical Reynolds number. It can be concluded that the value of the critical Reynolds number, Re D = 30, is non-dependent on geometric factors. The non-applicability of Levich equation for Re D 30 can be explained by the assumption of neglecting the edge effect taken by Levich [1] to establish Eq. (1). This assumption can be expressed by Eq. (2). For Re D =30, the ratio h /R D is equal to It means that the edge effects have to be taken into account as soon as the momentum boundary layer thickness is equal to 1/3 of the disc diameter. For non-newtonian CMC solutions, a critical Reynolds number could be also calculated by using the following generalised non-dimensional numbers: S cg = K (9) D 1 n and Re g = 2 n 2 R D K (10) The ratio Sh D /(Re g 1/2 Sc g 1/3 ) is reported in Fig. 13 in function of generalised Reynolds number. A critical Reynolds number of 30 is observed also for the validity of Levich equation (Fig. 13). Then, Re D =30 is valid for both Newtonian and power-law fluids.

10 1800 J. Legrand et al. / Electrochimica Acta 45 (2000) Fig. 8. Timeevolution of I d for HV45 solutions (potential step: 200 mv; 25 C). Fig. 9. Timeevolution of I d for 1.1% CMC solution (potential step: 300 mv; 25 C).

11 J. Legrand et al. / Electrochimica Acta 45 (2000) Fig. 10. Timeevolution of I d for 1.0% guar gum solution (potential step: 300 mv; 25 C). Fig. 11. Comparison of diffusion coefficients in HV45 solutions obtained with different methods. 5. Conclusion Three different experimental methods have been used for the determination of diffusion coefficient of ferricyanide ions in highly viscous Newtonian and non- Newtonian electrolyte solutions. Limitations of the rotating-disc electrode method have been emphasized for highly viscous fluids, due to the edge effects, and for viscoelastic fluids. In some cases, these limitations could be overcome by increasing the diameter of the rotating disc. Flow analysis will be needed in order to explain the decrease of mass transfer rate, for a given Reynolds number, when the momentum boundary layer thickness becomes of the same order of magnitude of the disc radius. A numerical simulation of rotating-disc flow will be further investigated. A critical Reynolds number, Re D =30, has been obtained. It corresponds to the lowest value of Re D for the application of Levich equation. Some limitations appear also with the Couette-flow cell for high concentrated polymer electrolytes, due to the fact that it is necessary to know the actual rheological law in the hydrodynamic conditions

12 1802 J. Legrand et al. / Electrochimica Acta 45 (2000) Fig. 12. Variation of Sh D /(Re D 1/2 Sc 1/3 ) with Re D for different rotating disc electrodes and for Newtonian fluids (EWPA, electrolyte without polymer added). Fig. 13. Variation of Sh D /(Re g 1/2 Sc g 1/3 ) with Re g for CMC solutions with a 2-mm diameter rotating electrode.

13 J. Legrand et al. / Electrochimica Acta 45 (2000) of the determination of D. The unsteady-diffusion method, obtained with a time evolution of the diffusional limiting current after a voltage step, is a very convenient and reliable method for the diffusion coefficient determination for all types of electrolyte. This method is not dependent on hydrodynamic conditions and on rheological behaviour of the fluids. For highly viscous Newtonian fluids, the decrease of diffusion coefficient is proportional to the root-square of the dynamic viscosity, and then the well-known equation, D /T=cst, cannot be applied. For CMC and gum guar aqueous solutions, the diffusion coefficient is of the same order of magnitude than the one obtained for electrolytes without polymer added. It can be also emphasized that the studied electrolytes are characterized by very high and non-usual values of Schmidt number, comprised between for HV45 45% water and for HV45 20% water solution. Appendix A. Nomenclature A numerical coefficient tabulated by Mitschka and Ulbrecht [14] A electrode area C concentration C b bulk concentration D diffusion coefficient d electrode diameter F Faraday number I d diffusional limiting current K consistency coefficient k d mass transfer coefficient n flow behaviour index R 1 inner cylinder radius of the Couette-flow cell R 2 outer cylinder radius of the Couette-flow cell R D rotating-disc radius Re D =R D2 / Reynolds number for rotating disc for Newtonian fluid Re g = 2 n R D2 /K generalized Reynolds number for rotating disc Sc= /D Schmidt number Sc g =K/ D 1 n generalised Schmidt number Sh D =R D k d /D Sherwood number for rotating disc S w wall shear rate T temperature t time 3/2 n Ta g = R 1 generalized Taylor number for (R 2 R 1 ) 1/2+n Couette-flow 2 n /K CO, N, NN, UD slopes of experimental curves d h e References shear rate diffusional boundary layer thickness momentum boundary layer thickness dynamic viscosity kinematic viscosity number of electrons transferred shear stress rotation speed [1] V.G. Levich, Physicochemical Hydrodynamics, Prentice Hall, Englewoods Cliffs, NJ, [2] C. Agbangla, P. Dumargue, P. Humeau, M.F. Morin, Electrochim. Acta 26 (1981) [3] B. Robertson, B. Tribollet, C. Deslouis, J. Electrochem. Soc. 135 (1988) [4] E. Dumont, F. Fayolle, D. Della Valle, J. Legrand, J. Food Process Eng. (1998) accepted. [5] A.J. Arvia, S.L. Marchiano, J.H. Podesta, Electrochim. Acta 12 (1967) 259. [6] A.J. Arvia, J.C. Bazan, J.S.W. Carrozza, Electrochim. Acta 13 (1968) 81. [7] J.R. Bourne, P. Dell Ava, O. Dossenbach, T. Post, J. Chem. Eng. Data 30 (1985) 160. [8] R.A. Mashelkar, A. Dutta, Chem. Eng. Sci. 37 (1982) 969. [9] C. Deslouis, B. Tribollet, J. Chim. Phys. 72 (1975) 224. [10] O. Wein, F.H. Assaf, Collect. Czech. Chem. Commun. 52 (1987) 626. [11] A.N. Pokryvaylo, O. Wein, N.D. Kovalevskaya, Electrodiffusion Diagnostics of Flows in Suspensions and Polymer Solutions (in Russian), Nauka I Tekhnika, Minsk, [12] R. Greif, J.A. Patterson, Phys. Fluids 16 (1813) [13] G.S. Hansford, M. Litt, Chem. Eng. Sci. 23 (1968) 849. [14] P. Mitschka, J. Ulbrecht, Collect. Czechoslov. Chem. Commun. 20 (1965) [15] T.J. Hanratty, J.A. Campbell, in: R.J. Goldstein (Ed.), Fluid Mechanics Measurements, Hemisphere, Washington, DC, 1983, p [16] G.I. Taylor, Proc. R. Soc. 223A (1923) 289. [17] V. Sobolik, J. Tihon, O. Wein, K. Wichterle, J. Appl. Electrochem. 28 (1998) 329. [18] C. Deslouis, I. Epelbouin, B. Tribollet, L. Viet, Proceedings of Euromech 90, 1977, INP Lorraine, Nancy, pp. III.C [19] M. Rabehanta, Thesis, INPL-Nancy (1986). [20] D. Béreiziat, Thesis, INPL-Nancy (1993). [21] C. Deslouis, B. Tribollet, Electrochim. Acta 23 (1978) 935..