**1. Introduction**

408 Computational Simulations and Applications

[8] Roe, P. L., 1984, "Generalized Formulation of TVD Lax-Wendroff Scheme, ICASE Report

[9] Rudman, M., 1997, "Volume-Tracking Methods for Interfacial Flow Calculations", International Journal for Numerical methods in Fluids, Vol. 24, 671-691.

[11] Sweby, P. K., 1984, "High Resolution Scheme Using Flux Limiter for Hyperbolic

[12] van Leer, B., 1979, "Towards the Ultimate Conservative Difference Scheme, V: A

[13] Vitola, M. A., Schettini, E. B. C., and Silvestrini, J. H., 2006, "Three Dimensional Wake

and Large-Eddy Simulation VI. : Springer Netherlands, 2006, p. 669-676. [14] Wanderley, J. B. V., and Levi, C., 2006, "Free Surface Viscous Flow around a Ship

[15] Wanderley, J. B. V., Souza, G. H. B., Sphaier, S. H., and Levi, C. A., 2008, "Vortex-

Second-Order Sequel to Godunov's Method, J. Comput. Phys., vol. 32, pp. 101-136.

Structure of Free Planar Shear Flow Around Horizontal Cylinder". In: Eric Lamballais, Rainer Friedrich, Bernard J. Geurts and Olivier Métais. (Org.). Direct

Model", Proceedings of the 25th International Conference on Offshore Mechanics

Induced Vibration of an Elastically Mounted Circular Cylinder using an Upwind TVD Two-Dimensional Numerical Scheme", Ocean Engineering, V. 35, pp. 1533-

[10] Schlichting, H., 1979, "Boundary Layer Theory". McGraw-Hill, New York.

Conservation Laws, SIAM J. Num. Anal., vol. 21, pp. 995-1011.

and Arctic Engineering, June 4-9, Hamburg, Germany.

84-53.

1544.

The systems in which an electrode immersed in a solution causes a chemical reaction have been studied for over a hundred years. It has long been known that the behavior of these systems is determined by two main factors: the rate with which the substance comes into contact with the electrode and the rate of the electrochemical reactions at the electrode.

During the first four decades of the twentieth century, many works were devoted to this subject; however, most of them were experimental works. Those works found that the limiting current increases with increasing the rate of stirring (Bircumshaw & Riddiford, 1952). Usually, the results were expressed by means of a power relation of the form:

$$j\_{\rm lim} \propto o^a \tag{1}$$

where *j*lim is the limiting current, is the rotation speed and *a* is the power such that 0 1 *a* .

The earliest theoretical studies of electrochemical cells with a rotating disk were reported by Nernst (Nernst, 1904; Nernst & Merriam, 1905). In that works, Nernst introduced the concept of diffusion layer. According to Nernst, the thickness of the diffusion layer is extremely small and the movement of fluid within it may be neglected.

In 1932, Eucken (Eucken, 1932) presented another theoretical study of electrochemical cells. The aim of that work was to provide an exact hydrodynamics theory of the diffusion towards a plane electrode submerged in a solution moving with a relatively high velocity.

Following a similar approach to that employed by Eucken, Levich (Levich, 1942) presented a theory describing accurately the hydrodynamics generated by an electrode with the shape of a flat disk of a sufficiently large area, rotating about an axis perpendicular to the plane of the disc with a constant angular velocity. The theory of Levich is restricted to the case of sufficiently small Reynolds numbers, so that the motion of the fluid might be considered as laminar. Levich used the transformation proposed by von Kármán (von Kármán, 1921), which allows writing the Navier-Stokes equations as a system of ordinary differential equations. Levich also used the solution proposed by Cochran (Cochran, 1934) to the resulting system of ordinary differential equations.

Although the theory developed by Levich was in a good agreement with most experimental results reported previously, at the end of his work Levich pointed out that: "… a precise experimental study of the phenomena of concentration polarization in a wide range of

Hydrodynamic Analysis of Electrochemical Cells 411

In the model of the ideal system considered by Levich (Levich, 1942), the analytical expressions describing the hydrodynamic behavior of the fluid in the vicinity of the electrode active surface is in accordance with the behavior described by von Kármán (von Kármán, 1921). Levich considered that the fluid velocity field inside the cell can be modeled as the steady-state motion of an incompressible Newtonian viscous fluid due to an infinite rotating plane disk whose thickness is equal to zero. A system like this was originally proposed by von Kármán (von Kármán, 1921). This model assumes that the fluid density and viscosity remain constant. Additionally, the fluid is infinite in extent and is located

Figure 1 shows the typical fluid flow pattern of the ideal system. To create this figure, a small value for the electrode rotation speed was employed. Several stream lines were included in this figure. The stream lines were colored in accordance with its radius value. Given that the rotating disc acts as a centrifugal fan, the fluid moves radially outwards near the disc. Therefore, to preserve continuity, an axial motion towards the lamina is generated (Cochran, 1934). Figure 1 shows that far from the disk the radial and angular velocity components are zero. On the contrary, near the disk, the behaviour of the fluid resembles logarithmic spirals. Figure 2 shows the fluid velocity field on a plane parallel to the disk.

This figure also shows that a stagnation point is formed at the center of the disk.

Fig. 1. Typical fluid flow pattern of the ideal system.

**2. Ideal system model 2.1 System description** 

above the disc.

Reynolds numbers and simple geometrical conditions, allowing to establish decisively the quantitative agreement between the theory and the experiment, is highly desirable" (Levich, 1942).

Several authors have analyzed experimentally some of the geometrical features of systems typically used in electrochemical studies in order to verify if these systems meet the basic theoretical requirements of the Levich theory.

The electrode shape, the electrode immersion depth and the cell volume were considered among the main sources of discrepancy between ideal and existent experimental systems. In accordance with Riddiford (Riddiford, 1966), a cylindrical shape for the electrode promotes an adverse fluid flow pattern and other electrode shapes should be used instead. Azim and Riddiford (Azim & Riddiford, 1962) recommended conical or bell-shaped electrode designs. The effect of the electrode shape upon the rate of mass transfer was widely analyzed by Blurton and Riddiford (Blurton & Riddiford, 1965) and by Prater and Adams (Prater & Adams, 1966). However, in accordance with Prater and Adams: "... the difficulty of fabricating the bell-shaped electrode probably outweighs its advantages...".

Prater and Adams (Prater & Adams, 1966) also studied the effect of electrode immersion depth upon limiting currents, but only a bell-shaped electrode was employed. In accordance with these authors: "... identical results may be obtained whether the electrode is immersed to a depth of 1.85 cm. or barely touching the solution...".

An accurate experimental determination of the velocity field near the electrode becomes difficult due to cell geometry and electrode dimensions. This task is hard to accomplish even using modern non-intrusive techniques such as Particle Imaging Velocimetry (PIV). Nevertheless, recently the flow field inside a confined electrochemical cell configuration was measured through Laser Doppler Anemometry (Mandin, Pauporte, Fanouillere, & Lincot, 2004).

Another alternative to obtain an accurate approximation of the velocity and pressure profiles due to a rotating disk electrode is to use the Computer Fluid Dynamics (CFD) technique in which, the three-dimensional Navier-Stokes equations are solved numerically.

The aim of this chapter is to review some of the works recently published in which the CFD technique has been used to characterize the hydrodynamic behavior of the liquid inside electrochemical cells with a rotating disk electrode (RDE). This chapter also discusses the assumptions stated in the revised works as well as the accurateness of their results. In addition, this paper discusses some of the implications of the results obtained with the CFD technique on the geometric characteristics of the electrochemical cell, the electrode geometry and the operating conditions.

This chapter is organized as follows. The properties as well as the main features of the ideal system considered in the theory developed by Levich (Levich, 1942) are discussed in Section 2. This section also presents a comparison between two approximate solutions to the problem stated by von Kármán (von Kármán, 1921). The main features of several works recently published in which the CFD technique has been used to characterize the hydrodynamics of electrochemical cells with a RDE are presented in Section 3. Based on the results obtained with CFD technique, Section 4 discusses some simple modifications that can be implemented in electrochemical cells that can significantly improve the accuracy of measurements made on these devices.

Reynolds numbers and simple geometrical conditions, allowing to establish decisively the quantitative agreement between the theory and the experiment, is highly desirable" (Levich,

Several authors have analyzed experimentally some of the geometrical features of systems typically used in electrochemical studies in order to verify if these systems meet the basic

The electrode shape, the electrode immersion depth and the cell volume were considered among the main sources of discrepancy between ideal and existent experimental systems. In accordance with Riddiford (Riddiford, 1966), a cylindrical shape for the electrode promotes an adverse fluid flow pattern and other electrode shapes should be used instead. Azim and Riddiford (Azim & Riddiford, 1962) recommended conical or bell-shaped electrode designs. The effect of the electrode shape upon the rate of mass transfer was widely analyzed by Blurton and Riddiford (Blurton & Riddiford, 1965) and by Prater and Adams (Prater & Adams, 1966). However, in accordance with Prater and Adams: "... the difficulty of

Prater and Adams (Prater & Adams, 1966) also studied the effect of electrode immersion depth upon limiting currents, but only a bell-shaped electrode was employed. In accordance with these authors: "... identical results may be obtained whether the electrode is immersed

An accurate experimental determination of the velocity field near the electrode becomes difficult due to cell geometry and electrode dimensions. This task is hard to accomplish even using modern non-intrusive techniques such as Particle Imaging Velocimetry (PIV). Nevertheless, recently the flow field inside a confined electrochemical cell configuration was measured through Laser Doppler Anemometry (Mandin, Pauporte, Fanouillere, & Lincot,

Another alternative to obtain an accurate approximation of the velocity and pressure profiles due to a rotating disk electrode is to use the Computer Fluid Dynamics (CFD) technique in which, the three-dimensional Navier-Stokes equations are solved

The aim of this chapter is to review some of the works recently published in which the CFD technique has been used to characterize the hydrodynamic behavior of the liquid inside electrochemical cells with a rotating disk electrode (RDE). This chapter also discusses the assumptions stated in the revised works as well as the accurateness of their results. In addition, this paper discusses some of the implications of the results obtained with the CFD technique on the geometric characteristics of the electrochemical cell, the electrode geometry

This chapter is organized as follows. The properties as well as the main features of the ideal system considered in the theory developed by Levich (Levich, 1942) are discussed in Section 2. This section also presents a comparison between two approximate solutions to the problem stated by von Kármán (von Kármán, 1921). The main features of several works recently published in which the CFD technique has been used to characterize the hydrodynamics of electrochemical cells with a RDE are presented in Section 3. Based on the results obtained with CFD technique, Section 4 discusses some simple modifications that can be implemented in electrochemical cells that can significantly improve the accuracy of

fabricating the bell-shaped electrode probably outweighs its advantages...".

to a depth of 1.85 cm. or barely touching the solution...".

1942).

2004).

numerically.

and the operating conditions.

measurements made on these devices.

theoretical requirements of the Levich theory.

Fig. 1. Typical fluid flow pattern of the ideal system.
