**3.1 Two-dimensional models**

Almost all the numerical simulations of the hydrodynamic behavior inside electrochemical cells with a RDE have used axisymmetric two-dimensional models, but there are significant differences in the extent of the electrochemical cell volume of the systems reported in literature. The entire cell volume was simulated by Mandin et al. (Mandin, et al., 2004) whereas only a small amount of liquid below a rotating disc ring electrode was considered by Dong et al. (Dong, Santhanagopalan, & White, 2007). In those works, only the electrode active face is in contact with the fluid. Nevertheless, a common practice of submerge the working electrode into the cell liquid was not considered in those works. Mandin et al. (Mandin, Fabian, & Lincot, 2006) show the significance of the submerged electrode side wall by means of two-dimensional numerical simulations of an electrochemical cell with a rotating cylinder electrode.

Dong et al. (Dong, Santhanagopalan, & White, 2008) carried out two-dimensional axisymmetric numerical simulations of a entire electrochemical cell with a RDE, where the electrode is partially submerged into the electrolyte. To include the existence of an air-liquid interface present in actual electrochemical cells, a slip wall was employed as boundary condition for the cell upper wall to represent numerically this interface. The system modeled by Dong et al. (Dong, et al., 2008) is schematically represented in Figure 6. In accordance with these authors, the simulated fluid velocity field and that obtained with the theoretical model proposed by von Kármán (von Kármán, 1921) and Cochran (Cochran, 1934) are in good agreement.

Mandin et al. (Mandin, et al., 2004) also carried out two-dimensional axisymmetric numerical simulations of a entire electrochemical cell with a RDE. The system studied by these authors is schematically represented in Figure 7. Their results were compared with the fluid flow pattern calculated with the analytical expressions obtained by Cochran (Cochran, 1934). These authors found that the velocity profiles calculated with these expressions are in accordance with their numerical simulations only in a narrow region close to the electrode active face. However, apparently these authors only made use of one of the two sets of equations that comprises the solution obtained by Cochran.

Figures 3, 4 and 5 show that the results of the approximate solutions developed by Cochran and by Ariel are very similar. Volgin and Davydov (Volgin & Davydov, 2007) shown that the error associated to the available approximate equations to calculate the velocity field range from 1 to 0.01%. In addition, they found that to provide an accuracy of about 0.1%, the length of computational region should be approximately twice the diffusion layer thickness. To reach higher accuracy, the extension of the computational region must be increased.

Computational Fluid Dynamics (CFD) is a technique that allows generating fluid flow simulations with by means of computers. CFD solves numerically the governing laws of fluid dynamics. The set of partial differential equations associated with the system under study are solved in a geometrical domain divided into small volumes, commonly known as a mesh (or grid). The accuracy and validity of simulation results depends on the choice of the CFD model, the physical features incorporated in the governing equations and the

Almost all the numerical simulations of the hydrodynamic behavior inside electrochemical cells with a RDE have used axisymmetric two-dimensional models, but there are significant differences in the extent of the electrochemical cell volume of the systems reported in literature. The entire cell volume was simulated by Mandin et al. (Mandin, et al., 2004) whereas only a small amount of liquid below a rotating disc ring electrode was considered by Dong et al. (Dong, Santhanagopalan, & White, 2007). In those works, only the electrode active face is in contact with the fluid. Nevertheless, a common practice of submerge the working electrode into the cell liquid was not considered in those works. Mandin et al. (Mandin, Fabian, & Lincot, 2006) show the significance of the submerged electrode side wall by means of two-dimensional numerical simulations of an electrochemical cell with a

Dong et al. (Dong, Santhanagopalan, & White, 2008) carried out two-dimensional axisymmetric numerical simulations of a entire electrochemical cell with a RDE, where the electrode is partially submerged into the electrolyte. To include the existence of an air-liquid interface present in actual electrochemical cells, a slip wall was employed as boundary condition for the cell upper wall to represent numerically this interface. The system modeled by Dong et al. (Dong, et al., 2008) is schematically represented in Figure 6. In accordance with these authors, the simulated fluid velocity field and that obtained with the theoretical model proposed by von Kármán (von Kármán, 1921) and Cochran (Cochran, 1934) are in

Mandin et al. (Mandin, et al., 2004) also carried out two-dimensional axisymmetric numerical simulations of a entire electrochemical cell with a RDE. The system studied by these authors is schematically represented in Figure 7. Their results were compared with the fluid flow pattern calculated with the analytical expressions obtained by Cochran (Cochran, 1934). These authors found that the velocity profiles calculated with these expressions are in accordance with their numerical simulations only in a narrow region close to the electrode active face. However, apparently these authors only made use of one of the two sets of

equations that comprises the solution obtained by Cochran.

boundary conditions (Ferziger & Peric, 1996; Tu, Yeoh, & Liu, 2008).

**3. Simulation with CFD** 

**3.1 Two-dimensional models** 

rotating cylinder electrode.

good agreement.

Fig. 6. Schematic representation of the cell simulated by Dong et al. (Dong, et al., 2008). The line in blue represents a slip wall. The line in red represents symmetry axis.

### **3.2 Three-dimensional models**

All the two-dimensional mathematical models of electrochemical cells with a RDE assumed that the fluid velocity field is axisymmetric. Nevertheless, the asymmetry of the fluid flow

Hydrodynamic Analysis of Electrochemical Cells 419

In the work of Real, et al. (Real, et al., 2008) only the hydrodynamic behavior of the electrolyte inside the cell was simulated. This means that the effect of the interface electrolyte-air in actual electrochemical cells is neglected. However, the significance of the free surface on the flow pattern inside a stationary cylinder with a rotating bottom has been recognized by several authors (Brøns, Shen, Sørensen, & Zhu, 2007). By comparing the flow inside two different container geometries, one with a rigid cover and the other with a free surface, significant differences in the resulting behavior were observed experimentally (Spohn, Mory, & Hopfinger, 1998). This result was reproduced numerically through three-

The effect of the liquid phase free surface on the flow pattern inside the electrochemical cell with a RDE has been studied recently by Real, et al. (Real, et al., 2008), Real-Ramirez, et al. (Real-Ramirez, et al., 2010) and Gonzalez, et al. (Gonzalez, et al., 2011). These works conducted several three-dimensional unsteady-state numerical simulations using biphasic systems. An example of the biphasic three-dimensional models employed in those works is presented in Figure 8. In this example, the volume occupied by the electrolyte was colored in blue and the electrode was colored in grey. The electrode submergence depth is equal to the electrode external diameter. The distance between the electrode active face and the cell

Fig. 9. Liquid velocity vectors near and below the electrode. A section of the liquid free-

Through the biphasic three-dimensional numerical simulations, the authors found that the fluid flow pattern inside the electrochemical cell is not symmetric respect the electrode rotation axis. Figure 9 presents the liquid velocity vectors near and below the electrode. This figure also shows a section of the liquid free-surface, which is colored in blue. A big value for the electrode rotation speed was employed to generate the results shown in Figure 9. By comparing Figures 1 and 9 it is clear that there is a difference between the behaviors of an actual electrochemical cell and the ideal model stated by von Kármán (von Kármán,

surface colored in blue is also presented.

1921).

dimensional numerical simulations (Serre & Bontoux, 2007).

bottom wall is equal to four times the electrode external diameter.

was observed in informal experiments reported by Adams (Adams, 1969) (see Figure 4-3 in that work). Despite the asymmetry is evident, no comments regarding this fact were raised; maybe because in accordance with the author, those experiments were done only to illustrate the general behavior of an electrochemical cell with a RDE.

Due to the size and inner geometry of most of the electrochemical cells, scarce measurements of the liquid velocities inside the cell are reported. However, measurements of liquid velocities inside a cell with a rotating disc electrode through the Doppler Laser Anemometry (DLA) technique were performed by Mandin, et al. (Mandin, et al., 2004). In that work, reproducible high amplitude oscillations that increased substantially with the electrode rotation speed were observed. These authors also compared their experimental measurements with the results of a two-dimensional axisymmetric mathematical model, but liquid velocities oscillations were not reproduced.

By using a three-dimensional model, the symmetry constraint can be avoided and therefore, it is possible to reproduce the hydrodynamic behavior of the internal flow of electrochemical cells under more realistic conditions. The correctness of this assumption can be evaluated by comparing the results of the numerical simulations against the physical experiments measurements.

Real, et al. (Real, et al., 2008) characterized the hydrodynamics inside electrochemical cells with several geometric features using three-dimensional models. These authors studied how the fluid flow pattern is affected by the electrode rotation speed, the cell volume, the electrode submergence depth and the distance between the electrode active face and the cell bottom wall. In that work, the authors found that the fluid flow pattern inside the electrochemical cell is not symmetric. Nonetheless, the asymmetry grade strongly depends on the geometrical configuration of the system.

Fig. 8. Example of the biphasic three-dimensional models employed by Gonzalez, et al. (Gonzalez, Real, Hoyos, Miranda, & Cervantes, 2011), Real-Ramirez, et al. (Real-Ramirez, Miranda-Tello, Hoyos-Reyes, & Gonzalez-Trejo, 2010) and Real, et al. (Real, et al., 2008).

was observed in informal experiments reported by Adams (Adams, 1969) (see Figure 4-3 in that work). Despite the asymmetry is evident, no comments regarding this fact were raised; maybe because in accordance with the author, those experiments were done only to

Due to the size and inner geometry of most of the electrochemical cells, scarce measurements of the liquid velocities inside the cell are reported. However, measurements of liquid velocities inside a cell with a rotating disc electrode through the Doppler Laser Anemometry (DLA) technique were performed by Mandin, et al. (Mandin, et al., 2004). In that work, reproducible high amplitude oscillations that increased substantially with the electrode rotation speed were observed. These authors also compared their experimental measurements with the results of a two-dimensional axisymmetric mathematical model, but

By using a three-dimensional model, the symmetry constraint can be avoided and therefore, it is possible to reproduce the hydrodynamic behavior of the internal flow of electrochemical cells under more realistic conditions. The correctness of this assumption can be evaluated by comparing the results of the numerical simulations against the physical experiments

Real, et al. (Real, et al., 2008) characterized the hydrodynamics inside electrochemical cells with several geometric features using three-dimensional models. These authors studied how the fluid flow pattern is affected by the electrode rotation speed, the cell volume, the electrode submergence depth and the distance between the electrode active face and the cell bottom wall. In that work, the authors found that the fluid flow pattern inside the electrochemical cell is not symmetric. Nonetheless, the asymmetry grade strongly depends

Fig. 8. Example of the biphasic three-dimensional models employed by Gonzalez, et al. (Gonzalez, Real, Hoyos, Miranda, & Cervantes, 2011), Real-Ramirez, et al. (Real-Ramirez, Miranda-Tello, Hoyos-Reyes, & Gonzalez-Trejo, 2010) and Real, et al. (Real, et al., 2008).

illustrate the general behavior of an electrochemical cell with a RDE.

liquid velocities oscillations were not reproduced.

on the geometrical configuration of the system.

measurements.

In the work of Real, et al. (Real, et al., 2008) only the hydrodynamic behavior of the electrolyte inside the cell was simulated. This means that the effect of the interface electrolyte-air in actual electrochemical cells is neglected. However, the significance of the free surface on the flow pattern inside a stationary cylinder with a rotating bottom has been recognized by several authors (Brøns, Shen, Sørensen, & Zhu, 2007). By comparing the flow inside two different container geometries, one with a rigid cover and the other with a free surface, significant differences in the resulting behavior were observed experimentally (Spohn, Mory, & Hopfinger, 1998). This result was reproduced numerically through threedimensional numerical simulations (Serre & Bontoux, 2007).

The effect of the liquid phase free surface on the flow pattern inside the electrochemical cell with a RDE has been studied recently by Real, et al. (Real, et al., 2008), Real-Ramirez, et al. (Real-Ramirez, et al., 2010) and Gonzalez, et al. (Gonzalez, et al., 2011). These works conducted several three-dimensional unsteady-state numerical simulations using biphasic systems. An example of the biphasic three-dimensional models employed in those works is presented in Figure 8. In this example, the volume occupied by the electrolyte was colored in blue and the electrode was colored in grey. The electrode submergence depth is equal to the electrode external diameter. The distance between the electrode active face and the cell bottom wall is equal to four times the electrode external diameter.

Fig. 9. Liquid velocity vectors near and below the electrode. A section of the liquid freesurface colored in blue is also presented.

Through the biphasic three-dimensional numerical simulations, the authors found that the fluid flow pattern inside the electrochemical cell is not symmetric respect the electrode rotation axis. Figure 9 presents the liquid velocity vectors near and below the electrode. This figure also shows a section of the liquid free-surface, which is colored in blue. A big value for the electrode rotation speed was employed to generate the results shown in Figure 9. By comparing Figures 1 and 9 it is clear that there is a difference between the behaviors of an actual electrochemical cell and the ideal model stated by von Kármán (von Kármán, 1921).

Hydrodynamic Analysis of Electrochemical Cells 421

Fig. 11. Fluid stream lines below the electrode when the distance between the electrode active face and the cell bottom wall is small. The blue circle defines the centre of the

Fig. 12. Fluid stream lines below the electrode when the distance between the electrode active face and the cell bottom wall is large. The blue line coincides with the axis of rotation

electrode.

of the electrode.

The asymmetry of the fluid flow pattern about the axis of rotation of the electrode causes a displacement of the stagnation point on the electrode active face. This displacement is shown in Figure 10.

The biphasic three-dimensional numerical simulations confirm that the asymmetry grade strongly depends on the geometrical configuration of the system. By analyzing the results of several physical and numerical simulations at different electrode rotation speeds and several cell sizes, those works found that there exist a synergetic effect of the cell internal walls, the submerged electrode side wall and the liquid free surface (Real-Ramirez, et al., 2010). Their numerical simulations showed that the asymmetry of the electrochemical cell flow pattern is intensified by the free surface asymmetry, which depends directly on the electrode rotation speed and the electrode submergence depth (Gonzalez, et al., 2011).

Several fluid stream lines below the electrode for two distinct values of the distance between the electrode active face and the cell bottom wall are shown in Figures 11 and 12. This distance for the cell shown in Figure 11 is small, whereas a large value for the distance between the electrode active face and the cell bottom wall was employed for the cell shown in Figure 12. The same value of the electrode rotation speed was used for the simulations shown in Figures 12 and 13.

Figures 11 and 12 clearly show that the offset from the center of the electrode of the stagnation point depends strongly on the distance between the electrode active face and the cell bottom wall. Based on the results of their numerical simulations, Real-Ramirez, et al. (Real-Ramirez, et al., 2010) stated as rule of thumb that the distance between the electrode active face and the cell bottom wall must be at least three times the electrode external diameter.

The asymmetry of the fluid flow pattern about the axis of rotation of the electrode causes a displacement of the stagnation point on the electrode active face. This displacement is

Fig. 10. Displacement of the stagnation point on the electrode active face originated by the asymmetry of the fluid flow pattern. The red circle denotes the centre of the electrode.

The biphasic three-dimensional numerical simulations confirm that the asymmetry grade strongly depends on the geometrical configuration of the system. By analyzing the results of several physical and numerical simulations at different electrode rotation speeds and several cell sizes, those works found that there exist a synergetic effect of the cell internal walls, the submerged electrode side wall and the liquid free surface (Real-Ramirez, et al., 2010). Their numerical simulations showed that the asymmetry of the electrochemical cell flow pattern is intensified by the free surface asymmetry, which depends directly on the electrode rotation speed and the electrode submergence depth

Several fluid stream lines below the electrode for two distinct values of the distance between the electrode active face and the cell bottom wall are shown in Figures 11 and 12. This distance for the cell shown in Figure 11 is small, whereas a large value for the distance between the electrode active face and the cell bottom wall was employed for the cell shown in Figure 12. The same value of the electrode rotation speed was used for the simulations

Figures 11 and 12 clearly show that the offset from the center of the electrode of the stagnation point depends strongly on the distance between the electrode active face and the cell bottom wall. Based on the results of their numerical simulations, Real-Ramirez, et al. (Real-Ramirez, et al., 2010) stated as rule of thumb that the distance between the electrode active face and the cell bottom wall must be at least three times the electrode external

shown in Figure 10.

(Gonzalez, et al., 2011).

shown in Figures 12 and 13.

diameter.

Fig. 11. Fluid stream lines below the electrode when the distance between the electrode active face and the cell bottom wall is small. The blue circle defines the centre of the electrode.

Fig. 12. Fluid stream lines below the electrode when the distance between the electrode active face and the cell bottom wall is large. The blue line coincides with the axis of rotation of the electrode.

Hydrodynamic Analysis of Electrochemical Cells 423

**1/2 (rad 1/2 s -1/2)**

The same analysis to that previously described but now for a concentration of NADH of 1.6 m*M* is presented in Figures 15 and 16. As in the as in the previous case, the goodness of the fit is superior by using the model proposed by Gonzalez, et al. (Gonzalez, et al., 2011).

The importance of electrode shape on the behavior of the electrochemical cell has been recognized for a long time. For instance, the inaccuracies caused by the geometry of the

**-1/2 (rad -1/2 s 1/2)**

**y = 1.3788x + 1.2275 R<sup>2</sup> = 0.9223**

0.095 0.1 0.105 0.11 0.115 0.12 0.125 0.13 1.35

Fig. 15. Koutecky-Levich plot of data reported by Unguresan & Gligor (Unguresan & Gligor,

Fig. 14. Fitting of the experimental data reported by Unguresan & Gligor (Unguresan &

Gligor, 2009) for a concentration of NADH of 1.2 m*M* by using Equation (21).

electrode are discussed in detail in Section 6 of the Levich work (Levich, 1942).

7.5 <sup>8</sup> 8.5 <sup>9</sup> 9.5 <sup>10</sup> 10.5 2.5

**y = 0.0981x + 1.8011 R<sup>2</sup> = 0.9929**

**- Ln ( 1/2 i -1**

2.55

**4. Effect of the electrode shape and future trends** 

**i -1**

2009) for a concentration of NADH of 1.6 m*M.*

**L ( A -1)**

1.355 1.36 1.365 1.37 1.375 1.38 1.385 1.39 1.395 1.4 1.405

2.6

2.65

2.7

2.75

2.8

2.85

**L )**

### **3.3 Comparison between mathematical and physical experiments**

Based on the results of their numerical simulations and following the functional form of the solutions proposed by Ackroyd (Ackroyd, 1978) and Ariel (Ariel, 1996), Gonzalez, et al. (Gonzalez, et al., 2011) argued that the fluid velocity component orthogonal to the electrode active face decreases almost exponentially with the square root of the electrode rotation speed. Gonzalez, et al. (Gonzalez, et al., 2011) proposed an empirical correction to the Levich equation as follows:

$$\mathbf{i}\_L = \boldsymbol{\wp} \,\,\alpha^{1/2} \,\,\mathbf{e}^{-\boldsymbol{\gamma}\,\alpha^{1/2}} \,\tag{21}$$

where 0 has the usual interpretation and is a constant such that 0 1 . The smaller the value of , the closer will be the system to the ideal one. The previous equation implies that 0 *<sup>L</sup> <sup>i</sup>* as 1/2 0 , nevertheless, the value of cannot be as large as desired because beyond a certain threshold, the Levich equation is no longer valid.

Fig. 13. Koutecky-Levich plot of data reported by Unguresan & Gligor (Unguresan & Gligor, 2009) for a concentration of NADH of 1.2 m*M.*

Gonzalez, et al. evaluated the validity of Equation (21) using the experimental results reported by Unguresan & Gligor (Unguresan & Gligor, 2009). The experiments correspond with the electrocatalytic NADH oxidation process taking place at graphite electrodes modified with a polymer of phenothiazine formaldehyde at various concentrations of NADH and for an electrolyte pH value of 6.0.

Figure 13 shows the traditional Koutecky-Levich plot of data reported by Unguresan & Gligor for a concentration of NADH of 1.2 m*M*. Figure 14 shows the fitting of the same experimental data by using Equation (21). The equations of the linear adjustment was included these figures. Similar values for the Levich constants are obtained with the model proposed by Gonzalez, et al. (Gonzalez, et al., 2011) and the Koutecky-Levich equation, but the goodness of the fit is significantly improved.

Based on the results of their numerical simulations and following the functional form of the solutions proposed by Ackroyd (Ackroyd, 1978) and Ariel (Ariel, 1996), Gonzalez, et al. (Gonzalez, et al., 2011) argued that the fluid velocity component orthogonal to the electrode active face decreases almost exponentially with the square root of the electrode rotation speed. Gonzalez, et al. (Gonzalez, et al., 2011) proposed an empirical correction to the Levich

1/2 1/2

**-1/2 (rad -1/2 s 1/2)**

0.095 0.1 0.105 0.11 0.115 0.12 0.125 0.13 1.58

Fig. 13. Koutecky-Levich plot of data reported by Unguresan & Gligor (Unguresan & Gligor,

Gonzalez, et al. evaluated the validity of Equation (21) using the experimental results reported by Unguresan & Gligor (Unguresan & Gligor, 2009). The experiments correspond with the electrocatalytic NADH oxidation process taking place at graphite electrodes modified with a polymer of phenothiazine formaldehyde at various concentrations of

Figure 13 shows the traditional Koutecky-Levich plot of data reported by Unguresan & Gligor for a concentration of NADH of 1.2 m*M*. Figure 14 shows the fitting of the same experimental data by using Equation (21). The equations of the linear adjustment was

model proposed by Gonzalez, et al. (Gonzalez, et al., 2011) and the Koutecky-Levich

, the closer will be the system to the ideal one. The previous equation

**y = 1.6153x + 1.4531**

are obtained with the

**R<sup>2</sup> = 0.8497**

 

(21)

is a constant such that 0 1

cannot be as large as desired

. The

*Li e*

 

0 , nevertheless, the value of

because beyond a certain threshold, the Levich equation is no longer valid.

**3.3 Comparison between mathematical and physical experiments** 

has the usual interpretation and

**i -1**

**L ( A -1)**

2009) for a concentration of NADH of 1.2 m*M.*

NADH and for an electrolyte pH value of 6.0.

included these figures. Similar values for the Levich constants

equation, but the goodness of the fit is significantly improved.

1.6

1.62

1.64

1.66

1.68

*<sup>i</sup>* as 1/2 

equation as follows:

smaller the value of

implies that 0 *<sup>L</sup>*

where 0 

Fig. 14. Fitting of the experimental data reported by Unguresan & Gligor (Unguresan & Gligor, 2009) for a concentration of NADH of 1.2 m*M* by using Equation (21).

The same analysis to that previously described but now for a concentration of NADH of 1.6 m*M* is presented in Figures 15 and 16. As in the as in the previous case, the goodness of the fit is superior by using the model proposed by Gonzalez, et al. (Gonzalez, et al., 2011).
