**2.1 System description**

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 above the disc.

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.

Hydrodynamic Analysis of Electrochemical Cells 413

von Kármán (von Kármán, 1921), introduced the following dimensionless independent

*z* 

 

and ( ) *<sup>p</sup> <sup>P</sup>*

2 2 *F HF F G* "' 0 (7)

(4)

 

0 , (10)

(11)

(12)

(13)

(14)

(5)

 , ( ) *<sup>w</sup> <sup>H</sup>* 

After applying the variable change, Equations (2) and (3) can be rewritten as a set of four

2 '0 *F H* (6)

" '2 0 *G HG FG* (8)

'' ' ' 0 *H HH P* (9)

0 :

Several authors have developed approximate solutions to the equations system (6)-(8) subject to boundary conditions given by Equations (10) and (11). The most famous approximate solution was obtained by Cochran (Cochran, 1934), which is composed by two infinite series, one a power series near the disk and the other a series in exponential

> 1 2 3 2 3

<sup>1</sup> <sup>3</sup> <sup>1</sup> 3 *G ba* 

<sup>234</sup> 1 3 6 *<sup>b</sup> H a* 

2 2 2 2

2 4 2 4 *A B AA B*

2 3

(15)

 

 

 

*<sup>b</sup> F a* 

> :

*F Ae e e*

 *r* 

**2.3 Approximate solutions** 

ordinary differential equations:

functions away from the disk.

and

(Equation 4) and dependent variables (Equation 5):

( ) *<sup>u</sup> <sup>F</sup> r* 

The boundary conditions for the system are:

0 *HFP* , 1 *G* at

0 *F G* at

The first set of equations is valid only near the disk

The second set is valid far from the disk

, ( ) *<sup>v</sup> <sup>G</sup>*

Fig. 2. Fluid velocity field on a plane parallel to the disk which resembles logarithmic spirals.

#### **2.2 Mathematical equations**

The mathematical equations of the ideal model are the continuity and the Navier-Stokes equations in cylindrical coordinates at the steady-state:

$$u + r\frac{\partial u}{\partial r} + r\frac{\partial w}{\partial z} = 0\tag{2}$$

$$u\frac{\partial u}{\partial r} - \frac{v^2}{r} + w\frac{\partial u}{\partial z} = -\frac{1}{\rho}\frac{\partial p}{\partial r} + \nu \left\{ \frac{\partial^2 u}{\partial r^2} + \frac{\partial}{\partial r} \left(\frac{u}{r}\right) + \frac{\partial^2 u}{\partial z^2} \right\}$$

$$u\frac{\partial v}{\partial r} - \frac{uv}{r} + w\frac{\partial v}{\partial z} = \nu \left\{ \frac{\partial^2 v}{\partial r^2} + \frac{\partial}{\partial r} \left(\frac{v}{r}\right) + \frac{\partial^2 v}{\partial z^2} \right\}\tag{3}$$

$$u\frac{\partial w}{\partial r} + w\frac{\partial w}{\partial z} = -\frac{1}{\rho}\frac{\partial p}{\partial z} + \nu \left\{ \frac{\partial^2 w}{\partial r^2} + \frac{1}{r}\frac{\partial w}{\partial r} + \frac{\partial^2 w}{\partial z^2} \right\}$$

This model assumes that the fluid flow pattern is axisymmetric respect the axial axis. In this equations, *r* and *z* are the radial and axial coordinates, *p* is the pressure, ρ is the density of the fluid and ν is the kinematic viscosity of the fluid. In (2) and (3), *u* , *v* and *w* are the radial, the angular and the axial velocity components respectively. This model also assumes that the flow regime is laminar.

The boundary conditions for (2) and (3) are as follows: At the disc surface *z* 0 *,* a non-slip condition is assumed, that is, 0 *u* , 0 *w* and *v r* , where is the electrode rotation speed. Far from the disc *z* , it is assumed that there is no flow in the radial and the angular directions, which can be expressed as *u* 0 and *v* 0 . In addition, the axial velocity reaches its limiting velocity *U*<sup>0</sup> , that is, *w U* <sup>0</sup> .

#### **2.3 Approximate solutions**

412 Computational Simulations and Applications

Fig. 2. Fluid velocity field on a plane parallel to the disk which resembles logarithmic

The mathematical equations of the ideal model are the continuity and the Navier-Stokes

*ur r*

1

*v uv v v v v u w r r z rr r z*

*uv u <sup>p</sup> u uu u w r r z r rr r z*

<sup>0</sup> *u w*

2 2 2

2 2 2 2

1 1

*w w <sup>p</sup> w ww u w r z z rr r z*

This model assumes that the fluid flow pattern is axisymmetric respect the axial axis. In this equations, *r* and *z* are the radial and axial coordinates, *p* is the pressure, ρ is the density of the fluid and ν is the kinematic viscosity of the fluid. In (2) and (3), *u* , *v* and *w* are the radial, the angular and the axial velocity components respectively. This model also assumes

The boundary conditions for (2) and (3) are as follows: At the disc surface *z* 0 *,* a non-slip

speed. Far from the disc *z* , it is assumed that there is no flow in the radial and the angular directions, which can be expressed as *u* 0 and *v* 0 . In addition, the axial

, where

(2)

(3)

is the electrode rotation

2 2

2 2 2 2

*r z* 

spirals.

**2.2 Mathematical equations** 

that the flow regime is laminar.

equations in cylindrical coordinates at the steady-state:

condition is assumed, that is, 0 *u* , 0 *w* and *v r*

velocity reaches its limiting velocity *U*<sup>0</sup> , that is, *w U* <sup>0</sup> .

von Kármán (von Kármán, 1921), introduced the following dimensionless independent (Equation 4) and dependent variables (Equation 5):

$$
\xi = z \sqrt{\frac{\alpha \nu}{\nu}} \tag{4}
$$

$$F(\xi) = \frac{\mathcal{U}}{r \, o\nu}, \quad G(\xi) = \frac{v}{r \, o\nu}, \quad H(\xi) = \frac{w}{\sqrt{\nu \, o\nu}} \quad \text{and} \quad P(\xi) = \frac{p}{\rho \, \nu \, o\nu} \tag{5}$$

After applying the variable change, Equations (2) and (3) can be rewritten as a set of four ordinary differential equations:

$$2F + H' = 0\tag{6}$$

$$F'' - HF' - F^2 + G^2 = 0\tag{7}$$

$$G'' - HG' - \mathcal{Z}FG = 0\tag{8}$$

$$H' - HH' - P' = 0\tag{9}$$

The boundary conditions for the system are:

$$H = F = P = 0 \quad G = 1 \quad \text{at} \quad \xi = 0 \,\, \tag{10}$$

and

$$F = G = 0 \quad \text{at} \quad \xi = \infty \tag{11}$$

Several authors have developed approximate solutions to the equations system (6)-(8) subject to boundary conditions given by Equations (10) and (11). The most famous approximate solution was obtained by Cochran (Cochran, 1934), which is composed by two infinite series, one a power series near the disk and the other a series in exponential functions away from the disk.

The first set of equations is valid only near the disk 0 :

$$F = a\xi - \frac{1}{2}\xi^2 - \frac{b}{3}\xi^3 + \dotsb \tag{12}$$

$$G = 1 + b\xi + \frac{1}{3}a\xi^{\varepsilon 3} + \dotsb \tag{13}$$

$$H = -a\xi^2 + \frac{1}{3}\xi^3 + \frac{b}{6}\xi^4 + \dotsb \tag{14}$$

The second set is valid far from the disk :

$$F = Ae^{-a\_{\circ}^{\circ}} - \frac{\left(A^2 + B^2\right)}{2\alpha^2} e^{-2a\_{\circ}^{\circ}} + \frac{A\left(A^2 + B^2\right)}{4\alpha^4} e^{-3a\_{\circ}^{\circ}} + \dotsb \tag{15}$$

Hydrodynamic Analysis of Electrochemical Cells 415

solution proposed by Ariel over that obtained by Cochran is that Equations (18)-(20) can be

These figures also shows that the trend of the two sets of functions that comprises the

Fig. 4. Comparison of the solutions developed by Cochran and Ariel for the angular velocity

0 0.5 1 1.5 2 2.5 3 3.5 4 -0.2

Fig. 5. Comparison of the solutions developed by Cochran and Ariel for the axial velocity

0 0.5 1 1.5 2 2.5 3 3.5 4 -0.2

.

 . Figures 3, 4 and 5 show a comparison of the solutions obtained by Cochran and Ariel for the radial, angular and axial velocity components, respectively. Roughly speaking, these figures show that the first of functions of the solution proposed by Cochran are valid when 1

0.9130294741 . The main advantage of the

.

Where

component.

component.

evaluated for any value of

is defined as in Equation (4) and

**G**

0 0.2 0.4 0.6 0.8 1 1.2 1.4

**H**

0

0.2

0.4

0.6

0.8

1

such that 0

**Ariel Cochran Set 1 Cochran Set 2**

**Ariel Cochran Set 1 Cochran Set 2**

solution proposed by Cochran are completely distinct for 1.5

$$G = Be^{-\alpha \xi} - \frac{B\left(A^2 + B^2\right)}{12a^4}e^{-3a\xi} + \dotsb \tag{16}$$

$$H = -a + \frac{2A}{a}e^{-a\xi} - \frac{\left(A^2 + B^2\right)}{2a^3}e^{-2a\xi} + \frac{A\left(A^2 + B^2\right)}{6a^5}e^{-3a\xi} + \dotsb \tag{17}$$

Where in both sets, *a* , *b* , *A* , *B* and are constants.

Fig. 3. Comparison of the solutions developed by Cochran and Ariel for the radial velocity component.

Contrary to the approach stated by Cochran, Ackroyd (Ackroyd, 1978) developed a solution composed by only one set of infinite series of exponential terms with negative exponents. Following Arckroy approach, Ariel (Ariel, 1996) presented an approximate solution in which is possible to obtain better results than that obtained with other approximate methods. The solution developed by Ariel is given by the following equations:

$$F = \beta^2 \left( \frac{1}{8} \left( 2e^{-\beta \cdot \xi} - e^{-2\beta \cdot \xi} \right) \ln \left( \frac{4e^{\beta \cdot \xi} - 1}{3} \right) + \frac{1}{2} \left( e^{-\beta \cdot \xi} - e^{-2\beta \cdot \xi} \right) \right) \tag{18}$$

$$G = \frac{1}{3} \left( 4e^{-\beta \frac{\varepsilon}{\sigma}} - e^{-2\beta \frac{\varepsilon}{\sigma}} \right) \tag{19}$$

$$H = \beta \left( \frac{1}{8} (4e^{-\beta \cdot \xi} - e^{-2\beta \cdot \xi}) \ln \left( \frac{4e^{\beta \cdot \xi} - 1}{3} \right) - \frac{1}{2} \left( 2 - 3e^{-\beta \cdot \xi} + e^{-2\beta \cdot \xi} \right) \right) \tag{20}$$

2 2

<sup>4</sup> 12 *BA B*

*A A B AA B He e e*

are constants.

0 0.5 1 1.5 2 2.5 3 3.5 4 -0.2

Fig. 3. Comparison of the solutions developed by Cochran and Ariel for the radial velocity

Contrary to the approach stated by Cochran, Ackroyd (Ackroyd, 1978) developed a solution composed by only one set of infinite series of exponential terms with negative exponents. Following Arckroy approach, Ariel (Ariel, 1996) presented an approximate solution in which is possible to obtain better results than that obtained with other approximate

<sup>2</sup> <sup>1</sup> 2 2 4 11 <sup>2</sup> ln

<sup>1</sup> <sup>2</sup> <sup>4</sup>

<sup>1</sup> 2 2 4 11 <sup>4</sup> ln 2 3

 

 

 

 

(19)

 

> 

(18)

(20)

methods. The solution developed by Ariel is given by the following equations:

 

3 *G ee* 

8 3 2 *<sup>e</sup> H ee e e* 

 

8 3 2 *<sup>e</sup> F ee e e*

2 2 2 2

2 6

3 5

*G Be e* 

**Ariel Cochran Set 1 Cochran Set 2**

2

Where in both sets, *a* , *b* , *A* , *B* and

**F**

component.

0

0.2

 3

2 3

(17)

 

(16)

 Where is defined as in Equation (4) and 0.9130294741 . The main advantage of the solution proposed by Ariel over that obtained by Cochran is that Equations (18)-(20) can be evaluated for any value of such that 0 .

Figures 3, 4 and 5 show a comparison of the solutions obtained by Cochran and Ariel for the radial, angular and axial velocity components, respectively. Roughly speaking, these figures show that the first of functions of the solution proposed by Cochran are valid when 1 . These figures also shows that the trend of the two sets of functions that comprises the solution proposed by Cochran are completely distinct for 1.5 .

Fig. 4. Comparison of the solutions developed by Cochran and Ariel for the angular velocity component.

Fig. 5. Comparison of the solutions developed by Cochran and Ariel for the axial velocity component.

Hydrodynamic Analysis of Electrochemical Cells 417

Fig. 6. Schematic representation of the cell simulated by Dong et al. (Dong, et al., 2008). The

Fig. 7. Schematic representation of the cell simulated by Mandin et al. (Mandin, et al., 2004). The line in red represents symmetry axis. The line in black represents the electroactive zone.

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

**3.2 Three-dimensional models** 

line in blue represents a slip wall. The line in red represents symmetry axis.

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.
