**2.1 Velocity gradient and frictional force measurement**

Reiss and Hanratty (1963) developed a model to obtain the velocity gradient Sw, in the vicinity of a wall of rectangular or cylindrical shape. By using a microelectrode mounted flush with a tube wall, the apparent mass transfer coefficient Ka could be obtained by electrochemical technique. This model was established in the cases where the thickness of the concentration boundary layer (c) was less than that of the hydrodynamic boundary layer (v) (figure 1) and was verified in cases of high Schmidt number.

The mass balance in the vicinity of the microelectrode can be developed as follow:

$$\frac{\partial \mathbf{C}}{\partial t} + \mathbf{S}\_{w} \cdot \mathbf{y} \frac{\partial \mathbf{C}}{\partial \mathbf{x}} = D \frac{\partial^2 \mathbf{C}}{\partial y^2} \tag{1}$$

Fig. 1. Concentration and velocity profiles over the electrode surface.

The instantaneous rate of mass transfer is proportional to the concentration gradient at the surface averaged over the electrode:

$$N = D < \frac{\partial \mathcal{C}}{\partial y} > \tag{2}$$

The mass transfer coefficient defined as : Ka = N/(C-Cw), Cw is the wall concentration.

The analytical solution of (1) allows to determine the local mass transfer coefficient::

$$\overline{K\_a} = \alpha \left(\frac{D^2 \overline{S\_w}}{L\_c}\right)^{1/3} \tag{3}$$

C : concentration, mol.m-3

46 Electrochemical Cells – New Advances in Fundamental Researches and Applications

Paragraph two develops this aspect, the electrochemical probe developed serves in the same time as a probe able to measure the velocity gradient at the wall of a spherical sphere and to

For the process aspect as the water and waste water treatment, the need of purifying water for human consumption is more and more required. Cleaning wastewater from industrial effluents before discharging is also a challenging work. In fact, innovative, cheap and

Electrocoagulation (EC) is an electrochemical method for treating polluted water which has been successfully applied for treatment of soluble or colloidal pollutants, such as wastewater containing heavy metals, emulsions, suspensions, etc., but also drinking water for lead or fluoride removal. A typical EC unit includes therefore an EC cell/reactor, a separator for settling or flotation, and often a filtration step. Indeed, the benefits of EC include simplicity, efficiency, environmental compatibility, safety, selectivity, flexibility and cost effectiveness. In particular, the main points involve the reduction of sludge generation, the minimization of the addition of chemicals and little space requirements due to shorter residence time. The main deficiency is the lack of dominant reactor design and modeling procedures. The literature reveals any systematic approach for design and scale-up purpose. The most papers use laboratory-scale EC cells in which magnetic stirring is adjusted

experimentally and the separation step by floatation/sedimentation is not studied.

That's why an innovative reactor is developed in order to optimize the cost of this process.

**2. Electrochemical method for measurements of frictional force and bubble** 

This part involves the measurement character of electrochemical methods that lead to measure the velocity gradient and frictional force on a particle in a bubble column and in a fluidized bed. The possibility of measuring bubble volume is also discussed. Thus, an electrochemical probe may be proposed to measure in the same time the frictional force and the bubble volume. The application of such electrochemical method is very important in

Reiss and Hanratty (1963) developed a model to obtain the velocity gradient Sw, in the vicinity of a wall of rectangular or cylindrical shape. By using a microelectrode mounted flush with a tube wall, the apparent mass transfer coefficient Ka could be obtained by electrochemical technique. This model was established in the cases where the thickness of the concentration boundary layer (c) was less than that of the hydrodynamic boundary

2

(1)

*w* 2 *C CC <sup>S</sup> y D t x <sup>y</sup>* 

measure the volume bubble in a bubble column.

effective techniques have to be developed.

This is the object of paragraph three.

biochemical engineering.

**2.1 Velocity gradient and frictional force measurement** 

layer (v) (figure 1) and was verified in cases of high Schmidt number.

The mass balance in the vicinity of the microelectrode can be developed as follow:

**size** 

D : coefficient diffusion, m2.s-1,

The method can be extended to spherical walls (sphere particle). A sphere is equipped with an inside channel, bent through 90°, in which a gold thread of 1 mm diameter was introduced, cut flush with the surface. A rigid tube serves as support. The microelectrode can be directed relative to the average direction of the liquid by rotating the support (figure 2).

Fig. 2. The electrochemical electrode.

The electrochemical technique adopted consisted of the determination of the apparent local transfer coefficient (and subsequently the velocity gradient at the wall), based on measurements of the diffusion – limiting current during the reduction of an electroactive species.

The relationship between the apparent mass transfer coefficient and the velocity gradient is then deduced:

$$\overline{K\_{a}} = 0.862 \left( \frac{D^2 \ \overline{S\_w}}{d\_{\epsilon}} \right)^{1/3} \tag{4}$$

Electrochemical Probe for Frictional Force and Bubble Measurements

1- Column 7- Temperature controller

Fig. 3. Experimental apparatus for the study of friction on a solid sphere.

potassium ferri-ferrocyanide system chosen consisted of a 0.5 mol.dm-3 sodium hydroxide solution with 5 x 10-3 mol.dm-3 each of potassium ferricyanide and potassium ferrocyanide. Sodium hydroxide is added as electrolyte support to minimize the migrational effects.

The physic-chemical properties of the solution are given in Table 1. The density of the solution was determined by pycnometry and the viscosity measured by a viscosmeter (Rheomat 15T-FC). The diffusion coefficient of the potassium ferricyanide is measured by means of a turning disc electrode, computed from the intensity of the limiting current on the

k = 1.61173 + 0.480306 Sc-1/3 + 0.23393 Sc-2/3 + 0.113151 Sc-1

This relationship is also used to determine the concentration of the solution once the

diffusion coefficient is known (assumed to be concentration independent).

J = D C0 Sc1/3.(/)1/2. 1/k (5)

2- Homogenization section 8- Cooling coil 3- Rotameter 9- Auxiliary electrode 4- Rotameters 10- Working electrode 5- Pump 11- Reference electrode

6- Liquid reservoir 12- Capillary

disc using the relationship (Levart & Schumann, 1974):

C0 : electroactive species concentration, mol.m-3

D: coefficient diffusion, m2.s−<sup>1</sup> J : molar flux species, mol.m−2 s−<sup>1</sup>

: kinematic velocityof liquid, m2.s-1 Sc : Schmidt number defined by /D

: angular velocity, rd.s-1

and Innovative Electrochemical Reactors for Electrocoagulation/Electroflotation 49

de : diameter of microelectrode, m

In order to test the validity of this model to predict the frictional force in sphere particles in bubble column and fluidized bed, the experimental apparatus is then designed (figure 3).

It is composed of a plexiglass fluidization column, 157 cm height (distance between the perforated stainless steel plate serving as liquid distributor and the point of liquid overflow) and 9.4 cm diameter. A column filled with glass sphere was placed underneath the liquid distributor; this serves as a homogenization section to avoid any preferential passage phenomena. A liquid reservoir is equipped with a temperature controller and a liquid cooling coil. Rotameters are used to meausure the flow rates of liquid and gas (nitrogen) which are introduced via a gas injector consisting of two concentric circular tubes with 90 regularly spaced holes (0.4 mm diameter). An auxiliary electrode (nickel ring), a reference electrode furnished with a Luggin's capillary and a working electrode are connected to a potentiometer which is linked to microcomputer performing the acquisition of electric current signals and computations of the average and fluctuating current intensities.

The method depends on creating a diffusional limitation at the transfer surface studied (working electrode) and simultaneously avoids such limitations at the counter electrode surface (i.e the counter electrode potential remained almost constant). This is achieved by using a counter electrode with a large surface area compared with the working electrode.

The intensity of the limiting current of the working electrode is obtained by the graph presenting the current intensity versus potential. In order to achieve this, the reference electrode (saturated calomel), has to be placed as closed as possible to the working electrode, to minimize the ohmic resistance of the solution. For this purpose, a capillary of about 1 mm diameter is introduced between the working and reference electrodes.

Several possible reactions for electrochemical determination of mass transfer coefficients are available, but the most frequently used system is potassium ferricyanide to potassium ferrocyanide. Normally the reaction at the working electrode is the reduction of ferricyanide, whereas the oxidation of ferrocyanide takes place at the counter electrode:

 Cathode: Fe(CN)63- + e- Fe(CN)6 4-

> Anode: Fe(CN)6 4- Fe(CN)6 4- + e-

The ferricyanide concentration should thus in principle remain constant during the experiment. This is verified by using a turning disc electrode, as explained later. The

The electrochemical technique adopted consisted of the determination of the apparent local transfer coefficient (and subsequently the velocity gradient at the wall), based on measurements of the diffusion – limiting current during the reduction of an electroactive

The relationship between the apparent mass transfer coefficient and the velocity gradient is

0.862 *<sup>w</sup> <sup>a</sup>*

*<sup>D</sup> <sup>S</sup> <sup>K</sup>*

current signals and computations of the average and fluctuating current intensities.

about 1 mm diameter is introduced between the working and reference electrodes.

Cathode: Fe(CN)63- + e- Fe(CN)64-

The method depends on creating a diffusional limitation at the transfer surface studied (working electrode) and simultaneously avoids such limitations at the counter electrode surface (i.e the counter electrode potential remained almost constant). This is achieved by using a counter electrode with a large surface area compared with the working electrode.

The intensity of the limiting current of the working electrode is obtained by the graph presenting the current intensity versus potential. In order to achieve this, the reference electrode (saturated calomel), has to be placed as closed as possible to the working electrode, to minimize the ohmic resistance of the solution. For this purpose, a capillary of

Several possible reactions for electrochemical determination of mass transfer coefficients are available, but the most frequently used system is potassium ferricyanide to potassium ferrocyanide. Normally the reaction at the working electrode is the reduction of ferricyanide, whereas the oxidation of ferrocyanide takes place at the counter electrode:

Anode: Fe(CN)64- Fe(CN)64- + e-The ferricyanide concentration should thus in principle remain constant during the experiment. This is verified by using a turning disc electrode, as explained later. The

1/3 <sup>2</sup>

(4)

*e*

*d* 

In order to test the validity of this model to predict the frictional force in sphere particles in bubble column and fluidized bed, the experimental apparatus is then designed (figure 3). It is composed of a plexiglass fluidization column, 157 cm height (distance between the perforated stainless steel plate serving as liquid distributor and the point of liquid overflow) and 9.4 cm diameter. A column filled with glass sphere was placed underneath the liquid distributor; this serves as a homogenization section to avoid any preferential passage phenomena. A liquid reservoir is equipped with a temperature controller and a liquid cooling coil. Rotameters are used to meausure the flow rates of liquid and gas (nitrogen) which are introduced via a gas injector consisting of two concentric circular tubes with 90 regularly spaced holes (0.4 mm diameter). An auxiliary electrode (nickel ring), a reference electrode furnished with a Luggin's capillary and a working electrode are connected to a potentiometer which is linked to microcomputer performing the acquisition of electric

species.

then deduced:

de : diameter of microelectrode, m

Fig. 3. Experimental apparatus for the study of friction on a solid sphere.

potassium ferri-ferrocyanide system chosen consisted of a 0.5 mol.dm-3 sodium hydroxide solution with 5 x 10-3 mol.dm-3 each of potassium ferricyanide and potassium ferrocyanide. Sodium hydroxide is added as electrolyte support to minimize the migrational effects.

The physic-chemical properties of the solution are given in Table 1. The density of the solution was determined by pycnometry and the viscosity measured by a viscosmeter (Rheomat 15T-FC). The diffusion coefficient of the potassium ferricyanide is measured by means of a turning disc electrode, computed from the intensity of the limiting current on the disc using the relationship (Levart & Schumann, 1974):

$$\mathbf{J} = \mathbf{D}' \mathbf{C}\_0 \mathbf{S} \mathbf{c}^{1/3} . \text{(co/v)} \\ \text{1/2. 1/k} \tag{5}$$

$$\mathbf{k} = \mathbf{1}.6117\mathbf{\hat{s}} + \mathbf{0}.480\mathbf{\hat{s}}06\mathbf{\hat{S}}\mathbf{c}^{1/3} + \mathbf{0}.2339\mathbf{\hat{S}}\mathbf{\hat{S}}\mathbf{c}^{2/3} + \mathbf{0}.113151\mathbf{\hat{S}}\mathbf{c}^{1/3}$$

D: coefficient diffusion, m2.s−<sup>1</sup>

J : molar flux species, mol.m−2 s−<sup>1</sup>

C0 : electroactive species concentration, mol.m-3

: angular velocity, rd.s-1

: kinematic velocityof liquid, m2.s-1

Sc : Schmidt number defined by /D

This relationship is also used to determine the concentration of the solution once the diffusion coefficient is known (assumed to be concentration independent).


Table 1. Physico-chemical properties of the solution used in the study.

The apparent mass transfer coefficient at any microelectrode position , is given by:

$$\mathbf{K}\_{a} = \frac{I\_{l}}{n.F.A\_{c}.C\_{0}} \tag{6}$$

Electrochemical Probe for Frictional Force and Bubble Measurements

and Innovative Electrochemical Reactors for Electrocoagulation/Electroflotation 51

Fig. 4. Surface velocity gradient as function of position for single-phase around a sphere.

In the case of three phase fluidization, glass spheres (2mm in diameter, s= 2532 kgm-3) and plastic spheres (5mm in diameter, s= 1388 kg.m-3) are used. This choice provides very different bubbly flows due to different balances of coalescence and break-up of bubbles.

The contribution of the frictional force is more important in "coalescent" fluidized beds than in "break-up" fluidized beds. The effect of gas injection is depending on the fluidized particle effect on bubble coalescence and break-up. Correlations are developed linking frictional force to gas hold-up. The correlations recommended for frictional force in

*F* is a dimensionless force defined by *F* = *Ff /Pa*, where *Pa* is the effective weight of the sphere. Re: Reynolds number defined by *dpUll/l*, dp is the particle diameter, l is the liquid density,

The purpose is to demonstrate that the electrochemical probe can be used as a means of measuring bubble sizes. First, calibrated bubbles are used by single tubes. Then, a gas injector is used in a bubble column with homogeneous bubbling regime. In this case, the average frequency of the fluctuations of diffusion limited current detected by the probe is postulated as being equal to the bubble frequency leading to an estimation of bubble size.

To test the possibility of measuring bubble size with the use of an electrochemical probe in a bubble column, a train of calibrated bubbles is generated in a tube in which gas is injected under closely controlled conditions. To obtain different bubble sizes, three tubes, T1, T2, and

fluidized beds for both systems, (i.e., coalescence and break-up) are as follows:

Ul is the superficial liquid velocity, l is the dynamic viscosity, g : gas holdup.

Glass spheres (2mm diameter, coalescence regime):

Plastic spheres (5mm diameter, break-up regime):

0.4 *,* standard deviation = 6%.

0.1 *,* standard deviation = 4%.

*F* = 2*.*43 Re0*.*<sup>052</sup> ε<sup>g</sup>

*F* = 0*.*123 Re0*.*<sup>3</sup> ε<sup>g</sup>

**2.2 Bubble size measurement** 

**2.2.1 Single orifice: bubble train** 

Il : Intensity of diffusion – limiting current (A)

F : Faraday constant (A s m-1)

n : number of electrons liberated during the course of the electrochemical reaction

Ae :surface of microelectrode (m2).

From the equation (4) we can deduce the velocity gradient:

$$S\_w(\theta) = \left(1.16 K\_a\right)^3 \cdot \frac{d\_\epsilon}{D^2} \tag{7}$$

As the velocity gradient at the wall is related to the local shear stress (w) through the relationship: w = l. Sw (Newtonian liquid), l is the dynamic viscosity (Pa.s).

Eq. (7) may be used to compute the frictional drag force (Ff) exerted on the whole sphere:

$$\sum\_{\sigma} \sum\_{\sigma}^{\sigma\_f}$$

$$F\_f = 2\pi R^2 \eta\_f \stackrel{\text{\(\sigma\)}}{\longleftrightarrow} 2\theta d\theta$$

#### **Method verification:**

The relationship of Reiss and Hanratty was established for the case of a flat surface and validity for a spherical surface should therefore first be verified.

Figure 4 shows the velocity gradient at the wall for five values of the Reynolds number and with different angles between the electrode surface and the mean fluid flow direction. The curves clearly show a maximum in velocity gradients around the 45° angle which is in agreement with calculations based on numerical solution of the momentum balance equation around a sphere. The figure also shows that the velocity gradient rises with the Reynolds number at all positions on the sphere. The influence is however far more noticeable in the front than in the rear of the sphere.

<sup>0</sup> .. . *<sup>l</sup> <sup>a</sup>*

 <sup>3</sup> <sup>2</sup> ( ) 1.16 . *<sup>e</sup> <sup>w</sup> <sup>a</sup> <sup>d</sup> <sup>S</sup> <sup>K</sup>*

2 2 0 *F R <sup>f</sup>* 2 () *<sup>l</sup> Sw* sin *d* 

The relationship of Reiss and Hanratty was established for the case of a flat surface and

Ul

d

Figure 4 shows the velocity gradient at the wall for five values of the Reynolds number and with different angles between the electrode surface and the mean fluid flow direction. The curves clearly show a maximum in velocity gradients around the 45° angle which is in agreement with calculations based on numerical solution of the momentum balance equation around a sphere. The figure also shows that the velocity gradient rises with the Reynolds number at all positions on the sphere. The influence is however far more

As the velocity gradient at the wall is related to the local shear stress (w) through the

Eq. (7) may be used to compute the frictional drag force (Ff) exerted on the whole sphere:

*D*

 

**w** 

*e*

*<sup>I</sup> <sup>K</sup> n F <sup>A</sup> <sup>C</sup>* (6)

(7)

(8)

Temperature (°C) 10-3 l (kg.m-3) 103 l (Pa.s) 1010 D (m2.s−1)

The apparent mass transfer coefficient at any microelectrode position , is given by:

n : number of electrons liberated during the course of the electrochemical reaction

relationship: w = l. Sw (Newtonian liquid), l is the dynamic viscosity (Pa.s).

R

 

validity for a spherical surface should therefore first be verified.

noticeable in the front than in the rear of the sphere.

25 1.02 1.17 5.17 Table 1. Physico-chemical properties of the solution used in the study.

Il : Intensity of diffusion – limiting current (A)

From the equation (4) we can deduce the velocity gradient:

F : Faraday constant (A s m-1)

**Method verification:** 

Ae :surface of microelectrode (m2).

Fig. 4. Surface velocity gradient as function of position for single-phase around a sphere.

In the case of three phase fluidization, glass spheres (2mm in diameter, s= 2532 kgm-3) and plastic spheres (5mm in diameter, s= 1388 kg.m-3) are used. This choice provides very different bubbly flows due to different balances of coalescence and break-up of bubbles.

The contribution of the frictional force is more important in "coalescent" fluidized beds than in "break-up" fluidized beds. The effect of gas injection is depending on the fluidized particle effect on bubble coalescence and break-up. Correlations are developed linking frictional force to gas hold-up. The correlations recommended for frictional force in fluidized beds for both systems, (i.e., coalescence and break-up) are as follows:


*F* is a dimensionless force defined by *F* = *Ff /Pa*, where *Pa* is the effective weight of the sphere.

Re: Reynolds number defined by *dpUll/l*, dp is the particle diameter, l is the liquid density, Ul is the superficial liquid velocity, l is the dynamic viscosity, g : gas holdup.
