1 Introduction

The need of purifying water for human consumption is more and more required. Thus, the management of the water cycle is essential for a sustainable water supply. An integrated solution is necessary to take into account all requirements and meet the global demand for water [1]. This may be achieved through the increased water reuse and recycling. Cleaning wastewater from industrial effluents before discharging is also a challenging work. In fact, innovative, cheap and effective techniques have to be developed. Chemical Coagulation-flocculation is one of the most used techniques for wastewater treatment [2] but certain drawbacks are induced by the presence of metal salts, acidification of the treated water and the large amount of sludge disposal. Biological treatment [3], advanced chemical oxidation such as UV, H2O2, O3 [4,5,6], adsorption on adsorbent materials such as activated carbon [7] or membrane processes were effective, but in most cases very costly. An interesting alternative to these methods is electrocoagulation (EC). The electrocoagulation (EC) process consists in producing, in situ, metal ions Mn+ (Fe2+, Fe3+, Al3+) by the electrochemical dissolution of electrodes under the effect of an electric current. The metal cations produced allow, in a first step, coagulation and, in a second step, flocculation of the polluting particles. The separation of the aqueous mass is carried out by flotation or by decantation. The major complication that limits the development of studies on these techniques is the lack of equipment for its application as well as the models related to this technology. Mollah et al. [8] describe six typical device configurations for the industrial application of electrocoagulation.

The published papers deal in general with electrocoagulation processes at the laboratory scale. Recent studies used semi pilot reactors [9]. For textile wastewater treatment, Ahmed Samir et al. (2016) used a rotated anode in reactor having 10 L as capacity. The hexavalent chromium removal was treated by EC in a cylindrical column reactor with 4 L capacity [10]. Treatment of table olive processing wastewaters using electrocoagulation in laboratory and also in pilot-scale reactors was investigated [11].

The proposed design does not in general emerge from a rigorous study resulting from a simulation or a chemical engineering approach.

The objective in our study is to design an innovative reactor that allows the electrochemical treatment with an optimal cost based on simulation results for flotation effluent recovery.

For this, the airlift reactors can offer this opportunity because the sludge recovery can be achieved by flotation in the disengagement section. Thus, settling time constitutes a drawback for the pollutants as mentioned by Khorram et al. in their paper [12] in which even with a relatively high efficiency of color removal the settling required more time.

Airlift reactors are widely used in biological processes such as fermentation [13] and for culturing fungi [14] and biochemical waste water treatment and other chemical process. The use of this kind of reactor is based on higher fluid circulation and mass transfer, lower shear stress and the lower energy consumption. The specific energy input to disperse the gas and to allow a good mixing in the reactor can be achieved with much less energy by using the kinetic energy of the gas into reactor in the pneumatic driven gas–liquid.

Airlift reactor consists of two main compartments: the riser in which the gas is introduced and the other is named downcomer. Due to the density gradient of the fluid between the two compartments (aerated zone: riser and non-aerated zone: downcomer), a liquid movement is induced [15].

In most papers the industrial applications concern chemical and biochemical slow reactions [15, 16].

Many studies were focused on liquid circulation, mass transfer and gas holdups with two or three phase flow [17, 18], but never as an electrocoagulation (EC) cells, as far as we know.

Not long ago, and for the first time, external loop airlift reactor was used as electrochemical reactor [19]. It was especially used for Electrocoagulation (EC)–Electroflotation (EF) to treat the liquid effluents. In order to transform this reactor to an electrochemical one, the gas phase was not injected, but electrochemically-generated.

An application of ectrocoagulation/electroflotation in external loop airlift was performed showing high efficiency in removing color from synthetic and real textile wastewater by using aluminum and iron electrodes [20]. The defluoridation by EC was also performed in external loop airlift reactor [21]. In another previous paper the designed reactor was operated with continuous mode flow of liquid [22]. The inlet volumetric liquid flow-rate QL varied between 0.1 and 2 L/min. A study of the residence time distribution (RTD) analysis of liquid phase has been performed. The liquid RTD is determined by means of the tracer response technique. The prior investigation [12] based on the residence time distribution measurements method had shown several troubles related to the liquid flow behavior. The inlet quantity is divided into two flows: one exit directly the rector by crossing the junction and the other cross the riser, the separator zone and the downcomer to exit. The percentage of flow that quits the reactor without reacting increased when the main flow increased and the current intensity decreased. The treated amount of effluent does not exceed 30% in continuous mode [22].

To resolve this problem, the Computational Fluid Dynamics (CFD) approach has been used, to validate the observed experimental anomalies and to propose an improvement of the airlift geometry and then the overall performance. The aim of this work is to investigate different hydrodynamic parameters in the previous airlift geometry and propose a new design to ensure a high treatment quantity of waste water.

CFD approach was used in internal loop airlift reactor, to produce cellulase by fungi [14, 23]. In external loop airlift reactors CFD was used to study the hydrodynamic, mixing and mass transfer [24]. CFD simulation was also used to determine local gas holdup in an external loop airlift reactor [25]. A recent study has exposed the experimental and modeling aspects in airlift reactors [26]. Numerical simulation method for flow process in airlift reactors are reviewed, which are classified into two series, i.e. mathematical mechanism models and CFD models. The review provides a comprehensive analysis of such efforts, with critical discussion of published standpoints. Another recent paper presented experimental and numerical study of two-phase gas–liquid flows in the external-loop airlift reactor [27]. Both phases are calculated by solving the steady-state Reynolds-Averaged Navier–Stokes (RANS) conservation equations with the high Reynolds number k–ε turbulence model. The aerodynamic drag, shear-lift and added mass forces, and the turbulent dispersion of the flow are considered.

As far we know no work has been investigated by using CFD in external loop airlift as electrochemical reactor in which gas phase is electrochemically generated.

In the present work, Navier–Stokes equations are coupled with k-ε model in order to take into account turbulence of the liquid flow. In this model, all the inter-phase forces (drag, lift and virtual masse) are used.

As the prime importance is to propose an adequate geometry in order to increase the quantity of water treatment with a good recirculation, only a mean bubble diameter is used. The population balance equations (PBE) need a high computational demand. In the next paper, the PBE equation will be used for the kinetic such as last work on fermentation [2].

2 Materials and methods

2.1 Airlift reactors as EC–EF cells

In a precursor study, Chisti et al. [16] developed a relationship between the induced overall liquid velocity and gas holdup difference between riser (εr) and downcomer (εd). This relationship is based on energy balance. The geometry and pressure drop were taking into account to give the following equation:

$$\mathop U\nolimits_{Lr} = \mathop {\left[ {\frac{{2g\mathop h\nolimits_{D} \left( {\mathop \varepsilon \nolimits_{r} - \mathop \varepsilon \nolimits_{d} } \right)}}{{\frac{{\mathop K\nolimits_{T} }}{{\mathop {\left( {1 - \mathop \varepsilon \nolimits_{r} } \right)}\nolimits^{2} }} + \mathop {\left( {\frac{{\mathop A\nolimits_{r} }}{{\mathop A\nolimits_{d} }}} \right)}\nolimits^{2} \frac{{\mathop K\nolimits_{B} }}{{\mathop {\left( {1 - \mathop \varepsilon \nolimits_{d} } \right)}\nolimits^{2} }}}}} \right]}\nolimits^{0.5}$$
(1)

KT is the pressure drop coefficient in the riser and the separator sections and KB is the pressure drop coefficient in the downcomer and the junction; Ad: cross-sectional area of the downcomer (m2) and Ar: cross sectional area of the riser (m2); hD is the dispersion height.

Taking into account the mass balance on liquid phase, the superficial liquid velocity in the riser (ULr) is deduced:

$$\mathop U\nolimits_{Lr} = \frac{{\mathop A\nolimits_{d} }}{{\mathop A\nolimits_{r} }}\mathop U\nolimits_{Ld}$$
(2)

The gas can be produced electrochemically by water electrolysis thus producing hydrogen at the cathode. The introduction of the electrodes (cathode and anode) in the riser section induced an overall liquid circulation because of the gas hold-up difference between the riser (εr) and the downcomer (εd) in which εr ≫ εd. The dispersion height and gas holdup in the riser depend on the axial position of electrodes in the riser.

In the case of aluminum electrodes, the reactions taking place at the electrodes are as follow:

At the anode, takes place oxidation:

$${\text{Al}} \to {\text{Al}}^{3 + } + 3{\text{e}}^{ - }$$
(3)

At the cathode, takes place reduction:

$$2{\text{H}}_{2} {\text{O}} + 2{\text{e}}^{ - } \to {\text{H}}_{2} + 2{\text{OH}}^{ - }$$
(4)

The bubbles of gas are produced in the vicinity of the cathode contributes strongly to the agitation of the medium. The bubble diameter is estimated at around 10–200 μm [19] depending on pH and current intensity.

The formation of metal hydroxides, metal oxy-hydroxides and polymeric hydroxides can destabilize the colloidal suspension, adsorb, neutralize or precipitate the pollutant species dissolved in the liquid. The flocs could be recovered by precipitation, filtration or flotation.

To realize a complete flotation, the hydrodynamic shear forces should be weak in the riser to avoid flock break-up and in the separator to limit break-up and erosion. This condition is reached with relatively low ULd values [19]. The design of an airlift reactor is mainly focused on hydrodynamics which is a key factor that plays a very important role in its performance.

2.2 Correspondence between current intensity, gas flow-rate and bubble velocity

The number and the size of bubbles are influenced by the current density [28]. Bubble behavior affects in turn the hydrodynamic regime of the separation process [29, 30].

Hydrodynamic parameters involve particularly the study of bubble rise velocity, bubble diameter effects.

According to the water electrolysis:

$${\text{H}}_{2} {\text{O}} + {\text{e}}^{ - } \to 1 /2{\text{H}}_{2} + {\text{OH}}^{ - }$$
(5)

One mole of H2 corresponds to 2 mol of electrons. Faraday’s law can therefore be used to deduce the mass flow-rate as follows:

$${\dot{m}_{G} = \frac{{I\,M_{{H_{2} }} }}{2\,F}}$$
(6)

where I(A), \({\dot{m}_{G} }\) (kg/s), MH2(g/mole) denotes respectively the current intensity, the mass flow rate of gas and the gas molar mass. F = 96,485 (C/mol) is the Faraday’s constant and MG = 2.016 g/mol the molar mass of hydrogen gas (H2). The volume flow-rate can be expressed as follow:

$${Q_{G} = \frac{{\dot{m}_{G} }}{{\rho_{{H_{2} }} }}}$$
(7)

ρH2 = 0.085 kg/m3 is the hydrogen density.

The superficial gas velocity (Ugr) is obtained by dividing the volume flow-rate to the surface of the electrode (Se):

$${U_{gr} = U_{g} = \frac{{Q_{G} }}{{S_{e} }}}$$
(8)

The relationship between superficial gas velocity and current intensity can be deduced:

$$\mathop U\nolimits_{g} = \frac{{I\,\mathop M\nolimits_{H2} \,}}{{2\,F\,\mathop \rho \nolimits_{G} \,\mathop S\nolimits_{e} }}$$
(9)

The surface of the electrode (Se) is chosen to be equal to 175 cm2 as in the previous paper [19].

Table 1 shows the correspondence between the superficial gas velocity and the current intensity.

Table 1 correspondence current intensity—superficial gas velocity
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2.3 Hydrodynamic model

The CFD model was developed and validated in our previous works [14, 31]. The flow model, gas holdup equations are based on Navier–Stokes equations for the Eulerian–Eulerian multiphase approach (Table 2). In the present work, all the inter-phase forces (drag, lift and virtual masse) are considered (see Table 2).

Table 2 Governing equations and interphase forces used in this work
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In order to take into account turbulence of the underlying flow, the same equation of k − ε [31] is used. Indeed, we are working with a modified high Reynolds number. The drag of the bubbles motion causes a small eddies and minimum turbulent channel which affect the overall velocity distribution. In our case the presence of the walls (reactor walls and anode/cathode walls) where the viscous effects dominate because of the viscous sub-layer so the turbulence in these regions became more important and the velocity distribution more affected. After some distance, small chaotic oscillations begin to develop in the boundary layer and the flow begins to transition to turbulence, eventually becoming fully turbulent. The most k − ω based models are low Reynolds number models and a standard k − ε model and other commonly encountered k − ε models are not low Reynolds number models. However, in our case, it supplemented with so-called damping functions that give the correct limiting behavior. They are then known as modified Reynolds number k- model which often gives a very accurate description at low Reynolds number and accurate description of the boundary layer because it improves near-wall stability

2.4 CFD package

In the present work, the Open source Field Operation and Manipulation (Open FOAM) C++ libraries are used as a free source. In previous works, this aspect is deeply described [14, 31].

The governing equations (Navier–Stokes equations) are discretized by the finite volume method that is based on the Weller approach [32]. This approach has proved very stable in both transient and steady state modes.

The non-linearity of the Navier–Stokes equations requires efficient methods. Indeed, these equations are quasi-nonlinear and weakly coupled. Nonlinearity is overcome by an iterative calculation. By choosing a stable numerical scheme, the errors introduced by the initial solution are damped and the procedure will converge easily to an acceptable final solution. The coupling problem is manifested by the appearance of velocity and pressure variables. The pressure gradient which appears as a source term in these equations plays the role of the driving force of flow.

One of the algorithms used is the SIMPLE algorithm. An initial pressure field is assumed and is injected into the momentum equation. The system is resolved to find an intermediate velocity field. The continuity equation is transformed to become a pressure correction equation. It is resolved to find a pressure correction that will reinject a new pressure in momentum equations. The cycle is repeated as many times as necessary until a zero pressure correction is obtained, a sign of the convergence of the algorithm.

PISO (Pressure Implicit with Splitting of Operators) is a procedure proposed by ISSA [33]. It is used to couple pressure to velocity via the continuity equation. The pressure equation is derived into a semi-discretized form of the momentum equation. This algorithm was initially developed for non-iterative calculations of non-stationary and compressible flows. It has been successfully adapted for iterative calculations of stationary problems. The algorithm is similar to SIMPLE with an improvement that consists of making two successive corrections instead of one [33].

The sequence of the solution procedure is summarized in Fig. 1. For the non-orhogonal mesh, Open Foam proposes an iterative procedure to correct the face fluxes by separating the flux into two parts known as the orthogonal and non-orthogonal distribution [34].

Fig. 1
figure 1

Numerical solution procedure

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The PBE was implemented in Open Foam as new solver and adapted to Two Phase Euler Foam handles incompressible two-phase flows [35] with the modification of k− ε equation as explained previously in our last paper [36]. In this work, the PBE equations are not solved as we need to study the influence and the effect of bubble diameters on the behavior of the circulation and the result comparing with the experimental data.

2.5 Geometries: First configuration and Proposed design for a new configuration

The first configuration consists of an external-loop airlift reactor. The reactor geometry is illustrated in Fig. 2. The detail was described in a previous work [19]. Figure 3 shows a photo of the reactor in which the electrocoagulation/electroflotation was performed to eliminate real textile effluent. The pollutant is recovered in floated sludge.

Fig. 2
figure 2

External loop airlift reactor (20 L) [6]. (1: downcomer section; 2: riser section; 3: conductivity probes; 4: conductimeter; 5: analog output/input terminal panel (UEIAC-1585-1); 6: 50-way ribbon cable kit; 7: data acquisition system; 8: electrodes; 9: separator; 10: electrochemically-generated bubbles)

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Fig. 3
figure 3

Photos showing decolourization by EC in external loop airlift reactor of real textile dye

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The simulation is carried out for this configuration to find the experimental results when this reactor is functioning with continuous mode. Thus, the residence time distribution study [9] revealed the quantity of the liquid that quits the reactor without reacting, is almost the same (between 70 and 90%).

A new reactor was then designed as a new external loop airlift reactor to overcome the problems encountered in the first configuration. The detail of the reactor is shown in Fig. 4. The design of this reactor was based on the simulations results as we can see later (Sect. 3.2).

Fig. 4
figure 4

Improved External-loop airlift reactor: 1: level of the effluent in the separator, 2: riser section, 3: downcomer section, 4: connecting column, 5: separation section, 6: slot, 7: electrodes, 8: generator, 9: H1: electrodes position, 10: scraping system

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2.6 Numerical solution and boundary conditions

In this paper, a QUICK scheme with deferred-correction method for the convection term discretization of gas holdup is adopted. The QUICK scheme takes the second order derivative into account, but ignores the third order derivative. The second order upwind scheme is adopted for all the other equations. The calculations are carried out for the two geometries presented above.

The problem of the precision of the results is related to the number of cells in the mesh which should be sufficient for an exact description of the phenomenon. On the other hand an excessive number of cells can cause a divergence of the program. In our case, tests were made to study the influence of the mesh on the numerical results. The adequate mesh was found ensuring convergence and accuracy. Concerning the numerical stability, a time step of 10−4 s was the appropriate time.

The boundary conditions are of considerable importance. They determine the actual profile of the flow.

A no-slip boundary is used and the standard wall functions are adopted for all phases at the solid wall [37].

At the inlet, the Dirichlet condition is adopted to the gas in which the gas fraction is equal to 1.

The inlet conditions for the turbulent kinetic energy and the dissipation rate are estimated by:

$$k = \frac{3}{2}\mathop {\left( {\mathop U\nolimits_{inlet} I} \right)}\nolimits^{\,2}$$
(10)

where I is the turbulent intensity.

$${\varepsilon = C_{\mu }^{0.75} \frac{{k^{1.5} }}{0.07L}}$$
(11)

where L is the characteristic length of the apparatus.

The Neuman condition was used for outflow boundary of the rector. The free surface is considered as free for liquid phase with zero normal velocity as suggested by Mudde and Van Den Akker [38].

The imposed superficial gas velocity, in the electrodes cell area, is presented on Table 1. The electrodes faces mesh use one as hydrogen volume fraction and zero for the liquid value.

The adopted method is dictated by the structure of the matrix. For symmetric matrices, the preconditioned conjugate gradient (ICCG) is adopted while for asymmetric matrices the preconditioned bi-conjugated gradient (PBICG) is the most appropriate [34]. ICCG with tolerance equal to 10−8 is used for pressure, while PBICG is used for velocities, dissipation, turbulent kinetic energy and phase.

2.7 Materials for experiments

Absorbance was measured every 10 min to follow the efficiency evolution. Colour removal efficiencies (YCOL) was expressed as follows:

$$\mathop Y\nolimits_{COL} (\% ) = \frac{{\mathop A\nolimits_{0} - A}}{{\mathop A\nolimits_{0} }}$$
(12)

In Eq. (12), A0 and A are respectively the initial absorbance and the absorbance at a certain time.

Spectrophotometer (UV–Visible spectrometer Helios γ, type Gamma) was used to determine dye concentration by its absorbance characteristics in the UV–vis range (200–800 nm). The maximum intensity of absorbance was obtained by using the wavelengths λmax = 436, 525 and 620 nm covering the absorbance of various types constituting the mixture. Initial pH measured with a pH meter (PHM 220 Radiometer analytical). Conductivity was determined by a conductivity meter (“Cyberscan 510-cond” “Conductimeter Consort K 120”).

Real wastewaters were provided from ITEX, Textile industry (Casablanca, Morocco). It consists of dispersee dye containing Yellow Terasil 4G, Red disperse and Blue Terasil 150%. The working temperature is 20 °C.

3 Results and discussion

3.1 First configuration (O.D.A) with continuous mode

The first step of the computation lies in the development of a mesh which ensures the convergence of the numerical procedure, and which is able to capture the different variations of the parameters in the computational domain. In addition, a mesh must ensure the independence of the solution of the refinement (increase of the number of meshes). In our study several meshes were tested, with different meshing scheme, proposed by Gambit software, near the walls. Figure 5 represents the study of three different meshes (mesh 1 = 250,514 elements, mesh 2 = 180,634 elements and mesh 3 = 154,605 elements). It shows that the numerical results of the induced liquid velocity versus the electrodes position are independent of the obtained results comparing with experiment results.

Fig. 5
figure 5

Influence of the mesh on the numerical results

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The mesh was chosen to ensure a minimum number of cells and provides efficiency in computing with good accuracy, and a short time of calculation. The mesh is fine in the zone of high velocity gradients (near the electrodes and walls). Figures 6 and 7 show a view of the used mesh which consisting of 154,605 elements. For the non-orthogonal cells, we use an iterative procedure to correct the face fluxes by separating the flux into two parts known as the orthogonal et non orthogonal distributions as proposed by Jasak [22].

Fig. 6
figure 6

Initial geometry and corresponding mesh for the O.D.A reactor with different outlet positions

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Fig. 7
figure 7

The mesh of O.D.A geometry with two outlet positions

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The previous study [12] revealed that a large amount of effluent to be treated exit directly to the outlet without reacting in the riser section. The goal of numerical simulation is to verify this result and find possible solutions.

To test the influence of the position of the liquid outlet on the hydrodynamic, two positions were tested as indicated in Figs. 6 and 7. An example of simulation is undertaken, for a current intensity with 2A value which induced a superficial gas velocity of 14.10−4 cm/s. The input liquid flow rate was chosen to be equal to 2 L/min.

The first position corresponds to the real position of the liquid outlet [12] of the reactor functioning with continuous mode. The second position corresponds to an upper position of the liquid outlet. Figure 8 shows the velocity vectors of the liquid medium phase.

Fig. 8
figure 8

Liquid velocity field for: O.D.A reactor: Lower position of the outlet: O.D.A reactor: Upper position of the outlet

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As observed in Fig. 8 for both outlet positions, a large amount of the liquid quits the rectors without reacting in the riser section in both cases. The troubles observed by using the Residence Time Distribution (RTD) [12] analysis were confirmed by the simulation results. The efficiency defined in continuous mode, as the percentage of the main flow that crosses the riser section (participating in electrochemical reaction) is weak in this kind of configuration.

The average liquid velocity in the downcomer, (ULd) was measured using the conductivity tracer technique [12]. Two conductivity probes placed in the downcomer section were used to record the tracer concentration resulting from the injection of 5 mL of a saturated NaCl solution at the top of the downcomer using a data acquisition system. Liquid velocity was estimated using the ratio of the mean transit time between the tracer peaks detected successively by the two electrodes and the distance between the probes.

The simulation results give the same tendency. The quantity of the liquid that quits the reactor without reacting, obtained by averaging the liquid velocities in junction section is almost the same (between 70 and 90%) as confirmed by the experimental results.

The average of velocity is averaged according to the section area:

$$\mathop U\nolimits_{Ld} = \frac{1}{{\mathop A\nolimits_{d} }}\int {u\,(r,\theta )\,r\,dr\,d\theta }$$
(13)

The flow pattern exhibits a turbulence flow as indicated by the vector representation of the liquid velocity field.

The flow pattern is the same for both cases in different compartment of the external loop airlift reactor (Fig. 8a, b). The calculation of the average liquid velocity in downcomer section shows slight different values for the two cases. Thus, the average liquid velocity reaches 1.125 cm/s in the case of upper outlet, whereas it reaches 1.28 cm/s in the case of lower position. The average liquid velocity decreases as the outlet moves upwards.

The experimental values of liquid velocity are around 1 cm/s in the case of continuous mode as found in the RTD analysis [12]. The velocity fields show relatively mild recirculation eddies.

The simulation indicated also that the gas is present only on the riser and disengagement sections as shown on Fig. 9. The same results are obtained for the two cases.

Fig. 9
figure 9

Distribution of the gas volume fraction in the O D A reactor

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3.2 Second configuration (N.D.A) with batch mode

The external loop airlift reactor has an operating volume of 20 L. As in the first configuration, the study of different meshes was studied. It shows also that the numerical results of the induced liquid velocity versus the electrodes position are independent of the obtained results comparing with experiment results. The computational domain for the model of the reactor is shown in Fig. 10. It contains 492,171 cells with 165,571 hexahedral cells, 170,058 tetrahedral cells, and 156,542 pyramidal cells.

Fig. 10
figure 10

Structured mesh: 165,571 cells in N D A reactor

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The purpose of the study of the functioning of this batch reactor is the evaluation of the impact of the position of the electrodes and bubble diameter on the superficial liquid velocity and gas holdups for a given current intensity.

3.2.1 Influence of electrodes position

The new external-loop airlift reactor is used as a batch reactor. The current intensity is 2A corresponding to 14.10−4 cm/s as a superficial gas velocity. The bubble diameter is chosen to be 0.2 mm. Three positions of electrodes were chosen: lower, middle and upper positions as indicated in Table 3 indicates the exact position.

Table 3 z Coordinates of electrodes (New design of external airlift)
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Figure 11 shows that the values obtained by simulation are very close to the experiments results showing that the average liquid velocity increases when the electrodes are positioned in depth. This result is in perfect accordance with the prevision of Eq. (1) in which the liquid velocity increased with the dispersion height hD i.e. when the electrodes occupy lower position in the downcomer.

Fig. 11
figure 11

Evolution of the mean liquid velocity following the electrodes position

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Figure 12 shows liquid velocity vectors. A maximum of liquid velocity is obtained near the electrodes. Contrary to the average value, the maximum value of the liquid velocity is almost the same for the three positions of the electrodes (31.1 cm/s). The maximum value is between five to ten times the values of the average liquid velocity depending on the electrodes position (Table 4).

Fig. 12
figure 12

Liquid velocity vector corresponding to three positions of electrodes

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Table 4 Average and maximum liquid velocities according to positions (I = 2A)
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The distribution of liquid velocity field is more presented in term of magnitude as indicated in Fig. 13.

Fig. 13
figure 13

Liquid velocity magnitude in three positions of electrodes

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The average liquid velocity is relatively high in the area located just above the electrodes, which is due to the upward movement of the hydrogen bubbles leading to high liquid velocity in this region. The liquid velocity is also relatively high in the case of the low position of the electrodes as indicated by the presence of green color.

Figure 14 represents a photo showing the electroflotation of the sludge. This indicates that a complete flotation of sludge is obtained allowing a high value of removal efficiency (detail in Sect. 3.3).

Fig. 14
figure 14

Photo showing the designed second configuration: the recoverd sludge by electroflotation

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The total flotation of sludge is obtained for the three positions of the electrodes, contrarily to the first configuration. Thus, the lower position in the first configuration induced an overall liquid circulation that produced an erosion of flotation layer in the separator [19]. This constitutes an improvement in reactor performance in the case of batch mode.

The distribution of the gas is presented in Fig. 15. In the steady state, the gas is localized only in the free surface of the reactor. The bubbles do not reach the downcomer which is an advantage for the effluent treatment. Thus, the gas bubbles generated should be used only to float the floc. The average gas holdup is then computed for different values of superficial gas velocities corresponding to different values of current intensities. Table 5 shows that the computations by CFD predict gas holdup values between 0.0065 (0.65%) and 0.072 (7.2%) when current intensity varied between 1 and 25 A. The corresponding values of average liquid velocity are also represented in Table 5.

Fig. 15
figure 15

Gas volume fraction for three positions of electrodes

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Table 5 Predicted gas hold-up (εg) and the corresponding average liquid velocity (Ul) versus current intensity (I) or current density (j)
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Figure 16 shows the evolution of average gas holdup versus current intensity in the case of the middle position of the electrodes. All the next results concerned the middle position of the electrodes. The results obtained for the superficial liquid velocity should be correlated to the average gas holdup. Indeed, the difference in gas holdups in the two compartments (εd and εr) is a driving force which induces the recirculation of the liquid.

Fig. 16
figure 16

Gas holdup versus superficial gas velocity (Ug = Qg/Se)

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The model based on Chisti’s equation [3, 16] can be developed to correlate ULd, hD and j (current density = I/Se). When gas hold-up in the downcomer is negligible, the energy balance model reduces to the following expression:

$${\text{U}}_{\text{Ld}} = {\text{K}}\left( {\upvarepsilon_{\text{r}}\, {\text{h}}_{\text{D}} } \right)^{0.5}$$
(14)

εd is equal to 0 as suggested by the simulations results and K is a constant.

On the basis of empirical correlations from the literature [27], εr should vary with superficial gas velocity UGr as U αGr with α between 0.5 and 0.6 in external-loop airlift reactors. As gas flow rate is proportional to current, this gives ULd ~ Iα/2.

Experimental data of Uld (or Ul) has to be presented versus the current density (j). Figure 17 shows the values of the superficial liquid velocity as a function of the current density (j = I/Se).

Fig. 17
figure 17

Superficial liquid velocity versus current density

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The values obtained show that the relationship between Ul and j follows the power law with a power coefficient of 0.35 as almost theoretically expected (0.3). CFD provides results which are supported both by the theoretical prediction of the balance of the dissipated energy balance gas in the reactor and the experimental results. The good agreement between predictions, experiments and simulations are illustrated by the fitted parameter α = 0.35 that is close to the literature [22] in which α is between 0.25 and 0.3.

3.2.2 Influence of bubbles diameter

At this stage, the middle position (z = 51.6 cm) is chosen to study the influence of bubble diameter. The superficial gas velocity is kept constant. The results of the investigation are represented in Table 6. It shows that the average liquid velocity increases strongly with the decreases of the bubbles diameter. When bubble diameter decreases from 0.25 to 0.05 mm the mean velocity of the liquid increases about 7%. However, numerical instabilities were observed in the case of a bubble diameter value lower than 0.05 mm (Table 6).The experiments in bubble column and airlifit reactor confirmed the simulations results. Thus, the increase in bubble diameter induces an increase in bubble rise velocity allowing a decrease in gas holdup in the riser. The high values of bubble diameter simulate the case of high viscosity in which coalescence phenomena of bubbles occur. It’s known that the high viscosity increases drag forces allowing to more coalescence. Consequently the bubble rise velocity increases leading to a reduction of gas holdup. Consequently the superficial liquid velocity decreases as gas holdup is the driving force for the liquid recirculation.

Table 6 Influence of bubbles diameter on the average liquid velocity
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3.3 Experimental application

Treatment of real effluent of disperse dyes was undertaken. Disperse dyes are used on the polyester. Discharges of disperse dyes have characteristics that do not facilitate treatment by electroflotation. These unfavorable parameters are acidity and conductivity of the medium (pH = 3.8 and low conductivity, κ = 1.1 mS/cm). Hence, the need to adjust these two parameters before treatment is crucial. This adjustment can make the conductive solution and a level of acidity that helps make favorable conditions for EC/EF. As shown in Fig. 18, the asymptotic value of decolourization is reached after an electrolysis time of 45 min. It reached 85% for wavelength corresponding to 436 nm whereas the decolourisation efficiency reached 75% in the case of a wavelength corresponding to 525 nm and 620 nm. This indicates that kinetics of decolourisation differs according to the component containing in the dye. This result is obtained with a total recovery of the sludge by electroflotation.

Fig. 18
figure 18

Evolution of the decolourization efficiency of real textile effluent versus time, pHi (initial) = 6.9, κ = 6.5 mS/cm, j = 31.4 mA/cm2)

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4 Conclusion

The first challenge of the present paper is the use of CFD to design an innovative reactor to promote a high performance allowing accomplishing the desired efficiency.

The numerical investigation had revealed the importance of current intensity and electrodes position to control the liquid velocity.

The simulations of the first configuration with continuous mode confirmed the experimental results in which only 10–30% of the main flow crosses the riser section.

The influence of outlet position concerns only the magnitude of liquid velocity: The average liquid velocity decreases slightly as the outlet moves upwards. However, continuous mode results are not improved by the output position.

Concerning the new design of external airlift reactor the main results of the simulations are:

  • Average liquid velocity increases when the electrodes are positioned in depth (lower position in the riser) but this position ensures a total flotation of sludge contrarily to the first configuration.

  • Average liquid velocity decreases when the bubble diameter increases.

As the filtration process is not needed the cost of electrocoagulation/electroflotation can be lowered.

Preliminary experiments for continuous mode are encouraging (not presented in this article) because 80% of the main flow crosses the riser section (presence of the electrodes) instead of 30% of the first configuration. The study of the inlet and the outlet of the reactor are in course to obtain 100% of the main flow that crosses the riser. This aspect and the use of population balance equations to take into account the bubbles diameter distribution will be the subject of the next paper.