# Parametric Mathematical Model of the Electrochemical Degradation of 2-Chlorophenol in a Flow-by Reactor under Batch Recirculation Mode

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

^{3}) was electrochemically treated in a flow-by reactor equipped with two boron-doped diamond electrodes (BDD) under batch recirculation mode for a period for 4 h, a current density of 0.14 A/cm

^{2}, a volumetric flow rate of 1 L/min, and pH = 7.3. In this work, a parametric mathematical model of the degradation efficiency of 2-CP was developed using an axial dispersion model and a continuous stirred tank for the flow-by reactor (FBR), which was constructed using a shell mass balance considering the dispersion and convection terms and the reservoir tank (CST), which was constructed using a mass balance of 2-CP. The parametric mathematic model of the electrochemical degradation of 2-chlorophenol was numerically resolved by employing the software package COMSOL Multiphysics

^{®}V. 5.3, where a mass transfer equation for diluted species and a global differential equation represents the FBR and CST, respectively. The results indicate that the parametric mathematical model proposed in this research fits the experimental results, and this is supported by the index performance values such as the determination coefficient (R

^{2}= 0.9831), the mean square error (MSE = 0.0307), and the reduced root-mean-square error (RMSE = 0.1754). Moreover, the degradation efficiency of 2-CP estimated by the proposed model achieves 99.06%, whereas the experimental degradation efficiency reached 99.99%, a comparative error of 0.93%. This corroborates the predictive ability of the developed mathematical model and the effectiveness of the employed electrooxidation process. Finally, a 0.143 USD/L total operating cost for the electrochemical plant was estimated.

## 1. Introduction

_{2}, SnO

_{2}, DSA (e.g., Ta

_{2}O

_{5}-IrO

_{2}and Nb

_{2}O

_{5}-IrO2), and BDD have been employed in many electrooxidation processes. The Pt electrode is expensive and has low oxidation efficiency, the PbO

_{2}electrode is susceptible to corrosion and dangerous for health and the environment, the SnO

_{2}electrode is cheap and has a low degradation rate, the DSA electrodes are not stable enough over an extended period, and the boron-doped diamond (BDD) electrode is stable for a long time and has high degradation efficiencies in comparison with all the electrodes mentioned above [9]. The high degradation efficiency of BDD electrodes is due to the generation of hydroxyl radicals (OH

^{●}) [10] that proceeds on the anode surface according to Equation (1) [11]. Additionally, OH

^{●}(oxidation potential of 2.8 V) is second on the list of the twelve principal oxidant species [12],

Process | Reactor Configuration | Environmental Conditions | Solution Via | Ref. |
---|---|---|---|---|

Electrooxidation of 2-CP | Hydroxyl radicals, volumetric flow rate of 1.0 L/min, pH of 3, volume treated of 2.5 L, current density of 140 mA/cm^{2}, 0.1 mol/L of Na_{2}SO_{4}, D_{ax} of 0.0005 m^{2}/s, temperature of 25 °C | Software COMSOL Multiphysics^{®} 5.3/Win64 interacting with the software MATLAB^{®} 2017a | [18] | |

Electrochemical oxidation of crystal violet dye | Hydroxyl radical, pH of 6.4, currents density of 10 mA/cm^{2}, 0.25 mol/L of Na_{2}SO_{4}, D_{ax} of 0.00042 m^{2}/s, volumetric flow rate of 3 L/min, temperature of 25 °C | Software FlexPDE academic version 6.36/W32 | [22] | |

Electro-peroxone degradation of orange reactive 16-dye | Hydrogen peroxide, pH of 3, volumetric flow rate of 1.135 L/min, D_{ax} of 0.00153 m^{2}/s, volume treated of 2 L, current density of 10 mA/cm^{2}, 0.05 mol/L of Na_{2}SO_{4}, and 0.5 L/min of O_{3}, temperature of 25 °C | Software COMSOL Multiphysics^{®} 5.3/Win64 | [20] | |

Sulfamethoxazole degradation | Active chlorine (HOCl), current density of 10 mA/cm^{2}, 0.02 mol/L of NaCl, volumetric flow rate of 5 L/min, temperature of 25 °C | Software COMSOL Multiphysics^{®} 5.2a commercial | [19] | |

Indigo carmine dye | Active chlorine (HOCl), 0.05 mol/L of NaCl, volumetric flow rate of 0.9 L/min, current density of 200 mA/cm^{2}, temperature of 25 °C | Software FlexPDE professional version 6.5/W64 3D | [23] | |

Reactive Black 5 | Active chlorine (HOCl), 0.05 mol/L of NaCl, volumetric flow rate of 0.8 L/min, current density of 200 mA/cm^{2}, temperature of 25 °C | Software COMSOL Mul-tiphysics^{®} | [14] |

^{®}V. 5.3, where the complete electrochemical plant is represented by a mass transfer equation for diluted species for the FBR and a global differential equation (EDOs and EDAs) for the CST that is coupled to the FBR. Additionally, this research intends to generate a better comprehension of the complete mathematical models of electrochemical plant treatments for wastewater that contains persistent organic compounds (e.g., 2-chlorophenol) through experimental validation.

## 2. Materials and Methods

#### 2.1. System Description

_{T}of 3.5 L, and that is equipped with two BDD electrodes ($\mathcal{l}=20\mathrm{cm}$). More detail about the FBR can be found in the author’s previous work [24]. The experimental system for the electrochemical degradation of 2-CP was comprised an FBR coupled to a polycarbonate reservoir tank (CST). The synthetic wastewater was supplied into the FBR by using a magnetic pump at 300 rpm, where the volumetric flow rate was controlled using a glass rotameter, the pipeline, valves, and connectors were made from PVC, and a power supply was employed to energize the electrodes. Figure 1 shows the flow chart of the complete electrochemical plant.

#### 2.2. RTD Analysis

^{®}Conductivity Probe every 0.33 s until the conductivity measurement was zero. Additionally, the RTD experiment was performed in triplicate and the calibration curve between the concentration (mg/L) and conductivity (µS/cm) was obtained.

#### 2.3. Electrolysis of 2-CP

^{2}, and a reaction time of 4 h.

#### 2.4. Parametric Mathematical Model and Solution Method

- The reservoir tank was simulated using a CST.
- The boundaries of the FBR inlet and outlet were both of the closed–closed vessel type [17].
- The electrooxidation of 2-CP took place on the electrode surface at a constant current.
- The hydroxyl radicals were uniformly formed on the surface of the electrode [26]. Hence, the reaction rate of 2-CP was modeled using a pseudo-first order.
- The FBR was isothermally operated.
- The electrooxidation of 2-CP was limited by the mass transfer of 2-CP from the bulk to the electrode surface [18].

#### 2.4.1. Axial Dispersion Mathematical Model for the Flow-By Reactor

_{2-CP}= 0 or $-{D}_{ax}\frac{{\partial}^{2}{C}_{\mathrm{2-}CP}^{FBR}}{\partial {x}^{2}}=0$). In addition, Equation (5) indicates the initial condition, which represents the initial concentration of the 2-CP (C

_{2-CP,0}= 1 mol/m

^{3}) at t = 0.

^{3}) is the local concentration of 2-CP at any time that feeds the CST, D

_{ax}is the axial dispersion coefficient (m

^{2}/s), x is the FBR length (cm), $u\left(\partial {C}_{\mathrm{2-}CP}^{FBR}/\partial x\right)$ is the convection transport term, ${D}_{ax}\left({\partial}^{2}{C}_{\mathrm{2-}CP}^{FBR}/\partial {x}^{2}\right)$ is the dispersion transport term, r

_{2-CP}is the reaction rate of 2-CP (mol/h), and k

_{app}is the apparent first-order reaction rate constant (1/h).

_{0}of 7.3, a current density of 0.14 A/cm

^{2}, and a volumetric flow rate of 1.0 L/min within 4 h of electrolysis time. Additionally, the axial dispersion coefficient (D

_{ax}) was calculated according to [24]. The D

_{ax}was computed when the FBR operates at a linear velocity of 0.0947 m/s, a volumetric flowrate of 1 L/min, a Reynolds number of 21,208.73, and a Peclet number of 39.93.

#### 2.4.2. Continuous Stirred Tank (CST) Model for the Reservoir Tank

^{3}) is the outlet concentration of 2-CP from the CST that feeds the FBR, V

_{T}is the reservoir tank volume (L), Q is the volumetric flow rate (L/min), and t is the electrolysis time (min).

#### 2.4.3. Numerical Solution Approach

^{®}5.3/win64, where the mass balance of the FBR (Equation (2)) was represented in the software by the mass transfer equation for diluted species and the mass balance of the CST (Equation (7)) was represented in the commercial software by a global differential equation (EDOs and EDAs). The numerical method to resolve the partial differential equation and the ordinary differential equation was the finite element, where the simulation mesh domain was controlled by the physics, the length of the elements was marked as normal, and the desired accuracy was 1 × 10

^{−5}.

^{®}Core i5 processor at 3.2 GHz with four nucleus and 16 GB of RAM.

_{2-CP}

_{, 0}(mol/m

^{3}) is the outlet concentration of 2-CP, and ${\left({C}_{\mathrm{2-}CP}\right|}_{t}$ (mol/m

^{3}) is the concentration of the 2-CP at any time.

#### 2.4.4. Performance of the Model

#### 2.4.5. Energy Balance and Total Operating Cost

^{2}), j is the applied current density (A/cm

^{2}), t is the reaction time (h), E

_{lectrodes}is the electricity used by the electrodes (kW h), E

_{pump,flow}is the electricity used by the recirculation pump, E

_{pump,heat}exchanger is the electricity used by the heat exchanger pump, P

_{n}is the supplier catalog nominal power (0.198 kW) for the recirculation pump, P

_{m}is the supplier catalog nominal power (0.123 kW) for the heat interchanger pump, Cost is the total operating cost (USD), α is the electricity price (0.046 USD/kW h, based on 1 USD/MXN 18.32), β is the price of the electrolyte (0.8 USD/kg), and m

_{electrolyte}is the consumed electrolyte mass (kg).

## 3. Results and Discussions

#### 3.1. ADM Validation

^{2}) of 0.9925.

^{®}5.3/win64.

^{2}is standard deviation, and t

_{m}is mean residence time in the FBR.

_{m}= 12.23 s) was computed using Equation (18) and the standard deviation (σ

^{2}= 7.29 s

^{2}) was computed by applying Equations (19) and (20) according to [31].

_{2-CP}= 0) can be used to model the hydrodynamics of the FBR.

_{a}

_{x}= 5 cm

^{2}/s) was determined by using Equation (21) in accordance with reference [31], at a volumetric flow rate (Q) of 1 L/min, a reactor longitude (l) of 20 cm, and a linear velocity (u) of 9.5 cm/s. The computed axial dispersion coefficient demonstrates a low deviation from the ideal plug-flow pattern (D

_{ax}= 0 cm

^{2}/s) since the value is less than 50 cm

^{2}/s, which agrees with that given in reference [17].

#### 3.2. Kinetic Reaction of 2-CP

^{●}), which are produced on the BDD anode surface via the oxidation of the water as reported in reference [11]. Complete data from the UHPLC analysis can be consulted in the Supplementary Material.

_{app}= 1.224 h

^{−1}) was determined by fitting a linear regression of Ln (C

_{2-CP}/C

_{2-CP, 0}) versus time with a determination coefficient (R

^{2}) of 0.9850 based on the integral method according to reference [32].

#### 3.3. Modeling of the Complete Degradation Efficiency of 2-CP

^{●}generated on the BDD anode surface is constant and sufficient to oxidize the 2-CP. Moreover, the parametric mathematical model proposed here predicts the concentration abatement of 2-CP due to the interaction of 2-CP with OH

^{●}(according to Equation (1)) in favor of an instantaneous reaction on the BDD anode. Upon this, the 2-CP concentration depletes rapidly to zero because the reaction rate is controlled by mass transfer. Hence, the 2-CP from the bulk solution goes rapidly to the BDD anode surface according to [22]. Moreover, as the mass transport resistance from the bulk solution towards the BDD anode surface here is low, a small diffusion layer thickness at the electrode interface is present [33,34].

^{2}) is close to 1.0, which is cataloged as an excellent model fitting since R

^{2}is greater than 0.9 [35]. Additionally, the reduced root-mean-square error value (RMSE of 0.174) is close to 0, indicating that the ADM model agrees very well with the experimental data [36]. The value of the mean square error (MSE = 0.0307) indicates that the degradation efficiency of 2-CP predicted by the ADM model fits very well with the experimental data of the degradation efficiency since the MSE value is close to 0 [37,38].

#### 3.4. Total Operating Cost

^{3}) used here, this work is highlights as a suitable electrochemical plant for the treatment of wastewater containing 2-CP with a treated volume of 2.5 L.

#### 3.5. Future Works

## 4. Conclusions

^{2}, MSE, and RMSE) classify the data prediction as very good model fitting concerning the experimental data. Additionally, the numerical solution strategy followed in this research to solve a partial differential equation (PDE) with an ordinary differential equation (ODE) was successful because the computing time solution was very short (8 s). Furthermore, the model established here is reliable for future electrochemical treatments when the electrochemical process is limited by mass transport and for scaling-up the same flow-by reactor.

## Supplementary Materials

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

**Figure A1.**(

**a**) Volume element of the FBR for shell mass balance. (

**b**) Diagram of the CST for macroscopic mass balance.

#### Appendix A.1. Axial Dispersion Mathematical Model for the Flow-By Reactor

_{ax}is constant, Equation (A3) reduces to:

#### Appendix A.2. Continuous Stirred Tank (CST) Model for the Reservoir Tank

_{T}is constant and the volumetric flow rates of the inflow and outflow streams are the same $\left({Q}_{inlet}={Q}_{outlet}=Q\right)$, Equation (A10) reduces to:

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**Figure 2.**Tracer concentration curve obtained in the FBR via the ADM model at a flow rate of 1.0 L/min.

**Figure 3.**Kinetic reaction model of the electrochemical degradation of 2-CP at a flow rate of 1.0 L/min, an initial pH of 7.3, a current density of 0.14 A/cm

^{2}, and a temperature of 25 °C.

**Figure 4.**(

**a**) Degradation efficiency of 2-CP. (

**b**) Parity diagram of the degradation efficiency of 2-CP.

Index Performance | Value | Remark |
---|---|---|

R^{2} | 0.9831 | Excellent model fit |

MSE | 0.0307 | Very good model fit |

RMSE | 0.1754 | Very good model fit |

Current Density (mA/cm ^{2}) | V_{treated}(L) | A (cm ^{2}) | C (mol/m ^{3}) | Electrodes | η (%) | Reference | ||
---|---|---|---|---|---|---|---|---|

Anode | Cathode | Exp. | Pred. | |||||

140.0 | 2.50 | 32 | 1.00 | BDD | BDD | 99.99 | 99.06 | [This work] |

100.0 | 0.025 | 2 | 5.00 | Carbon fiber | - | 98.00 | - | [40] |

32.7 | 0.030 | 14 | 10.00 | BDD | - | 83.60 | - | [41] |

90.0 | 1.00 | 70 | 15.56 | BDD | Stainless steel | 100.00 | - | [39] |

16.0 | 0.30 | - | 4.70 | Ti/SnO_{2} | - | 50.00 | - | [42] |

33.0 | 0.07 | 9 | 1.56 | Graphite felts | Pt | 58.91 | - | [43] |

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## Share and Cite

**MDPI and ACS Style**

Regalado-Méndez, A.; Ramos-Hernández, G.; Natividad, R.; Cordero, M.E.; Zárate, L.; Robles-Gómez, E.E.; Pérez-Pastenes, H.; Peralta-Reyes, E.
Parametric Mathematical Model of the Electrochemical Degradation of 2-Chlorophenol in a Flow-by Reactor under Batch Recirculation Mode. *Water* **2023**, *15*, 4276.
https://doi.org/10.3390/w15244276

**AMA Style**

Regalado-Méndez A, Ramos-Hernández G, Natividad R, Cordero ME, Zárate L, Robles-Gómez EE, Pérez-Pastenes H, Peralta-Reyes E.
Parametric Mathematical Model of the Electrochemical Degradation of 2-Chlorophenol in a Flow-by Reactor under Batch Recirculation Mode. *Water*. 2023; 15(24):4276.
https://doi.org/10.3390/w15244276

**Chicago/Turabian Style**

Regalado-Méndez, Alejandro, Guadalupe Ramos-Hernández, Reyna Natividad, Mario E. Cordero, Luis Zárate, Edson E. Robles-Gómez, Hugo Pérez-Pastenes, and Ever Peralta-Reyes.
2023. "Parametric Mathematical Model of the Electrochemical Degradation of 2-Chlorophenol in a Flow-by Reactor under Batch Recirculation Mode" *Water* 15, no. 24: 4276.
https://doi.org/10.3390/w15244276