# Quantitative Study of Non-Linear Convection Diffusion Equations for a Rotating-Disc Electrode

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## Abstract

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## 1. Introduction

- This study’s primary goals were to analyze a mathematical model for the reduction of ${H}^{+}$ ions and electrolysis of ${H}_{2}O$ in non-buffered aqueous electrolyte solutions and to investigate how specific parameters affect the e entropy of hydrogen (${H}^{+}$) and hydroxide ($O{H}^{-}$) ions in a rotating-disc electrode (RDE).
- The mathematical model of the convection-diffusion equation for the non-dimensional hydrogen (${H}^{+}$) and hydroxide ($O{H}^{-}$) ion concentrations on a rotating-disc electrode (RDE) has been solved for this problem.
- The behavior of the hydrogen (${H}^{+}$) and hydroxide ($O{H}^{-}$) ion concentrations are studied using the backpropagated Levenberg–Marquardt algorithm (BLMA) and neural networks (NNs).
- The reference data of target solutions were produced by the Runge–Kutta technique and were successfully used in the supervised learning phase of the NNs-BLMA.
- Convergence analysis based on curve fitting, mean-square error, error histograms, and regression analysis by reference data was used to verify the effectiveness of the designed NN-BLMA. The results establish that the suggested method is slick and straightforward, extending to more complex problems.

## 2. Mathematical Formulation of the Problem

- (i)
- At z = 0, the two species become$$at\phantom{\rule{0.277778em}{0ex}}z=0,\phantom{\rule{0.277778em}{0ex}}\frac{d{C}_{{H}^{+}}}{dz}={C}_{{H}^{+}}^{\infty}\phantom{\rule{0.277778em}{0ex}}and\phantom{\rule{0.277778em}{0ex}}\frac{d{C}_{O{H}^{-}}}{dz}=0,$$
- (ii)
- As z →∞, the concentration of ${H}^{+}$ ions (${C}_{{H}^{+}}$) equals the bulk concentration of ${H}^{+}$ ions (${C}_{{H}^{+}}^{\infty}$), and the concentration of $O{H}^{-}$ ions (${C}_{O{H}^{-}}$) approaches zero. That is,$$as\phantom{\rule{0.277778em}{0ex}}z\to \infty ,\phantom{\rule{0.277778em}{0ex}}{C}_{{H}^{+}}={C}_{{H}^{+}}^{\infty}\phantom{\rule{0.277778em}{0ex}}and\phantom{\rule{0.277778em}{0ex}}{C}_{O{H}^{-}}\to 0,$$
- (iii)
- The ${H}^{+}$ and $O{H}^{-}$ concentrations become$${C}_{{H}^{+}}(0,t)={e}^{\eta}{C}_{O{H}^{-}}(0,t),$$$${D}_{{H}^{+}}{\left(\frac{d{C}_{{H}^{+}}}{dz}\right)}_{z=0}=-{D}_{O{H}^{-}}{\left(\frac{d{C}_{O{H}^{-}}}{dz}\right)}_{z=0},$$$$\eta =\frac{F}{RT}(E-{E}^{{0}^{{}^{\prime}}}),$$

- In the first step, a numerical solution is computed using the Runge–Kutta technique of fourth order ($R{k}_{4}$) using Mathematica’s “ND Solve” module to create an initial dataset.
- In the second step, using the “nftool” from the MATLAB package, the BLM algorithm is run with the proper hidden neuron parameters and test data. Additionally, BLM employs the training, testing, and validation process and a reference solution to provide approximations for various nonlinear equation instances. Figure 2 and Figure 3 illustrate the NNs-LM technique using a single neuron model.

## 3. Comparison of Numerical Solutions

## 4. Results and Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Nomenclature

Symbol | Description | Unit |

${C}_{{H}^{+}}$ | ${H}^{+}$ ion concentration | mol cm^{−3} |

${C}_{O{H}^{-}}$ | $O{H}^{-}$ ion concentration | mol cm^{−3} |

${C}_{{H}^{+}}^{\infty}$ | ${H}^{+}$ ion bulk concentration | mol cm^{−3} |

${C}_{O{H}^{-}}^{\infty}$ | $O{H}^{-}$ ion bulk concentration | mol cm^{−3} |

${D}_{{H}^{+}}$ | coefficient of diffusion of ${H}^{+}$ ions | cm^{−2} s^{−1} |

${D}_{O{H}^{-}}$ | coefficient of diffusion of $O{H}^{-}$ ions | cm^{−2} s^{−1} |

$\upsilon $ | kinematic viscosity | cm^{2} s^{−1} |

$\Omega $ | rotation rate | s^{−1} |

${k}_{+3}$ | forward rate coefficient | mol^{−1} s^{−1} cm^{3} |

${k}_{-3}$ | backward rate coefficient | mol s^{−1} cm^{−3} |

${v}_{z}$ = −0.51 ${z}^{2}$ ${\Omega}^{\frac{3}{2}}$ ${v}^{\frac{-1}{2}}$ | velocity | cm s^{−1} |

a = 0.51023 ${v}^{\frac{-1}{2}}$ ${\Omega}^{\frac{3}{2}}$ | parameter | cm^{−1} s^{−1} |

${m}_{0}$ = $\frac{{k}_{-3}}{{D}^{\frac{1}{3}}{a}^{\frac{2}{3}}{C}_{{H}^{+}}^{\infty}}$ | dimensionless backward rate coefficient | none |

${m}_{1}$ = $\frac{{k}_{+3}{C}_{{H}^{+}}^{\infty}}{{D}^{\frac{1}{3}}{a}^{\frac{2}{3}}}$ | dimensionless forward rate coefficient | none |

m = $-{m}_{1}$ uv + ${m}_{0}$ | parameter | none |

$\psi $ | current | Ampere (or) C s^{−1} |

$\eta $ | potential | volt |

F | Faraday constant | C mol^{−1} |

A | area | cm^{−2} |

T | temperature | K |

$\zeta $ | dimensionless axial distance | none |

Z | axial distance | cm |

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**Figure 1.**Diagram illustrating the electrolysis of ${H}_{2}O$ and the reduction of ${H}^{+}$ ions in nonbuffered aqueous electrolyte solutions.

**Figure 5.**Results of dimensionless ${H}^{+}$ ion concentration with various rate constants, c. (

**a**) at c = 0.1; (

**b**) at c = 0.2; (

**c**) at c = 0.3; (

**d**) at c = 0.4.

**Figure 6.**Results of dimensionless $O{H}^{-}$ ion concentration with various rate constants, c. (

**a**) at c = 0.1; (

**b**) at c = 0.2; (

**c**) at c = 0.3; (

**d**) at c = 0.4.

**Figure 7.**Error histogram analysis of dimensionless ${H}^{+}$ ion concentration. (

**a**) At c = 0.1; (

**b**) At c = 0.2; (

**c**) At c = 0.3; (

**d**) At c = 0.4.

**Figure 8.**Error histogram analysis of dimensionless $O{H}^{-}$ ion concentration. (

**a**) At c = 0.1; (

**b**) At c = 0.2; (

**c**) At c = 0.3; (

**d**) At c = 0.4.

**Figure 9.**Fitting analysis of dimensionless ${H}^{+}$ ion concentration. (

**a**) At c = 0.1; (

**b**) At c = 0.2; (

**c**) At c = 0.3; (

**d**) At c = 0.4.

**Figure 10.**Fitting analysis of dimensionless $O{H}^{-}$ ion concentration. (

**a**) At c = 0.1; (

**b**) At c = 0.2; (

**c**) At c = 0.3; (

**d**) At c = 0.4.

**Figure 11.**Mean-square error of NN-LMT prediction of dimensionless ${H}^{+}$ ion concentration. (

**a**) At c = 0.1; (

**b**) At c = 0.2; (

**c**) At c = 0.3; (

**d**) At c = 0.4.

**Figure 12.**Meansquare error of NN-LMT prediction of dimensionless $O{H}^{-}$ ion concentration. (

**a**) At c = 0.1; (

**b**) At c = 0.2; (

**c**) At c = 0.3; (

**d**) At c = 0.4.

**Figure 13.**Regressionanalysis of dimensionless ${H}^{+}$ ion concentration. (

**a**) At c = 0.1; (

**b**) At c = 0.2; (

**c**) At c = 0.3; (

**d**) At c = 0.4.

**Figure 14.**Regression analysis of dimensionless $O{H}^{-}$ ion concentration. (

**a**) At c = 0.1; (

**b**) At c = 0.2; (

**c**) At c = 0.3; (

**d**) At c = 0.4.

**Figure 15.**Performance analysis of dimensionless ${H}^{+}$ ion concentration. (

**a**) At c = 0.1; (

**b**) At c = 0.2; (

**c**) At c = 0.3; (

**d**) At c = 0.4.

**Figure 16.**Performance analysis of dimensionless $O{H}^{-}$ ion concentration. (

**a**) At c = 0.1; (

**b**) At c = 0.2; (

**c**) At c = 0.3; (

**d**) At c = 0.4.

m($\mathit{\zeta}$) | ${\mathit{Rk}}_{4}$ | BLMA | Error | |
---|---|---|---|---|

At c = 0.1 | 0 | 0 | $2.43\times {10}^{-7}$ | $2.43\times {10}^{-7}$ |

0.1 | 0.112628 | 0.112628 | $7.65\times {10}^{-8}$ | |

0.2 | 0.224126 | 0.224126 | $7.99\times {10}^{-8}$ | |

0.3 | 0.334163 | 0.334163 | $3.91\times {10}^{-8}$ | |

0.4 | 0.442201 | 0.442201 | $2.97\times {10}^{-10}$ | |

0.5 | 0.547518 | 0.547518 | $5.56\times {10}^{-9}$ | |

0.6 | 0.649234 | 0.649234 | $6.17\times {10}^{-9}$ | |

0.7 | 0.74636 | 0.74636 | $6.25\times {10}^{-8}$ | |

0.8 | 0.837858 | 0.837858 | $1.15\times {10}^{-7}$ | |

0.9 | 0.922711 | 0.922711 | $1.09\times {10}^{-7}$ | |

1 | 1 | 1 | $2.20\times {10}^{-7}$ | |

At c = 0.2 | 0 | 0 | $5.34\times {10}^{-7}$ | $5.34\times {10}^{-7}$ |

0.1 | 0.117043 | 0.117043 | $1.60\times {10}^{-7}$ | |

0.2 | 0.231951 | 0.231951 | $1.57\times {10}^{-7}$ | |

0.3 | 0.344387 | 0.344387 | 4.47$\times {10}^{-8}$ | |

0.4 | 0.453808 | 0.453808 | $9.72\times {10}^{-8}$ | |

0.5 | 0.559495 | 0.559495 | $2.60\times {10}^{-8}$ | |

0.6 | 0.660585 | 0.660585 | $6.33\times {10}^{-8}$ | |

0.7 | 0.756128 | 0.756128 | $3.61\times {10}^{-8}$ | |

0.8 | 0.845145 | 0.845145 | $1.55\times {10}^{-7}$ | |

0.9 | 0.926705 | 0.926705 | $1.92\times {10}^{-7}$ | |

1 | 1 | 1 | $4.36\times {10}^{-7}$ | |

At c = 0.3 | 0 | 0 | $1.46\times {10}^{-7}$ | $1.46\times {10}^{-7}$ |

0.1 | 0.121457 | 0.121457 | $1.16\times {10}^{-7}$ | |

0.2 | 0.239776 | 0.239776 | $1.04\times {10}^{-7}$ | |

0.3 | 0.354611 | 0.354611 | $7.20\times {10}^{-8}$ | |

0.4 | 0.465415 | 0.465415 | $2.60\times {10}^{-8}$ | |

0.5 | 0.571471 | 0.571471 | 8.12$\times {10}^{-10}$ | |

0.6 | 0.671937 | 0.671937 | $2.42\times {10}^{-8}$ | |

0.7 | 0.765896 | 0.765896 | 5.86$\times {10}^{-8}$ | |

0.8 | 0.852433 | 0.852433 | $8.15\times {10}^{-8}$ | |

0.9 | 0.930699 | 0.930699 | $8.29\times {10}^{-8}$ | |

1 | 1 | 1 | $1.40\times {10}^{-7}$ | |

At c = 0.4 | 0 | 0 | $1.84\times {10}^{-7}$ | $1.84\times {10}^{-7}$ |

0.1 | 0.125872 | 0.125872 | $1.10\times {10}^{-7}$ | |

0.2 | 0.247601 | 0.247601 | $1.04\times {10}^{-7}$ | |

0.3 | 0.364835 | 0.364835 | $6.84\times {10}^{-8}$ | |

0.4 | 0.477022 | 0.477022 | $2.68\times {10}^{-8}$ | |

0.5 | 0.583448 | 0.583448 | $4.50\times {10}^{-9}$ | |

0.6 | 0.683288 | 0.683288 | $2.75\times {10}^{-8}$ | |

0.7 | 0.775665 | 0.775665 | $5.40\times {10}^{-8}$ | |

0.8 | 0.85972 | 0.85972 | $6.86\times {10}^{-8}$ | |

0.9 | 0.934694 | 0.934694 | $6.24\times {10}^{-8}$ | |

1 | 1 | 1 | $1.25\times {10}^{-7}$ |

n($\mathit{\zeta}$) | ${\mathit{Rk}}_{4}$ | BLMA | Error | |
---|---|---|---|---|

At c = 0.1 | 0 | 1 | 1 | $1.62\times {10}^{-7}$ |

0.1 | 0.896201 | 0.896201 | $1.10\times {10}^{-7}$ | |

0.2 | 0.791525 | 0.791524 | $9.64\times {10}^{-8}$ | |

0.3 | 0.686286 | 0.686286 | $5.69\times {10}^{-8}$ | |

0.4 | 0.581012 | 0.581012 | $2.07\times {10}^{-8}$ | |

0.5 | 0.476435 | 0.476435 | $4.28\times {10}^{-10}$ | |

0.6 | 0.373469 | 0.373469 | $1.36\times {10}^{-8}$ | |

0.7 | 0.273176 | 0.273176 | $4.27\times {10}^{-8}$ | |

0.8 | 0.176716 | 0.176716 | $7.91\times {10}^{-8}$ | |

0.9 | 0.085278 | 0.085278 | $8.91\times {10}^{-8}$ | |

1 | $-3.6\times {10}^{-9}$ | $1.3\times {10}^{-7}$ | $1.31\times {10}^{-7}$ | |

At c = 0.2 | 0 | 1 | 0.999998 | $1.94\times {10}^{-6}$ |

0.1 | 0.900616 | 0.900615 | $7.85\times {10}^{-7}$ | |

0.2 | 0.79935 | 0.799349 | $6.61\times {10}^{-7}$ | |

0.3 | 0.69651 | 0.69651 | $2.59\times {10}^{-7}$ | |

0.4 | 0.592619 | 0.592619 | $6.06\times {10}^{-7}$ | |

0.5 | 0.488411 | 0.488411 | $1.82\times {10}^{-7}$ | |

0.6 | 0.38482 | 0.384819 | $6.23\times {10}^{-7}$ | |

0.7 | 0.282944 | 0.282944 | $1.93\times {10}^{-7}$ | |

0.8 | 0.184004 | 0.184004 | $3.84\times {10}^{-8}$ | |

0.9 | 0.089272 | 0.089272 | 3.53$\times {10}^{-8}$ | |

1 | $1.99\times {10}^{-9}$ | $2.7\times {10}^{-6}$ | $2.72\times {10}^{-6}$ | |

At c = 0.3 | 0 | 1 | 1 | $2.53\times {10}^{-7}$ |

0.1 | 0.905031 | 0.90503 | $1.13\times {10}^{-7}$ | |

0.2 | 0.807175 | 0.807175 | $1.04\times {10}^{-7}$ | |

0.3 | 0.706734 | 0.706734 | $2.45\times {10}^{-8}$ | |

0.4 | 0.604225 | 0.604225 | $2.15\times {10}^{-8}$ | |

0.5 | 0.500388 | 0.500388 | $1.05\times {10}^{-8}$ | |

0.6 | 0.396172 | 0.396172 | $5.29\times {10}^{-10}$ | |

0.7 | 0.292713 | 0.292713 | $4.35\times {10}^{-8}$ | |

0.8 | 0.191291 | 0.191291 | $8.11\times {10}^{-8}$ | |

0.9 | 0.093267 | 0.093267 | $7.17\times {10}^{-8}$ | |

1 | $4.9\times {10}^{-9}$ | $2.3\times {10}^{-7}$ | $2.24\times {10}^{-7}$ | |

At c = 0.4 | 0 | 1 | 0.999999 | $7.11\times {10}^{-7}$ |

0.1 | 0.909445 | 0.909445 | 7.94$\times {10}^{-8}$ | |

0.2 | 0.815 | 0.815 | $8.51\times {10}^{-8}$ | |

0.3 | 0.716958 | 0.716958 | $6.50\times {10}^{-8}$ | |

0.4 | 0.615832 | 0.615832 | $1.58\times {10}^{-7}$ | |

0.5 | 0.512365 | 0.512365 | 1.35$\times {10}^{-8}$ | |

0.6 | 0.407523 | 0.407523 | $6.56\times {10}^{-8}$ | |

0.7 | 0.302481 | 0.302481 | $5.64\times {10}^{-8}$ | |

0.8 | 0.198578 | 0.198578 | $1.48\times {10}^{-7}$ | |

0.9 | 0.097261 | 0.097261 | $1.70\times {10}^{-7}$ | |

1 | $8.01\times {10}^{-9}$ | $5.7\times {10}^{-7}$ | $5.60\times {10}^{-7}$ |

Training | Testing | Validation | Max.iteration | Hidden Neurons | Performance Function |
---|---|---|---|---|---|

70% | 15% | 15% | 1000 | 10 | Mean square error |

**Table 4.**NN-BLMA’s performance measurement statistics for different rate constant values to obtain dimensionless ${H}^{+}$ ion concentration solutions.

Mean Square Error | ||||||||
---|---|---|---|---|---|---|---|---|

c | Neurons | Epochs | Gradient | Mu | Training | Testing | Validation | Regression |

0.1 | 10 | 141 | $9.99\times {10}^{-8}$ | 1.00$\times {10}^{-07}$ | $2.40\times {10}^{-14}$ | $2.31\times {10}^{-14}$ | $2.48\times {10}^{-14}$ | 1 |

0.2 | 10 | 211 | $9.93\times {10}^{-8}$ | $1.00\times {10}^{-12}$ | $1.25\times {10}^{-14}$ | $1.52\times {10}^{-14}$ | $1.53\times {10}^{-14}$ | 1 |

0.3 | 10 | 151 | $9.96\times {10}^{-8}$ | $1.00\times {10}^{-11}$ | $2.42\times {10}^{-13}$ | $2.42\times {10}^{-13}$ | $2.57\times {10}^{-13}$ | 1 |

0.4 | 10 | 150 | $9.99\times {10}^{-8}$ | $1.00\times {10}^{-11}$ | $2.50\times {10}^{-13}$ | $2.51\times {10}^{-13}$ | $2.65\times {10}^{-13}$ | 1 |

**Table 5.**NN-BLMA’s performance measurement statistics for different rate constant values to obtain dimensionless $O{H}^{-}$ ion concentration solutions.

Mean Square Error | ||||||||
---|---|---|---|---|---|---|---|---|

c | Neurons | Epochs | Gradient | Mu | Training | Testing | Validation | Regression |

0.1 | 10 | 166 | $9.93\times {10}^{-8}$ | $1.00\times {10}^{-12}$ | $2.09\times {10}^{-14}$ | $2.32\times {10}^{-14}$ | $2.25\times {10}^{-14}$ | 1 |

0.2 | 10 | 154 | $9.95\times {10}^{-8}$ | $1.00\times {10}^{-11}$ | $2.94\times {10}^{-13}$ | $3.40\times {10}^{-13}$ | $3.35\times {10}^{-13}$ | 1 |

0.3 | 10 | 376 | $9.98\times {10}^{-8}$ | $1.00\times {10}^{-09}$ | $7.54\times {10}^{-12}$ | $6.82\times {10}^{-12}$ | $1.20\times {10}^{-11}$ | 1 |

0.4 | 10 | 178 | $9.95\times {10}^{-8}$ | $1.00\times {10}^{-12}$ | $1.79\times {10}^{-14}$ | $2.06\times {10}^{-14}$ | $1.87\times {10}^{-14}$ | 1 |

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## Share and Cite

**MDPI and ACS Style**

Alshammari, F.S.; Jan, H.; Sulaiman, M.; Prathumwan, D.; Laouini, G. Quantitative Study of Non-Linear Convection Diffusion Equations for a Rotating-Disc Electrode. *Entropy* **2023**, *25*, 134.
https://doi.org/10.3390/e25010134

**AMA Style**

Alshammari FS, Jan H, Sulaiman M, Prathumwan D, Laouini G. Quantitative Study of Non-Linear Convection Diffusion Equations for a Rotating-Disc Electrode. *Entropy*. 2023; 25(1):134.
https://doi.org/10.3390/e25010134

**Chicago/Turabian Style**

Alshammari, Fahad Sameer, Hamad Jan, Muhammad Sulaiman, Din Prathumwan, and Ghaylen Laouini. 2023. "Quantitative Study of Non-Linear Convection Diffusion Equations for a Rotating-Disc Electrode" *Entropy* 25, no. 1: 134.
https://doi.org/10.3390/e25010134