# Mixing Performance Prediction of Detergent Mixing Process Based on the Discrete Element Method and Machine Learning

^{*}

## Abstract

**:**

## 1. Introduction

^{®}(Willy A. Bachofen AG, Muttenz, Switzerland) movement, and finally the machine provides the cup with the customized detergent.

^{®}movement, that has been used in the DIY4U project. An experimental analysis of the powder behaviour in a Turbula

^{®}mixer has been carried out in [7]. On the other hand, several works [8,9] have analysed the mixing performance of several mixers, as well as the influence of various process parameters on it, by using the DEM simulation [10,11,12]. Therefore, the DEM method was identified as the best option to model the mixing process of the powder detergent and to analyse its mixing performance.

## 2. Computational Model

^{®}movement with a constant speed of 45 rpm. The components utilized to obtain the formulation are surfactant particles, sodium sulphate, sodium carbonate, coloured speckle, and liquid nonionic surfactant. Initially, every component is poured into the cup individually, except surfactant particles and liquid nonionic that are premixed before being poured. The cup has a capacity of 565 mL, and it was defined as having a filling grade of 70%. The properties of formulation components are shown in Table 1.

^{®}movement is a three-dimensional movement based on the Schatz six-revolute mechanism [19], which has been employed to mixing powders. To replicate this movement, which was analysed in [20,21], an FEM model from the mixer geometry was created, and the parameterized position of every part during a cycle of movement was obtained by a rigid dynamics analysis. These position data were imported to the DEM software and applied to the DEM model. The movement described by the mixer model in each cycle is shown in Figure 3, representing the position of it every 45°.

## 3. Machine Learning Based Methodology

_{1}and β

_{2}are called as linear effect parameters, β

_{11}and β

_{22}are called as quadratic effect parameters, β

_{0}is the bias, and ε is the error in the normal distribution.

^{2}(coefficient of determination) metrics were calculated as in Equations (6)–(9):

_{i}is the actual value, ${\widehat{y}}_{i}$ is the predicted value, and n is the number of data points.

## 4. Results

#### 4.1. DEM Model Validation

#### 4.2. Mixing Performance Analysis

#### 4.3. Predictive Model

_{surfactant}is the mass fraction of surfactant particle, x

_{carbonate}is the mass fraction of sodium carbonate, x

_{sulphate}is the mass fraction of sodium sulphate, x

_{speckle}is the mass fraction of coloured speckle, x

_{nonionic}is the mass fraction of liquid nonionic surfactant, and x

_{time}is the time in seconds.

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Nomenclature

DIY | Do it yourself |

DEM | Discrete element method |

FEM | Finite element method |

ML | Machine learning |

MPR | Multivariate polynomial regression |

β_{i} | Linear effect parameter |

β_{ii} | Quadratic effect parameter |

ε | Error in the normal distribution |

MAPE | Mean absolute percentage error |

MAE | Mean absolute error |

RMSE | Root mean square error |

R^{2} | Coefficient of determination |

y_{i} | Actual value |

$\widehat{{y}_{i}}$ | Predicted value |

n | Number of data points |

M | Mixing index |

${\sigma}^{2}$ | Unbiased sample variance of the concentration in a multicomponent mixture |

${\sigma}_{0}^{2}$ | Variance for multicomponent mixtures in the completely segregated state |

${\sigma}_{r}^{2}$ | Variance for multicomponent mixtures in the completely mixed state |

CCD | Central composite design |

DOE | Design of experiments |

x_{surfactant} | Surfactant mass fraction |

x_{carbonate} | Carbonate mass fraction |

x_{sulphate} | Sulphate mass fraction |

x_{speckle} | Speckle mass fraction |

x_{nonionic} | Nonionic mass fraction |

x_{time} | Time, s |

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**Figure 6.**Mixing test. (

**a**) Initial filling of the components in the cup; (

**b**) mixture of the components after the mixing process.

**Figure 9.**A 6D scatter plot of the M index depending on the inputs surfactant, carbonate, speckle, nonionic and time.

Component | Particle Size Median (µm) | Bulk Density (kg/m^{3}) |
---|---|---|

Surfactant Particle | 443 | 830 |

Sodium Sulphate | 207 | 1550 |

Sodium Carbonate | 655 | 1150 |

Coloured Speckle | 1000 | 800 |

Liquid Nonionic Surfactant | - | 800 ^{1} |

^{1}Density of the liquid component.

Component | Percentage by Mass (%) | Particle Size (mm) |
---|---|---|

Surfactant Particle | 37.8 | 2.215 |

Sodium Sulphate | 14.0 | 1.035 |

Sodium Carbonate | 43.4 | 3.275 |

Coloured Speckle | 0.7 | 5.0 |

Liquid Nonionic Surfactant | 4.2 | - |

**Table 3.**Mechanical properties from [18].

Parameter | Value |
---|---|

Particle–particle static friction | 0.6 |

Particle–boundary static friction | 0.4 |

Restitution coefficient | 0.4 |

Rolling resistance | 0.001 |

Design Point | Surfactant Particle | Sodium Carbonate | Sodium Sulphate | Coloured Speckle | Liquid Nonionic Surfactant |
---|---|---|---|---|---|

1 | 25.0 | 0.9 | 72.1 | 1.0 | 1.0 |

2 | 25.0 | 0.9 | 68.7 | 1.0 | 4.4 |

3 | 25.0 | 0.9 | 73.1 | 0.0 | 1.0 |

4 | 25.0 | 0.9 | 69.7 | 0.0 | 4.4 |

5 | 25.0 | 36.7 | 36.3 | 1.0 | 1.0 |

6 | 25.0 | 36.7 | 32.9 | 1.0 | 4.4 |

7 | 25.0 | 36.7 | 37.3 | 0.0 | 1.0 |

8 | 25.0 | 36.7 | 33.9 | 0.0 | 4.4 |

9 | 40.0 | 0.9 | 57.1 | 1.0 | 1.0 |

10 | 40.0 | 0.9 | 53.7 | 1.0 | 4.4 |

11 | 40.0 | 0.9 | 58.1 | 0.0 | 1.0 |

12 | 40.0 | 0.9 | 54.7 | 0.0 | 4.4 |

13 | 40.0 | 36.7 | 21.3 | 1.0 | 1.0 |

14 | 40.0 | 36.7 | 17.9 | 1.0 | 4.4 |

15 | 40.0 | 36.7 | 22.3 | 0.0 | 1.0 |

16 | 40.0 | 36.7 | 18.9 | 0.0 | 4.4 |

17 | 32.5 | 18.8 | 45.5 | 0.5 | 2.7 |

18 | 25.0 | 18.8 | 53.0 | 0.5 | 2.7 |

19 | 40.0 | 18.8 | 38.0 | 0.5 | 2.7 |

20 | 32.5 | 0.9 | 63.4 | 0.5 | 2.7 |

21 | 32.5 | 36.7 | 27.6 | 0.5 | 2.7 |

22 | 32.5 | 18.8 | 46.0 | 0.0 | 2.7 |

23 | 32.5 | 18.8 | 45.0 | 1.0 | 2.7 |

24 | 32.5 | 18.8 | 47.2 | 0.5 | 1.0 |

25 | 32.5 | 18.8 | 43.8 | 0.5 | 4.4 |

Repose Angle | Value |
---|---|

Numerical simulation | 31.98° |

Experimental test | 32.00° |

Relative error | 0.063% |

**Table 6.**M index output for each design point at t = 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60 s.

Design Point | Time (s) | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

5 | 10 | 15 | 20 | 25 | 30 | 35 | 40 | 45 | 50 | 55 | 60 | |

1 | 0.776 | 0.870 | 0.885 | 0.918 | 0.968 | 0.976 | 0.979 | 0.988 | 0.989 | 0.992 | 0.991 | 0.992 |

2 | 0.780 | 0.892 | 0.885 | 0.953 | 0.970 | 0.985 | 0.978 | 0.989 | 0.989 | 0.987 | 0.989 | 0.991 |

3 | 0.788 | 0.871 | 0.879 | 0.916 | 0.968 | 0.975 | 0.976 | 0.984 | 0.988 | 0.989 | 0.991 | 0.989 |

4 | 0.716 | 0.868 | 0.857 | 0.943 | 0.965 | 0.976 | 0.967 | 0.987 | 0.987 | 0.985 | 0.987 | 0.991 |

5 | 0.853 | 0.909 | 0.929 | 0.946 | 0.973 | 0.978 | 0.980 | 0.981 | 0.976 | 0.986 | 0.983 | 0.986 |

6 | 0.833 | 0.912 | 0.912 | 0.964 | 0.975 | 0.977 | 0.981 | 0.985 | 0.983 | 0.984 | 0.984 | 0.992 |

7 | 0.843 | 0.905 | 0.925 | 0.952 | 0.972 | 0.975 | 0.979 | 0.986 | 0.979 | 0.988 | 0.984 | 0.986 |

8 | 0.829 | 0.911 | 0.909 | 0.960 | 0.971 | 0.980 | 0.977 | 0.985 | 0.979 | 0.986 | 0.985 | 0.992 |

9 | 0.721 | 0.854 | 0.886 | 0.914 | 0.964 | 0.974 | 0.970 | 0.983 | 0.982 | 0.984 | 0.985 | 0.988 |

10 | 0.695 | 0.862 | 0.876 | 0.934 | 0.941 | 0.970 | 0.973 | 0.981 | 0.982 | 0.988 | 0.983 | 0.990 |

11 | 0.721 | 0.852 | 0.886 | 0.926 | 0.968 | 0.976 | 0.976 | 0.985 | 0.979 | 0.987 | 0.988 | 0.987 |

12 | 0.684 | 0.853 | 0.862 | 0.936 | 0.950 | 0.972 | 0.970 | 0.986 | 0.984 | 0.984 | 0.983 | 0.989 |

13 | 0.742 | 0.855 | 0.899 | 0.935 | 0.964 | 0.970 | 0.964 | 0.979 | 0.974 | 0.988 | 0.986 | 0.990 |

14 | 0.684 | 0.853 | 0.862 | 0.936 | 0.950 | 0.972 | 0.970 | 0.986 | 0.984 | 0.984 | 0.983 | 0.989 |

15 | 0.750 | 0.847 | 0.895 | 0.926 | 0.969 | 0.971 | 0.963 | 0.981 | 0.969 | 0.986 | 0.985 | 0.989 |

16 | 0.705 | 0.845 | 0.870 | 0.928 | 0.957 | 0.964 | 0.971 | 0.977 | 0.978 | 0.987 | 0.988 | 0.994 |

17 | 0.785 | 0.897 | 0.906 | 0.940 | 0.966 | 0.975 | 0.979 | 0.983 | 0.987 | 0.984 | 0.983 | 0.989 |

18 | 0.794 | 0.907 | 0.908 | 0.951 | 0.972 | 0.982 | 0.980 | 0.985 | 0.987 | 0.986 | 0.985 | 0.987 |

19 | 0.762 | 0.887 | 0.898 | 0.941 | 0.971 | 0.970 | 0.976 | 0.979 | 0.983 | 0.985 | 0.986 | 0.990 |

20 | 0.727 | 0.868 | 0.895 | 0.933 | 0.955 | 0.974 | 0.981 | 0.984 | 0.990 | 0.987 | 0.988 | 0.988 |

21 | 0.798 | 0.884 | 0.894 | 0.941 | 0.970 | 0.974 | 0.980 | 0.983 | 0.981 | 0.987 | 0.991 | 0.991 |

22 | 0.798 | 0.884 | 0.894 | 0.941 | 0.970 | 0.974 | 0.980 | 0.983 | 0.981 | 0.987 | 0.991 | 0.991 |

23 | 0.836 | 0.893 | 0.912 | 0.949 | 0.971 | 0.976 | 0.980 | 0.982 | 0.987 | 0.984 | 0.985 | 0.988 |

24 | 0.799 | 0.893 | 0.914 | 0.935 | 0.968 | 0.974 | 0.976 | 0.983 | 0.982 | 0.985 | 0.985 | 0.988 |

25 | 0.783 | 0.892 | 0.902 | 0.954 | 0.965 | 0.977 | 0.973 | 0.986 | 0.984 | 0.986 | 0.985 | 0.991 |

Metric | MAPE (%) | MAE | RMSE | R^{2} |
---|---|---|---|---|

Training | 1.873 | 0.017 | 0.024 | 0.848 |

Validation | 1.503 | 0.014 | 0.017 | 0.9072 |

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**MDPI and ACS Style**

Cañamero, F.J.; Doraisingam, A.R.; Álvarez-Leal, M.
Mixing Performance Prediction of Detergent Mixing Process Based on the Discrete Element Method and Machine Learning. *Appl. Sci.* **2023**, *13*, 6094.
https://doi.org/10.3390/app13106094

**AMA Style**

Cañamero FJ, Doraisingam AR, Álvarez-Leal M.
Mixing Performance Prediction of Detergent Mixing Process Based on the Discrete Element Method and Machine Learning. *Applied Sciences*. 2023; 13(10):6094.
https://doi.org/10.3390/app13106094

**Chicago/Turabian Style**

Cañamero, Francisco J., Anand R. Doraisingam, and Marta Álvarez-Leal.
2023. "Mixing Performance Prediction of Detergent Mixing Process Based on the Discrete Element Method and Machine Learning" *Applied Sciences* 13, no. 10: 6094.
https://doi.org/10.3390/app13106094