# Identification of Granule Growth Regimes in High Shear Wet Granulation Processes Using a Physics-Constrained Neural Network

^{*}

## Abstract

**:**

## 1. Introduction

#### Objectives

## 2. Background

#### 2.1. Wet Granulation and Population Balance Model

#### 2.2. Artificial Neural Networks

#### 2.3. Previous ANN Studies in Granulation

#### 2.4. Physics-Constrained Neural Networks

## 3. Method and Implementation

#### 3.1. Data Generation

#### 3.2. Development of Artificial Neural Network

#### 3.2.1. Physical Constraints for Granule Growth

#### 3.2.2. Physics Constrained Neural Network for Granulation

#### 3.3. Input Parameter Sensitivity Analysis

## 4. Results and Discussion

#### 4.1. Comparing Artificial Neural Network and Physics-Constrained Neural Network Models

#### 4.2. Comparing Artificial Neural Networks and Physics-Constrained Neural Network at Growth Regime Boundary Conditions

#### 4.3. Input Parameters Sensitivity Analysis

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. PBM Equations

## Appendix B. Comparing Regime Predictions for PBM and PCNN

**Figure A1.**Comparing the PCNN predictions with PBM data with $S{t}_{def}>0.1$ and it can be observed that the PCNN is more accurate in identifying the undesired granule growth regime

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**Figure 2.**Comparing the training and test loss of the three ANNs and PCNN during training. (

**a**) The training loss of the three ANNs seems to be similar, which the normal ANN having a lower loss. (

**b**) The PCNN model has a higher loss compared to the other ANN model due the added regularization of the boundary condition loss. In both plots, the training loss is represented as solid lines while, the dotted lines illustrate the test loss at each epoch.

**Figure 3.**Each subplot corresponds to one simulated experimental result and the four model predictions are compared in the plot. A toal of six points at random were chosen for this plot from the test data set for comparison. The blue solid line represents the model prediction for the pure ANN model, the yellow line represents the model prediction for the $S{t}_{def}$ refined data model, and the green line represents the model prediction of the $S{t}_{def}$ and ${S}_{max}$. The blue solid line represents the prediction for the PCNN model. The blue dots represent the simulated experimental results. The performance of all four models seems to be similar.

**Figure 4.**Parity plot for the three ANN models and the PCNN model. The granule density and GSD predictions by each of these models is compared with the test dataset of 10,800 simulated experimental points. The data refined models in subfigure (

**b**,

**c**) seem to perform worse than the all data ANN model in subfigure (

**a**) and the PCNN model in subfigure (

**d**).

**Figure 5.**Comparing the predictions from all 4 model with the test data (PBM). This plot contains all the points where the predictions indicated $S{t}_{def}>0.1$. The PCNN performs better than the other data-driven models.

**Figure 6.**Input parameter sensitivity analysis for the physics-constrained neural network. Sub-figures (

**a**–

**h**) represent the effect of the input parameters on the various sieve cuts chosen. The interchangeable effect of RPM and impeller diameter has been identified by the PCNN. Sub-figure (

**i**) indicates that the effect of initial particle size affects the granule density the highest.

Input Parameter | Minimum Value | Maximum Value |
---|---|---|

Batch amount (kg) | 800 | 2000 |

Liquid amount (kg) | 600 | 1200 |

RPM | 100 | 600 |

Impeller diameter (m) | $0.075$ | $0.25$ |

Initial granule density (kg/m${}^{3}$) | 100 | 600 |

Initial porosity | $0.1$ | $0.5$ |

Hyperparameter | ANN Value | PCNN Value |
---|---|---|

No. of hidden layers | 2 | 3 |

Neurons in each hidden layer | 16 | 16 |

Optimizer algorithm | Adam | Adam |

Optimizer learning rate | $0.03$ | $0.03$ |

No. of epochs | 200 | 300 |

Regularization constant | $0.0$ | $0.0$ |

Hidden layer activation function | ‘tanh’ | ‘tanh’ |

Last layer activation function | ‘tanh’ | ‘tanh’ |

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**MDPI and ACS Style**

Sampat, C.; Ramachandran, R. Identification of Granule Growth Regimes in High Shear Wet Granulation Processes Using a Physics-Constrained Neural Network. *Processes* **2021**, *9*, 737.
https://doi.org/10.3390/pr9050737

**AMA Style**

Sampat C, Ramachandran R. Identification of Granule Growth Regimes in High Shear Wet Granulation Processes Using a Physics-Constrained Neural Network. *Processes*. 2021; 9(5):737.
https://doi.org/10.3390/pr9050737

**Chicago/Turabian Style**

Sampat, Chaitanya, and Rohit Ramachandran. 2021. "Identification of Granule Growth Regimes in High Shear Wet Granulation Processes Using a Physics-Constrained Neural Network" *Processes* 9, no. 5: 737.
https://doi.org/10.3390/pr9050737