# Characterization of Simultaneous Evolution of Size and Composition Distributions Using Generalized Aggregation Population Balance Equation

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## Abstract

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## 1. Introduction

## 2. Numerical Methods and System Analysis

#### 2.1. Cell Average Technique (CAT)

#### 2.2. Finite Volume Scheme (FVS)

#### 2.3. Kernel Selection

#### 2.3.1. Size-Independent Kernel

#### 2.3.2. Size-Dependent Kernel

#### 2.4. Model Initialization and Post-Processing

#### 2.5. Average Size Particles

#### 2.6. Quantification of Mixing

## 3. Results and Discussion

#### 3.1. Size-Independent Kernel

#### 3.1.1. Comparison of Moments and Number Density Prediction

#### 3.1.2. Comparison of Average Particle Size and Mixing State Prediction

#### 3.2. Size-Dependent Kernel

#### 3.2.1. Comparison of Moments and Number Density Prediction

#### 3.2.2. Comparison of Average Particle Size and Mixing State Prediction

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Nomenclature

Symbol | Description |

n | Particle property (size) distribution |

$\mathbf{u}$ | Particle property vector (size) |

t | time |

${N}_{i}$ | Number of particles in the cell i |

${\mu}_{i}$ | ith order moment |

$\mathbf{I}$ | Total number of cells |

${I}_{agg}$ | Degree of aggregation |

${l}_{j,k}$ | Index of the cell where $({\mathbf{u}}_{j}+{\mathbf{u}}_{k})$ falls |

$\phi $ | Weight function |

a | Aggregation kernel |

$\Delta $ | Measure of sectional error |

$\theta $ | Sum function |

$\overline{u}$ | Average size of particles along u axis |

$\overline{v}$ | Average size of particles along v axis |

${\chi}^{2}$ | Mixing of components |

## Abbreviations

PBE | Population balance equation |

FVS | Finite volume scheme |

CAT | Cell average technique |

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**Figure 4.**Average size particles formed in the system and ${\chi}^{2}$ parameter for mixing the components using constant kernel.

**Figure 6.**Average size particles formed in the system and ${\chi}^{2}$ parameter for mixing the components using sum kernel.

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

${N}_{0}$ | 1 |

${p}_{1},{p}_{2}$ | 1 |

${m}_{10},{m}_{20}$ | $0.04$ |

$n(u,v,t)$ | $\frac{4{N}_{0}}{{(t+2)}^{2}}\frac{{({p}_{1}+1)}^{({p}_{1}+1)}{({p}_{2}+1)}^{({p}_{2}+1)}}{{m}_{10}{m}_{20}}exp\left(\right)open="["\; close="]">\frac{-({p}_{1}+1)u}{{m}_{10}}+\frac{-({p}_{2}+1)v}{{m}_{20}}$ |

$\times {\sum}_{k=0}^{\infty}{\left(\frac{t}{t+2}\right)}^{k}\frac{{\left[{({p}_{1}+1)}^{({p}_{1}+1)}\right]}^{k}{\left[{({p}_{2}+1)}^{({p}_{2}+1)}\right]}^{k}{(u/{m}_{10})}^{(k+1)({p}_{1}+1)-1}{(v/{m}_{20})}^{(k+1)({p}_{2}+1)-1}}{\Gamma \left[({p}_{1}+1)(k+1)\right]\Gamma \left[({p}_{2}+1)(k+1)\right]}$ |

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

${N}_{0}$ | 1 |

${p}_{1},{p}_{2}$ | 1 |

${m}_{10},{m}_{20}$ | $0.04$ |

$\tau $ | $1-exp\left(\varphi t\right)$ |

s | $u+v$ |

${s}_{0}$ | ${m}_{10}+{m}_{20}$ |

$n(u,v,t)$ | ${N}_{0}(1-\tau )exp(\frac{-s\tau}{{s}_{0}})\frac{({p}_{1}+1)({p}_{2}+1)}{{m}_{10}{m}_{20}}exp\left(\right)open="["\; close="]">\frac{-({p}_{1}+1)u}{{m}_{10}}+\frac{-({p}_{2}+1)v}{{m}_{20}}$ |

$\times {\sum}_{k=0}^{\infty}\frac{1}{(k+1)!}{\left(\frac{-s\tau}{{s}_{0}}\right)}^{k}\frac{{[({p}_{1}+1)u/{m}_{10}]}^{(k+1)({p}_{1}+1)-1}{[({p}_{2}+1)v/{m}_{20}]}^{(k+1)({p}_{2}+1)-1}}{\Gamma \left[({p}_{1}+1)(k+1)\right]\Gamma \left[({p}_{2}+1)(k+1)\right]}$ | |

$\varphi $ | total mass of the particles in the system |

Moments | CAT | FVS | CAT | FVS |
---|---|---|---|---|

$\mathbf{20}\times \mathbf{20}$ | $\mathbf{20}\times \mathbf{20}$ | $\mathbf{25}\times \mathbf{25}$ | $\mathbf{25}\times \mathbf{25}$ | |

${\Delta}_{0,0}$ | 0.27110 | 0.14838 | 0.22142 | 0.10737 |

${\Delta}_{1,0}$ | 0.29518 | 0.21496 | 0.25672 | 0.14373 |

${\Delta}_{2,0}$ | 0.36367 | 0.32288 | 0.32813 | 0.25187 |

${\Delta}_{1,1}$ | 0.35038 | 0.30358 | 0.32345 | 0.24179 |

${\Delta}_{3,0}$ | 0.51598 | 0.46139 | 0.48715 | 0.42507 |

${\Delta}_{2,1}$ | 0.53205 | 0.51217 | 0.49247 | 0.45377 |

Moments | CAT | FVS | CAT | FVS |
---|---|---|---|---|

$\mathbf{20}\times \mathbf{20}$ | $\mathbf{20}\times \mathbf{20}$ | $\mathbf{25}\times \mathbf{25}$ | $\mathbf{25}\times \mathbf{25}$ | |

${\Delta}_{0,0}$ | 0.18180 | 0.15587 | 0.11325 | 0.08356 |

${\Delta}_{1,0}$ | 0.33015 | 0.14861 | 0.30735 | 0.10185 |

${\Delta}_{2,0}$ | 0.81804 | 0.29930 | 0.65429 | 0.27372 |

${\Delta}_{1,1}$ | 0.82524 | 0.27404 | 0.65840 | 0.26817 |

${\Delta}_{3,0}$ | 2.12448 | 0.59022 | 1.43660 | 0.37667 |

${\Delta}_{2,1}$ | 2.05260 | 0.67102 | 1.40687 | 0.38896 |

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**MDPI and ACS Style**

Singh, M.; Kumar, A.; Shirazian, S.; Ranade, V.; Walker, G.
Characterization of Simultaneous Evolution of Size and Composition Distributions Using Generalized Aggregation Population Balance Equation. *Pharmaceutics* **2020**, *12*, 1152.
https://doi.org/10.3390/pharmaceutics12121152

**AMA Style**

Singh M, Kumar A, Shirazian S, Ranade V, Walker G.
Characterization of Simultaneous Evolution of Size and Composition Distributions Using Generalized Aggregation Population Balance Equation. *Pharmaceutics*. 2020; 12(12):1152.
https://doi.org/10.3390/pharmaceutics12121152

**Chicago/Turabian Style**

Singh, Mehakpreet, Ashish Kumar, Saeed Shirazian, Vivek Ranade, and Gavin Walker.
2020. "Characterization of Simultaneous Evolution of Size and Composition Distributions Using Generalized Aggregation Population Balance Equation" *Pharmaceutics* 12, no. 12: 1152.
https://doi.org/10.3390/pharmaceutics12121152