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

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 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 |

## References

- Singh, M.; Ismail, H.Y.; Matsoukas, T.; Albadarin, A.B.; Walker, G. Mass-based finite volume scheme for aggregation, growth and nucleation population balance equation. Proc. R. Soc.
**2019**, 475, 20190552. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Ramkrishna, D. Population Balances: Theory and Applications to Particulate Systems in Engineering; Academic Press: Cambridge, MA, USA, 2000. [Google Scholar]
- Ahamed, F.; Singh, M.; Song, H.-S.; Doshi, P.; Ooi, C.W.; Ho, Y.K. On the use of sectional techniques for the solution of depolymerization population balances: Results on a discrete-continuous mesh. Adv. Powder Technol.
**2020**, 31, 2669–2679. [Google Scholar] [CrossRef] - Ho, Y.K.; Kirse, C.; Briesen, H.; Singh, M.; Chan, C.-H.; Kow, K.-W. Towards improved predictions for the enzymatic chain-end scission of natural polymers by population balances: The need for a non-classical rate kernel. Chem. Eng. Sci.
**2018**, 176, 329–342. [Google Scholar] [CrossRef] - Selomulya, C.; Bushell, G.; Amal, R.; Waite, T.D. Understanding the role of restructuring in flocculation: The application of a population balance model. Chem. Eng. Sci.
**2003**, 58, 327–338. [Google Scholar] [CrossRef] - Shen, X.; Hibiki, T. Bubble coalescence and breakup model evaluation and development for two-phase bubbly flows. Int. J. Multiph. Flow
**2018**, 109, 131–149. [Google Scholar] [CrossRef] - Matveev, S.; Stadnichuk, V.; Tyrtyshnikov, E.; Smirnov, A.; Ampilogova, N.; Brilliantov, N. Anderson acceleration method of finding steady-state particle size distribution for a wide class of aggregation–fragmentation models. Comput. Phys. Commun.
**2017**, 224, 154–163. [Google Scholar] [CrossRef] - Ismail, H.Y.; Shirazian, S.; Singh, M.; Whitaker, D.; Albadarin, A.B.; Walker, G.M. Compartmental approach for modelling twin-screw granulation using population balances. Int. J. Pharm.
**2020**, 576, 118737. [Google Scholar] [CrossRef] - Ismail, H.Y.; Singh, M.; Albadarin, A.B.; Walker, G.M. Complete two dimensional population balance modelling of wet granulation in twin screw. Int. J. Pharm.
**2020**, 591, 120018. [Google Scholar] [CrossRef] - Ismail, H.Y.; Singh, M.; Shirazian, S.; Albadarin, A.B.; Walker, G.M. Development of high-performance hybrid ann-finite volume scheme (ann-fvs) for simulation of pharmaceutical continuous granulation. Chem. Eng. Res. Des.
**2020**, 163, 320–326. [Google Scholar] [CrossRef] - Kumar, A.; Vercruysse, J.; Mortier, S.T.; Vervaet, C.; Remon, J.P.; Gernaey, K.V.; De Beer, T.; Nopens, I. Model-based analysis of a twin-screw wet granulation system for continuous solid dosage manufacturing. Comput. Chem. Eng.
**2016**, 89, 62–70. [Google Scholar] [CrossRef] - Shirazian, S.; Ismail, H.Y.; Singh, M.; Shaikh, R.; Croker, D.M.; Walker, G.M. Multi-dimensional population balance modelling of pharmaceutical formulations for continuous twin-screw wet granulation: Determination of liquid distribution. Int. J. Pharm.
**2019**, 566, 352–360. [Google Scholar] [CrossRef] [PubMed] - Kumar, A.; Gernaey, K.V.; De Beer, T.; Nopens, I. Model-based analysis of high shear wet granulation from batch to continuous processes in pharmaceutical production–a critical review. Eur. J. Pharm. Biopharm.
**2013**, 85, 814–832. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Chaudhury, A.; Armenante, M.E.; Ramachandran, R. Compartment based population balance modeling of a high shear wet granulation process using data analytics. Chem. Eng. Res. Des.
**2015**, 95, 211–228. [Google Scholar] [CrossRef] - Kaur, G.; Singh, M.; Matsoukas, T.; Kumar, J.; De Beer, T.; Nopens, I. Two-compartment modeling and dynamics of top-sprayed fluidized bed granulator. Appl. Math. Model.
**2019**, 68, 267–280. [Google Scholar] [CrossRef] - Iveson, S.M. Limitations of one-dimensional population balance models of wet granulation processes. Powder Technol.
**2002**, 124, 219–229. [Google Scholar] [CrossRef] - Matsoukas, T.; Lee, K.; Kim, T. Mixing of components in two-component aggregation. AIChE J.
**2006**, 52, 3088–3099. [Google Scholar] [CrossRef] - Singh, M.; Chakraborty, J.; Kumar, J.; Ramakanth, R. Accurate and efficient solution of bivariate population balance equations using unstructured grids. Chem. Eng. Sci.
**2013**, 93, 1–10. [Google Scholar] [CrossRef] - Matsoukas, T.; Marshall, C., Jr. Bicomponent aggregation in finite systems. EPL (Europhys. Lett.)
**2010**, 92, 46007. [Google Scholar] [CrossRef] - Bhutani, G.; Brito-Parada, P.R. Analytical solution for a three-dimensional non-homogeneous bivariate population balance equation—A special case. Int. J. Flow
**2017**, 89, 413–416. [Google Scholar] [CrossRef] - Fernández-Díaz, J.; Gómez-García, G. Exact solution of smoluchowski’s continuous multi-component equation with an additive kernel. Europhys. Lett.
**2007**, 78, 56002. [Google Scholar] [CrossRef] - Gelbard, F.; Seinfeld, J. Simulation of multicomponent aerosol dynamics. J. Colloid Interface Sci.
**1980**, 78, 485–501. [Google Scholar] [CrossRef] - Kaur, G.; Singh, R.; Singh, M.; Kumar, J.; Matsoukas, T. Analytical approach for solving population balances: A homotopy perturbation method. J. Phys. Math. Theor.
**2019**, 52, 385201. [Google Scholar] [CrossRef] [Green Version] - Nandanwar, M.N.; Kumar, S. A new discretization of space for the solution of multi-dimensional population balance equations. Chem. Eng. Sci.
**2008**, 63, 2198–2210. [Google Scholar] [CrossRef] - Chauhan, S.S.; Chakraborty, J.; Kumar, S. On the solution and applicability of bivariate population balance equations for mixing in particle phase. Chem. Eng. Sci.
**2010**, 65, 3914–3927. [Google Scholar] [CrossRef] - Vale, H.; McKenna, T. Solution of the population balance equation for two-component aggregation by an extended fixed pivot technique. Ind. Eng. Chem. Res.
**2005**, 44, 7885–7891. [Google Scholar] [CrossRef] [Green Version] - Wu, S.; Yapp, E.K.; Akroyd, J.; Mosbach, S.; Xu, R.; Yang, W.; Kraft, M. Extension of moment projection method to the fragmentation process. J. Comput. Phys.
**2017**, 335, 516–534. [Google Scholar] [CrossRef] [Green Version] - Sen, M.; Ramachandran, R. A multi-dimensional population balance model approach to continuous powder mixing processes. Adv. Powder Technol.
**2013**, 24, 51–59. [Google Scholar] [CrossRef] - Forestier-Coste, L.; Mancini, S. A finite volume preserving scheme on nonuniform meshes and for multidimensional coalescence. SIAM J. Sci. Comput.
**2012**, 34, B840–B860. [Google Scholar] [CrossRef] [Green Version] - Kumar, J.; Kaur, G.; Tsotsas, E. An accurate and efficient discrete formulation of aggregation population balance equation. Kinet. Relat. Model.
**2016**, 9, 373–391. [Google Scholar] - Qamar, S.; Warnecke, G. Solving population balance equations for two-component aggregation by a finite volume scheme. Chem. Eng. Sci.
**2007**, 62, 679–693. [Google Scholar] [CrossRef] - Singh, M.; Ismail, H.Y.; Singh, R.; Albadarin, A.B.; Walker, G. Finite volume approximation of nonlinear agglomeration population balance equation on triangular grid. J. Aerosol Sci.
**2019**, 137, 105430. [Google Scholar] [CrossRef] [Green Version] - Singh, M.; Kaur, G.; Thomas, D.B.; Nopens, I. Solution of bivariate aggregation population balance equation: A comparative study. React. Kinet. Mech. Catal.
**2018**, 123, 1–17. [Google Scholar] [CrossRef] - Singh, M.; Kumar, J.; Bück, A.; Tsotsas, E. A volume-consistent discrete formulation of aggregation population balance equations. Math. Methods Appl. Sci.
**2015**, 39, 2275–2286. [Google Scholar] [CrossRef] - Singh, M.; Kumar, J.; Bück, A.; Tsotsas, E. An improved and efficient finite volume scheme for bivariate aggregation population balance equation. J. Comput. Appl. Math.
**2016**, 308, 83–97. [Google Scholar] [CrossRef] - Singh, M.; Matsoukas, T.; Albadarin, A.B.; Walker, G. New volume consistent approximation for binary breakage population balance equation and its convergence analysis. ESAIM Math. Model. Numer. Anal.
**2019**, 53, 1695–1713. [Google Scholar] [CrossRef] - Singh, M.; Matsoukas, T.; Walker, G. Mathematical analysis of finite volume preserving scheme for nonlinear smoluchowski equation. Phys. D Nonlinear Phenom.
**2020**, 402, 132221. [Google Scholar] [CrossRef] - Singh, M.; Singh, R.; Singh, S.; Singh, G.; Walker, G. Finite volume approximation of multidimensional aggregation population balance equation on triangular grid. Math. Comput. Simul.
**2020**, 172, 191–212. [Google Scholar] [CrossRef] - Singh, M.; Singh, R.; Singh, S.; Walker, G.; Matsoukas, T. Discrete finite volume approach for multidimensional agglomeration population balance equation on unstructured grid. Powder Technol.
**2020**, 376, 229–240. [Google Scholar] [CrossRef] - Attarakih, M.M.; Drumm, C.; Bart, H.-J. Solution of the population balance equation using the sectional quadrature method of moments (sqmom). Chem. Eng. Sci.
**2009**, 64, 742–752. [Google Scholar] [CrossRef] - Kostoglou, M. On the constant kernel smoluchowski equation: Fast algorithm for solution with arbitrary initial conditions. Comput. Phys. Commun.
**2005**, 173, 34–40. [Google Scholar] [CrossRef] - Yu, M.; Liu, Y.; Lin, J.; Seipenbusch, M. Generalized temom scheme for solving the population balance equation. Aerosol Sci. Technol.
**2015**, 49, 1021–1036. [Google Scholar] [CrossRef] [Green Version] - Yuan, C.; Laurent, F.; Fox, R. An extended quadrature method of moments for population balance equations. J. Aerosol Sci.
**2012**, 51, 1–23. [Google Scholar] [CrossRef] [Green Version] - Ahmed, N.; Matthies, G.; Tobiska, L. Finite element methods of an operator splitting applied to population balance equations. J. Comput. Appl. Math.
**2011**, 236, 1604–1621. [Google Scholar] [CrossRef] - Zhu, Z.; Dorao, C.; Jakobsen, H. A least-squares method with direct minimization for the solution of the breakage–coalescence population balance equation. Math. Comput. Simul.
**2008**, 79, 716–727. [Google Scholar] [CrossRef] - Kumar, J.; Peglow, M.; Warnecke, G.; Heinrich, S. The cell average technique for solving multi-dimensional aggregation population balance equations. Comput. Chem. Eng.
**2008**, 32, 1810–1830. [Google Scholar] [CrossRef] - Kumar, R.; Kumar, J.; Warnecke, G. Numerical methods for solving two-dimensional aggregation population balance equations. Comput. Chem. Eng.
**2011**, 35, 999–1009. [Google Scholar] [CrossRef] - Mostafaei, P.; Rajabi-Hamane, M.; Salehpour, A. A modified cell average technique for the solution of population balance equation. J. Aerosol Sci.
**2015**, 87, 111–125. [Google Scholar] [CrossRef] - Singh, M.; Ghosh, D.; Kumar, J. A comparative study of different discretizations for solving bivariate aggregation population balance equation. Appl. Math. Comput.
**2014**, 234, 434–451. [Google Scholar] [CrossRef] - Chaudhury, A.; Kapadia, A.; Prakash, A.V.; Barrasso, D.; Ramachandran, R. An extended cell-average technique for a multi-dimensional population balance of granulation describing aggregation and breakage. Adv. Powder Technol.
**2013**, 24, 962–971. [Google Scholar] [CrossRef] - Hao, X.; Zhao, H.; Xu, Z.; Zheng, C. Population balance-monte carlo simulation for gas-to-particle synthesis of nanoparticles. Aerosol Sci. Technol.
**2013**, 47, 1125–1133. [Google Scholar] [CrossRef] - Lin, Y.; Lee, K.; Matsoukas, T. Solution of the population balance equation using constant-number monte carlo. Chem. Eng. Sci.
**2002**, 57, 2241–2252. [Google Scholar] [CrossRef] - Smith, M.; Matsoukas, T. Constant-number monte carlo simulation of population balances. Chem. Eng. Sci.
**1998**, 53, 1777–1786. [Google Scholar] [CrossRef] - Singh, M. Forward and Inverse Problems in Population Balances. Ph.D. Thesis, IIT, Kharagpur, India, 2015. [Google Scholar]
- Singh, M.; Kaur, G.; Kumar, J.; De Beer, T.; Nopens, I. A comparative study of numerical approximations for solving the smoluchowski coagulation equation. Braz. J. Chem. Eng.
**2018**, 35, 1343–1354. [Google Scholar] [CrossRef] - Singh, M.; Vuik, K.; Kaur, G.; Bart, H.-J. Effect of different discretizations on the numerical solution of 2D aggregation population balance equation. Powder Technol.
**2019**, 342, 972–984. [Google Scholar] [CrossRef] [Green Version] - Kaur, G.; Kumar, J.; Heinrich, S. A weighted finite volume scheme for multivariate aggregation population balance equation. Comput. Chem. Eng.
**2017**, 101, 1–10. [Google Scholar] [CrossRef] - Chakraborty, J.; Kumar, S. A new framework for solution of multidimensional population balance equations. Chem. Eng. Sci.
**2007**, 62, 4112–4125. [Google Scholar] [CrossRef]

**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 |

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

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

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

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

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

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

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

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

s | $u+v$ |

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

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

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

$\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 |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**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