# Particle-Scale Modeling to Understand Liquid Distribution in Twin-Screw Wet Granulation

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## Abstract

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## 1. Introduction

## 2. Particle Scale Modeling Approach

#### 2.1. Particle Flow Model

#### 2.2. Liquid Bridge Model

#### Liquid Loading, Bridge Volume Fraction & Liquid Bridge Coordination Number

#### 2.3. Simulation Set-Up and Input Parameters

#### 2.3.1. Simple Periodic Simulation Box

#### 2.3.2. Mixing Zone of a TSG

## 3. Results and Discussion

#### 3.1. Solid-Liquid Mixing in the Simple Periodic Simulation Box

#### 3.1.1. Effect of Change in Volume Fraction of Particles

#### 3.1.2. Effect of Change in Liquid Loading on Particles

#### 3.1.3. Effect of Change in Liquid Addition Zone Width

#### 3.2. Solid-Liquid Mixing in the Mixing Zone of a TSG

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

List of Acronyms | |

DEM | discrete element method. |

HSWG | high shear wet granulation. |

PBE | population balance equation. |

PBM | population balance model. |

TSG | twin-screw granulator. |

List of Symbols | |

${t}_{c}$ | Characteristic contact time $\left[\mathrm{s}\right]$. |

${c}_{f}$ | Dimensionless filling rate coefficient $[-]$. |

$\delta $ | Normal overlap between particles i and j [m]. |

${d}_{p}$ | Particle diameter $\left[\mathrm{m}\right]$. |

$\u03f5$ | Dimensionless reference film thickness $[-]$. |

${\eta}_{n}$ | Normal damping coefficient $[\mathrm{Ns}/\mathrm{m}]$. |

${e}_{n}$ | Normal coefficient of restitution $[-]$. |

${\eta}_{t}$ | Tangential damping coefficient $[\mathrm{Ns}/\mathrm{m}]$. |

${F}_{i}^{b}$ | body force of a particle i $\left[\mathrm{N}\right]$. |

${F}_{ij}^{coh}$ | Cohesion force between particle i and j $\left[\mathrm{N}\right]$. |

${F}_{ij}^{con}$ | Contact force between particle i and j $\left[\mathrm{N}\right]$. |

${\mathbf{F}}_{i}^{c,n}$ | Normal component of force acting on particle i $\left[\mathrm{N}\right]$. |

${\mathbf{F}}_{i}^{c,t}$ | Tangential component of force acting on particle i $\left[\mathrm{N}\right]$. |

$\gamma $ | Shear rate $[1/\mathrm{s}]$. |

${\gamma}^{*}$ | Scaled shear rate $[-]$. |

${h}_{0}$ | Dimensional reference film thickness [m]. |

${k}_{n}$ | Normal spring stiffness $[\mathrm{N}/\mathrm{m}]$. |

${k}_{t}$ | Tangential spring stiffness $[\mathrm{N}/\mathrm{m}]$. |

${L}_{p}$ | Volume of liquid present on the particle $\left(\right)$. |

${L}_{p,0}$ | Reference liquid content on the particles $[-]$. |

${m}_{\mathrm{eff}}$ | Effective mass of the particle [kg]. |

${\mu}_{l}$ | Dynamic viscosity of liquid $\left(\right)$. |

${n}_{b,i}$ | Number of liquid bridge connected to particle i $[-]$. |

${n}_{p}$ | Number of particles $[-]$. |

${n}_{p,liq}$ | Number of particles in the liquid addition region $[-]$. |

${\mathbf{n}}_{ij}$ | Unit normal vector $[-]$. |

$\omega $ | Eigen frequency of damped harmonic oscillator $[1/\mathrm{s}]$. |

$\varphi $ | Volume fraction of particles $[-]$. |

${\varphi}_{tr}$ | Fraction of liquid on the surface that is transferred into the bridge $[-]$. |

${Q}_{i}$ | Liquid addition rate to particle i in the liquid addition region $\left(\right)$. |

${Q}_{lod}$ | Dimensionless liquid load per particle $[-]$. |

${Q}_{tr}$ | Liquid transfer rate for particle $\left(\right)$. |

r | Radius of the particle $\left[\mathrm{m}\right]$. |

${\mathbf{r}}^{*}$ | Position vector of the particle $\left[\mathrm{m}\right]$. |

${r}_{\mathrm{eff}}$ | Effective radius of the particle $\left[\mathrm{m}\right]$. |

${\rho}_{p}$ | Density of the particles $\left(\right)$. |

${\sigma}_{l}$ | Surface tension of liquid $[\mathrm{N}/\mathrm{m}]$. |

${t}_{\mathrm{exp}}$ | Liquid addition time $\left[\mathrm{s}\right]$. |

${t}_{\mathrm{ref}}$ | Reference liquid bridge filling time [s]. |

${\mathbf{u}}_{ij}^{t}$ | Tangential overlap between particles i and j $\left[\mathrm{m}\right]$. |

${V}_{b}$ | Liquid bridge volume $\left(\right)$. |

${V}_{bf}$ | Liquid bridge fraction $[-]$. |

${\mathbf{v}}_{ij}^{n}$ | Normal relative particle velocity components $[\mathrm{m}/\mathrm{s}]$. |

${\mathbf{v}}_{ij}^{t}$ | Tangential relative particle velocity components $[\mathrm{m}/\mathrm{s}]$. |

${\overline{Z}}_{b}$ | Average number of liquid bridges per particle $[-]$. |

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**Figure 1.**Representation of normal and tangential contact forces using a spring, dash-pot and slider approach.

**Figure 2.**Shear box initial condition, front and side views. The red coloured particles indicate the liquid addition region (i.e., the wetting zone).

**Figure 4.**Transfer of liquid from particle surface (

**top left**) to other particles by convective transport (

**top right**) and transfer of liquid from particle surface to liquid bridges between particles by conductive transport before (

**bottom left**) and after shearing (

**bottom right**).

**Figure 5.**Changes in (

**a**) average number of liquid bridges per particle ${\overline{Z}}_{b}$ and (

**b**) the volume fraction of liquid in bridges ${V}_{bf}$ when the particle volume fraction $\varphi $ was increased from 0.3 to 0.5 for liquid loading of 7.5 × 10

^{−4}.

**Figure 6.**Changes in (

**a**) average number of liquid bridges per particle ${\overline{Z}}_{b}$ and (

**b**) volume fraction of liquid in bridges V

_{bf}when the liquid loading on particles Q

_{lod}was increased at particle volume fraction ϕ of 0.4.

**Figure 7.**Changes in (

**a**) average number of liquid bridges per particle ${\overline{Z}}_{b}$ and (

**b**) volume fraction of liquid in bridges V

_{bf}when the wetting zone width WZ

_{width}was increased, keeping the flux of liquid addition constant and particle volume fraction ϕ of 0.4.

**Figure 8.**Changes in volume of liquid on particle surface (left side snapshot in each sub-plot) and liquid in bridges (right side snapshot in each sub-plot) when two kneading discs were co-rotating at 100 rpm. The plots below every snapshot indicate the volume of liquid on particle and volume of liquid in bridge for each particle in the system.

**Figure 9.**Changes in (

**a**) volume of liquid on particle surface and (

**b**) liquid in bridges when two kneading discs were co-rotating at 100 rpm.

Quantity | Symbol | Value | Unit |
---|---|---|---|

Particle diameter | ${d}_{p}$ | 1.00E-03 | [m] |

Young’s modulus | G | 3.45E+9 | [N/m^{2}] |

Initial particle velocity | ${v}_{x}$, ${v}_{y}$ | 1, 0.1 | [m/s] |

Coefficient of restitution | ${e}_{n}$ | 0.9 | [–] |

Coefficient of friction | µ | 0.1 | [–] |

Poisson ratio | $\nu $ | 0.33 | [–] |

Film thickness | ${h}_{0}$/${r}_{\mathrm{eff}}$ | 1.00E-02 | [–] |

Dimensionless filling rate coefficient | ${c}_{f}$ | 1 | [–] |

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**MDPI and ACS Style**

Kumar, A.; Radl, S.; Gernaey, K.V.; De Beer, T.; Nopens, I.
Particle-Scale Modeling to Understand Liquid Distribution in Twin-Screw Wet Granulation. *Pharmaceutics* **2021**, *13*, 928.
https://doi.org/10.3390/pharmaceutics13070928

**AMA Style**

Kumar A, Radl S, Gernaey KV, De Beer T, Nopens I.
Particle-Scale Modeling to Understand Liquid Distribution in Twin-Screw Wet Granulation. *Pharmaceutics*. 2021; 13(7):928.
https://doi.org/10.3390/pharmaceutics13070928

**Chicago/Turabian Style**

Kumar, Ashish, Stefan Radl, Krist V. Gernaey, Thomas De Beer, and Ingmar Nopens.
2021. "Particle-Scale Modeling to Understand Liquid Distribution in Twin-Screw Wet Granulation" *Pharmaceutics* 13, no. 7: 928.
https://doi.org/10.3390/pharmaceutics13070928