# Numerical Modeling of Particle Dynamics Inside a Dry Powder Inhaler

^{1}

^{2}

^{3}

^{*}

^{†}

## Abstract

**:**

^{®}as a model DPI device. The methodology included standard fluid flow equations but also additional equations for the particle sticking mechanism, as well as particle behavior after contacting the DPI wall surface, including the particle detachment process. The results show that the coefficient of restitution between the particle and the impact surface does not have a high impact on the results, meaning that all tested combinations gave similar output efficiencies and particle behaviors. No sliding or rolling mechanisms were observed for the particle detachment process, meaning that simple bouncing off or deposition particle behavior is present inside DPIs. The developed methodology can serve as a basis for the additional understanding of the particles’ behavior inside DPIs, which is not possible using only in vitro experiments; this implies the possibility of increasing the efficiency of DPIs.

## 1. Introduction

#### Related Work

^{®}as a model DPI device, after which coupled CFD and DPM computational simulations were performed to determine both the fluid flow and particle behavior. The simulation results were compared with the results from the literature, specifically in terms of the total particle deposition presented in previous publications [26,27] based on in vitro experiments and the deposition obtained in the study of Milenkovic et al. [13], which came from numerical simulations.

## 2. Materials and Methods

#### 2.1. Geometry and Meshing

^{®}inhaler, which was used in our previous research [25]. The inhaler geometry was obtained using commercial CAD software designed for these purposes (i.e., CATIA version 5, Dassault Systems, France), and is shown in Figure 1.

^{5}to 2 × 10

^{7}and were composed of tetrahedral cells, modeled based on the data from Milenkovic [28]. The cells had a maximum skewness of 0.85. To determine mesh independence, total particle depositions for six different meshes (about 2 × 10

^{5}, 5 × 10

^{5}, 1 × 10

^{6}, 2 × 10

^{6}, 5 × 10

^{6}, and 1 × 10

^{7}) were compared to 100% deposition assumptions. According to these simulations, the 2 × 10

^{6}mesh gave enough resolution to obtain realistic particle simulation results. As a consequence, the mesh with the total number of nodes 349,460 and the number of cells 1,930,248 (≈2 × 10

^{6}) was employed to get the results presented in this study.

^{−4}. According to Milenkovic et al. [13] instantaneous volumetric flow rate increases rapidly and reaches PIFR, i.e., maximum value. Therefore, a steady-state airflow may be considered as a close approximation to dynamic airflow developed in DPI, because for most of the inhalation process duration the instantaneous flow rate is approximately equal to the PIFR. Consequently, steady-state airflow was considered in this paper. Particles were released from a height of 12.5 mm, which corresponds to the real position of the drug capsule inside the device. The velocity of particles was set to correspond to the fluid velocity in that region of the inhaler.

#### 2.2. CFD Modeling

_{t}, is introduced:

_{1}represents an empirically determined constant, S is defined by the strain rate tensor S

_{ij}and functions F

_{1}and F

_{2}give the connection between the k-ω and k-ε models.

#### 2.3. Particle Sticking Process

^{−3}J/m

^{2}.

#### 2.4. Particle Detachment Process

#### 2.4.1. Detachment by Rolling

_{C}, defined by Equation (16), is the composite Young’s modulus:

#### 2.4.2. Detachment by Sliding

#### 2.5. User Defined Functions (UDF)

## 3. Results

**,**it can be concluded that most of the particles will bounce off of or stick to the grid zone, while in the lower part of the inhaler (dispersion chamber) and mouthpiece, a smaller number of particles impact the wall. The rest of the particles do not impact the wall on the path toward the outlet. This figure has been created based on the Impact.txt file.

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

^{®}Vcaps

^{®}Plus DPI capsules and Plastiape

^{®}Spa (Osnago, LC, Italy) for kindly donating the RS01

^{®}Dry Powder Inhaler.

## Conflicts of Interest

## List of Used Symbols

Symbol | Parameter Name |

$V$ | Fluid velocity |

$u$ | Fluid velocity component parallel to the wall |

${\upsilon}_{n}$ | Fluid velocity component normal to the wall |

p | Pressure |

$\upsilon $ | Kinematic viscosity |

${\delta}_{ij}$ | Kronecker delta symbol |

k | Specific turbulent energy |

${F}_{st}$ | Sticking force |

${F}_{D}$ | Drag force |

${F}_{L}$ | Lift force |

$a$ | Deformation of the particle along the surface |

${d}_{p}$ | Particle diameter |

${v}_{n}$ | Normal (impact) velocity |

${v}_{cr}$ | Critical (capture) velocity |

${u}^{*}$ | Wall shear velocity |

${u}_{R}{}^{*}$ | Wall shear velocity for rolling |

${u}_{s}{}^{*}$ | Wall shear velocity for sliding |

${d}_{s}$ | Distance of the first grid point from the wall |

${C}_{D}$ | Drag coefficient |

${\mathrm{Re}}_{p}$ | Reynolds coefficient |

K_{c} | Composite Young’s modulus |

${E}_{s}$ | Young’s modulus for surface |

${E}_{p}$ | Young’s modulus for particle |

${W}_{A}$ | Work of adhesion |

${v}_{s}$ | Poisson’s ratio for surface |

${v}_{p}$ | Poisson’s ratio for particle |

${\rho}_{p}$ | Particle density |

${\rho}_{M}$ | Mixture density |

$\rho $ | Air density (at 1013.25 hPa (abs) and 15 °C) |

$\mu $ | Dynamic viscosity of fluid (air) |

$f$ | Correction factor for the near wall |

${C}_{u}$ | Cunningham correction factor |

${k}_{s}$ | Static coefficient of friction |

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**Figure 4.**Description of particle behavior—proposed methodology to track particle deposition and detachment.

**Figure 6.**Fluid velocity magnitude in characteristic cross-sections: (

**a**) outlet, (

**b**) grid zone, (

**c**) central chamber, and (

**d**) capsule chamber.

**Figure 7.**Percentage (%) of deposited particles for different variations of the tangential and normal COR values.

**Figure 10.**Relationship between the normal (${v}_{n}$) and critical (${v}_{cr}$) velocities for the same particle ID.

**Figure 11.**Dependence of critical (${v}_{cr}$) and normal (${v}_{n}$) particle velocities on z coordinate.

**Figure 12.**Dependence of critical (${v}_{cr}$) and normal (${v}_{n}$) velocities on the particle diameters.

Parameter Name | Symbol | Value | Unit | Reference |
---|---|---|---|---|

Young’s modulus for surface | ${E}_{s}$ | 4.1e^{9} | Pa | [28] |

Young’s modulus for particle | ${E}_{p}$ | 1e^{9} | Pa | [28] |

work of adhesion | ${W}_{A}$ | 0.039 | J/m^{2} | [9,32] |

Poisson’s ratio for surface | ${v}_{s}$ | 0.35 | / | [28] |

Poisson’s ratio for particle | ${v}_{p}$ | 0.4 | / | [28] |

particle density | ${\rho}_{p}$ | 1230 | kg/m^{3} | [25] |

air density (at 1013.25 hPa (abs) and 15 °C) | $\rho $ | 1.225 | kg/m^{3} | [28] |

dynamic viscosity of fluid (air) | $\mu $ | 1.7894e^{−5} | N s/m^{2} | [28] |

correction factor for the near wall | $f$ | 1.7 | / | [9,32] |

Cunningham correction factor | ${C}_{u}$ | 1 (for spherical particles) | / | [9,32] |

static coefficient of friction | ${k}_{s}$ | 0.5 | / | [9,32] |

**Table 2.**Total particle deposition (%) in the inhaler for different combinations of tangential and normal COR values.

COR_Normal | |||||
---|---|---|---|---|---|

COR_tangential | 0.20 | 0.25 | 0.50 | 0.75 | |

0.25 | 13.1 | 13.4 | 14.0 | 13.2 | |

0.50 | 15.1 | 15.2 | 14.4 | 15.0 | |

0.75 | 17.4 | 16.6 | 17.2 | 14.6 | |

0.80 | 18.4 | 17.0 | 18.1 | 16.8 |

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**MDPI and ACS Style**

Šušteršič, T.; Bodić, A.; Ignjatović, J.; Cvijić, S.; Ibrić, S.; Filipović, N. Numerical Modeling of Particle Dynamics Inside a Dry Powder Inhaler. *Pharmaceutics* **2022**, *14*, 2591.
https://doi.org/10.3390/pharmaceutics14122591

**AMA Style**

Šušteršič T, Bodić A, Ignjatović J, Cvijić S, Ibrić S, Filipović N. Numerical Modeling of Particle Dynamics Inside a Dry Powder Inhaler. *Pharmaceutics*. 2022; 14(12):2591.
https://doi.org/10.3390/pharmaceutics14122591

**Chicago/Turabian Style**

Šušteršič, Tijana, Aleksandar Bodić, Jelisaveta Ignjatović, Sandra Cvijić, Svetlana Ibrić, and Nenad Filipović. 2022. "Numerical Modeling of Particle Dynamics Inside a Dry Powder Inhaler" *Pharmaceutics* 14, no. 12: 2591.
https://doi.org/10.3390/pharmaceutics14122591