# Mathematical Modelling and Simulation of a Spray Fluidized Bed Granulator

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## Abstract

**:**

## 1. Introduction

## 2. Two-Compartment Model for Sprayed Fluidized Bed Granulation

- The system is divided into two compartments:
- (a)
- the first compartment, that is the wet zone, is represented by fraction $\eta $, and the volume is ${V}_{1}=\eta V$,
- (b)
- the second compartment, that is the dry zone, is represented by the fraction $1-\eta $, and the volume is ${V}_{2}=(1-\eta )V$, where V is the total volume of the reactor.

- Each compartment ($WZ$, as well as $DZ$) is considered to be a well-mixed system.
- The size of both compartments remains constant during the process.
- The mass in each compartment should be constant at each time step.
- The rate of exchange of volume flux between the compartments is constant during the process.

#### Finite Volume Scheme

## 3. Derivation of the Analytical Solutions for Moments

- Case 1: additive aggregation kernel, linear selection function with binary breakage kernel, that is ${\beta}_{WZ}(t,r,s)={\beta}_{0}(r+s),{S}_{DZ}(t,r)=r,{b}_{DZ}(r,s)=2/s$.
- Case 2: product aggregation kernel, linear selection function with binary breakage kernel, that is ${\beta}_{WZ}(t,r,s)={\beta}_{0}(rs),{S}_{DZ}(t,r)=r,{b}_{DZ}(r,s)=2/s$.

#### 3.1. Case 1

#### 3.2. Case 2

## 4. Results and Discussion

#### 4.1. Additive Aggregation Kernel and Binary Breakage with the Linear Selection Function

#### 4.2. Product Kernel and Binary Breakage with Linear Selection Function

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 3.**Different order moments and number density for the additive kernel. FVS, Finite Volume Scheme.

**Table 1.**Quantitative relative errors in the zeroth and first order moments for the additive kernel.

$\mathit{WZ}$ | $\mathit{DZ}$ | Total | $\mathit{WZ}$ | $\mathit{DZ}$ | Total | |
---|---|---|---|---|---|---|

Moments | $\mathit{\eta}=20\%$ | $\mathit{\eta}=20\%$ | $\mathit{\eta}=20\%$ | $\mathit{\eta}=30\%$ | $\mathit{\eta}=30\%$ | $\mathit{\eta}=30\%$ |

${\mu}_{0}$ | 0.0037 | 0.0037 | 0.0037 | 0.0033 | 0.0034 | 0.0034 |

${\mu}_{1}$ | $3.48\times {10}^{-10}$ | $3.32\times {10}^{-10}$ | $3.35\times {10}^{-10}$ | $1.60\times {10}^{-10}$ | $1.55\times {10}^{-9}$ | $1.57\times {10}^{-9}$ |

$\mathit{WZ}$ | $\mathit{DZ}$ | Total | $\mathit{WZ}$ | $\mathit{DZ}$ | Total | |
---|---|---|---|---|---|---|

Moments | $\mathit{\eta}=20\%$ | $\mathit{\eta}=20\%$ | $\mathit{\eta}=20\%$ | $\mathit{\eta}=30\%$ | $\mathit{\eta}=30\%$ | $\mathit{\eta}=30\%$ |

${\mu}_{0}$ | 0.0046 | 0.0044 | 0.0044 | 0.0056 | 0.0053 | 0.0054 |

${\mu}_{1}$ | $5.55\times {10}^{-8}$ | $2.05\times {10}^{-8}$ | $2.75\times {10}^{-8}$ | $1.62\times {10}^{-8}$ | $1.47\times {10}^{-8}$ | $1.52\times {10}^{-8}$ |

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**MDPI and ACS Style**

Kaur, G.; Singh, M.; Kumar, J.; De Beer, T.; Nopens, I.
Mathematical Modelling and Simulation of a Spray Fluidized Bed Granulator. *Processes* **2018**, *6*, 195.
https://doi.org/10.3390/pr6100195

**AMA Style**

Kaur G, Singh M, Kumar J, De Beer T, Nopens I.
Mathematical Modelling and Simulation of a Spray Fluidized Bed Granulator. *Processes*. 2018; 6(10):195.
https://doi.org/10.3390/pr6100195

**Chicago/Turabian Style**

Kaur, Gurmeet, Mehakpreet Singh, Jitendra Kumar, Thomas De Beer, and Ingmar Nopens.
2018. "Mathematical Modelling and Simulation of a Spray Fluidized Bed Granulator" *Processes* 6, no. 10: 195.
https://doi.org/10.3390/pr6100195