# Influence of Thermal Conditions on Particle Properties in Fluidized Bed Layering Granulation

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## Abstract

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## 1. Introduction

## 2. Mathematical Model

#### 2.1. Population Balance Model

#### 2.2. Heat and Mass Transfer

## 3. Results and Discussion

#### 3.1. Variations of the Size of Milled Particles

#### 3.2. Variations of Injection Rate

#### 3.3. Variations of the Temperature of the Fluidization Medium

#### 3.4. Disturbance of the Inlet Moisture Content of the Fluidization Medium

## 4. Conclusions and Outlook

- caloric parameters (e.g., evaporation enthalpy $\mathsf{\Delta}{h}_{evap}$ or heat capacities ${c}_{p,i}$),
- heat and mass transfer coefficients (${\alpha}_{gp}$ and ${\beta}_{gp}$),
- parameters related to the drying curve (p, ${x}_{crit}$, and ${x}_{eq}$),
- parameters of the relation between shell porosity and drying potential ($\mathsf{\Delta}{\u03f5}_{shell}$ and ${\u03f5}_{shell,0}$), and
- parameters of the periphery (mill and screen).

## Author Contributions

## Funding

## Conflicts of Interest

## Nomenclature

A | surface area (m^{2}) |

${c}_{p}$ | specific heat capacity (J/kg·K)) |

d_{32} | Sauter diameter (m) |

G | particle growth (m/s) |

h | mass specific enthalpy (J/kg) |

H | total enthalpy (J) |

$\dot{H}$ | enthalpy flow (J/$\mathrm{s})$ |

K | drain (1/s) |

L | particle size (m) |

m | mass (kg) |

$\dot{m}$ | mass flow (kg/$\mathrm{s})$ |

n | number-based particle size distribution (1/$\mathrm{m})$ |

$\dot{n}$ | particle flow (1/$\mathrm{m}\phantom{\rule{4.pt}{0ex}}\xb7\phantom{\rule{4.pt}{0ex}}\mathrm{s})$ |

p | material specific drying characteristics (-) |

q | normalized particle size distribution (1/m) |

$\dot{Q}$ | heat flow (J/s) |

t | simulation time (h) |

T | separation function (-) |

V | volume (m^{3}) |

x | (mass) fraction of solid (-) |

X | moisture content of solid (g_{wet}/kg_{wet}) |

Y | moisture content of fluid (g_{wet}/kg_{wet}) |

Greek letters | |

$\alpha $ | heat transfer coefficient (W/m^{2}·K) |

$\beta $ | mass transfer coefficient (m/$\mathrm{s})$ |

$\delta $ | normalized moisture content (-) |

$\eta $ | drying potential (-) |

$\theta $ | temperature (${}^{\circ}\mathrm{C}$) |

$\mu $ | mean diameter (m) |

$\dot{\nu}$ | normalized drying velocity (-) |

$\mathsf{\Pi}$ | parameter set (-) |

$\rho $ | mass density (kg/m^{3}) |

$\sigma $ | variance (m) |

$\tau $ | time constant (h) |

Subscripts | |

dry | dry part of particles or fluidization medium |

evap | evaporation |

f | fluidization medium |

fp | fluid phase to particle phase |

fine | particle fine fraction |

in | inlet |

inj | injection |

l | liquid |

mill | milled particles |

out | particle withdrawal or fluid exhaust |

oversized | particle oversized fraction |

p | particle phase |

prod | particle product fraction |

recycle | particle recycle |

s | suspension |

sat | saturation point |

screen | screen |

solvent | solvent on particles or in fluidization medium |

shell | (particle) shell |

v | vapor |

## Appendix A. Default Parameter Set

particle phase | ||

$\phantom{\rule{1.em}{0ex}}{m}_{p,dry}$ | 15 | $\left(\mathrm{kg}\right)$ |

$\phantom{\rule{1.em}{0ex}}{c}_{p,p}$ | 4200 | $\left(\right)$ |

fluidization medium | ||

$\phantom{\rule{1.em}{0ex}}{m}_{f,dry}$ | 1 | $\left(\mathrm{kg}\right)$ |

$\phantom{\rule{1.em}{0ex}}{\dot{m}}_{f,dry,in}$ | 1500 | $\left(\right)$ |

$\phantom{\rule{1.em}{0ex}}{Y}_{f,in}$ | 6 | $\left(\right)$ |

$\phantom{\rule{1.em}{0ex}}{\theta}_{f,in}$ | 95 | $\left(\right)$ |

injected suspension | ||

$\phantom{\rule{1.em}{0ex}}{\dot{m}}_{inj}$ | 40 | $\left(\right)$ |

$\phantom{\rule{1.em}{0ex}}{x}_{inj,s}$ | $0.35$ | (-) |

$\phantom{\rule{1.em}{0ex}}{\theta}_{inj}$ | 20 | $\left(\right)$ |

$\phantom{\rule{1.em}{0ex}}{\rho}_{inj,s}$ | 1440 | kg/m^{3} |

drying characteristics and porosity | ||

$\phantom{\rule{1.em}{0ex}}p$ | $0.1$ | (-) |

$\phantom{\rule{1.em}{0ex}}{x}_{eq}$ | 5 | $\left(\right)$ |

$\phantom{\rule{1.em}{0ex}}{x}_{crit}$ | 50 | $\left(\right)$ |

$\phantom{\rule{1.em}{0ex}}{\u03f5}_{shell,0}$ | $0.45$ | (-) |

$\phantom{\rule{1.em}{0ex}}\mathsf{\Delta}{\u03f5}_{shell}$ | $-0.33$ | (-) |

screen, mill, and recycle | ||

$\phantom{\rule{1.em}{0ex}}{\mu}_{screes,I}$ | $1.00$ | $\left(\mathrm{mm}\right)$ |

$\phantom{\rule{1.em}{0ex}}{\sigma}_{screes,I}$ | $0.065$ | $\left(\mathrm{mm}\right)$ |

$\phantom{\rule{1.em}{0ex}}{\mu}_{screes,II}$ | $1.40$ | $\left(\mathrm{mm}\right)$ |

$\phantom{\rule{1.em}{0ex}}{\sigma}_{screes,II}$ | $0.055$ | $\left(\mathrm{mm}\right)$ |

$\phantom{\rule{1.em}{0ex}}{\mu}_{mill}$ | $0.80$ | $\left(\mathrm{mm}\right)$ |

$\phantom{\rule{1.em}{0ex}}{\sigma}_{mill}$ | $0.10$ | $\left(\mathrm{mm}\right)$ |

$\phantom{\rule{1.em}{0ex}}{X}_{recycle}$ | 0 | $\left(\right)$ |

$\phantom{\rule{1.em}{0ex}}{\theta}_{recyclej}$ | 20 | $\left(\right)$ |

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**Figure 3.**Flow sheet of fluidized bed layering granulation (FBLG). (PBM: population balance model; ODE: ordinary differential equation).

**Figure 4.**Normalized drying velocity $\dot{\nu}$ over moisture content of particles X as introduced by van Meel [24] dependent on different p.

**Figure 5.**Bidirectional coupling of particulate phase and thermal conditions represented by PBM $n\left(\right)open="("\; close=")">t,L$ and the system of ODEs, respectively.

**Figure 7.**Normalized particle size distribution ${q}_{0}\left(\right)open="("\; close=")">t,L$ related to the first, second, and third simulation scenario.

**Figure 8.**Simulation results of the first scenario: the setpoint switch of mean size of milled particles ${\mu}_{mill}$ from $0.8\phantom{\rule{4.pt}{0ex}}\mathrm{mm}$ to $0.7\phantom{\rule{4.pt}{0ex}}\mathrm{mm}$ at ${t}_{1}=2\phantom{\rule{4.pt}{0ex}}\mathrm{h}$ leads to the arising of self-sustained oscillations. After the reset of ${\mu}_{mill}$ at ${t}_{2}$, the oscillations decay and the dynamic system reaches steady state again.

**Figure 9.**Simulation results of the second scenario: an increase of injection rate ${\dot{m}}_{inj}$ from $40\phantom{\rule{4pt}{0ex}}\mathrm{kg}\phantom{\rule{-1.111pt}{0ex}}/\phantom{\rule{-0.55542pt}{0ex}}\mathrm{h}$ to $50\phantom{\rule{4pt}{0ex}}\mathrm{kg}\phantom{\rule{-1.111pt}{0ex}}/\phantom{\rule{-0.55542pt}{0ex}}\mathrm{h}$ at ${t}_{1}$ induces variations of thermal conditions, resulting in an increased particle porosity ${\u03f5}_{p}$. After resetting ${\dot{m}}_{inj}$ to the default value, the dynamic states and particle properties settle at the (initial) steady state again. Regard the different time scales of the step response at ${t}_{1}$ and ${t}_{2}$ illustrated in Figure 10.

**Figure 10.**Normalized particle porosity $\mathsf{\Delta}{\u03f5}_{p}$ surrounding the set point switches of ${\dot{m}}_{inj}$ at ${t}_{1}$ and ${t}_{2}$ according to the second simulation scenario.

**Figure 11.**Simulation results of the third scenario: the reduction of the temperature of the fluidization medium ${\theta}_{f,in}$ from $95{\phantom{\rule{3.33333pt}{0ex}}}^{\circ}\mathrm{C}$ to $70{\phantom{\rule{3.33333pt}{0ex}}}^{\circ}\mathrm{C}$ at ${t}_{1}$ shifts the saturation point. This induces variations of thermal conditions and thus to a change of shell ${\u03f5}_{shell}$ and particle porosity ${\u03f5}_{p}$. After resetting ${\theta}_{f,in}$ to the default value at ${t}_{2}$, the dynamic states and particle properties settle at the (initial) steady state again.

**Figure 12.**Simulation results of fourth scenario: the disturbance of the moisture content of the fluid at inlet ${Y}_{in}$ at ${t}_{1}$ results in small variations of the thermal conditions and particle properties.

**Table 1.**Simulation scenarios according to the simulation results presented in Section 3.1, Section 3.2, Section 3.3 and Section 3.4.

${\mathit{\mu}}_{\mathit{mill}}$ | ${\dot{\mathit{m}}}_{\mathit{inj}}$ | ${\mathit{\theta}}_{\mathit{f},\mathit{in}}$ | ${\mathit{Y}}_{\mathit{f},\mathit{in}}$ | |
---|---|---|---|---|

1st | $0.70\phantom{\rule{4pt}{0ex}}\mathrm{mm}$ | - | - | - |

2nd | - | $50.0\mathrm{kg}/\mathrm{h}$ | - | - |

3rd | - | - | 65.0 °C | - |

4th | - | - | - | $15.0{\mathrm{g}}_{\mathrm{f},\mathrm{wet}}/{\mathrm{kg}}_{\mathrm{f},\mathrm{dry}}$ |

**Table 2.**Operating parameters related to Rieck et al. [9].

No. | ${\mathit{\theta}}_{\mathit{f},\mathit{in}}$ | ${\dot{\mathit{m}}}_{\mathit{inj}}$ | $\mathit{\eta}$ | ${\mathit{\u03f5}}_{\mathit{sh}}$ |
---|---|---|---|---|

1 | 50 °C | $0.504\mathrm{kg}/\mathrm{h}$ | 0.79 | 0.50 |

2 | 50 °C | $0.967\mathrm{kg}/\mathrm{h}$ | 0.56 | 0.64 |

3 | 95 °C | $0.512\mathrm{kg}/\mathrm{h}$ | 0.89 | 0.46 |

4 | 95 °C | $1.277\mathrm{kg}/\mathrm{h}$ | 0.72 | 0.50 |

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## Share and Cite

**MDPI and ACS Style**

Neugebauer, C.; Bück, A.; Palis, S.; Mielke, L.; Tsotsas, E.; Kienle, A.
Influence of Thermal Conditions on Particle Properties in Fluidized Bed Layering Granulation. *Processes* **2018**, *6*, 235.
https://doi.org/10.3390/pr6120235

**AMA Style**

Neugebauer C, Bück A, Palis S, Mielke L, Tsotsas E, Kienle A.
Influence of Thermal Conditions on Particle Properties in Fluidized Bed Layering Granulation. *Processes*. 2018; 6(12):235.
https://doi.org/10.3390/pr6120235

**Chicago/Turabian Style**

Neugebauer, Christoph, Andreas Bück, Stefan Palis, Lisa Mielke, Evangelos Tsotsas, and Achim Kienle.
2018. "Influence of Thermal Conditions on Particle Properties in Fluidized Bed Layering Granulation" *Processes* 6, no. 12: 235.
https://doi.org/10.3390/pr6120235