# A Population Balance Methodology Incorporating Semi-Mechanistic Residence Time Metrics for Twin Screw Granulation

^{*}

## Abstract

**:**

## 1. Introduction

#### 1.1. Twin Screw Granulation Population Balance Development

#### 1.2. Objectives

## 2. Materials and Methods

**FR**), processing screw speed (

**RPM**), liquid-to-solid percentage (

**LS**), screw configuration described by the number of kneading elements (

**NK**), and stagger angle (

**SA**) between them. The table also lists the available number of points sourced from each study.

## 3. Theory and Calculations

#### 3.1. Particle Grid Configuration

#### 3.2. PBE Configuration

#### 3.3. Aggregation and Breakage Rates

#### 3.4. PBE Compartmentalization

**NK**(two (2) compartments of six (6)

**NK**separated by a conveying section in each screw shaft) were modeled with seven (7) compartments. The remaining run cases with only a single kneading zone in the screw configuration were modeled as four (4) compartments.

#### 3.5. Axial Velocities and Dispersion Coefficients

#### 3.6. Numerical Techniques

#### 3.7. Output Metrics

#### 3.8. Parameter Estimation

## 4. Results and Discussion

#### 4.1. Quantitative Analysis—Parity Plots

#### 4.2. Qualitative Analysis—Correlation between RTD and PSD

**FR**10 kg/h, low

**NK**$1\times 4$, low

**SA**${30}^{\circ}$, and high

**RPM**of 900.

**FR**10 kg/h, low

**NK**$1\times 4$, low

**SA**${30}^{\circ}$, and high

**RPM**of 900).

**FR**25 kg/h, high

**NK**$2\times 6$, high

**SA**${90}^{\circ}$, and low

**RPM**of 900.

**FR**25 kg/h, high

**NK**$2\times 6$, high

**SA**${60}^{\circ}$, and low

**RPM**of 500).

#### 4.3. Qualitative Analysis—Compartmental Holdup

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

List of Acronyms | |

CFL condition | Courant–Freidrichs–Lewis condition |

DEM | discrete element method |

DOE | design of experiments |

FR | powder feed rate |

LS | liquid-to-solid ratio |

MRT | mean residence time |

NK | number of kneading elements |

PBE | population balance equation |

PEPT | positron emission particle tracking |

PSD | particle size distribution |

RMSE | root mean square error |

RPM | rotations per minute |

RTD | residence time distribution |

SA | stagger angle |

SSE | sum of square of errors |

TSG | twin screw granulation or twin screw granulator |

List of Symbols Used | ||

Symbol | Unit | Quantity |

z | unitless | external coordinate (spatial location) |

s | volume | internal coordinate (particle solid volume) |

t | time | time co-ordinate |

$N\left(z,s,t\right)$ | unitless, number | number of bulk particles |

$M\left(z,s,t\right)$ | unitless, number | number of tracer particles |

$\Re {\left(z,s,t\right)}_{Agg}$ | number/time | net aggregation rate of particles |

$\Re {\left(z,s,t\right)}_{Break}$ | number/time | net breakage rate of particles |

$\dot{F}{\left(z,s,t\right)}_{In}$ | number/time | rate of bulk particles coming into the |

compartment | ||

$\dot{F}{\left(z,s,t\right)}_{Out}$ | number/time | rate of bulk particles going out of the |

compartment | ||

$\dot{G}{\left(z,s,t\right)}_{In}$ | number/time | rate of tracer particles coming into the |

compartment | ||

$\dot{G}{\left(z,s,t\right)}_{Out}$ | number/time | rate of tracer particles going out of the |

compartment | ||

${v}_{bulk}\left(z\right)$ | length/time | axial velocity of bulk particles |

leaving the compartment | ||

${v}_{tracer}\left(z\right)$ | length/time | axial velocity of tracer particles |

leaving the compartment | ||

${D}_{bulk}\left(z\right)$ | length${}^{2}$/time | axial dispersion coefficient of bulk |

particles leaving the compartment | ||

${D}_{tracer}\left(z\right)$ | length${}^{2}$/time | axial dispersion coefficient of tracer |

particles leaving the compartment | ||

${l}_{bulk}\left(z,s,t\right)$ | volume | liquid volume associated with bulk particle |

having coordinates $z,s,t$ | ||

${l}_{tracer}\left(z,s,t\right)$ | volume | liquid volume associated with tracer particle |

having coordinates $z,s,t$ | ||

$\Re {\left(z,s,t\right)}_{Agg}^{form}$ | number/time | net formation rate of particles from aggregation |

$\Re {\left(z,s,t\right)}_{Agg}^{dep}$ | number/time | net depletion rate of particles due to aggregation |

$\beta (z,{s}^{\prime},s-{s}^{\prime},t)$ | number${}^{-1}$time${}^{-1}$ | specific aggregation rate between two chosen |

size classes of particles | ||

$\Re {\left(z,s,t\right)}_{Brk}^{form}$ | number/time | net formation rate of particles from breakage |

$\Re {\left(z,s,t\right)}_{Brk}^{dep}$ | number/time | net depletion rate of particles due to breakage |

$b(s,{s}^{\prime})$ | unitless | probability of a larger number particle of size class |

breaking into 2 smaller particles | ||

${K}_{break}(z,{s}^{\prime},t)$ | number/time | specific breakage rate of a particle |

${\beta}_{0}$ | time${}^{-1}$ | aggregation rate pre-constant |

$\alpha $ | unitless | aggregation liquid depndency |

enhancing parameter | ||

$\delta $ | unitless | aggregation liquid dependency |

diminishing parameter | ||

$vol(z,s,t)$ | volume | total volume of the particle |

${G}_{shear}$ | time${}^{-1}$ | shear rate imparted due to screw rotation |

${P}_{1}$ | unitless | breakage rate pre-constant |

${P}_{2}$ | unitless | breakage liquid dependency |

L | length | length of compartment of interest |

$MRT$ | time | mean residence time |

$F{R}_{vol,net}$ | volume/time | total volumetric flow rate |

of material into the system | ||

$Availvo{l}_{total}$ | volume | available vloume for particles to fill up |

inside the equipment | ||

$Dispvolrat{e}_{conv,total}$ | volme/time | volumetric dispense rate |

of materials per turn of screws | ||

${b}_{1}$ | time | scaling factor for the MRT |

${b}_{2}$ | unitless | effect of material throughput on Holdup factor |

${b}_{3}$ | unitless | effect of material throughput on Flow factor |

${b}_{4}$ | unitless | effect of volumetric dispense rate on flow factor |

$Pe$ | unitless | Péclet number |

$S{A}_{knead,deg}$ | unitless | stagger angle between kneading elements in degrees |

$N{K}_{knead}$ | unitless | number of neakding elements in kneading block of concern |

${b}_{5}$ | unitless | scaling factor for the Péclet number |

${b}_{6}$ | unitless | effect of volumetric dispense rate on mixing factor |

$MR{T}_{tracer}^{*}$ | unitless | scaling constant indicating ratio of MRT of tracer |

relative to MRT of bulk material | ||

$P{e}_{tracer}^{*}$ | unitless | scaling constant indicating ratio of Pe of tracer |

relative to Pe of bulk material | ||

SSE | unitless | sum of square of errors |

RMSE | unitless | root mean square of the errors |

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**Figure 1.**Schematic showing (

**a**) 7 compartment modeling of equipment for 12

**KE**configurations; and (

**b**) 4 compartment modeling of equipment for 2, 4, and 6

**KE**configurations.

**Figure 2.**Schematic showing the convective and dispersive fluxes entering and leaving a compartment with index ‘i’.

**Figure 3.**Experimental observations (X) vs. predicted model responses (Y) for RT metrics when model is trained on the Kumar et al., 2015 [18] dataset. (

**a**) MRT; (

**b**) variance.

**Figure 4.**Experimental observations (X) vs. predicted model responses (Y) for RT metrics when the model is trained trained Kumar et al., 2016 [19] dataset. (

**a**) MRT; (

**b**) variance.

**Figure 5.**Experimental observations (X) vs. predicted model responses (Y) for the Variance when the model is trained on the Kumar et al., 2016 [19] dataset without including the dispersion flow rates.

**Figure 6.**Experimental observations (X) vs. predicted model responses (Y) for outlet bulk granule sizes when the model is trained on the Kumar et al., 2016 [19] dataset.

**Figure 7.**Experimental observations (X) vs. predicted model responses (Y) for outlet fine and coarse granule sizes when the model is trained on the Kumar et al., 2016 [19] dataset. (

**a**) fines; (

**b**) coarse.

**Figure 8.**RTD trend obtained from a PBM pulse tracer study for an unfavorable granulation case from the experimental DOE of Kumar et al., 2016 [19].

**Figure 9.**PSD trend obtained from a PBM pulse tracer study for unfavorable granulation case from the DOE of Kumar et al., 2016 [19].

**Figure 10.**RTD trend obtained from a PBM pulse tracer study for a favorable granulation case from the experimental DOE of Kumar et al., 2016 [19].

**Figure 11.**Kumar et al., 2016 [19] PSD trend obtained from a PBM pulse tracer study for favorable granulation case.

**Figure 12.**Compartmental holdup obtained obtained from the PBE model when trained on the system as used by Kumar et al., 2016 dataset [19].

**Table 1.**Summary of the twin screw granulation RTD and sieve fractions available and collected from literature.

Data Source | Equipment Name | Process Material | Varied Parameters | Number of Data Points |
---|---|---|---|---|

Kumar et al., 2015 [18] | ConsiGma-25 | $\alpha $-Lactose MH | FR, RPM, NK & SA | 66 |

Kumar et al., 2016 [19] | ConsiGma-25 | $\alpha $-Lactose MH | FR, RPM, LS, NK & SA | 51 |

Total: 117 |

Constant | Kumar et al., 2015 [18] Value | Kumar et al., 2016 [19] Value | Unit |
---|---|---|---|

${\beta}_{0}$ | 9.38 | 7.91 | ×0.1 s${}^{-1}$ |

$\alpha $ | 0.13 | 0.10 | - |

$\delta $ | 0 | 0 | - |

${P}_{1}$ | 38.77 | 80.86 | - |

${P}_{2}$ | 0.28 | 0.24 | - |

${b}_{1,dry}$ | 1.20 | 0.97 | - |

${b}_{1,knead}$ | 0.16 | 0.15 | - |

${b}_{1,wet}$ | 0.88 | 0.81 | - |

${b}_{5,dry}$ | 7.19 | 1.13 | - |

${b}_{5,knead}$ | 6.53 | 0.38 | - |

${b}_{5,wet}$ | 5.18 | 1.14 | - |

$MR{T}_{tracer}^{*}$ | 0.55 | 0.45 | - |

$P{e}_{tracer}^{*}$ | 2.10 | 0.95 | - |

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**MDPI and ACS Style**

Muddu, S.V.; Ramachandran, R. A Population Balance Methodology Incorporating Semi-Mechanistic Residence Time Metrics for Twin Screw Granulation. *Processes* **2022**, *10*, 292.
https://doi.org/10.3390/pr10020292

**AMA Style**

Muddu SV, Ramachandran R. A Population Balance Methodology Incorporating Semi-Mechanistic Residence Time Metrics for Twin Screw Granulation. *Processes*. 2022; 10(2):292.
https://doi.org/10.3390/pr10020292

**Chicago/Turabian Style**

Muddu, Shashank Venkat, and Rohit Ramachandran. 2022. "A Population Balance Methodology Incorporating Semi-Mechanistic Residence Time Metrics for Twin Screw Granulation" *Processes* 10, no. 2: 292.
https://doi.org/10.3390/pr10020292