Predicting the Temperature Evolution during Nanomilling of Drug Suspensions via a Semi-Theoretical Lumped-Parameter Model
Abstract
:1. Introduction
2. Materials and Methods
2.1. Materials
2.2. Methods
2.2.1. Wet Stirred Media Milling
2.2.2. Formulation of the Lumped-Parameter Model (LPM)
2.2.3. Fits by the LPM and Predictions by the LPM Augmented with the PL and ML Models
3. Results and Discussion
3.1. Properties of the Milled Suspensions and Particles
3.2. Fitted LPM Parameters and the Origin of Temperature Rise during the Milling
3.3. LPM-Fitted Temperature Profiles and LPM–PL/LPM-ML Predictions in the Training Runs
3.4. Comparative Analysis of LPM and EBM Fits and Their Predictions for the Test Runs
3.5. The LPM and the EBM Comparison and the Limitations of the LPM
3.6. A holistic Perspective on the Impact of Process Parameters
4. Conclusions and Future Outlook
Supplementary Materials
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
Abbreviations
A | heat transfer surface area, m2 |
Am | heat transfer surface area of the mill chamber, m2 |
c | bead loading or fractional volumetric concentration of the beads, – |
Cp | specific heat capacity, J/g °C |
EBM | enthalpy balance model |
Db | bead size, µm |
LPM | lumped-parameter model |
m | mass in the mill chamber, g |
ML | machine learning |
PL | power law |
P | power consumption, J/min |
Pξ | heat generation rate during milling, retrieved from EBM study, J/min |
Qgen | apparent heat generation during milling, J/min |
RMSE | root-mean-squared error, °C |
T | temperature at the mill outlet, °C |
Tch | chiller temperature, °C |
Trise | temperature rise at 6 min of milling, °C |
T0 | initial temperature at the mill outlet, °C |
t | milling time, min |
U | overall heat transfer coefficient, W/m2 °C |
UA | apparent overall heat transfer coefficient times surface area, J/min °C |
UAm | apparent overall heat transfer coefficient times surface area in the mill chamber, J/min °C |
ξ | fraction of mechanical power converted into heat, – |
ω | stirrer (rotational) speed, rpm |
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Run No. | Stirrer Speed, ω (rpm) | Bead Loading, c (-) | Bead Size, Db (µm) |
---|---|---|---|
1 1 | 2000 | 0.4 | 200 |
2 1 | 2000 | 0.4 | 400 |
3 1 | 2000 | 0.4 | 800 |
4 1 | 2000 | 0.5 | 200 |
5 1 | 2000 | 0.5 | 400 |
6 1 | 2000 | 0.5 | 800 |
7 1 | 2000 | 0.6 | 200 |
8 1 | 2000 | 0.6 | 400 |
9 1 | 2000 | 0.6 | 800 |
10 1 | 3000 | 0.4 | 200 |
11 1 | 3000 | 0.4 | 400 |
12 1 | 3000 | 0.4 | 800 |
13 1 | 3000 | 0.5 | 200 |
14 1 | 3000 | 0.5 | 400 |
15 1 | 3000 | 0.5 | 800 |
16 1 | 3000 | 0.6 | 200 |
17 1 | 3000 | 0.6 | 400 |
18 1 | 3000 | 0.6 | 800 |
19 1 | 4000 | 0.4 | 200 |
20 1 | 4000 | 0.4 | 400 |
21 1 | 4000 | 0.4 | 800 |
22 1 | 4000 | 0.5 | 200 |
23 1 | 4000 | 0.5 | 400 |
24 1 | 4000 | 0.5 | 800 |
25 1 | 4000 | 0.6 | 200 |
26 1 | 4000 | 0.6 | 400 |
27 1 | 4000 | 0.6 | 800 |
28 2 | 2500 | 0.45 | 400 |
29 2 | 2500 | 0.55 | 400 |
30 2 | 3500 | 0.45 | 400 |
31 2 | 3500 | 0.55 | 400 |
32 2 | 4000 | 0.35 | 100 |
Run No: Identifier | Qgen (J/min) | UA (J/min °C) | RMSE (°C) |
---|---|---|---|
1: 2000 0.4 200 | 755.2 | 47.79 | 0.40 |
2: 2000 0.4 400 | 1616 | 103.5 | 0.46 |
3: 2000 0.4 800 | 787.4 | 48.08 | 0.56 |
4: 2000 0.5 200 | 441.4 | 25.82 | 0.32 |
5: 2000 0.5 400 | 720.8 | 40.18 | 0.33 |
6: 2000 0.5 800 | 1296 | 72.88 | 0.39 |
7: 2000 0.6 200 | 1343 | 68.63 | 0.44 |
8: 2000 0.6 400 | 2625 | 131.9 | 0.15 |
9: 2000 0.6 800 | 1822 | 89.95 | 0.50 |
10: 3000 0.4 200 | 1402 | 53.92 | 0.55 |
11: 3000 0.4 400 | 2938 | 107.0 | 0.57 |
12: 3000 0.4 800 | 2798 | 101.0 | 0.66 |
13: 3000 0.5 200 | 1837 | 61.49 | 0.40 |
14: 3000 0.5 400 | 3243 | 107.3 | 0.72 |
15: 3000 0.5 800 | 3307 | 107.4 | 0.83 |
16: 3000 0.6 200 | 2624 | 94.08 | 0.43 |
17: 3000 0.6 400 | 4598 | 128.2 | 0.76 |
18: 3000 0.6 800 | 4542 | 124.4 | 0.80 |
19: 4000 0.4 200 | 5075 | 135.1 | 0.42 |
20: 4000 0.4 400 | 6266 | 162.7 | 0.41 |
21: 4000 0.4 800 | 6245 | 162.5 | 0.51 |
22: 4000 0.5 200 | 6116 | 162.7 | 0.90 |
23: 4000 0.5 400 | 8490 | 220.0 | 0.30 |
24: 4000 0.5 800 | 9359 | 238.1 | 0.60 |
25: 4000 0.6 200 | 8383 | 208.9 | 0.62 |
26: 4000 0.6 400 | 10,600 | 261.7 | 0.28 |
27: 4000 0.6 800 | 10,740 | 171.9 | 0.34 |
Runs | Direct Fitting | PL Prediction | ML Prediction | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
Qgen (J/min) | UA (J/min °C) | LPM RMSE (°C) | EBM RMSE (°C) 1 | Qgen (J/min) | UA (J/min °C) | LPM RMSE (°C) | EBM RMSE (°C) 1 | Qgen (J/min) | UA (J/min °C) | LPM RMSE (°C) | EBM RMSE (°C) 1 | |
2500 0.45 400 | 1481 | 68 | 0.35 | 0.51 | 1634 | 78 | 0.74 | 0.93 | 1792 | 77 | 1.51 | 1.02 |
2500 0.55 400 | 2495 | 101 | 0.42 | 0.59 | 2118 | 91 | 1.49 | 1.02 | 2506 | 95 | 1.44 | 1.00 |
3500 0.45 400 | 4084 | 118 | 0.50 | 1.56 | 4519 | 137 | 1.56 | 2.22 | 4555 | 132 | 0.59 | 2.17 |
2500 0.55 400 | 5567 | 147 | 0.56 | 1.01 | 5856 | 160 | 1.02 | 1.17 | 5911 | 162 | 1.19 | 1.29 |
4200 0.35 100 | 3185 | 84 | 0.66 | 1.13 | 4180 | 129 | 4.13 | 4.17 | 4359 | 124 | 2.08 | 1.63 |
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Guner, G.; Yilmaz, D.; Yao, H.F.; Clancy, D.J.; Bilgili, E. Predicting the Temperature Evolution during Nanomilling of Drug Suspensions via a Semi-Theoretical Lumped-Parameter Model. Pharmaceutics 2022, 14, 2840. https://doi.org/10.3390/pharmaceutics14122840
Guner G, Yilmaz D, Yao HF, Clancy DJ, Bilgili E. Predicting the Temperature Evolution during Nanomilling of Drug Suspensions via a Semi-Theoretical Lumped-Parameter Model. Pharmaceutics. 2022; 14(12):2840. https://doi.org/10.3390/pharmaceutics14122840
Chicago/Turabian StyleGuner, Gulenay, Dogacan Yilmaz, Helen F. Yao, Donald J. Clancy, and Ecevit Bilgili. 2022. "Predicting the Temperature Evolution during Nanomilling of Drug Suspensions via a Semi-Theoretical Lumped-Parameter Model" Pharmaceutics 14, no. 12: 2840. https://doi.org/10.3390/pharmaceutics14122840