Global Sensitivity Analysis of a Spray Drying Process
Abstract
:1. Introduction
2. Materials and Methods
2.1. Spray Drying Model
2.2. Global Sensitivity Analysis
2.2.1. Variance-Based Sensitivity Analysis
2.2.2. Computation of Sensitivity Indices by Saltelli’s Method
- Generate a sample matrix of using the Sobol sequences. The sample matrix is split into two data matrices A (Equation (24)) and B (Equation (25)), each containing half of the samples. N is the number of samples to be used for computing the indices. The order of N can vary between a few hundreds to a few thousands.
- Define matrix C (Equation (26)) as the matrix with all columns of B except the column, which is taken from A.
- Compute the model output for all the input values in the three matrices .
- The first order sensitivity indices are estimated as:The total sensitivity indices are estimated as:
2.2.3. Computation of Sensitivity Indices Using Arbitrary Polynomial Chaos Expansions
2.3. GSA of the Spray Drying Process
3. Results
3.1. Computation of Sensitivity Indices
3.2. Global Sensitivity Analysis of Spray Dryer with a Pneumatic Nozzle
3.3. Global Sensitivity Analysis of the Spray Dryer with a Pressure Nozzle
4. Conclusions
Author Contributions
Funding
Conflicts of Interest
Abbreviations
ALR | Air liquid ratio | |
CMA | Critical material attributes | |
CPP | Critical process parameters | |
CQA | Critical quality attributes | |
GSA | Global sensitivity analysis | |
PCE | Polynomial chaos expansions | |
QbD | Quality by design | |
List of Symbols | ||
Heat transfer coefficient | ||
Mass transfer coefficient | ||
Viscosity | ||
Density | ||
- | Discharge coefficient | |
- | Drag coefficient | |
d | m | Diameter |
Effective diffusivity | ||
g | Acceleration due to gravity | |
Enthalpy of evaporation | ||
kg | Solid mass | |
Mass transfer rate | ||
M | Mass flow rate | |
Molecular weight | ||
- | Nusselt number | |
- | Ohnsorge number | |
P | Pa | Pressure |
Pa | Vapor pressure | |
Q | Volumetric flow rate | |
- | Reynolds number | |
- | First order sensitivity index | |
- | Total sensitivity index | |
T | K | Temperature |
U | Overall heat transfer coefficient | |
v | Velocity | |
W | - | Residual solvent content |
- | Weber number | |
- | Gas moisture content | |
- | Gas saturation moisture content | |
Z | m | Axial distance in the spray drying chamber |
Subscripts | ||
a | Air/gas | |
c | Critical | |
Equilibrium | ||
l | Feed liquid | |
n | Nozzle | |
p | Particle | |
s | Solids | |
w | Water/solvent |
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Variable | Symbol | Range of Variation | Units | ||
---|---|---|---|---|---|
Input variables | Flow of the liquid feed | – | |||
Flow of the drying gas | – | ||||
Initial Temp.of the gas | 350–420 | K | |||
Initial Temp. of the liquid | 300–350 | K | |||
Viscosity of the liquid | – | Pa·s | |||
Air-to-liquid ratio | 0.5–4 | - | |||
Output variables | Particle diameter | - | |||
Residual solvent content | - | - | |||
Particle density | - | ||||
Length for equilibrium | - | m | |||
Particle’s maximum temperature | - | K | |||
Upstream liquid pressure | P | - | bar |
Rank | dp | Wp | Zeq | |||||||
---|---|---|---|---|---|---|---|---|---|---|
Sal | PCE | Sal | PCE | Sal | PCE | Sal | PCE | Sal | PCE | |
1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | |
2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | |
- | - | 2 | 2 | 2 | 2 | 1 | 1 | 3 | 3 | |
- | - | 3 | 3 | 3 | 3 | 3 | 3 | 1 | 1 | |
- | - | - | - | - | - | - | - | 4 | 3 |
Parameter | PCE | PCE | PCE | PCE | PCE | Saltelli |
---|---|---|---|---|---|---|
0.3826 | 0.4481 | 0.4444 | 0.5030 | 0.5031 | 0.4920 | |
0.0706 | 0.1652 | 0.1789 | 0.2975 | 0.2352 | 0.2350 | |
0.1596 | 0.2315 | 0.3127 | 0.3825 | 0.3498 | 0.3582 | |
0.4883 | 0.4626 | 0.4463 | 0.5011 | 0.4728 | 0.5352 | |
0.2341 | 0.2514 | 0.2939 | 0.1907 | 0.2964 | 0.2289 | |
# of Samples | 200 | 400 | 600 | 1000 | 2000 | 23,000 |
# of Function Evaluations | 200 | 400 | 600 | 1000 | 2000 | 161,000 |
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Bhonsale, S.; Muñoz López, C.A.; Van Impe, J. Global Sensitivity Analysis of a Spray Drying Process. Processes 2019, 7, 562. https://doi.org/10.3390/pr7090562
Bhonsale S, Muñoz López CA, Van Impe J. Global Sensitivity Analysis of a Spray Drying Process. Processes. 2019; 7(9):562. https://doi.org/10.3390/pr7090562
Chicago/Turabian StyleBhonsale, Satyajeet, Carlos André Muñoz López, and Jan Van Impe. 2019. "Global Sensitivity Analysis of a Spray Drying Process" Processes 7, no. 9: 562. https://doi.org/10.3390/pr7090562
APA StyleBhonsale, S., Muñoz López, C. A., & Van Impe, J. (2019). Global Sensitivity Analysis of a Spray Drying Process. Processes, 7(9), 562. https://doi.org/10.3390/pr7090562