Parametrization of the NRTL Model with a Multiobjective Approach: Implications in the Process Simulation
Abstract
:1. Introduction
2. Parametrization of Thermodynamic Models
2.1. Conventional Procedure
A Practical Case of Parametrization with the NRTL Model
2.2. Multi-Property Approach
The Multi-Property Approach in the NRTL Parametrization
2.3. Multi-Objective Resolution of the Multi-Property Approach
2.3.1. The Optimum in MOP
2.3.2. Resolution of the MOP
2.3.3. Resolution by ε-Constraint Decomposition
2.3.4. Resolution of the Multi-Property Problem as an MOP
2.3.5. Multi-Objective Modelling Using NRTL
2.4. Comments on the Strategies of Analyzed Modelling
3. Influence of Modelling on the Simulation of a Separation Process
3.1. Description of the Separation Process
- -
- Distillation column C-1: recovery of pentane in bottom streams;
- -
- Distillate, stream 3: product under conditions close to azeotrope;
- -
- Distillates of C-1 and C-2 (x1 ≈ xaz) are blended and sent to E-1;
- -
- E-1 exchanger cools to form two immiscible liquid phases;
- -
- Decanter S-1 separates the immiscible liquid phases;
- -
- Stream 5 (x1 < xaz) is recirculated to column C-1;
- -
- Stream 6 (x1 > xaz) feeds column C-2;
- -
- The ester is obtained by the C-2 bottom.
3.2. Results of the Simulation
4. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
Glossary
A | linear constraint coefficient matrix |
b | RHS linear constraint vector |
excess heat capacity, J·mol−1K−1 | |
g | nonlinear inequality constraint (Equation (1)) /molar Gibbs function (J·mol−1) (Equation (A1)) |
gE | molar Gibbs excess function (J·mol−1) |
NRTL binary interaction parameters (Equation (A8)) | |
h | molar enthalpy J·mol−1/nonlinear equality constraint (Equation (1)) |
hE | molar excess enthalpy, J·mol−1 |
M/(Mx, My) | number of properties in the multi-property formulation/apply to ML model fitting (Equation (3)) |
P | domain of model parameters |
P* | Pareto set (Equation (13)) |
p | system pressure, kPa |
PF* | Pareto front (Equation (14)) |
R | universal gas constant, 8314 J·mol−1K−1/reflux ratio |
sY(·) | root mean squared error applied to generic property Y |
sML(·) | maximum likelihood objective function (Equation (1)) |
T | temperature, K |
TE-out, | heat exchanger outlet temperature, K |
ΔTE | temperature difference between cold inlet and hot outlet |
uY | Experimental uncertainty of generic property Y (Equation (3)) |
v | molar volume, m3mol−1 |
vE | molar excess volume, m3mol−1 |
wm | weighing factor of m-th generic property in error function calculation |
x, xi, xaz | liquid-phase molar fraction vector /i-th element/composition coordinate of the azeotropic point |
Y | generic property |
y | vapor-phase molar fraction vector |
Greek letters | |
α12/α | non-randomness parameter (Equation (A5))/1st liquid phase identifier (I) |
β | 2nd liquid phase identifier (II) |
δ(·) | generic error metric |
εY | ε-constraint boundary for generic property Y (Equation (15)) |
Γij | NRTL pairwise interaction potential (Equation (A6)) |
γ | activity coefficients |
Θ | model vector of parameters |
Ξ | thermodynamic canonical set |
NRTL pairwise interaction energies (Equations (A6) and (A8)) | |
Ω | feasible space |
Acronyms | |
APV88 | Aspen© NRTL parametrization using MLE |
D/F | Distillate to feed ratio |
FO | objective functions vector (Equation (10)) |
LLE | liquid–liquid equilibria |
NRTL | Non-Random Two Liquids model, Appendix A |
MOP | Multi-objective problem |
P1, P2, P3, P4 | NRTL parametrization obtaining after solving MOO, stated by Equation (15) |
SLE | solid–liquid equilibria |
UCST | upper critical solubility temperature |
VLE | vapor–liquid equilibria |
Appendix A
Non-Random Two Liquids (NRTL) Molecular Model
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−7.261/−6.811 | 2375.23/1878.42 | 0/0 | 0/0 | 0.2 |
s(LLE) | s(VLE) | s(hE/RT) | ||
0.0142 | 0.2994 | 0.0179 |
1.18 × 106/29.28 | −2.87 × 107/105.6 | −2.09 × 105/−5.37 | 3.93 × 102/9.40×10−3 | 0.002 |
s(LLE) | s(VLE) | s(hE/RT) | ||
0.098 | 0.815 | 0.017 |
P1 | 20.000/18.845 | 2376.45/1879.56 | −7.817/−6.099 | 0.0691/0.0247 | 0.0308 |
P2 | 12.746/24.428 | 2379.16/1882.88 | −8.113/−6.249 | 0.1145/−0.0038 | 0.0144 |
P3 | 34.675/30.078 | 2377.47/1880.99 | −10.000/−10.000 | 0.1381/−0.0095 | 0.0027 |
P4 | 30.764/31.139 | 2377.50/1881.07 | −9.999/−9.445 | 0.1325/−0.0053 | 0.0041 |
s(LLE) | s(ELV) | s(hE/RT) | |||
P1 | 0.0137 | 0.2995 | 0.0203 | ||
P2 | 0.0338 | 0.0310 | 0.0175 | ||
P3 | 0.0201 | 0.1079 | 0.0860 | ||
P4 | 0.0220 | 0.1079 | 0.0180 |
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Fernández, L.; Ortega, J.; Sosa, A. Parametrization of the NRTL Model with a Multiobjective Approach: Implications in the Process Simulation. Thermo 2022, 2, 267-288. https://doi.org/10.3390/thermo2030019
Fernández L, Ortega J, Sosa A. Parametrization of the NRTL Model with a Multiobjective Approach: Implications in the Process Simulation. Thermo. 2022; 2(3):267-288. https://doi.org/10.3390/thermo2030019
Chicago/Turabian StyleFernández, Luis, Juan Ortega, and Adriel Sosa. 2022. "Parametrization of the NRTL Model with a Multiobjective Approach: Implications in the Process Simulation" Thermo 2, no. 3: 267-288. https://doi.org/10.3390/thermo2030019
APA StyleFernández, L., Ortega, J., & Sosa, A. (2022). Parametrization of the NRTL Model with a Multiobjective Approach: Implications in the Process Simulation. Thermo, 2(3), 267-288. https://doi.org/10.3390/thermo2030019