An MINLP Model that Includes the Effect of Temperature and Composition on Property Balances for Mass Integration Networks
Abstract
:1. Introduction
2. Problem Statement
3. Model Formulation
3.1. Splitting of Process Streams
3.2. Splitting of Fresh Streams
3.3. Mass Balance at Inlet of Process Interceptors
3.4. Property Balance at Inlet of Property Interceptors
Property | Operator |
---|---|
Composition | ψz(z) = z |
Toxicity | ψTox(Tox) = Tox |
Chemical Oxygen Demand | ψCOD(COD) = COD |
pH | ψpH(pH) = 10pH |
Density | |
Viscosity | ψμ(μ) = log (μ) |
3.5. Energy Balances at the Inlet of Property Interceptors
3.6. Property Interceptors
3.7. Stream Splitting at the Outlet of Each Property Interceptor
3.8. Mass Balance for Process Sinks
3.9. Property Balance for Process Sinks
3.10. Energy Balance for Process Sinks
3.11. Mass Balance for Waste Stream
3.12. Property Balance in Waste Stream
3.13. Energy Balance in Waste Stream
3.14. Constraints
3.15. Modification of Properties
3.16. Properties as a Function of Temperature
4. Methodology for Estimation of Parameters
- (1)
- From the set of data, Equation (31) is used to obtain the parameter values. Temperature, , concentration ψZ(PX) and the uncorrected property operator are independent variables.
- (2)
- To estimate the parameters, takes the value for the property operator p(T) for the original process streams.
- (3)
- With the calculated parameters, the equation is implemented into the MINLP model.
4.1. Viscosity
4.2. Density
5. Case Study
Stream | Flow (kg/h) | Z | Tox (%) | COD (mgO2/L) | pH | ρ* (kg/m3) | µ* (cP) | T (K) | Cp** (kJ/kg K) |
---|---|---|---|---|---|---|---|---|---|
R1 | - | 0 | 0 | 0 | 7.0 | 994.0 | 0.913 | 298.15 | 4.1845 |
R2 | - | 0.010 | 0.1 | 0.010 | 7.1 | 986.1 | 0.743 | 308.15 | 4.1575 |
W1 | 3666 | 0.016 | 0.3 | 0.187 | 5.4 | 947.1 | 0.382 | 348.15 | 4.1601 |
W2 | 1769 | 0.024 | 0.5 | 48.450 | 5.1 | 958.8 | 0.442 | 338.15 | 4.1363 |
W3 | 1487 | 0.220 | 1.5 | 92.100 | 4.8 | 1022.1 | 0.745 | 313.15 | 3.7280 |
Stream | Flow (kg/h) | Zmax | Toxmax (%) | CODmax (mgO2/L) | pH max | ρmax (kg/m3) | µmax (cP) |
---|---|---|---|---|---|---|---|
G1 | 2721 | 0.013 | 2 | 100 | 8.0 | 1270 | 1.202 |
G2 | 1995 | 0.011 | 2 | 100 | 7.8 | 1113 | 2.230 |
G3 | 1129 | 0.100 | 2 | 100 | 8.2 | 1315 | 1.260 |
Waste | - | 0.005 | 0.001 | 75 | 9.0 | - | - |
Sink/Waste | Flow (kg/h) | pHmin | ρmin (kg/m3) | µmin (cP) |
---|---|---|---|---|
G1 | 2721 | 5.3 | 816 | 0.2 |
G2 | 1995 | 5.4 | 771 | 0.2 |
G3 | 1129 | 5.2 | 839 | 0.2 |
Waste | - | 5.5 | - | - |
Sink/Waste | Tmin (K) | Tmax (K) |
---|---|---|
G1 | 333.15 | 353.15 |
G2 | 303.15 | 348.15 |
G3 | 298.15 | 338.15 |
Waste | 290.15 | 308.15 |
Property | αu,m | ($/kg) |
---|---|---|
u(z1) | 0.98 | 0.0143 |
u(z2) | 0.85 | 0.0073 |
u(Tox1) | 1.00 | 0.0216 |
u(Tox2) | 0.90 | 0.0165 |
u(COD1) | 0.80 | 0.0143 |
u(COD2) | 0.55 | 0.0071 |
u(pH1) | 0.50 | 0.1389 |
u(pH2) | 0.30 | 0.0397 |
u(pH3) | −0.50 | 0.1433 |
u(pH4) | −0.30 | 0.0419 |
Compound | Ac | Bc | Cc |
---|---|---|---|
Water | 5.8221 | −0.01033 | 0.0000162 |
Phenol | 1.0809 | 0.003375 | 0 |
p(T) | C1,p(T) | C2,p(T) | C3,p(T) | C4,p(T) |
---|---|---|---|---|
Viscosity | −2.6816 | 786.50 | 0.18400 | −2.0699 × 10−5 |
Density | 6.8632 × 10−4 | 1.0715 × 10−6 | −1.9655 × 10−4 | −3.9471 × 10−4 |
Z | Tox (%) | COD (mgO2/L) | pH | ρ (kg/m3) | µ (cP) | T (K) | |
---|---|---|---|---|---|---|---|
G1 | 0.013 | 0.4978 | 22.549 | 5.30 | 953.3 | 0.4220 | 341.28 |
G2 | 0.011 | 0.2043 | 19.515 | 6.79 | 980.7 | 0.6740 | 313.57 |
G3 | 0.079 | 0.6707 | 28.577 | 5.29 | 968.3 | 0.4558 | 338.15 |
Waste | 0.005 | 0.0010 | 39.828 | 5.50 | 985.1 | 0.7441 | 308.15 |
Source | G1 | G2 | G3 | Waste |
---|---|---|---|---|
Aspen | 953.64 | 981.01 | 968.81 | 985.20 |
Model with p(z,T) | 953.29 | 980.68 | 968.34 | 985.12 |
Error, % | 0.04 | 0.033 | 0.049 | 0.01 |
Model, no p(z,T) | 958.72 | 980.36 | 969.10 | 956.22 |
Error, % | 0.53 | 0.07 | 0.03 | 2.94 |
Source | G1 | G2 | G3 | Waste |
---|---|---|---|---|
Aspen | 0.4208 | 0.6686 | 0.4506 | 0.7414 |
Model with p(T) | 0.4220 | 0.6740 | 0.4558 | 0.7441 |
Error, % | 0.29 | 0.80 | 1.17 | 0.37 |
Model, no p(T) | 0.4297 | 0.6898 | 0.4694 | 0.4526 |
Error, % | 2.12 | 3.17 | 4.18 | 38.95 |
6. Conclusions
Acknowledgments
Author Contributions
Nomenclature
A,B,C | empirical parameters |
COD | chemical oxygen demand |
CostFresh | cost of fresh sources |
CostInt | cost of property interceptor |
CP | heat capacity |
d | stream flowrate |
F, f | flowrate of fresh sources |
G, g | flowrate of process sinks |
HV | operating hours per year |
p, p' | any property |
q | flowrate from interceptors |
R, r | Flowrate of fresh sources |
sink | process sink |
source | fresh or stream source |
T | temperature |
TAC | total annual cost |
Tox | toxicity |
x | mole fraction |
W, w | flowrate of process streams |
waste | waste discharged to the environment |
Y | Boolean variable |
y | binary variable |
z | pollutant concentration |
Sets
I | process streams |
J | sinks |
M | properties to be treated |
R | fresh sources |
Up | treatment units for each property |
Subindices
i | process stream |
j | sink |
r | fresh source |
Superscripts
C | corrected |
In | inlet |
Int | interceptor |
max | maximum |
min | minimum |
Out | outlet |
UnC | uncorrected |
Greek symbols
α | separation efficiency of property interceptor |
Ψ | property operator |
ρ | density |
μ | viscosity |
Conflicts of Interest
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Jiménez-Gutiérrez, A.; Sandate-Trejo, M.D.C.; El-Halwagi, M.M. An MINLP Model that Includes the Effect of Temperature and Composition on Property Balances for Mass Integration Networks. Processes 2014, 2, 675-693. https://doi.org/10.3390/pr2030675
Jiménez-Gutiérrez A, Sandate-Trejo MDC, El-Halwagi MM. An MINLP Model that Includes the Effect of Temperature and Composition on Property Balances for Mass Integration Networks. Processes. 2014; 2(3):675-693. https://doi.org/10.3390/pr2030675
Chicago/Turabian StyleJiménez-Gutiérrez, Arturo, María Del Carmen Sandate-Trejo, and Mahmoud M. El-Halwagi. 2014. "An MINLP Model that Includes the Effect of Temperature and Composition on Property Balances for Mass Integration Networks" Processes 2, no. 3: 675-693. https://doi.org/10.3390/pr2030675