# An MINLP Model that Includes the Effect of Temperature and Composition on Property Balances for Mass Integration Networks

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Problem Statement

_{j}, and its contents have some property values ${P}_{j,p}^{InSink}$ that are constrained between minimum and maximum values. In addition, given is a set of process streams {i = 1, 2,…, I} with a flowrate W

_{i}that can be recycled and/or reused in process sinks. There is set of fresh sources {r = 1, 2,…, R} with costs $Cos{t}_{r}^{Fresh}$ ($/kg) and properties ${P}_{p,r}^{InFresh}$. There is a set of property interceptors {m = 1, 2,..., M}. Each property can be modified with the set of treatment units {u = 1, 2,…, U

_{p}}, with cost $Cos{t}_{u,m}^{Int}$, and given separation efficiencies α

_{u,m}. The waste stream must meet environmental constraints. The objective is synthesizing an optimal water network such that the total annual cost is minimized.

## 3. Model Formulation

#### 3.1. Splitting of Process Streams

#### 3.2. Splitting of Fresh Streams

_{r}

_{,j}that can be sent to different process sinks,

#### 3.3. Mass Balance at Inlet of Process Interceptors

_{m}, that treats property p' is the summation of the flows from process streams ${w}_{i,m}^{Int}$, and the flow from other interceptors q

_{m}

_{,m'},

#### 3.4. Property Balance at Inlet of Property Interceptors

_{m}is calculated through the following balance,

Property | Operator |
---|---|

Composition | ψ_{z}(z) = z |

Toxicity | ψ_{Tox}(Tox) = Tox |

Chemical Oxygen Demand | ψ_{COD}(COD) = COD |

pH | ψ_{pH}(pH) = 10pH |

Density | ${\psi}_{\rho}\left(\rho \right)=\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\rho $}\right.$ |

Viscosity | ψ_{μ}(μ) = log (μ) |

#### 3.5. Energy Balances at the Inlet of Property Interceptors

_{0}is a reference temperature, ${T}_{i}^{InSource}$ is the temperature of process stream i, ${T}_{m\text{'}}^{OutInt}$ is the outlet temperature from other interceptors m', ${T}_{m}^{InInt}$ is the inlet temperature to property interceptor d

_{m}, and Cps are heat capacities.

#### 3.6. Property Interceptors

_{u}

_{,m}is a Boolean variable, which when true implies the selection of the intercepting unit m with total cost $Cos{t}_{m}^{Int}$ operating with an efficiency separation factor α

_{u,m}; $\psi \left({p}_{m}^{InInt}\right)$ refers to the inlet value of property p, and $\psi \left({p}_{m}^{OutInt}\right)$ is the property value at the exit of the unit.

_{u}

_{,m}is equal to one, the unit m is used; otherwise, it is equal to zero. Since at most one type of interceptor is selected for the treatment of each property, the summation of the integer variables is bounded to one,

#### 3.7. Stream Splitting at the Outlet of Each Property Interceptor

_{m,m'}and ${g}_{m}^{Waste}$ that go to process sinks, other property interceptors, and waste,

#### 3.8. Mass Balance for Process Sinks

_{j}, is the summation of flows from process streams, ${w}_{i,j}^{Sink}$, from property interceptors, ${g}_{m,j}^{Sink}$, and from fresh streams, f

_{r,j},

#### 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

_{v}is an annualization factor.

#### 3.15. Modification of Properties

#### 3.16. Properties as a Function of Temperature

## 4. Methodology for Estimation of Parameters

_{1,p(T)}, C

_{2,p(T)}, C

_{3,p(T)}, C

_{4,p(T)}is carried out ahead of design. For instance, for the case when composition and temperature data are known, the following steps can be used to estimate such values.

- (1)
- From the set of data, Equation (31) is used to obtain the parameter values. Temperature, ${F}_{p(T)}\left({T}_{X}^{C}\right)$, concentration ψ
_{Z}(P_{X}) and the uncorrected property operator $\psi \left({p}_{T,X}^{UnC}\right)$ are independent variables. - (2)
- To estimate the parameters, $\psi \left({p}_{T,X}^{UnC}\right)$ 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 (mgO_{2}/L) | pH | ρ^{*} (kg/m^{3}) | 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 |

^{*}Values obtained from simulations with Aspen Plus;

^{**}Values obtained from reported sources.

Stream | Flow (kg/h) | Z^{max} | Tox^{max} (%) | COD^{max} (mgO_{2}/L) | pH ^{max} | ρ^{max} (kg/m^{3}) | µ^{max} (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) | pH^{min} | ρ^{min} (kg/m^{3}) | µ^{min} (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 | T^{min} (K) | T^{max} (K) |
---|---|---|

G1 | 333.15 | 353.15 |

G2 | 303.15 | 348.15 |

G3 | 298.15 | 338.15 |

Waste | 290.15 | 308.15 |

Property | α_{u,m} | ${\mathrm{Cost}}_{u,m}^{Unit}$ ($/kg) |
---|---|---|

u(z_{1}) | 0.98 | 0.0143 |

u(z_{2}) | 0.85 | 0.0073 |

u(Tox_{1}) | 1.00 | 0.0216 |

u(Tox_{2}) | 0.90 | 0.0165 |

u(COD_{1}) | 0.80 | 0.0143 |

u(COD_{2}) | 0.55 | 0.0071 |

u(pH_{1}) | 0.50 | 0.1389 |

u(pH_{2}) | 0.30 | 0.0397 |

u(pH_{3}) | −0.50 | 0.1433 |

u(pH_{4}) | −0.30 | 0.0419 |

Compound | A_{c} | B_{c} | C_{c} |
---|---|---|---|

Water | 5.8221 | −0.01033 | 0.0000162 |

Phenol | 1.0809 | 0.003375 | 0 |

p(T) | C_{1,p(T)} | C_{2,p(T)} | C_{3,p(T)} | C_{4,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} |

_{1}), another one for toxicity treatment, u(Tox

_{1}), and the third one for pH adjustment, u(pH

_{1}). In this case, some direct use of split process streams into process sinks are observed, with the use of fresh sources in the second sink for adjustment. Cooling of the waste stream in order to meet environmental constraints is also observed. The total annual cost for this case was $7.8686 × 10

^{5}/year.

^{5}/year. One can observe that the network structure for the two solutions is similar, but with differences in the flows for the streams. The difference in TCA is 4.14%, which is related to an improvement in the estimation of properties with the functionality implemented for temperature dependence.

${P}_{p,j}^{InSink}$ | Z | Tox (%) | COD (mgO_{2}/L) | pH | ρ (kg/m^{3}) | 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 |

Cost^{Fresh} | cost of fresh sources |

Cost^{Int} | cost of property interceptor |

C_{P} | heat capacity |

d | stream flowrate |

F, f | flowrate of fresh sources |

G, g | flowrate of process sinks |

H_{V} | 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 |

U_{p} | 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

## References

- Hohmann, E.C. Optimum Networks for Heat Exchange. Ph.D. Thesis, University of Southern California, Los Angeles, CA, USA, 1971. [Google Scholar]
- Linnhoff, B.; Flower, J.R. Synthesis of heat exchanger networks: I. Systematic generation of energy optimal networks. AIChE J.
**1978**, 24, 633–642. [Google Scholar] - Umeda, T.; Itoh, J.; Shiroko, K. Heat-exchanger system synthesis. Chem. Eng. Prog.
**1978**, 74, 70–76. [Google Scholar] - El-Halwagi, M.M.; Manousiouthakis, V. Synthesis of mass exchange networks. AIChE J.
**1989**, 35, 1233–1244. [Google Scholar] - Wang, Y.P.; Smith, R. Wastewater minimization. Chem. Eng. Sci.
**1994**, 49, 981–1006. [Google Scholar] [CrossRef] - Dhole, V.R.; Ramchandani, N.; Tainsh, R.A.; Wasilewski, M. Make your process Water pay for itself. Chem. Eng.
**1996**, 103, 100–103. [Google Scholar] - El-Halwagi, M.M.; Hamad, A.A.; Garrison,, G.W. Synthesis of waste interception and allocation networks. AIChE J.
**1996**, 42, 3087–3101. [Google Scholar] - El-Halwagi, M.M.; Spriggs, H.D. Solve design puzzles with mass integration. Chem. Eng. Prog.
**1998**, 94, 25–44. [Google Scholar] - Polley, G.T.; Polley, H.L. Design better water reuse networks. Chem. Eng. Prog.
**2000**, 96, 47–52. [Google Scholar] - Foo, D.C.Y. State-of-the-art review of pinch analysis techniques for water network synthesis. Ind. Eng. Chem. Res.
**2009**, 48, 5125–5159. [Google Scholar] [CrossRef] - Galan, B.; Grossmann, I.E. Optimal design of distributed wastewater treatment networks. Ind. Eng. Chem. Res.
**1998**, 37, 4036–4048. [Google Scholar] [CrossRef] - Lee, S.; Grossmann, I.E. Global optimization of nonlinear generalized disjunctive programming with bilinear equality constraints: Applications to process networks. Comput. Chem. Eng.
**2003**, 27, 1557–1575. [Google Scholar] [CrossRef] - Karuppiah, R.; Grossmann, I.E. Global optimization for the synthesis of integrated water systems in chemical processes. Comput. Chem. Eng.
**2006**, 20, 650–673. [Google Scholar] [CrossRef] - Savelski, M.J.; Bagajewicz, M.J. On the optimality conditions of water utilization systems in process plants with single contaminants. Chem. Eng. Sci.
**2000**, 55, 5035–5048. [Google Scholar] [CrossRef] - Savelski, M.J.; Bagajewicz, M.J. Algorithmic procedure to design water utilization systems featuring a single contaminant in process plants. Chem. Eng. Sci.
**2001**, 56, 1897–1911. [Google Scholar] [CrossRef] - Tan, R.R.; Ng, D.K.S.; Foo, D.C.Y.; Aviso, K.B. A superstructure model for the synthesis of single-contaminant water networks with partitioning regenerators. Process Saf. Environ. Prot.
**2009**, 87, 197–205. [Google Scholar] [CrossRef] - Teles, J.; Castro, P.M.; Novals, A.Q. LP-based solution strategies for the optimal design of industrial water networks with multiple contaminants. Chem. Eng. Sci.
**2008**, 63, 376–394. [Google Scholar] [CrossRef] - Shelley, M.D.; El-Halwagi, M.M. Component-less design of recovery and allocation systems: A functionality-based clustering approach. Comput. Chem. Eng.
**2000**, 24, 2081–2091. [Google Scholar] [CrossRef] - Qin, X.; Gabriel, F.; Harell, D.; El-Halwagi, M. Algebraic techniques for property integration via componentless design. Ind. Eng. Chem. Res.
**2004**, 43, 3792–3798. [Google Scholar] [CrossRef] - El-Halwagi, M.M.; Glasgow, I.M.; Qin, X.Y.; Eden, M.R. Property integration: Componentless design techniques and visualization tools. AIChE J.
**2004**, 50, 1854–1869. [Google Scholar] - Eljack, F.T.; Abdelhady, A.F.; Eden, M.R.; Gabriel, F.; Qin, X.; El-Halwagi, M.M. Targeting optimum resource allocation using reverse problem formulations and property clustering techniques. Comput. Chem. Eng.
**2005**, 29, 2304–2317. [Google Scholar] [CrossRef] - Eljack, F.; Eden, M.; Kazantzi, V.; Qin, X.; El-Halwagi, M.M. Simultaneous process and molecular design—A property based approach. AIChE J.
**2007**, 53, 1232–1239. [Google Scholar] [CrossRef] - Foo, D.C.Y.; Kazantzi, V.; El-Halwagi, M.M.; Manan, A. Surplus diagram and cascade analysis techniques for targeting property-based material reuse network. Chem. Eng. Sci.
**2006**, 61, 2626–2642. [Google Scholar] [CrossRef] - Ponce-Ortega, J.M.; Hortua, A.C.; El-Halwagi, M.M.; Jiménez-Gutiérrez, A. A property-based optimization of direct recycle networks and wastewater treatment processes. AIChE J.
**2009**, 55, 2329–2344. [Google Scholar] - Ponce-Ortega, J.M.; El-Halwagi, M.M.; Jiménez-Gutiérrez, A. Global optimization for the synthesis of property-based recycle and reuse networks including environmental constraints. Comput. Chem. Eng.
**2010**, 34, 318–330. [Google Scholar] - Ponce-Ortega, J.M.; Nápoles-Rivera, F.; El-Halwagi, M.M.; Jiménez-Gutierrez, A. An optimization approach for the synthesis of recycle and reuse water integration networks. Clean Technol. Environ. Policy
**2012**, 14, 133–151. [Google Scholar] - Nápoles-Rivera, F.; Ponce-Ortega, J.M.; El-Halwagi, M.M.; Jiménez-Gutiérrez, A. Global optimization of mass and property integration networks with in-plant property interceptors. Chem. Eng. Sci.
**2010**, 65, 4363–4377. [Google Scholar] - Nápoles-Rivera, F.; Ponce-Ortega, J.M.; El-Halwagi, M.M.; Jiménez-Gutiérrez, A. Global optimization of wastewater integration networks for processes with multiple contaminants. Environ. Prog. Sustain. Energy
**2012**, 31, 449–458. [Google Scholar] - El-Halwagi, M.M. Pollution Prevention through Process Integration: Systematic Design Tools; Elsevier Academic Press: San Diego, CA, USA, 1997. [Google Scholar]
- El-Halwagi, M.M. Process Integration; Elsevier Academic Press: San Diego, CA, USA, 2006. [Google Scholar]
- El-Halwagi, M.M. Sustainable Design through Process Integration: Fundamentals and Applications to Industrial Pollution Prevention, Resource Conservation, and Profitability Enhancement; Butterworth-Heinemann/Elsevier: Amsterdam, The Netherlands, 2012. [Google Scholar]
- Duhne, C.R. Viscosity-temperature correlations for liquids. Chem. Eng.
**1979**, 86, 83–91. [Google Scholar] - Kheireddine, H.; Dadmohammadi, Y.; Deng, C.; Feng, X.; El-Halwagi, M.M. Optimization of direct recycle networks with the simultaneous consideration of property, mass, and thermal effects. Ind. Eng. Chem. Res.
**2011**, 50, 3754–3762. [Google Scholar] [CrossRef]

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**MDPI and ACS Style**

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

**AMA Style**

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 Style**

Jiménez-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