# Fuel Gas Network Synthesis Using Block Superstructure

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Block-Based Representation of Fuel Gas Network

## 3. FGN Synthesis Problem Statement

## 4. MINLP Model Formulation for Block-Based Fuel Gas Network Synthesis

#### 4.1. Block Material Balance

#### 4.2. Flow Directions

#### 4.3. Block Energy Balance

#### 4.4. Product Stream Assignments and Logical Constraints

#### 4.5. Boundary Assignment

#### 4.6. Work Calculation

#### 4.7. Objective Function

## 5. Case Study

#### 5.1. Case Study Description

#### 5.2. Case 1: FGN Synthesis Without Pools

#### 5.3. Case 2: FGN Synthesis With Pools

## 6. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Nomenclature

**Sets and Indices**

I | Set of row numbers indexed by i |

J | Set of column numbers indexed by j |

D | Set of flow alignments indexed by d |

K | Set of components indexed by k |

$FS$ | Set of feed streams indexed by f |

$PS$ | Set of product streams indexed by p |

**Subsets**

$LN(i,j,{i}^{\prime},{j}^{\prime})$ | Set designating the connection between block ${B}_{i,j}$ and block ${B}_{{i}^{\prime},{j}^{\prime}}$ |

$kp(k,p)$ | Set relating the key component k with product stream p with purity specifications |

**Variables**

${F}_{i,j,k,d}$ | Flowrate of component k between block ${B}_{i,j}$ and ${B}_{i,j+1}$ in the flow alignment d |

${Q}_{i,j}$ | Amount of heat/cold utility consumed in block ${B}_{i,j}$ |

${W}_{i,j}$ | Amount of work energy added into or withdrawn from block ${B}_{i,j}$ |

$TAC$ | Total annual cost |

**Positive Continuous Variables**

${F}_{f,p}$ | Stream connecting source f and sink p in the classic superstructure |

${M}_{i,j,k,f}$ | Component flowrate for k in external feed stream f into block ${B}_{i,j}$ |

${M}_{i,j,k}^{f}$ | Component flowrate k of external feed stream into block ${B}_{i,j}$ |

${z}_{i,j,f}^{feedfrac}$ | Distribution of feed f into block ${B}_{i,j}$ |

${H}_{i,j,k,p}$ | Amount of component k in external product stream p withdrawn from block ${B}_{i,j}$ |

${H}_{i,j,k}^{p}$ | Component flowrate k of external product stream withdrawn from block ${B}_{i,j}$ |

${J}_{i,j,{i}^{\prime},{j}^{\prime},k}$ | Flowrate of component k from block ${B}_{i,j}$ to another block ${B}_{{i}^{\prime},{j}^{\prime}}$ |

${J}_{i,j,k}^{f}$ | Overall component flowrate k of jump connecting flow into block ${B}_{i,j}$ |

${J}_{i,j,k}^{p}$ | Overall component flowrate k of jump connecting flow withdrawn from block ${B}_{i,j}$ |

$F{P}_{i,j,k,d}$ | Positive component of flow ${F}_{i,j,k,d}$ |

$F{N}_{i,j,k,d}$ | Negative component of flow ${F}_{i,j,k,d}$ |

$F{P}_{i,j,d}^{T}$ | Total flowrate for flow $F{P}_{i,j,k,d}$ |

$F{N}_{i,j,d}^{T}$ | Total flowrate for flow $F{N}_{i,j,k,d}$ |

${J}_{i,j,{i}^{\prime},{j}^{\prime}}^{T}$ | Total flowrate for flow ${J}_{i,j,{i}^{\prime},{j}^{\prime},k}$ |

${M}_{i,j,f}^{T}$ | Total flowrate for flow ${M}_{i,j,k,f}$ |

${H}_{i,j,p}^{T}$ | Total flowrate for flow ${H}_{i,j,k,p}$ |

${y}_{i,j,k}^{b}$ | Block composition of component k in block ${B}_{i,j}$ |

${P}_{i,j}$ | Pressure designation in block ${B}_{i,j}$ |

${T}_{i,j}$ | Temperature designation in block ${B}_{i,j}$ |

${Q}_{i,j}^{h}$ | Heat amount supplied into block ${B}_{i,j}$ |

${Q}_{i,j}^{c}$ | Heat amount withdrawn from block ${B}_{i,j}$ |

$E{F}_{i,j,d}$ | Stream enthalpy carried by the material flow ${F}_{i,j,k,d}$ in flow direction d |

$E{M}_{i,j}$ | Overall enthalpy brought into block ${B}_{i,j}$ along with feed streams |

$E{P}_{i,j}$ | Overall enthalpy taken away by product streams at block ${B}_{i,j}$ |

$E{J}_{i,j}^{f}$ | Overall enthalpy carried into block ${B}_{i,j}$ through jump flow |

$E{J}_{i,j}^{p}$ | Overall enthalpy taken out from block ${B}_{i,j}$ through jump flow |

${W}_{i,j}^{com}$ | Work energy associated with compression operation |

${W}_{i,j}^{exp}$ | Work energy associated with expansion operation |

${W}_{i,j,d}^{comp,FP}$ | Compression work for positive component of flow ${F}_{i,j,k,d}$ |

${W}_{i,j,d}^{comp,FN}$ | Compression work for negative component of flow ${F}_{i,j,k,d}$ |

${W}_{i,j,f}^{comp,FS}$ | Compression work for feed stream f |

${W}_{{i}^{\prime},{j}^{\prime},i,j}^{comp,JF}$ | Compression work for jump flow ${J}_{i,j,{i}^{\prime},{j}^{\prime},k}$ |

${W}_{i,j,d}^{exp,FP}$ | Expansion work for positive component of flow ${F}_{i,j,k,d}$ |

${W}_{i,j,d}^{exp,FN}$ | Expansion work for negative component of flow ${F}_{i,j,k,d}$ |

${W}_{i,j,f}^{exp,FS}$ | Expansion work for feed stream f |

${W}_{{i}^{\prime},{j}^{\prime},i,j}^{exp,JF}$ | Expansion work for jump flow ${J}_{i,j,{i}^{\prime},{j}^{\prime},k}$ |

$P{R}_{i,j,d}^{F}$ | Pressure ratio between the block ${B}_{i,j+1}$ and ${B}_{i,j}$ for flow alignment $d=1$ or |

between the block ${B}_{i+1,j}$ and ${B}_{i,j}$ for flow alignment $d=2$ |

**Binary Variables**

${z}_{i,j,d}^{Plus}$ | 1 if ${F}_{i,j,k,d}$ is from block ${B}_{i,j}$ to ${B}_{i,j+1}$ ($d=1$) or from block ${B}_{i,j}$ to ${B}_{i+1,j}$ ($d=2$) |

${z}_{i,j,p}^{product}$ | 1 if product stream p is withdrawn from block ${B}_{i,j}$ |

${z}_{i,j,d}^{cr}$ | 1 if boundary between ${B}_{i,j}$ and ${B}_{i,j+1}$ for $d=1$ (between ${B}_{i,j}$ and ${B}_{i+1,j}$ for $d=2$) is completely restricted |

**Parameters**

${T}_{f}$ | Temperature of feed stream f |

${P}_{f}$ | Pressure of feed stream f |

${T}^{min}$ | Minimum temperature in the process |

${T}^{max}$ | Maximum temperature in the process |

$FL$ | Flowrate lower bound in the process |

$FU$ | Flowrate upper bound in the process |

${T}_{p}^{min}$ | Minimum temperature of product stream p |

${T}_{p}^{max}$ | Maximum temperature of product stream p |

${P}_{p}^{min}$ | Minimum pressure of product stream p |

${P}_{p}^{max}$ | Maximum pressure of product stream p |

${F}_{f}^{feed}$ | Available amount of feed stream f |

${y}_{k,f}^{feed}$ | Specification of component k in feed stream f |

${D}_{p}^{L}$ | Minimum amount requirement for product p |

${D}_{p}^{U}$ | Maximum amount requirement for product p |

$D{e}_{p}^{L}$ | Minimum energy demand for product p |

$D{e}_{p}^{U}$ | Maximum energy demand for product p |

${y}_{k,p}^{min,prod}$ | Minimum purity requirement of component k in product stream p |

${y}_{k,p}^{max,prod}$ | Maximum purity requirement of component k in product stream p |

${q}_{s,p}^{min,prod}$ | Minimum requirement of specification s in product stream p |

${q}_{s,p}^{max,prod}$ | Maximum requirement of specification s in product stream p |

${\pi}_{{k}^{{}^{\prime}},{k}^{{}^{\prime \prime}},p}^{prod}$ | Minimum product ratio requirement between component ${k}^{{}^{\prime}}$ and component ${k}^{{}^{\prime \prime}}$ for product p |

$LH{V}_{k}$ | Lower heating value for component k |

${q}_{s,k}$ | Specification s for component k |

$C{p}_{k}$ | Heat capacity of component k |

$MD{P}_{p}$ | Moisture dew-point temperature for the product p |

$HD{P}_{p}$ | Hydrocarbon dew-point temperature for the product p |

${R}_{gas}$ | Gas constant |

$\gamma $ | Adiabatic compression coefficient for process streams |

$\eta $ | Adiabatic compression efficiency |

${n}_{fs}$ | Adiabatic compression coefficient for feed f |

$UF{C}_{f}$ | Unit cost of different source streams |

$D{i}_{f}$ | Unit cost of treatment cost for the remaining source stream |

$Re{v}_{p}$ | Unit profit from excess energy in product stream p |

${\pi}_{f}$ | Unit transportation cost for source stream f |

$C{C}^{HU}$ | Unit cost of heaters |

$C{C}^{CU}$ | Unit cost of coolers |

$C{C}^{exp}$ | Unit cost of expansion operations |

$C{C}^{com}$ | Unit cost of compression operations |

**Abbreviations**

FGN | Fuel Gas Network |

MINLP | Mixed-integer nonlinear programming problem |

NLP | Nonlinear programming problem |

HEN | Heat exchange network |

MEN | Mass exchange network |

UPCS | unit-port-conditioning-stream approach |

L | Lower bound |

U | Upper bound |

min | Minimum |

max | Maximum |

prod | Product |

feed | Feed stream |

T | Total |

${B}_{i,j}$ | The block at row i and column j |

NG | Natural gas |

EFG | End flash gas |

HPFG | High-pressure fuel gas |

TBOG | Tankage boil-off gas |

FFF | Fuel from feed |

FFP | Fuel from products or byproducts |

GTG | Gas turbine generators |

GTD | Gas turbine drivers |

MR | Mixed refrigerant |

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**Figure 2.**Construction of superstructure for fuel gas synthesis problems: (

**a**) Block superstructure illustration; (

**b**) Block interaction via connecting streams (blue line: jump connecting flow from the block ${B}_{i,j}$; red line: jump connecting flow into the block ${B}_{i,j}$; blocks at diagonal positions are ignored for simplicity).

**Figure 3.**Block representation for fuel gas network problem: (

**a**) Classical superstructure for fuel gas network; (

**b**) Equivalent block superstructure for fuel gas network.

**Figure 4.**General superstructure for fuel gas network synthesis problem with intermediate pools: (

**a**) General superstructure for fuel gas network with intermediate pools; (

**b**) equivalent block superstructure.

**Figure 7.**Block representation and process flowsheet for the optimal solution of FGN without intermediate pools: (

**a**) Block representation for the optimal solution of FGN; (

**b**) process flowsheet for the optimal solution of FGN.

**Figure 8.**Block representation and process flowsheet for the optimal solution of FGN with intermediate pools: (

**a**) Block representation for the optimal solution of FGN; (

**b**) process flowsheet for the optimal solution of FGN.

Specification/Parameter | EFG | HPFG | TBOG | FFF |
---|---|---|---|---|

Adiabatic compression coefficient, ${n}_{fs}$ | 0.254 | 0.2 | 0.18 | 0.2 |

Availability, ${F}_{f}^{feed}$ (kmol/s) | 0.92938 | 0.05310 | 0.18255 | <7.30229 |

Temperature, ${T}_{f}^{feed}$ (K) | 240 | 325 | 113 | 298 |

Pressure, ${P}_{f}^{feed}$ (bar) | 1.72369 | 7.58423 | 1.72369 | 26.20007 |

Methane, CH${}_{4}$ (%) | 60.0 | 81.0 | 92.0 | 85.0 |

Ethane, C${}_{2}$H${}_{6}$ (%) | 0.0 | 6.0 | 0.0 | 5.0 |

Propane, C${}_{3}$H${}_{8}$ (%) | 0.0 | 5.0 | 0.0 | 4.0 |

C${}_{3+}$ (%) | 0.0 | 2.5 | 0.0 | 2.0 |

CO (%) | 0.0 | 0.0 | 0.0 | 0.05 |

N2 (%) | 40.0 | 5.5 | 8.0 | 3.95 |

Source unit cost, $UF{C}_{f}$ ($/kmol) | 0.0 | 0.0 | 0.0 | 4.184 |

Source disposal cost, $D{i}_{f}$ ($/kmol) | 0.209 | 0.292 | 0.209 | 0 |

Feed transporting cost, ${\pi}_{f}$ ($/kmol) | 0.0008 | 0.0008 | 0.0008 | 0.0008 |

**EFG**: end flash gas;

**HPFG**: high-pressure fuel gas;

**TBOG**: tankage boil-off gas;

**FFF**: fuel from feed.

Parameter | Methane | Ethane | Propane | C${}_{3+}$ | CO | N2 |
---|---|---|---|---|---|---|

LHV(MJ/kmol) | 800.234 | 1425.580 | 2041.113 | 2654.134 | 282.637 | 0 |

1/SG (28.96/mol wt) | 1.8060 | 0.9636 | 0.6571 | 0.4985 | 1.0344 | 1.0342 |

Cp [KJ/(kmol K)] | 37.16 | 57.40 | 80.30 | 114.93 | 29.20 | 29.15 |

Specification/Parameter | C1 | C2 | C3 | C4 | C5 |
---|---|---|---|---|---|

Energy demand, | 152.309 | 149.378 | 120.305 | 149.378 | 87.921 |

$D{e}_{p}$ (MJ/s) | |||||

Material demand, | 0.159–0.172 | 0.156–0.169 | 0.159–0.172 | 0.149–0.169 | 0.132–0.199 |

$\left[{D}_{p}^{L},{D}_{p}^{U}\right]$ (kmol/s) | |||||

Temperature range, | 113–1000 | 113–1000 | 113–1000 | 113–1000 | 113–1000 |

$\left[{T}_{p}^{min},{T}_{p}^{max}\right]$ (K) | |||||

Pressure range, | 1.72–24.82 | 1.72–24.82 | 1.72–24.82 | 1.72–24.82 | 1.72–24.82 |

$\left[{P}_{p}^{min},{P}_{p}^{max}\right]$ (bar) | |||||

$MD{P}_{p}$ (K) | 277 | 277 | 277 | 277 | 277 |

$HD{P}_{p}$ (K) | 277 | 277 | 277 | 277 | 277 |

$LHV$(MJ/kmol) | 264.885–8829.500 | 264.885–8829.500 | 264.885–8829.500 | 264.885–8829.500 | 264.885–8829.500 |

$1/SG$ (28.96/mol wt) | 1.0–2.4 | 1.0–2.4 | 1.0–2.4 | 1.0–2.4 | 1.0–2.4 |

Methane, CH${}_{4}$ (%) | >85.0 | >85.0 | >85.0 | >85.0 | >65.0 |

Ethane, C${}_{2}$H${}_{6}$ (%) | <15.0 | <15.0 | <15.0 | <15.0 | <15.0 |

Propane, C${}_{3}$H${}_{8}$ (%) | <15.0 | <15.0 | <15.0 | <15.0 | <15.0 |

C${}_{3+}$ (%) | <5.0 | <5.0 | <5.0 | <5.0 | <5.0 |

CO (%) | <10.0 | <10.0 | <10.0 | <10.0 | <10.0 |

N2 (%) | <15.0 | <15.0 | <15.0 | <15.0 | <15.0 |

Treatment factor, ${\psi}_{sp}$ | 1.0 | 1.0 | 1.0 | 1.0 | 1.0 |

Unit profit, $Re{v}_{p}$ ($/KJ) | 0 | 0 | 0 | 0 | 6.6347×10${}^{-6}$ |

Unit | CAPEX ($/KWh) | OPEX ($/KWh) | Total ($/KWh) |
---|---|---|---|

Compressor | 10 | 0.01 | $C{C}^{com}$ = 10.01 |

Expander | 1 | 0.05 | $C{C}^{exp}$ = 1.05 |

Heater | 5 | 0.01 | $C{C}^{HU}$ = 5.01 |

Cooler | 5 | 0.02 | $C{C}^{CU}$ = 5.02 |

Case 1 | Case 2 | |
---|---|---|

Continuous variable | 397 | 1741 |

Binary variable | 45 | 58 |

Bilinear terms | 849 | 9530 |

Signomial terms | 243 | 1321 |

CPU time (second) | 565 | 935 |

Solution (MM$/year) | 70.1 | 69.3 |

TAC Component ($/Year) | Case 1 | Case 2 |
---|---|---|

Source cost | 87,852,764 | 68,530,679 |

Revenue from excess energy in sinks | 17,736,117 | 16,329,735 |

Source transportation cost | 23,246 | 23,246 |

Heaters cost | 0 | 0 |

Coolers cost | 0 | 0 |

Expansion operation cost | 5850 | 3016 |

Compression operation cost | 3168 | 2744 |

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Li, J.; Demirel, S.E.; Hasan, M.M.F.
Fuel Gas Network Synthesis Using Block Superstructure. *Processes* **2018**, *6*, 23.
https://doi.org/10.3390/pr6030023

**AMA Style**

Li J, Demirel SE, Hasan MMF.
Fuel Gas Network Synthesis Using Block Superstructure. *Processes*. 2018; 6(3):23.
https://doi.org/10.3390/pr6030023

**Chicago/Turabian Style**

Li, Jianping, Salih Emre Demirel, and M. M. Faruque Hasan.
2018. "Fuel Gas Network Synthesis Using Block Superstructure" *Processes* 6, no. 3: 23.
https://doi.org/10.3390/pr6030023