# Single-Solution-Based Vortex Search Strategy for Optimal Design of Offshore and Onshore Natural Gas Liquefaction Processes

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## Abstract

**:**

## 1. Introduction

^{®}to model SMR process and subsequently optimized it using a hybrid strategy by combining Nelder–Mead Downhill Simplex and Tabu search algorithms. Khan et al. [16] decreased the overall compression energy requirement of the SMR process using Non-Linear Programming (NLP) and particle swarm paradigm. Wang et al. [17] designed the C3MR process in Aspen Plus

^{®}and presented an optimal design through Sequential Quadratic Programming (SQP). Hatcher et al. [18], Lee et al. [19], and Mortazavi et al. [20] also modeled the C3MR process and subsequently optimized it via the Box method, Successive Reduced Quadratic Programming (SRQP), and hybrid optimization (i.e., Genetic Algorithm (GA) and SQP). Lee et al. [21] applied multi-objective optimization (using SQP) via gProms process simulator to find an optimal design of SMR process. Furthermore, Lee and Moon [22] have also used the SRQP to find optimal designs of SMR and C3MR liquefaction processes. They applied a mathematical optimization model that includes the cost model, as well as thermodynamic model. However, Tsay and Baldea [23] reported that modeling, simulation, and optimization become more complicated when phase transition and recycle streams are incorporated into refrigeration cycles. Therefore, they used equation-oriented modeling and optimization strategies [24,25] to find the optimal design of SMR liquefaction processes. Moreover, Vikse et al. [26] presented a versatile simulation method to find an optimal design of complex SMR process. They investigated the capability of nonsmooth framework (i.e., equation-oriented modeling) for the optimal design of SMR process. Ali et al. [27] employed Generalized Polynomial Chaos (gPC) based surrogate modeling approach to study the operational reliability of the SMR process. Most recently, Zhu et al. [28] performed experimental investigation to analyze the flow distribution in plate-fin heat exchanger that is mostly used in the SMR process. Khan et al. [11] proposed a mixed refrigerant composition selection method based on the boiling point difference and specific refrigeration effect of individual components in mixed refrigerant. The proposed selection criteria were applied to the MR used in SMR and C3MR system; and a decrease in energy consumption for compression was observed. In another similar study, Xu et al. [29] developed a correlation between the ambient temperature and mixed refrigerant composition to evaluate energy utilization for PRICO LNG process. Qadeer et al. [30] adopted krill-herd optimization strategy for optimal solution of process conditions and MR composition of the SMR process.

^{®}v9. The VS algorithm is coded and modified in MATLAB version 2018b. The simulated LNG processes are linked with VS through ActiveX functionality. Optimal findings are analyzed and compared with the well-known algorithms including the GA and Particle Swarm Optimization (PSO). The VS algorithm can be applied to find the optimal design of any complex chemical process by simulating in Aspen Hysys.

## 2. Vortex Search Strategy

#### 2.1. Primary Estimation

_{0}) is given by Equation (1) as follows:

#### 2.2. Candidate Solutions

^{2}denotes the variance distribution and I denotes the identity matrix. The standard deviation (s

_{0}) under the initial conditions is given by Equation (4) as follows:

_{0}is considered as the initial radius (r

_{0}) and is differentiated to yield an entire covering of the weak vicinity in the neighborhood search area for a fully weak locality at the initial stage.

#### 2.3. Current Results Substitution

_{0}(X), wherein the current circle center µ

_{0}lies within the search space limits. If the new solutions are beyond the search space boundaries, then the candidate solutions are shifted into the specified bounds as given in Equation (5) as follows:

_{1}) of the circle decreases, and a new set of vectors C

_{1}(X) is obtained across the new center. In the second step of the selection phase, the new solution set C

_{1}(X) is evaluated via ${X}^{\prime}\in {C}_{1}\left(X\right)$. If the selected solution advances to the old solutions, then it is saved.

#### 2.4. Radius Reduction Methodology

## 3. Onshore and Offshore LNG Processes

^{®}) process is the most suitable candidate for small-scale and offshore applications compared with the expander-based LNG processes [37]. Although, for offshore applications, the nitrogen-expander-based liquefaction process have several dominant features such as safety and simplicity in operation but thermodynamically it is less attractive than the SMR process, due to high exergy destruction. Whereas, the C3MR process is considered as a promising candidate for onshore applications due to its relative high energy efficiency. Furthermore, this process is capable of producing 81% of the base-load LNG [38]. It is reported [39] that about 77% of the world’s LNG plants are using the C3MR technology. Nevertheless, the process exhibits a high degree of complexity. Detailed process descriptions of the SMR and C3MR liquefaction processes are described in the forthcoming sections.

#### 3.1. SMR LNG Process Description

#### 3.2. C3MR LNG Process Description

#### 3.3. Simulation Basis for LNG Process Modeling

## 4. Optimization Problem Formulation for SMR and C3MR Processes

- Target function(s).
- Constraint function(s) and their limits(s).
- Optimization variables.
- Optimizing variable bounds (search area).
- Other design parameters, if any.

#### 4.1. Constraint Handling Approach

#### 4.2. Exergy Destruction Analysis

## 5. Results and Discussion

#### Exergy Destruction Analysis and Figure of Merit

## 6. Conclusions

## Supplementary Materials

Supplementary File 1## Author Contributions

## Funding

## Conflicts of Interest

## Nomenclature

Abbreviations | |

V9 | Version 9 |

inf | Infinite |

ζ | Haphazardly yielded variable |

µ | Sample mean |

r | Radius |

JT | Joule–Thomson |

K | Compressor |

C | Cooler |

Q | Heat |

°C | Degree Celsius |

kg | Kilogram |

kW | Kilowatt |

Acronyms | |

C3MR | Propane precooled mixed refrigerant |

SMR | Single mixed refrigerant |

LNG | Liquified natural gas |

VS | Vortex search |

NG | Natural gas |

GHG | Greenhouse gas |

MR | Mixed refrigerant |

NLP | Nonlinear programing |

SQP | Sequential quadratic programming |

GA | Genetic algorithm |

SRQP | Successive reduced quadratic programming |

SCRS | Sequential coordinate random search |

PSO | Particle swarm optimization |

VSO | Vortex search optimization |

MFC | Mixed fluid cascade |

PRICO | Poly refrigerant integrated cycle operations |

MITA | Minimum internal temperature approach |

THCC | Temperature heat-flow composite curve |

TDCC | Temperature difference composite curve |

Subscripts | |

Xbest | Best solution |

o | Initialization |

So | Standard deviation |

N_{2} | Nitrogen |

C_{1} | Methane |

C_{2} | Ethane |

C_{3} | Propane |

W_{i} | Compressors work |

Ex_{des} | Exergy destruction |

Ex_{in} | Exergy in |

Ex_{out} | Exergy out |

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**Figure 2.**Representation of the working search process using a model of the VS in a two-dimensional nested circle form.

**Figure 6.**Base case (

**a**) temperature difference composite curve (TDCC) and (

**b**) temperature-heat flow composite curve (THCC) in comparison with the VS-optimized (

**c**) TDCC and (

**d**) THCC curves of the SMR process.

**Figure 7.**Base case (

**a**) TDCC and (

**b**) THCC curves in comparison with the VS-optimized (

**c**) TDCC and (

**d**) THCC curves of the C3MR process.

Name of Function and (Global Value) | Objective Function Equation | Minimum | Search Bound | Plot |
---|---|---|---|---|

Schafer, (0) | $f(x)=0.5+\frac{{\mathrm{sin}}^{2}\left(\sqrt{{x}_{1}^{2}+{x}_{2}^{2}}\right)-0.5}{{\left(1+0.001\left({x}_{1}^{2}+{x}_{2}^{2}\right)\right)}^{2}}$ | 0 | [−100,100] | |

Goldstein-Price, (3) | $\begin{array}{l}f(x)=\left[1+{\left({x}_{1}+{x}_{2}+1\right)}^{2}\left(19-14{x}_{1}+3{x}_{1}^{2}-14{x}_{2}+6{x}_{1}{x}_{2}+3{x}_{2}^{2}\right)\right]\cdot \\ \left[30+{\left(2{x}_{1}-3{x}_{2}\right)}^{2}\left(18-32{x}_{1}+12{x}_{1}^{2}+48{x}_{2}-36{x}_{1}{x}_{2}+27{x}_{2}^{2}\right)\right]\end{array}$ | 3 | [−2,2] | |

Six Hump Camel Back, (−1.0316) | $f(x)=4{x}_{1}^{2}-2.1{x}_{1}^{4}+\frac{1}{3}{x}_{1}^{6}+{x}_{1}{x}_{2}-4{x}_{2}^{2}+4{x}_{2}^{4}$ | −1.0316 | [−5,5] | |

Matyas, (0) | $f(x)=0.26\left({x}_{1}^{2}+{x}_{2}^{2}\right)-0.48{x}_{1}{x}_{2}$ | 0 | [−10,10] |

Property | Condition |
---|---|

NG feed condition | |

Flow rate | 1.0 kg/h |

Temperature | 32 °C |

Pressure | 50 bar |

NG feed composition | Mole fraction |

Methane | 0.9133 |

Ethane | 0.0536 |

Propane | 0.0214 |

n-Butane | 0.0047 |

i-Butane | 0.0046 |

n-Pentane | 0.0001 |

i-Pentane | 0.0001 |

Nitrogen | 0.0022 |

Intercooler outlet temperature | 40 °C |

Vapor fraction boil-off-gas | 8.0% |

Compressor isentropic efficiency | 0.75 |

Fluid package | Peng–Robinson |

Enthalpy/entropy calculation | Lee Kesler |

Pressure drops across LNG cryogenic exchanger in SMR process | |

“Stream-1” to “Stream-2” | 1.0 bar |

“Stream-14” to “Stream-4” | 1.0 bar |

“Stream-5” to “Stream-6” | 0.1 bar |

Pressure drops across LNG exchanger in C3MR process | |

“Stream-NG-1” to “Stream-NG-5” | 1.0 bar |

“Stream-4a” to “Stream-5a” | 1.0 bar |

“Stream-4b” to “Stream-5b” | 0.5 bar |

“Stream-6a” to “Stream-1a” | 0.05 bar |

“Stream-6b” to “Stream-1b” | 0.05 bar |

Heat loss to the environment in both SMR and C3MR processes | negligible |

Decision Variables | Lower Bound | Upper Bound |
---|---|---|

Design (decision) variables for SMR process | ||

High pressure of MR, P_{13} (bar) | 35.0 | 70.0 |

Evaporation pressure, P_{5} (bar) | 1.1 | 4.0 |

Flow rate of nitrogen, ${m}_{N2}$ (kg/h) | 0.1 | 0.65 |

Flow rate of methane, ${m}_{C1}$ (kg/h) | 0.25 | 0.85 |

Flow rate of ethane, ${m}_{C2}$ (kg/h) | 0.45 | 1.15 |

Flow rate of propane, ${m}_{C3}$ (kg/h) | 2.0 | 3.5 |

Design (decision) variables for C3MR processMR cycle | ||

High pressure of MR, P_{8} (bar) | 50.0 | 70.0 |

Evaporation pressure, P_{6a} (bar) | 2.5 | 10.0 |

Evaporation pressure, P_{6b} (bar) | 2.5 | 10.0 |

Flow rate of nitrogen, ${m}_{N2}$ (kg/h) | 0.05 | 0.45 |

Flow rate of methane, ${m}_{C1}$ (kg/h) | 0.25 | 0.85 |

Flow rate of ethane, ${m}_{C2}$ (kg/h) | 0.65 | 1.40 |

Flow rate of propane, ${m}_{C3}$ (kg/h) | 0.15 | 0.8 |

Propane cycle | ||

Propane 1st cooling stage (°C) | 15.0 | 30.0 |

Propane 2nd cooling stage (°C) | 0.0 | 10.0 |

Propane 3rd cooling stage (°C) | −20.0 | −5.0 |

Equipment | Exergy Destruction (kJ/h) |
---|---|

Compressor | $E{x}_{des}=\left(\mathsf{\u1e41}\right)\left(E{x}_{in}-E{x}_{out}\right)-\mathsf{\u1e86}$ |

Interstage coolers | $E{x}_{des}=\left(\mathsf{\u1e41}\right)\left(E{x}_{in}-E{x}_{out}\right)$ |

Multistream LNG heat exchanger | $E{x}_{des}=\sum \left(\mathsf{\u1e41}\right)E{x}_{in}-\sum \left(\mathsf{\u1e41}\right)E{x}_{out}$ |

JT valves | $E{x}_{des}=\left(\mathsf{\u1e41}\right)\left(E{x}_{in}-E{x}_{out}\right)$ |

SMR Process | C3MR Process | ||||
---|---|---|---|---|---|

Stream | T, (℃) | P, (bar) | Stream | T, (℃) | P, (bar) |

1 | 32.0 | 50.0 | 1a | −136.5 | 3.0 |

2 | −149.3 | 49 | 1b | −136.5 | 3.0 |

3 | −158.5 | 1.209 | 2 | −56.4 | 3.0 |

4 | −149.3 | 58.5 | 3 | −16.22 | 6.2 |

5 | −152.6 | 1.65 | 9 | 40.0 | 55.0 |

6 | 36.54 | 1.55 | 4a | −33.34 | 54.0 |

7 | 90.39 | 3.858 | 4b | −33.34 | 54.0 |

8 | 40.0 | 3.858 | 5a | −119.4 | 53.99 |

9 | 94.71 | 9.603 | 6a | −152.7 | 3.0 |

10 | 40.0 | 9.603 | 5b | −130.6 | 54.0 |

11 | 96.53 | 23.9 | 6b | −136.5 | 3.0 |

12 | 40.0 | 23.9 | NG-1 | 32.0 | 50.0 |

13 | 101.5 | 59.5 | NG-4 | −33.34 | 49.0 |

14 | 40.0 | 59.5 | NG-5 | −149.5 | 48.99 |

SMR-stream | 36.54 | 1.55 | LNG | −158.5 | 1.209 |

**Table 6.**Particle swarm optimization (PSO) and genetic algorithm (GA) parameters used to set the optimization framework for SMR and C3MR processes.

PSO | GA | ||
---|---|---|---|

Parameter | Value | Parameter | Value |

Number of particles | 30.0 | Number of population | 200.0 |

Cognition learning parameter | 2.0 | Selection method | Stochastic uniform |

Social learning parameter | 2.1 | Mutation | Adaptive feasible |

Maximum velocity of particle | 4.0 | Crossover function | Scatter |

Inertial weight | 0.9–0.2 | Fraction of migration | 0.2 |

Number of generations | 200.0 |

**Table 7.**Summary and comparison of the optimization results of the VS-optimized SMR with optimized SMR process using other well-known optimization algorithms.

Decision Variables | Base Case [44] | GA Optimized | PSO Optimized | VS Optimized |
---|---|---|---|---|

High pressure of MR, P_{13} (bar) | 48.0 | 45.72 | 54.50 | 59.50 |

Evaporation pressure of MR, P_{5} (bar) | 1.30 | 1.680 | 2.10 | 1.550 |

Flow rate of nitrogen, ${m}_{N2}$ (kg/h) | 0.2690 | 0.3300 | 0.2200 | 0.1650 |

Flow rate of methane, ${m}_{C1}$ (kg/h) | 0.5290 | 0.4510 | 0.5900 | 0.4630 |

Flow rate of ethane, ${m}_{C2}$ (kg/h) | 0.6190 | 0.7062 | 0.6740 | 0.6360 |

Flow rate of propane, ${m}_{C3}$ (kg/h) | 2.847 | 2.930 | 2.649 | 2.288 |

Constraints | ||||

MITA (°C) | 3.0 | 3.0 | 3.0 | 3.0 |

Liquefaction rate (%) | 92.0 | 92.0 | 92.0 | 92.0 |

Required specific power (kWh/kg-LNG) | 0.4400 | 0.4034 | 0.3862 | 0.3691 |

Relative energy saving (%) | - | 8.32 | 12.23 | 16.1 |

**Table 8.**Summary and comparison of the optimization results of the VS-optimized C3MR with optimized C3MR process using other well-known optimization algorithms.

Decision Variables | Base Case [44] | GA Optimized | PSO Optimized | VS Optimized |
---|---|---|---|---|

MR cycle | ||||

Condensation pressure of MR, P_{8} (bar) | 55.0 | 61.24 | 50.0 | 43.43 |

Evaporation pressure, (P_{6a}, P_{6b}) (bar) | 3.0 | 4.55 | 5.49 | 2.75 |

Flow rate of nitrogen, ${m}_{N2}$ (kg/h) | 0.30 | 0.2331 | 0.2519 | 0.08 |

Flow rate of methane, ${m}_{C1}$ (kg/h) | 0.75 | 0.5764 | 0.5959 | 0.4490 |

Flow rate of ethane, ${m}_{C2}$ (kg/h) | 0.95 | 0.9204 | 1.210 | 0.8430 |

Flow rate of propane, ${m}_{C3}$ (kg/h) | 0.70 | 0.4989 | 0.30 | 0.5690 |

Propane cycle | ||||

Propane 1st cooling stage (°C) | 20.00 | 19.48 | 15.54 | 18.40 |

Propane 2nd cooling stage (°C) | 3.5 | 1.919 | 4.50 | 0.0 |

Propane 3rd cooling stage (°C) | −16.0 | −15.27 | −14.70 | −16.75 |

Constraints | ||||

MITA (°C) | 3.0 | 3.0 | 3.0 | 3.0 |

Liquefaction rate (%) | 92.0 | 92.0 | 92.0 | 92.0 |

Required specific power (kWh/kg-LNG) | 0.3602 | 0.2778 | 0.2754 | 0.2600 |

Relative energy saving (%) | - | 22.9 | 23.5 | 27.8 |

FOM for C3MR Process | ||||||
---|---|---|---|---|---|---|

MFC Process | Precooling Section | MR Section | ||||

Min. Required Work | Total Energy | FOM | Min. Required Work | Total Energy | FOM | |

kJ/h | kJ/h | % | kJ/h | kJ/h | % | |

Base Case | 99.69 | 356.4 | 28.08 | 356.13 | 943.2 | 37.81 |

GA_C3MR | 86.03 | 306 | 28.00 | 356.13 | 691.2 | 51.40 |

PSO_C3MR | 101.19 | 349.2 | 28.96 | 356.13 | 640.8 | 55.48 |

VSO_C3MR | 83.63 | 302.4 | 27.68 | 356.14 | 633.6 | 56.18 |

FOM for SMR Process | ||||||

Base Case | 372.22 | 1584 | 23.50 | |||

GA_SMR | 371.17 | 1440 | 25.56 | |||

PSO_SMR | 371.17 | 1404 | 26.69 | |||

VSO_SMR | 371.20 | 1332 | 27.94 |

© 2020 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**

Qyyum, M.A.; Yasin, M.; Nawaz, A.; He, T.; Ali, W.; Haider, J.; Qadeer, K.; Nizami, A.-S.; Moustakas, K.; Lee, M. Single-Solution-Based Vortex Search Strategy for Optimal Design of Offshore and Onshore Natural Gas Liquefaction Processes. *Energies* **2020**, *13*, 1732.
https://doi.org/10.3390/en13071732

**AMA Style**

Qyyum MA, Yasin M, Nawaz A, He T, Ali W, Haider J, Qadeer K, Nizami A-S, Moustakas K, Lee M. Single-Solution-Based Vortex Search Strategy for Optimal Design of Offshore and Onshore Natural Gas Liquefaction Processes. *Energies*. 2020; 13(7):1732.
https://doi.org/10.3390/en13071732

**Chicago/Turabian Style**

Qyyum, Muhammad Abdul, Muhammad Yasin, Alam Nawaz, Tianbiao He, Wahid Ali, Junaid Haider, Kinza Qadeer, Abdul-Sattar Nizami, Konstantinos Moustakas, and Moonyong Lee. 2020. "Single-Solution-Based Vortex Search Strategy for Optimal Design of Offshore and Onshore Natural Gas Liquefaction Processes" *Energies* 13, no. 7: 1732.
https://doi.org/10.3390/en13071732