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Approximate solution of non-linear dynamic energy model for multiple effect evaporator using fourier series and metaheuristics

  • Process Systems Engineering, Process Safety
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Abstract

This article presents the approximate solution of non-linear dynamic energy model of multiple effect evaporator (MEE) using Fourier series and metaheuristics. The dynamic model of MEE involves first-order simultaneous ordinary differential equations (SODEs). Prior to solving the dynamic model, the non-linear steady-state model is solved to obtain the optimum steady-state process parameters. These process parameters serve as the initial conditions (constraints) for the SODEs. The SODEs are exemplified as an optimization problem by the weighted residual function to produce their approximate solutions. The optimization task is to find the best estimates of unknown coefficients in the Fourier series expansion using two preeminent metaheuristic approaches: Particle swarm optimization and harmony search. Besides, the influence of the number of approximation terms in Fourier series expansion on the accuracy of the approximate solutions has been investigated. The solution of the dynamic model assists in the investigation of open-loop dynamics of the MEE. Moreover, the acquired results may assist in designing suitable controllers to ensure energy-efficient performance of MEE and to monitor the product quality. The optimization results reveal that both the metaheuristic approaches offer minimum violation of the constraints and, therefore, validate their efficiency in solving such complex non-linear energy models.

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Abbreviations

A:

effective heat transfer area [m2]

BFFC:

backward feed flow configuration

BL:

black liquor

f:

feed

hL or h:

enthalpy of black liquor [kJ/kg]

H:

enthalpy of vapor [kJ/kg]

HEE:

heptads’ effect evaporator

HS:

harmony search

i:

effect number

J:

objective function for steady-state model solution

L:

liquor flow rate [kg/s]

MEE:

multiple effect evaporator

N:

no. of effects in the evaporator system

NT:

no. of approximation terms in Fourier series expansion

ODE:

ordinary differential equation

PF:

penalty function

PSO:

particle swarm optimization

R:

residual function

SC:

steam consumption [kg/s]

SE:

steam economy

SNLAEs:

simultaneous non-linear algebraic equations

SODEs:

simultaneous ordinary differential equations

T:

temperature of vapor produced [°C]

U:

overall heat transfer coefficient [kW/m2°C]

V:

vapor flow rate [kg/s]

W:

weight function

WRF:

weighted residual function

x:

liquor concentration

z :

decision variables

λ :

latent heat of vaporization [kJ/kg]

References

  1. O. P. Verma, G. Manik and S. K. Sethi, Renew. Sust. Energy Rev., 100, 90 (2019).

    Article  Google Scholar 

  2. O. P. Verma, G. Manik and T. H. Mohammed, Korean J. Chem. Eng., 4, 2570 (2017).

    Article  Google Scholar 

  3. R. Bhargava, S. Khanam, B. Mohanty and A. K. Ray, Comput. Chem. Eng., 32, 3213 (2008).

    Article  CAS  Google Scholar 

  4. O. P. Verma, T. H. Mohammed, S. Mangal and G. Manik, In: Adv. Intell. Syst. Comput., 437, 1011, Springer (2016).

  5. D. Kaya and H. Ibrahim Sarac, Energy, 32, 1536 (2007).

    Article  Google Scholar 

  6. G. Gautami and S. Khanam, Desalination, 288, 16 (2012).

    Article  CAS  Google Scholar 

  7. O. P. Verma, T. H. Mohammed, S. Mangal and G. Manik, Int. J. Syst. Assur. Eng. Manag., 9, 111 (2018).

    Article  Google Scholar 

  8. R. Bhargava, S. Khanam, B. Mohanty and A. K. Ray, Comput. Chem. Eng., 32, 2203 (2008).

    Article  CAS  Google Scholar 

  9. O. P. Verma, T. H. Mohammed, S. Mangal and G. Manik, Energy, 129, 148 (2017).

    Article  Google Scholar 

  10. O. P. Verma, Suryakant and G. Manik, Int. J. Syst. Assur. Eng. Manag., 8, 63 (2017).

    Article  Google Scholar 

  11. O. P. Verma, G. Manik, Suryakant, V. K. Jain, D. K. Jain and H. Wang, Sustain. Comput. Informatics Syst., 20, 130 (2018).

    Article  Google Scholar 

  12. S. Pati, D. Yadav and O. P. Verma, Neural Comput. Appl., S. I: Hybridization of Neural Computing with Nature Inspired Algorithms, 1 (2020).

  13. D. Yadav and O. P. Verma, Heliyon, 6, e04349 (2020).

    Article  PubMed  PubMed Central  Google Scholar 

  14. A. Olsson, Particle swarm optimization: theory, techniques and applications, Nova Science Publishers, Inc., US (2010).

    Google Scholar 

  15. M. Jaberipour, E. Khorram and B. Karimi, Comput. Math. Appl., 62, 566 (2011).

    Article  Google Scholar 

  16. Z. W. Geem, J. H. Kim and G. V. Loganathan, Simulation, 76, 60 (2001).

    Article  Google Scholar 

  17. Z. W. Geem, Music-inspired harmony search algorithm, Springer, Berlin Heidelberg (2009).

    Book  Google Scholar 

  18. X. S. Yang, Stud. Comput. Intell, 191, 1 (2009).

    Google Scholar 

  19. G. Ingram and T. Zhang, Stud. Comput. Intell, 191, 15 (2009).

    Google Scholar 

  20. O. P. Verma, G. Manik and V. K. Jain, J. Comput. Sci., 25, 238 (2018).

    Article  Google Scholar 

  21. O. P. Verma, T. H. Mohammed, S. Mangal and G. Manik, Trans. Inst. Meas. Control, 40, 2278 (2018).

    Article  Google Scholar 

  22. A. Sadollah, H. Eskandar, D. G. Yoo and J. H. Kim, Eng. Appl. Artif. Intell, 40, 117 (2015).

    Article  Google Scholar 

  23. M. Babaei, Appl. Soft Comput. J., 13, 3354 (2013).

    Article  Google Scholar 

  24. I. H. Osman and G. Laporte, Ann. Oper. Res., 63, 513 (1996).

    Article  Google Scholar 

  25. F. Glover and G. Kochenberger, Handbook of metaheuristics, Springer, US (2006).

    Google Scholar 

  26. X.-S. Yang, Engineering optimization an introduction with metaheuristic applications, John Wiley and Sons, Inc., Hoboken, New Jersey (2010).

    Book  Google Scholar 

  27. X.-S. Yang, Nature-inspired metaheuristic algorithms second edition, Luniver Press, UK (2010).

    Google Scholar 

  28. A. Beléndez, E. Arribas, M. Ortuño, S. Gallego, A. Márquez and I. Pascual, Comput. Math. Appl., 64, 1602 (2012).

    Article  Google Scholar 

  29. Z. Y. Lee, Appl. Math. Comput., 179, 779 (2006).

    Google Scholar 

  30. G. D. Mateescu, Romanian J. Econ. Forecasting, 3, 5 (2006).

    Google Scholar 

  31. N. Mastorakis and N. E. Mastorakis, WSEAS Trans. on Mathematics, 5, 1276 (2006).

    Google Scholar 

  32. H. Cao, L. Kang, Y. Chen and J. Yu, Genet. Program. Evolvable Mach., 1, 309 (2000).

    Article  Google Scholar 

  33. C. L. Karr and E. Wilson, Appl. Intell, 19, 147 (2003).

    Article  Google Scholar 

  34. C. Reich, In: Proc. ACM Symp. Appl. Comput., USA, 1, 428 (2000).

    Google Scholar 

  35. J. C. Bansal, Evolutionary and swarm intelligence algorithms, Springer International Publishing, Switzerland (2019).

    Book  Google Scholar 

  36. T. Zhang and Z. W. Geem, Swarm Evol. Comput., 48, 31 (2019).

    Article  Google Scholar 

  37. A. Sadollah, Y. Choi, D. G. Yoo and J. H. Kim, Appl. Soft Comput. J., 33, 360 (2015).

    Article  Google Scholar 

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Acknowledgements

The first and second author would like to thank the Ministry of Human Resource Development, New Delhi, India for providing the Research Fellowship for carrying out this work. The first author gratefully acknowledges her family without whose help and support this work would ever have been possible.

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Authors and Affiliations

  1. Faculty of Informatics, Technische Universität Wien, 1040, Vienna, Austria

    Drishti Yadav

  2. Department of Electrical Engineering, Indian Institute of Technology, Roorkee, India

    Saurav Kumar

  3. Department of Instrumentation and Control Engineering, Dr. B. R. Ambedkar National Institute of Technology, Jalandhar, India

    Drishti Yadav, Saurav Kumar & Om Prakash Verma

  4. School of Electrical and Electronics, SASTRA University, Thanjavur, Tamilnadu, 613401, India

    Nikhil Pachauri

  5. Department of Industrial and Production Engineering, Dr. B. R. Ambedkar National Institute of Technology, Jalandhar, India

    Varun Sharma

Authors
  1. Drishti Yadav
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  2. Saurav Kumar
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  3. Om Prakash Verma
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  4. Nikhil Pachauri
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  5. Varun Sharma
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Corresponding author

Correspondence to Om Prakash Verma.

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This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.

Declaration of Competing Interest

No author associated with this paper has disclosed any potential or pertinent conflicts which may be perceived to have impending conflict with this work.

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Yadav, D., Kumar, S., Verma, O.P. et al. Approximate solution of non-linear dynamic energy model for multiple effect evaporator using fourier series and metaheuristics. Korean J. Chem. Eng. 38, 906–923 (2021). https://doi.org/10.1007/s11814-021-0787-3

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  • DOI: https://doi.org/10.1007/s11814-021-0787-3

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