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Research Article

Analyses of entropy generation and Hall influence in the Cattaneo–Christov double diffusive radiative Non-Darcy flow of a Casson nanofluid within rotating disks

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Received 12 Jan 2023, Accepted 19 Jul 2023, Published online: 08 Aug 2023

References

  • T. V. Kármán, “Über laminare und turbulente Reibung,” Z. Angew. Math. Mech, vol. 1, no. 4, pp. 233–252, 1921. DOI: 10.1002/zamm.19210010401.
  • M. Mustafa, “MHD nanofluid flow over a rotating disk with partial slip effects: Buongiorno model,” Int. J. Heat Mass Transfer, vol. 108, pp. 1910–1916, 2017. DOI: 10.1016/j.ijheatmasstransfer.2017.01.064.
  • S. Suresh Kumar Raju, M. Jayachandra Babu and C. S. K. Raju, “Irreversibility analysis in hybrid nanofluid flow between two rotating disks with activation energy and cross-diffusion effects,” Chin. J. Phys., vol. 72, pp. 499–529, 2021. DOI: 10.1016/j.cjph.2021.03.016.
  • A. Ayub, Z. Sabir, S. Z. H. Shah, H. A. Wahab, R. Sadat and M. R. Ali, “Effects of homogeneous-heterogeneous and Lorentz forces on 3-D radiative magnetized cross nanofluid using two rotating disks,” Int. Commun. Heat Mass Transfer, vol. 130, pp. 105778, 2022. DOI: 10.1016/j.icheatmasstransfer.2021.105778.
  • C. Cattaneo, “Sulla conduzione del calore,” Atti Semin. Mat. Fis. Univ. Modena Reggio Emilia, vol. 3, pp. 83–101, 1948.
  • C. I. Christov, “On frame indifferent formulation of the Maxwell–Cattaneo model of finite-speed heat conduction,” Mech. Res. Commun., vol. 36, no. 4, pp. 481–486, 2009. DOI: 10.1016/j.mechrescom.2008.11.003.
  • T. Hayat, S. Qayyum, M. Imtiaz and A. Alsaedi, “Flow between two stretchable rotating disks with Cattaneo-Christov heat flux model,” Results Phys., vol. 7, pp. 126–133, 2017. DOI: 10.1016/j.rinp.2016.12.007.
  • Y.-M. Chu, B. M. Shankaralingappa, B. J. Gireesha, F. Alzahrani, M. I. Khan and S. U. Khan, “Combined impact of Cattaneo-Christov double diffusion and radiative heat flux on bio-convective flow of Maxwell liquid configured by a stretched nano-material surface,” Appl. Math. Comput., vol. 419, pp. 126883, 2022. DOI: 10.1016/j.amc.2021.126883.
  • A. Hafeez, M. Khan, A. Ahmed and J. Ahmed, “Features of Cattaneo-Christov double diffusion theory on the flow of non-Newtonian Oldroyd-B nanofluid with Joule heating,” Appl. Nanosci., vol. 12, no. 3, pp. 265–272, 2022. DOI: 10.1007/s13204-020-01600-x.
  • I. Ahmad, M. Faisal, T. Javed and I. L. Animasaun, “Insight into the relationship between unsteady Cattaneo-Christov double diffusion, random motion and thermo-migration of tiny particles,” Ain Shams Eng. J., vol. 13, no. 1, pp. 101494, 2022. DOI: 10.1016/j.asej.2021.05.008.
  • F. Mabood, A. Rauf, B. C. Prasannakumara, M. Izadi and S. A. Shehzad, “Impacts of Stefan blowing and mass convention on flow of Maxwell nanofluid of variable thermal conductivity about a rotating disk,” Chin. J. Phys., vol. 71, pp. 260–272, 2021. DOI: 10.1016/j.cjph.2021.03.003.
  • H. Waqas, U. Manzoor, T. Muhammad and S. Hussain, “Thermo-bioconvection transport of nanofluid over an inclined stretching cylinder with Cattaneo–Christov double-diffusion,” Commun. Theor. Phys., vol. 73, no. 7, pp. 075006, 2021. DOI: 10.1088/1572-9494/abfcb9.
  • A. Chinna Venkata Ramudu, K. Anantha Kumar, V. Sugunamma and N. Sandeep, “Heat and mass transfer in MHD Casson nanofluid flow past a stretching sheet with thermophoresis and Brownian motion,” Heat Transfer, vol. 49, no. 8, pp. 5020–5037, 2020. DOI: 10.1002/htj.21865.
  • S. M. Abo-Dahab, M. A. Abdelhafez, F. Mebarek-Oudina and S. M. Bilal, “MHD Casson nanofluid flow over nonlinearly heated porous medium in presence of extending surface effect with suction/injection,” Indian J. Phys., vol. 95, no. 12, pp. 2703–2717, 2021. DOI: 10.1007/s12648-020-01923-z.
  • N. Shaheen, M. Ramzan and M. K. Alaoui, “Impact of Hall Current on a 3D Casson nanofluid flow past a rotating deformable disk with variable characteristics,” Arab. J. Sci. Eng., vol. 46, no. 12, pp. 12653–12666, 2021. DOI: 10.1007/s13369-021-06060-1.
  • U. Khan, S. Bilal, A. Zaib, O. D. Makinde and A. Wakif, “Numerical simulation of a nonlinear coupled differential system describing a convective flow of Casson gold–blood nanofluid through a stretched rotating rigid disk in the presence of Lorentz forces and nonlinear thermal radiation,” Numer. Methods Partial Differ. Equ., vol. 38, no. 3, pp. 308–328, 2022.
  • M. Jawad, A. Saeed, A. Khan and S. Islam, “MHD bioconvection Darcy-Forchheimer flow of Casson nanofluid over a rotating disk with entropy optimization,” Heat Transfer, vol. 50, no. 3, pp. 2168–2196, 2021. DOI: 10.1002/htj.21973.
  • B. Khatoon Siddiqui, S. Batool, M. Y. Malik, Q. Mahmood Ul Hassan and A. S. Alqahtani, “Darcy Forchheimer bioconvection flow of Casson nanofluid due to a rotating and stretching disk together with thermal radiation and entropy generation,” Case Stud. Thermal Eng., vol. 27, pp. 101201, 2021. DOI: 10.1016/j.csite.2021.101201.
  • M. Jawad, et al., “Insight into the dynamics of second grade hybrid radiative nanofluid flow within the boundary layer subject to Lorentz force,” Sci. Rep., vol. 11, no. 1, pp. 4894, 2021. DOI: 10.1038/s41598-021-84144-6.
  • M. Jawad, A. Saeed, P. Kumam, Z. Shah and A. Khan, “Analysis of boundary layer MHD darcy-forchheimer radiative nanofluid flow with soret and dufour effects by means of marangoni convection,” Case Stud. Thermal Eng., vol. 23, pp. 100792, 2021. DOI: 10.1016/j.csite.2020.100792.
  • M. Khan, W. Ali and J. Ahmed, “A hybrid approach to study the influence of Hall current in radiative nanofluid flow over a rotating disk,” Appl. Nanosci., vol. 10, no. 12, pp. 5167–5177, 2020. DOI: 10.1007/s13204-020-01415-w.
  • P. Rana, J. Mackolil, B. Mahanthesh and T. Muhammad, “Cattaneo-Christov theory to model heat flux effect on nanoliquid slip flow over a spinning disk with nanoparticle aggregation and Hall current,” Waves Random Complex Media, pp. 1–23, 2022. DOI: 10.1080/17455030.2022.2048127.
  • N. S. Khan, Q. Shah, A. Bhaumik, P. Kumam, P. Thounthong and I. Amiri, “Entropy generation in bioconvection nanofluid flow between two stretchable rotating disks,” Sci. Rep., vol. 10, no. 1, pp. 4448, 2020. DOI: 10.1038/s41598-020-61172-2.
  • N. Khan, et al., “Aspects of chemical entropy generation in flow of Casson nanofluid between radiative stretching disks,” Entropy, vol. 22, no. 5, pp. 495, 2020. DOI: 10.3390/e22050495.
  • K. Hosseinzadeh, A. R. Mogharrebi, A. Asadi, M. Sheikhshahrokhdehkordi, S. Mousavisani and D. D. Ganji, “Mohammadamin Sheikhshahrokhdehkordi, Seyedmohammad Mousavisani, and DD Ganji. Entropy generation analysis of mixture nanofluid (H2O/C2H6O2)−Fe3O4 flow between two stretching rotating disks under the effect of MHD and nonlinear thermal radiation,” Int. J. Ambient Energy, vol. 43, no. 1, pp. 1045–1057, 2022. DOI: 10.1080/01430750.2019.1681294.
  • A. Abbasi, et al., “Entropy generation applications in flow of viscoelastic nanofluid past a lubricated disk in presence of nonlinear thermal radiation and Joule heating,” Commun. Theor. Phys., vol. 73, no. 9, pp. 095004, 2021. DOI: 10.1088/1572-9494/ac0c75.
  • A. Mahesh, S. V. K. Varma, C. S. K. Raju, M. J. Babu, K. Vajravelu and W. Al-Kouz, “Significance of non-Fourier heat flux and radiation on PEG–Water based hybrid Nanofluid flow among revolving disks with chemical reaction and entropy generation optimization,” Int. Commun. Heat Mass Transfer, vol. 127, pp. 105572, 2021. DOI: 10.1016/j.icheatmasstransfer.2021.105572.
  • A. Wakif, A. Chamkha, T. Thumma, I. L. Animasaun and R. Sehaqui, “Thermal radiation and surface roughness effects on the thermo-magneto-hydrodynamic stability of alumina–copper oxide hybrid nanofluids utilizing the generalized Buongiorno’s nanofluid model,” J. Thermal Anal. Calorim., vol. 143, no. 2, pp. 1201–1220, 2021. DOI: 10.1007/s10973-020-09488-z.
  • A. Wakif, I. L. Animasaun, P. V. Satya Narayana and G. Sarojamma, “Meta-analysis on thermo-migration of tiny/nano-sized particles in the motion of various fluids,” Chin. J. Phys., vol. 68, pp. 293–307, 2020. DOI: 10.1016/j.cjph.2019.12.002.
  • G. Rasool and A. Wakif, “Numerical spectral examination of EMHD mixed convective flow of second-grade nanofluid towards a vertical Riga plate using an advanced version of the revised Buongiorno’s nanofluid model,” J. Thermal Anal. Calorim., vol. 143, no. 3, pp. 2379–2393, 2021. DOI: 10.1007/s10973-020-09865-8.
  • A. Wakif, M. Zaydan, A. Saleh Alshomrani, T. Muhammad and R. Sehaqui, “New insights into the dynamics of alumina-(60% ethylene glycol+ 40% water) over an isothermal stretching sheet using a renovated Buongiorno’s approach: a numerical GDQLLM analysis,” Int. Commun. Heat Mass Transfer, vol. 133, pp. 105937, 2022. DOI: 10.1016/j.icheatmasstransfer.2022.105937.
  • A. Wakif and N. A. Shah, “Hydrothermal and mass impacts of azimuthal and transverse components of Lorentz forces on reacting Von Kármán nanofluid flows considering zero mass flux and convective heating conditions,” Waves Random Complex Media, pp. 1–22, 2022. DOI: 10.1080/17455030.2022.2136413.
  • A. Wakif, I. L. Animasaun, U. Khan, N. A. Shah and T. Thumma, “Dynamics of radiative-reactive Walters-b fluid due to mixed convection conveying gyrotactic microorganisms, tiny particles experience haphazard motion, thermo-migration, and Lorentz force,” Phys. Scr., vol. 96, no. 12, pp. 125239, 2021. DOI: 10.1088/1402-4896/ac2b4b.
  • A. Wakif, et al., “Importance of exponentially falling variability in heat generation on chemically reactive von kármán nanofluid flows subjected to a radial magnetic field and controlled locally by zero mass flux and convective heating conditions: a differential quadrature analysis,” Front. Phys., vol. 10, pp. 768, 2022. DOI: 10.3389/fphy.2022.988275.
  • A. Wakif, I. L. Animasaun and R. Sehaqui, “A brief technical note on the onset of convection in a horizontal nanofluid layer of finite depth via Wakif-Galerkin weighted residuals technique (WGWRT),” DDF, vol. 409, pp. 90–94, 2021. DOI: 10.4028/www.scientific.net/DDF.409.90.
  • T. Gul, et al., “The study of the entropy generation in a thin film flow with variable fluid properties past over a stretching sheet,” Adv. Mech. Eng., vol. 10, no. 11, pp. 168781401878952, 2018. DOI: 10.1177/1687814018789522.
  • A. Saeed, et al., “Darcy-Forchheimer hybrid nanofluid flow over a stretching curved surface with heat and mass transfer,” PLoS One, vol. 16, no. 5, pp. e0249434, 2021. DOI: 10.1371/journal.pone.0249434.
  • T. Gul, M. Khan, W. Noman, I. Khan, T. Abdullah Alkanhal and I. Tlili, “Fractional order forced convection carbon nanotube nanofluid flow passing over a thin needle,” Symmetry, vol. 11, no. 3, pp. 312, 2019. DOI: 10.3390/sym11030312.
  • T. Gul, W. Noman, M. Sohail and M. A. Khan, “Impact of the Marangoni and thermal radiation convection on the graphene-oxide-water-based and ethylene-glycol-based nanofluids,” Adv. Mech. Eng., vol. 11, no. 6, pp. 168781401985677, 2019. DOI: 10.1177/1687814019856773.
  • T. Gul, M. Altaf Khan, A. Khan and S. Muhammad, “Fractional-order three-dimensional thin-film nanofluid flow on an inclined rotating disk,” Eur. Phys. J. Plus, vol. 133, no. 12, pp. 500, 2018. DOI: 10.1140/epjp/i2018-12315-4.
  • M. Bilal, A. Saeed, T. Gul, I. Ali, W. Kumam and P. Kumam, “Numerical approximation of microorganisms hybrid nanofluid flow induced by a wavy fluctuating spinning disc,” Coatings, vol. 11, no. 9, pp. 1032, 2021. DOI: 10.3390/coatings11091032.
  • G. R. Ganesh, W. Sridhar, K. Al-Farhany and S. E. Ahmed, “Electrically MHD Casson nanofluid flow and entropy exploration under the influence of the viscous dissipation, radiation, and higher-order chemical reaction,” Phys. Scr., vol. 97, no. 6, pp. 065208, 2022. DOI: 10.1088/1402-4896/ac6e51.
  • A. S. Sabu, A. Mathew, T. S. Neethu and K. Anil George, “Statistical analysis of MHD convective ferro-nanofluid flow through an inclined channel with hall current, heat source and Soret effect,” Thermal Sci. Eng. Prog., vol. 22, pp. 100816, 2021. DOI: 10.1016/j.tsep.2020.100816.
  • H. Waqas, U. Farooq, T. Muhammad, S. Hussain and I. Khan, “Thermal effect on bioconvection flow of Sutterby nanofluid between two rotating disks with motile microorganisms,” Case Stud. Thermal Eng., vol. 26, pp. 101136, 2021. DOI: 10.1016/j.csite.2021.101136.
  • T. Hayat, M. Javed, M. Imtiaz and A. Alsaedi, “Convective flow of Jeffrey nanofluid due to two stretchable rotating disks,” J. Mol. Liq., vol. 240, pp. 291–302, 2017. DOI: 10.1016/j.molliq.2017.05.024.
  • T. Hussain and H. Xu, “Time-dependent squeezing bio-thermal MHD convection flow of a micropolar nanofluid between two parallel disks with multiple slip effects,” Case Stud. Thermal Eng., vol. 31, pp. 101850, 2022. DOI: 10.1016/j.csite.2022.101850.
  • J. Buongiorno, “Convective transport in nanofluids,” J. Heat Transfer, vol. 128, no. 3, pp. 240–250, 2006. DOI: 10.1115/1.2150834.
  • A. Sahoo and R. Nandkeolyar, “Entropy generation in convective radiative flow of a Casson nanofluid in non-Darcy porous medium with Hall current and activation energy: the multiple regression model,” Appl. Math. Comput., vol. 402, pp. 125923, 2021.
  • S. K. Som, G. Biswas and S. Chakraborty, Introduction to Fluid Mechanics and Fluid Machines, 3rd ed. Tata McGraw-Hill Education, New Delhi; 2012.
  • A. Bhat and N. N. Katagi, “Magnetohydrodynamic flow of viscous fluid and heat transfer analysis between permeable discs: Keller-box solution,” Case Stud. Thermal Eng., vol. 28, pp. 101526, 2021. DOI: 10.1016/j.csite.2021.101526.
  • B. Jalili, H. Roshani, P. Jalili, M. Jalili, P. Pasha and D. D. Ganji, “The magnetohydrodynamic flow of viscous fluid and heat transfer examination between permeable disks by AGM and FEM,” Case Stud. Thermal Eng., vol. 45, pp. 102961, 2023. DOI: 10.1016/j.csite.2023.102961.
  • N. Khan, M. S. Hashmi, S. U. Khan, F. Chaudhry, I. Tlili and M. S. Shadloo, “Effects of homogeneous and heterogeneous chemical features on Oldroyd-b fluid flow between stretching disks with velocity and temperature boundary assumptions,” Math. Probl. Eng., vol. 2020, pp. 1–13, 2020. DOI: 10.1155/2020/5284906.
  • K. Das, S. Jana and N. Acharya, “Slip effects on squeezing flow of nanofluid between two parallel disks,” Int. J. Appl. Mech. Eng., vol. 21, no. 1, pp. 5–20, 2016. DOI: 10.1515/ijame-2016-0001.
  • S. Qayyum, M. Ijaz Khan, T. Hayat and A. Alsaedi, “Entropy generation and thermo-diffusion effects on unsteady chemically reactive slip flow between two rotating disks,” Int. J. Numer. Methods Heat Fluid Flow, vol. 29, pp. 3795-3821, 2019.
  • M. Sathish Kumar, N. Sandeep and B. Rushi Kumar, “Unsteady MHD nonlinear radiative squeezing slip-flow of Casson fluid between parallel disks,” J. Comput. Appl. Res. Mech. Eng., vol. 7, no. 1, pp. 35–45, 2017.
  • H. Tasawar, Q. Sumaira, I. Maria, A. Faris and A. Alsaedi, “Partial slip effect in flow of magnetite-Fe3O4 nanoparticles between rotating stretchable disks,” J. Magn. Magn. Mater., vol. 413, pp. 39–48, 2016.
  • Z. Tie-Hong, M. I. Khan and Y.-M. Chu, “Artificial neural networking (ANN) analysis for heat and entropy generation in flow of non-Newtonian fluid between two rotating disks,” Math. Methods Appl. Sci., vol. 46, no. 3, pp. 3012–3030, 2023. DOI: 10.1002/mma.7310.
  • F. Alzahrani and M. Ijaz Khan, “Entropy generation and Joule heating applications for Darcy Forchheimer flow of Ree-Eyring nanofluid due to double rotating disks with artificial neural network,” Alex. Eng. J., vol. 61, no. 5, pp. 3679–3689, 2022. DOI: 10.1016/j.aej.2021.08.071.
  • I. Ali, T. Gul and A. Khan, “Unsteady hydromagnetic flow over an inclined rotating disk through neural networking approach,” Mathematics, vol. 11, no. 8, pp. 1893, 2023. DOI: 10.3390/math11081893.
  • T. Hayat, T. Nasir, M. I. Khan and A. Alsaedi, “Non-darcy flow of water-based single (SWCNTs) and multiple (MWCNTs) walls carbon nanotubes with multiple slip conditions due to rotating disk,” Results Phys., vol. 9, pp. 390–399, 2018. DOI: 10.1016/j.rinp.2018.02.044.
  • N. Acharya, R. Bag and P. K. Kundu, “Influence of Hall current on radiative nanofluid flow over a spinning disk: a hybrid approach,” Physica E, vol. 111, pp. 103–112, 2019. DOI: 10.1016/j.physe.2019.03.006.
  • M. Abdel-Wahed and M. Akl, “Effect of hall current on MHD flow of a nanofluid with variable properties due to a rotating disk with viscous dissipation and nonlinear thermal radiation,” AIP Adv., vol. 6, no. 9, pp. 095308, 2016. DOI: 10.1063/1.4962961.
  • T. Hayat, M. I. Khan, A. Alsaedi and M. I. Khan, “Joule heating and viscous dissipation in flow of nanomaterial by a rotating disk,” Int. Commun. Heat Mass Transfer, vol. 89, pp. 190–197, 2017. DOI: 10.1016/j.icheatmasstransfer.2017.10.017.
  • M. Shoaib, et al., “Numerical analysis of 3-D MHD hybrid nanofluid over a rotational disk in presence of thermal radiation with joule heating and viscous dissipation effects using Lobatto IIIA technique,” Alex. Eng. J., vol. 60, no. 4, pp. 3605–3619, 2021. DOI: 10.1016/j.aej.2021.02.015.
  • M. Kumar and P. Kumar Mondal, “Irreversibility analysis of hybrid nanofluid flow over a rotating disk: effect of thermal radiation and magnetic field,” Colloids Surf. A: Physicochem. Eng. Asp., vol. 635, pp. 128077, 2022. DOI: 10.1016/j.colsurfa.2021.128077.
  • M. Ramzan, et al., “Significance of hall current and viscous dissipation in the bioconvection flow of couple-stress nanofluid with generalized Fourier and Fick laws,” Sci. Rep., vol. 12, no. 1, pp. 21812, 2022. DOI: 10.1038/s41598-022-22572-8.
  • S. Kumar Rawat, H. Upreti and M. Kumar, “Numerical study of activation energy and thermal radiation effects on Oldroyd-B nanofluid flow using the Cattaneo–Christov double diffusion model over a convectively heated stretching sheet,” Heat Transfer, vol. 50, no. 6, pp. 5304–5331, 2021. DOI: 10.1002/htj.22125.
  • A. Hafeez, M. Khan, A. Ahmed and J. Ahmed, “Rotational flow of Oldroyd-B nanofluid subject to Cattaneo-Christov double diffusion theory,” Appl. Math. Mech.-Engl. Ed., vol. 41, no. 7, pp. 1083–1094, 2020. DOI: 10.1007/s10483-020-2629-9.
  • A. S. Dogonchi and D. D. Ganji, “Impact of Cattaneo–Christov heat flux on MHD nanofluid flow and heat transfer between parallel plates considering thermal radiation effect,” J. Taiwan Inst. Chem. Eng., vol. 80, pp. 52–63, 2017. DOI: 10.1016/j.jtice.2017.08.005.
  • T. Rafiq and M. Mustafa, “Computational analysis of unsteady swirling flow around a decelerating rotating porous disk in nanofluid,” Arab. J. Sci. Eng., vol. 45, no. 2, pp. 1143–1154, 2020. DOI: 10.1007/s13369-019-04257-z.
  • F. G. Awad, P. Sibanda and A. A. Khidir, “Thermodiffusion effects on magneto-nanofluid flow over a stretching sheet,” Bound. Value Probl., vol. 2013, no. 1, pp. 1–13, 2013.
  • T. Hayat, S. Qayyum, M. I. Khan and A. Alsaedi, “Entropy generation in magnetohydrodynamic radiative flow due to rotating disk in presence of viscous dissipation and joule heating,” Phys. Fluids, vol. 30, no. 1, pp. 017101, 2018. DOI: 10.1063/1.5009611.
  • T. Hayat, M. I. Khan, S. Qayyum, M. I. Khan and A. Alsaedi, “Entropy generation for flow of Sisko fluid due to rotating disk,” J. Mol. Liq., vol. 264, pp. 375–385, 2018. DOI: 10.1016/j.molliq.2018.05.022.
  • M. Ijaz, M. Ayub and H. Khan, “Entropy generation and activation energy mechanism in nonlinear radiative flow of Sisko nanofluid: rotating disk,” Heliyon, vol. 5, no. 6, pp. e01863, 2019. DOI: 10.1016/j.heliyon.2019.e01863.
  • T. Barman, S. Roy and A. J. Chamkha, “Entropy generation analysis of MHD hybrid nanofluid flow due to radiation with non-erratic slot-wise mass transfer over a rotating sphere,” Alex. Eng. J., vol. 67, pp. 271–286, 2023. DOI: 10.1016/j.aej.2022.12.051.
  • M. I. Khan, S. Qayyum, T. Hayat and A. Alsaedi, “Entropy generation minimization and statistical declaration with probable error for skin friction coefficient and Nusselt number,” Chin. J. Phys., vol. 56, no. 4, pp. 1525–1546, 2018. DOI: 10.1016/j.cjph.2018.06.023.
  • M. K. Siddiq and M. Ashraf, “Bioconvection of micropolar nanofluid with modified Cattaneo–Christov theories,” Adv. Mech. Eng., vol. 12, no. 5, pp. 168781402092521, 2020. DOI: 10.1177/1687814020925217.

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