Mathematical Investigation of Non-Linear Reaction-Diffusion Equations on Multiphase Flow Transport in the Entrapped-Cell Photobioreactor Using Asymptotic Methods

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DOI: https://doi.org/10.28924/2291-8639-22-2024-74
Published: May 6, 2024
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A. Reena, R. Swaminathan, Mathematical Investigation of Non-Linear Reaction-Diffusion Equations on Multiphase Flow Transport in the Entrapped-Cell Photobioreactor Using Asymptotic Methods, Int. J. Anal. Appl., 22 (2024), 74.

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A. Reena, R. Swaminathan

Abstract

The mathematical investigation of a multiphase flow transport model intended to clarify the interaction mechanism between reactions and diffusion processes in the gel granules containing the entrapped-cell photobioreactor Rhodopseudomonas palustris CQK 01 is presented in this research. The model uses two pertinent non-linear reaction-diffusion equations for biochemical interactions in the photobioreactor under steady-state circumstances to reflect the substrate and product concentration within the gel granules. The solid phase and liquid phase fluxes within the gel granules and their concentrations are analytically calculated using the asymptotic methods of the Akbari-Ganji method and the homotopy perturbation approach. Our analytical results and the numerical data obtained by MATLAB software are compared to determine accuracy. The analytical results agreed with the simulated results for all possible reaction-diffusion and saturation parameter values. In addition, the impacts of applying two limiting cases—saturated and unsaturated enzyme kinetics—were examined. The close correspondence between the simulated and analytical data demonstrates that the parameters in our suggested solution can be used to simulate the dynamic performance of a system.

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References

  1. M.Y. Azwar, M.A. Hussain, A.K. Abdul-Wahab, Development of Biohydrogen Production by Photobiological, Fermentation and Electrochemical Processes: A Review, Renew. Sustain. Energy Rev. 31 (2014), 158-173. https://doi.org/10.1016/j.rser.2013.11.022.
  2. M.D. Redwood, R.L. Orozco, A.J. Majewski, L.E. Macaskie, An Integrated Biohydrogen Refinery: Synergy of Photofermentation, Extractive Fermentation and Hydrothermal Hydrolysis of Food Wastes, Bioresource Technol. 119 (2012), 384-392. https://doi.org/10.1016/j.biortech.2012.05.040.
  3. Y. Luo, W. Guo, H.H. Ngo, L.D. Nghiem, F.I. Hai, J. Zhang, S. Liang, X.C. Wang, A Review on the Occurrence of Micropollutants in the Aquatic Environment and Their Fate and Removal During Wastewater Treatment, Sci. Total Environ. 473–474 (2014), 619–641. https://doi.org/10.1016/j.scitotenv.2013.12.065.
  4. P. Prachanurak, C. Chiemchaisri, W. Chiemchaisri, K. Yamamotob, Biomass Production from Fermented Starch Wastewater in Photo-Bioreactor with Internal Overflow Recirculation, Bioresource Technol. 165 (2014), 129–136. https://doi.org/10.1016/j.biortech.2014.03.119.
  5. Q. Li, B. Guo, J. Yu, J. Ran, B. Zhang, H. Yan, J.R. Gong, Highly Efficient Visible-Light-Driven Photocatalytic Hydrogen Production of CdS-Cluster-Decorated Graphene Nanosheets, J. Amer. Chem. Soc. 133 (2011), 10878–10884. https://doi.org/10.1021/ja2025454.
  6. Y.Z. Wang, Q. Liao, X. Zhu, R. Chen, C.L. Guo, J. Zhou, Bioconversion Characteristics of Rhodopseudomonas Palustris CQK 01 Entrapped in a Photobioreactor for Hydrogen Production, Bioresource Technol. 135 (2013), 331–338. https://doi.org/10.1016/j.biortech.2012.09.105.
  7. Y.Z. Wang, Q. Liao, X. Zhu, X. Tian, C. Zhang, Characteristics of Hydrogen Production and Substrate Consumption of Rhodopseudomonas Palustris CQK 01 in an Immobilized-Cell Photobioreactor, Bioresource Technol. 101 (2010), 4034–4041. https://doi.org/10.1016/j.biortech.2010.01.045.
  8. C. Zhang, A.J. Wang, Q.G. Zhang, A Two-Dimensional Mass Transfer Model for an Annular Bioreactor Using Immobilized Photosynthetic Bacteria for Hydrogen Production, Biotechnol. Lett. 35 (2013), 1579–1587. https://doi.org/10.1007/s10529-013-1250-2.
  9. C. Zhang, H. Zhang, Z. Zhang, Y. Jiao, Q. Zhang, Effects of Mass Transfer and Light Intensity on Substrate Biological Degradation by Immobilized Photosynthetic Bacteria Within an Annular Fiber-Illuminating Biofilm Reactor, J. Photochem. Photobiol. B: Biol. 131 (2014), 113–119. https://doi.org/10.1016/j.jphotobiol.2014.01.015.
  10. Y. Yang, Q. Liao, X. Zhu, H. Wang, R. Wu, D.J. Lee, Lattice Boltzmann Simulation of Substrate Flow Past a Cylinder with PSB Biofilm for Bio-Hydrogen Production, Int. J. Hydrogen Energy. 36 (2011), 14031–14040. https://doi.org/10.1016/j.ijhydene.2011.04.026.
  11. Q. Liao, D.-M. Liu, D.-D. Ye, X. Zhu, D.-J. Lee, Mathematical modeling of two-phase flow and transport in an immobilized-cell photobioreactor, Int. J. Hydrogen Energy. 36 (2011), 13939–13948. https://doi.org/10.1016/j.ijhydene.2011.03.088.
  12. C.L. Guo, H.S. Pei, H.X. Cao, F.Q. Guo, D.M. Liu, Y.M. Li, Simulation on Characteristics of Photo-Hydrogen Production and Substrate Degradation under Various Stacking Types, Int. J. Hydrogen Energy. 40 (2015), 10401–10409. https://doi.org/10.1016/j.ijhydene.2015.06.152.
  13. D.D. Ganji, N. Ranjbar Malidarreh, M. Akbarzade, Comparison of Energy Balance Period with Exact Period for Arising Nonlinear Oscillator Equations, Acta Appl. Math. 108 (2008), 353–362. https://doi.org/10.1007/s10440-008-9315-2.
  14. S.Z. Shirejini, M. Fattahi, Mathematical Modeling and Analytical Solution of Two-Phase Flow Transport in an Immobilized-Cell Photo Bioreactor Using the Homotopy Perturbation Method (HPM), Int. J. Hydrogen Energy. 41 (2016), 18405–18417. https://doi.org/10.1016/j.ijhydene.2016.08.055.
  15. T. Praveen, L. Rajendran, Theoretical Analysis Through Mathematical Modeling of Two-Phase Flow Transport in an Immobilized-Cell Photobioreactor, Chem. Phys. Lett. 625 (2015), 193–201. https://doi.org/10.1016/j.cplett.2015.01.007.
  16. V.S. Sajja, B. Sripathy, A Spectral Method to Reaction-Diffusion Equations in Gel Granules Using Bernoulli Wavelets, Int. J. Pure Appl. Math. 118 (2018), 301-310.
  17. A. Meena, L. Rajendran, Analytical Solution of System of Coupled Non-Linear Reaction Diffusion Equations. Part I: Mediated Electron Transfer at Conducting Polymer Ultramicroelectrodes, J. Electroanal. Chem. 647 (2010), 103–116. https://doi.org/10.1016/j.jelechem.2010.06.013.
  18. P. Jeyabarathi, M. Abukhaled, M. Kannan, L. Rajendran, M.E.G. Lyons, New Analytical Expressions of Concentrations in Packed Bed Immobilized-Cell Electrochemical Photobioreactor, Electrochem. 4 (2023), 447–459. https://doi.org/10.3390/electrochem4040029.
  19. C.L. Guo, H.X. Cao, H.S. Pei, F.Q. Guo, D.-M. Liu, A Multiphase Mixture Model for Substrate Concentration Distribution Characteristics and Photo-Hydrogen Production Performance of the Entrapped-Cell Photobioreactor, Bioresource Technol. 181 (2015), 40–46. https://doi.org/10.1016/j.biortech.2015.01.022.
  20. H. Koku, Aspects of the Metabolism of Hydrogen Production by Rhodobacter Sphaeroides, Int. J. Hydrogen Energy. 27 (2002), 1315–1329. https://doi.org/10.1016/s0360-3199(02)00127-1.
  21. L.C. Mary, R.U. Rani, A. Meena, L. Rajendran, Nonlinear Mass Transfer at the Electrodes with Reversible Homogeneous; Reactions: Taylor’s Series and Hyperbolic Function Method, Int. J. Electrochem. Sci. 16 (2021), 151037. https://doi.org/10.20964/2021.01.73.
  22. K. Nirmala, B. Manimegalai, L. Rajendran, Steady-State Substrate and Product Concentrations for Non-Michaelis-Menten Kinetics in an Amperometric Biosensor –Hyperbolic Function and Padé Approximants Method, Int. J. Electrochem. Sci. 15 (2020), 5682–5697. https://doi.org/10.20964/2020.06.09.
  23. R. Usha Rani, L. Rajendran, M.E.G. Lyons, Steady-State Current in Product Inhibition Kinetics in an Amperometric Biosensor: Adomian Decomposition and Taylor Series Method, J. Electroanal. Chem. 886 (2021), 115103. https://doi.org/10.1016/j.jelechem.2021.115103.
  24. S. Vinolyn Sylvia, R. Joy Salomi, L. Rajendran, M. Abukhaled, Solving Nonlinear Reaction–Diffusion Problem in Electrostatic Interaction with Reaction-Generated pH Change on the Kinetics of Immobilized Enzyme Systems Using Taylor Series Method, J. Math. Chem. 59 (2021), 1332–1347. https://doi.org/10.1007/s10910-021-01241-7.
  25. A.M. Wazwaz, A Reliable Modification of Adomian Decomposition Method, Appl. Math. Comp. 102 (1999), 77–86. https://doi.org/10.1016/s0096-3003(98)10024-3.
  26. K. Ranjani, R. Swaminathan, S.G. Karpagavalli, Mathematical Modelling of a Mono-Enzyme Dual Amperometric Biosensor for Enzyme-Catalyzed Reactions Using Homotopy Analysis and Akbari-Ganji Methods, Int. J. Electrochemi. Sci. 18 (2023), 100220. https://doi.org/10.1016/j.ijoes.2023.100220.
  27. M. Abukhaled, Variational Iteration Method for Nonlinear Singular Two-Point Boundary Value Problems Arising in Human Physiology, J. Math. 2013 (2013), 720134. https://doi.org/10.1155/2013/720134.
  28. J.H. He, Homotopy Perturbation Technique, Comp. Meth. Appl. Mech. Eng. 178 (1999), 257–262. https://doi.org/10.1016/s0045-7825(99)00018-3.
  29. J.H. He, A Coupling Method of a Homotopy Technique and a Perturbation Technique for Non-Linear Problems, Int. J. Non-Linear Mech. 35 (2000), 37–43. https://doi.org/10.1016/s0020-7462(98)00085-7.
  30. R. Swaminathan, K. Venugopal, M. Rasi, M. Abukhaled, L. Rajendran, Analytical Expressions for the Concentration and Current in the Reduction of Hydrogen Peroxide at a Metal-Dispersed Conducting Polymer Film, Quim. Nova. 43 (2020), 58–65. https://doi.org/10.21577/0100-4042.20170454.
  31. R. Swaminathan, M. Chitra Devi, L. Rajendran, K. Venugopal, Sensitivity and Resistance of Amperometric Biosensors in Substrate Inhibition Processes, J. Electroanal. Chem. 895 (2021), 115527. https://doi.org/10.1016/j.jelechem.2021.115527.
  32. S.V. Sylvia, R.J. Salomi, L. Rajendran, M.E.G. Lyons, Amperometric Biosensors and Coupled Enzyme Nonlinear Reactions Processes: A Complete Theoretical and Numerical Approach, Electrochimica Acta. 415 (2022), 140236. https://doi.org/10.1016/j.electacta.2022.140236.
  33. A. Reena, SG. Karpagavalli, L. Rajendran, B. Manimegalai, R. Swaminathan, Theoretical Analysis of Putrescine Enzymatic Biosensor with Optical Oxygen Transducer in Sensitive Layer Using Akbari–Ganji Method, Int. J. Electrochem. Sci. 18 (2023), 100113. https://doi.org/10.1016/j.ijoes.2023.100113.
  34. A. Nebiyal, R. Swaminathan, SG. Karpagavalli, Reaction Kinetics of Amperometric Enzyme Electrode in Various Geometries Using the Akbari-Ganji Method, Int. J. Electrochem. Sci. 18 (2023), 100240. https://doi.org/10.1016/j.ijoes.2023.100240.
  35. A. Reena, SG. Karpagavalli, R. Swaminathan, Theoretical Analysis and Steady-State Responses of the Multienzyme Amperometric Biosensor System for Nonlinear Reaction-Diffusion Equations, Int. J. Electrochem. Sci. 18 (2023), 100293. https://doi.org/10.1016/j.ijoes.2023.100293.