Mathematical model to estimate volumetric oxygen transfer coefficient in bioreactors using conformable calculus

  • R. Melgarejo-Torres
  • D. Rosales-Mercado
  • M.A. Polo-Labarrios
  • G. Fernández-Anaya
  • M. Morales-Ibarría
  • S.B. Pérez-Vega
  • M.B. Arce-Vázquez
  • D.M. Palmerín-Carreño
Keywords: fractional calculus; conformable derivative; mathematical model; volumetric oxygen transfer coefficient; Akaike and Bayesian information criterion


This work proposes a novel mathematical model based on time conformable derivative convective mass transfer equation to calculate the volumetric oxygen transfer coefficient (kLa) in a bioreactor. To validate the novel model, a full mixed-level experimental design was proposed with two factors: agitation speed and dispersed phase. The model employs the conformable derivate order operator (α) and the electrode constant (kp), which changes with electrode use and the operating conditions of the bioreactor. The results show that when the viscosity increases and the agitation decreases, the value α increases, and vice versa. Therefore, α is a parameter that has a physical meaning in the process. The correlation coefficient of the proposed model with the experimental data (R2 > 0.985) is higher than the one obtained with conventional models. The Akaike information criterion determined that the proposed conformal model describes the experimental data by 59%, while the conventional models describe the experimental data by 25% and 15%. There are no reports of similar mathematical models that determine mass transfer coefficients in bioprocesses.


Aiba S, Humphrey A.E, Millis N.F. (1973) Biochemical Engineering, University of Tokyo Press, Tokyo,
Akaike H, (1973) Information theory as an extension of the maximum likelihood principle, in: B.N. Petrov, F. Csaki (Eds.), Second International Symposium on Information Theory, Akademiai Kiado, Budapest.
Almeida R, (2017) A Caputo fractional derivative of a function with respect to another function. Communications in Nonlinear Science and Numerical Simulation. 44:460-481.
Ascanio G, B. Castro, E. Galindo, (2004) Measurement of power consumption in stirred vessels: a review, Trans IChemE, Part A: Chem. Eng. Res. Des. 82:1282–1290.
Babakhani A, V. Daftardar–Gejji, (2002) On calculus of local fractional derivatives, J. Math. Anal. Appl. 270:66–79.
Cerri M, Esperança M, Badino and M. (2016) Perencin de Arruda Ribeiro. A new approach for kLa determination by gassing-out method in pneumatic bioreactors. J Chem Technol Biotechnol. 91:3061–3069. DOI 10.1002/jctb.4937
Chen Z, H.W. Liu, H. Zhang, W.Ying, D. Fang, (2013) Oxygen mass transfer coefficient in bubble column slurry reactor with ultrafine suspended particles and neural network prediction, Can. J. Chem. Eng. 91:532–541.
Diethelm K (1997) An algorithms for numerical solution of differential equations of fractional order. Electron. Trans. Numer. Anal. 5:97–126.
Fang S, P.W. Todd, T.R. Hanley (2017) Enhanced oxygen delivery to a continuous multiphase bioreactor, Chem. Eng. Sci.170:597–605.
Fernández-Anaya G, Quezada-García S, Polo-Labarrios M.A., Quezada-Téllez L.A., (2021). Novel solution to the fractional neutron point kinetic equation using conformable derivatives. Annals of Nuclear Energy 160, 108407.
Fuchs, D. Dewey, A. Humphrey (1971) Effect of surface aeration on scale-up producers for fermentation processes, Ind. Eng. Chem. Proc. Des. Dev. 10(2):1990.
Garcia-Ochoa F, E. Gomez, V.E. Santos, J.C. Merchuk (2010) Oxygen uptake rate in microbial processes: An overview, Biochem. Eng. J. 49:289–307.
Garcia-Ochoa G and E. Gomez (2009) Bioreactor scale-up and oxygen transfer rate in microbial processes: an overview. Biotechnol Adv. 27:153–176. DOI: 10.1016/j.biotechadv.2008.10.006
Gorenflo R, F. Mainardi (1997) Fractional calculus: integral and differential equations of fractional order, in: A. Carpinteri, F. Mainardi (Eds.), Fractals and Fractional Calculus in Continuum Mechanics. Springer, Berlin.
Hadjiev D, Sabiri E, Zanati A, (2006) Mixing time in bioreactors under aerated conditions. Biochem. Eng. J. 27:323–330.
Hilfer R (2000) Applications of Fractional Calculus in Physics. World Scientific, Singapore.
Ingdal M, Johnsen R, Harrington D, (2019) The Akaike information criterion in weighted regression of immittance data. Electrochimica Acta. 317:648.
Khalil R, M. Al Horani, A. Yousef, M. Sababheh (2014) A new definition of fractional derivative, Journal of Computational and Applied Mathematics, 264:65-70.
Kilbas A, Srivastava H, Trujillo J, (2006) Theory and Applications of Fractional Fifferential Equations. ISBN: 978-0-444-51832-3, Elsevier, Amsterdam.
Koizumi J, Aiba S, (1984) Reassessment of the dynamic kLa method, Biotechnol. Bioeng. 26:1131–1133.
Li S, C. Zhu, T. Fu, Y. Ma (2012) Study on the mass transfer of bubble swarms in three different rheological fluids, Int. J. Heat Mass Tran. 55:6010–6016.
Linek V and Vacek V (1981) Chemical engineering use of catalyzed sulfute oxidation kinetics for the determination ofmass transfer characteristics of gas–liquid contactors. Chem Eng Sci. 36:1747–1768.
Magin R (2010) Fractional calculus models of complex dynamics in biological tissues, Comput. Math. App. 56:1586–1593.
Melgarejo-Torres R, Castillo-Araiza C, López-Ordaz P, Torres Martínez D, Gutiérrez-Rojas M, G.J. Lye, S. Huerta-Ochoa, (2014) Kinetic mathematical model for ketone bioconversion using Escherichia coli TOP10 pQR239. Chem. Eng. J. 240:1-9.
Mendes CE and Badino AC (2015) Oxygen transfer in different pneumatic bioreactors containing viscous Newtonian fluids. Chem Eng Res Des. 94:456–465.
Montgomery D, (2013) Design and Analysis of Experiments. Eighth Edition. John Wiley & Sons Inc.
Nauman EB, (2008) Chemical Reactor Design, Optimization, and Scale-Up, New York: John Wiley & Sons
Oldham KB, J. Spanier, (1974) The fractional calculus theory and applications of differentiation and integration to arbitrary order. Academic Press, New York and London.
Palmerín-Carreño D, Castillo-Araiza C, Rutiaga-Quiñones O, Verde-Calvo J, Huerta-Ochoa S, (2016) Kinetic, oxygen mass transfer and hydrodynamic studies in a three-phase stirred tank bioreactor for the bioconversion of (+)-valencene on Yarrowia lipolytica 2.2ab. Biochem. Eng. J. 113:37-46.
Podlubny I (1998) Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications. Academic Press, New York.
Sohail RL, K. Vimar, J.R. Seay, D.L. Englert, H.T. Hwang (2008) Evaluation of volumetric mass transfer coefficient in a stirred tank bioreactor using response surface methodology, Environ. Prog. Sustain. 38(2):387–401.
Sun HG, Y. Zhan, D.Baleanu, W. Chen, Y.O. Chen (2018) A new collection of real-World applications of fractional calculus in science and engineering, Commun. Non-Linear Sci. 64:213–231.
Suresh S, V.C. Srivastava, I.M. Mishra (2009) Techniques for oxygen transfer measurement in bioreactors: a review, J. Chem. Technol. Biot. 84:1091–1103.
Zuluaga-Bedoya M, M. Ruíz-Botero, M. Ospina-Alarcón, J. García-Tirado (2018) A dynamical model of an aeration plant for wastewater treatment using a phenomenological based semi-physical modeling methodology, Comput. Chem. Eng. 117:420–432.
How to Cite
Melgarejo-Torres, R., Rosales-Mercado, D., Polo-Labarrios, M., Fernández-Anaya, G., Morales-Ibarría, M., Pérez-Vega, S., Arce-Vázquez, M., & Palmerín-Carreño, D. (2022). Mathematical model to estimate volumetric oxygen transfer coefficient in bioreactors using conformable calculus. Revista Mexicana De Ingeniería Química, 21(2), Bio2701.

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