Approximate analytical solution of non-linear reaction diffusion equation in fluidized bed biofilm reactor

Affiliation(s)

Department of Mathematics, The Madura College, Madurai, India.

Department of Mathematics, The Madura College, Madurai, India;.

Department of Mathematics, The Madura College, Madurai, India.

Department of Mathematics, The Madura College, Madurai, India;.

ABSTRACT

A mathematical model for the fluidized bed biofilm reactor (FBBR) is discussed. An approximate analytical solution of concentration of phenol is obtained using modified Adomian decomposition method (MADM). The main objective is to propose an analytical method of solution, which do not require small parameters and avoid linearization and physically unrealistic assumptions. Theoretical results obtained can be used to predict the biofilm density of a single bioparticle. Satisfactory agreement is obtained in the comparison of approximate analytical solution and numerical simulation.

A mathematical model for the fluidized bed biofilm reactor (FBBR) is discussed. An approximate analytical solution of concentration of phenol is obtained using modified Adomian decomposition method (MADM). The main objective is to propose an analytical method of solution, which do not require small parameters and avoid linearization and physically unrealistic assumptions. Theoretical results obtained can be used to predict the biofilm density of a single bioparticle. Satisfactory agreement is obtained in the comparison of approximate analytical solution and numerical simulation.

Cite this paper

Usha, S. , Anitha, S. and Rajendran, L. (2012) Approximate analytical solution of non-linear reaction diffusion equation in fluidized bed biofilm reactor.*Natural Science*, **4**, 983-991. doi: 10.4236/ns.2012.412127.

Usha, S. , Anitha, S. and Rajendran, L. (2012) Approximate analytical solution of non-linear reaction diffusion equation in fluidized bed biofilm reactor.

References

[1] Zobell, C.E. and Anderson, D.Q. (1936) Observationson the multiplication of bacteria in different volumes of stored sea water and the influence of oxygen tension and solid surfaces. Biological Bulletin, 71, 324-342. doi:10.2307/1537438

[2] Williamson, K. and McCarty, P.L. (1976) A model of substrate utilization by bacterial films. Journal of the Water Pollution Control Federation, 48, 9-24.

[3] Harremo?s, P. (1976) The significance of pore diffusion to filter denitrification. Journal of the Water Pollution Control Federation, 48, 377-388.

[4] Rittmann, B.E and McCarty, P.L. (1980) Model of steady- state-biofilm kinetics. Biotechnology and Bioengineering, 22, 2343-2357. doi:10.1002/bit.260221110

[5] Rittmann, B.E and McCarty, P.L. (1981) Substrate flux into biofilms of any thickness. Journal of Environmental Engineering, 107, 831-849.

[6] Rittman, B.E and McCarty, P.L. (1978) Variable-order model of bacterial-film kinetics. American Society of Civil Engineers. Environmental Engineering Division, 104, 889-900.

[7] Choi, J.W., Min, J., Lee, W.H and Lee, S.B. (1999) Mathematical model of a three-phase fluidized bed biofilm reactor in wastewater treatment. Biotechnology and Bioprocess Engineering, 4, 51-58. doi:10.1007/BF02931914

[8] Meikap, B.C and Roy, G.K. (1995) Recent advances in biochemical reactors for treatment of wastewater. International Journal of Environmental Protection, 15, 44-49.

[9] Vinod, A.V. and Reddy, G.V. (2003) Dynamic behaviour of a fluidised bed bioreactor treating waste water. Indian Chemical Engineer Section A, 45, 20-27.

[10] Sokol, W. (2003) Treatment of refinery wastewater in a three-phase fluidized bed bioreactor with a low-density biomass support. Biochemical Engineering Journal, 15, 1-10. doi:10.1016/S1369-703X(02)00174-2

[11] Gonzalez, G., Herrera, M.G., Garcia, M.T and Pena, M.M. (2001) Biodegradation of phenol in a continuous process: Comparative study of stirred tank and fluidized-bed bioreactors. Bioresource Technology, 76, 245-251. doi:10.1016/S0960-8524(00)00092-4

[12] Sokol, W. and Korpal, W. (2004) Determination of the optimal operational parameters for a three-phase fluidised bed bioreactor with a light biomass support when used intreatment of phenolic wastewaters. Biochemical Engineering Journal, 20, 49-56. doi:10.1016/j.bej.2004.02.009

[13] Tanyolac, A. and Beyenal, H. (1996) Predicting average biofilm density of a fully active spherical bioparticle. Journal of Biotechnology, 52, 39-49. doi:10.1016/S0168-1656(96)01624-0

[14] Beyenal, H. and Tanyolac, A. (1998) The effects of biofilm characteristics on the external mass transfercoefficient in a fluidized bed biofilm reactor. Biochemical Engineering Journal, 1, 53-61. doi:10.1016/S1369-703X(97)00010-7

[15] Adomian, G. (1976) Nonlinear stochastic differential equations. Journal of Mathematical Analysis and Applications, 55, 441-452. doi:10.1016/0022-247X(76)90174-8

[16] Adomian, G. and Adomian, G.E. (1984) A global method for solution of complex systems. Mathematical Model, 5, 521-568. doi:10.1016/0270-0255(84)90004-6

[17] Adomian, G. (1994) Solving frontier problems of physics: The decomposition method. Kluwer Academic Publishers, Boston, 1994.

[18] Hasan, Y.Q and Zhu, L.M. (2008) Modified adomian decomposition method for singular initial value problems in the second-order ordinary differential equations. Surveys in Mathematics and Its Applications, 3, 183-193.

[19] Hosseini, M.M. (2006) Adomian decomposition method with Chebyshev polynomials. Applied Mathematics and Computation, 175, 1685-1693. doi:10.1016/j.amc.2005.09.014

[20] Wazwaz, A.M. (1999) A reliable modifications of Adomian decomposition method. Applied Mathematics and Computation, 102, 77-86. doi:10.1016/S0096-3003(98)10024-3

[21] Wazwaz, A.M. (1999) Analytical approximations and Pade approximants for Volterra’s population model. Applied Mathematics and Computation, 100, 13-25. doi:10.1016/S0096-3003(98)00018-6

[22] Wazwaz, A.M. (2002) A new method for solving singular initial value problems in the second-order ordinary differential equations. Applied Mathematics and Computation, 128, 45-57. doi:10.1016/S0096-3003(01)00021-2

[1] Zobell, C.E. and Anderson, D.Q. (1936) Observationson the multiplication of bacteria in different volumes of stored sea water and the influence of oxygen tension and solid surfaces. Biological Bulletin, 71, 324-342. doi:10.2307/1537438

[2] Williamson, K. and McCarty, P.L. (1976) A model of substrate utilization by bacterial films. Journal of the Water Pollution Control Federation, 48, 9-24.

[3] Harremo?s, P. (1976) The significance of pore diffusion to filter denitrification. Journal of the Water Pollution Control Federation, 48, 377-388.

[4] Rittmann, B.E and McCarty, P.L. (1980) Model of steady- state-biofilm kinetics. Biotechnology and Bioengineering, 22, 2343-2357. doi:10.1002/bit.260221110

[5] Rittmann, B.E and McCarty, P.L. (1981) Substrate flux into biofilms of any thickness. Journal of Environmental Engineering, 107, 831-849.

[6] Rittman, B.E and McCarty, P.L. (1978) Variable-order model of bacterial-film kinetics. American Society of Civil Engineers. Environmental Engineering Division, 104, 889-900.

[7] Choi, J.W., Min, J., Lee, W.H and Lee, S.B. (1999) Mathematical model of a three-phase fluidized bed biofilm reactor in wastewater treatment. Biotechnology and Bioprocess Engineering, 4, 51-58. doi:10.1007/BF02931914

[8] Meikap, B.C and Roy, G.K. (1995) Recent advances in biochemical reactors for treatment of wastewater. International Journal of Environmental Protection, 15, 44-49.

[9] Vinod, A.V. and Reddy, G.V. (2003) Dynamic behaviour of a fluidised bed bioreactor treating waste water. Indian Chemical Engineer Section A, 45, 20-27.

[10] Sokol, W. (2003) Treatment of refinery wastewater in a three-phase fluidized bed bioreactor with a low-density biomass support. Biochemical Engineering Journal, 15, 1-10. doi:10.1016/S1369-703X(02)00174-2

[11] Gonzalez, G., Herrera, M.G., Garcia, M.T and Pena, M.M. (2001) Biodegradation of phenol in a continuous process: Comparative study of stirred tank and fluidized-bed bioreactors. Bioresource Technology, 76, 245-251. doi:10.1016/S0960-8524(00)00092-4

[12] Sokol, W. and Korpal, W. (2004) Determination of the optimal operational parameters for a three-phase fluidised bed bioreactor with a light biomass support when used intreatment of phenolic wastewaters. Biochemical Engineering Journal, 20, 49-56. doi:10.1016/j.bej.2004.02.009

[13] Tanyolac, A. and Beyenal, H. (1996) Predicting average biofilm density of a fully active spherical bioparticle. Journal of Biotechnology, 52, 39-49. doi:10.1016/S0168-1656(96)01624-0

[14] Beyenal, H. and Tanyolac, A. (1998) The effects of biofilm characteristics on the external mass transfercoefficient in a fluidized bed biofilm reactor. Biochemical Engineering Journal, 1, 53-61. doi:10.1016/S1369-703X(97)00010-7

[15] Adomian, G. (1976) Nonlinear stochastic differential equations. Journal of Mathematical Analysis and Applications, 55, 441-452. doi:10.1016/0022-247X(76)90174-8

[16] Adomian, G. and Adomian, G.E. (1984) A global method for solution of complex systems. Mathematical Model, 5, 521-568. doi:10.1016/0270-0255(84)90004-6

[17] Adomian, G. (1994) Solving frontier problems of physics: The decomposition method. Kluwer Academic Publishers, Boston, 1994.

[18] Hasan, Y.Q and Zhu, L.M. (2008) Modified adomian decomposition method for singular initial value problems in the second-order ordinary differential equations. Surveys in Mathematics and Its Applications, 3, 183-193.

[19] Hosseini, M.M. (2006) Adomian decomposition method with Chebyshev polynomials. Applied Mathematics and Computation, 175, 1685-1693. doi:10.1016/j.amc.2005.09.014

[20] Wazwaz, A.M. (1999) A reliable modifications of Adomian decomposition method. Applied Mathematics and Computation, 102, 77-86. doi:10.1016/S0096-3003(98)10024-3

[21] Wazwaz, A.M. (1999) Analytical approximations and Pade approximants for Volterra’s population model. Applied Mathematics and Computation, 100, 13-25. doi:10.1016/S0096-3003(98)00018-6

[22] Wazwaz, A.M. (2002) A new method for solving singular initial value problems in the second-order ordinary differential equations. Applied Mathematics and Computation, 128, 45-57. doi:10.1016/S0096-3003(01)00021-2