Analysis of One Dimensional Advection Diffusion Model Using Finite Difference Method

Authors: Vijay Kumar Kukreja, Ravneet Kaur

Abstract:

In this paper, one dimensional advection diffusion model is analyzed using finite difference method based on Crank-Nicolson scheme. A practical problem of filter cake washing of chemical engineering is analyzed. The model is converted into dimensionless form. For the grid Ω × ω = [0, 1] × [0, T], the Crank-Nicolson spatial derivative scheme is used in space domain and forward difference scheme is used in time domain. The scheme is found to be unconditionally convergent, stable, first order accurate in time and second order accurate in space domain. For a test problem, numerical results are compared with the analytical ones for different values of parameter.

Keywords: Consistency, Crank-Nicolson scheme, Gerschgorin circle, Lax-Richtmyer theorem, Peclet number, stability.

Digital Object Identifier (DOI): doi.org/10.5281/zenodo.3566325

References:

[1] H. Kopka and P. W. Daly, A Guide to LATEX, 3rd ed. Harlow, England: Addison-Wesley, 1999.
[2] H. Brenner, The diffusion model of longitudinal mixing in beds of finite length. Numerical values, Chem. Eng. Sci. 17(4) (1962) 229-243.
[3] J. Crank, P. Nicolson, A practical method for numerical evaluation of solutions of partial differential equations of the heat conduction type, Proc. Cambridge Philosophy Soc. 43(1) (1947) 50-67.
[4] M.J. Cocero, J. Garcia, Mathematical model of supercritical extraction applied to oil seed extraction by CO2+ saturated alcohol I.Desorption model, J. Supercritical Fluids 20(3) (2001) 229-243.
[5] P.V. Danckwerts, Continuous flow systems distribution of residence times, Chem. Eng. Sci. 2(1) (1953) 1-44.
[6] S. Farooq, I.A. Karimi, Dispersed plug flow model for steady-state laminar flow in a tube with a first order sink at the wall, Chem. Eng. Sci. 58(1) (2003) 7180.
[7] M. Feiz, A 1-D multigroup diffusion equation nodal model using the orthogonal collocation method, Annals of Nuclear Energy 24(3) (1997) 187-196.
[8] L. Gardini, A. Servida, M. Morbidelli, S. Carra, Use of orthogonal collocation on finite elements with moving boundaries for fixed bed catalytic reactor simulation, Comp. Chem. Eng. 9(1) (1985) 1-17.
[9] B. Giojelli, C. Verdier, J.Y. Hihn, J.F. Beteau, A. Rozzi, Identification of axial dispersion coefficients by model method in gas/liquid/solid fluidised beds, Chem. Eng. P. 40(2) (2001) 159-166.
[10] C. Grossman, H.G. Roos, M. Stynes, Numerical Treatment of Partial Differential Equations, Springer-Verlag, Heidelberg 2007.
[11] S. Karacan, Y. Cabbar, M. Alpbaz, H. Hapoglu, The steady-state and dynamic analysis of packed distillation column based on partial differential approach, Chem Eng. P. 37(5) (1998) 379-388.
[12] I.A. Khan, K.F. Loughlin, Kinetics of sorption in deactivated zeolite crystal adsorbents, Comp. Chem. Eng. 27(5) (2003) 689-696.
[13] J.H. Koh, P.C. Wankat, N.H.L. Wang, Pore and surface diffusion and bulk-phase mass transfer in packed and fluidized beds, Ind. Eng. Chem. Res. 37(1) (1998) 228-239.
[14] V.K. Kukreja, A.K. Ray, V.P. Singh, N.J. Rao, A mathematical model for pulp washing on different zones of a rotary vacuum filter, Indian Chem. Eng., Sec- A 37(3) (1995) 113-124.
[15] V.K. Kukreja, A.K. Ray, Mathematical modeling of a rotary vacuum washer used for pulp washing: A case study of a lab scale washer, Cell. Chem. Tech. 43(1-3) (2009) 25-36.
[16] L. Lefevre, D. Dochain, S.F. Azevedo, A. Magnus, Optimal selection of orthogonal polynomials applied to the integration of chemical reactor equations by collocation methods, Comp. Chem. Eng. 24(12) (2000) 2571-2588.
[17] J.R. LeVeque, R. Bali, Finite Difference Methods for Ordinary and Partial Differential Equations Steady-State and Time-Dependent Problems, SIAM, Philadelphia, 2007.
[18] W.S. Long, S. Bhatia, A. Kamaruddin, Modeling and simulation of enzymatic membrane reactor forkinetic resolution of ibuprofen ester, J. Membrane Sci. 219(1-2) (2003) 69-88.
[19] C.G. Mingham, D.M. Causon, Introductory Finite Difference Methods for PDEs, Ventus Publishing, 2010.
[20] F. Potucek, Washing of pulp fibre beds, Collect. Czech. Chem. Commun. 62(4) (1997) 626-644.
[21] A.K. Ray, V.K. Kukreja, Solving pulp washing problems through mathematical models, AIChE Symposium Series, 96(324) (2000) 42-47.
[22] R.D. Richtmyer, K.W. Morton, Difference Methods for Initial Value Problems, Interscience Publishers, John Wiley & Sons, New York, 1967.
[23] L. Sajc, G.V. Novakovic, Extractive bioconversion in a four-phase external-loop airlift bioreactor, AIChE J. 46(7) (2000) 1368-1375.
[24] N.V. Saritha, G. Madras, Modeling the chromatographic response of inverse size-exclusion chromatography, Chem. Eng. Sci. 56(23) (2001) 6511-6524.
[25] G.D. Smith, Numerical Solutions of Partial Differential Equations: Finite Difference Methods, Clarendon press-Oxford, New York, 1985.
[26] P. Sridhar, Implementation of the one point orthogonal collocation method to an affinity packed bed model, Ind. Chem. Eng., Sec. A 41(1) (1999) 39-46.
[27] J.C. Strikwerda, Finite Difference Schemes and Partial Differential Equations, SIAM, Philadelphia 2004.
[28] L.M. Sun, F. Meunier, An improved finite difference method for fixed bed multicomponent sorption, AIChE J. 37(2) (1991) 244-254.
[29] M.K. Szukiewicz, New approximate model for diffusion and reaction in a porous catalyst, AIChE J. 46(3) (2000) 661-665.