Analysis of One Dimensional Advection Diffusion Model Using Finite Difference Method

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.

[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)
[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)
[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)
[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.