Efficient Numerical Solution of Diffusion Convection Problem of Chemical Engineering

Transcription

1 Cheical and Process Engineering Research ISSN (Paper) ISSN 5-9 (Online) Vol, 5 Efficient Nuerical Solution of Diffusion Convection Proble of Cheical Engineering Bharti Gupta VK Kukreja * Departent of Matheatics, Sant Longowal Institute of Engineering and Technology Longowal 486, Punjab, India * E-ail of the corresponding author: vkkukreja@gailco Abstract In this paper, the cubic B-spline collocation schee is ipleented to find nuerical solution of diffusion convection proble of cheical engineering The schee is based on the Crank Nicolson forulation for tie integration and cubic B-spline functions for space integration The nuerical results are found to be in good agreeent with the exact solutions Results are also shown graphically and are copared with results given in the literature Keywords: Diffusion; Cubic B-spline; Collocation; Tridiagonal syste; Thoas algorith Introduction Consider a parabolic diffusion convection proble with ixed boundary conditions and initial condition which is encountered in different branches of cheical engineering: c c c Lc = + =, in Ω (,), () t 4Pe c c Bc = { 4Pec =, atx = ; =, at x = }, () Ic = c( x,) =, for all x () The proble is used to describe the displaceent of an initial hoogeneous solute fro a ediu (called bed in cheistry) of finite length x (in this casex = ) at tie t ( t ), by the introduction of a solvent Here c( xt, ) represents the concentration profile of the solute and Pe is a constant paraeter called the Peclet Partial differential equations can be solved analytically by using techniques like separation of variables, Laplace transfor and also nuerically by using a verity of ethods However, soe of these techniques have liited applications In particular, the Laplace transfor and nuerical techniques, which is a useful tool for solving partial differential equations in cheical engineering, could involve the solution of coplicated and transcendental equations which are tie consuing [Al-Jabari et al (994); Brenner (96); Grahs (975); Lapidus and Aundson (95); Potucek ()] In this paper, nuerical solution of syste (-) by using the cubic B-spline collocation ethod (CSCM) is proposed The collocation ethod together with B-spline approxiations represents an econoical alternative, since it only requires the evaluation of the unknown paraeters at the grid points Nuerical results thus obtained are copared with analytic results of Brenner (96) and pdepe solver results of Singh et al (8) A good agreeent is found with analytic ones whereas significant difference is found with the results obtained using pdepe solver Nuerical Schee Consider, a partition of the doain x at knot x, =,, N, such that = x < x < < x N = and hi = xi xi = h, ie, unifor partition i =,, N has been taken Each spline can be written as linear cobination of basis function of given spline by Prenter (975) and de Boor (978) Cubic B-splines B, =,, N + are defined over the interval [, ] by: ( x x ),[ x, x ] ( ) ( ) ( ),[, ] h + h x x + h x x x x x x ( ) = + ( + ) + ( + ) ( + ),[, + ] h x+ x x+ x+ B x h h x x h x x x x x x ( ),[, ], otherwise Each basis function B( x ) is twice continuously differentiable, and the values of B( x ), B ( x), B ( x) ay be tabulated as in Table 7

2 Cheical and Process Engineering Research ISSN (Paper) ISSN 5-9 (Online) Vol, 5 The global approxiation c N to the exact solution c can be given by: N + cn ( xt, ) = δ( t) B( x), = where δ are tie dependent paraeters to be deterined using cubic B-spline collocation ethod The collocation points are selected in order to coincide with knots x Using the approxiate solution and cubic B- splines B( x ), nodal values c, c (first derivative) and c (second derivative) at knots x are obtained in ters of the eleent paraeters by: c = δ + 4δ + δ, + c ( ) = h δ δ +, = 6 ( δ h δ + δ+ ) c Discretization of Model The solution of given odel Eqs (-) can be obtained by assuing that the paraeters δ and its tie derivatives are linearly interpolated between two tie levels n and n +, n c n n+ c c = θf ( c, xt, ) + ( θ) F ( c, xt, ), where F( c, xt, ) = t 4Pe x The tie derivative is discretized in the usual finite difference way and applying Crank Nicolson schee to Eq (), it becoes: n+ n n+ n n+ n c c ( cxx) + ( cxx) ( cx) + ( cx) = t 4Pe Separating the ters of advanced and initial level: n+ n+ ( ) n+ ( ) n n n δ α + δ 4 + α + δ+ ( α ) = δ ( + α ) + δ ( 4 α ) + δ+ ( + α ), (4) where α 6 6 ; ; 8Peh h t α 8Peh t α = + = = 8Peh h t Boundary conditions are: 4 Pe( δ + 4 δ + δ) ( δ δ ) =, at x =, (5) h ( N N ) h δ + δ =, at x = (6) Initial condition is: δ + 4δ + δ+ =, at t =, =,,, N (7) Now using initial condition (7) and eliinating δ and δ N + fro boundary conditions (5) and (6), we can find initial approxiation δ fro the tri-diagonal atrix: 4 α5 + α4 δ 4 δ 4 =, 4 δn 4 δ N 4Peh where α 4 = + 4Peh ; 6Peh α5 = 4Peh + Also Eq (4) is reduced to a ( N + ) ( N + ) atrix syste as: 7

3 Cheical and Process Engineering Research ISSN (Paper) ISSN 5-9 (Online) Vol, 5 By using δ δ n n δ + α -α + α α α + α -α α n α 4 α α + δ + α 4 α α n+ + δn α α 4 α - + n+ δn n δ -α -α α + -α α + α α + + α n α 4 α α δ + + = α 4 α α n + + δn α α 4 α + + n δn n, the above atrix is solved for δ s in the approxiate solution, exit concentration c N can be deterined s by using Thoas algorith (see Appendix A) By putting 4 Results and Discussion Cheical engineers are particularly interested in the exit solute concentration Stiff syste of Eqs (-) have been solved analytically up to acceptable accuracy by Brenner (96) A coparison of analytic results of Brenner (96), results of Singh et al (8) and results obtained using CSCM for wide range of Pe are given in Table The Exit concentration profiles are drawn for different values of Peclet nuber, as shown in Figure, and For Pe, the transport by axial dispersion becoes stronger than convective transport, which causes ore back ixing, resulting in ore dispersion of solute, leading to longer period for solute to leach out fro fibers For large Pe ( = 4), solution profile is converging to steady-state condition sharply, because dispersion becoes saller thereby increasing the solute reoval rate However, no significant effect is observed on solution profiles for Pe > 4 Al-Jabari et al (994) have established that efficient washing operations can be achieved for Pe 4 5 Conclusions CSCM is used due to its conceptual siplicity, wide applicability and ease of ipleentation Breakthrough curves indicates, that an increase in the Pe causes the area under the curve to increase, resulting in less dispersion and lesser back ixing which leads to ore reoval of solute adsorbed on the particle surface CSCM gave good results for large values of Pe also where as Brenner (96) has given asyptotic expression for such large values of Pe Results are copared with Singh et al (8) also and fro error analysis it is observed that CSCM is giving uch ore accurate results than MATLAB pdepe solver References Al-Jabari, M, Van Heiningen, A R P and Van de Ven, T G M (994), Modeling the flow and the deposition of fillers in packed beds of pulp fibers, J Pulp Paper Sci (9), J49 J5 Brenner, H (96), The diffusion odel of longitudinal ixing in beds of finite length, Nuerical values, Che Eng Sci 7, 9 4 de Boor, C (978), A practical guide to splines, Springer Verlag, NewYork Grahs, L E (975), Displaceent washing of packed beds of cellulose fibers, Part-: Matheatical odel, Svensk Papperstid 78(), Lapidus, L and Aundson, N R (95), Matheatics of adsorption in beds, Part-VI: The effect of longitudinal diffusion in ion exchange and chroatographic coluns, J Phys Che 56, Potucek, F (), Displaceent washing of pulp : Dispersion odel for non flow, Papir Celulóza 56(), 8 Prenter, P M (975), Splines and variational ethods, Wiley interscience publication, New York Singh, V P, Kuar, V and Kuar, D (8), Nuerical solution of diffusion odel of brown stock washing 7

4 Cheical and Process Engineering Research ISSN (Paper) ISSN 5-9 (Online) Vol, 5 beds of finite length using MATLAB, nd UKSIM, 95 Appendix A For the tri-diagonal atrix equation, b c L L y a b c L y a b c L y M M O O O M M L an bn c n y n L L an bn yn the steps of Thoas algorith to obtain the solutions are as follows: Step : β = b Step : γ = r /β Step : For i =,, n β i = b i (a i c i- /β i- ) γ i = (r i a i γ i- )/β i End Step 4: y n = γ n Step 5: For j =,, n y n-j = γ n-j c n-j y n-j+ /β n-j End = r r r, M r n r n Table Values of B( x ), B ( x), B ( x) xi xi x i x i + x i + B ( x ) 4 B ( x ) -/h /h B ( x ) 6/h -/h 6/h t (Tie) Table Coparison of CSCM with results of Brenner (96) and Singh et al (8) Brenner (96) Pe = Pe = 4 Pe = Singh et al (8) CSCM Brenner (96) Singh et al (8) CSCM Brenner (96) Singh et al (8) CSCM

5 Cheical and Process Engineering Research ISSN (Paper) ISSN 5-9 (Online) Vol, 5 Nuerical solution for Pe = 5 Exit concentration c Tie t Distance x Figure Solution profile for Peclet nuber 5 Nuerical solution for Pe = Exit concentration c Tie t Distance x Figure Solution profile for Peclet nuber 4 Nuerical solution for Pe = Exit concentration c Distance x Tie t Figure Solution profile for Peclet nuber 74

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