In this paper, the cubic B-spline collocation scheme is implemented to find numerical solution of diffusion convection problem of chemical engineering. The scheme is based on the Crank–Nicolson formulation for time integration and cubic B-spline functions for space integration. The numerical results are found to be in good agreement with the exact solutions. Results are also shown graphically and are compared with results given in the literature. Keywords: Diffusion; Cubic B-spline; Collocation; Tridiagonal system; Thomas algorithm

V.K. Kukreja, Bharti Gupta

English

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Chemical and Process Engineering Research                                                                                                                                    www.iiste.org ISSN 2224-7467 (Paper) ISSN 2225-0913 (Online) Vol.31, 2015  70  Efficient Numerical Solution of Diffusion Convection Problem of Chemical Engineering   Bharti Gupta      V.K. Kukreja * Department of Mathematics, Sant Longowal Institute of Engineering and Technology Longowal–148106, Punjab, India * E-mail of the corresponding author: vkkukreja@gmail.com  Abstract In this paper, the cubic B-spline collocation scheme is implemented to find numerical solution of diffusion convection problem of chemical engineering. The scheme is based on the Crank–Nicolson formulation for time integration and cubic B-spline functions for space integration. The numerical results are found to be in good agreement with the exact solutions. Results are also shown graphically and are compared with results given in the literature.     Keywords: Diffusion; Cubic B-spline; Collocation; Tridiagonal system; Thomas algorithm.  1. Introduction Consider a parabolic diffusion convection problem with mixed boundary conditions and initial condition which is encountered in different branches of chemical engineering:  Lc = 2214c c ct x Pe x∂ ∂ ∂+ =∂ ∂ ∂, in (0,1)Ω ≡ ,          (1)  Bc = { 4 0cPecx∂− =∂, at 0x = ;  0cx∂=∂, at 1x = },   (2)  Ic = ( ,0) 1c x = , for all x .   (3) The problem is used to describe the displacement of an initial homogeneous solute from a medium (called bed in chemistry) of finite length x  (in this case 1x = ) at time t  ( 0)t ≥ , by the introduction of a solvent. Here ( , )c x t  represents the concentration profile of the solute and Pe  is a constant parameter called the Peclet. Partial differential equations can be solved analytically by using techniques like separation of variables, Laplace transform and also numerically by using a verity of methods. However, some of these techniques have limited applications. In particular, the Laplace transform and numerical techniques, which is a useful tool for solving partial differential equations in chemical engineering, could involve the solution of complicated and transcendental equations which are time consuming [Al-Jabari et al. (1994); Brenner (1962); Grahs (1975); Lapidus and Amundson (1952); Potucek (2001)]. In this paper, numerical solution of system (1-3) by using the cubic B-spline collocation method (CSCM) is proposed. The collocation method together with B-spline approximations represents an economical alternative, since it only requires the evaluation of the unknown parameters at the grid points. Numerical results thus obtained are compared with analytic results of Brenner (1962) and ‘pdepe’ solver results of Singh et al. (2008). A good agreement is found with analytic ones whereas significant difference is found with the results obtained using ‘pdepe’ solver.  2. Numerical Scheme Consider, a partition of the domain 0 1x≤ ≤  at knot , 0,..., ,mx m N=  such that 0 10 ... 1Nx x x= < < < = and1, . .,i i ih x x h i e−= − =  uniform partition 1,...,i N= has been taken. Each spline can be written as linear combination of basis function of given spline by Prenter (1975) and de Boor (1978). Cubic B-splines mB , 1,..., 1m N= − +  are defined over the interval [0, 1] by: 32 2 13 2 2 31 1 1 13 2 2 31 1 1 1332 1 2( ) ,[ , ]3 ( ) 3 ( ) 3( ) ,[ , ]1( ) 3 ( ) 3 ( ) 3( ) ,[ , ]( ) ,[ , ]0 ,m m mm m m m mm m m m m mm m mx x x xh h x x h x x x x x xB x h h x x h x x x x x xhx x x xotherwise− − −− − − −+ + + ++ + + −+ − + − − −= + − + − − −−. Each basis function ( )mB x  is twice continuously differentiable, and the values of ( )mB x , ( )mB x′ , ( )mB x′′  may be tabulated as in Table 1. Chemical and Process Engineering Research                                                                                                                                    www.iiste.org ISSN 2224-7467 (Paper) ISSN 2225-0913 (Online) Vol.31, 2015  71  The global approximation Nc  to the exact solution c can be given by: 11( , ) ( ) ( )NN m mmc x t t B xδ+= −= ∑ , where mδ  are time dependent parameters to be determined using cubic B-spline collocation method. The collocation points are selected in order to coincide with knots mx . Using the approximate solution and cubic B-splines ( )mB x , nodal values c , c′  (first derivative) and c′′ (second derivative) at knots mx  are obtained in terms of the element parameters by: 1 14m m m mc δ δ δ− += + + , 1 13( )m m mchδ δ+ −′ = − , 26(mch′′ = 1 12 )m m mδ δ δ− +− + .  3. Discretization of Model The solution of given model Eqs. (1-3) can be obtained by assuming that the parameters mδ  and its time derivatives are linearly interpolated between two time levels n and n +1, 1( , , ) (1 ) ( , , )nn nc F c x t F c x ttθ θ +∂ = + −∂, where 221( , , )4c cF c x tPe xx∂ ∂= −∂∂. The time derivative is discretized in the usual finite difference way and applying Crank–Nicolson scheme to Eq. (1), it becomes: 1 1 1( ) ( ) ( ) ( )14 2 2n n n n n nm m xx m xx m x m x mc c c c c ct Pe+ + +   − + += −      ∆    . Separating the terms of advanced and initial level: ( ) ( ) ( ) ( ) ( ) ( )1 1 11 1 2 1 3 1 1 2 1 31 4 1 1 4 1n n n n n nm m m m m mδ α δ α δ α δ α δ α δ α+ + +− + − +− + + + − = + + − + + ,   (4) where 1 2 32 2 26 3 12 6 3; ;2 28 8 8t t th hPeh Peh Pehα α α     = + ∆ = ∆ = − ∆          . Boundary conditions are:  1 0 1 1 134 ( 4 ) ( ) 0Pehδ δ δ δ δ− −+ + − − = , at x = 0,   (5)  1 13( ) 0N Nhδ δ+ −− = , at x =1.   (6) Initial condition is:  1 14 1m m mδ δ δ− ++ + = , at t = 0, 0,1,...,m N= .   (7) Now using initial condition (7) and eliminating 1δ−  and 1Nδ +  from boundary conditions (5) and (6), we can find initial approximation 0mδ  from the tri-diagonal matrix: 5 44 1 0 . . . .1 4 1 . . . .0 1 4 1 . . .. . . . . . .. . . . . . .. . . 0 1 4 1. . . . 0 2 4α α− +           011...NNδδδδ−           =11...11           , where 43 43 4PehPehα− =  + ; 5164 3PehPehα =  + . Also Eq. (4) is reduced to a ( 1) ( 1)N N+ × +  matrix system as: Chemical and Process Engineering Research                                                                                                                                    www.iiste.org ISSN 2224-7467 (Paper) ISSN 2225-0913 (Online) Vol.31, 2015  72  102 5 5 1 3 4 4 111 2 3 111 2 3 111 3 24 1 0 . . . .1 4 1 . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . 0 1 4 1. . . . 0 2 4nnnNnNδα α α α α α α αα α α δα α α δα α α δ+++−+ + + − +      − + −                  − + −      − +  - -- 05 5 1 2 4 4 1 31 2 3 11 2 3 11 3 24 1 0 . . . .1 4 1 . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . 0 1 4 1. . . . 0 2 4nnnNnNδα α α α α α α αα α α δα α α δα α α δ− + + + +      + − +         =          + − +      + + −  - - -. By using 0mδ , the above matrix is solved for nmδ ’s by using Thomas algorithm (see Appendix A.). By putting nmδ ’s in the approximate solution, exit concentration Nc  can be determined.   4. Results and Discussion Chemical engineers are particularly interested in the exit solute concentration. Stiff system of Eqs. (1-3) have been solved analytically up to acceptable accuracy by Brenner (1962). A comparison of analytic results of Brenner (1962), results of Singh et al. (2008) and results obtained using CSCM for wide range of Pe are given in Table 2. The Exit concentration profiles are drawn for different values of Peclet number, as shown in Figure. 1, 2 and 3. For 10Pe ≤ , the transport by axial dispersion becomes stronger than convective transport, which causes more back mixing, resulting in more dispersion of solute, leading to longer period for solute to leach out from fibers. For large ( 40)Pe = , solution profile is converging to steady-state condition sharply, because dispersion becomes smaller thereby increasing the solute removal rate. However, no significant effect is observed on solution profiles for 40Pe > . Al-Jabari et al. (1994) have established that efficient washing operations can be achieved for Pe ≤ 40.  5. Conclusions CSCM is used due to its conceptual simplicity, wide applicability and ease of implementation. Breakthrough curves indicates, that an increase in the Pe  causes the area under the curve to increase, resulting in less dispersion and lesser back mixing which leads to more removal of solute adsorbed on the particle surface. CSCM gave good results for large values of Pe also where as Brenner (1962) has given asymptotic expression for such large values of Pe . Results are compared with Singh et al. (2008) also and from error analysis it is observed that CSCM is giving much more accurate results than MATLAB ‘pdepe’ solver.  References  Al-Jabari, M., Van Heiningen, A. R. P. and Van de Ven, T. G. M. (1994), “Modeling the flow and the deposition of fillers in packed beds of pulp fibers”, J. Pulp Paper Sci. 20(9), J249–J253. Brenner, H. (1962), “The diffusion model of longitudinal mixing in beds of finite length, Numerical values”, Chem. Eng. Sci. 17, 229–243.  de Boor, C. (1978), “A practical guide to splines”, Springer Verlag, NewYork. Grahs, L. E. (1975), “Displacement washing of packed beds of cellulose fibers, Part-1: Mathematical model”, Svensk Papperstid. 78(12), 446–450. Lapidus, L. and Amundson, N. R. (1952), “Mathematics of adsorption in beds, Part-VI: The effect of longitudinal diffusion in ion exchange and chromatographic columns”, J. Phys. Chem. 56, 984–988.  Potucek, F. (2001), “Displacement washing of pulp 1: Dispersion model for non flow”, Papir Celulóza 56(1), 8–11. Prenter, P. M. (1975), “Splines and variational methods”, Wiley interscience publication, New York. Singh, V. P., Kumar, V. and Kumar, D. (2008), “Numerical solution of diffusion model of brown stock washing Chemical and Process Engineering Research                                                                                                                                    www.iiste.org ISSN 2224-7467 (Paper) ISSN 2225-0913 (Online) Vol.31, 2015  73  beds of finite length using MATLAB”, 2nd UKSIM, 295–300.   Appendix A.  For the tri-diagonal matrix equation, 1 12 2 23 3 31 1 10 00 00 00 00 0n n nn nb ca b ca b ca b ca b− − −          L LLLM M O O O MLL L1231nnyyyyy−          M = 1231nnrrrrr−          M, the steps of Thomas algorithm to obtain the solutions are as follows:  Step 1: β1 = b1  Step 2: γ1 = r1/β1  Step 3: For i = 2, …, n   βi = bi − (ai ci-1/βi-1)   γi = (ri − ai γi-1)/βi  End  Step 4: yn = γn  Step 5: For j = 1, …, n−1   yn-j = γn-j − cn-j yn-j+1/βn-j    End  Table 1. Values of ( )mB x , ( )mB x′ , ( )mB x′′ .  2ix −  1ix −  ix  1ix +  2ix +  ( )mB x  0 1 4 1 0 ( )mB x′  0 -3/h 0 3/h 0 ( )mB x′′  0 6/h2 -12/h2 6/h2 0  Table 2. Comparison of CSCM with results of Brenner (1962) and Singh et al. (2008). t (Time)Pe = 0 Pe = 40 Pe = 100  Brenner (1962) Singh et al. (2008) CSCM Brenner (1962) Singh et al. (2008) CSCM Brenner (1962) Singh et al.(2008) CSCM 0.0 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.1 0.9048 0.9815 0.9054 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.2 0.8187 0.9398 0.8192 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.3 0.7408 0.8977 0.7412 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.4 0.6703 0.8572 0.6706 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.5 0.6065 0.8186 0.6067 1.0000 0.9998 1.0000 1.0000 1.0000 1.0000 0.6 0.5488 0.7817 0.5490 1.0000 0.9978 1.0000 1.0000 0.9991 1.0000 0.7 0.4966 0.7465 0.4967 0.9992 0.9837 0.9993 1.0000 0.9920 1.0000 0.8 0.4493 0.7128 0.4494 0.9746 0.9264 0.9750 0.9991 0.9552 0.9992 0.9 0.4066 0.6807 0.4066 0.8137 0.7785 0.8151 0.9276 0.8355 0.9303 1.0 0.3679 0.6500 0.3679 0.4778 0.5254 0.4800 0.4859 0.5815 0.4946     Chemical and Process Engineering Research                                                                                                                                    www.iiste.org ISSN 2224-7467 (Paper) ISSN 2225-0913 (Online) Vol.31, 2015  74   Figure. 1. Solution profile for Peclet number 0.5.   Figure. 2. Solution profile for Peclet number 40.   Figure. 3. Solution profile for Peclet number 100. 00.20.40.60.8101234500.10.20.30.40.50.60.70.80.91Time tNumerical solution for Pe = 0.5Distance xExit concentration c00.20.40.60.8100.511.5200.10.20.30.40.50.60.70.80.91Time tNumerical solution for Pe = 40Distance xExit concentration c00.5100.20.40.60.811.21.41.61.8200.10.20.30.40.50.60.70.80.91Time tNumerical solution for Pe = 100Distance xExit concentration cThe IISTE is a pioneer in the Open-Access hosting service and academic event management.  The aim of the firm is Accelerating Global Knowledge Sharing.  More information about the firm can be found on the homepage:  http://www.iiste.org  CALL FOR JOURNAL PAPERS There are more than 30 peer-reviewed academic journals hosted under the hosting platform.   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Efficient Numerical Solution of Diffusion Convection Problem of Chemical Engineering

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Efficient Numerical Solution of Diffusion Convection Problem of Chemical Engineering

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