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Analytical Solution of the Advection–Diffusion Transport Equation Using a Change-Of-Variable and Integral Transform Technique

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Analytical Solution of the Advection–Diffusion Transport Equation Using a Change-Of-Variable and Integral Transform Technique International Journal of Heat and Mass Transfer 52 (2009) 3297–3304 Contents lists available at ScienceDirect International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt Analytical solution of the advection–diffusion transport equation using a change-of-variable and integral transform technique J.S. Pérez Guerrero a, L.C.G. Pimentel b, T.H. Skaggs c,*, M.Th. van Genuchten d a Radioactive Waste Division, Brazilian Nuclear Energy Commission, DIREJ/DRS/CNEN, R. General Severiano 90, 22290-901 RJ-Rio de Janeiro, Brazil b Department of Meteorology, Federal University of Rio de Janeiro, Rio de Janeiro, Brazil c U.S. Salinity Laboratory, USDA-ARS, 450 W. Big Springs Rd, Riverside, CA 92507, USA d Department of Mechanical Engineering, LTTC/COPPE, Federal University of Rio de Janeiro, UFRJ, Rio de Janeiro, Brazil article info abstract Article history: This paper presents a formal exact solution of the linear advection–diffusion transport equation with con- Received 21 July 2008 stant coefficients for both transient and steady-state regimes. A classical mathematical substitution Received in revised form 13 January 2009 transforms the original advection–diffusion equation into an exclusively diffusive equation. The new dif- Available online 11 March 2009 fusive problem is solved analytically using the classic version of Generalized Integral Transform Tech- nique (GITT), resulting in an explicit formal solution. The new solution is shown to converge faster Keywords: than a hybrid analytical–numerical solution previously obtained by applying the GITT directly to the Analytical solution advection–diffusion transport equation. Transport equation Published by Elsevier Ltd. Integral transforms 1. Introduction guishes one integral transform from another. Exact solutions of lin- ear diffusion problems by classical integral transform techniques Analytical solutions of advective–diffusive transport problems were reviewed and classified by Mikhailov and Ozisik [12]. They continue to be of interest in many areas of science and engineering, identified and unified seven classes of problems and demonstrated such as heat and mass transfer and pollutant dispersion in air, soils, many applications in heat and mass diffusion. Cotta [13] general- and water. They are useful for a variety of applications [1–5], such ized and extended the classical integral transform method pre- as providing initial or approximate analyses of alternative pollu- sented by Mikhailov and Ozisik [12], thereby creating a new tion scenarios, conducting sensitivity analyses to investigate the systematic procedure referred to as the Generalized Integral Trans- effects of various parameters or processes on contaminant trans- form Technique (GITT). port, extrapolation over large times and distances where numerical The literature also features the use of mathematical substitu- solutions may be impractical, serving as screening models or tions in obtaining analytical solutions to partial differential equa- benchmark solutions for more complex transport processes that tions. Mathematical substitutions can simplify the structure of an cannot be solved exactly, and for validating more comprehensive equation, thereby facilitating more flexible applications of certain numerical solutions of the governing transport equations. solution methods. Compilations of transformations and substitu- The literature presents several methods to analytically solve the tions are presented by Zwillinger [14] and Polyanin [15]. Most partial differential equations governing transport phenomena existing analytical solutions for advection–diffusion transport [6–10]. For example, the method of separation-of-variables is one problems [3,16–18], including problems with growth and decay of the oldest and most widely used techniques. Similarly, the clas- terms, are for semi-infinite or infinite regions, with solutions for sical Green’s function method can be applied to problems with finite domains being mostly limited to one-dimensional problems. source terms and inhomogeneous boundary conditions on finite, The aim of this paper is to present an analytical methodology to semi-infinite, and infinite regions [10,11]. solve advection–diffusion transport problems in a finite domain for Integral transform techniques, such as the Laplace and Fourier both transient and steady-state regimes. The proposed methodol- transform methods, employ a mathematical operator that pro- ogy uses change-of-variables in combination with the classic ver- duces a new function by integrating the product of an existing sion of the Generalized Integral Transform Technique (GITT). function and a kernel function between suitable limits. The kernel of an integral transform, along with the integration limits, distin- 2. Problem formulation We study transport in a finite domain and consider a three- * Corresponding author. Tel.: +1 951 369 4853; fax: +1 951 342 4964. E-mail address: [email protected] (T.H. Skaggs). dimensional linear problem with decay and source terms. The 0017-9310/$ - see front matter Published by Elsevier Ltd. doi:10.1016/j.ijheatmasstransfer.2009.02.002 3298 J.S. Pérez Guerrero et al. / International Journal of Heat and Mass Transfer 52 (2009) 3297–3304 Nomenclature a generic position on the boundary u, v,w advection (convection) components C dimensionless solute concentration v symbolic coordinates for integration Dx, Dy, Dz dispersion–diffusion constants V generic finite volume F filter function x, y, z spatial coordinates f i integral transform of q G source term Greek symbols G integral transform of G bi eigenvalue H1, H2 coefficients c dimensionless coefficient i,j indices dij Kronecker delta L mathematical operator h function in purely diffusive equation L0 domain length hi integral transform of the function h M function with homogeneous boundary conditions k generic decay constant (Eq. (2)) * Ni norm k decay constant for the second test case Pe Peclet number li eigenvalue p1, p2, p3 constants for the algebraic substitution p number Pi q1, q2, q3 constants for the algebraic substitution q generic initial condition R coefficient s auxiliary variable S source term r2 Laplacian operator T scalar quantity wi eigenfunction ~ t time wi normalized eigenfucntion boundary conditions can be any combination of the first, second, or as systematized by Cotta [13]. In this technique, termed the Gener- third kinds. We begin with formal mathematical definitions for alized Integral Transform Technique (GITT), the unknown function both transient and steady-state problems. is represented in terms of an eigenfunction series expansion. Basi- cally, the GITT has the following steps: 2.1. Transient problem (a) Choose an appropriate auxiliary eigenvalue problem and The transient transport equation for a scalar quantity T(x,y,z,t) find the associated eigenvalues, eigenfunctions, norm, and undergoing constant advection (or convection) and dispersion (or orthogonalization property. diffusion) is given by: (b) Develop the integral and inverse transforms. (c) Transform the partial differential equation into a system of @Tðx; y; z; tÞ R þ LTðx; y; z; tÞþSðx; y; z; tÞ¼r2Tðx; y; z; tÞð1Þ ordinary differential or algebraic equations. @t (d) Solve the ordinary differential or algebraic system. 2 where the operators L and r are defined by: (e) Use the inverse transform to obtain the unknown function. @ @ @ L u þ v þ w þ kR ð2Þ @x @y @z 2 2 2 3.2. Transient regime 2 @ @ @ r Dx 2 þ Dy 2 þ Dz 2 ð3Þ @x @y @z Considering a more general case we assume that the boundary in which the parameters u, v, and w are constant velocity coeffi- conditions associated with Eq. (1) are non-homogeneous. To homogenize the boundary conditions, we rewrite Eq. (1) using cients, k is a constant decay term, Dx, Dy, and Dz are constant disper- sion (or diffusion) coefficients, and S(x,y,z,t) is a source term. The the following filter strategy: coefficient R in Eq. (1) is also a constant parameter (e.g., the retar- Tðx; y; z; tÞ¼Mðx; y; z; tÞþFðx; y; z; tÞð6Þ dation factor in many subsurface contaminant transport problems). A general initial condition can be established as: where F(x,y,z;t) is any function satisfying exactly the original boundary conditions of T(x,y,z,t). Eq. (1) now becomes: Tðx; y; z; 0Þ¼qðx; y; zÞð4Þ @Mðx; y; z; tÞ R þ LMðx; y; z; tÞ¼r2Mðx; y; z; tÞþGðx; y; z; tÞð7Þ @t 2.2. Steady-state problem where G(x,y,z,t) is the new source term which includes the original source term as well as terms containing the filtering function. The The transport equation for a steady-state transport regime inde- filtered initial condition is given by: pendent of time, t, follows from Eq. (1) as: Mðx; y; z; 0Þ¼qðx; y; zÞFðx; y; z; 0Þð8Þ r2Tðx; y; zÞ¼LTðx; y; zÞþSðx; y; zÞð5Þ Using a change-of-variable, similar to that used by [19,20,10] we define M(x,y,z,t) in terms of a new function, h(x,y,z,t): 3. Mathematical analysis Mðx; y; z; tÞ¼hðx; y; z; tÞ exp½p1x þ p2y þ p3z þ tðq1 þ q2 þ q3Þ ð9Þ 3.1. Generalized Integral Transform Technique (GITT) where p1, p2, p3, q1, q2, and q3 are constants that need to be determined. The transient and steady-state problems given by Eqs. (1) and A comparable substitution was used by Brenner [19] to obtain a (5), respectively, can be solved using integral transform methods transient, one-dimensional solution for miscible fluid displace- J.S. Pérez Guerrero et al. / International Journal of Heat and Mass Transfer 52 (2009) 3297–3304 3299 ment in finite length beds. Selim and Mansell [20] similarly pre- To implement the GITT, we selected an eigenvalue problem sented an analytical solution for reactive solutes with linear with the same kind of boundary conditions as specified for adsorption, a sink/source term, a finite domain, and continuous h(x,y,z,t). and flux-plug type inlet conditions at the inlet boundary. Recently, In that case the problem is given by: Goltz [21] used an equivalent substitution studying convective– dispersive solute transport with constant production, first-order r2wðx; y; zÞþl2wðx; y; zÞ¼0 ð13Þ decay, and equilibrium sorption in a porous medium.
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