Analytical Solution To One-dimensional Advection-diffusion .

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J. Agr. Sci. Tech. (2013) Vol. 15: 1231-1245Downloaded from jast.modares.ac.ir at 4:15 IRDT on Thursday May 6th 2021Analytical Solution to One-dimensional Advection-diffusionEquation with Several Point Sources through ArbitraryTime-dependent Emission Rate PatternsM. Mazaheri1 , J. M. V. Samani1, and H. M. V. Samani2ABSTRACTAdvection-diffusion equation and its related analytical solutions have gained wideapplications in different areas. Compared with numerical solutions, the analyticalsolutions benefit from some advantages. As such, many analytical solutions have beenpresented for the advection-diffusion equation. The difference between these solutions ismainly in the type of boundary conditions, e.g. time patterns of the sources. Almost all theexisting analytical solutions to this equation involve simple boundary conditions. Mostpractical problems, however, involve complex boundary conditions where it is verydifficult and sometimes impossible to find the corresponding analytical solutions. In thisresearch, first, an analytical solution of advection-diffusion equation was initially derivedfor a point source with a linear pulse time pattern involving constant-parameterscondition (constant velocity and diffusion coefficient). Hence, using the superpositionprinciple, the derived solution can be extended for an arbitrary time pattern involvingseveral point sources. The given analytical solution was verified using four hypotheticaltest problems for a stream. Three of these test problems have analytical solutions given byprevious researchers while the last one involves a complicated case of several pointsources, which can only be numerically solved. The results show that the proposedanalytical solution can provide an accurate estimation of the concentration; hence it issuitable for other such applications, as verifying the transport codes. Moreover, it can beapplied in applications that involve optimization process where estimation of the solutionin a finite number of points (e.g. as an objective function) is required. The limitations ofthe proposed solution are that it is valid only for constant-parameters condition, and isnot computationally efficient for problems involving either a high temporal or a highspatial resolution.Keywords: Advection-diffusion equation, Analytical solution, Laplace transformation, Pointsource, Solute transport.as environmental engineering, mechanicalengineering, heat transfer, soil science,petroleum engineering, chemical engineeringAdvection-DiffusionEquation(ADE)and as well in biology. ADE interprets thedescribes the transport of solute under thespreading of scalar or non-scalar quantitiescombined effects of advection and diffusion.under specified initial and boundaryThis equation is a parabolic partial differentialconditions. This equation can be solved eitherone derived as based upon the conservation ofanalytically or numerically. There are severalmass and Fick’s first law. ADE benefits fromschemes for solving ADE numerically whichwide applications in such different disciplinesdo not fall within the scope of this paper.INTRODUCTION1Department of Water Structures, Faculty of Agriculture, Tarbiat Modares University, Tehran, IslamicRepublic of Iran. Corresponding author’s email: m.mazaheri@modares.ac.ir2Department of Civil Engineering, Faculty of Agriculture, Shahid Chamran University, Ahvaz, IslamicRepublic of Iran.1231

Downloaded from jast.modares.ac.ir at 4:15 IRDT on Thursday May 6th 2021Mazaheri et al.Analytical solutions are as useful tools inmany areas (van Genuchten, 1981; Jury et al.,1983; Leij et al., 1991; Quezada et al., 2004).They can be applied for providing preliminaryor approximate analyses of alternativepollution scenarios, conducting sensitivityanalyses to investigate the effects of variousparameters or processes on contaminanttransport, extrapolation over large times anddistances where numerical solutions may beimpractical, serving as screening models orbenchmark solutions for more complextransport processes that cannot be accuratelysolved, and for verifying more comprehensivenumerical solutions of the governing transportequations (Guerrero et al., 2009).Formerly, the analytical solutions of ADEwere obtained by reducing the original ADEinto a diffusion equation by omitting theadvective terms (Kumar et al., 2009); thisbeing carried out either by applying movingcoordination (Ogata and Banks, 1961;Harleman and Rumer, 1963; Bear, 1972;Guvanasen and Volker, 1983; Aral and Liao,1996; Marshal et al., 1996); or by introducinganother dependent variable (Banks and Ali,1964; Ogata, 1970; Lai and Jurinak, 1971;Marino, 1974; Al-Niami and Rushton, 1977;Kumar et al., 2009). In more recent works,analytical solutions of ADE have beenobtained using such integral transformtechniques as either Laplace or Fouriertransforms (Govindaraju and Bhabani, 2007).In fact these transformations can act aspowerful tools in solving differentialequations. Van Genuchten and Alves (1982)provided a comprehensive manual of ADEanalytical solutions for different initial andboundary conditions, utilizing Laplacetransform technique. Other researchers havewidely applied this technique in solving ADEanalytically (Smedt, 2007; KazezyılmazAlhan, 2008; Kumar et al., 2009; Kumar et al.,2010). In addition to either Laplace or Fouriertransform methods, Hankel transform method,Aris moment method, perturbation approach,methods using Green’s function, superpositionmethod and mathematical substitutions havealso been applied to provide analyticalsolutions for ADE (Courant and Hilbert, 1953;Morse and Feshbach, 1953; Carslaw andJaeger, 1959; Sneddon, 1972; Ozisik, 1980;Zwillinger, 1998; Leij and van Genuchten,2000; Polyanin, 2002; Kumar et al., 2009).Guerrero et al. (2009) have presented somealgorithms to derive formal exact solutions ofADE using change-of-variable and integraltransform techniques. Cotta (1993) developedDeneral Integral Transform Technique(GITT). This method allows derivation ofanalytical or semi-analytical solution of themore general form of ADE’s. In the GITT, thesolution is written in terms of eigenfunctionseries expansions. There exist some researchesregarding analytical solutions of ADE usingGITT (Chongxuan et al., 2000; Barros et al.,2006; Guerrero and Skaggs, 2010).Most of the mathematically closed formanalytical solutions of ADE are limited tosimple initial and boundary conditions. Themore complexity of boundary conditions leadsto complicated mathematical formulae thatmay involve numerically solvable integrals,special functions or infinite summations, andin many cases an analytical solution cannot befound. In this paper, an analytical solution ofone-dimensional constant-parameter ADE isinitially derived for linear pulse timedependent boundary conditions. Using thesuperposition principle, the solution can beextended to an analytical one for any desiredtime-dependentboundaryconditions.Applying this method, the mathematicallyclosed form formula can be derived for anynumber of point sources of the desired timedependent emission rate patterns.Analytical Solution of 1D ADE for LinearPulse Time-dependent Boundary ConditionHere, the analytical solution of the onedimensional ADE for linear pulse timedependent boundary condition is derived usingLaplace transform. The ADE is consideredwith constant parameters where the decay isalso taken into account. The ADE with theseproperties is as follows: C C 2C U D 2 kC , 0 x ,(1) t x x0 t In which, C is the solute concentration, Uthe constant flow velocity, D is the constantdiffusion coefficient, k the coefficient of first-1232

Downloaded from jast.modares.ac.ir at 4:15 IRDT on Thursday May 6th 2021Analytical Solution to One-Dimensional ADEorder reaction, while t and x representing thevariables of time and space respectively.Equation (1) is solved with respect to thefollowing initial and boundary conditions:C ( x, 0) C0(2)C (0, t ) (at b) [u (t t1 ) u (t t2 )](3a) C x 0(3b)x Where, C0 is the initial concentration, a andb are the parameters of the linear pulseboundary condition at x 0 , t1 and t2 are thebeginning and ending times of the sourceactivation, respectively, while u(t ti ) beingthe shifted Heaviside function, defined to be 0for t ti and 1 for t ti . Equation (3a) isdepicted in Figure 1.texp( t1 s )exp( t1 s ) a ss2exp( t2 s )exp( t2 s ) C0(at2 b) a ss ks2β x 2 D and γ Ux (2 D) , and replacing theEquations (9a) and (9b) in Equation (8), thisequation is simplified to:(4)C ( x, s ) (5a)exp( t1 s )exp( t1 s) a (at1 b)ss2 exp( t2 s)exp( t2 s ) C0 (10) a (at2 b)ss k s2C exp ( γ ) exp α β s 0s k (5b) L ( 0)x Where, L is the Laplace transform operator.Using the forward Laplace transform formulas(Abramowitz and Stegun, 1970), Equations (4)and (5) can be written as an ordinarydifferential equation in the Laplace domain:dCd 2C D 2 kCdxdx(9b)Assuming α U 2 x 2 (4 D 2 ) kx 2 D , CL xsC C ( x, 0) U(8)c2 (at1 b)2L ( (at b) [u (t t1 ) u(t t2 )]) C 0 s k Where, c1 and c2 are arbitrary constants,that can be specified using Equations (7a) and(7b). Using these equations, c 1 and c2 wouldbe as follows:c1 0(9a)Applying the Laplace transform to Equation(1) and its boundary conditions yields thecorresponding problem in the Laplace domain;that is:L ( C (0, t ) ) (7b) 0x UxU 2 x 2 kx 2 sx 2c1 exp 2D4D2DD UxU 2 x 2 kx 2 sx 2c2 exp 2DDD4D2 Figure 1. Depiction of the input boundarycondition. C C C L D 2 kC L U x x t dCdxC ( x, s) C at bt2(7a)In which C C ( x, s) is the correspondingdependent variable of Equation (1) in theLaplace domain and s representing theLaplace transform variable. Equation (6) is alinear inhomogeneous ordinary differentialequation that is of the following generalsolution:Ct1exp( t1 s ) sexp( t1 s )exp( t2 s )a (at2 b) s2sexp( t2 s )as2C (0, s ) (at1 b)()Now, the analytical solution to Equation (1)with respect to (2) and (3) can be achieved byreturning Equation (10) back from the Laplace(6)1233

Downloaded from jast.modares.ac.ir at 4:15 IRDT on Thursday May 6th 2021Mazaheri et al.domain to the time domain. This can be doneby obtaining the inverse Laplace transform ofEquation (10). Using the linearity property ofinverse Laplace transform operator, the inverseLaplace transform of Equation (10) can bewritten as: 1L ( C ( x, s ) ) (at1 b) exp ( γ ) ( exp α β sL 1 exp ( t1 s ) s a exp ( γ ) ( exp α β sL 1 exp ( t1 s ) s2 (at2 b) exp ( γ ) ( exp α β sL 1 exp ( t2 s ) s a exp ( γ ) ( exp α β sL 1 exp ( t2 s ) s2 C0 exp ( γ ) ( t t1β C ( x, t ) exp γ α a 4 α 2 β 2(t t1 ) α at b 1erfc 2 β (t t ) 2 1 t t1β at1 b exp γ α a 2 4 α 2(() ) β 2(t t1 ) α erfc 2 β (t t )1 C 0 2 (12b) β 2γ t exp( kt ) erfc exp ( 2γ kt ) 2 βt ) β 2γ t erfc C0 exp( kt ) 2 β t ) ) t t1β C ( x, t ) exp γ α a 4 α 2 β 2(t t1 ) α at b erfc 1 2 β (t t ) 2 1 t t1β exp γ α a 2α 4 (11)( ) ( )) β 2(t t1 ) αat1 b erfc 2 β (t t )2 1 ) exp α β s C L 1 1 L 1 0 s k s k Where, L 1 is the inverse Laplace transform t t2β exp γ α a 4 α 2 β 2(t t2 ) α at2 b erfc 2 β (t t ) 2 2 t t2β exp γ α a 2α 4 (operator. The inverse Laplace transformsappearing in Equation (11) could be solvedaccording to the inverse Laplace transformformulas (Abramowitz and Stegun, 1970).Therefore, the final solution for differentvalues of t would be as the following: β 2γ t CC ( x, t ) 0 exp ( kt ) erfc 2 β t 2 β 2γ t C(12a) 0 exp ( 2γ kt ) erfc 2 β t 2 C0 exp( kt ),0 t t1()) β 2(t t2 ) αat2 b erfc 2 β (t t )2 2 C02(12c) β 2γ t exp ( kt ) erfc 2 βt β 2γ t exp ( 2γ kt ) erfc 2 β t C0 exp( kt )Where, erfc(.) is the complementary errorfunction. Equations (12a), (12b) and (12c) areestablished for a point source at x 0 . For ageneral point source at location x xs , variable1234

Analytical Solution to One-Dimensional ADEDownloaded from jast.modares.ac.ir at 4:15 IRDT on Thursday May 6th 2021x must be replaced by x xs . In this caseCEquations (12a), (12b) and (12c) can beinterpreted as a unified equation using theshifted Heaviside function. Simplifying andperformingthenecessaryalgebraicmanipulations would yield to:C ( x, t , xs , t1 , t2 ) A( x xs , t , t1 , 1) A( x xs , t , t1 ,1) A( x xs , t , t2 , 1) (13)A( x xs , t , t2 ,1) B( x xs , t , 1) B( x xs , t ,1) 2 B(0, t , 0)In which, xs is the source location and A(.)and B (.) are functions defined according tothe following:A( x, t , ti , m) u ( x)u (t ti ) UxU 2 x 2 kx 2 exp m 2D4D 2D t ti ati b mx a 2 2 U 2 4 Dk 2 x m(t t ) U 2 4 Dkierfc Dt ti2 t1t2 t3t4t5t6t7tFigure 2. A typical piecewise linear timepattern.Figure 2.This can be done according to thesuperposition principle, meaning that a sourcewith a piecewise linear time pattern may beconsidered as several linear units and theeffects of these units being additive. In thiscase the solution can be written as:(14)nl C ( x, t , xs ) [ A( x xs , t , ti , 1) i 1A( x xs , t , ti ,1) A( x xs , t , ti 1 , 1) CB ( x, t , m ) 0 u ( x ) 2(15)Ux x mUt exp kt (1 m) erfc 2D 2 Dt Equation (13) is the analytical solution for alinear pulse point source at x xs . Note thatEquation (13) yields to 0 for x xs . Equations(13), (14) and (15) in this form are suitable forcomputer implementation. In Equation (13),the terms expressed, using functionA( x, t , ti , m) consider the effects of advection,diffusion and decay of pollutant released froma point source while the terms expressed, usingfunction B( x, t , m) consider the effects ofadvection, diffusion and decay of initiallyexisting concentration at t 0 .(16)A( x xs , t , ti 1 ,1)] 2 B(0, t , 0) B( x xs , t , 1) B( x xs , t ,1)Where, C ( x, t , xs ) is the concentration valueat ( x, t ) due to a point source at location xs ,nl is the number of linear units in the sourcetime pattern while ti and ti 1 are the beginningand ending times of the ith linear unitrespectively, and functions A(.) and B (.)having been described before. It is clear that incomputing the values of function A(.) for eachi in Equation (16), the correspondingparameters ( ai and bi ) must be applied. Notethat in Equation (16) the terms expressed byfunction B (.) are out of the summation as theeffects of initial conditions must be consideredas only once. For a point source with emissiontime pattern that does not follow a piecewiselinear pattern, it is possible to fit anapproximate piecewise linear function andthen use Equation (16). The number of linearsegments in the fitting function could beincreased for an accurate capturing of theoriginal pattern.Extending the Solution for a MoreRealistic Point Source Emission TimePatternIn this section the derived analytical solutionis extended for any such desired piecewiselinear time pattern as the one depicted in1235

Mazaheri et al.with the numerical solution. Here, thesolutions are given for a stream flow byvelocity, U , of 0.7 m s-1, width, B , of 20 m,depth, h , of 1.5 m and longitudinal slope, S0 ,of 0.0005 . The dispersion coefficient, D , iscalculated from Fischer’s formula (1979)( D 0.011U 2 B 2 ( h 9.81hS0 ) ) equal to 16.8m2 s-1.Solution for Several Point SourcesDownloaded from jast.modares.ac.ir at 4:15 IRDT on Thursday May 6th 2021Using the superposition principle for severalpoint sources, it is possible to extend Equation(16) for several point sources as follows:nsnlC ( x, t ) A( x xsj , t , t j ,i , 1) j 1 i 1A( x xsj , t , t j ,i ,1) A( x xsj , t , t j ,i 1 , 1) A( x xsj , t , t j ,i 1 ,1) A( x xsj , t , t j ,i 1 ,1) (17)Test Problem I: Constant Pulse PointSource2 B(0, t , 0) B( x, t , 1) B( x, t ,1)Where, C ( x, t ) is the concentration at ( x, t )due to all point sources, ns is the number ofpoint sources and j the subscript thatencounters the jth source. Once again theterms in B (.) function are out of thesummation in Equation (17) and the variablex in this function must be considered withrespect to the minimum available value in xcoordinate.In fact using the property of linearity ofEquation (1) and superposition principle intime and space, the solution can be wellconstructed. The final solution for severalpoint sources with different arbitrary timepatterns can be expressed as a simple functionthat is straight forward for computerimplementation. The only possible difficulty incomputer implementation these formulas iscomputing the complementary error function( erfc(.) ). However, most of computerpackages and libraries in relevant disciplinesare capable of computing the function.Figure 3 shows a constant pulse point sourcevs. time with a duration of 1 hour starting fromt1 1 to t2 2 hr and intensity of Ws 5 kg s-1.The values of a and b for this source are 0and 0.24 kg m-3, respectively. Note that, therelationship between point source massdischarge, Ws , and its concentration, Cs , isCs Ws Q where Q UBh is the streamvolumetric discharge.Other parameters are: xs 1 km, C0 0 ,k 0 and the whole stream length isconsidered as equal to 4 km. Figure 4 showsthe results of the proposed analytical solutionand a classic analytical solution (vanGenuchten and Alves, 1982) for the wholestream length within different times.As depicted in Figure 4, the results of theproposed analytical solution and the one givenby van Genuchten and Alves (1982) areexactly the same. In fact the classic solutioncan be derived from Equation (13) by lettinga 0 and b equal to the source intensity.RESULTS AND DISCUSSIONIn this section, the obtained analyticalsolution is verified. Verification of analyticalsolutions developed by other researchers hasbeen carried out using hypothetical examplesand numerical methods (Smedt, 2007;Kazezyılmaz-Alhan, 2008; Williams andTomasko, 2008; Kumar et al., 2010). In thispaper verification is conducted usinganalytical solutions proposed in the literaturein addition to some more complicatedhypothetical examples. In the latter case, thepresented analytical solution will be comparedFigure 3. Constant pulse point source emissiontime pattern (test problem I).1236

Downloaded from jast.modares.ac.ir at 4:15 IRDT on Thursday May 6th 2021Analytical Solution to One-Dimensional ADEFigure 4. Results of the present analytical and classic solutions for test problem I.which is caused by the fact that the classicanalytical solution to this problem has beenderived for some ideal conditions. In otherwords the mass is spilled instantaneously. It isevident that this assumption is somehowunrealistic and in the real situations theduration must be duly considered.Test Problem II: Instantaneous SpillThe solution of instantaneous spill of a massin a stream has been given by Fischer et al.(1979). The point source emission time patternin this condition could be considered as aconstant pulse with a very short duration. It isclear that the area under mass discharge timepattern curve must be equal to the total massspilled (Figure 5). Figure 6 shows the resultsof the proposed analytical and classic solutionsfor a mass spill problem. The magnitude of1000 kg of a conservative material is spilledinto the stream at t 0. Other conditions aresimilar to those in the test problem I. Theduration for this problem ( t ) is considered asequal to 0.1 second. As depicted in Figure 6,there exists a reasonable agreement betweenthe two solutions. A small difference occurred,Figure 5. Instantaneous spill of a mass as avery short constant pulse

Advection-Diffusion Equation (ADE) describes the transport of solute under the combined effects of advection and diffusion. This equation is a parabolic partial differential one derived as based upon the conservation of mass and Fick s first law. ADE benefits

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