Analytical solution of the advection–diffusion transport equation using a change-of-variable and integral transform technique

Guerrero, J.S. Pérez; Pimentel, Leonardo Duarte; Skaggs, Todd H.; Genuchten, Martinus Th. van

doi:10.1016/j.ijheatmasstransfer.2009.02.002

Cited by 131 publications

(67 citation statements)

References 18 publications

(16 reference statements)

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“…Analytical methodologies to solve the advectiondiffusion equation by mathematical substitutions and transformations (Mikhailov and Ozisik, 1984;Cotta, 2005) are widely applied to many experimental applications in heat and mass diffusion (Leij and van Genuchten, 2000;Guerrero et al, 2009). Nevertheless, a large number of parameters, coefficients, constants and functions are used in such methodologies, which may reduce their range of applications to systems under strictly controlled external conditions.…”

Section: F F Pereira Et Al: Assessment Of Numerical Schemesmentioning

confidence: 99%

Assessment of numerical schemes for solving the advection–diffusion equation on unstructured grids: case study of the Guaíba River, Brazil

Pereira

Fragoso

Uvo

et al. 2013

Nonlin. Processes Geophys.

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Abstract. In this work, a first-order upwind and a high-order flux-limiter schemes for solving the advection-diffusion equation on unstructured grids were evaluated. The numerical schemes were implemented as a module of an unstructured two-dimensional depth-averaged circulation model for shallow lakes (IPH-UnTRIM2D), and they were applied to the Guaíba River in Brazil. Their performances were evaluated by comparing mass conservation balance errors for two scenarios of a passive tracer released into the Guaíba River. The circulation model showed good agreement with observed data collected at four water level stations along the Guaíba River, where correlation coefficients achieved values up to 0.93. In addition, volume conservation errors were lower than 1 % of the total volume of the Guaíba River. For all scenarios, the higher order flux-limiter scheme has been shown to be less diffusive than a first-order upwind scheme. Accumulated conservation mass balance errors calculated for the flux limiter reached 8 %, whereas for a first-order upwind scheme, they were close to 18 % over a 15-day period. Although both schemes have presented mass conservation errors, these errors are assumed negligible compared with kinetic processes such as erosion, sedimentation or decay rates.

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Section: F F Pereira Et Al: Assessment Of Numerical Schemesmentioning

confidence: 99%

Assessment of numerical schemes for solving the advection–diffusion equation on unstructured grids: case study of the Guaíba River, Brazil

Pereira

Fragoso

Uvo

et al. 2013

Nonlin. Processes Geophys.

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“…The different numbers of terms required to achieve convergence for CONFIG1 and CONFIG2 can be explained from the influence of the diffusive and the advective terms in the atmospheric diffusion equation. According to Cotta [20] and Pérez Guerrero et al [32], a faster convergence corresponds to a smaller Peclet (Pe) number. This can be verified defining Pe =ŪH/K, wherē H = z 0 − z 1 , andŪ andK are the vertical average of the mean wind velocity and the vertical eddy diffusivity, respectively.…”

Section: (B) Convergence Analysis Of the Simulated Resultsmentioning

confidence: 99%

Assessment of the unified analytical solution of the steady-state atmospheric diffusion equation for stable conditions

et al. 2014

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In this work, the performance of a unified formal analytical solution for the simulation of atmospheric diffusion problems under stable conditions is evaluated. The eigenquantities required by the formal analytical solution are obtained by solving numerically the associated eigenvalue problem based on a newly developed algorithm capable of being used in high orders and without missing eigenvalues. The performance of the formal analytical solution is evaluated by comparing the converged predicted results against the observed values in the stable runs of the Prairie Grass experiment as well as the simulated results available in the literature. It was found that the developed algorithm was efficient and that the convergence rate depends on the stability condition and the considered parametrizations for wind speed and turbulence. The comparisons among predicted and observed concentrations showed a good agreement and indicate that the considered dispersion formulations are appropriate to simulate dispersion under slightly to moderate atmospheric stable conditions.

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“…Perez Guerrero et al (2009) presents an analytical solution of dispersion equation with constant coefficients in both transient and steady state conditions by using integral transform and change of variable techniques in a finite spatial domain. Hunt (2006) has given solution for instantaneous, continuous and steady pollution sources in uniform groundwater and Zoppou and Knight (1997) obtained an analytical solutions for the spatially dependent dispersion Yates (1990Yates ( , 1992 obtained analytical solutions of advection dispersion equation in one-dimension considering dispersion coefficient as linear and exponential increasing.…”

mentioning

confidence: 99%

One-dimensional spatially dependent solute transport in semi-infinite porous media: an analytical solution

Yadav¹,

Kumar²

2017

Int. J. Eng. Sci. Tech

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The present study is an attempt to describe analytical solution of spatially dependent solute transport in one-dimensional semiinfinite homogeneous porous domain. In this mathematical model the dispersion coefficient is considered spatially dependent while seepage velocity is considered exponentially decreasing function of space. Dispersion parameter and velocity are directly proportional to each other. Space dependent retardation factor is also taken. The nature of porous media and solute pollutant are considered chemically non-reactive. Initially porous domain is considered solute free and the input source condition is considered uniformly continuous. A new transformation is introduced to solve the advection dispersion equation. The analytical solution is obtained by using Laplace Transformation Techniques (LTT). The effects of spatial dependence on the solute concentration dispersion of various physical parameters are explained with help of different graphs..

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Analytical solution of the advection–diffusion transport equation using a change-of-variable and integral transform technique

Cited by 131 publications

References 18 publications

Assessment of numerical schemes for solving the advection–diffusion equation on unstructured grids: case study of the Guaíba River, Brazil

Assessment of numerical schemes for solving the advection–diffusion equation on unstructured grids: case study of the Guaíba River, Brazil

Assessment of the unified analytical solution of the steady-state atmospheric diffusion equation for stable conditions

One-dimensional spatially dependent solute transport in semi-infinite porous media: an analytical solution

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