Analytical solution of a spatially variable coefficient advection–diffusion equation in up to three dimensions

Zoppou, Christopher; Knight, J. H.

doi:10.1016/s0307-904x(99)00005-0

Cited by 99 publications

(59 citation statements)

References 17 publications

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“…Further, this exact form of partitioning is assumed to be valid in this instance as concentration profiles under diffusive flow are commonly based on Gaussian distributions. 16,17 This assumes kinetic inhibition is a consequence of diffusion, which is justified later. The ERF is defined as the cumulative probability function (or the integral) of the normal distribution.…”

Section: Psd Reconstruction Modelsmentioning

confidence: 99%

The Pivotal Role of Alumina Pore Structure in HF Capture and Fluoride Return in Aluminum Reduction

McIntosh

Agbenyegah

Hyland

et al. 2016

JOM

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Fluoride emissions during primary aluminum production are mitigated by dry scrubbing on alumina which, as the metal feedstock, also returns fluoride to the pots. This ensures stable pot operation and maintains process efficiency but requires careful optimization of alumina for both fluoride capture and solubility. The Brunauer-Emmett-Teller (BET) surface area of 70-80 m 2 g À1 is currently accepted. However, this does not account for pore accessibility. We demonstrate using industry-sourced data that pores <3.5 nm are not correlated with fluoride return. Reconstructing alumina pore size distributions (PSDs) following hydrogen fluoride (HF) adsorption shows surface area is not lost by pore diameter shrinkage, but by blocking the internal porosity. However, this alone cannot explain this 3.5 nm threshold. We show this is a consequence of surface diffusion-based inhibition with surface chemistry probably playing an integral role. We advocate new surface area estimates for alumina which account for pore accessibility by explicitly ignoring <3.5 nm pores.

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Section: Psd Reconstruction Modelsmentioning

confidence: 99%

The Pivotal Role of Alumina Pore Structure in HF Capture and Fluoride Return in Aluminum Reduction

McIntosh

Agbenyegah

Hyland

et al. 2016

JOM

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“…Using the Online Encyclopedia of Integer Sequences [27], the numbers 1, 5, 61, 1385 appearing in the numerators in Equation (39) are the second to fifth Euler numbers (http://oeis.org/A000364) while the numbers 2, 24, 360, 6720 are the first four numbers in the integer sequence, (2k+2)!/k!, for k = 0, 1, 2, 3 (http://oeis.org/A126804). Therefore, we propose that…”

Section: A Rigorous Connection To Scaling Results For Flow In Homogenmentioning

confidence: 99%

Accurate and efficient calculation of response times for groundwater flow

Carr¹,

Simpson²

2018

Journal of Hydrology

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We study measures of the amount of time required for transient flow in heterogeneous porous media to effectively reach steady state, also known as the response time. Here, we develop a new approach that extends the concept of mean action time. Previous applications of the theory of mean action time to estimate the response time use the first two central moments of the probability density function associated with the transition from the initial condition, at t = 0, to the steady state condition that arises in the long time limit, as t → ∞. This previous approach leads to a computationally convenient estimation of the response time, but the accuracy can be poor. Here, we outline a powerful extension using the first k raw moments, showing how to produce an extremely accurate estimate by making use of asymptotic properties of the cumulative distribution function. Results are validated using an existing laboratory-scale data set describing flow in a homogeneous porous medium. In addition, we demonstrate how the results also apply to flow in heterogeneous porous media. Overall, the new method is: (i) extremely accurate; and (ii) computationally inexpensive. In fact, the computational cost of the new method is orders of magnitude less than the computational effort required to study the response time by solving the transient flow equation. Furthermore, the approach provides a rigorous mathematical connection with the heuristic argument that the response time for flow in a homogeneous porous medium is proportional to L 2 /D, where L is a relevant length scale, and D is the aquifer diffusivity. Here, we extend such heuristic arguments by providing a clear mathematical definition of the proportionality constant.

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“…Other methodologies are possible to tackle the LAD problem on finite or infinite domains. Without the pretension of being exhaustive, we can cite Bosen [3], Kumar et al [8], Pérez Guérrero et al [11] and Zoppou and Knight [14].…”

Section: Introductionmentioning

confidence: 99%

One-dimensional linear advection–diffusion equation: Analytical and finite element solutions

Mojtabi

Deville

2015

Computers & Fluids

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OATAO is an open access repository that collects the work of Toulouse researchers and makes it freely available over the web where possible. This is an author-deposited version published in : http://oatao.univ-toulouse.fr/ Eprints ID : 15920 a b s t r a c tIn this paper, a time dependent one-dimensional linear advection-diffusion equation with Dirichlet homogeneous boundary conditions and an initial sine function is solved analytically by separation of variables and numerically by the finite element method. It is observed that when the advection becomes dominant, the analytical solution becomes ill-behaved and harder to evaluate. Therefore another approach is designed where the solution is decomposed in a simple wave solution and a viscous perturbation. It is shown that an exponential layer builds up close to the downstream boundary. Discussion and comparison of both solutions are carried out extensively offering the numericist a new test model for the numerical integration of the Navier-Stokes equation.

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Analytical solution of a spatially variable coefficient advection–diffusion equation in up to three dimensions

Cited by 99 publications

References 17 publications

The Pivotal Role of Alumina Pore Structure in HF Capture and Fluoride Return in Aluminum Reduction

The Pivotal Role of Alumina Pore Structure in HF Capture and Fluoride Return in Aluminum Reduction

Accurate and efficient calculation of response times for groundwater flow

One-dimensional linear advection–diffusion equation: Analytical and finite element solutions

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