Analytical Solution to the One-Dimensional Advection-Diffusion Equation with Temporally Dependent Coefficients

Kumar, Jaiswal Dilip; Kumar, Ashok; Ram, Yadav Raja

doi:10.4236/jwarp.2011.31009

Cited by 75 publications

(48 citation statements)

References 19 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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“…Thus, (FADE) may be introduced to provide a good simulation for diffusion. Some special forms of the fractional (ADE) were solved analytically in [7]. Rocca et al [8] considered the fractional diffusion-advection equation for solar cosmicray transport and gave its general solution.…”

Section: Introductionmentioning

confidence: 99%

On Adomian's Decomposition Method for Solving a Fractional Advection-Dispersion Equation

Hikal¹,

Ibrahim²

2015

Int. J. of Pure and Appl. Math.

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In this paper, the Adomian's decomposition method (ADM) is considered to solve a fractional advection-dispersion model. This model can be represented if the first order derivative in time is replaced by the Caputo fractional derivative of order α (0 < α ≤ 1). In addition, the space derivative orders are replaced by the alternative orders 0 < β ≤ 1 and 1 < γ ≤ 2. The obtained solutions are formulated in a convergent infinite series in terms of Mittage-Leffler functions. Finally, two illustrative examples are introduced to ensure the effectiveness of the used method.

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“…The advective-diffusion equation for the temperature front given by equation 52, has the well-known analytical solution given by Ogata and Banks (1961) which is also used in various other literature (e.g., Lake 1989; Geiger et al 2006;Jaiswal et al 2010):…”

Section: Analytical 1d Modelmentioning

confidence: 99%

Optimization of Geothermal-Well-Doublet Placement

Smit¹,

Salimi

Wolf

2014

Proceedings

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The lifetime of geothermal projects mainly depends on the thermal breakthrough (thermal breakthrough occurs when a cold water front reaches the producer). Currently, geothermal-energy production is marginally economical because of its uncertainties and risks associated with the subsurface such as lifetime, flow rate, temperature. Lifetime of a geothermal reservoir plays the most important role in the use of geothermal energy because it mainly determines whether or not geothermal-energy production is economically viable.Through optimization of the well positions from one or more geothermal doublets in a homogeneous or heterogeneous reservoir, the profitability of the project, which is largely dependent on the time of compositional breakthrough, temperature breakthrough and the rate of temperature decline, can be improved. This thesis studies optimization of the well positions such that the Net Present Value (NPV) of a project is maximized in a 2D geothermal reservoir for the selected heterogeneity structure. For this purpose an automated, gradient-based optimization method is used. The approach is based on the concept to surround the wells, whose locations have to be optimized, by so-called pseudo-wells. The reservoir simulations are performed using the Finite Element Method in the program COMSOL Multiphysics 4.2a. The major features of the simulation results are discussed in detail.The compositional front moves faster than the thermal front (the ratio of these two is the thermal retardation factor). Breakthrough of water with altered composition will therefore occur at an early stage in the doublet lifetime. Reservoir heterogeneities influence the time at which thermal and compositional breakthrough occur and also determine the rate at which temperature and composition decline after breakthrough. The temperature and compositional decline curves after breakthrough are generally steeper in a homogeneous reservoir than in a heterogeneous reservoir. Therefore, the thermal breakthrough does not necessarily mean the end of the lifetime of a doublet. It is also shown that the effect of heterogeneities on the thermal retardation factor is small.Three successful optimization sequences in two different reservoirs are described in this thesis. It is shown that, from an economical standpoint, is makes little sense to assume a doublet lifetime of more than 30 years. Furthermore, the effectiveness at which a geothermal doublet is able to deplete a reservoir (recovery factor) and profitability of a geothermal doublet are closely interlinked. However, a higher recovery factor does not necessarily mean that the doublet is more profitable and vice versa. There exists an optimum well spacing for doublets positioned in homogeneous reservoirs, such that additional gain of later breakthrough (when placing the production well further away from the injector) is negated by the loss in pressure support of the injection well. This optimum well spacing is found to be an important factor, influencing the profitability in homogeneous...

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Analytical Solution to the One-Dimensional Advection-Diffusion Equation with Temporally Dependent Coefficients

Cited by 75 publications

References 19 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

On Adomian's Decomposition Method for Solving a Fractional Advection-Dispersion Equation

Optimization of Geothermal-Well-Doublet Placement

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