scholarly journals Analytical model of reactive transport processes with spatially variable coefficients

2015 ◽  
Vol 2 (5) ◽  
pp. 140348 ◽  
Author(s):  
Matthew J. Simpson ◽  
Liam C. Morrow
Keyword(s):  
Reactive Transport ◽  
Initial Conditions ◽  
Perturbation Technique ◽  
Variable Coefficients ◽  
Transport Processes ◽  
Screening Tools ◽  
Analytical Models ◽  
Solution Technique ◽  
Regular Perturbation ◽  
Contaminant Fate And Transport

Analytical solutions of partial differential equation (PDE) models describing reactive transport phenomena in saturated porous media are often used as screening tools to provide insight into contaminant fate and transport processes. While many practical modelling scenarios involve spatially variable coefficients, such as spatially variable flow velocity, v ( x ), or spatially variable decay rate, k ( x ), most analytical models deal with constant coefficients. Here we present a framework for constructing exact solutions of PDE models of reactive transport. Our approach is relevant for advection-dominant problems, and is based on a regular perturbation technique. We present a description of the solution technique for a range of one-dimensional scenarios involving constant and variable coefficients, and we show that the solutions compare well with numerical approximations. Our general approach applies to a range of initial conditions and various forms of v ( x ) and k ( x ). Instead of simply documenting specific solutions for particular cases, we present a symbolic worksheet, as supplementary material, which enables the solution to be evaluated for different choices of the initial condition, v ( x ) and k ( x ). We also discuss how the technique generalizes to apply to models of coupled multispecies reactive transport as well as higher dimensional problems.

scholarly journals Numerical Solutions of Odd Order Linear and Nonlinear Initial Value Problems Using a Shifted Jacobi Spectral Approximations

2012 ◽  
Vol 2012 ◽  
pp. 1-25 ◽  
Author(s):  
A. H. Bhrawy ◽  
M. A. Alghamdi
Keyword(s):  
Galerkin Method ◽  
Initial Value Problems ◽  
Numerical Solutions ◽  
Initial Conditions ◽  
Variable Coefficients ◽  
Solution Technique ◽  
Initial Value ◽  
Spectral Approximations ◽  
Base Functions ◽  
And Performance

A shifted Jacobi Galerkin method is introduced to get a direct solution technique for solving the third- and fifth-order differential equations with constant coefficients subject to initial conditions. The key to the efficiency of these algorithms is to construct appropriate base functions, which lead to systems with specially structured matrices that can be efficiently inverted. A quadrature Galerkin method is introduced for the numerical solution of these problems with variable coefficients. A new shifted Jacobi collocation method based on basis functions satisfying the initial conditions is presented for solving nonlinear initial value problems. Through several numerical examples, we evaluate the accuracy and performance of the proposed algorithms. The algorithms are easy to implement and yield very accurate results.

An analytical model for multispecies reactive transport through a heterogeneous formation consisting multiple layers of differing physical and chemical properties

2020 ◽  
Author(s):  
Jui-Sheng Chen ◽  
Ching-Ping Liang ◽  
Cheng-Hung Chang
Keyword(s):  
Analytical Model ◽  
Reactive Transport ◽  
Chlorinated Solvents ◽  
Chemical Properties ◽  
Transport Processes ◽  
Analytical Models ◽  
Physical And Chemical Properties ◽  
Heterogeneous Formation ◽  
Physical And Chemical ◽  
And Chemical Properties

<p>Transport behaviors of contaminants through a heterogeneous formation consisting multiple layers are complicated because of the different physical and chemical properties for each individual layer. Few analytical solutions for single-species contaminant transport in a multi-layer heterogeneous formation have been reported in the literature. Some contaminants of concern such as radionuclide, nitrogen and chlorinated solvents can decay or degrade to form new successor products during their transport processes, thus making migration of these contaminants much complicated. Clearly, analytical models for multispecies transport coupled by a series of decay reactions in a multi-layer formation are useful tools for synchronous determination of the fate and transport of the predecessor and successor species of decaying or degradable contaminants. This study attempts to develop an analytical model for the multispecies reactive transport of degradable or decaying contaminants through a multi-layer heterogeneous formation. The derived analytical model is shown to be correct and accurate as the consistent results of comparisons between the derived analytical model and the numerical model. The developed analytical model will provide a more reliable predicting tool for real world application.</p><p> </p>

scholarly journals Laminar condensation on a moving drop. Part 1. Singular perturbation technique

1984 ◽  
Vol 139 ◽  
pp. 105-130 ◽  
Author(s):  
J. N. Chung ◽  
P. S. Ayyaswamy ◽  
S. S. Sadhal
Keyword(s):  
Singular Perturbation ◽  
Perturbation Technique ◽  
Continuous Phase ◽  
Transport Processes ◽  
Forced Flow ◽  
Saturated Steam ◽  
Singular Perturbation Technique ◽  
Spherical Drop ◽  
Coupled Equations ◽  
Induced Velocity

In this paper, laminar condensation on a spherical drop in a forced flow is investigated. The drop experiences a strong, radial, condensation-induced velocity while undergoing slow translation. In view of the high condensation velocity, the flow field, although the drop experiences slow translation, is not in the Stokes-flow regime. The drop environment is assumed to consist of a mixture of saturated steam (condensable) and air (non-condensable). The study has been carried out in two different ways. In Part 1 the continuous phase is treated as quasi-steady and the governing equations for this phase are solved through a singular perturbation technique. The transient heat-up of the drop interior is solved by the series-truncation numerical method. The solution for the total problem is obtained by matching the results for the continuous and dispersed phases. In Part 2 both the phases are treated as fully transient and the entire set of coupled equations are solved by numerical means. Validity of the quasi-steady assumption of Part 1 is discussed. Effects due to the presence of the non-condensable component and of the drop surface temperature on transport processes are discussed in both parts. A significant contribution of the present study is the inclusion of the roles played by both the viscous and the inertial effects in the problem treatment.

Porous media as a canvas for hydro-bio-geo-chemical processes: Facing the challenges

2021 ◽  
Author(s):  
Xavier Sanchez-Vila
Keyword(s):  
Porous Media ◽  
Reactive Transport ◽  
Field Work ◽  
Open Science ◽  
Transport Processes ◽  
Aquifer Recharge ◽  
Music Production ◽  
Emerging Compounds ◽  
Technological Developments ◽  
Large Research

<p>The more we study flow and transport processes in porous media, the larger the number of questions that arise. Heterogeneity, uncertainty, multidisciplinarity, and interdisciplinarity are key words that make our live as researchers miserable… and interesting. There are many ways of facing complexity; this is equivalent as deciding what colors and textures to consider when being placed in front of a fresh canvas, or what are the sounds to include and combine in a music production. You can try to get as much as you can from one discipline, using very sophisticated state-of-the-art models. On the other hand, you can choose to bring to any given problem a number of disciplines, maybe having to sacrifice deepness in exchange of the better good of yet still sophisticated multifaceted solutions. There are quite a number of examples of the latter approach. In this talk, I will present a few of those, eventually concentrating in managed aquifer recharge (MAR) practices. This technology involves water resources from a myriad of perspectives, covering from climate change to legislation, from social awareness to reactive transport, from toxicological issues to biofilm formation, from circular economy to emerging compounds, from research to pure technological developments, and more. All of these elements deserve our attention as researchers, and we cannot pretend to master all of them. Integration, development of large research groups, open science are words that will appear in this talk. So does mathematics, and physics, and geochemistry, and organic chemistry, and biology. In any given hydrogeological problem you might need to combine equations, statistics, experiments, field work, and modeling; expect all of them in this talk. As groundwater complexity keeps amazing and mesmerizing me, do not expect solutions being provided, just anticipate more and more challenging research questions being asked.</p>

Assessment of groundwater contamination by chlorpyrifos using the PWC model in Valencia Region (Spain)

2021 ◽  
Author(s):  
Ricardo Pérez Indoval ◽  
Javier Rodrigo-Ilarri ◽  
Eduardo Cassiraga
Keyword(s):  
Reactive Transport ◽  
Management Practices ◽  
Fate And Transport ◽  
Environmental Modeling ◽  
Transport Processes ◽  
Negative Effects ◽  
Advection Dispersion ◽  
Agricultural Applications ◽  
Transformation Processes ◽  
Mass Transport Equation

<p>Chlorpyrifos is commoly used as an pesticide to control weeds and prevent nondesirable grow of algae, fungi and bacteria in many agricultural applications. Despite its highly negative effects on human health, environmental modeling of this kind of pesticide in the groundwater is not commonly done in real situations. Predicting the fate of pesticides released into the natural environment is necessary to anticipate and minimize adverse effects both at close and long distances from the contamination source. A number of models have been developed to predict the behavior, mobility, and persistence of pesticides. These models should account for key hydrological and agricultural processes, such as crop growth, pesticide application patterns, transformation processes and field management practices.</p><p>This work shows results obtained by the Pesticide Water Calculator (PWC) model to simulate the behavior of chlorpyrifos. PWC model is used as a standard pesticide simulation model in USA and in this work it has been used to  simulate the fate and transport of chlorpyrifos in the unsaturated zone of the aquifer. The model uses a whole set of parameters to solve a modified version of the mass transport equation considering the combined effect of advection, dispersion and reactive transport processes. PWC is used to estimate the daily concentrations of chlorpyrifos in the Buñol-Cheste aquifer in Valencia Region(Spain).</p><p>A whole set of simulation scenarios have been designed to perform a parameter sensitivity analysis. Results of the PWC model obtained in this study represents a crucial first step towards the development of a pesticide risk assessment in Valencia Region. Results show that numerical simulation is a valid tool for the analysis and prediction of the fate  and transport of pesticides in the groundwater.</p>

scholarly journals Stellar mass spectrum within massive collapsing clumps

2018 ◽  
Vol 611 ◽  
pp. A89 ◽  
Author(s):  
Yueh-Ning Lee ◽  
Patrick Hennebelle
Keyword(s):  
Large Scale ◽  
Equations Of State ◽  
Initial Conditions ◽  
Peak Position ◽  
Density Fluctuations ◽  
Initial Mass Function ◽  
Analytical Models ◽  
Numerical Resolution ◽  
Tidal Forces ◽  
The Imf

Context. Understanding the origin of the initial mass function (IMF) of stars is a major problem for the star formation process and beyond. Aim. We investigate the dependence of the peak of the IMF on the physics of the so-called first Larson core, which corresponds to the point where the dust becomes opaque to its own radiation. Methods. We performed numerical simulations of collapsing clouds of 1000 M⊙ for various gas equations of state (eos), paying great attention to the numerical resolution and convergence. The initial conditions of these numerical experiments are varied in the companion paper. We also develop analytical models that we compare to our numerical results. Results. When an isothermal eos is used, we show that the peak of the IMF shifts to lower masses with improved numerical resolution. When an adiabatic eos is employed, numerical convergence is obtained. The peak position varies with the eos, and using an analytical model to infer the mass of the first Larson core, we find that the peak position is about ten times its value. By analyzing the stability of nonlinear density fluctuations in the vicinity of a point mass and then summing over a reasonable density distribution, we find that tidal forces exert a strong stabilizing effect and likely lead to a preferential mass several times higher than that of the first Larson core. Conclusions. We propose that in a sufficiently massive and cold cloud, the peak of the IMF is determined by the thermodynamics of the high-density adiabatic gas as well as the stabilizing influence of tidal forces. The resulting characteristic mass is about ten times the mass of the first Larson core, which altogether leads to a few tenths of solar masses. Since these processes are not related to the large-scale physical conditions and to the environment, our results suggest a possible explanation for the apparent universality of the peak of the IMF.

scholarly journals Analytical Models for Gravitating Radiating Systems

2015 ◽  
Vol 2015 ◽  
pp. 1-7 ◽  
Author(s):  
B. P. Brassel ◽  
S. D. Maharaj ◽  
G. Govender
Keyword(s):  
Consistency Condition ◽  
Fluid Pressure ◽  
Nonlinear Ordinary Differential Equation ◽  
Variable Coefficients ◽  
Analytical Models ◽  
Heat Conducting ◽  
Elementary Functions ◽  
Complex Transformation ◽  
All Solutions ◽  
Metric Functions

We analyse the gravitational behaviour of a relativistic heat conducting fluid in a shear-free spherically symmetric spacetime. We show that the isotropy of pressure is a consistency condition which realises a second order nonlinear ordinary differential equation with variable coefficients in the gravitational potentials. Several new classes of solutions are found to the governing equation by imposing various forms on one of the potentials. Interestingly, a complex transformation leads to an exact solution with only real metric functions. All solutions are written in terms of elementary functions. We demonstrate graphically that the fluid pressure, energy density, and heat flux are well behaved for the model, and the model is consistent with a core-envelope framework.

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