Critical times in multilayer diffusion. Part 1: Exact solutions

Hickson, Roslyn I.; Barry, Steven; Mercer, G. N.

doi:10.1016/j.ijheatmasstransfer.2009.08.013

Cited by 74 publications

(50 citation statements)

References 27 publications

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“…Thus, considering a 10‐m thick uncontaminated clay or shale unit with a diffusion coefficient D = 10 −10 m 2 /s, we could calculate how long it takes for solute contaminants accumulating at the top surface of the unit to diffuse across this 10‐m thick clay aquitard and come to a new equilibrium. Although there are multiple definitions of response time, or critical time, because mathematically an infinite amount of time is required for diffusive processes to reach steady state (Landman and McGuinness 2000; Hickson et al 2009), a common approximation of critical time is found using the exact solution of for a single layer of material with boundary conditions of solute concentration, C = C 0 (a constant) at the upper surface of the layer, and C = 0 at the lower surface, and initial concentration of the contaminant solute of zero everywhere (Crank 1975, 51–52) as which corresponds to approximately 84% of the average steady‐state concentration at equilibrium (Hickson et al 2009). Based on , it will take more than 5000 years for the solutes to diffuse across that aquitard, demonstrating the exceedingly large response times of systems dominated by chemical diffusion.…”

Section: Understanding Flow and Transport Problems Through Scaling Anmentioning

confidence: 99%

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Retracted: On Understanding and Predicting Groundwater Response Time

Sophocleous¹

2011

Groundwater

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An aquifer system, when perturbed, has a tendency to evolve to a new equilibrium, a process that can take from just a few seconds to possibly millions of years. The time scale on which a system adjusts to a new equilibrium is often referred to as "response time" or "lag time." Because groundwater response time affects the physical and economic viability of various management options in a basin, natural resource managers are increasingly interested in incorporating it into policy. However, the processes of how groundwater responds to land-use change are not well understood, making it difficult to predict the timing of groundwater response to such change. The difficulty in estimating groundwater response time is further compounded because the data needed to quantify this process are not usually readily available. This article synthesizes disparate pieces of information on aquifer response times into a relatively brief but hopefully comprehensive review that the community of water professionals can use to better assess the impact of aquifer response time in future groundwater management investigations. A brief exposition on dimensional/scaling analysis is presented first, followed by an overview of aquifer response time for simplified aquifer systems. The aquifer response time is considered first from a water-quantity viewpoint and later expanded to incorporate groundwater age and water-quality aspects. Monitoring programs today, as well as water policies and regulations, should address this issue of aquifer response time so that more realistic management expectations can be reached.

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Section: Understanding Flow and Transport Problems Through Scaling Anmentioning

confidence: 99%

“…which corresponds to approximately 84% of the average steady-state concentration at equilibrium (Hickson et al 2009). Based on Equation 6, it will take more than 5000 years for the solutes to diffuse across that aquitard, demonstrating the exceedingly large response times of systems dominated by chemical diffusion.…”

Section: R E T R a C T E Dmentioning

confidence: 99%

Retracted: On Understanding and Predicting Groundwater Response Time

Sophocleous¹

2011

Groundwater

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“…To determine whether it is appropriate to work with such steady state solutions, we must decide whether a sufficient amount of time has passed so that the transient solution has effectively reached steady state. Several definitions of critical time have been proposed for this purpose [1][2][3][4][5][6][7]. One such definition, the mean action time (MAT) [3][4][5][6][7][8], also known as the local accumulation time [9][10][11][12], is the mean of a probability density function (PDF) associated with the linear reaction-diffusion problem of interest.…”

Section: Introductionmentioning

confidence: 99%

Simplified approach for calculating moments of action for linear reaction-diffusion equations

et al. 2013

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The mean action time is the mean of a probability density function that can be interpreted as a critical time, which is a finite estimate of the time taken for the transient solution of a reaction-diffusion equation to effectively reach steady state. For high-variance distributions, the mean action time underapproximates the critical time since it neglects to account for the spread about the mean. We can improve our estimate of the critical time by calculating the higher moments of the probability density function, called the moments of action, which provide additional information regarding the spread about the mean. Existing methods for calculating the nth moment of action require the solution of n nonhomogeneous boundary value problems which can be difficult and tedious to solve exactly. Here we present a simplified approach using Laplace transforms which allows us to calculate the nth moment of action without solving this family of boundary value problems and also without solving for the transient solution of the underlying reaction-diffusion problem. We demonstrate the generality of our method by calculating exact expressions for the moments of action for three problems from the biophysics literature. While the first problem we consider can be solved using existing methods, the second problem, which is readily solved using our approach, is intractable using previous techniques. The third problem illustrates how the Laplace transform approach can be used to study coupled linear reaction-diffusion equations.

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“…with T and F defined in Eqs. (20), (22), respectively, in which λ is replaced by s. To obtain the propagator in time domain, one needs to perform an inverse Laplace transform. This is done by looking for the poles s = λ n ofG and the above formula shows that they are given by the zeros of F (s), as expected.…”

Section: Computation Of the Normmentioning

confidence: 99%

Diffusion Across Semi-permeable Barriers: Spectral Properties, Efficient Computation, and Applications

Moutal

2019

J Sci Comput

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We present an efficient method to compute the eigenvalues and eigenmodes of the diffusion operator ∇(D∇) on one-dimensional heterogeneous structures with multiple semi-permeable barriers. This method allows us to calculate the diffusion propagator and related quantities such as diffusion MRI signal or first exit time distribution analytically for regular geometries and numerically for arbitrary ones. The effect of the barriers and the transition from infinite permeability (no barriers) to zero permeability (impermeable barriers) are investigated.

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Critical times in multilayer diffusion. Part 1: Exact solutions

Cited by 74 publications

References 27 publications

Retracted: On Understanding and Predicting Groundwater Response Time

Retracted: On Understanding and Predicting Groundwater Response Time

Simplified approach for calculating moments of action for linear reaction-diffusion equations

Diffusion Across Semi-permeable Barriers: Spectral Properties, Efficient Computation, and Applications

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