515 research outputs found
Calculating how long it takes for a diffusion process to effectively reach steady state without computing the transient solution
Mathematically, it takes an infinite amount of time for the transient
solution of a diffusion equation to transition from initial to steady state.
Calculating a \textit{finite} transition time, defined as the time required for
the transient solution to transition to within a small prescribed tolerance of
the steady state solution, is much more useful in practice. In this paper, we
study estimates of finite transition times that avoid explicit calculation of
the transient solution by using the property that the transition to steady
state defines a cumulative distribution function when time is treated as a
random variable. In total, three approaches are studied: (i) mean action time
(ii) mean plus one standard deviation of action time and (iii) a new approach
derived by approximating the large time asymptotic behaviour of the cumulative
distribution function. The new approach leads to a simple formula for
calculating the finite transition time that depends on the prescribed tolerance
and the th and th moments () of the distribution.
Results comparing exact and approximate finite transition times lead to two key
findings. Firstly, while the first two approaches are useful at characterising
the time scale of the transition, they do not provide accurate estimates for
diffusion processes. Secondly, the new approach allows one to calculate finite
transition times accurate to effectively any number of significant digits,
using only the moments, with the accuracy increasing as the index is
increased.Comment: 17 pages, 2 figures, accepted version of paper published in Physical
Review
Rear-surface integral method for calculating thermal diffusivity from laser flash experiments
The laser flash method for measuring thermal diffusivity of solids involves
subjecting the front face of a small sample to a heat pulse of radiant energy
and recording the resulting temperature rise on the opposite (rear) surface.
For the adiabatic case, the widely-used standard approach estimates the thermal
diffusivity from the rear-surface temperature rise history by calculating the
half rise time: the time required for the temperature rise to reach one half of
its maximum value. In this article, we develop a novel alternative approach by
expressing the thermal diffusivity exactly in terms of the area enclosed by the
rear-surface temperature rise curve and the steady-state temperature over time.
Approximating this integral numerically leads to a simple formula for the
thermal diffusivity involving the rear-surface temperature rise history. Using
synthetic experimental data we demonstrate that the new formula produces
estimates of the thermal diffusivity - for a typical test case - that are more
accurate and less variable than the standard approach. The article concludes by
briefly commenting on extension of the new method to account for heat losses
from the sample.Comment: 7 pages, 1 figure, accepted versio
Drug delivery from microcapsules: how can we estimate the release time?
Predicting the release performance of a drug delivery device is an important
challenge in pharmaceutics and biomedical science. In this paper, we consider a
multi-layer diffusion model of drug release from a composite spherical
microcapsule into an external surrounding medium. Based on this model, we
present two approaches that provide useful indicators of the release time, i.e.
the time required for the drug-filled capsule to be depleted. Both approaches
make use of temporal moments of the drug concentration versus time curve at the
centre of the capsule, which provide useful insight into the timescale of the
process and can be computed exactly without explicit calculation of the full
transient solution of the multi-layer diffusion model. The first approach,
which uses the zeroth and first temporal moments only, provides simple
algebraic expressions involving the various parameters in the model (e.g. layer
diffusivities, mass transfer coefficients, partition coefficients) to
characterize the release time while the second approach yields an asymptotic
estimate of the release time that depends on consecutive higher moments.
Through several test cases, we show that both approaches provide a
computationally-cheap and useful measure to compare \textit{a priori} the
release time of different composite microcapsule configurations.Comment: 15 pages, 4 figures, submitte
Krylov subspace approximations for the exponential Euler method: error estimates and the harmonic Ritz approximant
We study Krylov subspace methods for approximating the matrix-function vector product φ(tA)b where φ(z) = [exp(z) - 1]/z. This product arises in the numerical integration of large stiff systems of differential equations by the Exponential Euler Method, where A is the Jacobian matrix of the system. Recently, this method has found application in the simulation of transport phenomena in porous media within mathematical models of wood drying and groundwater flow. We develop an a posteriori upper bound on the Krylov subspace approximation error and provide a new interpretation of a previously published error estimate. This leads to an alternative Krylov approximation to φ(tA)b, the so-called Harmonic Ritz approximant, which we find does not exhibit oscillatory behaviour of the residual error
Accurate and efficient calculation of response times for groundwater flow
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
, to the steady state condition that arises in the long time limit, as . 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 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
, where is a relevant length scale, and is the aquifer
diffusivity. Here, we extend such heuristic arguments by providing a clear
mathematical definition of the proportionality constant.Comment: 22 pages, 3 figures, accepted version of paper published in Journal
of Hydrolog
New homogenization approaches for stochastic transport through heterogeneous media
The diffusion of molecules in complex intracellular environments can be
strongly influenced by spatial heterogeneity and stochasticity. A key challenge
when modelling such processes using stochastic random walk frameworks is that
negative jump coefficients can arise when transport operators are discretized
on heterogeneous domains. Often this is dealt with through homogenization
approximations by replacing the heterogeneous medium with an
homogeneous medium. In this work, we present a new class
of homogenization approximations by considering a stochastic diffusive
transport model on a one-dimensional domain containing an arbitrary number of
layers with different jump rates. We derive closed form solutions for the th
moment of particle lifetime, carefully explaining how to deal with the internal
interfaces between layers. These general tools allow us to derive simple
formulae for the effective transport coefficients, leading to significant
generalisations of previous homogenization approaches. Here, we find that
different jump rates in the layers gives rise to a net bias, leading to a
non-zero advection, for the entire homogenized system. Example calculations
show that our generalized approach can lead to very different outcomes than
traditional approaches, thereby having the potential to significantly affect
simulation studies that use homogenization approximations.Comment: 9 pages, 2 figures, accepted version of paper published in The
Journal of Chemical Physic
Semi-analytical solution of multilayer diffusion problems with time-varying boundary conditions and general interface conditions
We develop a new semi-analytical method for solving multilayer diffusion
problems with time-varying external boundary conditions and general internal
boundary conditions at the interfaces between adjacent layers. The convergence
rate of the semi-analytical method, relative to the number of eigenvalues, is
investigated and the effect of varying the interface conditions on the solution
behaviour is explored. Numerical experiments demonstrate that solutions can be
computed using the new semi-analytical method that are more accurate and more
efficient than the unified transform method of Sheils [Appl. Math. Model.,
46:450-464, 2017]. Furthermore, unlike classical analytical solutions and the
unified transform method, only the new semi-analytical method is able to
correctly treat problems with both time-varying external boundary conditions
and a large number of layers. The paper is concluded by replicating solutions
to several important industrial, environmental and biological applications
previously reported in the literature, demonstrating the wide applicability of
the work.Comment: 24 pages, 8 figures, accepted version of paper published in Applied
Mathematics and Computatio
Rear-surface integral method for calculating thermal diffusivity: finite pulse time correction and two-layer samples
We study methods for calculating the thermal diffusivity of solids from laser
flash experiments. This experiment involves subjecting the front surface of a
small sample of the material to a heat pulse and recording the resulting
temperature rise on the opposite (rear) surface. Recently, a method was
developed for calculating the thermal diffusivity from the rear-surface
temperature rise, which was shown to produce improved estimates compared with
the commonly used half-time approach. This so-called rear-surface integral
method produced a formula for calculating the thermal diffusivity of
homogeneous samples under the assumption that the heat pulse is instantaneously
absorbed uniformly into a thin layer at the front surface. In this paper, we
show how the rear-surface integral method can be applied to a more physically
realistic heat flow model involving the actual heat pulse shape from the laser
flash experiment. New thermal diffusivity formulas are derived for handling
arbitrary pulse shapes for either a homogeneous sample or a heterogeneous
sample comprising two layers of different materials. Presented numerical
experiments confirm the accuracy of the new formulas and demonstrate how they
can be applied to the kinds of experimental data arising from the laser flash
experiment.Comment: 10 pages, 3 figures, accepted versio
Advection improves homogenized models of continuum diffusion in one-dimensional heterogeneous media
We propose an alternative method for one-dimensional continuum diffusion
models with spatially variable (heterogeneous) diffusivity. Our method, which
extends recent work on stochastic diffusion, assumes the constant-coefficient
homogenized equation takes the form of an advection-diffusion equation with
effective (diffusivity and velocity) coefficients. To calculate the effective
coefficients, our approach involves solving two uncoupled boundary value
problems over the heterogeneous medium and leads to coefficients depending on
the spatially-varying diffusivity (as usual) as well as the boundary conditions
imposed on the heterogeneous model. Computational experiments comparing our
advection-diffusion homogenized model to the standard homogenized model
demonstrate that including an advection term in the homogenized equation leads
to improved approximations of the solution of the original heterogeneous model.Comment: 7 pages, 2 figures, accepted version of paper published in Physical
Review
- …