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Calculating the thermal diffusivity of solid materials is commonly carried out using the laser flash experiment. This classical experiment considers a small (usually thin disc-shaped) sample of the material with parallel front and rear surfaces, applying a heat pulse to the front surface and recording the resulting rise in temperature over time on the rear surface. Recently, Carr and Wood [Int J Heat Mass Transf, 144 (2019) 118609] showed that the thermal diffusivity can be expressed analytically in terms of the heat flux function applied at the front surface and the temperature rise history at the rear surface. In this paper, we generalise this result to radial unidirectional heat flow, developing new analytical formulas for calculating the thermal diffusivity for cylindrical shell and spherical shell shaped samples. Two configurations are considered: (i) heat pulse applied on the inner surface and temperature rise recorded on the outer surface and (ii) heat pulse applied on the outer surface and temperature rise recorded on the inner surface. Code implementing and verifying the thermal diffusivity formulas for both configurations is made available., Comment: 10 pages, 3 figures, accepted version

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., 7 pages, 1 figure, accepted version

In this work we consider a recent experimental data set describing heat conduction in living porcine tissues. Understanding this novel data set is important because porcine skin is similar to human skin. Improving our understanding of heat conduction in living skin is relevant to understanding burn injuries, which are common, painful and can require prolonged and expensive treatment. A key feature of skin is that it is layered, with different thermal properties in different layers. Since the experimental data set involves heat conduction in thin living tissues of anesthetised animals, an important experimental constraint is that the temperature within the living tissue is measured at one spatial location within the layered structure. Our aim is to determine whether this data is sufficient to reliably infer the heat conduction parameters in layered skin, and we use a simplified two-layer mathematical model of heat conduction to mimic the generation of experimental data. Using synthetic data generated at one location in the two-layer mathematical model, we explore whether it is possible to infer values of the thermal diffusivity in both layers. After this initial exploration, we then examine how our ability to infer the thermal diffusivities changes when we vary the location at which the experimental data is recorded, as well as considering the situation where we are able to monitor the temperature at two locations within the layered structure. Overall, we find that our ability to parameterise a model of heterogeneous heat conduction with limited experimental data is very sensitive to the location where data is collected. Our modelling results provide guidance about optimal experimental design that could be used to guide future experimental studies.NomenclatureA brief description of all variables used in the document are given in Table 1.Table 1:Variable nomenclature and description.

Accurately determining the timber properties for products prior to cutting the tree is difficult. In this work we discuss a method for reconstructing a timber billet virtually, including internal features, after it has been peeled into a full veneer (ribbon). This reconstruction process is the first stage in developing a mathematical model for the variation in timber properties within a given tree. The reconstruction of internal timber features is typically achieved through the use of computed tomography (CT) scanning. However, this requires the use of equipment that may be cost-prohibitive. Here we discuss an approach that utilises more readily available equipment for timber processors, including a spindleless lathe and digital SLR camera. In comparison to conventional scanning methods, this reconstruction method based on a destructive process has the key advantage of delivering high-resolution colour images. This reconstruction serves two purposes. Firstly, we are able to generate three-dimensional visualisations of the timber billet, to uncover internal structures such as knots, defects, insect or fungi attack, discoloration, resin etc. Secondly, the reconstruction allows us to map timber properties measured on the veneer to their original location within the billet. This allows us to locally inform the mapping with wood properties and subsequently derive their distribution throughout the billet. From this information it is then possible to extract any part of the billet and obtain the appearance and wood properties of any processed products. To validate our reconstruction process we show that we can obtain reasonable agreement between our predicted billet modulus of elasticity and that measured on the original billet.

International audience; Numerous problems involving gradient-driven transport processes—e.g., Fourier's and Darcy's law—in heterogeneous materials concern a physical domain that is much larger than the scale at which the coefficients vary spatially. To overcome the prohibitive computational cost associated with such problems, the well-established Distributed Microstructure Model (DMM) provides a two-scale description of the transport process that produces a computationally cheap approximation to the fine-scale solution. This is achieved via the introduction of sparsely distributed micro-cells that together resolve small patches of the fine-scale structure: a macroscopic equation with an effective coefficient describes the global transport and a microscopic equation governs the local transport within each micro-cell. In this paper, we propose a new formulation, the Extended Distributed Microstructure Model (EDMM), where the macroscopic flux is instead defined as the average of the microscopic fluxes within the micro-cells. This avoids the need for any effective parameters and more accurately accounts for a non-equilibrium field in the micro-cells. Another important contribution of the work is the presentation of a new and improved numerical scheme for performing the two-scale computations using control volume, Krylov subspace and parallel computing techniques. Numerical tests are carried out on two challenging test problems: heat conduction in a composite medium and unsaturated water flow in heterogeneous soils. The results indicate that while DMM is more efficient, EDMM is more accurate and is able to capture additional fine-scale features in the solution.

Diffusion in a composite slab consisting of a large number of layers provides an ideal prototype problem for developing and analysing two-scale modelling approaches for heterogeneous media. Numerous analytical techniques have been proposed for solving the transient diffusion equation in a one-dimensional composite slab consisting of an arbitrary number of layers. Most of these approaches, however, require the solution of a complex transcendental equation arising from a matrix determinant for the eigenvalues that is difficult to solve numerically for a large number of layers. To overcome this issue, in this paper, we present a semi-analytical method based on the Laplace transform and an orthogonal eigenfunction expansion. The proposed approach uses eigenvalues local to each layer that can be obtained either explicitly, or by solving simple transcendental equations. The semi-analytical solution is applicable to both perfect and imperfect contact at the interfaces between adjacent layers and either Dirichlet, Neumann or Robin boundary conditions at the ends of the slab. The solution approach is verified for several test cases and is shown to work well for a large number of layers. The work is concluded with an application to macroscopic modelling where the solution of a fine-scale multilayered medium consisting of two hundred layers is compared against an “up-scaled” variant of the same problem involving only ten layers.

We assess the performance of an exponential integrator for advancing stiff, semidiscrete formulations of the unsaturated Richards equation in time. The scheme is of second order and explicit in nature but requires the action of the matrix function φ ( A ) = A −1 ( e A − I ) on a suitability defined vector v at each time step. When the matrix A is large and sparse, φ ( A ) v can be approximated by Krylov subspace methods that require only matrix–vector products with A . We prove that despite the use of this approximation the scheme remains second order. Furthermore, we provide a practical variable-stepsize implementation of the integrator by deriving an estimate of the local error that requires only a single additional function evaluation. Numerical experiments performed on two-dimensional test problems demonstrate that this implementation outperforms second-order, variable-stepsize implementations of the backward differentiation formulae.