Analytical formulas for calculating the thermal diffusivity of cylindrical shell and spherical shell samples

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: 12 pages, 5 figure

Simplified models of diffusion in radially-symmetric geometries

We consider diffusion-controlled release of particles from $d$-dimensional radially-symmetric geometries. A quantity commonly used to characterise such diffusive processes is the proportion of particles remaining within the geometry over time, denoted as $P(t)$. The stochastic approach for computing $P(t)$ is time-consuming and lacks analytical insight into key parameters while the continuum approach yields complicated expressions for $P(t)$ that obscure the influence of key parameters and complicate the process of fitting experimental release data. In this work, to address these issues, we develop several simple surrogate models to approximate $P(t)$ by matching moments with the continuum analogue of the stochastic diffusion model. Surrogate models are developed for homogeneous slab, circular, annular, spherical and spherical shell geometries with a constant particle movement probability and heterogeneous slab, circular, annular and spherical geometries, comprised of two concentric layers with different particle movement probabilities. Each model is easy to evaluate, agrees well with both stochastic and continuum calculations of $P(t)$ and provides analytical insight into the key parameters of the diffusive transport system: dimension, diffusivity, geometry and boundary conditions.Comment: 22 pages, 3 figures, submitte

Analytical formulas for calculating the thermal diffusivity of cylindrical shell and spherical shell samples

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.</p

Simplified models of diffusion in radially-symmetric geometries

We consider diffusion-controlled release of particles from d-dimensional radially-symmetric geometries. A quantity commonly used to characterise such diffusive processes is the proportion of particles remaining within the geometry over time, denoted as P(t). The stochastic approach for computing P(t) is time-consuming and lacks analytical insight into key parameters while the continuum approach yields complicated expressions for P(t) that obscure the influence of key parameters and complicate the process of fitting experimental release data. In this work, to address these issues, we develop several simple surrogate models to approximate P(t) by matching moments with the continuum analogue of the stochastic diffusion model. Surrogate models are developed for homogeneous slab, circular, annular, spherical and spherical shell geometries with a constant particle movement probability and heterogeneous slab, circular, annular and spherical geometries, comprised of two concentric layers with different particle movement probabilities. Each model is easy to evaluate, agrees well with both stochastic and continuum calculations of P(t) and provides analytical insight into the key parameters of the diffusive transport system: dimension, diffusivity, geometry and boundary conditions.</p