The Stefan problem with variable thermophysical properties and phase change temperature

In this paper we formulate a Stefan problem appropriate when the thermophysical properties are distinct in each phase and the phase-change temperature is size or velocity dependent. Thermophysical properties invariably take different values in different material phases but this is often ignored for mathematical simplicity. Size and velocity dependent phase change temperatures are often found at very short length scales, such as nanoparticle melting or dendrite formation; velocity dependence occurs in the solidification of supercooled melts. To illustrate the method we show how the governing equations may be applied to a standard one-dimensional problem and also the melting of a spherically symmetric nanoparticle. Errors which have propagated through the literature are highlighted. By writing the system in non-dimensional form we are able to study the large Stefan number formulation and an energy-conserving one-phase reduction. The results from the various simplifications and assumptions are compared with those from a finite difference numerical scheme. Finally, we briefly discuss the failure of Fourier's law at very small length and time-scales and provide an alternative formulation which takes into account the finite time of travel of heat carriers (phonons) and the mean free distance between collisions.Comment: 39 pages, 5 figure

The one-dimensional Stefan problem with non-Fourier heat conduction

We investigate the one-dimensional growth of a solid into a liquid bath, starting from a small crystal, using the Guyer-Krumhansl and Maxwell-Cattaneo models of heat conduction. By breaking the solidification process into the relevant time regimes we are able to reduce the problem to a system of two coupled ordinary differential equations describing the evolution of the solid-liquid interface and the heat flux. The reduced formulation is in good agreement with numerical simulations. In the case of silicon, differences between classical and non-classical solidification kinetics are relatively small, but larger deviations can be observed in the evolution in time of the heat flux through the growing solid. From this study we conclude that the heat flux provides more information about the presence of non-classical modes of heat transport during phase-change processes.Comment: 29 pages, 6 figures, 2 tables + Supplementary Materia

Enhancing engagement in evidence-based tobacco cessation treatment for smokers with mental illness: A pilot randomized trial.

ObjectiveTo evaluate the efficacy of a brief telephone-delivered Motivational Interviewing (MI)-based intervention to facilitate engagement in evidence-based cessation treatment for Veterans with mental illness referred to smoking cessation treatment.Methods86 military Veteran smokers with mental illness were recruited from a tobacco cessation consult clinic and randomized to receive either a MI-based treatment engagement intervention (TE; n = 48) or a non-MI assessment and information control (CON; n = 38) condition. Intervention was delivered during a single brief telephone contact. Primary engagement outcomes were 1) attending a treatment session within 30 days and 2) combination treatment (attending session plus using pharmacotherapy). Cessation outcomes included self-reported 24 h cessation attempts and 7 day point abstinence at 3 months post-intervention. Outcomes were assessed at 1 and 3 months post intervention.ResultsOutcome analyses included 85 participants (47 TE, 38 CON) using an intent-to-treat analytic approach. Participants were on average 49.5 (13.4) years old, 88% Male, 59% white, 18% African American and 14% Hispanic/Latino(a). Following intervention delivery TE and CON participants did not differ on likelihood of attending a treatment session during the subsequent 30 days (47% vs 45%, respectively). A significant difference was observed when classified as utilizing combination treatment, 40% of TE versus 18% of CON reported use of smoking cessation medication and behavioral counseling (p = 0.04). No statistical differences were observed for cessation outcomes, although more TE than CON participants reported 7 day point abstinence at 3 months post-intervention (30% vs 18%).ConclusionsThe present pilot study provides initial evidence for the feasibility, acceptability and efficacy of a telephone delivered TE intervention for enhancing engagement in combinationevidence evidence-based treatment in a sample of Veteran smokers with mental illness referred to smoking cessation treatment. Smokers with mental illness typically have greater difficulty stopping smoking than those without mental illness. Increased engagement in combination treatment thus has the potential to increase quit rates and ultimately reduce the burden of tobacco use for this population

Una Anàlisi matemàtica del moviment d'una pilota de futbol durant el vol

En aquest article es presenta un estudi analític i numèric de les equacions tridimensionals que descriuen el moviment d'una pilota que gira a l'aire. L'anàlisi inicial considera coeficients de fregament constants i després s'estén al cas d'un fregament que depèn de la velocitat de rotació de la pilota. S'observa una coincidència excel.lent entre els resultats numèrics, els analítics i els experimentals. La solució analítica ens mostra de manera explícita com el moviment de la pilota depèn de paràmetres com la rugositat, la velocitat i les condicions atmosfèriques. Es demostra la importància d'aplicar models tridimensionals en comptes d'aproximacions bidimensionals.In this paper an analytical and numerical study of the three-dimensional equations describing the motion through the air of a spinning ball is presented. The initial analysis involves constant drag coefficients but is later extended to involve drag varying with the spin ratio. Excellent agreement is demonstrated between numerical and analytical results. The analytical solution shows explicitly how the balls motion depends on parameters such as ball roughness, velocity and atmospheric conditions. The importance of applying three-dimensional models, rather than two-dimensional approximations, is demonstrated

A mathematical model for the energy stored in green roofs

A simple mathematical model to estimate the energy stored in a green roof is developed. Analytical solutions are derived corresponding to extensive (shallow) and intensive (deep) substrates. Results are presented for the surface temperature and energy stored in both green roofs and concrete during a typical day. Within the restrictions of the model assumptions the analytical solution demonstrates that both energy and surface temperature vary linearly with fractional leaf coverage, albedo and irradiance, while the effect of evaporation rate and convective heat transfer is non-linear. It is shown that a typical green roof is significantly cooler and stores less energy than a concrete one even when the concrete has a high albedo coating. Evaporation of even a few millimetres per day from the soil layer can reduce the stored energy by a factor of more than three when compared to an equivalent thickness concrete roof

Asymptotic reduction of a porous electrode model for lithium-ion batteries

We present a porous electrode model for lithium-ion batteries using Butler--Volmer reaction kinetics. We model lithium concentration in both the solid and fluid phase along with solid and liquid electric potential. Through asymptotic reduction, we show that the electric potentials are spatially homogeneous which decouples the problem into a series of time-dependent problems. These problems can be solved on three distinguished time scales, an early time scale where capacitance effects in the electrode dominate, a mid-range time scale where a spatial concentration gradient forms in the electrolyte, and a long-time scale where each of the electrodes saturate and deplete with lithium respectively. The solid-phase concentration profiles are linear functions of time and the electrolyte potential is everywhere zero, which allows the model to be reduced to a system of two uncoupled ordinary differential equations. Analytic and numerical results are compared with full numerical simulations and experimental discharge curves demonstrating excellent agreement.Comment: Accepted in SIAM Journal on Applied Mathematic

Three-Dimensional Magnetohydrodynamics Simulations Of Counter-Helicity Spheromak Merging In The Swarthmore Spheromak Experiment

Recent counter-helicity spheromak merging experiments in the Swarthmore Spheromak Experiment (SSX) have produced a novel compact torus (CT) with unusual features. These include a persistent antisymmetric toroidal magnetic field profile and a slow, nonlinear emergence of the n = 1 tilt mode. Experimental measurements are inconclusive as to whether this unique CT is a fully merged field-reversed configuration (FRC) with strong toroidal field or a partially merged doublet CT configuration with both spheromak- and FRC-like characteristics. In this paper, the SSX merging process is studied in detail using three-dimensional resistive MHD simulations from the Hybrid Magnetohydrodynamics (HYM) code. These simulations show that merging plasmas in the SSX parameter regime only partially reconnect, leaving behind a doublet CT rather than an FRC. Through direct comparisons, we show that the magnetic structure in the simulations is highly consistent with the SSX experimental observations. We also find that the n = 1 tilt mode begins as a fast growing linear mode that evolves into a slower-growing nonlinear mode before being detected experimentally. A simulation parameter scan over resistivity, viscosity, and line-tying shows that these parameters can strongly affect the behavior of both the merging process and the tilt mode. In fact, merging in certain parameter regimes is found to produce a toroidal-field-free FRC rather than a doublet CT. (C) 2011 American Institute of Physics. [doi:10.1063/1.3660533

Effective thermal conductivity of rectangular nanowires based on phonon hydrodynamics

A mathematical model is presented for thermal transport in nanowires with rectangular cross sections. Expressions for the effective thermal conductivity of the nanowire across a range of temperatures and cross-sectional aspect ratios are obtained by solving the Guyer--Krumhansl hydrodynamic equation for the thermal flux with a slip boundary condition. Our results show that square nanowires transport thermal energy more efficiently than rectangular nanowires due to optimal separation between the boundaries. However, circular nanowires are found to be even more efficient than square nanowires due to the lack of corners in the cross section, which locally reduce the thermal flux and inhibit the conduction of heat. By using a temperature-dependent slip coefficient, we show that the model is able to accurately capture experimental data of the effective thermal conductivity obtained from Si nanowires, demonstrating that phonon hydrodynamics is a powerful framework that can be applied to nanosystems even at room temperature

Contaminant Removal by Adsorption

We develop a mathematical model for filtration in a cylindrical column packed with a porous material. The base model involves coupling an advection-diffusion equation to a sink term which represents the sorption and is appropriate when trace quantities are removed from the fluid. This is then extended to include the variation of velocity and pressure, which is appropriate for the removal of significant quantities, and leads to a system of five coupled equations. For the case of CO2 removal we are able to reduce the complexity of the equations and to derive an analytical expression for the breakthrough curve. This expression is then verified against experimental data for the adsorption of CO2 from gas and antibiotics from water. Finally, we show how the work may be modified to deal with certain extraction processes, where a clean fluid is used to remove material from the porous matrix, such as lanolin from wool

A mathematical model for the energy stored in green roofs

A simple mathematical model to estimate the energy stored in a green roof is developed. Analytical solutions are derived corresponding to extensive (shallow) and intensive (deep) substrates. Results are presented for the surface temperature and energy stored in both green roofs and concrete during a typical day. Within the restrictions of the model assumptions the analytical solution demonstrates that both energy and surface temperature vary linearly with fractional leaf coverage, albedo and irradiance, while the effect of evaporation rate and convective heat transfer is non-linear. It is shown that a typical green roof is significantly cooler and stores less energy than a concrete one even when the concrete has a high albedo coating. Evaporation of even a few millimetres per day from the soil layer can reduce the stored energy by a factor of more than three when compared to an equivalent thickness concrete roof.Peer ReviewedPostprint (published version