A Lagrangian approach to modeling heat flux driven close-contact melting

Close-contact melting refers to the process of a heat source melting its way into a phase-change material. Of special interest is the close-contact melting velocity, or more specifically the relative velocity between the heat source and the phase-change material. In this work, we present a novel numerical approach to simulate quasi-steady, heat flux driven close-contact melting. It extends existing approaches found in the literature, and, for the first time, allows to study the impact of a spatially varying heat flux distribution. We will start by deriving the governing equations in a Lagrangian reference frame fixed to the heat source. Exploiting the narrowness of the melt film enables us to reduce the momentum balance to the Reynolds equation, which is coupled to the energy balance via the velocity field. We particularize our derivation for two simple, yet technically relevant geometries, namely a 3d circular disc and a 2d planar heat source. An iterative solution procedure for the coupled system is described in detail and discussed on the basis of a convergence study. Furthermore, we present an extension to allow for rotational melting modes. Various test cases demonstrate the proficiency of our method. In particular, we will utilize the method to assess the efficiency of the close-contact melting process and to quantify the model error introduced if convective losses are neglected. Finally, we will draw conclusions and present an outlook to future work

Finite element formulation to study thermal stresses in nanoencapsulated phase change materials for energy storage

Nanoencapsulated phase change materials (nePCMs) – which are composed of a core with a phase change material and of a shell that envelopes the core – are currently under research for heat storage applications. Mechanically, one problem encountered in the synthesis of nePCMs is the failure of the shell due to thermal stresses during heating/cooling cycles. Thus, a compromise between shell and core volumes must be found to guarantee both mechanical reliability and heat storage capacity. At present, this compromise is commonly achieved by trial and error experiments or by using simple analytical solutions. On this ground, the current work presents a thermodynamically consistent and three-dimensional finite element (FE) formulation considering both solid and liquid phases to study thermal stresses in nePCMs. Despite the fact that there are several phase change FE formulations in the literature, the main novelty of the present work is its monolithic coupling – no staggered approaches are required – between thermal and mechanical fields. Then, the FE formulation is implemented in a computational code and it is validated against one-dimensional analytical solutions. Finally, the FE model is used to perform a thermal stress analysis for different nePCM geometries and materials to predict their mechanical failure by using Rankine’s criterion

A two-scale Stefan problem arising in a model for tree sap exudation

The study of tree sap exudation, in which a (leafless) tree generates elevated stem pressure in response to repeated daily freeze-thaw cycles, gives rise to an interesting multi-scale problem involving heat and multiphase liquid/gas transport. The pressure generation mechanism is a cellular-level process that is governed by differential equations for sap transport through porous cell membranes, phase change, heat transport, and generation of osmotic pressure. By assuming a periodic cellular structure based on an appropriate reference cell, we derive an homogenized heat equation governing the global temperature on the scale of the tree stem, with all the remaining physics relegated to equations defined on the reference cell. We derive a corresponding strong formulation of the limit problem and use it to design an efficient numerical solution algorithm. Numerical simulations are then performed to validate the results and draw conclusions regarding the phenomenon of sap exudation, which is of great importance in trees such as sugar maple and a few other related species. The particular form of our homogenized temperature equation is obtained using periodic homogenization techniques with two-scale convergence, which we investigate theoretically in the context of a simpler two-phase Stefan-type problem corresponding to a periodic array of melting cylindrical ice bars with a constant thermal diffusion coefficient. For this reduced model, we prove results on existence, uniqueness and convergence of the two-scale limit solution in the weak form, clearly identifying the missing pieces required to extend the proofs to the fully nonlinear sap exudation model. Numerical simulations of the reduced equations are then compared with results from the complete sap exudation model.Comment: 35 pages, 8 figures. arXiv admin note: text overlap with arXiv:1411.303

Experimental validation of the exact analytical solution to the steady periodic heat transfer problem in a PCM layer

Phase change materials (PCM) are used in many industrial and residential applications for their advantageous characteristic of high capacity of latent thermal storage by means of an isothermal process. In this context, it is very useful to have predictive mathematical models for the analysis of the thermal performance and for the thermal design of these layers. In this work, an experimental validation of an analytical model that resolves the steady periodic heat transfer problem in a finite layer of PCM is presented. The experimental investigation was conducted employing a PCM with thermophysical and thermochemical behavior very close to those hypothesized in the formulation of the analytical model. For the evaluation of the thermophysical properties of the PCM sample used, an experimental procedure created by the authors was employed. In all tests realized in a sinusoidal and non-sinusoidal periodic regime, the comparison between the measured and calculated trends of the temperature at different sample heights and of the surface heat fluxes show an excellent agreement. Moreover, also having verified the analytical total stored energy, the analytical model constitutes a valid instrument for the evaluation of the latent and sensible contribution and the trend in time of the position of the bi-phase interface.The work was partially funded by the Spanish government (ENE2015-64117-C5-1-R (MINECO/FEDER), ENE2015-64117-C5-3-R (MINECO/FEDER), and ULLE10-4E-1305). GREA is certified agent TECNIO in the category of technology developers from the Government of Catalonia. The authors would like to thank the Catalan Government for the quality accreditation given to their research group (2014 SGR 123). This project has received funding from the European Commission Seventh Framework Programme (FP/2007-2013) under Grant agreement Nº PIRSES-GA-2013-610692 (INNOSTORAGE) and from European Union's Horizon 2020 research and innovation programme under grant agreement Nº 657466 (INPATH-TES). Alvaro de Gracia would like to thank Ministerio de Economia y Competitividad de España for Grant Juan de la Cierva, FJCI-2014-19940. Julià Coma would like to thank the Departament d'Universitats, Recerca i Societat de la Informació de la Generalitat de Catalunya for his research fellowship (2016FI_B2 00147). Aran Solé would like to thank Ministerio de Economía y Competitividad de España for Grant Juan de la Cierva, FJCI-2015-25741

The pear-shaped fate of an ice melting front

A fluid-structure interaction problem with the melting of water around a heated horizontal circular cylinder is analysed with numerical simulations. Dynamic meshing was used for evolving the flow domain in time as the melting front extended radially outward from the cylinder; a node shuffle algorithm was used to retain mesh quality across the significant mesh deformation. We simulated one case above the density inversion point of water and one case below, yielding pear-shaped melting fronts due to thermal plumes either rising or falling from the cylinder, respectively. Results were compared with previous experimental studies and the melting front profiles matched reasonably well and melting rates were in agreement. We confirm that natural convection plays a significant role in the transport of energy as the melt zone increases, and needs to be considered for accurately modelling phase change under these conditions.Comment: Accepted for the 12th International Conference on CFD in Oil & Gas, Metallurgical and Process Industries. SINTEF, Trondheim, Norway. May 30th - June 1st, 201

An adjusted analytical solution for thermal design in artificial ground freezing

Artificial ground freezing is a widely used, reliable method for excavation in water-bearing ground. The questions posed in the thermal design of ground freezing projects require solving moving boundary (Stefan) problems. Approximate analytical solutions, such as the ones by St¨ ander1 and Sanger and Sayles,2 have been developed for thermal engineering design and are used by practitioners across the industry. For instance, Sanger & Sayles’ solution is widely used for the single-freeze-pipe problem, but it has proven to be of limited accuracy.3 In the present paper, an adjustment to this formula is proposed based on the re-evaluation of their empirical assumption that the ratio between the temperature penetration depth and the phase-change radius equals a constant value of 3 regardless the conditions. A sensitivity study is performed using a verified numerical model as a benchmark to study several problems with different initial and boundary conditions (initial, phase change and freeze pipe temperatures) and thermal properties of the ground (water content, thermal conductivity and heat capacity). This is done for the freezing times of 10 and 365 days, in order to consider the potential change of the ratio with the freezing time. In this way, a calibrated formula is proposed to find appropriate values of this ratio and a suitable adjustment to Sanger & Sayles’ solution is determined. Adjusting Sanger & Sayles’ solution in this manner, a significantly higher and more consistent accuracy is achieved for different boundary and initial conditions. This accuracy improvement was checked for real conditions from an engineering project, which shows that the adjustment can be useful for thermal problems in engineering design of ground freezing

Radiation hydrodynamics integrated in the code PLUTO

The transport of energy through radiation is very important in many astrophysical phenomena. In dynamical problems the time-dependent equations of radiation hydrodynamics have to be solved. We present a newly developed radiation-hydrodynamics module specifically designed for the versatile MHD code PLUTO. The solver is based on the flux-limited diffusion approximation in the two-temperature approach. All equations are solved in the co-moving frame in the frequency independent (grey) approximation. The hydrodynamics is solved by the different Godunov schemes implemented in PLUTO, and for the radiation transport we use a fully implicit scheme. The resulting system of linear equations is solved either using the successive over-relaxation (SOR) method (for testing purposes), or matrix solvers that are available in the PETSc library. We state in detail the methodology and describe several test cases in order to verify the correctness of our implementation. The solver works in standard coordinate systems, such as Cartesian, cylindrical and spherical, and also for non-equidistant grids. We have presented a new radiation-hydrodynamics solver coupled to the MHD-code \PLUTO that is a modern, versatile and efficient new module for treating complex radiation hydrodynamical problems in astrophysics. As test cases, either purely radiative situations, or full radiation-hydrodynamical setups (including radiative shocks and convection in accretion discs) have been studied successfully. The new module scales very well on parallel computers using MPI. For problems in star or planet formation, we have added the possibility of irradiation by a central source.Comment: 13 pages, 11 figures, accepted by Astronomy & Astrophysic

A mathematical model for the dissolution of particles in multi-component alloys

AbstractDissolution of stoichiometric multi-component particles is an important process occurring during the heat treatment of as-cast aluminum alloys prior to hot extrusion. A mathematical model is proposed to describe such a process. In this model equations are given to determine the position of the particle interface in time, using a number of diffusion equations which are coupled by nonlinear boundary conditions at the interface. This problem is known as a vector valued Stefan problem. A necessary condition for existence of a solution of the moving boundary problem is proposed and investigated using the maximum principle for the parabolic partial differential equation. Furthermore, for an unbounded domain and planar co-ordinates an asymptotic approximation based on self-similarity is derived. The asymptotic approximation is used to gain insight into the influence of all components on the dissolution. Subsequently, a numerical treatment of the vector valued Stefan problem is described. The numerical solution is compared with solutions obtained by the analytical methods. Finally, an example is shown

Scale/Analytical Analyses of Freezing and Convective Melting With Internal Heat Generation

Using a scale/analytical analysis approach, we model phase change (melting) for pure materials which generate constant internal heat generation for small Stefan numbers (approximately one). The analysis considers conduction in the solid phase and natural convection, driven by internal heat generation, in the liquid regime. The model is applied for a constant surface temperature boundary condition where the melting temperature is greater than the surface temperature in a cylindrical geometry. The analysis also consider constant heat flux (in a cylindrical geometry).We show the time scales in which conduction and convection heat transfer dominate