Explicit solutions for one-dimensional two-phase free boundary problems with either shrinkage or expansion

Natale, María Fernanda; Marcus, Eduardo A. Santillan; Tarzia, Domingo A.

doi:10.1016/j.nonrwa.2009.04.014

Cited by 13 publications

(4 citation statements)

References 21 publications

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“…In Section 3 we give the necessary and/or sufficient conditions for the three real parameters involved in the trascendental equation in order to obtain an instantaneous phase-change process (1)-( 7) with the corresponding similarity solution. We generalize results obtained for particular cases given in [13,20]. Finally, in Section 4 we analize the relationship between the problems (1)-( 7) and ( 1)-( 6) and ( 8), and we obtain conditions for data under which both problem became equivalents.…”

Section: Introductionmentioning

confidence: 66%

Similarity solutions for thawing processes with a convective boundary condition

Ceretani,

Tarzia

2014

Preprint

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Similarity solutions for a one-dimensional mathematical model for thawing in a saturated semi-infinite porous media is considered when change of phase induces a density jump and a convective boundary condition is imposed at the fixed face x = 0. Different cases depending on physical parameters are analysed and an explicit solution of a similarity type is obtained if and only if an inequality for data is verified. Moreover, a monotony property respect to the coefficient which characterizes the heat transfer at the fixed face x = 0 is obtained for the coefficient involved in the definition of the free boundary.Relationship between the Stefan problem with convective condition at x = 0 considered in this paper and the Stefan problem with temperature condition at the same face studied in (Fasano-Primicerio-Tarzia, Math. Models Meth. Appl. Sci., 9 (1999), 1-10) is analized and conditions for physical parameters under which both problems became equivalents are obtained. Furthermore, an inequality to be satisfied for the coefficient which characterizes the free boundary of the problem with a temperature or a convective boundary condition at the fixed face x = 0 is also obtained.

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Section: Introductionmentioning

confidence: 66%

Similarity solutions for thawing processes with a convective boundary condition

Ceretani,

Tarzia

2014

Preprint

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“…Stefan () began the study of this specific type of system by tracking the heat conduction front, present in phase change problems in ice‐water systems. Since then, many studies have focused on obtaining both analytical solutions (Barry & Caunce, ; Cherniha & Kovalenko, ; Natale, Santillan Marcus, & Tarzia, ; Salva & Tarzia, ; Voller & Falcini, ), and numerical solutions (Caldwell & Chan, ; Kutluay, Bahadir, & Özdeş, ; Mitchell & O'Brien, ; Mitchell, Vynnycky, Gusev, & Sazhin, ; Sadoun, Si‐Ahmed, Colinet, & Legrand, ; Sadoun, Si‐Ahmed, & Legrand, ; Savovic & Caldwell, ). The works of Caldwell and Kwan () bring together the most common numerical methods for solving Stefan problems.…”

Section: Introductionmentioning

confidence: 99%

Modeling of soybean hydration as a Stefan problem: Boundary immobilization method

Nicolin

Silva

Jorge

et al. 2018

J Food Process Engineering

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In the present work, the mathematical modeling of the soybean hydration process is presented, considering the increase in the grain radius by Stefan problem approach. The boundary immobilization method was used in the model to take into account the spatial mesh variation caused by the diffusion of moisture in the grains. The results provided by the model show its quality when compared with experimental data, both for the adequacy of the predictions for moisture data over time and for the adequacy of grain radius increase predictions over time, for five different temperatures. The presented methodology showed improvements on results already reported in the literature. Practical applications The approach presented in this article allows one to fit diffusivity values for soybean hydration by taking into account a phenomenon commonly observed in this process: the increase of the grains during water absorption. Thus, it is expected the diffusivities to be more realistic and precise, improving the project of equipment. Since the model proposed provide the increase of the grains along the hydration process, an estimation of the increase of a higher amount of the grains could be obtained as well. Such estimation is relevant due to the considerable increase that grains suffer after absorbing water.

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“…Consequently, in this chapter we will focus on the effect of density change. Our work is motivated by the melting of spherical nanoparticles however the model could be applied to more general situations of practical interest, such as pipe bursting, cryopreservation, phase change microvalves and metal casting, see [3,28,76,77,80].…”

Section: Introductionmentioning

confidence: 99%

“…However, they later neglect this term to permit exact similarity solutions. In [80] a similar approach is taken, again to find similarity solutions. In [13] the cubic term is neglected altogether and seek small time and similarity solutions for the freezing of a liquid layer and phase change in a porous half-space.…”

Section: Introductionmentioning

confidence: 99%

Beyond the classical Stefan problem

Font Martínez

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In this thesis we develop and analyse mathematical models describing phase change phenomena linked with novel technological applications. The models are based on modifications to standard phase change theory. The mathematical tools used to analyse such models include asymptotic analysis, similarity solutions, the Heat Balance Integral Method and standard numerical techniques such as finite differences. In chapters 2 and 3 we study the melting of nanoparticles. The Gibbs-Thomson relation, accounting for melting point depression, is coupled to the heat equations for the solid and liquid and the associated Stefan condition. A perturbation approach, valid for large Stefan numbers, is used to reduce the governing system of partial differential equations to a less complex one involving two ordinary differential equations. Comparison between the reduced system and the numerical solution shows good agreement. Our results reproduce interesting behaviour observed experimentally such as the abrupt melting of nanoparticles. Standard analyses of the Stefan problem impose constant physical properties, such as density or specific heat. We formulate the Stefan problem to allow for variation at the phase change and show that this can lead to significantly different melting times when compared to the standard formulation. In chapter 4 we study a mathematical model describing the solidification of supercooled liquids. For Stefan numbers larger than unity the classical Neumann solution provides an analytical expression to describe the solidification. For Stefan numbers equal or smaller than unity, the Neumann solution is no longer valid. Instead, a linear relationship between the phase change temperature and the front velocity is often used. This allows solutions for all values of the Stefan number. However, the linear relation is an approximation to a more complex, nonlinear relationship, valid for small amounts of supercooling. We look for solutions using the nonlinear relation and demonstrate the inaccuracy of the linear relation for large supercooling. Further, we show how the classical Neumann solution significantly over-predicts the solidification rate for values of the Stefan number near unity. The Stefan problem is often reduced to a 'one-phase' problem (where one of the phases is neglected) in order to simplify the analysis. When the phase change temperature is variable it has been claimed that the standard reduction loses energy. In chapters 5 and 6, we examine the one-phase reduction of the Stefan problem when the phase change temperature is time-dependent. In chapter 5 we derive a one-phase reduction of the supercooled Stefan problem, and test its performance against the solution of the two-phase model. Our model conserves energy and is based on consistent physical assumptions, unlike one-phase reductions from previous studies. In chapter 6 we study the problem from a general perspective, and identify the main erroneous assumptions of previous studies leading to one-phase reductions that do not conserve energy or, alternatively, are based on non-physical assumptions. We also provide a general one-phase model of the Stefan problem with a generic variable phase change temperature, valid for spherical, cylindrical and planar geometries. En aquesta tesi desenvolupem i analitzem models matemàtics que descriuen processos de transició de fase vinculats a noves tecnologies. Els models es basen en modificacions de la teoria estàndard de canvis de fase. Les tècniques matemàtiques per resoldre els models es basen en l'anàlisi asimptòtic, solucions autosimilars, el mètode de la integral del balanç de la calor i mètodes numèrics estàndards com ara el mètode de les diferències finites. En els capítols 2 i 3 estudiarem la transició solid-liquid d'una nanopartícula, acoblant la relació de Gibbs-Thomson, que descriu la depressió de la temperatura de fusió en una superfície corba, amb l'equació de la calor per la fase sòlida i líquida, i la condició de Stefan. Mitjançant el mètode de pertorbacions, per a valors grans del nombre de Stefan, el problema d'equacions en derivades parcials inicial és reduït a un sistema més senzill de dues equacions diferencials ordinàries. La solució del sistema reduït concorda perfectament amb la solució numèrica del sistema inicial d'equacions en derivades parcials. Els resultats confirmen la transició ultra ràpida de sòlid a líquid observada en experiments amb nanopartícules. Els anàlisis estàndards del problema de Stefan consideren propietats físiques com la densitat i el calor específic constants en la fase sòlida i líquida. En aquesta tesi, formularem el problema de Stefan relaxant la condició de densitat constant, el que portarà a diferències molt significatives en els temps totals de fusió al comparar-los amb els temps obtinguts mitjançant la formulació habitual del problema de Stefan. En el capítol 4 estudiarem un model matemàtic que descriu la solidificació de líquids sota-refredats. Per nombres de Stefan més grans que la unitat la solució clàssica de Neumann dona una expressió analítica que descriu el procés. Per valors del nombre de Stefan més petits o iguals que la unitat, la solució de Neumann no es vàlida i, habitualment, per tal de trobar solucions en aquest règim, s'estableix una relació lineal entre la temperatura de canvi de fase i la velocitat del front de solidificació. Aquesta relació lineal però, és una aproximació per sota-refredaments moderats d'una relació no lineal més complexa. En aquest capítol, buscarem solucions del problema incorporant la relació no lineal al model, i demostrarem la poca precisió a l'utilitzar l'aproximació lineal. A més a més, veurem com la solució de Neumann sobreestima de manera significativa la velocitat del procés per valors propers a la unitat. Habitualment, el problema de Stefan és simplificat a un problema d'una fase (on una de les fases es desestimada) per tal de reduir la dificultat de l'anàlisi. En casos on la temperatura de transició de fase és variable s'ha reclamat que la simplificació d'una fase del problema no conserva l'energia. En els capítols 5 i 6, examinarem les diferents reduccions d'una fase del problema de Stefan en el cas on la temperatura de transició depèn del temps. En el capítol 5 derivarem el problema de Stefan d'una fase associat a la solidificació de líquids sota-refredats, i compararem la solució del sistema resultant amb la solució del problema de Stefan estàndard de dues fases. A diferència dels models d'una fase descrits en estudis previs, el nostre model reduït conserva l'energia i està basat en suposicions físiques consistents. En el capítol 6 estudiarem el problema des d'una perspectiva més general i identificarem les suposicions errònies d'estudis previs que porten a la no conservació de l'energia o que, alternativament, estan basades en suposicions físiques poc consistents. A més a més, derivarem un model d'una fase amb temperatura de canvi de fase variable, vàlid per geometries esfèriques, cilíndriques i planes.

show abstract

Explicit solutions for one-dimensional two-phase free boundary problems with either shrinkage or expansion

Cited by 13 publications

References 21 publications

Similarity solutions for thawing processes with a convective boundary condition

Similarity solutions for thawing processes with a convective boundary condition

Modeling of soybean hydration as a Stefan problem: Boundary immobilization method

Beyond the classical Stefan problem

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