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At least 19 records
All sourcesNASA NTRSDOE OSTI
DOE OSTI2020

Deep learning of free boundary and Stefan problems

Free boundary problems appear naturally in numerous areas of mathematics, science and engineering. These problems present a great computational challenge because they necessitate numerical methods that can yield an accurate approximation of free boundaries and complex dynamic interfaces. In this work, we propose a multi-network model based on physics-informed neural networks to tackle a general class of forward and inverse free boundary problems called Stefan problems. Specifically, we approximate the unknown solution as well as any moving boundaries by two deep neural networks. Besides, we formulate a new type of inverse Stefan problems that aim to reconstruct the solution and free boundaries directly from sparse and noisy measurements. We demonstrate the effectiveness of our approach in a series of benchmarks spanning different types of Stefan problems, and illustrate how the proposed framework can accurately recover solutions of partial differential equations with moving boundaries and dynamic interfaces. All code and data accompanying this manuscript are publicly available at https://github.com/PredictiveIntelligenceLab/DeepStefan.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2023

A general non-Fourier Stefan problem formulation that accounts for memory effects

The Stefan problem is the classical model of a melting phase change. In heterogeneous systems, such phase changes can exhibit non-Fourier (anomalous) behaviors, where the advance of the melt interface does not follow the expected time scaling. These situations can be modeled by replacing the derivatives, in the governing partial differential equations, with fractional order derivatives. In particular, replacing the time derivatives leads to non-Fourier models that account for memory effects in the system. In this work, by using appropriate time convolution integrals, a general thermodynamic balance statement for melting phase problems, explicitly accounting for memory effects, is developed. From this balance, a general model formulation applicable to problems involving melting over a temperature range (i.e., a mushy region) is derived. A key component in this model is the representation of memory effects through the use of fractional derivative based constitutive models of the enthalpy and heat flux. Further, on shrinking the mushy region to a single isotherm, a general sharp interface melting model is obtained. Here, in contrast to the classic Stefan problem, the fractional derivatives induce a natural regularization, such that the constitutive models for enthalpy and heat flux are continuous at the melt interface; a result confirmed through numerical simulation. To further support the theoretical findings, a physical example of a non-Fourier Stefan problem is presented. Overall the development and results in this paper underscore the importance of explicitly relating the development of fractional calculus models to the appropriate thermodynamic balance statements.

42 ENGINEERING↗
DOE OSTI2024

Development of Modified Perturbation Solutions to the One-Phase Stefan Problems With a Convective Boundary

The classical Stefan problem is used to track the moving solid-liquid interface during the freezing process. Perturbation theory has often been applied to find an approximate analytical solution due to the nonlinearity of the moving interface. However, the Stefan number (i.e., the sensible over latent heat) must be small and usually less than 0.01 to assume the perturbation expansion, which in turn limits the thermal engineering applications. In this study, a modified perturbation solution is developed by adding a correction term after the leading-order solution to be valid for a much wider range of Stefan numbers (i.e., 0.01 less than or equal to Ste less than or equal to 1). Specifically, a one-phase Stefan problem is first formulated subjected to a convective boundary in the Cartesian, cylindrical, and spherical coordinate systems for inward solidification. The leading-order solution is calculated based on the regular perturbation theory, while the correction term is obtained using the Monte-Carlo method and a multi-variant regression. Results show that the correction term has a linear relationship with the Stefan number and is not significantly influenced by the Biot number. The proposed modified perturbation solution can accurately and rapidly predict the nonlinear moving interface motion for the freezing process.

DIRECT ENERGY CONVERSION,MATHEMATICS AND COMPUTING↗
DOE OSTI2018

Explicit solution for a two-phase fractional Stefan problem with a heat flux condition at the fixed face

A generalized Neumann solution for the two-phase fractional Lamé–Clapeyron–Stefan problem for a semi-infinite material with constant initial temperature and a particular heat flux condition at the fixed face is obtained, when a restriction on data is satisfied. The fractional derivative in the Caputo sense of order α∈(0,1) respect on the temporal variable is considered in two governing heat equations and in one of the conditions for the free boundary. Furthermore, we find a relationship between this fractional free boundary problem and another one with a constant temperature condition at the fixed face and based on that fact, we obtain an inequality for the coefficient which characterizes the fractional phase–change interface obtained in Roscani and Tarzia (Adv Math Sci Appl 24(2):237–249, 2014). We also recover the restriction on data and the classical Neumann solution, through the error function, for the classical two-phase Lamé–Clapeyron–Stefan problem for the case α=1.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2018

Determination of unknown thermal coefficients in a Stefan problem for Storm’s type materials

We consider a nonlinear one-dimensional Stefan problem for a semi-infinite material x>0, with phase change temperature T{sub f}. We assume that the heat capacity and the thermal conductivity satisfy a Storm’s condition. A convective boundary condition and a heat flux over-specified condition on the fixed face x=0 are considered. Unknown thermal coefficients are determined for the free boundary problem and for the associate moving boundary problem and we give sufficient conditions to obtain a parametric representation of a similarity type solution. Moreover, we give formulae for the thermal coefficients in both cases.

42 ENGINEERING↗
DOE OSTI2020

An implementation of the phase-field model based on coupled thermomechanical finite element solvers for large-strain twinning, explicit dynamic fracture and the classical Stefan problem

The implementation of a phase-field model in finite elements usually requires significant expertise and involves the development of a user element with additional degrees of freedom. An alternative implementation of the phase-field model within a thermo-mechanical finite element simulation package was presented in (Cho et al 2012 Int. J. Solids Struct. 49 1973–1992), where the phase-field variable is treated as the temperature degree of freedom. However, this approach has only been used for small strain phase-field modelling of martensitic transformations and quasistatic phase-field modelling of fracture. Here, we present a phase-field finite element implementation via the temperature degree of freedom for several additional cases from the literature: (i) the large-strain phase-field description of deformation twinning presented in (Clayton and Knap 2011 Physica D 240 841–858), (ii) phase-field description of brittle fracture with inertial effects based on the theory from (Molnár and Gravouil 2017, Finite Elem. Anal. Des. 130 27–38) and (Miehe et al 2010 Int. J. Numer. Methods Eng. 83 1273–1311) and (iii) the classical Stefan problem of solidification presented in (Mackenzie and Robertson 2002 J. Comput. Phys. 181 526–544). The last problem involves the temperature and phase-field variables as unknowns.

36 MATERIALS SCIENCE↗
DOE OSTI2018

Similarity solution for a two-phase one-dimensional Stefan problem with a convective boundary condition and a mushy zone model

A two-phase solidification process for a one-dimensional semi-infinite material is considered. It is assumed that it is ensued from a constant bulk temperature present in the vicinity of the fixed boundary, which it is modelled through a convective condition (Robin condition). The interface between the two phases is idealized as a mushy region and it is represented following the model of Solomon, Wilson, and Alexiades. An exact similarity solution is obtained when a restriction on data is verified, and it is analysed the relation between the problem considered here and the problem with a temperature condition at the fixed boundary. Moreover, it is proved that the solution to the problem with the convective boundary condition converges to the solution to a problem with a temperature condition when the heat transfer coefficient at the fixed boundary goes to infinity, and it is given an estimation of the difference between these two solutions. Results in this article complete and improve the ones obtained in Tarzia (Comput Appl Math 9:201–211, 1990).

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2018

Classical Solvability of the Radial Viscous Fingering Problem in a Hele–Shaw Cell with Surface Tension

We consider the one-phase problem on radial viscous fingering structures in a Hele–Shaw cell with surface tension. This problem is a nonlinear free-boundary problem for elliptic equations. Unlike the Stefan problem for the heat equation, we deal with a problem of hydrodynamic type. We establish the classical solvability of the one-phase Hele–Shaw problem with radial geometry by using the same method as that used for the Stefan problem and justifying the vanishing coefficient of the time-derivative in the parabolic equation.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2021

Modeling polymer crystallisation induced by a moving heat sink

Recent experimental work has shown that polymer crystallisation can be used to “move” and organize nanoparticles (NP). As a first effort at modeling this situation, we consider the classical Stefan problem but with the modification that polymer crystallisation does not occur at a single temperature. Rather, the rate of crystallisation is proportional to its subcooling, and here we employ a form inspired by the classical Avrami model to describe this functional form. Our results for the movement of the polymer crystallisation front, as defined as the point where the crystallinity is 50%, closely track the results of the classical Stefan problem. Thus, at this level of approximation, the crystallisation kinetics of the polymer do not cause qualitative changes to the physics of this situation. Inspired by this fact we study the more interesting situation where the directional recrystallisation of a polymer melt is considered, e.g. , through the application of a moving heat sink over an initially molten polymer, reminiscent of a processing technique termed zone annealing. The polymer crystallisation shows that a steady state exists for a range of sink velocities. The solid–melt interface moves slightly ahead of the sink but at the same velocity. The steady-state distance between the sink and the interface decreases with increasing sink velocity – this is a consequence of the excess cooling provided by the sink over what is required to crystallise the melt. The most interesting new result is that the temperature of the crystal–melt interface decreases with increasing sink velocity. This is in line with the ansatz of Lovinger and Gryte who suggested that larger zone annealing velocities correspond to progressively larger effective undercoolings at which polymer crystallisation occurs.

36 MATERIALS SCIENCE↗
DOE OSTI2012

Solid–Liquid Phase Change Driven by Internal Heat Generation

This article presents results of solid-liquid phase change, the Stefan Problem, where melting is driven internal heat generation, in a cylindrical geometry. The comparison between a quasi-static analytical solution for Stefan numbers less than one and numerical solutions shows good agreement. The computational results of phase change with internal heat generation show how convection cells form in the liquid region. A scale analysis of the same problem shows four distinct regions of the melting process.

99 GENERAL AND MISCELLANEOUS↗
DOE OSTI2015

Numerical simulation of phase transition problems with explicit interface tracking

Phase change is ubiquitous in nature and industrial processes. Started from the Stefan problem, it is a topic with a long history in applied mathematics and sciences and continues to generate outstanding mathematical problems. For instance, the explicit tracking of the Gibbs dividing surface between phases is still a grand challenge. Our work has been motivated by such challenge and here we report on progress made in solving the governing equations of continuum transport in the presence of a moving interface by the front tracking method. The most pressing issue is the accounting of topological changes suffered by the interface between phases wherein break up and/or merge takes place. The underlying physics of topological changes require the incorporation of space-time subscales not at reach at the moment. Therefore we use heuristic geometrical arguments to reconnect phases in space. This heuristic approach provides new insight in various applications and it is extensible to include subscale physics and chemistry in the future. We demonstrate the method on applications such as simulating freezing, melting, dissolution, and precipitation. The later examples also include the coupling of the phase transition solution with the Navier-Stokes equations for the effect of flow convection.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2021

Physics-Informed Neural Networks for Heat Transfer Problems

Abstract Physics-informed neural networks (PINNs) have gained popularity across different engineering fields due to their effectiveness in solving realistic problems with noisy data and often partially missing physics. In PINNs, automatic differentiation is leveraged to evaluate differential operators without discretization errors, and a multitask learning problem is defined in order to simultaneously fit observed data while respecting the underlying governing laws of physics. Here, we present applications of PINNs to various prototype heat transfer problems, targeting in particular realistic conditions not readily tackled with traditional computational methods. To this end, we first consider forced and mixed convection with unknown thermal boundary conditions on the heated surfaces and aim to obtain the temperature and velocity fields everywhere in the domain, including the boundaries, given some sparse temperature measurements. We also consider the prototype Stefan problem for two-phase flow, aiming to infer the moving interface, the velocity and temperature fields everywhere as well as the different conductivities of a solid and a liquid phase, given a few temperature measurements inside the domain. Finally, we present some realistic industrial applications related to power electronics to highlight the practicality of PINNs as well as the effective use of neural networks in solving general heat transfer problems of industrial complexity. Taken together, the results presented herein demonstrate that PINNs not only can solve ill-posed problems, which are beyond the reach of traditional computational methods, but they can also bridge the gap between computational and experimental heat transfer.

Engineering↗
DOE OSTI2013

Non-isothermal melting of ice in the gas-diffusion layer of a proton-exchange-membrane fuel cell

Non-isothermal ice melting in the fibrous gas-diffusion layer (GDL) of a proton-exchange-membrane fuel cell (PEMFC) is investigated using differential scanning calorimetry (DSC). Non-isothermal ice-melting rates and ice-melting times are obtained from heat-flow measurements in water-saturated Toray GDLs at heating rates of 1, 2.5, 5, 10, and 25 K/min. In all cases, ice-melting times decrease nonlinearly with increasing heating rate. Nevertheless, melting temperatures remain at 272.9 ± 0.5 and 272.7 ± 0.4 K for bulk ice and ice within the GDL, respectively, reiterating that melting is thermodynamic-based at a rate limited by heat transfer. The slight GDL ice melting-point depression is consistent with the Gibbs-Thomson equation for equilibrium melting using an average pore diameter of 30 μm. Ice-melting endotherms are predicted from overall DSC energy balances coupled with a moving-boundary Stefan problem, where an ice-melting front within a GDL propagates with volume-averaged properties through an effective medium. Agreement between DSC experiment and theory is excellent. The proposed model accurately predicts ice-melting endotherms for Toray GDLs with two ice saturations and for bulk ice. Further, a pseudo-steady-state analysis obtains an analytical expression for ice-melting time, which is controlled by the time for heat addition to the propagating solid/liquid interface. Significantly, the new expression elucidates parameters controlling ice melting and allows for better design of both GDL materials and heating strategies to enhance the success of PEMFC cold-start.

25 ENERGY STORAGE↗
DOE OSTI2022

Measurement of Thermal Conductivity in a Supercooled Hydrogel-Salt Complex Near Its Phase Transition

Solid-liquid phase transitions, i.e. solidification processes, have applications in data storage, development of novel thermoelectric materials, cooling of microelectronic substrates and air conditioning condensers. Standard analyses of solidification (Stefan problem) assume constant thermal properties of the solid and liquid sides. It is not known how these properties change across the spatial transition interface, though most studies report a discontinuity in the solid and liquid properties through the transition temperature [1]. When the phase transition releases enthalpy, recent research has shown if the phonon or electron transport time is of the same order of magnitude as the time scale of the atomic transformation, this increases the heat capacity of the solid material at temperatures near the phase transition temperature [2]. A fundamental understanding of the phase transition may help shed light on the molecular origins of supercooling and spontaneous nucleation, which will help with applications involving them. We report studies of supercooling and nucleation of sodium sulfate decahydrate, a salt hydrate which is of recent interest in thermal storage, and of a hydrogel-sodium sulfate complex which shows limited supercooling. We will also report the measurement of their thermal properties during the phase transition process. This will be done with a hot-wire setup which produces small temperature changes of the order of ~1 C.

crystallization, phase transformation, thermal con↗
DOE OSTI2021

Boundary layer description of directional polymer crystallisation

Nearly fifty years ago Lovinger and Gryte suggested that the directional crystallization of a polymer was analogous to the quiescent isothermal crystallization experiment but at a supercooling where the crystal growth velocity was equal to the velocity of the moving front. Our experiments showed that this equivalence holds in a detailed manner at low directional velocities. To understand the underlying physics of these situations, we modeled the motion of a crystallization front in a liquid where the left side boundary is suddenly lowered below the melting point (Stefan's problem) but with the modification that the crystallization kinetics follow a version of the Avrami model. Our numerical results surprisingly showed that the results of the polymer analog track with the Stefan results which were derived for a simple liquid that crystallizes completely at its melting point; in particular, the position of the crystal growth-front evolved with time exactly as in the Stefan problem. The numerical solution also showed that the temperature in the immediate vicinity of the growth-front decreased with increasing front velocity, which is in line with Lovinger and Gryte's ansatz. To provide a clear theoretical understanding of these numerical results we derive a boundary layer solution to the governing coupled differential equations of the polymer problem. The analytical results are in agreement with our observations from experiments and numerical computations but show that this equivalence between the small molecule and polymer analog only holds in the limit where the crystallization enthalpy is much larger than the rate at which heat is conducted away in the polymer. In particular, in the context of the temperature profile, the enthalpy generated by the crystallisation process which is spread out over a narrow spatial region can be approximated as a point source whose location and temperature correspond to the Lovinger–Gryte ansatz.

36 MATERIALS SCIENCE↗
DOE OSTI2020

An implicit, sharp numerical treatment of viscous terms at arbitrarily shaped liquid-gas interfaces in evaporative flows

We report a novel methodology for accurate treatment of viscous terms in evaporation problems. The proposed scheme is an extension of the sharp viscous treatment of Kang et al. (2000) [7] to 3D phase change problems. To ensure accuracy and grid converging solutions, a new implicit approach to computing viscous fluxes across the phase interfaces is proposed, which was previously unavailable in fixed grid numerical schemes. Analytical relations were derived for the jump in velocity gradients across the 2D and 3D phase interfaces, an important constituent of the proposed scheme. The relations show a non-vanishing jump in the tangential gradients across the phase interface that are associated with evaporative flux and interfacial curvature. The proposed methodology demonstrated first order accuracy in canonical test cases. It is general and applicable to arbitrarily oriented interfaces, and can be readily implemented in existing evaporation flow solvers.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2016

Amplified charge and discharge rates in phase change materials for energy storage using spatially-enhanced thermal conductivity

Composites made with high thermal conductivity meshes embedded in phase change materials (PCMs) increase charge/discharge rates of latent heat energy storage systems. Here we study the benefits of spatially-dependent enhancements to thermal conductivity on the charge/discharge rates of PCMs in both one-dimensional Cartesian and one-dimensional cylindrical coordinates. Our nondimensionalized quasi-steady (Stefan number ≲ 0.1) solution indicates that the average charge (discharge) rate in a spatially-enhanced PCM outperforms the uniformly-enhanced case by maximizing the enhancement near the heat source and therein reducing the time averaged thermal resistance to melting (solidifying). Relative to a uniformly-enhanced thermal conductivity, the optimal charge/discharge rate enhancement is a modest 12% in one-dimensional Cartesian coordinates but as high as 140% in one-dimensional cylindrical coordinates. Our analytical solutions are a design guide for graded mesh structures that can be realized by advanced fabrication techniques such as additive manufacturing and applied in applications ranging from telecommunications to buildings, where PCMs are employed to harness rapidly varying energy sources.

25 ENERGY STORAGE↗