Abstract We analyze the consequences of the nonlinear terms in the heat-transport equation of the thermomass theory on heat pulses propagating in a nanowire in nonequilibrium situations. As a consequence of the temperature dependence of the speeds of propagation, in temperature ranges wherein the specific heat shows negligible variations, heat pulses will shrink (or extend) spatially, and will increase (or decrease) their average temperature when propagating along a temperature gradient. A comparison with the results predicted by a different theoretical proposal on the shape of a propagating heat pulse is made, too.
{"title":"Heat-pulse propagation along nonequilibrium nanowires in thermomass theory","authors":"A. Sellitto, P. Rogolino, I. Carlomagno","doi":"10.1515/caim-2016-0005","DOIUrl":"https://doi.org/10.1515/caim-2016-0005","url":null,"abstract":"Abstract We analyze the consequences of the nonlinear terms in the heat-transport equation of the thermomass theory on heat pulses propagating in a nanowire in nonequilibrium situations. As a consequence of the temperature dependence of the speeds of propagation, in temperature ranges wherein the specific heat shows negligible variations, heat pulses will shrink (or extend) spatially, and will increase (or decrease) their average temperature when propagating along a temperature gradient. A comparison with the results predicted by a different theoretical proposal on the shape of a propagating heat pulse is made, too.","PeriodicalId":37903,"journal":{"name":"Communications in Applied and Industrial Mathematics","volume":"7 1","pages":"39 - 55"},"PeriodicalIF":1.3,"publicationDate":"2016-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/caim-2016-0005","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"67374926","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract Complex macroscopic systems (like for instance those encountered in nanotechnology and biology) need to be investigated in a family of mesoscopic theories involving varying amount of details. In this paper we formulate a general thermodynamics providing a universal framework for such multiscale viewpoint of mesoscopic dynamics. We then discuss its role in making extensions (i.e. in lifting a mesoscopic theory to a more microscopic level that involves more details).
{"title":"Role of thermodynamics in extensions of mesoscopic dynamical theories","authors":"M. Grmela","doi":"10.1515/caim-2016-0006","DOIUrl":"https://doi.org/10.1515/caim-2016-0006","url":null,"abstract":"Abstract Complex macroscopic systems (like for instance those encountered in nanotechnology and biology) need to be investigated in a family of mesoscopic theories involving varying amount of details. In this paper we formulate a general thermodynamics providing a universal framework for such multiscale viewpoint of mesoscopic dynamics. We then discuss its role in making extensions (i.e. in lifting a mesoscopic theory to a more microscopic level that involves more details).","PeriodicalId":37903,"journal":{"name":"Communications in Applied and Industrial Mathematics","volume":"7 1","pages":"56 - 80"},"PeriodicalIF":1.3,"publicationDate":"2016-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/caim-2016-0006","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"67375496","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract In this paper, a generalized heat transport equation including relaxational, nonlocal and nonlinear effects is provided, which contains diverse previous phenomenological models as particular cases. The aim of the present work is to establish an extended irreversible thermodynamic framework, with generalized expressions of entropy and entropy flux. Nonlinear thermodynamic force-flux relation is proposed as an extension of the usual linear one, giving rise to the nonlinear terms in the heat transport equation and ensuring compatibility with the second law. Several previous results are recovered in the linear case, and some additional results related to nonlinear terms are also obtained.
{"title":"Thermodynamic framework for a generalized heat transport equation","authors":"Yangyu Guo, Moran Wang","doi":"10.1515/caim-2016-0012","DOIUrl":"https://doi.org/10.1515/caim-2016-0012","url":null,"abstract":"Abstract In this paper, a generalized heat transport equation including relaxational, nonlocal and nonlinear effects is provided, which contains diverse previous phenomenological models as particular cases. The aim of the present work is to establish an extended irreversible thermodynamic framework, with generalized expressions of entropy and entropy flux. Nonlinear thermodynamic force-flux relation is proposed as an extension of the usual linear one, giving rise to the nonlinear terms in the heat transport equation and ensuring compatibility with the second law. Several previous results are recovered in the linear case, and some additional results related to nonlinear terms are also obtained.","PeriodicalId":37903,"journal":{"name":"Communications in Applied and Industrial Mathematics","volume":"7 1","pages":"167 - 176"},"PeriodicalIF":1.3,"publicationDate":"2016-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/caim-2016-0012","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"67375232","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract The purpose of this work is to present a study on heat conduction in systems that are composed out of spherical and cylindrical micro- and nanoparticles dispersed in a bulk matrix. Special emphasis is put on the dependence of the effective heat conductivity on various selected parameters as particle size and also its shape, surface specularity and density, including particle-matrix interaction. The heat transfer at nanometric scales is modelled using extended irreversible thermodynamics, whose main feature is to elevate the heat flux vector to the status of independent variable. The model is illustrated by a Copper-Silicium (Cu-Si) system. It is shown that all the investigated parameters have a considerable influence, the particle size being especially useful to either increase or decrease the effective thermal conductivity.
{"title":"Heat transfer at nanometric scales described by extended irreversible thermodynamics","authors":"H. Machrafi","doi":"10.1515/caim-2016-0013","DOIUrl":"https://doi.org/10.1515/caim-2016-0013","url":null,"abstract":"Abstract The purpose of this work is to present a study on heat conduction in systems that are composed out of spherical and cylindrical micro- and nanoparticles dispersed in a bulk matrix. Special emphasis is put on the dependence of the effective heat conductivity on various selected parameters as particle size and also its shape, surface specularity and density, including particle-matrix interaction. The heat transfer at nanometric scales is modelled using extended irreversible thermodynamics, whose main feature is to elevate the heat flux vector to the status of independent variable. The model is illustrated by a Copper-Silicium (Cu-Si) system. It is shown that all the investigated parameters have a considerable influence, the particle size being especially useful to either increase or decrease the effective thermal conductivity.","PeriodicalId":37903,"journal":{"name":"Communications in Applied and Industrial Mathematics","volume":"7 1","pages":"177 - 195"},"PeriodicalIF":1.3,"publicationDate":"2016-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/caim-2016-0013","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"67375328","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract The thermal diode is the fundamental device for phononics. There are various mechanisms for thermal rectification, e.g. different temperature dependent thermal conductivity of two ends, asymmetric interfacial resistance, and nonlocal behavior of phonon transport in asymmetric structures. The phonon hydrodynamics and thermomass theory treat the heat conduction in a fluidic viewpoint. The phonon gas flowing through the media is characterized by the balance equation of momentum, like the Navier-Stokes equation for fluid mechanics. Generalized heat conduction law thereby contains the spatial acceleration (convection) term and the viscous (Laplacian) term. The viscous term predicts the size dependent thermal conductivity. Rectification appears due to the MFP supersession of phonons. The convection term also predicts rectification because of the inertia effect, like a gas passing through a nozzle or diffuser.
{"title":"Thermal rectification based on phonon hydrodynamics and thermomass theory","authors":"Yuan Dong","doi":"10.1515/caim-2016-0004","DOIUrl":"https://doi.org/10.1515/caim-2016-0004","url":null,"abstract":"Abstract The thermal diode is the fundamental device for phononics. There are various mechanisms for thermal rectification, e.g. different temperature dependent thermal conductivity of two ends, asymmetric interfacial resistance, and nonlocal behavior of phonon transport in asymmetric structures. The phonon hydrodynamics and thermomass theory treat the heat conduction in a fluidic viewpoint. The phonon gas flowing through the media is characterized by the balance equation of momentum, like the Navier-Stokes equation for fluid mechanics. Generalized heat conduction law thereby contains the spatial acceleration (convection) term and the viscous (Laplacian) term. The viscous term predicts the size dependent thermal conductivity. Rectification appears due to the MFP supersession of phonons. The convection term also predicts rectification because of the inertia effect, like a gas passing through a nozzle or diffuser.","PeriodicalId":37903,"journal":{"name":"Communications in Applied and Industrial Mathematics","volume":"7 1","pages":"26 - 38"},"PeriodicalIF":1.3,"publicationDate":"2016-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/caim-2016-0004","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"67374952","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract We provide an overview on the problem of modeling heat transport at nanoscale and in far-from-equilibrium processes. A survey of recent results is summarized, and a conceptual discussion of them in the framework of Extended Irreversible Thermodynamics is developed.
{"title":"Constitutive equations for heat conduction in nanosystems and nonequilibrium processes: an overview","authors":"D. Jou, V. Cimmelli","doi":"10.1515/caim-2016-0014","DOIUrl":"https://doi.org/10.1515/caim-2016-0014","url":null,"abstract":"Abstract We provide an overview on the problem of modeling heat transport at nanoscale and in far-from-equilibrium processes. A survey of recent results is summarized, and a conceptual discussion of them in the framework of Extended Irreversible Thermodynamics is developed.","PeriodicalId":37903,"journal":{"name":"Communications in Applied and Industrial Mathematics","volume":"7 1","pages":"196 - 222"},"PeriodicalIF":1.3,"publicationDate":"2016-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/caim-2016-0014","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"67376010","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract Transport phenomena in silicon nanowires with different cross-section are investigated using an Extended Hydrodynamic model, coupled to the Schrödinger-Poisson system. The model has been formulated by closing the moment system derived from the Boltzmann equation on the basis of the maximum entropy principle of Extended Thermodynamics, obtaining explicit closure relations for the high-order fluxes and the production terms. Scattering of electrons with acoustic and non polar optical phonons have been taken into account. The bulk mobility is evaluated for square and equilateral triangle cross-sections of the wire.
{"title":"Electron transport in silicon nanowires having different cross-sections","authors":"O. Muscato, T. Castiglione","doi":"10.1515/caim-2016-0003","DOIUrl":"https://doi.org/10.1515/caim-2016-0003","url":null,"abstract":"Abstract Transport phenomena in silicon nanowires with different cross-section are investigated using an Extended Hydrodynamic model, coupled to the Schrödinger-Poisson system. The model has been formulated by closing the moment system derived from the Boltzmann equation on the basis of the maximum entropy principle of Extended Thermodynamics, obtaining explicit closure relations for the high-order fluxes and the production terms. Scattering of electrons with acoustic and non polar optical phonons have been taken into account. The bulk mobility is evaluated for square and equilateral triangle cross-sections of the wire.","PeriodicalId":37903,"journal":{"name":"Communications in Applied and Industrial Mathematics","volume":"7 1","pages":"25 - 8"},"PeriodicalIF":1.3,"publicationDate":"2016-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/caim-2016-0003","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"67374705","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract The paper deals with the meaning of non-equilibrium temperatures in nanosystems with an internal variable, describing defects inside them, and implications on heat transport. In equilibrium all definitions of temperature lead to the same value, but in nonequilibrium steady states they lead to different values, giving information on different degrees of freedom. We discuss the caloric and entropic non-equilibrium temperatures and the relations among them, in defective nanosystems (crystals with dislocations or porous channels, carbon nanotubes in a solid matrix and so on), crossed by an external energy flux. Here, we present a model for nanocrystals with dislocation defects submitted to an external energy flux. The dislocations may have a strong influence on the effective thermal conductivity, and their own dynamics may be coupled in relevant way to the heat flux dynamics. In the linear case the constitutive relations, the rate equations for the internal variable and the heat flux are worked out and a generalized telegraphic heat equation is derived in the anisotropic and isotropic case, describing the thermal disturbances with finite velocity.
{"title":"Non-equilibrium temperatures and heat transport in nanosystems with defects, described by a tensorial internal variable","authors":"L. Restuccia","doi":"10.1515/caim-2016-0007","DOIUrl":"https://doi.org/10.1515/caim-2016-0007","url":null,"abstract":"Abstract The paper deals with the meaning of non-equilibrium temperatures in nanosystems with an internal variable, describing defects inside them, and implications on heat transport. In equilibrium all definitions of temperature lead to the same value, but in nonequilibrium steady states they lead to different values, giving information on different degrees of freedom. We discuss the caloric and entropic non-equilibrium temperatures and the relations among them, in defective nanosystems (crystals with dislocations or porous channels, carbon nanotubes in a solid matrix and so on), crossed by an external energy flux. Here, we present a model for nanocrystals with dislocation defects submitted to an external energy flux. The dislocations may have a strong influence on the effective thermal conductivity, and their own dynamics may be coupled in relevant way to the heat flux dynamics. In the linear case the constitutive relations, the rate equations for the internal variable and the heat flux are worked out and a generalized telegraphic heat equation is derived in the anisotropic and isotropic case, describing the thermal disturbances with finite velocity.","PeriodicalId":37903,"journal":{"name":"Communications in Applied and Industrial Mathematics","volume":"13 2 1","pages":"81 - 97"},"PeriodicalIF":1.3,"publicationDate":"2016-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/caim-2016-0007","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"67375190","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract In this paper we address the problem of optimization of the so called supercooling effect in thermoelectric nanoscaled layers. The effect arises when a short term electric pulse is applied to the layer. The analysis is based on constitutive equations of the Maxwell-Cattaneo type describing the time evolution of dissipative flows with the thermal and electric conductivities depending on the width of the layer. This introduces memory and nonlocal effects and consequently a wave-like behaviour of system’s temperature. We study the effects of the shape of the electric pulse on the maximum diminishing of temperature by applying pulses of the form ta with a a power going from 0 to 10. Pulses with a a fractionary number perform better for nanoscaled devices whereas those with a bigger than unity do it for microscaled ones. We also find that the supercooling effect is improved by a factor of 6.6 over long length scale devices in the best performances and that the elapsed supercooling time for the nanoscaled devices equals the best of the microscaled ones. We use the spectral methods of solution which assure a well representation of wave behaviour of heat and electric charge in short time scales given their spectral convergence.
{"title":"Optimization of supercooling effect in nanoscaled thermoelectric layers","authors":"I. Rivera, A. Figueroa, F. Vázquez","doi":"10.1515/caim-2016-0008","DOIUrl":"https://doi.org/10.1515/caim-2016-0008","url":null,"abstract":"Abstract In this paper we address the problem of optimization of the so called supercooling effect in thermoelectric nanoscaled layers. The effect arises when a short term electric pulse is applied to the layer. The analysis is based on constitutive equations of the Maxwell-Cattaneo type describing the time evolution of dissipative flows with the thermal and electric conductivities depending on the width of the layer. This introduces memory and nonlocal effects and consequently a wave-like behaviour of system’s temperature. We study the effects of the shape of the electric pulse on the maximum diminishing of temperature by applying pulses of the form ta with a a power going from 0 to 10. Pulses with a a fractionary number perform better for nanoscaled devices whereas those with a bigger than unity do it for microscaled ones. We also find that the supercooling effect is improved by a factor of 6.6 over long length scale devices in the best performances and that the elapsed supercooling time for the nanoscaled devices equals the best of the microscaled ones. We use the spectral methods of solution which assure a well representation of wave behaviour of heat and electric charge in short time scales given their spectral convergence.","PeriodicalId":37903,"journal":{"name":"Communications in Applied and Industrial Mathematics","volume":"7 1","pages":"110 - 98"},"PeriodicalIF":1.3,"publicationDate":"2016-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/caim-2016-0008","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"67375535","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract In this paper the optimal control of alignment models composed by a large number of agents is investigated in presence of a selective action of a controller, acting in order to enhance consensus. Two types of selective controls have been presented: an homogeneous control filtered by a selective function and a distributed control active only on a selective set. As a first step toward a reduction of computational cost, we introduce a model predictive control (MPC) approximation by deriving a numerical scheme with a feedback selective constrained dynamics. Next, in order to cope with the numerical solution of a large number of interacting agents, we derive the mean-field limit of the feedback selective constrained dynamics, which eventually will be solved numerically by means of a stochastic algorithm, able to simulate effciently the selective constrained dynamics. Finally, several numerical simulations are reported to show the effciency of the proposed techniques.
{"title":"Selective model-predictive control for flocking systems","authors":"G. Albi, L. Pareschi","doi":"10.2478/caim-2018-0009","DOIUrl":"https://doi.org/10.2478/caim-2018-0009","url":null,"abstract":"Abstract In this paper the optimal control of alignment models composed by a large number of agents is investigated in presence of a selective action of a controller, acting in order to enhance consensus. Two types of selective controls have been presented: an homogeneous control filtered by a selective function and a distributed control active only on a selective set. As a first step toward a reduction of computational cost, we introduce a model predictive control (MPC) approximation by deriving a numerical scheme with a feedback selective constrained dynamics. Next, in order to cope with the numerical solution of a large number of interacting agents, we derive the mean-field limit of the feedback selective constrained dynamics, which eventually will be solved numerically by means of a stochastic algorithm, able to simulate effciently the selective constrained dynamics. Finally, several numerical simulations are reported to show the effciency of the proposed techniques.","PeriodicalId":37903,"journal":{"name":"Communications in Applied and Industrial Mathematics","volume":"9 1","pages":"21 - 4"},"PeriodicalIF":1.3,"publicationDate":"2016-03-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"69187232","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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