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Temperature and heat conduction beyond Fourier law Peter Ván HAS, RIPNP, Department of Theoretical Physics and BME, Department of Energy Engineering 1.Introduction 2.Theories −Cattaneo-Vernotte −Guyer-Krumhansl −Jeffreys type 3.Experiments Common work with B. Czél, T. Fülöp, Gy. Gróf and J. Verhás

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Heat exchange experiment

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Model 1 (Newton, 3 parameters): EstimateStd. Error l T0T T ini

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Model 1 (Newton, 3 parameters): Model 2 (Extended Newton, 5 parameters): EstimateStd. Error l T0T T ini

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Model 1 (Newton, 3 parameters): Model 2 (Extended Newton, 5 parameters): EstimateStd. Error l T0T T ini EstimateStd. Error l T0T T ini vT ini

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Why? − two step process T0T0 T1T1 T αβ

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Oscillations in heat exchange: − parameters model - values micro - interpretation − macro-meso mechanism

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Fourier – local equilibrium (Eckart, 1940) Entropy production: Constitutive equations (isotropy): Fourier law

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Constitutive equations (isotropy): Cattaneo-Vernotte Cattaneo-Vernotte equation (Gyarmati, 1977, modified) Entropy production:

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Heat conduction constitutive equations Fourier (1822) Cattaneo (1948), (Vernotte (1958)) Guyer and Krumhansl (1966) Jeffreys type (Joseph and Preziosi, 1989)) Green-Naghdi type (1991) there are more…

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Thermodynamic approach vectorial internal variable and current multiplier (Nyíri 1990, Ván 2001) Entropy production:

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Constitutive equations (isotropy):

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1+1 D: Fourier Cattaneo-Vernotte Guyer and Krumhansl and Jeffreys type Green-Naghdi type

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Calculations

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1+1D: a) Jeffreys-type equation – heat separation Jeffreys ‘Meso’ models

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Taylor series: b) Jeffreys-type equation – dual phase lag Jeffreys This is unacceptable.

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1+1D: c) Jeffreys-type equation – two steps Jeffreys

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Waves +… in heat conduction: − thermodynamic frame nonlocal hierarchy (length scales) − macro-meso mechanisms frame is satisfied

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MemoryNonlocalityObjectivityThermo- dynamics Fourier no researchok Cattaneo-Vernotte yesnoresearchok Guyer-Krumhansl Gyarmati-Nyíri, (linearized Boltzmann) yes ?ok Jeffreys type 1. internal variable, 2.heat separation 3. dual phase lag, 4. two steps yes ?ok Ballistic-diffusive (Boltzmann split) yes ?? Heat conduction equations

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Experiments Homogeneous inner structure – metals - typical relaxation times: = s - Cattaneo-Vernotte is accepted: ballistic phonons (nano- and microtechnology?) Inhomogeneous inner structure - typical relaxation times: = s - experiments are not conclusive

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Resistance wireThermocouple Kaminski, 1990 Particulate materials: sand, glass balottini, ion exchanger = s

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Mitra-Kumar-Vedavarz-Moallemi, 1995 Processed frozen meat: = s

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Scott-Tilahun-Vick, 2009 − repeating Kaminski and Mitra et. al. − no effect Herwig-Beckert, 2000 ˗ sand, different setup ˗ no effect Roetzel-Putra-Das, 2003 − similar to Kaminski and Mitra et. al. − small effect

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Summary and conclusions − Inertial and gradient effects in heat conduction − Internal variables versus substructures (macro – micro) black box – universality − No experiments for gradient effects (supressed waves?)

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1D: Ballistic, analytic solution Ballistic-diffusive equation (Chen, 2001) Boltzmann felbontás diffusive ?

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