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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.

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Presentation on theme: "Temperature and heat conduction beyond Fourier law Peter Ván HAS, RIPNP, Department of Theoretical Physics and BME, Department of Energy Engineering 1.Introduction."— Presentation transcript:

1 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

2 Heat exchange experiment

3

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

5

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

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8 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

9 Why? − two step process T0T0 T1T1 T αβ

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

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

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

13 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…

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

15 Constitutive equations (isotropy):

16 1+1 D: Fourier Cattaneo-Vernotte Guyer and Krumhansl and Jeffreys type Green-Naghdi type

17 Calculations

18 1+1D: a) Jeffreys-type equation – heat separation Jeffreys ‘Meso’ models

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

20 1+1D: c) Jeffreys-type equation – two steps Jeffreys

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

22 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

23 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

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

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

26 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

27 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?)

28 1D: Ballistic, analytic solution Ballistic-diffusive equation (Chen, 2001) Boltzmann felbontás diffusive ?


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