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Temperature and he at conduction beyond Fourier law Peter Ván

Temperature and he at conduction beyond Fourier law Peter Ván HAS, RIPNP, Department of Theoretical Physics and BME, Department of Energy Engineering. Introduction Theories Cattaneo-Vernotte Guyer-Krumhansl Jeffreys type Experiments.

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Temperature and he at conduction beyond Fourier law Peter Ván

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  1. Temperature and heatconduction beyond Fourier law Peter Ván HAS, RIPNP, Department of TheoreticalPhysics and BME, Department of EnergyEngineering • Introduction • Theories • Cattaneo-Vernotte • Guyer-Krumhansl • Jeffreys type • Experiments Common work with B. Czél, T. Fülöp , Gy. Gróf and J. Verhás

  2. Heat exchange experiment http://remotelab.energia.bme.hu/index.php?page=thermocouple_remote_desc&lang=en

  3. Model 1 (Newton, 3 parameters):

  4. Model 1 (Newton, 3 parameters): Model 2 (Extended Newton, 5 parameters):

  5. Model 1 (Newton, 3 parameters): Model 2 (Extended Newton, 5 parameters):

  6. Why? −two step process α β T0 T1 T

  7. Oscillationsinheatexchange: • − parameters • model - values • micro - interpretation • − macro-mesomechanism

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

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

  10. 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…

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

  12. Constitutive equations (isotropy):

  13. 1+1 D: Fourier Cattaneo-Vernotte Guyer and Krumhansl and Jeffreys type Green-Naghdi type

  14. Calculations

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

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

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

  18. Waves+…inheatconduction: • −thermodynamicframe • nonlocalhierarchy (lengthscales) • −macro-mesomechanisms • frame is satisfied

  19. Heat conduction equations

  20. Experiments Homogeneous inner structure – metals - typical relaxation times:  = 10-13- 10-17 s - Cattaneo-Vernotte is accepted: ballistic phonons (nano- and microtechnology?) Inhomogeneousinnerstructure - typical relaxation times:  = 10-3- 100s - experiments are not conclusive

  21. Kaminski, 1990 Resistancewire Thermocouple Particulatematerials: sand, glassbalottini, ion exchanger  = 20-60 s

  22. Mitra-Kumar-Vedavarz-Moallemi, 1995 Processed frozen meat:  = 20-60 s

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

  24. Summary and conclusions • Inertial and gradient effects • in heat conduction • Internal variables versus substructures • (macro – micro) • blackbox– universality • No experiments for gradient effects • (supressedwaves?)

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

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