Transient Heat Flow in a One-component Lennard-Jones/spline Fluid. A Non-equilibrium Molecular Dynamics Study
Department of Chemistry, Norwegian University of Science and Technology
N-7491 Trondheim, Norway
E-mail: bjorn.hafskjold@ntnu.no
Received:
Received: 20 February 2017; revised: 27 March 2017; accepted: 02 April 2017; published online: 09 September 2017
DOI: 10.12921/cmst.2017.00000012
Abstract:
A one-component Lennard-Jones/spline fluid at equilibrium was perturbed by a sudden change of the temperature at one of the system’s boundaries. The system’s response was determined by non-equilibrium molecular dynamics (NEMD). The results show that heat was transported by two mechanisms: (1) Heat diffusion and conduction, and (2) energy dissipation associated with the propagation of a pressure (shock) wave. These two processes occur at different time scales, which makes it possible to separate them in one single NEMD run. The system was studied in gas, liquid, and supercritical states with various forms and strengths of the thermal perturbation. Near the heat source, heat was transported according to the transient heat equation. In addition, there was a much faster heat transport, correlated with a pressure wave. This second mechanism was similar to the thermo-mechanical “piston effect” in near-critical fluids and could not be explained by the Joule-Thomson effect. For strong perturbations, the pressure wave travelled faster than the speed of sound, turning it into a shock wave. The system’s local measurable heat flux was found to be consistent with Fourier’s law near the heat source, but not in the wake of the shock. The NEMD results were, however, consistent with the Cattaneo-Vernotte model. The system was found to be in local equilibrium in the transient phase, even with very strong perturbations, except for a low-density gas. For dense systems, we did not find that the local equilibrium assumption used in classical irreversible thermodynamics is inconsistent with the Cattaneo-Vernotte model.
Key words:
fluids, Joule-Thomson effect, molecular dynamics, piston effect, shock waves, transient heat flow
References:
[1] B.J. Alder and T. Wainwright, Phase Transition for a Hard Sphere System, J. Chem. Phys. 27, pp. 1208–1209, 1959.
[2] W.G. Hoover, Nonequilibrium Molecular Dynamics, Ann. Rev. Phys. Chem. 34, pp. 103–127, 1983.
[3] R.K. Eckhoff, Water vapour explosions – A brief review, J. Loss Prevention in the Process Industries 40, pp. 188–198, 2016.
[4] L.A. Dombrovsky, Steam explosions in nuclear reactors: Droplets of molten steel vs core melt droplets, Int. J. Heat and Mass Transfer 107, pp. 432–438, 2017.
[5] Y. Liu, T. Olewski and L. N. Véchot, Modeling of a cryogenic liquid pool boiling by CFD simulation, J. Loss Prevention in the Process Industries 35, pp. 125–134, 2015.
[6] K.A. Bhaskaran and P. Roth, The shock tube as wave reactor for kinetic studies and material systems, Progress in energy and combustion science 28, pp. 151–192, 2002.
[7] K. Thoma, U. Hornemann, M. Sauer and E. Schneider, Shock waves – Phenomenology, experimental, and numerical simulation, Meteorites & Planetary Science 40, pp. 1283–1298, 2005. [8] L.D. Landau and E. M. Lifshitz, Fluid Mechanics, Oxford: Pergamon Press, 1959.
[9] J.O. Hirschfelder, C. F. Curtiss and R. B. Bird, Molecular theory of Gases and Liquids, New York: John Wiley & Sons, 1954.
[10] W.G. Hoover and C. G. Hoover, Simulations and Control of Chaotic Nonequilibrium Systems, Advanced Series in Nonlinear Dynamics, Vol 27, New Jersey: World Scientific, 2015.
[11] W.G. Hoover, Structure of a Shock-Wave Front in a Liquid, Phys. Rev. Lett. 42, pp. 1531–1534, 1979.
[12] W.G. Hoover, H. C. G. and F. J. Uribe, Flexible Macroscopic Models for Dense-Fluid Shockwaves: Partitioning Heat and Work; Delaying Stress and Heat Flux; Two-Temperature Thermal Relaxation, arcXiv, p. 1005.1525v1, 2010. [13] W.G. Hoover and C. G. Hoover, Tensor Temperature and Shockwave Stability in a Strong Two-Dimensional Shockwave, arcXiv, p. 0905.1913v2, 2013.
[14] W.G. Hoover, C. G. Hoover and K. Travis, Shock-Wave Compression and Joule-Thomson Expansion, Phys. Rev. Lett. 112, p. 144504, 2014.
[15] B.L. Holian, Atomistic computer simulations of shock waves, Shock Waves 5, pp. 149–157, 1995.
[16] B.L. Holian and M. Mareschal, Heat-flow equation motivated by the ideal-gas shock wave, Phys. Rev. E 82, p. 026707, 2010.
[17] D.-S. Tang, Y.-C. Hua, B.-D. Nie and B.-Y. Cao, Phonon wave propagation in ballistic-diffusive regime, J. Applied Phys. 119, p. 124301, 2016.
[18] B.L. Holian, W. G. Hoover, B. Moran and G. K. Straub, Shock-wave structure via nonequilibrium molecular dynamics and Navier-Stokes continuum mechanics, Phys. Rev. A 22, pp. 2798–2808, 1980.
[19] W.J. M. Rankine, On the thermodynamic theory of waves of finite longitudinal distrubances, Phil. Trans. Roy. Soc. London 160, pp. 277–288, 1870.
[20] H.Hugoniot,Mémoiresurlapropagationdesmouvementsdanslescorpsetspécialementdanslesgazparfaits(premicˇrepartie),J.ÉcolePolytechnique58,pp.1–125,1887.
[21] B. Zappoli, D. Beysens and Y. Garrabos, Heat Transfers and Related Effects in Supercritical Fluids, London: Springer, 2015.
[22] L. Chen, Microchannel Flow Dynamics and Heat Transfer of Near-Critical Fluid, Singapore: Springer, 2017.
[23] A. Onuki, H. Hao and R. A. Ferrell, Fast adiabatic equilibration in a single-component fluid near the liquid-vapor critical point, Phys. Rev. A 41, pp. 2256–2259, 1990.
[24] T. Ikeshoji and B. Hafskjold, Non-equilibrium molecular dynamics calculation of heat conduction in liquid and through liquid-gas interface, Mol. Phys. 81, pp. 251–261, 1994.
[25] C. Cattaneo, Sulla conduzione del calore, Atti del Semin. Mat. e Fis. Uni. Modena 3, pp. 83–101, 1948.
[26] P. Vernotte, Les paradoxes de la théorie continue de l’équation de la chaleur, C. R. Hebd. Sceances Acad. Sci. 246, pp. 3154–3155, 1958.
[27] D. D. Joseph and L. Preziosi, Heat waves, Rev. Mod. Phys. 61, pp. 41–73, 1989.
[28] B.L. Holian and D. J. Evans, Shear viscosities away from the melting line: A comparison of equilibrium and nonequilibrium molecular dynamics, J. Chem. Phys. 78, pp. 5147–5150, 1983.
[29] D.M. Heyes, The Lennard-Jones Fluid i the Liquid-Vapour Region, CMST 21, no. 4, pp. 169–179, 2015.
[30] B. Hafskjold, T. Ikeshoji and S. K. Ratkje, On the molecular mechanism of thermal diffusion in liquids, Mol. Phys. 80, pp. 1389–1412, 1993.
[31] S. Kjelstrup, D. Bedeaux, E. Johannesen and J. Gross, Non-Equilibrium Thermodynamics for Engineers, New Jersey: World Scientific, 2010.
[32] D. J. Evans and G. P. Morriss, Statistical Mechanics of Nonequilibrium Liquids, Canberra: ANU E Press, 2007.
[33] Y. Huang and H. H. Bau, Thermoacoustic waves in a semi-infinite medium, Int. J. Heat and Mass Transfer 38, no. 8, pp. 1329–1345, 1995.
[34] A. Nakano, Studies on piston and soret effects in a binary mixture supercritical fluid, Int. J. Heat and Mass Transfer 50, pp. 4678–4687, 2007.
[35] B. L. Holian, Modeling shock-wave deformation via molecular dynamics, Phys. Rev. A 37, pp. 2562–2568, 1988.
[36] B. Vick and M. N. Özisik, Growth and Decay of a Thermal Pulse Predicted by the Hyperbolic Heat Conduction Equation, Trans. ASME 105, pp. 902–907, 1983.
[37] B. Hafskjold and S. K. Ratkje, Criteria for Local Equilibrium in a System with Transport of Heat and Mass, J. Stat. Phys. 78, pp. 463–494, 1995.
[38] Y.-K. Guo, Z.-Y. Guo and X.-G. Liang, Three-Dimensional Molecular Dynamics Simulation on Heat Propagation in Liquid Argon, Chin. Phys. Lett. 18, pp. 71–73, 2001.
[39] Z. Guo, D. Xiong and Z. Li, Molecular Dynamics Simulation of Heat Propagation in Liquid Argon, Tsinghua Sci. Tech. 2, no. 2, pp. 613–618, 1997.
[40] H. Boukari, J. N. Shaumeyer, M. E. Briggs and R. W. Gammon, Critical speeding up in pure fluids, Phys. Rev. A 41, no. 4, pp. 2260–2263, 1990.
[41] A. Onuki, Thermoacoustic effects in supercritical fluids near the critical point: Resonance, piston effect, and acoustic emission and reflection, Phys. Rev. E 76, p. 061126, 2007.
[42] G. Lebon and D. Jou, Early history of extended irreversible termodynamics (1953–1983): An exploration beyond local equilibrium and classical transport theory, Eur. Phys. J. H 40, pp. 205–240, 2015.
[43] K. S. Glavatsky, Local equilibrium and the second law of thermodynamics for irreversible systems with thermodynamic inertia, J. Chem. Phys. 143, pp. 164101-1–11, 2015.6
A one-component Lennard-Jones/spline fluid at equilibrium was perturbed by a sudden change of the temperature at one of the system’s boundaries. The system’s response was determined by non-equilibrium molecular dynamics (NEMD). The results show that heat was transported by two mechanisms: (1) Heat diffusion and conduction, and (2) energy dissipation associated with the propagation of a pressure (shock) wave. These two processes occur at different time scales, which makes it possible to separate them in one single NEMD run. The system was studied in gas, liquid, and supercritical states with various forms and strengths of the thermal perturbation. Near the heat source, heat was transported according to the transient heat equation. In addition, there was a much faster heat transport, correlated with a pressure wave. This second mechanism was similar to the thermo-mechanical “piston effect” in near-critical fluids and could not be explained by the Joule-Thomson effect. For strong perturbations, the pressure wave travelled faster than the speed of sound, turning it into a shock wave. The system’s local measurable heat flux was found to be consistent with Fourier’s law near the heat source, but not in the wake of the shock. The NEMD results were, however, consistent with the Cattaneo-Vernotte model. The system was found to be in local equilibrium in the transient phase, even with very strong perturbations, except for a low-density gas. For dense systems, we did not find that the local equilibrium assumption used in classical irreversible thermodynamics is inconsistent with the Cattaneo-Vernotte model.
Key words:
fluids, Joule-Thomson effect, molecular dynamics, piston effect, shock waves, transient heat flow
References:
[1] B.J. Alder and T. Wainwright, Phase Transition for a Hard Sphere System, J. Chem. Phys. 27, pp. 1208–1209, 1959.
[2] W.G. Hoover, Nonequilibrium Molecular Dynamics, Ann. Rev. Phys. Chem. 34, pp. 103–127, 1983.
[3] R.K. Eckhoff, Water vapour explosions – A brief review, J. Loss Prevention in the Process Industries 40, pp. 188–198, 2016.
[4] L.A. Dombrovsky, Steam explosions in nuclear reactors: Droplets of molten steel vs core melt droplets, Int. J. Heat and Mass Transfer 107, pp. 432–438, 2017.
[5] Y. Liu, T. Olewski and L. N. Véchot, Modeling of a cryogenic liquid pool boiling by CFD simulation, J. Loss Prevention in the Process Industries 35, pp. 125–134, 2015.
[6] K.A. Bhaskaran and P. Roth, The shock tube as wave reactor for kinetic studies and material systems, Progress in energy and combustion science 28, pp. 151–192, 2002.
[7] K. Thoma, U. Hornemann, M. Sauer and E. Schneider, Shock waves – Phenomenology, experimental, and numerical simulation, Meteorites & Planetary Science 40, pp. 1283–1298, 2005. [8] L.D. Landau and E. M. Lifshitz, Fluid Mechanics, Oxford: Pergamon Press, 1959.
[9] J.O. Hirschfelder, C. F. Curtiss and R. B. Bird, Molecular theory of Gases and Liquids, New York: John Wiley & Sons, 1954.
[10] W.G. Hoover and C. G. Hoover, Simulations and Control of Chaotic Nonequilibrium Systems, Advanced Series in Nonlinear Dynamics, Vol 27, New Jersey: World Scientific, 2015.
[11] W.G. Hoover, Structure of a Shock-Wave Front in a Liquid, Phys. Rev. Lett. 42, pp. 1531–1534, 1979.
[12] W.G. Hoover, H. C. G. and F. J. Uribe, Flexible Macroscopic Models for Dense-Fluid Shockwaves: Partitioning Heat and Work; Delaying Stress and Heat Flux; Two-Temperature Thermal Relaxation, arcXiv, p. 1005.1525v1, 2010. [13] W.G. Hoover and C. G. Hoover, Tensor Temperature and Shockwave Stability in a Strong Two-Dimensional Shockwave, arcXiv, p. 0905.1913v2, 2013.
[14] W.G. Hoover, C. G. Hoover and K. Travis, Shock-Wave Compression and Joule-Thomson Expansion, Phys. Rev. Lett. 112, p. 144504, 2014.
[15] B.L. Holian, Atomistic computer simulations of shock waves, Shock Waves 5, pp. 149–157, 1995.
[16] B.L. Holian and M. Mareschal, Heat-flow equation motivated by the ideal-gas shock wave, Phys. Rev. E 82, p. 026707, 2010.
[17] D.-S. Tang, Y.-C. Hua, B.-D. Nie and B.-Y. Cao, Phonon wave propagation in ballistic-diffusive regime, J. Applied Phys. 119, p. 124301, 2016.
[18] B.L. Holian, W. G. Hoover, B. Moran and G. K. Straub, Shock-wave structure via nonequilibrium molecular dynamics and Navier-Stokes continuum mechanics, Phys. Rev. A 22, pp. 2798–2808, 1980.
[19] W.J. M. Rankine, On the thermodynamic theory of waves of finite longitudinal distrubances, Phil. Trans. Roy. Soc. London 160, pp. 277–288, 1870.
[20] H.Hugoniot,Mémoiresurlapropagationdesmouvementsdanslescorpsetspécialementdanslesgazparfaits(premicˇrepartie),J.ÉcolePolytechnique58,pp.1–125,1887.
[21] B. Zappoli, D. Beysens and Y. Garrabos, Heat Transfers and Related Effects in Supercritical Fluids, London: Springer, 2015.
[22] L. Chen, Microchannel Flow Dynamics and Heat Transfer of Near-Critical Fluid, Singapore: Springer, 2017.
[23] A. Onuki, H. Hao and R. A. Ferrell, Fast adiabatic equilibration in a single-component fluid near the liquid-vapor critical point, Phys. Rev. A 41, pp. 2256–2259, 1990.
[24] T. Ikeshoji and B. Hafskjold, Non-equilibrium molecular dynamics calculation of heat conduction in liquid and through liquid-gas interface, Mol. Phys. 81, pp. 251–261, 1994.
[25] C. Cattaneo, Sulla conduzione del calore, Atti del Semin. Mat. e Fis. Uni. Modena 3, pp. 83–101, 1948.
[26] P. Vernotte, Les paradoxes de la théorie continue de l’équation de la chaleur, C. R. Hebd. Sceances Acad. Sci. 246, pp. 3154–3155, 1958.
[27] D. D. Joseph and L. Preziosi, Heat waves, Rev. Mod. Phys. 61, pp. 41–73, 1989.
[28] B.L. Holian and D. J. Evans, Shear viscosities away from the melting line: A comparison of equilibrium and nonequilibrium molecular dynamics, J. Chem. Phys. 78, pp. 5147–5150, 1983.
[29] D.M. Heyes, The Lennard-Jones Fluid i the Liquid-Vapour Region, CMST 21, no. 4, pp. 169–179, 2015.
[30] B. Hafskjold, T. Ikeshoji and S. K. Ratkje, On the molecular mechanism of thermal diffusion in liquids, Mol. Phys. 80, pp. 1389–1412, 1993.
[31] S. Kjelstrup, D. Bedeaux, E. Johannesen and J. Gross, Non-Equilibrium Thermodynamics for Engineers, New Jersey: World Scientific, 2010.
[32] D. J. Evans and G. P. Morriss, Statistical Mechanics of Nonequilibrium Liquids, Canberra: ANU E Press, 2007.
[33] Y. Huang and H. H. Bau, Thermoacoustic waves in a semi-infinite medium, Int. J. Heat and Mass Transfer 38, no. 8, pp. 1329–1345, 1995.
[34] A. Nakano, Studies on piston and soret effects in a binary mixture supercritical fluid, Int. J. Heat and Mass Transfer 50, pp. 4678–4687, 2007.
[35] B. L. Holian, Modeling shock-wave deformation via molecular dynamics, Phys. Rev. A 37, pp. 2562–2568, 1988.
[36] B. Vick and M. N. Özisik, Growth and Decay of a Thermal Pulse Predicted by the Hyperbolic Heat Conduction Equation, Trans. ASME 105, pp. 902–907, 1983.
[37] B. Hafskjold and S. K. Ratkje, Criteria for Local Equilibrium in a System with Transport of Heat and Mass, J. Stat. Phys. 78, pp. 463–494, 1995.
[38] Y.-K. Guo, Z.-Y. Guo and X.-G. Liang, Three-Dimensional Molecular Dynamics Simulation on Heat Propagation in Liquid Argon, Chin. Phys. Lett. 18, pp. 71–73, 2001.
[39] Z. Guo, D. Xiong and Z. Li, Molecular Dynamics Simulation of Heat Propagation in Liquid Argon, Tsinghua Sci. Tech. 2, no. 2, pp. 613–618, 1997.
[40] H. Boukari, J. N. Shaumeyer, M. E. Briggs and R. W. Gammon, Critical speeding up in pure fluids, Phys. Rev. A 41, no. 4, pp. 2260–2263, 1990.
[41] A. Onuki, Thermoacoustic effects in supercritical fluids near the critical point: Resonance, piston effect, and acoustic emission and reflection, Phys. Rev. E 76, p. 061126, 2007.
[42] G. Lebon and D. Jou, Early history of extended irreversible termodynamics (1953–1983): An exploration beyond local equilibrium and classical transport theory, Eur. Phys. J. H 40, pp. 205–240, 2015.
[43] K. S. Glavatsky, Local equilibrium and the second law of thermodynamics for irreversible systems with thermodynamic inertia, J. Chem. Phys. 143, pp. 164101-1–11, 2015.6
