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NUMERICAL COMPARISON BETWEEN S-MODEL KINETIC EQUATION AND MOLECULAR
DYNAMICS SIMULATIONS FOR HEAT TRANSFER THROUGH ARGON VAPOR
Moritz C.W. Wolf, Alexey Polikarpov, Arjan Frijns, Irina A. Graur, Silvia Nedea, Ryan Enright
To cite this version:
Moritz C.W. Wolf, Alexey Polikarpov, Arjan Frijns, Irina A. Graur, Silvia Nedea, et al.. NUMERICAL COMPARISON BETWEEN S-MODEL KINETIC EQUATION AND MOLECULAR DYNAMICS SIMULATIONS FOR HEAT TRANSFER THROUGH ARGON VAPOR. The 10th International Conference on Boiling & Condensation Heat Transfer, Mar 2018, Nagasaki, Japan. �hal-02407088�
10th International Conference on Boiling and Condensation Heat Transfer 12th-15th March 2018 in Nagasaki, Japan www.icbcht2018.org
NUMERICAL COMPARISON BETWEEN
S-MODEL KINETIC EQUATION AND MOLECULAR DYNAMICS SIMULATIONS FOR HEAT TRANSFER THROUGH ARGON VAPOR
Moritz C.W. Wolf*1,3, Alexey Ph. Polikarpov2, Arjan J.H. Frijns3, Irina A. Graur4, Silvia V. Nedea3, Ryan Enright1
1 Nokia Bell Labs, Blanchardstown Business & Technology Park Dublin, D15 Y6NT, Ireland
2 Ural Federal University, 51 str. Lenina, 620000 Yekaterinburg, Russia
3 Eindhoven University of Technology, PO Box 513, 5600MB Eindhoven, the Netherlands
4 Aix-Marseille Université, CNRS, IUSTI UMR 7343, 13013 Marseille, France ABSTRACT
During the last decades, the miniaturization of electrical devices has grown considerably. Because of this, the heat transfer density within those devices has increased. Therefore, the need of advanced microscale cooling systems has become important. These systems need to be more energy efficient, i.e. to evacuate higher heat fluxes than the existing cooling systems [6]. This can be achieved by new generation two-phase flow evaporative systems. Within these systems, a liquid evaporates through a nanopores membrane. The latent heat of
vaporization is the dominant mode of heat transfer and the nanopores geometry generates the requisite capillary pressure to drive the liquid flow to the heat source. It is important to understand the evaporation and
condensation process as well as the corresponding vapor flow behaviors for the development of these two-phase cooling systems. The first step is to focus on the liquid-vapor phase and to be able to capture any non-
equilibrium effects, e.g. temperature and pressure jumps at the liquid-vapor interface by performing Molecular Dynamics (MD) simulations. However, MD simulations become computationally expensive when simulating one or multiple nanopores. Therefore, it is inevitable to use different methods such as the S-model kinetic equation [1]. This model is less computationally demanding, but its applicability needs to be investigated by comparing its results with those of the MD simulation.
The numerical set-up consists of a steady-state heat transfer through Argon vapor between its condensed phase as shown in Fig.1. The liquid temperature ratio is T1/T2=1.045 with T2=100K. The rarefaction parameter
δ=7.9 which is defined as the length of the vapor phase Lvdivided by the mean free path λ. The dimensions of the numerical simulation box are: Lx=Lz=5.75 nm and Ly=75.0nm with periodic boundary conditions in each direction. For the MD simulations, the Lennard-Jones 12-6 potential is used to calculate the intermolecular forces between the Argon particles with parameters: ε=0.24036 Kcal/mol, σ=0.34 nm and cut-off distance
rc=2.55 nm (7.5σ). The time step is Δt=4.0fs. A Nosé-Hoover thermostat is applied to the liquid Argon layers to keep the average temperature fixed at T1 and T2 respectively. The liquid and vapor boundaries are determined by following the procedure given by Meland [2]. To obtain a steady-state MD simulation, the atoms are shifted in y-direction during the simulation [2]. This will keep the average number of atoms in the liquid layers equal. The final results are obtained by averaging from 1ns to 20ns.
The steady-state dimensionless S-model kinetic equations [3] is defined as,
c∂ f(y , c)
∂ y =δn
√
T(
fs−f)
where, f(y , c) is the one particle molecular velocity distribution function, c is the molecular velocity vector, y
is the distance between the liquid layers and fs the equilibrium distribution function [3]. The S-model kinetic equation is solved by the discrete velocity method in one-dimension in physical space and two-dimension in molecular velocity space. When the numerical values of f(y , c) are obtained, the macroscopic quantities such as temperature, pressure are obtained by integrating it over the molecular velocities. At the liquid-vapor interface, it is assumed that all the incident molecules evaporate immediately which corresponds to complete evaporation.
Fig. 1 Numerical simulation box for MD with Nosé-Hoover(NH) thermostat applied to a part of the liquid phase.
RESULTS
All the results are obtained by using the peculiar velocity V=c−u, with u the vapor bulk velocity. In Fig. 2, the normalized vapor temperature profiles are shown. Both the S-model and MD predict an inverted temperature gradient profile which occurs when Δ P/ΔT>4.7723 [4,5] which is the case here. The temperature jumps can be found near the liquid-vapor interfaces. At y=0, the vapor temperature TV/T2≈1.005 is lower than the temperature of the hot liquid layer T1/T2=1.0 4 5. Contrarily, at y=1, the vapor temperature
TV/T2≈1.035 is higher than the temperature of the cold liquid layer T2/T2=1.0. The heat flux profiles are shown in Fig.3.
For the S-model, the results of two different temperature ratios are depicted. This change in temperature (+0.3K) corresponds to a small shift of the position of the liquid-vapor boundary at the evaporation side (hot liquid layer). The agreement between the MD and S-model results improves considerably (Fig.3+4, black line).
Furthermore, the evaporation/condensation coefficients do have a large influence as well (not depicted).
Therefore, it is important to obtain the evaporation/condensation coefficients and the position of the liquid-vapor boundary as accurate as possible.
The next step is to investigate, how the coefficients and the position of the liquid-vapor boundary can be extracted from MD simulation more accurately compared to Meland [2].
Fig. 2 Normalized temperature profile T¿=T1(y)/T2
Fig. 3 Normalized heat flux
q¿y= qy(y)
Psat(T2)
√
2RspecificT2ACKNOWLEDGEMENTS
This project has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No. 643095.
REFERENCES
[1] Shakhov, E.M., Generalization of the Krook kinetic relaxation equation. Fluid Dyn. 3(5), 95-96, 1968.
[2] Meland, R., Molecular Effects on Evaporation and Condensation. Doctoral thesis, Norwegian University of Science and Technology, 2002.
[3] Graur, I.A. and Polikarpov, A.Ph., Comparison of different kinetic models for the heat transfer problem.
J. Heat and Mass transfer, 46, 237-244, 2009
[4] Sone, Y., Ohwada, T. & Aoki, K. Evaporation and condensation of a rarefied gas between its two parallel condensed phases with different temperatures and negative temperature-gradient phenomenon - Numerical analysis of the Boltzmann equation for hard-sphere molecules, Math. Aspects Fluid Plasma Dyn. Springer Verlag, 186–202, 1991.
[5] Gatapova, E.Ya., Graur, I.A., Sharipov, F., Kabov O.A. The temperature and pressure jumps at the vapor- liquid interface: Application to a two-phase cooling system. J. Heat and Mass Transfer, 83, 235-243, 2015.
[6] Gatapova, E.Ya, Graur, I.A., Sharipiv, F., Kabov, O.A., The temperature and pressure jumps at the vapor- liquid interface: Application to a two-phase cooling system. Int. J. of Heat and Mass Transfer, 83, 235-243, 2015.
