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Efficient moment method for modeling nanoporous evaporation

Thomas C. De Fraja1,2,*, Anirudh S. Rana3, Ryan Enright4, Laura J. Cooper1, Duncan A. Lockerby2,†, and James E. Sprittles1,‡

  • 1Mathematics Institute, University of Warwick, Coventry CV4 7AL, United Kingdom
  • 2School of Engineering, University of Warwick, Coventry CV4 7AL, United Kingdom
  • 3Department of Mathematics, Birla Institute of Technology and Science, Pilani, Rajasthan 333031, India
  • 4Nokia Bell Labs, New Providence, New Jersey 07974, USA

  • *tcd2000@hw.ac.uk
  • †d.lockerby@warwick.ac.uk
  • ‡j.e.sprittles@warwick.ac.uk
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Phys. Rev. Fluids 7, 024201 – Published 3 February, 2022

DOI: https://doi.org/10.1103/PhysRevFluids.7.024201

Abstract

Thin-film-based nanoporous membrane technologies exploit evaporation to efficiently cool microscale and nanoscale electronic devices. At these scales, when domain sizes become comparable to the mean-free path in the vapor, traditional macroscopic approaches such as the Navier-Stokes-Fourier (NSF) equations become less accurate, and the use of higher-order moment methods is called for. Two higher-order moment equations are considered; the linearized versions of the Grad 13 and Regularized 13 equations. These are applied to the problem of nanoporous evaporation, and results are compared to the NSF method and the method of direct simulation Monte Carlo (i.e., solutions to the Boltzmann equations). Linear and nonlinear versions of the boundary conditions are examined, with the latter providing improved results, at little additional computational expense, compared to the linear form. The outcome is a simultaneously accurate and computationally efficient method, which can provide simulation-for-design capabilities at the nanoscale.

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Corrections

15 February, 2022

Correction: The third affiliation was presented incorrectly and has been fixed.

Article Text

References (74)

  1. K. R. Bagnall, Device-level thermal analysis of GaN-based electronics, Ph.D. thesis, Massachusetts Institute of Technology, 2013.
  2. Z. Lu, T. R. Salamon, S. Narayanan, K. R. Bagnall, D. F. Hanks, D. S. Antao, B. Barabadi, J. Sircar, M. E. Simon, and E. N. Wang, Design and modeling of membrane-based evaporative cooling devices for thermal management of high heat fluxes, IEEE Trans. Comp. Packag. Manufact. Technol. 6, 1056 (2016).
  3. D. F. Hanks, Z. Lu, J. Sircar, T. R. Salamon, D. S. Antao, K. R. Bagnall, B. Barabadi, and E. N. Wang, Nanoporous membrane device for ultra high heat flux thermal management, Microsyst. Nanoeng. 4, 1 (2018).
  4. Z. Lu, Design and modeling of a high flux cooling device based on thin film evaporation from thin nanoporous membranes, Ph.D. thesis, Massachusetts Institute of Technology, 2014.
  5. C. G. Gray, K. E. Gubbins, and C. G. Joslin, Theory of Molecular Fluids: Volume 2: Applications (Oxford University Press, Oxford, UK, 2011), Vol. 10.
  6. Y. Akkuş, Modeling of evaporation from nanoporous membranes using molecular dynamics simulation, Isı Bilimi ve Tekniği Dergisi 39, 91 (2019).
  7. G. A. Bird, Molecular Gas Dynamics and the Direct Simulation of Gas Flows, Oxford Engineering Science Series, Vol. 42 (Clarendon Press, Oxford; Oxford University Press, New York, 1994).
  8. G. Dimarco, R. Loubère, J. Narski, and T. Rey, An efficient numerical method for solving the Boltzmann equation in multidimensions, J. Comput. Phys. 353, 46 (2018).
  9. W. Su, L. Zhu, and L. Wu, Fast convergence and asymptotic preserving of the general synthetic iterative scheme, SIAM J. Sci. Comput. 42, B1517 (2020).
  10. H. Struchtrup, Macroscopic transport equations for rarefied gas flows, in Macroscopic Transport Equations for Rarefied Gas Flows (Springer, Berlin, 2005).
  11. H. Struchtrup and M. Torrilhon, Higher-order effects in rarefied channel flows, Phys. Rev. E 78, 046301 (2008).
  12. X. Gu and D. Emerson, A computational strategy for the regularized 13 moment equations with enhanced wall-boundary conditions, J. Comput. Phys. 225, 263 (2007).
  13. H. Grad, On the kinetic theory of rarefied gases, Commun. Pure Appl. Math. 2, 331 (1949).
  14. H. Struchtrup and M. Torrilhon, Regularization of Grad's 13 moment equations: Derivation and linear analysis, Phys. Fluids 15, 2668 (2003).
  15. Y. Fan, J. Koellermeier, J. Li, R. Li, and M. Torrilhon, Model reduction of kinetic equations by operator projection, J. Stat. Phys. 162, 457 (2016).
  16. J. Koellermeier, R. P. Schaerer, and M. Torrilhon, A framework for hyperbolic approximation of kinetic equations using quadrature-based projection methods, Kinetic Related Models 7, 531 (2014).
  17. P. Taheri, A. S. Rana, M. Torrilhon, and H. Struchtrup, Macroscopic description of steady and unsteady rarefaction effects in boundary value problems of gas dynamics, Contin. Mech. Thermodyn. 21, 423 (2009).
  18. P. Taheri and H. Struchtrup, Effects of rarefaction in microflows between coaxial cylinders, Phys. Rev. E 80, 066317 (2009).
  19. M. Torrilhon, Slow gas microflow past a sphere: Analytical solution based on moment equations, Phys. Fluids 22, 072001 (2010).
  20. R. Claydon, A. Shrestha, A. S. Rana, J. E. Sprittles, and D. A. Lockerby, Fundamental solutions to the regularised 13-moment equations: Efficient computation of three-dimensional kinetic effects, J. Fluid Mech. 833, R4(2017).
  21. A. Rana, M. Torrilhon, and H. Struchtrup, A robust numerical method for the R13 equations of rarefied gas dynamics: Application to lid driven cavity, J. Comput. Phys. 236, 169 (2013).
  22. M. Torrilhon, Two-dimensional bulk microflow simulations based on regularized grad's 13-moment equations, Multiscale Model. Simul. 5, 695 (2006).
  23. L. Theisen and M. Torrilhon, Fenicsr13: A tensorial mixed finite element solver for the linear R13 equations using the Fenics computing platform, ACM Trans. Math. Softw. 47, 1 (2021).
  24. A. Westerkamp and M. Torrilhon, Finite element methods for the linear regularized 13-moment equations describing slow rarefied gas flows, J. Comput. Phys. 389, 1 (2019).
  25. A. Westerkamp, A continuous interior penalty method for the linear regularized 13-moment equations describing rarefied gas flows, Ph.D. thesis, Universitätsbibliothek der RWTH Aachen, 2017.
  26. A. S. Rana, S. Saini, S. Chakraborty, D. A. Lockerby, and J. E. Sprittles, Efficient simulation of non-classical liquid–vapour phase-transition flows: A method of fundamental solutions, J. Fluid Mech. 919, A35 (2021).
  27. M. Torrilhon, Modeling nonequilibrium gas flow based on moment equations, Annu. Rev. Fluid Mech. 48, 429 (2016).
  28. B. John, R. Enright, J. E. Sprittles, L. Gibelli, D. R. Emerson, and D. A. Lockerby, Numerical investigation of nanoporous evaporation using direct simulation Monte Carlo, Phys. Rev. Fluids 4, 113401(2019).
  29. B. John, L. Gibelli, R. Enright, J. E. Sprittles, D. A. Lockerby, and D. R. Emerson, Evaporation from arbitrary nanoporous membrane configurations: An effective evaporation coefficient approach, Phys. Fluids 33, 032022 (2021).
  30. H. Grad, Principles of the kinetic theory of gases, in Thermodynamik der Gase/Thermodynamics of Gases (Springer, Berlin, 1958), pp. 205–294.
  31. H. Struchtrup, Stable transport equations for rarefied gases at high orders in the Knudsen number, Phys. Fluids 16, 3921 (2004).
  32. X.-j. Gu and D. R. Emerson, A high-order moment approach for capturing non-equilibrium phenomena in the transition regime, J. Fluid Mech. 636, 177 (2009).
  33. M. Torrilhon and H. Struchtrup, Regularized 13-moment equations: Shock structure calculations and comparison to Burnett models, J. Fluid Mech. 513, 171 (1999).
  34. M. Y. Timokhin, H. Struchtrup, A. Kokhanchik, and Y. A. Bondar, Different variants of R13 moment equations applied to the shock-wave structure, Phys. Fluids 29, 037105 (2017).
  35. A. Bobylev, Instabilities in the Chapman-Enskog expansion and hyperbolic Burnett equations, J. Stat. Phys. 124, 371 (2006).
  36. H. Struchtrup, A. Beckmann, A. Rana, and A. Frezzotti, Evaporation boundary conditions for the R13 equations of rarefied gas dynamics, Phys. Fluids 29, 092004 (2017).
  37. M. Bond and H. Struchtrup, Mean evaporation and condensation coefficients based on energy dependent condensation probability, Phys. Rev. E 70, 061605 (2004).
  38. C. Navier, Mémoire sur les lois du mouvement des fluides, Mémoires de l'Académie Royale des Sciences de l'Institut de France 6, 389 (1823).
  39. K. Wang, Z. Chai, G. Hou, W. Chen, and S. Xu, Slip boundary condition for lattice Boltzmann modeling of liquid flows, Comput. Fluids 161, 60 (2018).
  40. M. Torrilhon and H. Struchtrup, Boundary conditions for regularized 13-moment-equations for micro-channel-flows, J. Comput. Phys. 227, 1982 (2008).
  41. M. Gad-el Hak, The fluid mechanics of microdevices—The Freeman scholar lecture, J. Fluids Eng. 121, 5 (1999).
  42. A. S. Rana, D. A. Lockerby, and J. E. Sprittles, Lifetime of a Nanodroplet: Kinetic Effects and Regime Transitions, Phys. Rev. Lett. 123, 154501 (2019).
  43. X.-J. Gu, D. R. Emerson, and G.-H. Tang, Kramers' problem and the knudsen minimum: A theoretical analysis using a linearized 26-moment approach, Continuum Mech. Thermodyn. 21, 345 (2009).
  44. Z. Cai and M. Torrilhon, Numerical simulation of microflows using moment methods with linearized collision operator, J. Sci. Comput. 74, 336 (2018).
  45. J. Koellermeier and M. Torrilhon, Numerical study of partially conservative moment equations in kinetic theory, Commun. Comput. Phys. 21, 981 (2017).
  46. J. Koellermeier and M. Torrilhon, Two-dimensional simulation of rarefied gas flows using quadrature-based moment equations, Multiscale Model. Simul. 16, 1059 (2018).
  47. D. A. Lockerby and B. Collyer, Fundamental solutions to moment equations for the simulation of microscale gas flows, J. Fluid Mech. 806, 413 (2016).
  48. T. Ytrehus, Molecular-flow effects in evaporation and condensation at interfaces, Multiphase Sci. Technol. 9, 205 (1997).
  49. COMSOL Multiphysics, Comsol multiphysics v5.6 (2020).
  50. P. M. Gresho and R. L. Sani, Incompressible Flow and The Finite Element Method, Volume 2: Incompressible flow and finite element (Wiley, 1998).
  51. A. S. Rana and H. Struchtrup, Thermodynamically admissible boundary conditions for the regularized 13 moment equations, Phys. Fluids 28, 027105 (2016).
  52. N. Sarna and M. Torrilhon, On stable wall boundary conditions for the hermite discretization of the linearised Boltzmann equation, J. Stat. Phys. 170, 101 (2018).
  53. S. J. Plimpton, and M. A. Gallis, SPARTA direct simulation Monte Carlo (DSMC) simulator (Sandia National Laboratories, USA, 2015), http://sparta.sandia.gov.
  54. J. Fan, in Preface: 29th International Symposium on Rarefied Gas Dynamics, AIP Conf. Proc. No. 1628 (AIP, New York, 2014), pp. 1–2.
  55. M. A. Gallis, T. P. Koehler, J. R. Torczynski, and S. J. Plimpton, Direct simulation Monte Carlo investigation of the Rayleigh-Taylor instability, Phys. Rev. Fluids 1, 043403 (2016).
  56. K. Nanbu, Direct simulation scheme derived from the Boltzmann equation. II. Multicomponent gas mixtures, J. Phys. Soc. Jpn. 49, 2050 (1980).
  57. H. Babovsky and R. Illner, A convergence proof for Nanbu's simulation method for the full Boltzmann equation, SIAM J. Numer. Anal. 26, 45 (1989).
  58. W. Wagner, A convergence proof for Bird's direct simulation monte carlo method for the Boltzmann equation, J. Stat. Phys. 66, 1011 (1992).
  59. D. Labuntsov and A. Kryukov, Analysis of intensive evaporation and condensation, Int. J. Heat Mass Transf. 22, 989 (1979).
  60. C. Cercignani, Rarefied Gas Dynamics: From Basic Concepts to Actual Calculations (Cambridge University Press, Cambridge, UK, 2000), Vol. 21.
  61. N. Hadjiconstantinou and A. Garcia, Statistical Error in Particle Simulations of Low Mach number Flows (Elsevier, Amsterdam, 2000).
  62. F. Sharipov, Modeling and calculations of rarefied gas flows: DSMC vs kinetic equation, Invited talk (2011), Vol. 45.
  63. D. W. Mackowski, D. H. Papadopoulos, and D. E. Rosner, Comparison of Burnett and DSMC predictions of pressure distributions and normal stress in one-dimensional, strongly nonisothermal gases, Phys. Fluids 11, 2108 (1999).
  64. I. Müller and T. Ruggeri, Rational Extended Thermodynamics (Springer Science & Business Media, Berlin, 2013), Vol. 37.
  65. H. Struchtrup, Heat transfer in the transition regime: Solution of boundary value problems for Grad's moment equations via kinetic schemes, Phys. Rev. E 65, 041204 (2002).
  66. J. Koellermeier, Error estimators for adaptive simulation of rarefied gases using hyperbolic moment models, in Proceedings: 31st International Symposium on Rarefied Gas Dynamics: RGD31, AIP Conf. Proc. No. 2132 (AIP Publishing LLC, 2019), p. 120004.
  67. M. Abdelmalik and E. van Brummelen, Error estimation and adaptive moment hierarchies for goal-oriented approximations of the Boltzmann equation, Comput. Methods Appl. Mech. Eng. 325, 219 (2017).
  68. M. Torrilhon, J. Au, and H. Struchtrup, Explicit fluxes and productions for large systems of the moment method based on extended thermodynamics, Continuum Mech. Thermodyn. 15, 97 (2003).
  69. M. Torrilhon and N. Sarna, Hierarchical Boltzmann simulations and model error estimation, J. Comput. Phys. 342, 66 (2017).
  70. J. Young, The condensation and evaporation of liquid droplets in a pure vapour at arbitrary Knudsen number, Int. J. Heat Mass Transf. 34, 1649 (1991).
  71. I. Müller, Thermodynamics (Pitman, Boston, MA, 1985).
  72. G. Taylor, On scraping viscous fluid from a plane surface, in The Scientific Papers of Sir Geoffrey Ingram Taylor, edited by G. K. Batchelor (Cambridge University Press, 1962), pp. 410–413.
  73. H. K. Moffatt, Viscous and resistive eddies near a sharp corner, J. Fluid Mech. 18, 1 (1964).
  74. L. C. Nitsche and P. Parthasarathi, Stokes flow singularity at the junction between impermeable and porous walls, J. Fluid Mech. 713, 183 (2012).

Outline

Information

Phys. Rev. Fluids 7, 024201– Published 3 February, 2022
  • Received 8 September 2021
  • Accepted 20 January 2022
  • Corrected 15 February 2022
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