Evaporation-driven vapour microflows: analytical solutions from moment methods (original) (raw)
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Journal of Fluid Mechanics, 2021
Classical continuum-based liquid vapour phase-change models typically assume continuity of temperature at phase interfaces along with a relation which describes the rate of evaporation at the interface (Hertz-Knudsen-Schrage, for example). However, for phase transitions processes at small scales, such as the evaporation of nanodroplets, the assumption that the temperature is continuous across the liquid-vapour interface leads to significant inaccuracies (McGaughey & Ward 2002; Rana et al. 2019), as may the adoption of classical constitutive relations that lead to the Navier-Stokes-Fourier equations (NSF). In this article, to capture the notable effects of rarefaction at small scales, we adopt an extended continuum-based approach utilizing the coupled constitutive relations (CCR). In CCR theory, additional terms are invoked in the constitutive relations of NSF equations originating from the arguments of irreversible thermodynamics as well as consistent with kinetic theory of gases. The modelling approach allows us to derive new fundamental solutions for the linearised CCR model and to develop a numerical framework based upon the method of fundamental solutions (MFS) and enables threedimensional multiphase micro-flow simulations to be performed at remarkably low computational cost. The new framework is benchmarked against classical results and then explored as an efficient tool for solving three-dimensional phase-change events involving droplets.
Kinetic modelling of evaporation and condensation phenomena around a spherical droplet
International Journal of Heat and Mass Transfer, 2021
The steady evaporation and condensation phenomena around a spherical droplet of the condensed phase of a vapor are investigated with basis on a kinetic model to the linearized Boltzmann equation. The kinetic equation is solved via a discrete velocity method which takes into account the discontinuity of the distribution function of molecular velocities on the spherical interface. The calculations are carried out in a wide range of the gas rarefaction and evaporation-condensation coefficient. The results obtained via solution of the linearized Navier-Stokes equations with temperature and pressure jump boundary conditions are also presented and compared to those obtained via kinetic equation. A comparison between the linearized and nonlinear solutions of the kinetic model is also presented to show the limit of applicability of the linearized approach.
Evaporation of Micro-Droplets: the "Radius-Square-Law" Revisited
Acta Physica Polonica A
The range of applicability of a fundamental tool for studying the evolution of droplets, the radius-square-law, was investigated both analytically and numerically, on the basis of the experimental results of our own as well as of other authors. Standard issues were briey discussed. Departures from the radius-square-law caused by the inuence of impurities encountered in non-ideal liquids, by the kinetic and surface tension eects encountered for small droplets or by thermal imbalance encountered in light-absorbing droplets were analysed. The entanglement between the kinetic and impurities eects was studied numerically yielding a possible explanation to evaporation coecient discrepancies found in the literature. An unexpected radius-square-law persistence in case of non-isothermal evolutions of very small droplets in atmosphere nearly saturated with vapour was analysed. The coexistence of the kinetic eects and the strong eects of surface tension was found responsible for this eect.
The Journal of chemical physics, 2016
In the present study, we investigate the process of evaporative cooling of nanometer-sized droplets in vacuum using molecular dynamics simulations with the TIP4P/2005 water model. The results are compared to the temperature evolution calculated from the Knudsen theory of evaporation which is derived from kinetic gas theory. The calculated and simulation results are found to be in very good agreement for an evaporation coefficient equal to unity. Our results are of interest to experiments utilizing droplet dispensers as well as to cloud micro-physics.
Reports on Progress in Physics, 2013
Evaporation is ubiquitous in nature. This process influences the climate, the formation of clouds, transpiration in plants, the survival of arctic organisms, the efficiency of car engines, the structure of dried materials and many other phenomena. Recent experiments discovered two novel mechanisms accompanying evaporation: temperature discontinuity at the liquid-vapour interface during evaporation and equilibration of pressures in the whole system during evaporation. None of these effects has been predicted previously by existing theories despite the fact that after 130 years of investigation the theory of evaporation was believed to be mature. These two effects call for reanalysis of existing experimental data and such is the goal of this review. In this article we analyse the experimental and the computational simulation data on the droplet evaporation of several different systems: water into its own vapour, water into the air, diethylene glycol into nitrogen and argon into its own vapour. We show that the temperature discontinuity at the liquid-vapour interface discovered by Fang and Ward (1999 Phys. Rev. E 59 417-28) is a rule rather than an exception. We show in computer simulations for a single-component system (argon) that this discontinuity is due to the constraint of momentum/pressure equilibrium during evaporation. For high vapour pressure the temperature is continuous across the liquid-vapour interface, while for small vapour pressures the temperature is discontinuous. The temperature jump at the interface is inversely proportional to the vapour density close to the interface. We have also found that all analysed data are described by the following equation: da/dt = P 1 /(a + P 2), where a is the radius of the evaporating droplet, t is time and P 1 and P 2 are two parameters. P 1 = −λ T /(q eff ρ L), where λ is the thermal conductivity coefficient in the vapour at the interface, T is the temperature difference between the liquid droplet and the vapour far from the interface, q eff is the enthalpy of evaporation per unit mass and ρ L is the liquid density. The P 2 parameter is the kinetic correction proportional to the evaporation coefficient. P 2 = 0 only in the absence of temperature discontinuity at the interface. We discuss various models and problems in the determination of the evaporation coefficient and discuss evaporation scenarios in the case of single-and multi-component systems.
A kinetic model for vapor-liquid flows
AIP Conference Proceedings, 2005
The evaporation of a liquid slab into vacuum is studied by numerical solutions of the Enskog-Vlasov equation for a fluid of spherical molecules interacting by Sutherland potential. The main aim of this work is to obtain the structure of the vapor-liquid interface in non-equilibrium conditions as well as the distribution function of evaporating molecules. The results show that the distribution function of molecules crossing a properly defined vapor-liquid boundary is almost Maxwellian and that the vapor phase is reasonably well described by the Boltzmann equation with diffusive boundary condition. CP762, Rarefied Gas Dynamics: 24 th International Symposium, edited by M. Capitelli © 2005 American Institute of Physics 0-7354-0247-7/05/$22.50
Numerical investigation of the evaporation of two-component droplets
2011
A numerical model for the complete thermo-fluid-dynamic and phase-change transport processes of twocomponent hydrocarbon liquid droplets consisting of n-heptane, n-decane and mixture of the two in various compositions is presented and validated against experimental data. The Navier-Stokes equations are solved numerically together with the VOF methodology for tracking the droplet interface, using an adaptive local grid refinement technique. The energy and concentration equations inside the liquid and the gaseous phases for both liquid species and their vapor components are additionally solved, coupled together with a model predicting the local vaporization rate at the cells forming the interface between the liquid and the surrounding gas. The model is validated against experimental data available for droplets suspended on a small diameter pipe in a hot air environment under convective flow conditions; these refer to droplet's surface temperature and size regression with time. An extended investigation of the flow field is presented along with the temperature and concentration fields. The equilibrium position of droplets is estimated together with the deformation process of the droplet. Finally, extensive parametric studies are presented revealing the nature of multi-component droplet evaporation on the details of the flow, the temperature and concentration fields.
Journal of Physics: Condensed Matter, 2013
For a one-component fluid on a solid substrate, a thermal singularity may occur at the contact line where the liquid-vapor interface intersects the solid surface. Physically, the liquid-vapor interface is almost isothermal at the liquid-vapor coexistence temperature in one-component fluids while the solid surface is almost isothermal for solids of high thermal conductivity. Therefore, a temperature discontinuity is formed if the two isothermal interfaces are of different temperatures and intersect at the contact line. This leads to the so-called thermal singularity. The localized hydrodynamics involving evaporation/condensation near the contact line leads to a contact angle depending on the underlying substrate temperature. This dependence has been shown to lead to the motion of liquid droplets on solid substrates with thermal gradients (Xu and Qian 2012 Phys. Rev. E 85 061603). In the present work, we carry out molecular dynamics (MD) simulations as numerical experiments to further confirm the predictions made from our previous continuum hydrodynamic modeling and simulations, which are actually semi-quantitatively accurate down to the small length scales in the problem. Using MD simulations, we investigate the motion of evaporative droplets in one-component Lennard-Jones fluids confined in nanochannels with thermal gradients. The droplet is found to migrate in the direction of decreasing temperature of solid walls, with a migration velocity linearly proportional to the temperature gradient. This agrees with the prediction of our continuum model. We then measure the effect of droplet size on the droplet motion. It is found that the droplet mobility is inversely proportional to a dimensionless coefficient associated with the total rate of dissipation due to droplet movement. Our results show that this coefficient is of order unity and increases with the droplet size for the small droplets (∼10 nm) simulated in the present work. These findings are in semi-quantitative agreement with the predictions of our continuum model. Finally, we measure the effect of liquid-vapor coexistence temperature on the droplet motion. Through a theoretical analysis on the size of the thermal singularity, it can be shown that the droplet mobility decreases with decreasing coexistence temperature. This is observed in our MD simulations.
Evaporation and condensational growth of tiny droplets
Journal of Aerosol Science, 1980
The one-speed transport equation normally used in neutron transport and radiative transfer studies is applied to the process of vapour (mass) transport in a gaseous media to a spherical droplet. The rate of mass transfer has been obtained in the quasi-stationary approximation. The treatment is quite general and valid for any "accommodation coefficient" and any "saturated vapour pressure". A method for incorporating these concepts in the transport equation is outlined. It is shown that the use of Kelvin's equation in this theory leads to results quite different from those given by conventional diffusion theory.