Irina N. Shishkova

A numerical algorithm for kinetic modelling of evaporation processes

A numerical algorithm for kinetic modelling of droplet evaporation processes is suggested. This algorithm is focused on the direct numerical solution of the Boltzmann equations for two gas components: vapour and air. The physical and velocity spaces are discretised, and the Boltzmann equations are presented in discretised forms. The solution of these discretised equations is performed in two steps. Firstly, molecular displacements are calculated ignoring the effects of collisions. Secondly, the collisional relaxation is calculated under the assumption of spatial homogeneity. The conventional approach to calculating collisional integrals is replaced by the integration based on random cubature formulae. The distribution of molecular velocities after collisions is found based on the assumption that the total impulse and energy of colliding molecules are conserved. The directions of molecular impulses after the collisions are random, but the values of these impulses belong to an a priori chosen set. A new method of finding the matching condition for vapour mass fluxes at the outer boundary of the Knudsen layer of evaporating droplets and at the inner boundary of the hydrodynamic region is suggested. The numerical algorithm is applied to the analysis of three problems: the relaxation of an initially non-equilibrium distribution function towards the Maxwellian one, the analysis of the mixture of vapour and inert gas confined between two infinite plates and the evaporation of a diesel fuel droplet into a high pressure air. The solution of the second problem showed an agreement between the results predicted by the widely used Birds algorithm and the algorithm described in this paper. In the third problem the difference of masses and radii of vapour and air molecules is taken into account. The kinetic effects predicted by the numerical algorithm turned out to be noticeable if the contribution of air in the Knudsen layer is taken into account.

A solution of the Boltzmann equation in the presence of inelastic collisions

The effect of inelastic collisions between two molecules on the solution of the Boltzmann equation is taken into account by presenting the change of state of molecules after collisions as a random (with uniform probability distribution) movement along a surface of an N-dimensional sphere, the squared radius of which is equal to the total energy of the molecules before and after the collision in the reference system of the centre of mass. The projection of a point on the surface of this sphere in each of N directions gives the root square of the kinetic energy in one of three directions in the physical space, or the internal energy of one of degrees of freedom, of one of two molecules. The kinetic energies of two molecules are described by the first six dimensions of the system, and the remaining (N-6) dimensions describe the internal energies. This approach is applied to three test problems: shock wave structure in nitrogen, one-dimensional heat transfer through a mixture of n-dodecane and nitrogen and one-dimensional evaporation of n-dodecane into nitrogen. In the first problem, the predictions of the model are shown to be close to experimental data and also to the predictions of the earlier developed model, based on a different approach to taking into account the effects of inelastic collisions. The predicted heat flux for the second problem and mass flux for the third problem are shown to be very weak functions of the number of internal degrees of freedom when this number exceeds about 15. These results open the way for considering systems with arbitrarily large numbers of internal degrees of freedom by reducing the analysis of these systems to the analysis of systems with relatively small numbers of internal degrees of freedom.

Kinetic modelling of fuel droplet heating and evaporation: calculations and approximations

Simple approximate formulae describing temporal evolution of diesel fuel droplet radii and temperatures predicted by the kinetic model are suggested. These formulae are valid in the range of gas temperatures relevant to diesel engine-like conditions and fixed values of initial droplet radii, or in the range of initial droplet radii relevant to diesel engine-like conditions and fixed values of gas temperature. During the time period before the hydrodynamic model predicts complete droplet evaporation, these approximations are based on the calculation of the correction to the prediction of the hydrodynamic model. At longer times, the approximations at the earlier times are extrapolated up until the total evaporation of the droplet, using quadratic fittings. The new approximations are shown to be reasonably accurate for predicting the temporal evolution of droplet radii and droplet evaporation times. The predictions of droplet temperature turned out to be less accurate than those of droplet radii, but this accuracy is believed to be sufficient for many practical applications.

Fuel droplet heating and evaporation: new hydrodynamic and kinetic models

Recently developed approaches to the hydrodynamic and kinetic modelling of fuel droplet heating and evaporation are reviewed. Two new solutions to the heat conduction equation, taking into account the effect of the moving boundary during transient heating of an evaporating droplet, are discussed. The first solution is the explicit analytical solution to this equation, while the second one reduces the solution of the differential transient heat conduction equation to the solution of the Volterra integral equation of the second kind. It has been pointed out that the new approach predicts lower droplet surface temperatures and slower evaporation rates compared with the traditional approach. A simplified model for multi-component droplet heating and evaporation, based on the analytical solution of the species diffusion equation inside droplets, is discussed. A new algorithm, based on simple approximations of the kinetic results for droplet radii and temperatures, suitable for engineering applications, is discussed.© 2010 ASME

Vapor Flow with Evaporation–Condensation on Solid Particles

Strongly nonequilibrium vapor (gas) flows in a region filled by solid particles are considered with allowance for particle‐size variation due to evaporation–condensation on the particle surface. The study is performed by directly solving the kinetic Boltzmann equation with allowance for the transformation of the distribution function of gas molecules due to their interaction with dust particles.

Selective water vapor cryopumping through argon

A selective cryopumping process for water vapor control takes place in vacuum systems for web coating or plasma operations, such as sputter deposition, etching, etc. Excessive water vapor content will affect the quality of the processes and final products. These vacuum systems typically operate at pressures corresponding to transitional or viscous flow regimes, and water vapor cryopumping is highly limited by diffusion of water vapor molecules through a noncondensable process gas (argon, air). An analytical model was created to describe water vapor condensing process through a noncondensable gas diffusion barrier. The model accounts for the collisions of different molecules by means of Boltzmann kinetic equations for two-component rarefied gas. It was assumed that water vapor content is about three orders of magnitude lower than that of the noncondensable gas (argon). Cryopumping process was analyzed for two simplified cases when water vapor source and cryosurface are parallel plates and coaxial cylinders...

Modelling of droplet heating and evaporation: recent results and unsolved problems

The most recent results referring to the hydrodynamic and kinetic modelling of droplet heating and evaporation are briefly summarised. Two new solutions to the heat conduction equation, taking into account the effect of the moving boundary during transient heating of an evaporating droplet, are discussed. The first solution is the explicit analytical solution to this equation, while the second one reduces the solution of the differential transient heat conduction equation to the solution of the Volterra integral equation of the second kind. It has been pointed out that the new approach predicts lower droplet surface temperatures and slower evaporation rates compared with the traditional approach. A simplified model for multi-component droplet heating and evaporation, based on the analytical solution of the species diffusion equation inside droplets, is reviewed. A new algorithm, based on simple approximations of the kinetic results for droplet radii and temperatures, suitable for engineering applications, is discussed.

Mass and heat transfer between evaporation and condensation surfaces: Atomistic simulation and solution of Boltzmann kinetic equation

Boundary conditions required for numerical solution of the Boltzmann kinetic equation (BKE) for mass/heat transfer between evaporation and condensation surfaces are analyzed by comparison of BKE results with molecular dynamics (MD) simulations. Lennard–Jones potential with parameters corresponding to solid argon is used to simulate evaporation from the hot side, nonequilibrium vapor flow with a Knudsen number of about 0.02, and condensation on the cold side of the condensed phase. The equilibrium density of vapor obtained in MD simulation of phase coexistence is used in BKE calculations for consistency of BKE results with MD data. The collision cross-section is also adjusted to provide a thermal flux in vapor identical to that in MD. Our MD simulations of evaporation toward a nonreflective absorbing boundary show that the velocity distribution function (VDF) of evaporated atoms has the nearly semi-Maxwellian shape because the binding energy of atoms evaporated from the interphase layer between bulk phase and vapor is much smaller than the cohesive energy in the condensed phase. Indeed, the calculated temperature and density profiles within the interphase layer indicate that the averaged kinetic energy of atoms remains near-constant with decreasing density almost until the interphase edge. Using consistent BKE and MD methods, the profiles of gas density, mass velocity, and temperatures together with VDFs in a gap of many mean free paths between the evaporation and condensation surfaces are obtained and compared. We demonstrate that the best fit of BKE results with MD simulations can be achieved with the evaporation and condensation coefficients both close to unity.

Kinetic and MD modelling of automotive fuel droplets heating and evaporation: recent results

Recent results of the investigation of kinetic and molecular dynamics (MD) models for automotive fuel droplet heating and evaporation are summarised. The kinetic model is based on the consideration of the kinetic region in the close vicinity of the surface of the heated and evaporating droplets, where the motion of molecules is described in terms of the Boltzmann equations for vapour components and air, and the hydrodynamic region away from this surface. The effects of finite thermal conductivity and species diffusivity inside the droplets and inelastic collisions in the kinetic region are taken into account. A new self-consistent kinetic model for heating and evaporation of Diesel fuel droplets is briefly described. The values of temperature and vapour densities at the outer boundary of the kinetic region are inferred from the requirement that both heat flux and mass flux of vapour components in the kinetic and hydrodynamic regions in the vicinity of the interface between these regions are equal. At first, the heat and mass fluxes in the hydrodynamic region are calculated based on the values of temperature and vapour density at the surface of the droplet. Then the values of temperature and vapour density at the outer boundary of the kinetic region, obtained following this procedure, are used to calculate the corrected values of hydrodynamic heat and species mass fluxes. The latter in their turn lead to new corrected values of temperature and vapour density at the outer boundary of the kinetic region. It is shown that this process quickly converges and leads to self-consistent values for both heat and mass fluxes. Boundary conditions at the surface of the droplet for kinetic calculations are inferred from the MD calculations. These calculations are based on the observation that methyl (CH 3 ) or methylene (CH 2 ) groups in n-dodecane (approximation of Diesel fuel) molecules can be regarded as separate atom-like structures in a relatively simple United Atom Model. Some results of the application of quantum chemical methods to the estimation of the evaporation/condensation coefficient are discussed. DOI: http://dx.doi.org/10.4995/ILASS2017.2017.4593

Fuel Droplet Heating and Evaporation: Analysis of Liquid and Gas Phase Models

Recently developed liquid and gas phase models for fuel droplet heating and evaporation, suitable for implementation into computational fluid dynamics (CFD) codes, are reviewed. The analysis is focused on the liquid phase model based on the assumption that the liquid thermal conductivity is infinitely large (infinite thermal conductivity (ITC) model), and the so called effective thermal conductivity (ETC) model. Seven gas phase models are compared. It is pointed out that the gas phase model, taking into account the finite thickness of the thermal boundary layer around the droplet predicts the evaporation time closest to the one based on the approximation of experimental data. In most cases, the droplet evaporation time depends strongly on the choice of the gas phase model. The dependence of this time on the choice of the liquid phase model, however, is weak if the droplet break-up processes are not taken into account. Corrections to Newtons law for droplet transient heating are discussed. For the values of parameters relevant to diesel engines, the values of these corrections were shown to be significant. Recent kinetic models for droplet evaporation into a high pressure background gas are reviewed. It is recommended that the kinetic effects are taken into account when accurate analysis of diesel fuel droplet evaporation is essential. A new dynamic decomposition technique for a system of ordinary differential equations is reviewed.