Condensation by heterogeneous nucleation in a thermal boundary layer

Condensation can occur by the nucleation and growth of water droplets on particles in the air. This mode of mass transport can readily occur in a thermal boundary layer. The primary condition required for droplet nucleation is the existence of supersaturation in the boundary layer. Equations are developed to described droplet nucleation an d transport for the boundary layer on a horizontal, upward facing surface. These equations allow diffusion and nucleation mass fluxes and droplet concentration at a point in the boundary layer to be calculated. Experimental data were collected and compared to the theoretical calculations. Accurately predicting droplet concentration is difficult, but the presence of nucleation condensation can be readily predicted.     

Experimental determination of droplet size and density field in condensing flows

 We report a detailed experimental characterization of the process of homogeneous condensation in supersonic expanding flow. In our experiments, the supersaturated mixture expands in a Laval nozzle, where, depending on the initial conditions, a steady or periodically oscillating flow may evolve due to the non-linear interaction of nucleation and droplet growth rate with the flow field. Two experimental techniques are utilized: holographic interferometry for the determination of the density field and a time-resolved white-light extinction method. The latter is employed to derive the evolution in time of the droplet cloud (i.e., modal radius, number density, and relative width) and to measure the frequency of oscillations. In combination with the wide-field density data, droplet size measurements provide additional physical insights in the mechanism of interaction in condensing flows and serve as an excellent test case for the critical assessment of nucleation and droplet growth theories. To this purpose, the accuracy of the measurements is carefully reviewed due to the difficulties of characterizing dense sub-micron droplet clouds by means of light-scattering techniques. An important byproduct of this analysis is an evaluation of the applicability of single-scattering approximations, i.e., Lambert-Beer law, for a variety of experimental configurations. Received: 24 April 2001 / Accepted: 29 August 2001     

Effect of confinement on droplet deformation in shear flow

The dynamics of a single droplet under shear flow between two parallel plates is investigated by using the immersed boundary method. The immersed boundary method is appropriate for simulating the drop-ambient fluid interface. We apply a volume-conserving method using the normal vector of the surface to prevent mass loss of the droplet. In addition, we present a surface remeshing algorithm to cope with the distortion of droplet interface points caused by the shear flow. This mesh quality improvement in conjunction with the volume-conserving algorithm is particularly essential and critical for long time evolutions. We study the effect of wall confinement on the droplet dynamics. Numerical simulations show good agreement with previous experimental results and theoretical models.     

A random simulation of droplet distribution in nozzle and plume flows

A simple entrainment model is used to estimate droplet streamlines, velocity and mass flux in rocket exhaust plumes. Since droplet mass flux constitutes only about 1% of the exhaust mass flux, the effect of droplet entrainment on the gas flow is neglected. The novelty of the present model is in obtaining the droplet distribution within the nozzle by assuming a small radial random velocity component for droplets at the throat. Gas flow in the nozzle is approximated as isentropic plus a correction for the boundary layer. The computed distribution of droplet mass flux is found to be in good agreement with experimental data. Received 15 January 1996 / Accepted 11 September 1996     

Boundary layer and heat transfer in a two-phase gas-evaporating droplet medium

An asymptotic model of the flow in the laminar boundary layer of a gas-evaporating droplet mixture is constructed within the framework of the two-continuum approximation. The case of evaporation of the droplets into an atmosphere of their own vapor is examined in detail with reference to the example of longitudinal flow over a hot flat plate. Numerical and asymptotic solutions of the boundary layer equations constructed are found for a number of limiting situations (low droplet concentration, no droplet deposition, significant droplet deposition). The development of the flow with respect to the longitudinal coordinate is studied and it is shown that in the absence of droplet deposition a region of pure vapor may be formed near the surface. Similarity criteria are established and the mechanism of surface heat transfer enhancement is studied for a low evaporating droplet concentration in the boundary layer. In the inertial deposition regime the results of calculating the integral heat transfer coefficient are found to correspond with the experimental data [1].Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.3, pp. 42–50, May–June, 1992.     

Zur Beschreibung homogener unärer und binärer Kondensationsvorgänge

The modified theoretical analysis of the binary nucleation and the subsequent heterogeneous droplet growth presented includes the limiting unary condensation. By introducing a new parameterζ, the ratio of the reduced partial molecular volumes, it is possible to illustrate the changing binary droplet composition during their growth from the metastable equilibrium. A complete calculation of the droplet growth confirms the assumptions. As a typical result it is found, that the binary droplets will obtain their equilibrium composition within 10^?5s — starting with their metastable value. The comparison of the droplet growth model and parallel measurements with water-ethanol mixtures in the expansion fan of a shock tube confirms the results.     

Origin of droplet size underprediction in modeling of low pressure nucleating flows of steam

This article examines deficiencies in nucleation rate and droplet growth models that impair modeling of low pressure (LP) nucleating flows of steam. It is shown that classical nucleation theory (CNT) exhibits excessive dependence on supersaturation in the operating range of LP condensing (wet-steam) flows. The complex mechanisms of the nucleation-growth model are explained with regard to discrepancies in the modeling results. The discrepancy between modeling results and LP wet-steam experiments is attributed to imprecision in CNT and inadequacies in the employed droplet growth equation. The link between the excessive dependence of CNT on supersaturation and underprediction of the mean droplet size is explained. Two examples are given demonstrating that the inverse correlation between mean droplet size and nucleation rate can be moderated by rectifying and reducing the dependence of CNT on supersaturation. Moreover, prediction of mean droplet size is improved without modifying the location and magnitude of the condensation shock.     

Explosive boiling of a liquid droplet

A mathematical model describing growth of an internal vapor bubble produced by homogeneous nucleation within a liquid droplet during explosive boiling is presented. Existing experimental results for explosive boiling of superheated droplets confirm the predictions of the model. The difference between the present model and the classical theories of bubble growth is discussed.     

Study on heat transfer performance of spray cooling: model and analysis

In consideration of droplet–film impaction, film formation, film motion, bubble boiling (both wall nucleation bubbles and secondary nucleation bubbles), droplet–bubble interaction, bulk air convection and radiation, a model to predict the heat and mass transfer in spray cooling was presented in this paper. The droplet–film impaction was modeled based on an empirical correlation related with droplet Weber number. The film formation, film motion, bubble growth, and bubble motion were modeled based on dynamics fundamentals. The model was validated by the experimental results provided in this paper, and a favorable comparison was demonstrated with a deviation below 10%. The film thickness, film velocity, and non-uniform surface temperature distribution were obtained numerically, and then analyzed. A parameters sensitivity analysis was made to obtain the influence of spray angle, surface heat flux density, and spray flow rate on the surface temperature distribution, respectively. It can be concluded that the heat transfer induced by droplet–film impaction and film-surface convection is dominant in spray cooling under conditions that the heated surface is not superheated. However, the effect of boiling bubbles increases rapidly while the heated surface becomes superheated.     

Mathematical modeling of near-wall flows of two-phase mixture with evaporating droplets

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A Hybrid Boundary/Finite Element Method for Simulating Viscous Flows and Shapes of Droplets in Electric Fields

A mixed boundary element and finite element numerical algorithm for the simultaneous prediction of the electric fields, viscous flow fields, thermal fields and surface deformation of electrically conducting droplets in an electrostatic field is described in this paper. The boundary element method is used for the computation of the electric potential distribution. This allows the boundary conditions at infinity to be directly incorporated into the boundary integral formulation, thereby obviating the need for discretization at infinity. The surface deformation is determined by solving the normal stress balance equation using the weighted residuals method. The fluid flow and thermal fields are calculated using the mixed finite element method. The computational algorithm for the simultaneous prediction of surface deformation and fluid flow involves two iterative loops, one for the electric field and surface deformation and the other for the surface tension driven viscous flows. The two loops are coupled through the droplet surface shapes for viscous fluid flow calculations and viscous stresses for updating the droplet shapes. Computing the surface deformation in a separate loop permits the freedom of applying different types of elements without complicating procedures for the internal flow and thermal calculations. Tests indicate that the quadratic, cubic spline and spectral boundary elements all give approximately the same accuracy for free surface calculations; however, the quadratic elements are preferred as they are easier to implement and also require less computing time. Linear elements, however, are less accurate. Numerical simulations are carried out for the simultaneous solution of free surface shapes and internal fluid flow and temperature distributions in droplets in electric fields under both microgravity and earthbound conditions. Results show that laser heating may induce a non-uniform temperature distribution in the droplets. This non-uniform thermal field results in a variation of surface tension along the surface of the droplet, which in turn produces a recirculating fluid flow in the droplet. The viscous stresses cause additional surface deformation by squeezing the surface areas above and below the equator plane.     

竖直振动无黏液滴的法拉第不稳定性分析

由于外部周期性的振动而在液滴表面产生的Faraday不稳定效应广泛存在于超声雾化、喷涂加工等应用中,对Faraday不稳定性进行分析对研究振动液滴的表面动力学有着重要意义.本文将Faraday不稳定性问题从径向振动拓展到竖直振动,研究了竖直振动无黏液滴表面波的不稳定性.竖直方向的振动使得液滴动量方程为含有空间相关项和时...     

Low-Reynolds-number droplet motion in a square microfluidic channel

In this study, we investigate computationally the low-Reynolds-number droplet motion in a square micro-channel, a problem frequently encountered in microfluidic devices, enhanced oil recovery and coating processes. The droplet deformation and motion are determined via a three-dimensional spectral boundary element method for wall-bounded flows. The effects of the flow rate, viscosity ratio and droplet size on the interfacial dynamics are identified for droplets smaller and larger than the channel size and for a wide range of viscosity ratio. Owing to the stronger hydrodynamic forces in the thin lubrication film between the droplet interface and the solid walls, large droplets exhibit larger deformation and smaller velocity. Under the same average velocity, a droplet in a channel shows a significantly smaller deformation and higher velocity than in a cylindrical tube with the same size, owing to the existence of the corners’ area in the channel which permits flow of the surrounding fluid. A suitable periodic boundary implementation for our spectral element method is developed to study the dynamics of an array of identical droplets moving in the channel. In this case, the droplet deformation and velocity are reduced as their separation decreases; the reduction is influenced by the flow rate, viscosity ratio and more significantly the droplet size.     

Asymptotic analysis of the nucleation rate for rapidly varying gas dynamic parameters

An asymptotic method is proposed for reducing the order of the equations of condensation kinetics not only in the region of gradual droplet growth but also where diffusion in the space of sizes is important. This method makes it possible to extend the class of problems of determining the unsteady nucleation rate that have analytic solutions and to weaken the constraints imposed on the rate of variation of the gas dynamic parameters. Effects associated with the rapid variation of the critical droplet size are investigated. For this purpose the concept of a threshold dimension, in whose neighborhood the clusters go over into the droplet growth regime, is introduced. In the highly nonstationary case the threshold size differs from the critical size. Examples of simple calculations are presented.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 163–171, January-February, 1988.The author wishes to thank A. B. Vatazhin for numerous discussions.     

COMPARISON OF ANALYTICAL SOLUTIONS FOR CMSMPR CRYSTALLIZER WITH QMOM POPULATION BALANCE MODELING IN FLUENT

Fluent version 6.2 computational fluid dynamics environment has been enhanced with a population balance capability that operates in conjunction with its multiphase calculations to predict the particle size distribution within the flow field. The population balance is solved by the quadrature method of moments （QMOM）. Fluent＇s prediction capabilities are tested by using a 2-dimensional analogy of a constantly stirred tank reactor with a fluid flow compartment that mixes the fluid quickly and efficiently using wall movement and has a feed stream and a product stream. The results of these Fluent simulations using QMOM population balance solver are compared to steady state analytical solutions for the population balance in a stirred tank where 1） growth, 2） aggregation, and 3） breakage, take place separately and 4） combined nucleation and growth and 5） combined nucleation, growth and aggregation take place. The results of these comparisons show that the moments of the population balance are accurately predicted for nucleation, growth, aggregation and breakage when the flow field is turbulent. With laminar flow the mixing is not ideal and as a result the steady state well mixed solutions are not accurately simulated.     

Numerical study of the spontaneous nucleation of self-rotational moist gas in a converging–diverging nozzle

Spontaneous nucleation is the primary way of droplet formation in the supersonic gas separation technology, and the converging–diverging nozzle is the condensation and separation unit of supersonic gas separation devices. A three-dimensional geometrical model for the generation of self-rotational transonic gas flow is set up, based on which, the spontaneous nucleation of self-rotational transonic moist gas in the converging–diverging nozzle is carried out using an Eulerian multi-fluid model. The simulated results of the main flow and nucleation parameters indicate that the spontaneous nucleation can occur in the diverging part of the nozzle. However, different from the nucleation flow without self-rotation, the distributions of these parameters are unsymmetrical about the nozzle axis due to the irregular flow form caused by the self-rotation of gas flow. The nucleation region is located on the position where gas flows with intense rotation and the self-rotation impacts much on the nucleation process. Stronger rotation delays the onset of spontaneous nucleation and yields lower nucleation rate and narrow nucleation region. In addition, influences of other factors such as inlet total pressure p ₀, inlet total temperature T ₀, the nozzle-expanding ratio ? and the inlet relative humidity ф ₀ on the nucleation of self-rotational moist gas flow in the nozzle are also discussed.     

Heat and mass transfer of a fuel droplet evaporating in oscillatory flow

A numerical study of the heat and mass transfer from an evaporating fuel droplet in oscillatory flow was performed. The flow was assumed to be laminar and axisymmetric, and the droplet was assumed to maintain its spherical shape during its lifetime. Based on these assumptions, the conservation equations in a general curvilinear coordinate were solved numerically. The behaviors of droplet evaporation in the oscillatory flow were investigated by analyzing the effects of flow oscillation on the evaporation process of a n-heptane fuel droplet at high pressure.The response of the time history of the square of droplet diameter and space-averaged Nusselt numbers to the main flow oscillation were investigated in frequency band of 1–75 Hz with various oscillation amplitudes. Results showed that, depending on the frequency and amplitude of the oscillation, there are different modes of response of the evaporation process to the flow oscillation. One response mode is synchronous with the main flow oscillation, and thus the quasi-steady condition is attained. Another mode is asynchronous with the flow oscillation and is highly unsteady. As for the evaporation rate, however, in all conditions is more greatly enhanced in oscillatory flow than in quiescent air.To quantify the conditions of the transition from quasi-steady to unsteady, the response of the boundary layer around the droplet surface to the flow oscillation was investigated. The results led to including the oscillation Strouhal number as a criteria for the transition. The numerical results showed that at a low Strouhal number, a quasi-steady boundary layer is formed in response to the flow oscillation, whereas by increasing the oscillation Strouhal number, the phenomena become unsteady.     

Design and performance of a rapid piston expansion tube for the investigation of droplet condensation

 A piston expansion tube (pex-tube) is described which has been developed for the investigation of homogeneous nucleation in a supersaturated vapor and subsequent droplet growth. The design of the tube with periphery is discussed as well as the underlying gasdynamics. Examples for the measurement of nucleation and growth rates in a binary mixture of n-propanol/water carried in nitrogen demonstrate performance and range. The focus of the paper is on the determination of the nucleation state. It is shown that this state may not be based on an adiabatic expansion when helium is used as carrier gas instead of argon or nitrogen. Received: 14 May 1997 / accepted: 24 August 1997