Mathematical modeling of the isothermal impingement of liquid droplets in spraying processes
01 Dec 1991-Metallurgical and Materials Transactions B-process Metallurgy and Materials Processing Science (Springer US)-Vol. 22, Iss: 6, pp 901-914
TL;DR: In this article, a mathematical representation has been developed and computed results are presented describing the spreading of droplets impacting onto a solid substrate, which is of major practical interest in plasma spraying (PS) and in spray forming (SF) operations.
Abstract: A mathematical representation has been developed and computed results are presented describing the spreading of droplets impacting onto a solid substrate. Problems of this type are of major practical interest in plasma spraying (PS) and in spray forming (SF) operations. While the present study was confined to the fluid flow aspects of the process, information has been generated on both the final splat dimensions and on the time required to complete the spreading process. Through this treatment, it is possible to relate these quantities (the splat size and the spreading time) to the operating conditions,i.e., droplet size and droplet velocity, and material properties. The theoretical predictions were found to be in good agreement with both Madejski’s asymptotic solution[17] and with available experimental results. For typical SF conditions (droplet sizes in the 100-µm range and droplet velocities in the 100 m/s range), the spreading times were of the order of microseconds,i.e., significantly shorter than the estimated solidification time.
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TL;DR: The fluid dynamic phenomena of liquid drop impact are described and reviewed in this article, and specific conditions under which the above phenomena did occur in experiments are analyzed and the characteristics of drop impact phenomena are described in detail.
1,081 citations
TL;DR: In this article, a numerical solution of the Navier-Stokes equation using a modified SOLA-VOF method was used to model the impact of water droplets on a flat, solid surface using both experiments and numerical simulation.
Abstract: Impact of water droplets on a flat, solid surface was studied using both experiments and numerical simulation. Liquid–solid contact angle was varied in experiments by adding traces of a surfactant to water. Impacting droplets were photographed and liquid–solid contact diameters and contact angles were measured from photographs. A numerical solution of the Navier–Stokes equation using a modified SOLA‐VOF method was used to modeldroplet deformation. Measured values of dynamic contact angles were used as a boundary condition for the numerical model. Impacting droplets spread on the surface until liquid surface tension and viscosity overcame inertial forces, after which they recoiled off the surface. Adding a surfactant did not affect droplet shape during the initial stages of impact, but did increase maximum spread diameter and reduce recoil height. Comparison of computer generated images of impacting droplets with photographs showed that the numerical model modeled droplet shape evolution correctly. Accurate predictions were obtained for droplet contact diameter during spreading and at equilibrium. The model overpredicted droplet contact diameters during recoil. Assuming that dynamic surface tension of surfactant solutions is constant, equaling that of pure water, gave predicted droplet shapes that best agreed with experimental observations. When the contact angle was assumed constant in the model, equal to the measured equilibrium value, predictions were less accurate. A simple analytical model was developed to predict maximum droplet diameter after impact. Model predictions agreed well with experimental measurements reported in the literature. Capillary effects were shown to be negligible during droplet impact when We≫Re1/2.
1,049 citations
TL;DR: In this paper, a rebound model was proposed to predict the tendency of a droplet to deposit or to rebound on flat surfaces at room temperature at impact velocities, viscosities, and surface roughness.
Abstract: The spread and rebound of droplets upon impact on flat surfaces at room temperature were studied over a wide range of impact velocities (0.5–6 m/s), viscosities (1–100 mPa.s), static contact angles (30–120°), droplet sizes (1.5–3.5 mm), and surface roughnesses using a fast-shutter-speed CCD camera. The maximum spread of a droplet upon impact depended strongly on the liquid viscosity and the impact velocity. The tendency of a droplet to deposit or to rebound is determined primarily by the liquid viscosity and the liquid/substrate static contact angle. A model more broadly applicable than existing models was developed to predict maximum spread as a function of the Reynolds number, the Weber number, and the static contact angle. Based on the conservation of energy, a rebound model is proposed that predicts the tendency to rebound as a function of maximum spread and static contact angle.
The maximum-spread model prediction agrees to within 10% with more than 90% of the experimental data from different sources. In the current study, the rebound model successfully predicts the tendency of a droplet to rebound.
561 citations
TL;DR: In this article, an experimental and theoretical study of the deformation of a spherical liquid droplet colliding with a flat surface is presented, which accounts for the presence of inertia, viscous, gravitation, surface tension, and wetting effects, including the phenomenon of contact angle hysteresis.
Abstract: In this paper an experimental and theoretical study of the deformation of a spherical liquid droplet colliding with a flat surface is presented. The theoretical model accounts for the presence of inertia, viscous, gravitation, surface tension, and wetting effects, including the phenomenon of contact‐angle hysteresis. Experiments with impingement surfaces of different wettability were performed. The study showed that the maximum splat radius decreased as the value of the advancing contact angle increased. The effect of impact velocity on droplet spreading was more pronounced when the wetting was limited. The experimental results were compared to the numerical predictions in terms of droplet deformation, splat radius, and splat height. The theoretical model predicted well the deformation of the impacting droplet, not only in the spreading phase, but also during recoiling and oscillation. The wettability of the substrate upon which the droplet impinges was found to affect significantly all phases of the spre...
480 citations
TL;DR: In this article, the impact and flattening of single particles on smooth or rough substrates with different tilting is summarized and different diagnostic methods, including imaging, are briefly described.
Abstract: This paper summarizes our knowledge at the beginning of 2003 about splat formation. First, the analytical and numerical models related to the impact and flattening of single particles on smooth or rough substrates with different tilting are recalled. Then, the different diagnostic methods, including imaging, are briefly described. The last part of the paper is devoted to the results and their discussion. Studies are related to the effect of various parameters on particle flattening. They include the characteristics of particles prior to impact: normal impact velocity, temperature, molten state, oxidation state, etc.; the parameters related to the substrate: tilting angle, roughness, oxide layer composition, thickness and crystallinity, desorption of adsorbates and condensates, wetting properties between impacting particle and substrate, etc.; and, finally, the parameters related to the heat exchange between the flattening particle and the substrate. They depend on previous parameters and control the propagation of the solidification front within the flattening particle, eventually modifying its liquid flow. It is obvious from this review that, if our understanding of the involved phenomena has been drastically improved during the last years, many points have still to be clarified. This is of primary importance because all the coating properties are linked to the particle flattening, splat formation, and layering.
428 citations
References
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TL;DR: In this paper, the concept of a fractional volume of fluid (VOF) has been used to approximate free boundaries in finite-difference numerical simulations, which is shown to be more flexible and efficient than other methods for treating complicated free boundary configurations.
11,567 citations
TL;DR: In this paper, a new technique is described for the numerical investigation of the time-dependent flow of an incompressible fluid, the boundary of which is partially confined and partially free The full Navier-Stokes equations are written in finite-difference form, and the solution is accomplished by finite-time step advancement.
Abstract: A new technique is described for the numerical investigation of the time‐dependent flow of an incompressible fluid, the boundary of which is partially confined and partially free The full Navier‐Stokes equations are written in finite‐difference form, and the solution is accomplished by finite‐time‐step advancement The primary dependent variables are the pressure and the velocity components Also used is a set of marker particles which move with the fluid The technique is called the marker and cell method Some examples of the application of this method are presented All non‐linear effects are completely included, and the transient aspects can be computed for as much elapsed time as desired
5,841 citations
TL;DR: In this paper, a model for the movement of a small viscous droplet on a surface is constructed that is based on the lubrication equations and uses the dynamic contact angle to describe the forces acting on the fluid at the contact line.
Abstract: A model for the movement of a small viscous droplet on a surface is constructed that is based on the lubrication equations and uses the dynamic contact angle to describe the forces acting on the fluid at the contact line. The problems analysed are: the spreading or retraction of a circular droplet; the advance of a thin two-dimensional layer; the creeping of a droplet or cell on a coated surface to a region of greater adhesion; the distortion of droplet shape owing to surface contamination. Relevant biological problems concerning cell movement and adhesion are described.
613 citations
TL;DR: In this paper, a simple model of two-dimensional radial flow has been used, and the degree of flattening ξm of a droplet depends upon the Weber, Reynolds and Peclet numbers, and upon the freezing constant U, taken from the solution of a Stefan problem.
512 citations
TL;DR: In this article, the full Navier-Stokes equations are solved numerically in cylindrical coordinates in order to investigate the splash of a liquid drop onto a flat plate, into a shallow pool, or into a deep pool.
Abstract: The full Navier‐Stokes equations are solved numerically in cylindrical coordinates in order to investigate the splash of a liquid drop onto a flat plate, into a shallow pool, or into a deep pool. Solution is accomplished with the Marker‐and‐Cell technique using a high‐speed computer. Results include data on pressures, velocities, oscillations, droplet rupture, and the effects of compressibility. They also show how the technique can be applied to a wide variety of other complicated fluid flow problems involving the transient behavior of a free surface.
367 citations