J. Mostaghimi

Capillary effects during droplet impact on a solid surface

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.

A three-dimensional model of droplet impact and solidification

Abstract A three-dimensional model has been developed to simulate the fluid dynamics, heat transfer and phase-change that occur when a molten metal droplet falls onto a flat substrate. The model is an extension of one developed by Bussmann et al. [Phys. Fluids 11 (1999) 1406] and combines a fixed-grid control volume discretization of the fluid flow and energy equations with a volume-tracking algorithm to track the droplet free surface, and an improved fixed velocity method to track the solidification front. Surface tension is modeled as a volume force acting on fluid near the free surface. Contact angles are applied as a boundary condition at liquid–solid contact lines. The energy equations in both the liquid and solid portions of the droplet are solved using the Enthalpy method. Heat transfer within the substrate is by conduction alone. Thermal contact resistance at the droplet–substrate interface is included in the model. We studied the deposition of tin droplets on a stainless steel surface using both experiments and numerical simulations. The results of two different scenarios are presented: the normal impact of a 2.7 mm tin droplet at 1 m/s, and of the oblique impact of a 2.2 mm tin droplet at 2.35 m/s onto a surface inclined at 45° to the horizontal. Images obtained from numerical model were compared with experimental photographs and found to agree well.

Deposition of tin droplets on a steel plate: simulations and experiments

Abstract Impact and solidification of tin droplets on a flat stainless steel plate was studied using both experiments and numerical simulation. In the experiments, tin droplets (2.1 mm diameter) were formed and dropped onto a stainless steel surface whose temperature was varied from 25 to 240°C. Impact of droplets was photographed, and evolution of droplet spread diameter and liquid-solid contact angle measured from photographs. Substrate temperature variation under an impinging droplet was measured. A complete numerical solution of the Navier-Stokes and energy equations, based on a modified SOLA-VOF method, was used to model droplet deformation and solidification and heat transfer in the substrate. Measured values of liquid-solid contact angle were used as a boundary condition for the numerical model. The heat transfer coefficient at the droplet-substrate interface was estimated by matching numerical predictions of the variation of substrate temperature with measurements. Comparison of computer generated images of impacting droplets with photographs showed that the numerical model correctly modelled droplet shape during impact as it simultaneously deformed and solidified. A simple analytical model was developed to predict the maximum spread diameter of a droplet freezing during impact.

Splat shapes in a thermal spray coating process: Simulations and experiments

We studied the deposition of nickel particles in a plasma spray on a stainless steel surface using both experiments and numerical simulations. We developed a three-dimensional computational model of free-surface fluid flow that includes heat transfer and solidification and used it to simulate the impact of nickel partcles. In our experiments, particles landing on a polished stainless steel surface at a temperature below 300 °C splashed and formed irregular splats, whereas those deposited on substrates heated above 400 °C formed round disk splats. Simulations showed that formation of fingers around the periphery of a spreading drop is caused by the presence of a solid layer. Droplets that spread completely before the onset of solidification will not splash. To sufficiently delay the instant at which solidification started in our simulations to obtain disk splats, we had to increase the thermal contact resistance between the droplet and the substrate by an order of magnitude. We measured the thickness of the oxide layer on the test surfaces used in our experiments and confirmed that heating them creates an oxide layer on the surface that increases the thermal contact resistance. We demonstrated that the numerical model could be used to simulate the deposition of multiple droplets on a surface to build up a coating.

Cooling effectiveness of a water drop impinging on a hot surface

Abstract We studied, using both experiments and a numerical model, the impact of water droplets on a hot stainless steel surface. Initial substrate temperatures were varied from 50°C to 120°C (low enough to prevent boiling in the drop) and impact velocities from 0.5 to 4 m/s. Fluid mechanics and heat transfer during droplet impact were modelled using a “Volume-of-Fluid” (VOF) code. Numerical calculations of droplet shape and substrate temperature during impact agreed well with experimental results. Both simulations and experiments show that increasing impact velocity enhances heat flux from the substrate by only a small amount. The principal effect of raising droplet velocity is that it makes the droplet spread more during impact, increasing the wetted area across which heat transfer takes place. We also developed a simple model of heat transfer into the droplet by one-dimensional conduction across a thin boundary layer which gives estimates of droplet cooling effectiveness that agree well with results from the numerical model. The analytical model predicts that for fixed Reynolds number ( Re ) cooling effectiveness increases with Weber number ( We ). However, for large Weber numbers, when We ≫ Re 0.5 , cooling effectiveness is independent of droplet velocity or size and depends only on the Prandtl number.

Dynamics of Splat Formation in Plasma Spray Coating Process

The paper describes recent developments in modeling formation of plasma spray coatings. Specific attention is paid to the three-dimensional simulation of droplet impact and solidification under plasma spraying conditions. It is shown that the extent of maximum spread is primarily determined by the Reynolds number and, to a lesser degree, by the Weber number. Splashing and break-up is shown to be the result of solidification; fluid instabilities do not play a significant role in this regard. Finally, the effect of solidification on droplet spreading is insignificant when the ratio of Stefan number to Prandtl number is much smaller than unity.

On the spreading and solidification of molten particles in a plasma spray process effect of thermal contact resistance

The spreading and simultaneous solidification of a liquid droplet upon its impingement onto a substrate permitting thermal contact resistance has been numerically simulated; the effect of contact resistance and the importance of solidification on droplet spreading are investigated. The numerical solution for the complete Navier-Stokes equations is based on the modified SOLA-VOF method using rectangular mesh in axisymmetric geometry. The solidification of the deforming droplet is considered by a one-dimensional heat conduction model. The predictions are in good agreement with the available experimental data and the model may be well suited for investigating droplet impact and simultaneous solidification permitting contact resistance at the substrate. We found that the final splat diameter could be extremely sensitive to the magnitude of the thermal contact resistance. The results also show that for the condition of higher Reynolds and/or higher Stefan numbers the effect of solidification on the final splat diameter is more important.

Splat formation in plasma-spray coating process*

The paper describes recent developments in modeling formation of plasma-spray coatings. This is a stochastic process in which particle-impact conditions, thermophysical properties, substrate topology, and temperature all play important roles in determining the structure of coating. Specific attention is paid to the simulation of droplet impact and solidification under plasma-spraying conditions. Results of 3-dimensional models show that on a flat surface solidification may cause splashing and break-up of the impacting particle. Undercooling effects result in faster solidification and, thus, smaller extent of spreading. More recent works investigate the effect of surrounding gas on dynamics of impact. It is shown that this could result in entrapment of a bubble under the droplet.

The generalized Laplace equation of capillarity I. Thermodynamic and hydrostatic considerations of the fundamental equation for interfaces

Abstract Theories of capillarity are reviewed. The limitations of the classical theory as developed by Gibbs are demonstrated. The generalized theories of capillarity initiated by Buff, and later by Murphy, Kondo, Kralchevsky, and Boruvka and Neumann are scrutinized by considering the basic requirements of formulating thermodynamic fundamental equations. The different generalized Laplace equations of different theories stem from the different setups of the fundamental equation. It is concluded that only Boruvka and Neumanns (BN) generalized theory satisfies all the requirements of thermodynamics and mathematics. Further, to test the BN theory, a hydrostatic treatment of a two phase capillary system is presented. This non-thermodynamic approach is based on the concept of virtual work as the condition for equilibrium of a capillary system, and on the concept of parallel surfaces for evaluating the stress tensor field and excess properties within the interfacial region. Following a straight-forward procedure, it is shown that the hydrostatic results for surface tension γ and two bending moments, C1, and C2, agree with the results of the BN generalized thermodynamic theory of capillarity. This agreement indicates that the form of the BN fundamental equation for surfaces with the extensive geometric curvatures ( J and K total mean and total Gaussian curvatures, respectively) is the proper expression required to generalize the theory of capillarity.

Simulation of Semi-Molten Particle Impacts Including Heat Transfer and Phase Change

A numerical method is proposed in this article which is capable of modeling the impact and deposition of semi-molten particles. By attributing a high viscosity to the solid region, simulating the motion of solid cores including solidification/melting is made possible inside the particle. The Navier-Stokes equations are solved in combination with the volume-of-fluid technique to track the free surface of the particles. In addition, the heat transfer including phase change is modeled using the enthalpy method. The viscous terms are treated implicitly to relax the restrictions associated with small computational time-steps in explicit formulations. Several case studies with operating conditions of a typical thermal spray process are simulated. These cases include the impact of molten and semi-molten nickel particles in an atmospheric plasma spray process. The effects of various parameters such as particle solid-core diameter, initial velocity, and temperature are investigated. The simulations show that the size of the solid core has an important effect on the amount of particle splashing during the impact.