A CCURATE representations of the drag coefficients of -^Vspheres over a wide range of flow conditions are a necessary prerequisite to the calculation of gas-particle flows. For greatest utility these representations should be in a form suitable for computer calculations. Two such correlations, both still used in computer programs, were published prior to the collection of a considerable body of experimental data, and, consequently are outdated. A third representation, published subsequently, requires the use of tabular data, and is inaccurate in some flow regimes of interest. This paper presents a new correlation, suitable for utilization in computer calculations, which simplifies in the limit to certain equations derived from theory, and which offers significantly improved agreement with the experimental data. The flow regimes of interest include continuum, slip, transition, and molecular flow at Mach numbers up to 6, and at Reynolds numbers up to the laminar-turbulent transition. The effect on drag of a temperature difference between the sphere and the gas is incorporated.

Drag Coefficients of Spheres in Continuum and Rarefied Flows

A review of compressibility and rarefaction effects on spherical particle drag was conducted based on existing experimental data, theoretical limits, and direct simulation Monte Carlo method results. The data indicated a nexus point with respect to effects of Mach number and Knudsen number. In particular, it was found that a single drag coefficient (of about 1.63) is obtained for all particle conditions when the particle Reynolds number is about 45, and is independent of compressibility or rarefaction effects. At lower Reynolds numbers, the drag is dominated by rarefaction, and at higher Reynolds numbers, it is dominated by compressibility. The nexus, therefore, allows construction of two separate models for these two regimes. The compression-dominated regime is obtained using a modification of the Clift-Gauvin model to specifically incorporate Mach number effects. The resulting model was based on a wide range of experimental data and showed superior prediction robustness compared with previous models. For the rarefaction-dominated regime, the present model was constructed to directly integrate the theoretical creeping flow limits, including the incompressible continuum flow limit (Stokes drag), the incompressible weak rarefaction limit (Basset-Knudsen correction), and the incompressible free-molecular flow limit (Epstein theory). Empirical correlations are used to extend this model to finite particle Reynolds numbers within the rarefaction-dominated regime.

Compressibility and Rarefaction Effects on Drag of a Spherical Particle

Cold spraying is increasingly attracting attentions from both scientific and industrial communities due to its unique ‘low-temperature’ coating build-up process and its potential applications in the additive manufacturing across a variety of industries. The existing studies mainly focused on the following subjects: particle acceleration and heating, coating build-up, coating formation mechanism, coating properties, and coating applications, among which particle acceleration and heating can be regarded as the premise of the other subjects because it directly determines whether particles have sufficient energy to deposit and form the coating. Investigations on particle acceleration and heating behavior in cold spraying have been widely conducted both numerically and experimentally over decades, where many valuable conclusions were drawn. However, existing literature on this topic is vast; a systematical summery and review work is still lack so far. Besides, some curtail issues involved in modeling and experiments are still not quite clear, which needs to be further clarified. Hence, a comprehensive summary and review of the literature are very necessary. In this paper, the gas flow, particle acceleration, and heat transfer behavior in the cold spray process are systematically reviewed. Firstly, a brief introduction is given to introduce the early analytical models for predicting the gas flow and particle velocity in cold spraying. Subsequently, special attention is directed towards the application of computational fluid dynamics technique for cold spray modeling. Finally, the experimental observations and measurements in cold spraying are summarized.

Gas Flow, Particle Acceleration, and Heat Transfer in Cold Spray: A review

In this study, analysis of flow properties around a sphere and its aerodynamic coefficients in the high-Mach-and-low-Reynolds-numbers conditions is carried out by direct numerical simulations solving the three-dimensional compressible Navier–Stokes equations. The calculation is performed on a boundary-fitted coordinate system with a high-order scheme of sufficient accuracy. The analysis is conducted by assuming a rigid sphere with a Reynolds number of between 50 and 300, based on the diameter of the sphere and the freestream velocity and a freestream Mach number of between 0.3 and 2.0, together with the adiabatic wall boundary condition. The calculation shows the following yields: (1) unsteady fluctuation of hydrodynamic forces become smaller as the Mach number increases under the same Reynolds number condition, (2) the drag coefficient increases with the Mach number due to an increase in the pressure drag by the shock wave, and (3) an accurate prediction of the drag coefficient in the supersonic regime using traditional models might be difficult.

Investigation on subsonic to supersonic flow around a sphere at low Reynolds number of between 50 and 300 by direct numerical simulation

The two-phase flow properties of copper particle laden nitrogen are computationally modeled and compared with the data obtained from the experiments, determining the achievable degree of consistency between model and reality. Two common, commercial nozzles are studied. A two-way coupled Lagrangian scheme along with the RSM turbulence model is used to track the particles and to model the interactions between the gas and the particulate phase. Significant agreement is found for the geometrical gas flow structure, the resulting particle velocities, and the dependence of the two-phase flow on the particulate phase mass loading. The particle velocities decrease with increasing mass loading, even for modest powder feed rates of <3 g/s. The velocity drop occurs even when the gas flow rate is kept constant. Adiabatic gas flow models neglecting the energy consumption by the particles are thus inaccurate, except for very dilute suspensions with low technical relevance. For the cases modeled, the experiments evidence the high predictive power of the chosen CFD approach.

C. T. Crowe

Papers

Drag coefficient of particles in a rocket nozzle.