Mathematical modeling of jet flow around of bodies when applying polymer powder coatings

Mathematical modeling of processes occurring during the spraying of polymer powder compositions on the surface of parts and products is considered. Relations describing the flow around processed bodies, movement of the powder particles in a stream of the carrying medium, dynamics of their deposition on the body surface are presented.


Introduction
At the present time, coatings based on the polymer powder compositions (PPC) [1] are becoming more common in many industries, construction and transport. Promising methods of PPC spraying are jet methods such as flameless spraying on a heated surface of parts and products, hot spraying, spraying under the influence of centrifugal forces in an electrostatic field. Internal gas-dynamic, thermal problems in the cavities of spraying guns [2], other spraying devices and external problems when considering jet nonisothermal inleakage of the carrying medium (air, combustion products) containing the powder particles on the surface of parts and products are distinguished at mathematical modeling of processes occurring in the implementation of the named methods. It should be noted that in these problems apart from the dynamics of gas and particles the heat exchange of powder particles with the carrying medium, thermal behavior of the powder coating material layer and in some cases the heat exchange of a jet with the environment are of great importance. Similar problems are encountered at jet painting, abrasive machining of parts and products, work of ventilation systems of industrial and residential premises, reversing devices, when studying the processes occurring during the takeoff of the aerospace objects and in other cases [3][4][5][6].

Relations describing the flow around processed bodies, movement of the powder particles, dynamics of their deposition on the body surface
Models of an ideal either viscous compressible or incompressible fluid can be used for a mathematical description of the medium movement when spraying the polymer powder compositions on parts and products according to the classification of jet streams [7]. In this case, the medium can be considered as a homogeneous fluid. However, in the considered processes it definitely represents a multiphase gas suspension or depending on the particles sizes the aerosol. Wherein disperse particles may or may not have an electric charge. Furthermore, as mentioned above, it is necessary to take into consideration the interaction of the jet with the environment. A jet can spread in a stationary or moving medium in the absence or presence of limitations (straitened jets). When the jet flows around parts and products the latter also represent limitations moreover they can be of very different shapes, sizes, they can move, rotate relatively to the source of the jet or be stationary. Finally, plane, axisymmetric, circular, fan, nonswirled or swirled jet can be identified by the shape and nature of medium movement in the jet. In this work it is assumed that the jet fluid is ideal, incompressible and contains small sized uncharged particles of a polymer powder. There are no limitations of a jet flow with the exception of a streamlined body. Thermal interaction of a jet with environment and particles with the carrying medium is disregarded. Since in practice mainly circular or slotted nozzles are used when spraying only plane and axisymmetric gas powder jets are considered below.
For a plane-parallel fluid flow solutions are obtained as a rule by the conformal mapping method, for example, the problems of jet inleakage on the plate of finite length, semi-infinite and infinitely stretched plate [8-10 and others]. Peculiarities of the jet flow around the bodies of arbitrary crosssection, circular cylinder, in particular, rotating one are discussed in [11]. Free jet, zone of flow reversal and jets spreading over the surface, probably, separating from it can be marked out in the flow region when a fluid or gas jet flows at a moderate speed on the body. Specificity of studies of jet dynamics, movement of powder particles in the flow when spraying is that from the practical point of view the greatest interest represents the zone of flow reversal in the vicinity of the body front point where particles deposit on the surface.
In the case when the width of the jet cross-section near the body is greater than its transverse size it can be expected that the jet flow in the reversal zone differs only slightly from the flow in the same region when unbounded stream flows around a body. Here, solutions to the problem of an unbounded stream flowing around a body can be used in the said zone to assess the behavior of the powder particles. Thus, for flow around a stationary circular cylinder of radius a velocity vector components of a fluid in the cylindrical coordinate system   Here 0 v -fluid velocity at a distance from the body directed along the r-axis.
If the cylinder rotates around the Oz -axis counterclockwise with an angular velocity  then flow function is: Further let us consider the spatial flow around a body. If the body is a sphere of radius a , the flow is axisymmetric and the symmetry axis coincides with the Oz -axis of cylindrical coordinate system   ,, zr with the origin at the center of the sphere then according to [12] flow function is: In the case of axisymmetric flow around a semi-infinite body of rotation the flow function is where r  -radius of the body cross-section removed far from the front point located at a distance of 0.5r  from the origin.
In addition to the mentioned above cylindrical coordinates we introduce Cartesian coordinates Ox -axis is directed along the velocity vector of the oncoming flow. At a distance from the surface of the subject body, upstream, in the transition region from the zone of a free jet to the zone of flow reversal we single out section 0 S perpendicular to the velocity vector of the oncoming flow. We assume that in this section polymer powder particles are thrown into the flow. Confining ourselves to the case of two-dimensional flow of the medium (plane-parallel or axisymmetric flow) the initial coordinates of the particles are denoted as Following the Lagrangian continual approach, neglecting the forces of gravity, added masses and Bosse, to describe the motion of powder particles in a fluid we use the following relations: Here  -time, Considering that in equations (5) Hence the particles coordinates Under condition that 1 R    these dependencies can be simplified: Relations (7,8) are equations of particles trajectories in the parametric form. Using them it is possible to find the intersection points of these trajectories with the streamlined body surface, estimate the mass distribution of the powder on its surface.