3 Analytical model of thermally deposited spray spot pattern
3.5 Model of spray spot from double and multiple injectors
The coating pattern produced by spraying with multiple injectors can be represented as a superposition of the multiple spots produced by single injectors. Typically a double and triple injection is
used for APS spray processes. In the case of double injection the coating powder is fed from two external injectors placed at the opposite positions of one diameter line on the spray gun. This placement makes
possible a
symmetrical
distribution of powder flow around the central axis of
the jet, which consequently leads to a symmetrical spray spot. As experimentally investigated and discussed in Chapter 1.3.2, the distribution of mass flux is practically controlled by carrier gas flow. In particular, with an increase in carrier gas flow rate the radial distributions of the powder particles’ sizes, concentration, velocities and, as a result, the particles’ flux, changes from a single bell-like to a double bell-like shape. The shape of the spray spot produced by a double injection can be modeled based on the approach used for single injector spraying. As we showed in Chapter 3.3, a single spray spot can be modeled as an elliptic spot with a Gaussian thickness distribution even for spraying onto an inclined substrate surface. Let us assume that the thickness distribution in a single spot is known by definition of the basic process parameters summarized in Table 3.1. Typical spray spots produced with 8YSZ coating by the APS process are presented in Figure 3.8. In particular, the shape of the spot is defined by a maximal height calculated according to the relation (3.20), standard deviations x, y and the angle of rotation 0. In the case of inclined substrate, as shown in Chapter 3.3, these parameters are functions of spray angle and
Figure 3.8: Optical photographs and models of APS 8YSZ spray spots produced with a) single and b) double injection process.
a) b)
x, mm x, mm
1 cm 1 cm
57 direction of the substrate inclination, defined by the angle
. Let us assume that the displacements of the individual spots produced by one or two injectors are known and equal to x10 x0 and y10 y0 . Assuming a symmetrical placement of the injectors, whose positions are offset angularly by 180°, the displacement of the second spot can be calculated as x20 x0 and y20 y0. The total coating thickness produced by these two injectors can be calculated as a sum of corresponding individual spots:) relation (3.20) in the case of normal spraying and by (3.42) in the general case.
Let us consider the case of spraying with more than two injectors. In practice, the triple and modeled, similarly to the double injection,
as a superposition of multiple single spots, displaced by certain distances from each other.
According to the equation (3.20), for each individual spot we can write following expression:
Assuming a symmetrical placement of injectors, the displacements can be calculated as:
spot and corresponding model according to the equation (3.45) is shown in Figure 3.9.
Figure 3.9: Spray spot produced with triple injection:
a) photo of a real spot, b) model of the spray spot.
a) b)
1 cm
58 From the theoretical point of view, it is important to analyze a contour of the multiple spot in dependence on the displacements of the individual spots. For example, let us perform analytical analysis for the simplest case of a double injection spot. It can be shown that a double spray spot has one of the basic configurations presented in Figure 3.10.
Figure 3.10: Basic configurations of thickness distribution in double injection spots: a) coinciding single spots, b) overlapping single spots with common maximum, c) overlapping double spot with two maximums, d) two displaced almost not interacting spots.
The configuration of the double spray spot is defined by a relative displacement of the single spray spots from each other. In order to investigate a spot configuration analytically, let us assume that each single spot is displaced along the x axis, furthermore that this axis coincides with the major axis of the spot ellipse. In this case, the coating thickness distribution in the cross section of the spot can be described by the following equation:
) 2 (
) ,
( 2
2 0 2
2 0
2 ) ( 2
) ( 0 0
eff eff
x x x
x
e h e
x x
h
. (3.47)
As presented in Figure 3.10, the spray spot contour can vary from the configuration with one single peak in the middle to the configuration with two peaks corresponding to the
a) b)
c) d)
59 displacements of the spots. Let us find the position of the maximum of the spot. The local maximum position can be calculated using a requirement:
0 injection on the displacement 2x0 of the single spots. As can be seen from the graph, the spray profile can have three possible configurations according to the positions of the double spot maximums.
The configuration of type 1 defines a spot with a single common maximum in the
geometrical centre of the double spot, achieved due to a sufficient overlapping of two single spots. This configuration exists in the range of displacements with x0 eff . Cases a) and b) from Figure 3.9 are examples of this configuration. An intermediate configuration of type 2 exists in the displacement range with eff x0 2eff , and defines a double spot produced by partial overlapping of two single spots with two maximums as in case c) from Figure 3.9.
The configuration of type 3 is formed in the range of high displacements x0 2eff and defines an extreme case of two non-interacting spray spots, with an example d) in Figure 3.9.
For practical use, the configuration of type 1 with a single maximum in the middle of the spot is more preferable due to the high level of spot symmetry. The exact modeling of the spray spots of different configurations, taking into account variations of the process parameters and geometry relations, such as variation of spray distance and angle, are the necessary requirement for a realistic simulation of more complex coating patterns such as a spray profile and complete coating layer.
Figure 3.10: Dependence of the common maximum position of the double spray spot on the relative displacement of the single spots.
0,00
0,00 0,50 1,00 1,50 2,00 2,50 3,00
a/sigma xmax/sigma
Type 1
Type 2
Type 3
60
Figure 4.1: Modeling of the coating deposition process.