## 1. Introduction

Thermal spraying is a technique in which small molten particles of metals, alloys or ceramics (tens of μm) are sprayed at high speed (above 100 ms

^{−1}) onto a substrate creating a protective coating. One of these thermal spraying processes is the atmospheric plasma spray (APS). The mechanisms involved in the flattening are described in several studies where the different variables that can affect the final form of splat are analysed [

1,

2,

3].

After impact, the particles solidify on the substrate producing thin and dense lamellae or layers, called splats, whose shape depends on the size, shape and powder material properties, flying particle temperature and impact velocity and roughness and temperature of the substrate, in short, heat transfer and flow fluid. The coating’s main characteristics, the adhesion strength on the substrate and the residual stresses generated at the interface depend on these properties [

1,

4,

5,

6,

7,

8,

9,

10,

11,

12,

13,

14,

15,

16].

Therefore, it is necessary to develop a deep understanding of the physical principles involved in the spread and solidification of the droplet, in order for the spraying process parameters, coating adhesion strength and splat morphology to be correlated and, as a result, the coating quality can be controlled [

17].

With regard to the solidification, the cooling rate (CR) determines the phase in the solidified splat. Thus, the models developed by de Levi et al. provided the following results: an amorphous phase with a cooling rate around 10

^{5} K/s; a gamma phase with slower cooling rates; and finally, an alpha phase with a cooling rate in the range from 1 to 100 K/s [

18]. On the other hand, several unidimensional analytical models show that the CR depends on [

19,

20,

21,

22,

23]: the contact between the splat and the substrate, the melting latent heat, the relationship between the thermal diffusivities of the substrate and the splat and the inverse of splat thickness. Sobolev et al. obtained similar results with a simplified model [

24,

25].

The interface, which is usually defined by the inverse of the heat transfer coefficient in the interface, called contact thermal resistance, R

_{th}, also depends on the absorbates and condensates in the substrate surface and also has an influence on the CR [

26,

27,

28,

29,

30,

31,

32,

33]. These products can disappear if preheating is applied on the substrate surface before the coating process. Yang et al. [

34] have also evaluated the effect of substrate preheating on the splat grain structure. Fukumoto et al. [

35] investigated the fractional change of the disk splat with an increase in the substrate temperature as well as the microstructure of the cross section of the splats using Y

_{2}O

_{3,} YSZ and Al

_{2}O

_{3} powder materials that were plasma sprayed on an AISI304 stainless steel substrate.

After establishing that the phases in the splat are the consequence of the CR, it is possible to induce the CR from the phases and use experimental information in order to verify the results of a simulation. Thus, using the nucleation theory at a steady regime and the heterogeneous nucleation from the interface in APS, it is possible to verify the high values of the CR, up to 10

^{9} K/s [

23,

36,

37,

38].

The main aim is the design of a network model for the solidification problem that is reliable and efficient and simulated in open source software for circuit analysis like NgSpice [

39], which also provides the transitory solution. This code allows the user to introduce easily and efficiently non-linear effects which are not available in commercial software. Furthermore, it can be customized to a certain application in order to reduce the computational time. The code is flexible enough to allow the calculation of new material, boundary conditions and more complex geometry with a simple change in a few lines. It should be noted that the code provides the temperature gradient, an interesting parameter to determine the phase. The aforementioned model is based on the partial differential equations (PDEs) that form the governing equations and that are spatially discretized. The first step to build it is to establish the equivalence between the electrical and thermal variables, in this case, voltage and temperature. Then, the PDEs linear terms are implemented using electrical devices like coils, resistors and capacitors, while coupled and nonlinear terms are implemented using controlled sources or auxiliary circuits. In addition, suitable electrical components are used to implement the boundary and initial conditions.

The network simulation method has demonstrated its efficiency in many engineering and science problems, such as electrochemistry, heat transfer, inverse problems, deformable solid mechanics, oxidation processes, solute and fluid flow transport, dry friction problems, etc. [

40,

41,

42,

43,

44,

45,

46,

47,

48,

49,

50].

## 3. The Network Model

The basic rules for designing a network model can be found in González-Fernández [

52] and Sánchez-Pérez et al. [

53]. However, to understand it better the steps for its design are described below. The circuit’s simulation software NgSpice [

39], our choice to simulate the network model, uses the most powerful computational algorithms to solve strongly coupled and non-lineal mathematical models as described above. Based on the thesis of Nagel [

54], the algorithms are described in the NgSpice’s manual [

55], include Gear’s methods [

56] and trapezoidal integration [

57]. The reduction of the local truncation error and the stability in the convergence of these methods provide greater accuracy to the solution.

As mentioned in

Section 2, the splat is represented as a rectangle. In order to consider the thermal inertia of the substrate, a portion of substrate, large enough in comparison with the splat, is added,

Figure 1.

This model can be applied to more complex geometries as in

Figure 2, where (a) corresponds to a slow impact, while (b) represents a fast impact.

From Equations (1) and (2), the mathematical model can be expressed as two coupled networks. Each equation represents the heat transfer in each material phase.

The temporal discretization is implemented directly by the open source software used, NgSpice [

39], which sets the time step up automatically in order to obtain the convergence faster. The spatial discretization must be defined by the user of this program, considering that the balances are applied in the cell and not in the central node of the cell.

Figure 3 represents the geometry of each cell used in the discretization.

The first term in Equation (1) is transformed in:

and the second and third ones as:

Figure 4 represents a cell inside the network model. In this circuit, the terms in Equations (8) and (9) are equivalent to electric currents, I

_{B} or I

_{R}, respectively, which are balanced in a common node.

The term dT

_{k}/dt is equivalent to a current passing through the condenser C

_{Tk}. The condenser voltage, V

_{CTk} = C

_{Tk}^{−1}·∫(dT

_{rk}/dt)·dt is simply the variable T

_{rk} if C

_{Tk} = 1F. To fulfil the requirement of continuity in direct current analysis established by NgSpice code, a resistor with a high value of resistance, R

_{Inf}, is used in parallel with some elements [

39]. The initial conditions are applied directly by the definition of the condenser.

The term in Equation (8) is represented by a current controlled source, I_{Bk}, and the four terms in Equation (9), which correspond to the currents I_{R1Tk}, I_{R2Tk}, I_{R3Tk} and I_{R4Tk}, are implemented by simple resistances, R_{1Tk}, R_{2Tk}, R_{3Tk} and R_{4Tk}, respectively, as the constitutive equation for these electrical elements is i_{R} = V_{R}/R. Thus, the resistances of these elements are defined by R_{1Tk} = R_{3Tk} = ∆z^{2}/2α_{l} and R_{4Tk} = R_{2Tk} = ∆r^{2}/2α_{l}. Finally, Equation (2) is discretized in an analogous way.

The circuit in

Figure 4 shows several branches with two resistors, one for the liquid phase and another for the solid phase. Each resistor is connected to the cell centre by a switch, which is closed as the cell has the corresponding phase. Initially, the currents go through the resistors associated to the liquid phase. When the solidification finishes, the resistor associated to the solid phase is connected to the common node and the other one is switched off.

The cell temperature during the solidification is constant and equal to the melting temperature. This is ensured by a battery connected at that moment. After the solidification, a second switch disconnects the battery.

The Stefan equation, Expression (4), whose physical meaning is the advance of the border, can be discretized after an integration during the solidification process, and a division by Ω

_{cell}:

The first and second terms in Equation (10) can be discretized by:

Each term in Expression (11) is represented by a current controlled source. The integration is implemented by a condenser connected to the current controlled sources during the solidification process, which starts when the cell reaches the solidification temperature, and disconnects when the condenser voltage reaches the unit value.

Figure 5 represents this circuit.

The convective heat transfer, Equation (5), can be represented as:

The radioactive heat transfer, Expression (6), can be represented by:

These terms corresponds to a current controlled sources.

Equation (7) must be transformed in order to make it easier to use in the network model:

where k

_{gap}= Δz

_{gap}/R

_{th}.

After this transformation, this term is represented in the circuit by a resistor which joins the bottom of the splat cells with the top of the substrate cells. Thus, the resistance of these electrical devices is R_{1Tk} = ∆z^{2}/2α_{gap}, which is equivalent to R_{1Tk} = ∆z^{2}·ρ_{salp}·c_{p,splat} /2k_{gap} = ∆z·ρ_{splat}·c_{p,splat}·R_{th}/2.

## 5. Conclusions

A numerical model has been designed for the solidification of alumina splat, without making assumptions or linearizing the equations, based on the network simulation method. The code is enough flexible to allow the calculation of new material, boundary conditions and more complex geometry with a simple change in a few lines.

The model solves the problem in negligible calculation times and shows accurate results. The average splat dimension has been obtained from experimental tests and the influence of preheating the substrate has been considered. The results, to be more precise the temperature of tin solidification and the forecast of alumina phase by temperature gradient, an unusual representation which implies additional calculation, are consistent with the experience. It is worth highlighting the study of the cooling rate, an interesting parameter to determine the phase, where in some cases it is higher than 10^{5} K/s, a value that establishes the beginning of the amorphous phase. In contrast to other studies, the proposed method is implemented in open source software and is flexible enough to enable it to be integrated in a greater application to monitor a manufacturing process.

A sensitive study of the mesh was included in order to optimize the computational time. This study shows the difficulty of choosing the correct discretization because of the opposing effect of the error sources.