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Contributions to heat and mass transfer between a plasma jet and droplets in suspension plasma spraying
SCT-20026; No of Pages 6 Surface & Coatings Technology xxx (2015) xxx–xxx
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Contributions to heat and mass transfer between a plasma jet and droplets in suspension plasma spraying F. Girard a,b, E. Meillot a, S. Vincent c,⁎, J.P. Caltagirone b, L. Bianchi a a b c
CEA-DAM, Le Ripault, F-37260 Monts, France University of Bordeaux, I2M-TREFLE UMR CNRS 5295, 16 Avenue Pey-Berland, 33607 Pessac, France Univ. Marne-La-Vallée, MSME UMR 8208, 5 Bd Descartes 77454 Marne-la-Vallée Cedex 2, France
a r t i c l e
i n f o
Available online xxxx Keywords: suspension plasma spraying droplet computational fluid dynamics heat and mass transfer
a b s t r a c t In suspension plasma spraying, nano-sized particles are injected via a liquid or a solvent into a plasma jet. Due to shear by the plasma jet, the liquid is fragmented into small droplets flying into the gas flow. The present work has focused on a single droplet evaporating into the plasma environment, and considers, on the one hand, the physical analysis of the phenomena and, on the other hand, the characterization of the solvent’s thermal and motion characteristics of evaporation by means of direct numerical simulation of a multiphase flow. Specific parameters such as the evaporation time of the droplet have been estimated. In addition, a validation study has been proposed by comparing numerical simulation and physical analysis. The direct numerical simulation of multiphase flow was based on a compressible 1-fluid model, modified for handling phase change. The employed numerical methods included the volume of fluid (VOF) approach for interface tracking and finite volumes on structured grids. © 2015 Published by Elsevier B.V.
1. Introduction As exposed in [1], plasma spraying is commonly used for depositing ceramic coatings on the surfaces of industrial parts. Its principle is to use a plasma jet to accelerate and melt sub-millimeter particles that will coat the treated surface. As explained in [2], the coating properties depend highly on the spraying parameters such as particle size and density, plasma enthalpy, momentum, viscosity, chemical composition, and spraying distance. A pertinent way to improve the coating properties is through particle size reduction from the micrometer to the nanometer scale. Introducing nanoparticles in a plasma core requires putting them in a solution or suspension due to their low inertia [3]. So, from this starting point, understanding suspension plasma spraying implies different stages of analysis: - The first step involves hydrodynamic interactions between the charged liquid with the plasma jet leading to the breakup of the liquid suspension jet into droplets. - The second step is the heat treatment of these droplets and the influence of added liquid fluid, with its phase changing from liquid to gas, in the surrounding plasma. - The third step includes the behavior of the nanoparticles trapped in an unsteady environment. ⁎ Corresponding author at: Université UPEM - Paris-Est, Laboratoire Modélisation et Simulation Multi Echelle (MSME), UMR CNRS 8208, 5 boulevard Descartes 77454 Marne-La-Vallée.
- The final step takes into account the behavior of free nanoparticles in the mixture constituted by a plasma atmosphere and steam before impacting the substrate. The first step of hydrodynamic interaction has been previously investigated [4]. Consequently, this paper is focused on the second step, i.e., the heat treatment of as-fragmented droplets. The way the droplets transform in steam has a high influence on the behavior of the nanoparticles trapped in the droplets. We thus ask ourselves, can the particles agglomerate with each other or do they, on the contrary, remain separated? This work focuses on the heat transfer of post-fragmented water droplets flying in a plasma jet in order to determine which phenomena, upon convection and radiation, has the most significant effect on the heat transfer. As the experimental approach is very complex to apply to thermal plasma spraying because of the extreme temperature and velocity of the process, an alternative means of investigation by analytical and numerical analysis was performed. Few studies have been devoted to the analytical or numerical exploration of droplet evaporation in plasma at small scales. Among them, we can cite [4,5]. The present paper is therefore organized as follows: - The first section details the description of the considered setup under investigation with related assumptions. - The second section is dedicated to the analysis of a droplet’s trajectories in the plasma leading to its in-flight time. - Section three presents an analytical study underlining the main phenomena occurring in the boiling process of a droplet.
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- The last section presents direct numerical simulations of droplet motion in the plasma. - Finally, conclusions and perspectives are given.
2. Experimental and numerical setup Fig. 1 presents a scheme of the industrial process of suspension plasma spray coating. The plasma torch creates an electric arc between a cathode and an anode, which ionizes a variety of gases such as argon, hydrogen, and helium. The high-temperature plasma gases flow outside the torch with a high velocity before impacting the surface to be coated. The colloidal suspension is injected in this jet, fragmented and evaporated, in such a way that the nanoparticles can be heated and accelerated by the jet until they impact the surface and form a coating. As explained previously, the analysis is dedicated to free water droplets, in a first approximation without any ceramic nanoparticles. A suspension can be slightly more viscous, and generally presents a lower surface tension [6], but water droplets were nonetheless considered in this study for the sake of simplicity. Despite that the surface tension might vary with temperature [7], the fixed value of 0.061 N.m− 1 from [9] was retained. No break-up was taken into account: the analyzed droplets originated from the upstream destruction of the injected jet and were deduced from [4].
Fig. 2. Plasma velocity (m.s−1) and investigation zone with a continuous liquid jet (Fig. 19 from [8]).
2.2. Physical properties Table 1 lists data deduced from [4] and describes the average state of the different fluids at the appearance of the first fragmented droplets. 2.3. In -flight time of a droplet and average relative velocity
2.1. Local assumptions The forthcoming analyses were based upon the injection of a continuous water jet into an Ar/H2 plasma plume. The operating conditions and an explanation of these choices can be found in [8,9] and are not detailed here. As shown in Fig. 2, the continuous liquid jet was completely disintegrated into droplets 3 mm downstream of the torch exit. The asfragmented droplets were immersed in a plasma atmosphere at a high temperature, around 10,000 K, while the surrounding gas flowed at more than 500 m.s−1. The PDF (probability density function) distribution of the droplet revealed an average diameter of 20 μm for the droplet population prior to the evaporation. Due to the inertia of the continuous liquid shape, at the initial moment when fragmentation started, the asfragmented droplets had no velocity. They were accelerated quickly by the plasma drag force. In the forthcoming text, only 20-μm mean droplets were considered with a null initial velocity. At this point, the relative speed between a droplet and the plasma flow was 500 m.s−1. With the acceleration due to the plasma, the droplet velocity increased to 500 m.s−1 and the relative speed tended toward 0. Nevertheless, to investigate larger or smaller penetration of the continuous liquid jet, other plasma flow velocities were also taken into account: 1000, 600 and 400 m.s−1. For the sake of clarity, the use of “droplet” in the forthcoming text implies water liquid entities while “particle” refers to nanoparticles inside the droplets.
First of all, the in-flight time of the droplet was a necessary parameter in order to determine the hierarchy of the phenomena, and to obtain a limit for considering or not the different possible physical contributions to droplet evaporation. When taking into account a single droplet of 20 μm diameter in a plasma flow, Newton’s law of conservation of momentum was employed to follow the droplet as a single Lagrangian point and to compute an approximate flying time between the fragmentation point and the target substrate at 40 mm. For the sake of simplicity, we calculated the Reynolds number of an isolated droplet, Re, which showed that the flow was very laminar. Using Re, the drag coefficient CD can be approximated by the Schiller and Naumann formula [10]. Re ¼
ρpla V rel ϕdrop ¼ 1:14 μ pla
ð1Þ
where ρpla is the plasma density, φdrop is the droplet diameter, Vrel is the relative speed of the plasma flow, and μplasma is the plasma viscosity at 10,000 K. This gives: CD ¼
24 0:687 1 þ 0:15Re Re
ð2Þ
Fig. 3 shows the droplet velocity when it reaches different plasma states represented by various velocities (1000, 600 or 400 m.s− 1)
Fig. 1. Principle of a plasma spray process.
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Table 1 Physical properties of the different fluids for the studied case. Physical properties
Plasma (10,000 K)
Steam (400 K)
Liquid water (373 K)
Density ρ (kg.m−3) Dynamic viscosity μ (kg.(m.s)−1) Thermal conductivity λ (W.(m.K)−1) Specific heat cp(J.K−1.kg−1) Molecular mass m (kg.mol−1)
0.0291 2.56 10−4 1.69 2.34 103 0.0305
0.6 1.43 10−5 3.89 10−2 1.9 103 0.018
1000 2.82 10−4 2.76 10−2 4.185 103 0.018
versus time. The same graph displays a plot of the relative speed, between the plasma and the droplets, for these different cases. So, the order of magnitude of the flight time was 10−4 s. Regarding the curves of the relative velocity, it is obvious that it was higher than 100 m.s−1 over the entire flight. Consequently, to observe a relative flow rather representative of the relative speed order of magnitude during the inflight time of the droplet, the mean relative speed chosen in next steps was set to 500 m.s−1. As the experiments showed that no liquid water arrived to the substrate, this means that the water was evaporated in less than 10−4 s. This time is an upper limit: in fact, due to the vaporization of the water leading to a loss of matter, the kinetic inertia of the droplets decreased, resulting in an acceleration of the droplet and this in a shorter dwell time. The next part presents the analytical calculation to determine which type of heat transfer evaporated the post-fragmentation droplet.
2.4. Characteristic conductive heat transfer speed versus characteristic speed of steam ejection This section assumes that the droplet is heated in any way. The consequences of this heating were analyzed through the speed of steam ejection during the first instant when the phenomenon occurred. During these first moments, the Kolmogorov scale was computed by [4] to be on the order of 10− 4 m while the droplet diameter was 20 10−6 m. So, from the point of view of the droplet, in first approximation, there was no turbulence, only a laminar flow. First of all, the calculation of the lower limit of the interface motion speed is presented. The evaporation of a 20-μm droplet of water in less than the in-flight time of 10−4 s implies that the interface consumption had a minimal mean speed (Vinterface) = −0.1 m/s, provided that the evaporation time was equal to the in-flight time. Two assumptions were possible here. The first implied that the environment was at atmospheric pressure (that of the plasma flow) and that the vapor density thus remained at the one described in Table 1.
From the mass flow rate balance at the interface, where the water liquid transformed into vapor, the vapor speed could be calculated and approximated to 160 m.s− 1 at the droplet interface. This was a lower limit of the boiling phenomenon if the solvent was completely evaporated in less than 10−4 s. The second assumption involved the fast volume change of liquid into vapor at the moment of evaporation leading to an increase in the local pressure around the droplet and thus to a change in vapor density. In this case, there was a decrease in the vapor speed. However, due to the conservative momentum equation balance between the plasma flow and the vapor, and to the open space field in the plasma flow, the vapor pressure decreased to that of the surroundings so that there was a balance with the mechanical barrier representing the plasma flow. This could lead to a smaller boundary layer of vapor around the droplet. However, as the vapor density decreased with the pressure, the vapor volume increased because of mass conservation. To conclude this part, the two assumptions (with or without pressure evolution) led to the same result: the existence of a vapor boundary layer around the droplet which had a lower limit of ejected vapor speed around 160 m.s−1. Of course, there could be a mixture of the vapor boundary layer with the plasma but the molecular diffusion was a very slow phenomenon (with a characteristic time of 10−2 s calculated by Fick’s law) and we previously noticed that the droplet scale was smaller than the Kolmogorov scale of turbulence, wherefore the droplet was considered to be in a laminar flow. This laminar hypothesis was made in the first run for simplicity reasons, but needs to be checked again. On the contrary, the upper limit of the heat transfer by conductive heating must be calculated. Consequently, one can compare the speed of the ejected steam to the speed of sound in the steam layer at the boiling temperature and atmospheric pressure, making the assumption of γ = 1.12. The sound speed in vapor was 4.4 102 m.s−1. So, if the steam ejection was supersonic, using the mass flow rate balance would lead to an interface speed Vinterface of 0.263 m.s−1, and an evaporation time for a 20-μm droplet of 38 10− 6 s. If the droplet
Fig. 3. Droplet speed and relative plasma/droplet speed computed before impacting of the substrate at 40 mm.
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was evaporated in less than this time, the steam ejected would be supersonic, and as a shockwave, become a boundary. Indeed, the speed of sound is a limit that mechanical information cannot overcome. As conductive heat transfer corresponds to macroscopic propagation by successive shocks and thermal kinetic energy transmission, it cannot be faster than the sound of speed. As a consequence, the droplet cannot be evaporated in less than 38 10−6 s by conductive heat transfer. During this lapse of time, the plasma jet and steam interacted with each other. Calculations of their volumetric momentum revealed that the steam momentum was higher than that of the plasma (respectively, 90 and 15 kg m s−1 m−3) near the droplet interface. This implies that if the boiling was uniform, the steam would push the plasma, and prevent it from touching the droplet. Consequently, the conductive heat transfer did not occur between the hot plasma and the cold droplet, but through a layer of steam, from a hot heat source to a cold target. The next point was to determine the thickness of the vapor layer. At the scale of the process, the evaporation took between 38 and 100 μs. Considering the droplet scale, we assumed that in an arbitrary virtual spherical surrounding of the droplet, most of the mass was in the droplet. If we observed the phenomenon in an infinitely small time scale, the variation of the density in this virtual sphere was not high, whereas the ejected steam moved a lot. In other words, a mass flow rate of steam with a volumetric momentum comparable to that of plasma did not change the average density of a virtual sphere around the droplet because 99.9% of the mass remained liquid, while the steam already moved over a great distance. The characteristic time of the mechanic steam/plasma interaction was very small compared to the evaporation in 38.10−6 s. As the interface moved so slowly compared to the steam ejection, the radius of the droplet could be taken as constant during the rapid advection of the steam. We can thus consider a permanent regime around the droplet. Consequently, the divergence of the flux ρV through a closed surface was zero, i.e., ∇ · (ρV) = 0. This closed surface can be either the interface between the liquid and the steam, or any concentric virtual sphere of radius r. The flux ρV that crossed any of these spheres was the same: 2
2
ρstmðR dropÞ V stmðR dropÞ 4πRdrop er ¼ ρstmðrÞ V stmðrÞ 4πr er
ð3Þ
Finally, when applying the mass flow rate conservation, we get: 2
ρstmðrÞ V stmðrÞ 4π er ¼ ρstmðR dropÞ V stmðR dropÞ 4π
Rdrop 2 er r
ð4Þ
The quantity ρstm(r)Vstm(r) is the volumetric momentum of the steam pstm(r), thus: pstmðrÞ ¼ ρstmðrÞ V stmðrÞ
ð5Þ
2 R drop pstmðrÞ er ¼ ρstmðR dropÞ V stmðR dropÞ er r
ð6Þ
where er is the radial unitary vector, pstm is the steam volumetric momentum at radius r, the quantity ρstm(R drop)(Vstm(R drop).er) is the mass flow rate per unit area at the droplet interface, Rdrop is the radius of the droplet, and r is the radius of the virtual sphere on which the steam volumetric momentum is computed. On the other hand, the momentum balance between plasma and steam, considering that the thickness of the steam layer has a minimal value at the upstream side where the two momentums are in opposition, can be written as pstmðrÞ er ¼ −ppla er
ð7Þ
where pstm (r) and ppla are, respectively, the volumetric momentum of
steam at a distance r from the center of the droplet and the global volumetric momentum of the plasma. Evaluating r reveals the minimal thickness of the steam layer. So, from Eqs. (6) and (7), we have: 2 R drop ppla er ¼ −pstmðrÞ er ¼ −ρstmðR dropÞ V stmðrÞ er r
ð8Þ
and so r, the radius of the virtual sphere where the volumetric momentum of plasma and steam are equal at the upstream side of the droplet, is equal to: vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi u uρstmðR dropÞ V stmðR dropÞ u −5 r¼t Rdrop ¼ 1:83 10 m ρpla V pla
ð9Þ
where ρstm(R drop)|Vstm(R drop)| is the absolute value of mass flow rate per unit area at the droplet interface, ρpla|Vpla| the plasma volumetric momentum, Rdrop is the radius of the droplet, and r is the radius of the virtual sphere on which the steam volumetric momentum is computed to be equal to the one of the plasma, i.e., the radius of the steam sphere surrounding the droplet. In our case, the minimal thickness of the steam has an order of magnitude of r − Rdrop = 8.3 μm, which is not negligible compared to the droplet diameter. From that, we need to know if the plasma molecule can diffuse inside the steam layer. The mean free path of the ejected steam is: λstm R drop
μ ¼ ρ
sffiffiffiffiffiffiffiffiffiffiffiffiffiffi πmH2 O −7 ≈10 m 2kb T
ð10Þ
This is very short compared to the mean free path of the plasma far from the droplet (T = 10,000 K): λpla∞ ¼
μ ρ
sffiffiffiffiffiffiffiffiffiffiffiffi πmpla −6 ≈5 10 m 2kb T
ð11Þ
Consequently, the steam is quite impervious to the plasma, and the heat transfer occurs through the steam. This means that the thermal transfer does not result from a flow sliding around a spherical droplet, i.e., the use of a Nusselt number as in [4,5] is no longer representative of evaporation in this case. It is thus necessary to calculate the characteristic speed of the heat transfer Vtherm through the steam by using thermal diffusivity αsteam: α stm ¼
λstm −5 −2 ¼ 3:27 10 m:s ρstm cpðstmÞ
ð12Þ
where λstm, ρstm, cp(stm) are, respectively, the thermal conductivity, the density, and the specific heat at constant pressure of the steam. Therefore, V therm er ¼ −
α stm −1 −1 ¼ −3:94 10 m s r−Rdrop
ð13Þ
Again, this velocity is very small compared to the steam’s ejection speed |Vstm| = 160 m.s−1. Boiling by conductive heat transfer through the steam layer thus seems impossible with a spherical symmetry. Further research is necessary to estimate the heat and mass transfer in the vicinity of the droplet (see Section 3). 2.5. Orders of magnitude of momentum transfer and pressure field As the Reynolds number is quite small (Eq. (1)), since the plasma density and droplet diameter are very low, the plasma flow around the droplet is a mix between a full viscous regime (called “Stokes regime”) and a full inertial regime. The motion of the
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droplet is driven both by the hydrodynamic pressure on the upstream side, but also by the interfacial shear stress due to viscosity. To know which phenomena occurs, it is necessary to compare the terms of the Navier–Stokes equation: Inertial contribution ρ(V ∇)V: 2
ρpla jV jpla 8 −1 ¼ 3:53 10 Pa m ϕ
ð14Þ
Viscous contribution μ∇2(V) μ pla jV jpla
−1
8
¼ 3:18 10 Pa m
ϕ2
ð15Þ
Capillary contribution: σ 8 −1 ¼ 1:53 10 Pa m ϕ2
ð16Þ
These are all in the same range. It is also interesting to compare the pressure inside the jet downstream of the torch exit (here, the atmospheric pressure) with the total pressure (physically, it is a theoretical pressure inside the torch that generates the momentum and the motion of the plasma jet), to take into account the compressibility of the flow (Mach number M = 0.5). Using a formula available with the ideal gas law in an isentropic flow, assuming the ratio of specific heat γ = 1.2:
P 0 −P ¼ ϕ
P
γ 2 γ−1 1 þ γ−1 −P 2 M ϕ
9
−1
¼ 1:46 10 Pa m
In other words, through a stream trace, assuming an isentropic flow, from the torch to the stop point upstream of the droplet, the pressure of the flow increased by P0 − P = 2.92 104 Pa. As the flow was not isentropic (due to radiative losses and viscous dissipation), this contribution was perhaps less important. All these contributions are of the same order of magnitude, so none of them are negligible, and the order of importance between them depends on the variation in relative speed and on the droplet diameter. To determine an order of magnitude of the pressure field around the droplet, analytical formula are available in Clift et al. [11] for the Stokes regime, taking into account only the viscous contribution. P ext ðr; θÞ ¼ P 0 −μ pla jV rel j
2 þ 3φ ϕ=2 cosðθÞ 1 þ φ 2r 2
5
3. Direct numerical simulation of droplet motion in a plasma flow 3.1. Numerical model The Thetis CFD code, specially designed for multi-phase flows, has been used. The compressibility of the plasma was taken into account with the equations exposed in [12]. A volume of fluid [13] piecewise linear interface calculation (VOF-PLIC) [14] method was utilized to follow the liquid phase. In this work, neither the boiling of the droplet nor the steam generation were computed. Computations have been done with both the ideal gas law and thermodynamic and physical local properties. After having compared 2D simulations with the case of study in 3D, we skipped back to a 2D case, which showed the same behavior. This was done to increase the accuracy of the mesh without having a too high number of cells. The mesh size around the droplet was 50 nm, which means that there were 400 cells in each droplet diameter. In order to have a CFL condition lower than 0.3 (required for stability reasons by the interface tracking algorithm), the time step was 2.5 10−11 s. The domain size was 200 μm long and 100 μm wide. After 20 μm in the y-direction, the mesh became exponential to reduce the computational time. The left side was the input side, with a 500 m.s−1 condition for the gas velocity, and the right side was the output side. The top boundary had a sliding condition, and the bottom boundary had a planar symmetry, so as to divide the number of cells by 2. In order to properly calculate the Laplace pressure, an accurate representation of the interface properties was implemented for the local normal and curvature by [15]. Indeed, without this method and even with a quite thin mesh, spurious oscillations of the interface could be induced, thus disturbing the interface in an artificial and non-physical way. 3.2. Numerical results 3.2.1. Numerical proof of the invalidation of a spherical evaporation by conductive heat transfer Fig. 4 represents the momentum field of the plasma. On the upstream part of the droplet, the momentum was much lower than the value of 90 kg.m.s−1 m−3 of the steam calculated in Section 2.3. Consequently, if the droplet evaporated in 10−4 s, the plasma could not reach the droplet on the upstream side, causing a steam layer to form on the droplet interface. As the evaporation temperature depended on the pressure and temperature, the pressure difference between the
ð17Þ
μ
where ϕ ¼ μ liq ¼ 1:75 pla
The pressure step between upstream and downstream is: ΔðP Þ ¼ 2 μ pla jV rel j
2 þ 3φ ϕ=2 ≈1:2 bar 1 þ φ 2r 2
ð18Þ
As the flow is a combination of Stokes flow and inertial flow, the pressure step around the droplets should be between 3 104 Pa and 105 Pa. Since the Laplace pressure is one order of magnitude below (4σ/ϕ = 1.2 104 Pa), the droplet deforms in the flow. This will influence the heat transfer. The next section presents direct numerical simulations to take into account all contributions in the multi-phase flow describing the interaction of the droplet motion with the plasma flow (without evaporation) and with the spherical shape. These small-scale simulations are compared to previous orders of magnitudes for parameters of heat, mass, and momentum transfer.
Fig. 4. Volumetric momentum of the plasma (kg.m.s−1m−3) at t = 2.5 10−7 s (the plasma flows from left to right).
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was thus deformed under the plasma stress. This fact will be analyzed in future studies to determine the influence of the deformation. 4. Conclusions and perspectives
Fig. 5. Pressure (Pa) at t = 2.5 10−7 s (the plasma flows from left to right).
upstream and downstream parts of the droplet induced different evaporation rates at these points so that the spherical shape during evaporation would no longer occur.
3.2.2. Preferential boiling sites due to a high-pressure gradient Fig. 5 represents the pressure field in the plasma and in the droplet. The pressure field was not symmetric: it was very high in the upstream part and very low in the downstream part, from 7 104 Pa to 1.5 105 Pa. Consequently, the boiling temperature of the surface was not homogeneous, and varied from 363 K to 383 K, according to [16]. This was not negligible because if the initial temperature of the droplet was 300 K, the ratio between the temperature increasing downstream and the temperature increasing upstream was around 25%. Thus, the water on the upstream side had to elevate its temperature by 25% more to reach the boiling point. However, with shear stress, and the viscous dissipation in the injector, the water was probably at a higher temperature when it was injected in the plasma jet. As a result, the previous ratio increased. In this case, the steam was first ejected from places where the pressure of the plasma were lower, which is another argument against a spherical evaporation. In addition, the pressure gradient between the upstream and downstream parts of the droplet was on the order of 105 Pa, one order of magnitude more than the capillary pressure as previously analyzed in Section 2.4. The droplet
The heat and motion of suspension droplet evaporation in a plasma flow has been studied both analytically and numerically through direct numerical simulation. It has been demonstrated that a steam layer formed at the droplet interface and that the droplet shape did not remain spherical during the evaporation. By assuming a fixed spherical shape of the droplet, the pressure gradient along the particle diameter led to a difference in boiling temperature, which resulted in preferential boiling sites. This was a first step to understanding how droplet evaporation starts. Future studies will include investigating droplet deformation due to shear stress, steam creation due to phase change, and completely solving the boiling of a water droplet in a plasma jet. In the last step, the nanoparticle behavior during the boiling might be computed using a multiphase model [17] for an explicit representation of the nanoparticle tracking. The wettability phenomena when the nanoparticles reach the interface should then be important. With the solving of the complete phenomenon, the understanding of the potential agglomeration of nanoparticles during the boiling of the solvent should be achieved. References [1] P. Fauchais, A. Vardelle, B. Dussoubs, J. Therm. Spray Tech. 10 (1) (2001) 44–66. [2] P. Fauchais, J. Appl. Phys. D Appl. Phys. 37 (2004) 86–108. [3] P. Fauchais, A. Vardelle, Solution and Suspension Plasma Spraying of Nanostructure Coatings, Advanced Plasma Spray Applications, Global Analysis, Pure and Applied, InTech, 2012. [4] C. Caruyer, Modélisation de nanomatériaux injectés par voie liquide dans un jet de plasma pour la fabrication de nanostructures(Ph. D. thesis, in French) Université de Bordeaux, 2011. 1. [5] K. Wittman-Teneze, Etude de l'élaboration de couches minces par projection plasma(Ph.D. thesis, in French) Universite de Limoges, 2001. [6] L. Dong, D. Johnson, Adv. Space Res. 32 (2) (2003) 149–153. [7] E. Mezger, J. Phys. Radium 7 (10) (1946). [8] E. Meillot, S. Vincent, C. Caruyer, D. Damiani, J.P. Caltagirone, J. Phys. D. Appl. Phys. 46 (22) (2013). [9] E. Meillot, S. Vincent, C. Caruyer, J.-P. Caltagirone, D. Damiani, J. Therm. Spray Tech. 18 (5–6) (2009) 875–886. [10] J.A. Dean, Lange's Handbook of Chemistry, R.R. Donnelley & Sons Company, 1999. [11] R. Clift, J. Grace, M. Weber, Bubbles, Drops and Particles, Academic Press, 1978. [12] J.-P. Caltagirone, S. Vincent, C. Caruyer, Comput. Fluids 50 (2009) 875–886. [13] C.W. Hirt, B.D. Nichols, J. Comput. Phys. 39 (1981) 201–225. [14] D. Youngs, K. Morton, M. Baines, Numer. Methods Fluid Dyn. (1982) 273–285. [15] G. Pianet, S. Vincent, J. Leboi, J. Caltagirone, M. Anderhuber, Int. J. Multiphase Flow 36 (2010) 273–283. [16] E. Schmidt, Properties of water and steam in SI units 0-800 c 0-1000 bar, SpringerVerlag, 1982. [17] S. Vincent, J.B. de Motta, A. Sarthou, J. Estivalezes, O. Simonin, E. Climent, J. Comput. Phys. 256 (2014) 582–614.
Please cite this article as: F. Girard, et al., Surf. Coat. Technol. (2015), http://dx.doi.org/10.1016/j.surfcoat.2015.01.011
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Surface & Coatings Technology journal homepage: www.elsevier.com/locate/surfcoat
Contributions to heat and mass transfer between a plasma jet and droplets in suspension plasma spraying F. Girard a,b, E. Meillot a, S. Vincent c,⁎, J.P. Caltagirone b, L. Bianchi a a b c
CEA-DAM, Le Ripault, F-37260 Monts, France University of Bordeaux, I2M-TREFLE UMR CNRS 5295, 16 Avenue Pey-Berland, 33607 Pessac, France Univ. Marne-La-Vallée, MSME UMR 8208, 5 Bd Descartes 77454 Marne-la-Vallée Cedex 2, France
a r t i c l e
i n f o
Available online xxxx Keywords: suspension plasma spraying droplet computational fluid dynamics heat and mass transfer
a b s t r a c t In suspension plasma spraying, nano-sized particles are injected via a liquid or a solvent into a plasma jet. Due to shear by the plasma jet, the liquid is fragmented into small droplets flying into the gas flow. The present work has focused on a single droplet evaporating into the plasma environment, and considers, on the one hand, the physical analysis of the phenomena and, on the other hand, the characterization of the solvent’s thermal and motion characteristics of evaporation by means of direct numerical simulation of a multiphase flow. Specific parameters such as the evaporation time of the droplet have been estimated. In addition, a validation study has been proposed by comparing numerical simulation and physical analysis. The direct numerical simulation of multiphase flow was based on a compressible 1-fluid model, modified for handling phase change. The employed numerical methods included the volume of fluid (VOF) approach for interface tracking and finite volumes on structured grids. © 2015 Published by Elsevier B.V.
1. Introduction As exposed in [1], plasma spraying is commonly used for depositing ceramic coatings on the surfaces of industrial parts. Its principle is to use a plasma jet to accelerate and melt sub-millimeter particles that will coat the treated surface. As explained in [2], the coating properties depend highly on the spraying parameters such as particle size and density, plasma enthalpy, momentum, viscosity, chemical composition, and spraying distance. A pertinent way to improve the coating properties is through particle size reduction from the micrometer to the nanometer scale. Introducing nanoparticles in a plasma core requires putting them in a solution or suspension due to their low inertia [3]. So, from this starting point, understanding suspension plasma spraying implies different stages of analysis: - The first step involves hydrodynamic interactions between the charged liquid with the plasma jet leading to the breakup of the liquid suspension jet into droplets. - The second step is the heat treatment of these droplets and the influence of added liquid fluid, with its phase changing from liquid to gas, in the surrounding plasma. - The third step includes the behavior of the nanoparticles trapped in an unsteady environment. ⁎ Corresponding author at: Université UPEM - Paris-Est, Laboratoire Modélisation et Simulation Multi Echelle (MSME), UMR CNRS 8208, 5 boulevard Descartes 77454 Marne-La-Vallée.
- The final step takes into account the behavior of free nanoparticles in the mixture constituted by a plasma atmosphere and steam before impacting the substrate. The first step of hydrodynamic interaction has been previously investigated [4]. Consequently, this paper is focused on the second step, i.e., the heat treatment of as-fragmented droplets. The way the droplets transform in steam has a high influence on the behavior of the nanoparticles trapped in the droplets. We thus ask ourselves, can the particles agglomerate with each other or do they, on the contrary, remain separated? This work focuses on the heat transfer of post-fragmented water droplets flying in a plasma jet in order to determine which phenomena, upon convection and radiation, has the most significant effect on the heat transfer. As the experimental approach is very complex to apply to thermal plasma spraying because of the extreme temperature and velocity of the process, an alternative means of investigation by analytical and numerical analysis was performed. Few studies have been devoted to the analytical or numerical exploration of droplet evaporation in plasma at small scales. Among them, we can cite [4,5]. The present paper is therefore organized as follows: - The first section details the description of the considered setup under investigation with related assumptions. - The second section is dedicated to the analysis of a droplet’s trajectories in the plasma leading to its in-flight time. - Section three presents an analytical study underlining the main phenomena occurring in the boiling process of a droplet.
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- The last section presents direct numerical simulations of droplet motion in the plasma. - Finally, conclusions and perspectives are given.
2. Experimental and numerical setup Fig. 1 presents a scheme of the industrial process of suspension plasma spray coating. The plasma torch creates an electric arc between a cathode and an anode, which ionizes a variety of gases such as argon, hydrogen, and helium. The high-temperature plasma gases flow outside the torch with a high velocity before impacting the surface to be coated. The colloidal suspension is injected in this jet, fragmented and evaporated, in such a way that the nanoparticles can be heated and accelerated by the jet until they impact the surface and form a coating. As explained previously, the analysis is dedicated to free water droplets, in a first approximation without any ceramic nanoparticles. A suspension can be slightly more viscous, and generally presents a lower surface tension [6], but water droplets were nonetheless considered in this study for the sake of simplicity. Despite that the surface tension might vary with temperature [7], the fixed value of 0.061 N.m− 1 from [9] was retained. No break-up was taken into account: the analyzed droplets originated from the upstream destruction of the injected jet and were deduced from [4].
Fig. 2. Plasma velocity (m.s−1) and investigation zone with a continuous liquid jet (Fig. 19 from [8]).
2.2. Physical properties Table 1 lists data deduced from [4] and describes the average state of the different fluids at the appearance of the first fragmented droplets. 2.3. In -flight time of a droplet and average relative velocity
2.1. Local assumptions The forthcoming analyses were based upon the injection of a continuous water jet into an Ar/H2 plasma plume. The operating conditions and an explanation of these choices can be found in [8,9] and are not detailed here. As shown in Fig. 2, the continuous liquid jet was completely disintegrated into droplets 3 mm downstream of the torch exit. The asfragmented droplets were immersed in a plasma atmosphere at a high temperature, around 10,000 K, while the surrounding gas flowed at more than 500 m.s−1. The PDF (probability density function) distribution of the droplet revealed an average diameter of 20 μm for the droplet population prior to the evaporation. Due to the inertia of the continuous liquid shape, at the initial moment when fragmentation started, the asfragmented droplets had no velocity. They were accelerated quickly by the plasma drag force. In the forthcoming text, only 20-μm mean droplets were considered with a null initial velocity. At this point, the relative speed between a droplet and the plasma flow was 500 m.s−1. With the acceleration due to the plasma, the droplet velocity increased to 500 m.s−1 and the relative speed tended toward 0. Nevertheless, to investigate larger or smaller penetration of the continuous liquid jet, other plasma flow velocities were also taken into account: 1000, 600 and 400 m.s−1. For the sake of clarity, the use of “droplet” in the forthcoming text implies water liquid entities while “particle” refers to nanoparticles inside the droplets.
First of all, the in-flight time of the droplet was a necessary parameter in order to determine the hierarchy of the phenomena, and to obtain a limit for considering or not the different possible physical contributions to droplet evaporation. When taking into account a single droplet of 20 μm diameter in a plasma flow, Newton’s law of conservation of momentum was employed to follow the droplet as a single Lagrangian point and to compute an approximate flying time between the fragmentation point and the target substrate at 40 mm. For the sake of simplicity, we calculated the Reynolds number of an isolated droplet, Re, which showed that the flow was very laminar. Using Re, the drag coefficient CD can be approximated by the Schiller and Naumann formula [10]. Re ¼
ρpla V rel ϕdrop ¼ 1:14 μ pla
ð1Þ
where ρpla is the plasma density, φdrop is the droplet diameter, Vrel is the relative speed of the plasma flow, and μplasma is the plasma viscosity at 10,000 K. This gives: CD ¼
24 0:687 1 þ 0:15Re Re
ð2Þ
Fig. 3 shows the droplet velocity when it reaches different plasma states represented by various velocities (1000, 600 or 400 m.s− 1)
Fig. 1. Principle of a plasma spray process.
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Table 1 Physical properties of the different fluids for the studied case. Physical properties
Plasma (10,000 K)
Steam (400 K)
Liquid water (373 K)
Density ρ (kg.m−3) Dynamic viscosity μ (kg.(m.s)−1) Thermal conductivity λ (W.(m.K)−1) Specific heat cp(J.K−1.kg−1) Molecular mass m (kg.mol−1)
0.0291 2.56 10−4 1.69 2.34 103 0.0305
0.6 1.43 10−5 3.89 10−2 1.9 103 0.018
1000 2.82 10−4 2.76 10−2 4.185 103 0.018
versus time. The same graph displays a plot of the relative speed, between the plasma and the droplets, for these different cases. So, the order of magnitude of the flight time was 10−4 s. Regarding the curves of the relative velocity, it is obvious that it was higher than 100 m.s−1 over the entire flight. Consequently, to observe a relative flow rather representative of the relative speed order of magnitude during the inflight time of the droplet, the mean relative speed chosen in next steps was set to 500 m.s−1. As the experiments showed that no liquid water arrived to the substrate, this means that the water was evaporated in less than 10−4 s. This time is an upper limit: in fact, due to the vaporization of the water leading to a loss of matter, the kinetic inertia of the droplets decreased, resulting in an acceleration of the droplet and this in a shorter dwell time. The next part presents the analytical calculation to determine which type of heat transfer evaporated the post-fragmentation droplet.
2.4. Characteristic conductive heat transfer speed versus characteristic speed of steam ejection This section assumes that the droplet is heated in any way. The consequences of this heating were analyzed through the speed of steam ejection during the first instant when the phenomenon occurred. During these first moments, the Kolmogorov scale was computed by [4] to be on the order of 10− 4 m while the droplet diameter was 20 10−6 m. So, from the point of view of the droplet, in first approximation, there was no turbulence, only a laminar flow. First of all, the calculation of the lower limit of the interface motion speed is presented. The evaporation of a 20-μm droplet of water in less than the in-flight time of 10−4 s implies that the interface consumption had a minimal mean speed (Vinterface) = −0.1 m/s, provided that the evaporation time was equal to the in-flight time. Two assumptions were possible here. The first implied that the environment was at atmospheric pressure (that of the plasma flow) and that the vapor density thus remained at the one described in Table 1.
From the mass flow rate balance at the interface, where the water liquid transformed into vapor, the vapor speed could be calculated and approximated to 160 m.s− 1 at the droplet interface. This was a lower limit of the boiling phenomenon if the solvent was completely evaporated in less than 10−4 s. The second assumption involved the fast volume change of liquid into vapor at the moment of evaporation leading to an increase in the local pressure around the droplet and thus to a change in vapor density. In this case, there was a decrease in the vapor speed. However, due to the conservative momentum equation balance between the plasma flow and the vapor, and to the open space field in the plasma flow, the vapor pressure decreased to that of the surroundings so that there was a balance with the mechanical barrier representing the plasma flow. This could lead to a smaller boundary layer of vapor around the droplet. However, as the vapor density decreased with the pressure, the vapor volume increased because of mass conservation. To conclude this part, the two assumptions (with or without pressure evolution) led to the same result: the existence of a vapor boundary layer around the droplet which had a lower limit of ejected vapor speed around 160 m.s−1. Of course, there could be a mixture of the vapor boundary layer with the plasma but the molecular diffusion was a very slow phenomenon (with a characteristic time of 10−2 s calculated by Fick’s law) and we previously noticed that the droplet scale was smaller than the Kolmogorov scale of turbulence, wherefore the droplet was considered to be in a laminar flow. This laminar hypothesis was made in the first run for simplicity reasons, but needs to be checked again. On the contrary, the upper limit of the heat transfer by conductive heating must be calculated. Consequently, one can compare the speed of the ejected steam to the speed of sound in the steam layer at the boiling temperature and atmospheric pressure, making the assumption of γ = 1.12. The sound speed in vapor was 4.4 102 m.s−1. So, if the steam ejection was supersonic, using the mass flow rate balance would lead to an interface speed Vinterface of 0.263 m.s−1, and an evaporation time for a 20-μm droplet of 38 10− 6 s. If the droplet
Fig. 3. Droplet speed and relative plasma/droplet speed computed before impacting of the substrate at 40 mm.
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was evaporated in less than this time, the steam ejected would be supersonic, and as a shockwave, become a boundary. Indeed, the speed of sound is a limit that mechanical information cannot overcome. As conductive heat transfer corresponds to macroscopic propagation by successive shocks and thermal kinetic energy transmission, it cannot be faster than the sound of speed. As a consequence, the droplet cannot be evaporated in less than 38 10−6 s by conductive heat transfer. During this lapse of time, the plasma jet and steam interacted with each other. Calculations of their volumetric momentum revealed that the steam momentum was higher than that of the plasma (respectively, 90 and 15 kg m s−1 m−3) near the droplet interface. This implies that if the boiling was uniform, the steam would push the plasma, and prevent it from touching the droplet. Consequently, the conductive heat transfer did not occur between the hot plasma and the cold droplet, but through a layer of steam, from a hot heat source to a cold target. The next point was to determine the thickness of the vapor layer. At the scale of the process, the evaporation took between 38 and 100 μs. Considering the droplet scale, we assumed that in an arbitrary virtual spherical surrounding of the droplet, most of the mass was in the droplet. If we observed the phenomenon in an infinitely small time scale, the variation of the density in this virtual sphere was not high, whereas the ejected steam moved a lot. In other words, a mass flow rate of steam with a volumetric momentum comparable to that of plasma did not change the average density of a virtual sphere around the droplet because 99.9% of the mass remained liquid, while the steam already moved over a great distance. The characteristic time of the mechanic steam/plasma interaction was very small compared to the evaporation in 38.10−6 s. As the interface moved so slowly compared to the steam ejection, the radius of the droplet could be taken as constant during the rapid advection of the steam. We can thus consider a permanent regime around the droplet. Consequently, the divergence of the flux ρV through a closed surface was zero, i.e., ∇ · (ρV) = 0. This closed surface can be either the interface between the liquid and the steam, or any concentric virtual sphere of radius r. The flux ρV that crossed any of these spheres was the same: 2
2
ρstmðR dropÞ V stmðR dropÞ 4πRdrop er ¼ ρstmðrÞ V stmðrÞ 4πr er
ð3Þ
Finally, when applying the mass flow rate conservation, we get: 2
ρstmðrÞ V stmðrÞ 4π er ¼ ρstmðR dropÞ V stmðR dropÞ 4π
Rdrop 2 er r
ð4Þ
The quantity ρstm(r)Vstm(r) is the volumetric momentum of the steam pstm(r), thus: pstmðrÞ ¼ ρstmðrÞ V stmðrÞ
ð5Þ
2 R drop pstmðrÞ er ¼ ρstmðR dropÞ V stmðR dropÞ er r
ð6Þ
where er is the radial unitary vector, pstm is the steam volumetric momentum at radius r, the quantity ρstm(R drop)(Vstm(R drop).er) is the mass flow rate per unit area at the droplet interface, Rdrop is the radius of the droplet, and r is the radius of the virtual sphere on which the steam volumetric momentum is computed. On the other hand, the momentum balance between plasma and steam, considering that the thickness of the steam layer has a minimal value at the upstream side where the two momentums are in opposition, can be written as pstmðrÞ er ¼ −ppla er
ð7Þ
where pstm (r) and ppla are, respectively, the volumetric momentum of
steam at a distance r from the center of the droplet and the global volumetric momentum of the plasma. Evaluating r reveals the minimal thickness of the steam layer. So, from Eqs. (6) and (7), we have: 2 R drop ppla er ¼ −pstmðrÞ er ¼ −ρstmðR dropÞ V stmðrÞ er r
ð8Þ
and so r, the radius of the virtual sphere where the volumetric momentum of plasma and steam are equal at the upstream side of the droplet, is equal to: vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi u uρstmðR dropÞ V stmðR dropÞ u −5 r¼t Rdrop ¼ 1:83 10 m ρpla V pla
ð9Þ
where ρstm(R drop)|Vstm(R drop)| is the absolute value of mass flow rate per unit area at the droplet interface, ρpla|Vpla| the plasma volumetric momentum, Rdrop is the radius of the droplet, and r is the radius of the virtual sphere on which the steam volumetric momentum is computed to be equal to the one of the plasma, i.e., the radius of the steam sphere surrounding the droplet. In our case, the minimal thickness of the steam has an order of magnitude of r − Rdrop = 8.3 μm, which is not negligible compared to the droplet diameter. From that, we need to know if the plasma molecule can diffuse inside the steam layer. The mean free path of the ejected steam is: λstm R drop
μ ¼ ρ
sffiffiffiffiffiffiffiffiffiffiffiffiffiffi πmH2 O −7 ≈10 m 2kb T
ð10Þ
This is very short compared to the mean free path of the plasma far from the droplet (T = 10,000 K): λpla∞ ¼
μ ρ
sffiffiffiffiffiffiffiffiffiffiffiffi πmpla −6 ≈5 10 m 2kb T
ð11Þ
Consequently, the steam is quite impervious to the plasma, and the heat transfer occurs through the steam. This means that the thermal transfer does not result from a flow sliding around a spherical droplet, i.e., the use of a Nusselt number as in [4,5] is no longer representative of evaporation in this case. It is thus necessary to calculate the characteristic speed of the heat transfer Vtherm through the steam by using thermal diffusivity αsteam: α stm ¼
λstm −5 −2 ¼ 3:27 10 m:s ρstm cpðstmÞ
ð12Þ
where λstm, ρstm, cp(stm) are, respectively, the thermal conductivity, the density, and the specific heat at constant pressure of the steam. Therefore, V therm er ¼ −
α stm −1 −1 ¼ −3:94 10 m s r−Rdrop
ð13Þ
Again, this velocity is very small compared to the steam’s ejection speed |Vstm| = 160 m.s−1. Boiling by conductive heat transfer through the steam layer thus seems impossible with a spherical symmetry. Further research is necessary to estimate the heat and mass transfer in the vicinity of the droplet (see Section 3). 2.5. Orders of magnitude of momentum transfer and pressure field As the Reynolds number is quite small (Eq. (1)), since the plasma density and droplet diameter are very low, the plasma flow around the droplet is a mix between a full viscous regime (called “Stokes regime”) and a full inertial regime. The motion of the
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droplet is driven both by the hydrodynamic pressure on the upstream side, but also by the interfacial shear stress due to viscosity. To know which phenomena occurs, it is necessary to compare the terms of the Navier–Stokes equation: Inertial contribution ρ(V ∇)V: 2
ρpla jV jpla 8 −1 ¼ 3:53 10 Pa m ϕ
ð14Þ
Viscous contribution μ∇2(V) μ pla jV jpla
−1
8
¼ 3:18 10 Pa m
ϕ2
ð15Þ
Capillary contribution: σ 8 −1 ¼ 1:53 10 Pa m ϕ2
ð16Þ
These are all in the same range. It is also interesting to compare the pressure inside the jet downstream of the torch exit (here, the atmospheric pressure) with the total pressure (physically, it is a theoretical pressure inside the torch that generates the momentum and the motion of the plasma jet), to take into account the compressibility of the flow (Mach number M = 0.5). Using a formula available with the ideal gas law in an isentropic flow, assuming the ratio of specific heat γ = 1.2:
P 0 −P ¼ ϕ
P
γ 2 γ−1 1 þ γ−1 −P 2 M ϕ
9
−1
¼ 1:46 10 Pa m
In other words, through a stream trace, assuming an isentropic flow, from the torch to the stop point upstream of the droplet, the pressure of the flow increased by P0 − P = 2.92 104 Pa. As the flow was not isentropic (due to radiative losses and viscous dissipation), this contribution was perhaps less important. All these contributions are of the same order of magnitude, so none of them are negligible, and the order of importance between them depends on the variation in relative speed and on the droplet diameter. To determine an order of magnitude of the pressure field around the droplet, analytical formula are available in Clift et al. [11] for the Stokes regime, taking into account only the viscous contribution. P ext ðr; θÞ ¼ P 0 −μ pla jV rel j
2 þ 3φ ϕ=2 cosðθÞ 1 þ φ 2r 2
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3. Direct numerical simulation of droplet motion in a plasma flow 3.1. Numerical model The Thetis CFD code, specially designed for multi-phase flows, has been used. The compressibility of the plasma was taken into account with the equations exposed in [12]. A volume of fluid [13] piecewise linear interface calculation (VOF-PLIC) [14] method was utilized to follow the liquid phase. In this work, neither the boiling of the droplet nor the steam generation were computed. Computations have been done with both the ideal gas law and thermodynamic and physical local properties. After having compared 2D simulations with the case of study in 3D, we skipped back to a 2D case, which showed the same behavior. This was done to increase the accuracy of the mesh without having a too high number of cells. The mesh size around the droplet was 50 nm, which means that there were 400 cells in each droplet diameter. In order to have a CFL condition lower than 0.3 (required for stability reasons by the interface tracking algorithm), the time step was 2.5 10−11 s. The domain size was 200 μm long and 100 μm wide. After 20 μm in the y-direction, the mesh became exponential to reduce the computational time. The left side was the input side, with a 500 m.s−1 condition for the gas velocity, and the right side was the output side. The top boundary had a sliding condition, and the bottom boundary had a planar symmetry, so as to divide the number of cells by 2. In order to properly calculate the Laplace pressure, an accurate representation of the interface properties was implemented for the local normal and curvature by [15]. Indeed, without this method and even with a quite thin mesh, spurious oscillations of the interface could be induced, thus disturbing the interface in an artificial and non-physical way. 3.2. Numerical results 3.2.1. Numerical proof of the invalidation of a spherical evaporation by conductive heat transfer Fig. 4 represents the momentum field of the plasma. On the upstream part of the droplet, the momentum was much lower than the value of 90 kg.m.s−1 m−3 of the steam calculated in Section 2.3. Consequently, if the droplet evaporated in 10−4 s, the plasma could not reach the droplet on the upstream side, causing a steam layer to form on the droplet interface. As the evaporation temperature depended on the pressure and temperature, the pressure difference between the
ð17Þ
μ
where ϕ ¼ μ liq ¼ 1:75 pla
The pressure step between upstream and downstream is: ΔðP Þ ¼ 2 μ pla jV rel j
2 þ 3φ ϕ=2 ≈1:2 bar 1 þ φ 2r 2
ð18Þ
As the flow is a combination of Stokes flow and inertial flow, the pressure step around the droplets should be between 3 104 Pa and 105 Pa. Since the Laplace pressure is one order of magnitude below (4σ/ϕ = 1.2 104 Pa), the droplet deforms in the flow. This will influence the heat transfer. The next section presents direct numerical simulations to take into account all contributions in the multi-phase flow describing the interaction of the droplet motion with the plasma flow (without evaporation) and with the spherical shape. These small-scale simulations are compared to previous orders of magnitudes for parameters of heat, mass, and momentum transfer.
Fig. 4. Volumetric momentum of the plasma (kg.m.s−1m−3) at t = 2.5 10−7 s (the plasma flows from left to right).
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was thus deformed under the plasma stress. This fact will be analyzed in future studies to determine the influence of the deformation. 4. Conclusions and perspectives
Fig. 5. Pressure (Pa) at t = 2.5 10−7 s (the plasma flows from left to right).
upstream and downstream parts of the droplet induced different evaporation rates at these points so that the spherical shape during evaporation would no longer occur.
3.2.2. Preferential boiling sites due to a high-pressure gradient Fig. 5 represents the pressure field in the plasma and in the droplet. The pressure field was not symmetric: it was very high in the upstream part and very low in the downstream part, from 7 104 Pa to 1.5 105 Pa. Consequently, the boiling temperature of the surface was not homogeneous, and varied from 363 K to 383 K, according to [16]. This was not negligible because if the initial temperature of the droplet was 300 K, the ratio between the temperature increasing downstream and the temperature increasing upstream was around 25%. Thus, the water on the upstream side had to elevate its temperature by 25% more to reach the boiling point. However, with shear stress, and the viscous dissipation in the injector, the water was probably at a higher temperature when it was injected in the plasma jet. As a result, the previous ratio increased. In this case, the steam was first ejected from places where the pressure of the plasma were lower, which is another argument against a spherical evaporation. In addition, the pressure gradient between the upstream and downstream parts of the droplet was on the order of 105 Pa, one order of magnitude more than the capillary pressure as previously analyzed in Section 2.4. The droplet
The heat and motion of suspension droplet evaporation in a plasma flow has been studied both analytically and numerically through direct numerical simulation. It has been demonstrated that a steam layer formed at the droplet interface and that the droplet shape did not remain spherical during the evaporation. By assuming a fixed spherical shape of the droplet, the pressure gradient along the particle diameter led to a difference in boiling temperature, which resulted in preferential boiling sites. This was a first step to understanding how droplet evaporation starts. Future studies will include investigating droplet deformation due to shear stress, steam creation due to phase change, and completely solving the boiling of a water droplet in a plasma jet. In the last step, the nanoparticle behavior during the boiling might be computed using a multiphase model [17] for an explicit representation of the nanoparticle tracking. The wettability phenomena when the nanoparticles reach the interface should then be important. With the solving of the complete phenomenon, the understanding of the potential agglomeration of nanoparticles during the boiling of the solvent should be achieved. References [1] P. Fauchais, A. Vardelle, B. Dussoubs, J. Therm. Spray Tech. 10 (1) (2001) 44–66. [2] P. Fauchais, J. Appl. Phys. D Appl. Phys. 37 (2004) 86–108. [3] P. Fauchais, A. Vardelle, Solution and Suspension Plasma Spraying of Nanostructure Coatings, Advanced Plasma Spray Applications, Global Analysis, Pure and Applied, InTech, 2012. [4] C. Caruyer, Modélisation de nanomatériaux injectés par voie liquide dans un jet de plasma pour la fabrication de nanostructures(Ph. D. thesis, in French) Université de Bordeaux, 2011. 1. [5] K. Wittman-Teneze, Etude de l'élaboration de couches minces par projection plasma(Ph.D. thesis, in French) Universite de Limoges, 2001. [6] L. Dong, D. Johnson, Adv. Space Res. 32 (2) (2003) 149–153. [7] E. Mezger, J. Phys. Radium 7 (10) (1946). [8] E. Meillot, S. Vincent, C. Caruyer, D. Damiani, J.P. Caltagirone, J. Phys. D. Appl. Phys. 46 (22) (2013). [9] E. Meillot, S. Vincent, C. Caruyer, J.-P. Caltagirone, D. Damiani, J. Therm. Spray Tech. 18 (5–6) (2009) 875–886. [10] J.A. Dean, Lange's Handbook of Chemistry, R.R. Donnelley & Sons Company, 1999. [11] R. Clift, J. Grace, M. Weber, Bubbles, Drops and Particles, Academic Press, 1978. [12] J.-P. Caltagirone, S. Vincent, C. Caruyer, Comput. Fluids 50 (2009) 875–886. [13] C.W. Hirt, B.D. Nichols, J. Comput. Phys. 39 (1981) 201–225. [14] D. Youngs, K. Morton, M. Baines, Numer. Methods Fluid Dyn. (1982) 273–285. [15] G. Pianet, S. Vincent, J. Leboi, J. Caltagirone, M. Anderhuber, Int. J. Multiphase Flow 36 (2010) 273–283. [16] E. Schmidt, Properties of water and steam in SI units 0-800 c 0-1000 bar, SpringerVerlag, 1982. [17] S. Vincent, J.B. de Motta, A. Sarthou, J. Estivalezes, O. Simonin, E. Climent, J. Comput. Phys. 256 (2014) 582–614.
Please cite this article as: F. Girard, et al., Surf. Coat. Technol. (2015), http://dx.doi.org/10.1016/j.surfcoat.2015.01.011