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Suspension Plasma Spraying is a complex process in which several physical mechanisms play a part. So the modeling and understanding of the interaction between a high-velocity and thermal flow and a liquid precursor phase is of major importance concerning the control and characterization of the process. The liquid droplet size distribution has a high influence on the kinetic properties of the as-sprayed nanometer particles before impacting on a target substrate. An overview of existing models is provided dealing with the penetration of the liquid phase into the thermal flame and the resulting fragmentation and vaporization of this phase before impact. The physical characteristics of the flow as well as existing Lagrangian and Eulerian modeling strategies are briefly discussed while paying attention to the physical parameters characterized and measured by numerical simulation. The potential of the various models and also their limits are intended to be highlighted. Future coupled Eulerian-Lagrangian modeling strategies are also proposed for a global and more exhaustive representation of the injection, fragmentation and dispersion part of the two-phase gas-liquid flow before particle impact on the substrate.

Increasing higher efficiency rates or lifetimes of functional industrial parts requires the development of new materials such as ceramic coating. For many years now, the industry has relied upon surface treatment processes to improve them by numerous materials deposition. In the ceramic deposition field, plasma spraying or High Velocity Oxy Fuel processes are the main one to deposit ceramics. Since the end of the 1990’s, the processing of nanostructured materials has been performed using them [

Now, thermal spraying is up-to-date to build nanometre-scale structure. Nevertheless, modelling and simulation of the interaction between a thermal jet and a liquid precursor phase continue to be of major importance concerning the control and characterization in terms of size of

− Carrier liquid droplet (size, distribution, position and velocity),

− Distribution of nanometer particles before impact on a target substrate,

− And thermokinetic properties of these solid particles carried by the liquid phase. The typical problem of interest is illustrated in

Decreasing the deposit width (<100 µm) with nano-structured coatings leads to improve as-manufactured properties, but due to their low size and low inertia, nanometer particles are injected in the jet with a liquid precursor [

From a modelling point of view, several major informations can be extracted from this figure:

− the heat and mass transfers are multi-scale in time and space,

− the multi-physic characters of the flow as chemical, thermal, turbulent, multi-phase and electromagnetic features,

− The coupling between fluid and solid mechanics.

These mechanisms have to be modelled depending on the considered zone of the problem (from hot flame inlet to impact of particles). General reviews have

been published recently [

Among the numerous contributions leading to models for suspension plasma spraying, those dealing with multi-phase heat and mass transfers can be classified in three categories:

− Experiments and macroscopic behaviour laws that provide a global description of specific parameters of the process. For example, the flow and thermal transfer characteristics have been widely measured for both HVOF and plasma gun configurations. Thanks to these experiments, the chemical composition of the gas as well as the temperatures or velocities of the related flow have been obtained by numerous authors [

All these parameters are of primary importance when evaluating the type of liquid/flame interaction that will occur according to existing classifications of

Breakup mode | Weber number |
---|---|

Vibrational breakup | We < 12 |

Bag breakup | 12 < We < 50 |

Bag and Stamen breakup | 50 < We < 100 |

Sheet stripping | 100 < We < 350 |

Wave crest stripping | We < 350 |

Catastrophic breakup | We > 350 |

break-up phenomena [

− The heat and fluid flow models for the flame without liquid injection. Many works have been devoted to the modelling and simulation of the flame as a preliminary step to injection of powders or carrier liquid droplets. If reference is made to the Mach Ma (Ma = V/c with c is the speed of sound in the medium) and Reynolds Re numbers (Re = (ρvD)/µ with D is the diameter of the torch exit) which belong respectively to the ranges 0.3 ≤ Ma ≤ 2 and 1500 ≤ Re ≤ 8000 depending on the plasma and HVOF characteristics, it can be deduced that turbulence and compressible effects have to be taken into account. A majority of the flow models assumes that a continuum medium is representative of heat and mass transfers and that the flow is turbulent, compressible (except [

− The plume generator models: specific characters of this part of the suspension spraying process require the use of dedicated models for accounting of electromagnetic effects in a plasma torch [

• A global approach, integrating the physical phenomena through the distinct equations of electromagnetic and fluid mechanics. These studies employed the same set of equations, based on the mass, momentum and energy conservation. This was coupled with the Maxwell equations for electromagnetism effects based on the local thermodynamic equilibrium (LTE) assumption for the gases, which supposes that all the species are at the same temperature. Last recent works are from [

• A simple approach based on a Joule effect in the arc column including correlations with experimental time-dependent voltage measurements [

The turbulence models and energy conservation models are the same as those described previously.

The article provides an overview of existing models for dealing with the penetration and the transport of the liquid phase within a thermal flow and the resulting fragmentation and vaporization of this phase before impact on the target substrate. In the two first sections, existing Lagrangian and Eulerian modelling strategies are briefly discussed while paying attention to the physical characteristics obtained by numerical simulation. Comparisons with existing experiments are provided in order to highlight the potential of the various models and also their limits. The final part is devoted to future coupled Eulerian-Lagrangian modelling strategies for a global and more exhaustive representation of the injection, fragmentation and dispersion of the two phase flame-liquid flow before impact on the substrate. Conclusions are finally drawn.

For both HVOF [

∂ ρ ∂ t + ∇ ⋅ ( ρ u ) = S m (1)

ρ [ ∂ u ∂ t + ∇ ⋅ ( u ⊗ u ) ] = − ∇ p + ∇ ⋅ ( μ + μ t ) [ ∇ u + ∇ t u ] + S M (2)

ρ [ ∂ h ∂ t + ∇ ⋅ ( h u ) ] = ∇ ⋅ ( λ + λ t ) ∇ T + S h (3)

ρ [ ∂ ξ i ∂ t + ∇ ⋅ ( ξ i u ) ] = ∇ ⋅ ( D i + D i , t ) ∇ ξ i + S ξ (4)

where i denote the different species and S_{m}, S_{M}, S_{h} and S_{ξ} are source terms accounting for radiative effects, impact of liquid precursor on thermal flow or chemical reaction sources. According to how the turbulent viscosity, conductivity and species diffusion coefficient are modeled with a RANS or a LES approach, the variable u, p, h and e must be understood as statistical mean values for RANS turbulence models whereas these variables correspond to the large scale resolved filtered unknowns in the framework of LES. The turbulent coefficients μ_{t}, λ_{t} and D_{i,t} are generally obtained by means of a RANS statistical modeling. Reynolds decomposition is introduced for each variable which is split into an averaged statistical quantity and a fluctuating part. The closure assumptions are often based on the analogy of the Kolmogorov energy cascade [

m p d V p d t = F d + F a + F b (5)

m p C p d T p d t = π d λ N u ( T f − T p ) + d m p d t | V p | + Q r (6)

d r p d t = − d m p d t 1 4 π ρ p r p 2 (7)

where F_{D}, F_{b} and F_{a} are respectively drag, buoyancy and additional mass forces whereas Q_{r} is the radiative flux to which are subjected the liquid precursor particles. The model (1) - (7) has been used by many authors to characterize the behavior of various precursor droplets in HVOF and plasma suspension processes [

(1) - (4) to also report the interaction between liquid precursor and thermal flow. For example, Basu and co-workers [

The main limitation of the Lagrangian modeling of liquid precursor/thermal flow interaction is a priori definition of interaction laws for the liquid particles with the surrounding fluid in terms of drag law, heat transfer coefficient or Nusselt number and vaporization mass transfer coefficients. These laws do not generally account for the presence of other droplets in the vicinity of the considered particle or their modification of the thermal flow (cooling effect, liquid break-up mode, deceleration, air engulfment…). In addition, only isolated liquid droplets are followed by the Lagrangian model, so that no coalescence or explicit secondary break-up is solved. The Lagrangian models can be improved by introducing two- or four-way coupling models for accounting of the effect of particles on the flame and also interaction between particles themselves [

The building of an Eulerian modeling for the interaction between a liquid precursor jet and a thermal flow aims at describing all the time and length scales of the liquid primary and secondary break-up under thermal flow shearing with thermal transfers and vaporization being solved at the same time. These objectives intrinsically rely on a deterministic description of the problem concerning liquid deformation and rupture, as well as turbulence structures. As a consequence, the previously presented RANS modeling of the thermal flow cannot be used in this description of the problem as each space scale of droplet generation and modification will be solved at each time step. The only work which reports on a compressible model for the simulation at small scale of the interaction between a precursor liquid jet with a plasma flow is based on the following model [

∂ p ∂ t + τ χ T ∇ ⋅ u = 0 (8)

ρ [ ∂ u ∂ t + u ⋅ ∇ u ] = − ∇ ( p − τ χ T ∇ ⋅ u ) + ∇ ⋅ ( μ + μ t ) [ ∇ u + ∇ t u ] + S S T (9)

ρ C p [ ∂ T ∂ t + u ⋅ ∇ T ] = ∇ ⋅ ( λ + λ t ) ∇ T + S h (10)

∂ χ ∂ t + u ⋅ ∇ χ = ∇ ⋅ ( D + D t ) ∇ χ + S ξ (11)

∂ C ∂ t + u ⋅ ∇ C = 0 (12)

where τ is a characteristic time of the problem chosen equal to the numerical time step Δt used to discretize the time derivatives. The liquid volume fraction C is representative of the volume of liquid precursor in each elementary volume or grid cell. By definition, C = 1 in the liquid and C = 0 elsewhere. This approach is termed Volume of Fluid (VOF) method in the literature [_{p} of the fluids (air, plasma gas and liquid) are built according to the pressure, temperature, plasma gas concentration χ and liquid volume fraction C. The surface tension forces are determined according to C and integrated as a local source term in the cells cut by the interface through a volume force S_{ST} [_{t} [_{t} is known, the turbulent conductivity λ_{t} and plasma gas diffusion coefficient D_{t} are obtained by means of a turbulent Prandtl and Lewis analogy. With model (8) - (12), it is assumed that all the time and space scales of the liquid precursor interface are solved, meaning that the discretization grid has to be refined enough to capture all the interfacial structures generated by the interaction of the liquid jet with the thermal flow. Due to heat exchange between water and plasma, the phase change cannot be neglected for long time simulations. A phase change model is integrated; it is an adaptation of a Lagrangian model of [

The droplet diameter decreases according to the following law:

d p n + 1 = d p n − Δ t 4 λ g ρ p c p , g d p n N u ln ( 1 + B T ) (13)

where d_{p} is the drop diameter, Δt the time step, λ the thermal conductivity, ρ the density, c_{p} the specific heat, Nu the Nusselt number, B_{T} the thermal Biot number and n the discretization t index.

Each cell cut by the interface is considered as an equivalent sphere whose tradius is calculated as the local curvature radius ∇ ⋅ ∇ C ‖ ∇ C ‖ . The Equation (1) is

applied on these equivalent spheres. The evaporated water volume is calculated and removed in the concerned cells and finally the phase function C is updated. All variables are local and known in each cell of the mesh.

Typical results obtained with the Eulerian LES model are presented in ^{−8} s. The computations were performed with a Sulzer Metco PTF4 torch, with a nozzle diameter of 6 mm, discharging Ar H_{2} into ambient air. The flow rate is 45/15 slm. The torch was assumed to operate at 500 A and 65 V, with a thermal efficiency of 52%. The velocity and temperature profiles are imposed at the domain entry, according to previous calculations of plasma flow with time-depending boundary conditions [^{−1} and the temperature profile of the cross-flow is included between 3000 and 9000 K for ArH_{2} flow [

The primary and secondary fragmentations of the precursor jet can be observed and typical probability density functions of droplet size can be extracted from this type of simulation. The main drawback of the Eulerian modeling is that it is not possible to use a grid refined enough on existing parallel computers (512 processors are used in

First of all, the interactions between the different fluids can be defined by the Weber number. Classically, the gas Weber number (We_{g}) is defined as the ratio of the disrupting inertial forces to the restorative surface tension forces. Previous works (ref-CC2) were dedicated to the calculation of the Weber number versus the plasma flow radius. These ones show different break-up modes depending on the investigation zone. It was found that instability waves developed along the liquid shape in interaction with the plasma flow. These waves were the roots of threads which rapidly broke into big droplets in the first layers of the flow. Analyses of the Weber number versus the plasma flow radius showed its high

evolution due to the velocity and density gradients which, in contrast, moved one by one. This variation explains the complexity of the fragmentation and the unusual modes observed.

The evolution of the jet injection into ArH_{2} plasma can be seen in

An overpressure zone is well observed in the upstream direction to the jet (

The plasma temperature decreases strongly with the liquid injection testifying from air engulfment inside the plasma (

Two calculations types have been done with and without phase change. To differentiate the droplet behavior and the impact of the phase change, the droplet number has been calculated at the same time (100 µs) in the whole field. With and without the phase change model, the jet behavior does not seem different, but the droplet number decreases due to the water vaporization (

If the biggest size droplets stay in the same number range (more than 38 µm) due to their thermal inertia, the lowest diameter droplets diminish in number because of the evaporation. The lost mass can be estimated to 10% that is not negligible in a so short time, less than 50 µs, required time to get the system in balance.

The domain is cut in several parts from the center line to the border of the plasma jet (

It is observed that smaller droplets (10 μm) are especially present in the zones close to the torch axis. In this place, temperature and velocity are higher; thus droplets are more evaporated or broken-up. On the contrary, droplets with a diameter of the order of 30 μm are more numerous in the peripheral zone (

As recommended by Cetegen and Basu [

envisaged as efficient tools in understanding the thermal spray process in the near future. In this section, a proposition is made to build efficient and realistic multi-scale models for thermal spray processes. By considering the general structure of a typical plasma-liquid jet interaction, as presented in

It is proposed to use a deterministic LES compressible model (8) - (12) everywhere on a computational grid whose finest grid cell is of the size of the smaller droplets in the secondary break-up zone (namely d_{d1} in

soon as a liquid droplet diameter is smaller than d_{d1}, namely range d_{d2}, the local VOF function C is chosen equal to 0 in the Eulerian model and these particles are then considered as a Lagrangian object to which Lagrangian macroscopic physical models for velocity, radius, temperature and vaporization, similar to those described in (5) - (7), are applied. The correct numerical method for implementing such a multi-scale Eulerian-Lagrangian model is the VOF-Sub Mesh approach [_{d2}), has been first presented by Caruyer et al. [

Thermal spray technologies, such as HVOF or plasma spraying, are able to produce nano structured coatings by introducing nano material suspensions into the high temperature and high velocity gas flow. The high radiation and velocity of the flow complicate the analysis of the phenomenon. Hence, numerical investigations can give numerous information of interest. From this review, the following aspects should be integrated in a Computational Fluid Dynamics code to achieve elementary or global simulations:

− Compressible effects of the flow,

− High resolution of the turbulence with LES turbulence models and large interfacial scales,

− High number of physical identities (solid particle, droplet) treated by Eulerian methods.

In fact, the interactions between the liquid and high velocity and high temperature gas flows, such as those generated by HVOF or DC plasma guns, lead to such a large range of droplet sizes, size which evolves with time during evaporation, that the taking-into-account of numerous and small physical entities can only be treated by mixed Eulerian-Lagrangian models. That is the main strategy proposed here as conclusion and advocated by the authors to analyze in depth the behavior of nanoparticles injected in thermal spray processes.

The authors thank the Aquitaine Regional Council for the financial support dedicated to a 256-processor cluster investment. We are grateful for access to the computational facilities of the French CINES (National computing center for higher education) under project number A0012b06115 and CCRT (Research and Technology Computing Center) under project number gen6115. The authors also thank the CEA for its support. The ANR Modemi is also associated to this project concerning multi-scale modeling of multi-phase flows.

Vincent, S., Meillot, E., Caruyer, C. and Caltagirone, J.-P. (2018) Modeling the Interaction between a Thermal Flow and a Liquid: Review and Future Eulerian-Lagrangian Approaches. Open Journal of Fluid Dynamics, 8, 264-285. https://doi.org/10.4236/ojfd.2018.83017

χ Plasmagen gas concentration

χ_{T} Isothermal compressibility (Pa^{−1})

μ Dynamic viscosity (Pa∙s)

μ_{t} Turbulent viscosity (Pa∙s)

λ Thermal conductivity (W∙m^{−1}∙K^{−1})

λ_{t} Turbulent conductivity (W∙m^{−1}∙K^{−1})

ρ Density (Kg∙m^{−3})

τ Characteristic time (s)

ξ Mass fraction of species

C Liquid volume fraction

C_{p} Heat capacity (J∙kg^{−1})

d Liquid precursor diameter (m)

D_{i} Molecular diffusion coefficient (m^{2}∙s^{−1})

D_{t} Turbulent diffusion coefficient (m^{2}∙s^{−1})

H Enthalpy (J∙kg^{−1}∙K^{−1})

m_{p} Mass of liquid precursor particle (Kg)

Ma Mach number

Nu Nusselt number

p Pressure (Pa)

t Time (s)

r_{p} Radius of a liquid precursor particle (m)

Re Reynolds number

T Temperature (K)

T_{f} Thermal flow temperature (K)

T_{p} Liquid precursor temperature (K)

U Velocity (m∙s^{−1})

V_{p} Liquid precursor velocity (m∙s^{−1})

We Weber number