Introduction

As indicated by many experimental and numerical work of cold spray (Ref 1-4), the spray particle is required to have a higher impact velocity over a material-dependent critical velocity to be deposited on a substrate. Since the critical velocity depends on the particle temperature at impact, the particle velocity and particle temperature are two of the most important parameters. In the research on cold spray, we could predict the particle velocity and temperature by numerical simulation. From the powder-material point of view, the particle velocity depends on the material density, and the particle temperature depends on the material density as well as the specific heat. Since the density and specific heat varies from one material to another, we normally need to simulate the particle velocity and temperature for each powder-material independently (Ref 5).

From the theoretical point of view, however, we can recognize two parameters which combine particle diameter and material properties. One is particle-velocity related parameter, which is the particle diameter multiplied by particle density (Ref 6), as in the equation of particle motion. The other one is, particle-temperature related parameter, which is the squared particle-diameter multiplied by the particle density and specific heat, as in the equation of particle energy.

This article investigates the particle velocity and temperature of cold spray for a wide range of both diameters and material properties of particle by numerical simulation. When the simulated particle impact-velocity is plotted against the particle-velocity related parameter, the distributions of the particle impact velocity are almost identical regardless of the material type. This is also true for the particle impact-temperature against the particle-temperature related parameter. In addition, the maximum impact velocity of particle, and corresponding temperature and diameter for any powder-material is graphically predicted by using numerical data for the specific powder-material. These results will improve our understanding of both acceleration and heating processes of particle in cold spray.

Numerical Methods

Gas Flow

The assumptions used in the present computational fluid dynamics (CFD) model are listed as follows:

  1. (1)

    The gas flow is two-dimensional axisymmetric.

  2. (2)

    The gas is thermally perfect and calorically perfect.

  3. (3)

    The gas species inside and outside the nozzle is the same.

  4. (4)

    The momentum transfer and heat transfer from the particle to the gas flow are neglected.

The above assumptions are normally used in the CFD model of the cold spray. The governing equations of the gas flow are given by the two-dimensional axisymmetric, time-dependent Navier-Stokes equations along with the k-ε turbulence model. The governing equations are solved sequentially in an implicit, iterative manner using a finite difference formulation. The governing equations are solved with the third-order, upwind, total variation diminishing scheme (Ref 7). The nitrogen and helium were used as a working gas independently in the CFD model. For N2 and He, the stagnation pressure p os and temperature T os upstream of the nozzle were set at the same values of 3 MPa and 572 K, respectively. The atmospheric condition was set outside the nozzle as an initial condition.

Figure 1 shows an axisymmetric nozzle used in the CFD model. The nozzle has a throat diameter of 2 mm, an exit diameter of 7 mm, and a conically diverging length of 100 mm. The distance from the nozzle exit to the substrate is 10 mm. The size of the computational grid used in this simulation is 150 × 40 grids inside the nozzle, and 60 × 90 grids outside the nozzle. This grid size was found to be enough to obtain an almost mesh-size-independent solution; the finer computational grids, 230 × 60 and 90 × 135 for inside and outside the nozzle, respectively, showed negligible change in the gas velocity along the center line.

Fig. 1
figure 1

Nozzle geometry (unit:mm)

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The steady one-dimensional isentropic theory (Ref 8) calculates the Mach number of the gas flow at the nozzle exit, the design Mach number M d, of 4.15 for N2. According to the theories of steady one-dimensional isentropic flow and normal shock wave (NSW), the M d = 4.15 nozzle discharging N2 gas generates an over-expanded flow at p os = 3 MPa. In this case, we theoretically expect that the flow separates from the nozzle inner-wall by shock-wave/boundary-layer interaction (Ref 9). The similar flow pattern is also expected for the He case.

Particle Flow

The following assumptions are used to simulate the particle velocity and temperature to simplify the computation.

  1. (1)

    The particles are spherical in shape.

  2. (2)

    The particles travel along the nozzle axis.

  3. (3)

    The interaction between the particles is negligible.

  4. (4)

    The particle is accelerated by only gasdynamic drag force.

  5. (5)

    The particle is heated by the gas through heat transfer.

  6. (6)

    The temperature inside the particle is uniform.

Then, the equation of particle motion is written as;

$$ m_{{\text{p}}} u_{{\text{p}}} \frac{{du_{{\text{p}}} }} {{dx}} = \frac{1} {2}c_{{\text{d}}} \uprho _{{\text{g}}} {\left( {u_{{\text{g}}} - u_{{\text{p}}} } \right)} \cdot {\left| {u_{{\text{g}}} - u_{{\text{p}}} } \right|}A_{p} $$
(1)

where x is the axial distance along the center line from the nozzle exit, ρg the gas density, u g the gas velocity, m p the mass of the particle, u p the particle velocity, A p the projected area of the particle, c d the drag coefficient of the particle, respectively. The value of c d was calculated by using a database made from the experimental data (Ref 10), along with a correction given by Eq 2 due to high temperature of the gas (Ref 11).

$$ c_{{\text{d}}} = c_{{{\text{d}},\exp }} f^{{ - 0.45}}_{{{\text{prop}}}} $$
(2)

where c d,exp is the drag coefficient obtained from the database, f prop the correction factor given by Eq 9 later in this section.

The particle temperature can be calculated by the following equation;

$$ m_{{\text{p}}} Cu_{{\text{p}}} \frac{{dT_{{\text{p}}} }} {{dx}} = \upalpha \cdot {\left( {T_{{\text{g}}} - T_{{\text{p}}} } \right)} \cdot A_{{\text{s}}} $$
(3)

where T g is the gas temperature, T p the particle temperature, C the specific heat of the particle, A s the surface area of the particle, α the heat transfer coefficient which can be given as;

$$ \upalpha = k_{{{\text{g,f}}}} N_{{\text{u}}} /d_{{\text{p}}} $$
(4)

Here, N u is the Nusselt number, k g the thermal conductivity of the gas, d p the particle diameter. The subscript f in Eq 4 means the value at the film temperature, T f, defined as;

$$ T_{{\text{f}}} = {\left( {T_{{\text{g}}} + T_{{\text{p}}} } \right)}/2 $$
(5)

The Nusselt number, N u, in Eq 4 was computed by Ranz-Marshall correlation along with the correction factors (Ref 11).

$$ N_{{\text{u}}} = {\left( {2 + 0.6Re^{{1/2}}_{{\text{p}}} P^{{1/3}}_{{\text{r}}} } \right)} \cdot {\left( {\frac{{c_{{\text{p}}} }} {{c_{{{\text{p,w}}}} }}} \right)}^{{0.38}} f^{{0.6}}_{{{\text{prop}}}} $$
(6)

where c p is the specific heat of the gas at constant pressure, and the subscript w indicates the surface of the particle. In Eq 6, the particle Reynolds number Re p and the Prandtl number of the gas P r are defined as;

$$ Re_{{\text{p}}} \equiv \uprho _{{{\text{g,f}}}} d_{{\text{p}}} {\left| {u_{{\text{g}}} - u_{{\text{p}}} } \right|}/\upmu _{{{\text{g,f}}}} $$
(7)
$$ P_{{\text{r}}} \equiv \upmu _{{{\text{g,f}}}} c_{{{\text{p,f}}}} /k_{{{\text{g,f}}}} $$
(8)

where μg is the gas viscosity. The factors f prop and f Kn in Eq 6 represent the effect of variation in the gas temperature in the boundary layer on the particle surface and non-continuum effect, respectively. They are given as follows (Ref 11);

$$ f_{{{\text{prop}}}} = \frac{{\uprho _{{\text{g}}} \upmu _{{\text{g}}} }} {{\uprho _{{{\text{g,w}}}} \upmu _{{{\text{g,w}}}} }} $$
(9)

The particle velocity and temperature were determined from a step-wise integration of Eq 1 and 3, respectively.

The powder-materials used in this simulation are WC-12Co, Cu and Ti, covering large to small material density and specific heat, as shown in Table 1. The range of the particle diameter is 0.1-30 μm for each material.

Table 1 Powder material
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Results and Discussion

Gas/Particle Flows

The simulated Mach number contour for N2 gas is shown in Fig. 2. The axial distance measured from the nozzle exit, x, is shown in the bottom side of the figure. Figure 2 shows that the gas flow is accelerated to supersonic flow through the throat in the downstream direction. Then, the Mach number reaches the maximum value of M g = 3.7 at x ∼ −10 mm, around where the flow separates from the nozzle-wall, as was theoretically predicted in the previous chapter. The Mach number right before the NSW outside the nozzle is 2.8.

Fig. 2
figure 2

Mach number contour for N2 gas

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Figure 3a shows the velocities of the N2 gas and 10 μm-Cu particle on the center line. The gas sharply accelerates through the nozzle throat, and then it gradually accelerates until the flow separates from the wall at x ∼ −10 mm. After exiting the nozzle, the gas flow is suddenly decelerated through the NSW at x = 9 mm. The 10 μm-Cu particle is propelled by the N2 gas, and then, it impinges the substrate at 631 m/s.

Fig. 3
figure 3

Gas and 10 μm-Cu particle velocity on center line for (a) N2, and (b) He

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Figure 3b shows the velocities of the He gas and 10 μm-Cu particle on the center line. The He gas accelerates more sharply through the throat compared to the N2 case. This is because the speed of sound of the He gas at the throat, for example, is theoretically 2.7 times the N2. The gas gradually accelerates after the throat until the separation point at x ∼ −20 mm. The location of the NSW is at x = 9 mm, the same as the N2 case. Although the 10 μm-Cu particle is accelerated more by He than N2, the particle velocity relative to the gas velocity, at arbitrary x, is smaller for He than N2.

The reason for the difference in the particle velocity for N2 and He gases needs to be clarified to understand the gasdynamics of the cold spray. According to Eq 1, if we assume \( u_{{\text{p}}} \ll u_{{\text{g}}} \), the gasdynamic drag force acting on the spray particle is approximately proportional to the momentum flux (MF); ρg u 2g . However, according to the steady one-dimensional isentropic theory,

$$ \frac{{{\left[ {\uprho _{\rm g} u^{2}_{\rm g} } \right]}_{{{\text{He}}}} }} {{{\left[ {\uprho _{\rm g} u^{2}_{\rm g} } \right]}_{{{\text{N}}_{2} }} }} = \frac{{{\left[ {\upgamma /{\left( {1 + \frac{{\upgamma - 1}} {2}M^{2}_{\rm g} } \right)}^{{\frac{\upgamma } {{\upgamma - 1}}}} } \right]}_{{{\text{He}}}} }} {{{\left[ {\upgamma /{\left( {1 + \frac{{\upgamma - 1}} {2}M^{2}_{\rm g} } \right)}^{{\frac{\upgamma } {{\upgamma - 1}}}} } \right]}_{{{\text{N}}_{2} }} }} $$
(10)

where, the specific heat ratio γ is 1.67 for He and 1.40 for N2. The value of Eq 10 lies between 1.10 and 1.17 for 1.0 < M g < 2.5. The closeness of the values of MF for N2 and He was also confirmed by the results of the CFD model, which is not shown here. Based on the discussion made above, the assumption \( u_{{\text{p}}} \ll u_{{\text{g}}} \) should not be used when considering the dependency of particle accelerating process on gas type.

When, we do not employ the assumption \( u_{{\text{p}}} \ll u_{{\text{g}}} \), the gasdynamic drag force acting on the particle is proportional to the relative momentum flux (RMF); ρg(u g − u p)2. Figure 4 shows the RMF on the center line for the 10 μm-Cu particle. It is clear that the difference of the RMF for He and N2 is much larger than that of the theoretical MF along the horizontal axis. This means that the larger particle velocity propelled by He compared to N2 is due to the larger RMF, not MF.

Fig. 4
figure 4

Relative momentum flux for 10 μm-Cu particle on center line

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Particle Velocity and Temperature on Substrate

It is generally expected that the axial distributions of the simulated particle velocity u p and temperature T p differ from one powder-material to another. However, by investigating the equations of particle motion and temperature, we can see two parameters which are independent of the material type. The equation of particle motion, Eq 1, and that of particle temperature, Eq 3, can be modified as follows;

$$ \frac{{du_{{\text{p}}} }} {{dx}} = \frac{{3c_{{\text{d}}} \uprho _{{\text{g}}} }} {{4u_{{\text{p}}} }}{\left( {u_{{\text{g}}} - u_{{\text{p}}} } \right)} \cdot {\left| {u_{{\text{g}}} - u_{{\text{p}}} } \right|} \cdot \frac{1} {{d_{{\text{p}}} \uprho _{{\text{p}}} }} \propto \frac{1} {{d_{{\text{p}}} \uprho _{{\text{p}}} }} $$
(11)
$$ \frac{{dT_{{\text{p}}} }} {{dx}} = \frac{{6k_{{{\text{g,f}}}} N_{{\text{u}}} {\left( {T_{{\text{g}}} - T_{{\text{p}}} } \right)}}} {{u_{{\text{p}}} }} \cdot \frac{1} {{d^{2}_{{\text{p}}} \uprho _{{\text{p}}} C}} \propto \frac{1} {{d^{2}_{{\text{p}}} \uprho _{{\text{p}}} C}} $$
(12)

Equation 11 shows that the combination parameter with respect to the particle d pρp, not d p and ρp separately, affects the particle velocity. The parameter d pρp is also pointed out by Jodoin (Ref 6), who calculated the velocity of particle flying through a NSW in cold spray. However, the parameter’s usefulness in the non-uniform flow of the cold spray has not yet been fully verified. Hereafter, the product d pρp is called particle-velocity-parameter (PVP) in this article. Likewise, Eq 12 indicates that the combination parameter with respect to the particle is d 2p ρp C, which affects the particle temperature. The product d 2p ρp C is called particle-temperature-parameter (PTP) hereafter in this paper.

In Fig. 5, the particle velocity right before the NSW (open symbols) and particle impact velocity on the substrate, u pi (solid symbols) on center line are plotted against the PVP for WC-12Co, Cu and Ti. The horizontal dash-dotted line shows the nozzle-exit gas velocity on the center line, u ge, as a reference value. Figure 5(a) for N2 shows that the particle velocity right before the NSW and that on the substrate are almost independent of the material type. It is also quite valuable to note in the figure that the maximum impact velocity, \(u^{ * }_{{{\text{pi}}}} = 735\,{\text{m/s}} \) = 735 m/s, is common for all three materials. (The asterisk as in \(u^{ * }_{{{\text{pi}}}}\) shows the condition of the maximum impact velocity.) Furthermore, we can see in Fig. 5(a) that when the particle diameter is decreased, \(u^{ * }_{{{\text{pi}}}}\) is obtained at the condition that the particle-velocity right before the NSW first reaches the nozzle-exit gas velocity at

$$ {\text{PVP}}^{{\text{*}}} \cong 0.020\,{\text{kg}}/{\text{m}}^{{\text{2}}} \quad {\text{for N}}_{{\text{2}}} {\text{ at }}u^{ * }_{{{\text{pi}}}} = 735\,{\text{m/s}} $$
(13)
Fig. 5
figure 5

Particle velocity vs. PVP for (a) N2, and (b) He

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For PVP < PVP*, the smaller the PVP, the smaller the particle impact velocity. The same trend can be observed in Fig. 5(b) for He. In the He case,

$$ {\text{PVP}}^{{\text{*}}} \cong 0.0065\,{\text{kg}}/{\text{m}}^{{\text{2}}} \quad {\text{for He at }}u^{ * }_{{{\text{pi}}}} = 1900\,{\text{m/s}} $$
(14)

where the particle velocity before the NSW reaches the gas velocity u ge. The value of PVP* of He (0.0065 kg/m2) is about one-third of that of N2 (0.020 kg/m2). This means that the NSW of He decelerates the particle less than the NSW of N2.

In Fig. 6, the particle temperature right before the NSW (open symbols) and impact temperature on the substrate, T pi (solid symbols) on the center line are plotted against PTP. The horizontal dash-dotted line in the figure shows the nozzle-exit gas temperature on the center line, T ge. Figure 6(a) for N2 shows that the particle temperature before the NSW and that on the substrate are independent of the material type. In the range of PTP > ∼ 3 × 10−5 J/(m K) of Fig. 6(a), the particle temperature before/after the NSW decreases with decreasing PTP. This trend can be explained qualitatively by using Eq 11 and 12, along with the gas temperature on the center line in the nozzle (not shown in this paper) as follows. The gas flow expands so rapidly after passing through the throat that the gas temperature is below the particle temperature from around the throat to the exit of the nozzle. The smaller particle has larger particle velocity; Eq 11 shows that du p/dx is proportional to 1/d p, resulting in shorter residence time in the cooler supersonic gas flow. As for particle temperature, however, Eq 12 shows that dT p/dx is proportional to 1/d 2p . As a result, smaller heat capacity dominates the particle temperature for smaller particles, instead of smaller residence time in the nozzle, in the range of PTP > ∼3 × 10−5 J/(m K). In addition, we can see in Fig. 6(a) that the particle temperature on the substrate increases for PTP < ∼3 × 10−5 J/(m K). This is because the smaller particle has a smaller heat capacity, then the particle temperature easily follows the raised stagnant gas temperature on the substrate. The same trend can be seen in Fig. 6(b) of the He gas for PTP < ∼1 × 10−4 J/(m K).

Fig. 6
figure 6

Particle temperature vs. PTP for (a) N2, and (b) He

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Finally, by using the results of Fig. 5 and 6, we can deduce the relationships between the maximum impact velocity \(u^{ * }_{{{\text{pi}}}}\), corresponding particle temperature \(T^{ * }_{{{\text{pi}}}}\), particle diameter d *p , and material properties of the particle as shown in Fig. 7. The graph consists of the partial fitting curves of solid symbols in Fig. 6, and curves of Eq 15 and 16 derived from Eq 13 and 14, respectively as;

$$ d^{*}_{{\text{p}}} C \cong 0.020{\left( {C \mathord{\left/ {\vphantom {C {\uprho _{{\text{p}}} }}} \right. \kern-\nulldelimiterspace} {\uprho _{{\text{p}}} }} \right)}\quad {\text{for N}}_{{\text{2}}} {\text{ at }}u^{ * }_{{{\text{pi}}}} = 735\,{\text{m/s}} $$
(15)
$$ d^{*}_{{\text{p}}} C \cong 0.0065{\left( {C \mathord{\left/ {\vphantom {C {\uprho _{{\text{p}}} }}} \right. \kern-\nulldelimiterspace} {\uprho _{{\text{p}}} }} \right)}\quad {\text{for He at }}u^{ * }_{{{\text{pi}}}} = 1900\,{\text{m/s}} $$
(16)
Fig. 7
figure 7

Relationships between material properties, particle diameter, particle temperature for the maximum impact velocity of (a) 735 m/s for N2 and (b) 1900 m/s for He

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The horizontal axis of Fig. 7, C/ρp, is obtained from that of Fig. 6 by putting d p = d *p , then C/ρp = (d *2p ρp C)/(d *p ρp)2 = (d *2p ρp C)/(PVP*)2. The horizontal range of the curves in Fig. 7 was determined by considering a wide variety of the powder-materials normally used in the thermal spray process. Once the material type (value on the horizontal axis, C/ρp) is specified, \(T^{ * }_{{{\text{pi}}}}\) and d *p are obtained from the left- and right-vertical axis of Fig. 7, respectively, to obtain \(u^{ * }_{{{\text{pi}}}}\) of 735 m/s for N2 in Fig. 7(a) and 1900 m/s for He in Fig. 7(b). Therefore, for the nozzle geometry shown in Fig. 2 and p os = 3 MPa, T os = 572 K, Fig. 7 can be used to estimate the conditions of the maximum impact-velocity for any powder-materials. In order to confirm this, the maximum impact-velocity condition of aluminium (ρ p  = 2641 kg/m3, C = 1040 J/(kg·K)), which is not used to construct the curves in Fig. 7, is calculated for the He gas. Figure 7(b) yields d *p  = 2.65 μm and \(T^{ * }_{{{\text{pi}}}}\) = 235 K for Al, C/ρp = 0.39 J·m3/(kg2·K), to obtain \(u^{ * }_{{{\text{pi}}}}\) = 1900 m/s. For comparison, the numerical integration of Eq 1 and 3 for d p = 2.65 μm for the Al particle flying in the He gas flow obtained by CFD yield u pi = 1863 m/s and T pi = 230 K, respectively, which are almost identical to the graphical estimation from Fig. 7(b).

Concluding Remarks

The numerical simulation of the cold spray was performed to investigate the usefulness of the combination parameters, which are independent of the material type, that affect the particle velocity and temperature. A special attention was paid to the particle impact velocity. The N2 and He were used as the process gas, respectively. The stagnation pressure and temperature of the gas upstream of the nozzle throat was set at 3 MPa and 576 K, respectively. The materials of the spray particle selected were WC-12Co, Cu and Ti. The particle-diameter range was 0.1-30 μm. The results are summarized as follows:

  1. (1)

    The particle velocity right before the normal shock wave increases with decreasing the particle diameter until the particle velocity reaches the gas velocity at the nozzle exit at a specific particle diameter, where the impact velocity takes the maximum value. The maximum impact velocity is almost independent of the material type when the gasdynamic and geometrical spray conditions are fixed.

  2. (2)

    The particle-diameter multiplied by the material density is the material-independent combination parameter, which affects the particle velocity. The maximum impact velocity is obtained for the combination-parameter of 0.020 kg/m2 for nitrogen and that of 0.0065 kg/m2 for helium, regardless of the material type.

  3. (3)

    The squared particle-diameter multiplied by the material density and specific heat is the material-independent combination parameter which affects the particle temperature.

  4. (4)

    The maximum impact velocity and corresponding particle temperature and diameter can be estimated by using the two combination parameters for any powder-materials conventionally used in the thermal spray process.