Simulation of jet-flow solid fraction during spray forming

A numerical model was developed to simulate the jet-flow solid fraction of W18Cr4V high-speed steel during spray forming. The whole model comprises two submodels: one is an individual droplet model, which describes the motion and thermal behaviors of individual droplets on the basis of Newton’s laws of motion and the convection heat transfer mechanism; the other is a droplet distribution model, which is used to calculate the droplet size distribution. After being verified, the model was used to analyze the effects of parameters, including the initial gas velocity, deposition distance, superheat degree, and the ratio of gas-to-metal mass flow rates, on the jet-flow solid fraction. Finally, an equation to predict the jet-flow solid fraction directly and conveniently according to the parameters was presented. The values predicted by the equation show good agreement with those calculated by the numerical model.


Introduction
As a novel rapid solidification and near-net-shape technique, spray forming has various advantages, such as elimination of macrosegregation compared to casting technology and fewer process steps compared to conventional powder metallurgy. As a result, spray forming has received tremendous attention in the materials manufacturing field. Mesquita and Barbosa [1] reported that spray-formed AISI M3:2 high-speed steel has a finer and more uniform microstructure and greater isotropy than conventionally cast steel. Schulz et al. [2] reported that laboratory-scale spray-formed high-alloy steels were competitive with conventional products in terms of toughness and strength. Zhang et al. [3] observed that, in the case of spray-formed T15 high-speed steel, M 6 C and MC carbides were spherically refined and uniformly distributed after thermomechanical processing. In recent years, spray forming has been widely used to design and prepare various high-performance alloys. For instance, FGH95 superalloy [4], AA7050 alloy [5], Al50Si (wt%) alloy [6], and Cu11.85Al3.2Ni3Mn (wt%) shape memory alloy [7] exhibit fine and uniform microstructures as well as good mechanical properties when prepared by spray forming.
During the spray-forming process, the surface solid fraction of the deposited preform is a key factor determining the properties and qualities of the products. In the case of an excessively high solid fraction, high density of the deposited preform can hardly be achieved; by contrast, the spray yield will be reduced in the case of a low solid fraction. However, the surface solid faction of the deposit is very difficult to measure directly. Given that the surface solid fraction is determined by the droplets' cooling process, an individual droplet's solid fraction and the distribution of the droplet size are usually combined and simulated to reflect the surface fraction. Thus far, research on the simulation technology of spray forming has led to substantial advancements. Grant et al. [8] established an individual droplet model to investigate the motion and thermal history of a droplet. Using this model, the influential parameters for spray forming, including the droplet size distribution, initial axial gas velocity, melt mass flow rate, melt superheat, and the alloy composition, were investigated [9]. On the basis of the results of these investigations, Grant and Cantor [10] analyzed complex equations to give an approximate expression for the spray solid fraction as a function of process parameters. Mi and Grant [1112] later modified the model and presented a companion model to simulate the heat flow and solidification of Ni superalloy rings during spray forming. Cai and Lavernia [1314] developed a porosity model based on droplet thermal history, particle-packing theory, and flow mechanics to estimate the porosity in a deposited IN718 alloy billet. Kang and Chang [15] established a theoretical model to predict the shape of the billet and the effects of the most dominant processing conditions. In addition, Hattel et al. [16] developed an integrated model to predict the shape of a Gaussian or a billet deposit. Cui and Schulz [17] developed a shape model for clad deposits. Mi et al. [18] developed an integrated, multiphysics numerical model that consisted of an atomization model, a droplet spray model, a droplet deposition model, and a porosity model. Jiang et al. [19] reviewed the physical models and advanced methods used in simulations of gas-liquid two-phase jet flows during atomization and spray processes. A gas jet superposition model, coupled with a series of newly developed numerical-parameter-independent submodels, including the radius-of-influence collision model, mean collision time model, and the polar interpolation model, was proposed by Gao et al. [20] for computing group-hole nozzle sprays. In another work, Du and Wei [21] established a 3D numerical model of flow and heat-transfer to analyze the appropriate forming conditions. However, few simulation studies concerning spray-formed high-speed steel have been reported. Therefore, the previous simulation results cannot be directly applied to the spray forming of high-speed steels. Additionally, few of the previously reported simulation results have been verified in an industrial production process.
In this paper, the motion and heat-transfer mechanisms of W18Cr4V high-speed steel droplets during spray forming were investigated and the optimal production parameters were discussed.

Gas motion and cooling behaviors
The scheme of the spray-forming process is shown in Fig.  1. The axial gas velocity shows an exponential decay with increasing distance and can be described as where v g is the gas velocity, v gi is the gas velocity at the exit from the atomizer or the initial gas velocity, Z is the deposition distance, and 0 gi n v a v   is the exponential decay coefficient, where a 0 = 3.04  10 -4 m 0.24 s 1.24 and n=1.24 [8]. The gas temperature T g increases exponentially from an initial temperature T gi = 25°C at the atomizer exit to a final temperature T gf = 75°C on the surface of the deposited preform [8]: where λ T is a constant and equals 0.1 m.

Droplet motion behaviors
Droplets were considered to be essentially stationary. A drag force, which results from the difference in velocity between the droplets and the atomizing gas, can cause the droplets to accelerate or decelerate. The acceleration of the droplets is expressed on the basis of Newton's second law [22]: where ρ d and v d are the droplet density and droplet velocity, respectively, ρ g and v g are the gas density and velocity, respectively, t is droplet flight time, g is the acceleration of gravity, d is the droplet diameter, and C D is the drag coefficient, which can be obtained by where Re = (ρ g d|v d -v g |)/μ g is the Reynolds number (0.1  Re  4000) and μ g is the gas viscosity.

Droplet cooling behaviors
The typical W18Cr4V droplet cooling process comprises four stages. First, the molten droplets cool. Because of a lack of heterogeneous nucleation sites, droplets cool rapidly below the alloy liquidus temperature T 1 from the initial droplet temperature T di = T 1 + ΔT until reaching the nucleation temperature T n , where ΔT is the superheat degree. Thus, the supercooled droplets gain sufficient energy for nucleation. In this stage, the droplet temperature is calculated by where T d and C d are the temperature and specific heat of individual droplets, respectively, ρ 1 is the density of the melt metal, T g is the temperature of gas, and h is the heat-transfer coefficient, which is given by where k g is the thermal conductivity of gas, Pr = C g μ g /k g is the Prandtl number of gas, and C g is the specific heat of gas. Second, nucleation and recoalescence occur. At the beginning of nucleation, the high undercooling degree, which provides a strong driving force, leads to rapid nucleation. During the nucleation, the latent heat is released within an extremely short period. As a consequence, the droplet temperature rapidly increases to the liquidus temperature. The frequency of heterogeneous nucleation is described as Eq. (7) [23]: where σ SL is the interfacial energy between the solid and the liquid interface, θ is the wetting angle of a nucleus on the wheel surface, ΔT d is the undercooling degree, k B is the Boltzmann constant, and ΔH f is the latent heat per unit mass. The catalytic efficiency of heterogeneous nucleation F(θ) is given by [8]     where a 1 = 5.025 × 10 −3 and a 2 = 1.005 m [24][25].
The droplet solid fraction f d = ΔT d C 1 /ΔH f , where C 1 is specific heat of the melt metal.
Third, the phase transition occurs. After recoalescence, the undercooling degree of droplets decreases; hence, the solidification rate slows. When the latent heat cannot afford the heat loss through convection, the droplet temperature decreases again. Until the melting point T m is reached, the equivalent specific heat C equ = C d + ΔH f /(T 1 -T m ) is used to replace the latent heat; thus, the droplet temperature can be described as The change of the droplet solid fraction in phase transition can be calculated by the Scheil equation: where k 0 is the equilibrium partition coefficient. Fourth, the solid droplets cool. Solidification is completed when the melting temperature T m is reached, then the full solid-state droplets continue cooling. The temperature can be described by

Jet-flow solid fraction
The jet-flow solid fraction f s is calculated by combining the individual droplets solid fraction and the droplet size distribution.
where P(d i ) is the droplet probability density with diameter d i and is given by Eq. (13) [26]: where σ = (d 50 /13) 1/3 is the geometric standard deviation and d 50 is the average diameter, which can be calculated by Eq. (14), as proposed by Lubanska [27].
where D is the diameter of the molten metal nozzle,  is a constant and equals 40, κ m is the kinematic viscosity of the molten metal, κ g is the kinematic viscosity of the gas, R is the ratio of gas-to-metal mass flow rates, and σ m is the surface tension of the molten metal.

Motion and thermal behaviors of individual droplets
The values of thermophysical properties of N 2 atomizing gas and W18Cr4V high-speed steel used in the calculation are listed in Table 1. The velocities, heat-transfer coefficients, temperatures, and solid fractions of individual droplets are calculated from the model described in the paper; the results are shown in Figs. 2-5 for droplet sizes of 20 to 200 m, an initial gas velocity of 300 m/s, a deposition distance of 0.5 m, a superheat degree of 100 K, and a ratio of gas-to-metal mass flow rates of 0.55.
As shown in Fig. 2, the droplet velocity with different diameters increases rapidly to a maximum value and then decreases gradually. The droplet velocity is strongly affected by the droplet diameter. The smaller droplets show sharp acceleration or deceleration; however, the larger ones gradually accelerate and decelerate slowly due to their larger inertia. For instance, droplets with a diameter of 20 m reach the maximum velocity of 205 m/s at a distance of 0.13 m, whereas the 130-m ones reach the maximum velocity of 110 m/s at a distance of 0.34 m.   Fig. 3 shows the heat-transfer coefficient as a function of deposition distance. The initial heat-transfer coeffi-cients of all the droplets are extremely high because of the large relative velocity between the droplets and the gas. The gas then accelerates the droplets sharply, which reduces the relative velocity; as a consequence, the heat-transfer coefficients decrease rapidly. When the droplet velocity is equal to the gas velocity, the heat-transfer coefficients momentarily decrease to a minimum value. Subsequently, the heat-transfer coefficients again increase because the decay of the gas velocity increases the relative velocity between the droplets and the gas molecules. Fig. 4 shows droplet temperature as a function of deposition distance. Four different cooling stages are observed, including rapid cooling of the melt droplets, nucleation and recoalescence, phase transition, and cooling of the solid droplets. The 20-m droplets are clearly observed to undergo four cooling stages. Because smaller droplets favor higher cooling rates, the smaller droplets nucleate at a shorter distance and have a lower temperature when they reach the deposition surface.

Motion and thermal behaviors of jet flow
The jet-flow solid fraction was calculated by combining the solid fraction of individual droplets and the droplet size distribution. The effects of initial gas velocity, deposition distance, superheat, and the ratio of gas-to-metal mass flow rates were investigated. Fig. 6 shows the heat transfer coefficient, velocity, flight time, temperature, and solid fraction of 80-m droplets as a function of the deposition distance, with the same superheat degree of 100 K and the same ratio of gas-to-metal mass flow rate of 0.55, but with an initial gas velocity increasing from 250 to 350 m/s. As depicted in Fig. 6(a), the heat transfer coefficient increases with increasing initial gas velocity. Moreover, for droplets with a larger initial gas velocity, the point where the heat transfer coefficient decreases to the minimum value is farther from the nozzle than the corresponding point for droplets with a smaller initial gas velocity; the mean heat transfer coefficient is also evidently larger.
The acceleration of droplets depends on the gas; therefore, a larger initial gas velocity results in a larger droplet velocity, as shown in Fig. 6(b). Additionally, we observed that the relative velocity between droplets and gas decreases to 0 m/s at a point and then increases. For droplets with a larger initial gas velocity, the 0 m/s point is farther from the nozzle than the corresponding points for droplets with a smaller initial gas velocity. Compared with Fig. 6(a), the relative velocity and the heat transfer coefficient have a similar variation trend because the latter has a positive relationship with the former, as shown in Eq. (5). Fig. 6(c) shows the flight time as a function of deposition distance. This figure clearly shows that the increase in initial gas velocity will shorten the droplet flight time. For example, the flight time over a 0.5-m deposition distance decreases from 0.0052 to 0.0035 s when the initial gas velocity increases from 250 to 350 m/s. The droplet temperature is determined by its cooling rate and time. We concluded that the mean heat transfer coefficient increases with increasing gas velocity, as shown in Fig.  6(a). Consequently, the cooling rate also increases. However, droplets will require less time to reach the surface of the deposit, as shown in Fig. 6(c). The result of these two antagonistic factors is that the droplet temperature with a larger initial gas velocity is slightly higher compared to that with a lower initial gas velocity; consequently, the solid fraction of droplets is smaller, as shown in Figs. 6(d) and 6(f), respectively. Fig. 6(e) shows that the increase in gas velocity causes an increase in the percentage of smaller droplets because the gas has more energy to atomize the molten metal better.
On the basis of the analysis for the parameters how to affect the solid fraction of individual droplets, the variations in jet-flow solid fraction as functions of the parameters were investigated. Fig. 7(a) shows the variation of jet-flow solid fraction with the deposition distances of 0.4, 0.5, and 0.6 m in the range of initial gas velocities from 200 to 400 m/s at 50 m/s intervals, a gas-to-metal mass flow rate ratio of 0.55, and a superheat degree of 100 K. The jet-flow solid fraction increases linearly with increasing initial gas velocity. This behavior is attributable to two aspects: (1) a decrease in solid fraction of individual droplets, as shown in Fig. 3(f), which decreases the jet-flow solid fraction, and (2) a narrower particle size distribution, which has the opposite effect of the first aspect. However, the second aspect has a greater influence; therefore, when the initial gas velocity increases, the jet-flow solid fraction also increases. Fig. 7(b) shows a solid fraction of jet flow as a function of the ratio of the gas-to-metal mass flow rate from 0.35 to 0.75, with deposition distances of 0.4, 0.5, and 0.6 m, with an initial gas velocity of 300 m/s, and with a superheat degree of 100 K. When the ratio increases, the droplet size distribution becomes narrower because of the gas having more power to atomize the liquid metal to smaller droplets. Accordingly, the solid fraction of jet flow increases. Fig. 7(c) shows the jet-flow solid fraction as a function of superheat degree from 50 to 250 K, with deposition distances of 0.4, 0.5, and 0.6 m, an initial gas velocity of 300 m/s, and a gas-to-metal mass flow rate ratio of 0.55. We concluded that the jet-flow solid fraction decreases with increasing superheat degree because each individual droplet contains more heat compared to those with a lower superheat degree at the same deposition distance.
To verify the model, an infrared thermometer was used to measure the surface temperature of the preform during spray forming, with an initial gas velocity of 300 m/s, a gas-to-metal mass flow rate ratio of 0.55, deposition distances of 0.4, 0.5, and 0.6 m, and a superheat degree of 100 K. As exhibited in Figs. 8(a)8(c), the values calculated by the numerical model are certainly consistent with the measured values; thus, the correctness of the numerical model has been demonstrated.

Prediction of the jet-flow solid fraction
During a sufficiently short period of time Δt, the spray process can be divided into three steps, as shown in Fig. 9. First, the alloy is melted. Second, the molten alloy is atomized into tiny droplets. Third, the droplets crash into the substrate and the billet are formed. In the entire process, the molten alloy has the same weight m = MΔt but different temperatures as the newly formed billet, where m is the weight of the alloy depositing on the substrate during the period of time Δt and M is the mass flow rate of the molten alloy. Thus, as shown by steps 1 to 3 in Fig. 9, the change in heat of the newly deposited alloy Q 1 can be calculated on the basis of the thermal physical properties of the metals as where T surface and f s are the average temperature and the solid fraction of the jet flow on the surface of the billet, respectively. Further analysis reveals that the different temperatures of molten metals and the newly formed billet are mainly caused by Q 2 , which is the heat dissipation of droplets. Therefore, Q 2 can be calculated by accumulating the heat dissipation of each single droplet as where Q m is the heat released by the jet flow per unit mass and Q m is calculated according to the numerical model in this paper with a wide range of parameters; details of the analysis are shown in Figs. 10-12. Specifically, Fig. 10 shows the variation in Q m as a function of v 0.22 gi over the range 200 ≤ v gi ≤ 400 m/s at 50-m/s intervals, with different deposition distances Z, and with gas-to-metal mass flow rate ratios R. The best-fit line is where k is the slope of the best-fit line. The values of k and R-square were calculated; the results are listed in Table 2.
The minimum R-square value of 0.9763 means that the points agree well with the best-fit lines.
where the value of k and R-square are listed in Table 3.
The points show reasonable agreement with the best-fit lines.
where the value of k and R-square are listed in Table 3. The points show good agreement with the best-fit lines.  As a consequence, a relatively simple equation can be obtained from Eqs. (19), (20), and (21)