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5.3.4Modeling of Multiphase Flows and Heat Transfer in Sprays

A metal spray is comprised of a dispersion of metal droplets of different sizes in atomization gas. Droplet-gas interactions in the spray determine droplet velocity, temperature, and solidification history, and consequently affect the spray enthalpy and radial mass distribution. These factors in-turn critically influence the microstructural evolution and the ultimate microstructural characteristics, such as average grain size and porosity, in the sprayed materials. Variations in microstructure at different locations within a sprayed material have been observed in experiments, suggesting the importance of the spatial distributions of thermal and solidification histories of the droplets in the spray. To elucidate the microstructural evolution of the droplets, it is necessary to systematically analyze the multidimensional flow, momentum and heat transfer, as well as rapid solidification phenomena, which occur simultaneously during drop- let-gas interactions. Accordingly, the modeling of droplet-gas interactions and rapid solidification kinetics in the spray is aimed at controlling the spray mass distribution to achieve the required spray pattern, and controlling the thermal and solidification histories to obtain the desired spray enthalpy and microstructural characteristics.

Generally, 3-D models are essential for calculating the radial distributions of spray mass, spray enthalpy, and microstructural characteristics. In some applications, axisymmetry conditions may be assumed, so that 2-D models are adequate. Similarly to normal liquid sprays, the momentum, heat and mass transfer processes between atomization gas and metal droplets may be treated using either an Eulerian or a Lagrangian approach.

Crowe[576] classified the numerical models for dilute sprays as two-fluid model and discrete element model. The two-fluid model treats the dispersed phase as another fluid with appropriate constitutive relationships for effective viscosity and thermal conductivity. The advantage of the two-fluid model is that the same algorithm used for the gas phase can be applied directly to the droplet phase. The drawback is the lack of information on the dispersion coefficient and the effective

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viscosity of droplets that must be estimated empirically. The discrete element model is similar to the stochastic particle method suggested by Dukowicz.[549] In this model, droplets are simulated by the motion of discrete droplet parcels. The parcels of identical droplets are released in each time step. It is assumed that a parcel moves through computational cells in the flow field as the motion of an individual droplet and the temperature history of the droplets in the parcel is the same as that of an individual droplet. The model accounts for the full two-way coupling between droplets and gas due to mass, momentum, and energy exchanges. The dispersion of droplets due to turbulence is accomplished by adding a fluctuation velocity component to the gas phase velocity. There is no need to select an effective droplet viscosity. The approach is suitable for dilute, unsteady flows because it captures the essence of the Lagrangian nature of the dispersed phase. It can also be used for dense flows where collisions dominate the particle motion. The disadvantage of this approach is the number of droplets needed to represent the droplet field. The viability of the predictions by both the two-fluid model and discrete element model depends on the reliability of initial conditions, i.e., the flow properties at the end of the dense flow region or after the completion of secondary breakup.

A simpler version of the discrete element model is the socalled trajectory model. In the trajectory approach, droplet field is modeled as a series of trajectories that emanate from the atomizer or a starting point. The coupling effects are included by summing the heat release to and the drag force on the gas phase. This approach can be used for steady dilute flows.

Numerical modeling of multiphase flows, heat transfer and droplet-gas interactions in metal sprays has been reported by a number of investigators.[154][495]-[497][576][617]-[620] In limited studies,[617][621][622] 2-D experimental measurements of atomization gas and droplet velocities were conducted. A few complex 2-D Eulerian models[615][618] were used to calculate the droplet-gas interactions in metal sprays. Most of the previous studies, however, were confined to 1-D, and only the droplet trajectory and thermal history along the spray centerline were addressed using uncoupled trajectory approach.

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The spatial motion of a droplet during spray forming, not only determines its residence time in the spray, but also critically influences its cooling and solidification by changing the boundary conditions around the droplet surface. Therefore, the knowledge of the velocity fields of both atomization gas and droplets is a precondition for numerical calculations of droplet temperature and solidification histories. The momentum and energy exchanges between the gas phase and the discrete droplet phase not only determine the droplet size, velocity, cooling rate, and solidification time, but also affect the gas flow. Thus, the two- (or three-) phase flows in the spray during spray atomization or spray forming should be solved in a fully coupled way. However, some experimental studies[169] showed that the droplets have little effect on the gas flow, as supported by the nearly identical shock structures and shear layer shapes in a gas-only flow field and a droplet-gas two-phase flow field under the same operation conditions. Hence, reasonably accurate results may be obtained from an uncoupled solution. This simplification not only makes the problem tractable but also leads to much less computational cost and time relative to a fully coupled solution. In the uncoupled approach, the gas flow field is calculated separately and its solution is used for modeling liquid metal breakup and droplet dynamics during atomization and spray. There is evidence, however, that a relatively large discrepancy may occur between experimental data and the predictions by the uncoupled solution approach. The spray predicted by the uncoupled approach was narrower than that measured experimentally, while reasonably good agreement was obtained between experimental data and the prediction by the coupled solution approach.[576]

In the Lagrangian frame, droplet trajectories in the spray may be calculated using Thomas’ 2-D equations of motion for a sphere[579] or the simplified forms.[154][156] The gas velocity distribution in the spray can be determined by either numerical modeling or direct experimental measurements. Using the uncoupled solution approach, many CFD software packages or Navier-Stokes solvers can be used to calculate the gas velocity distribution for various process parameters and atomizer geometries/configurations. On the other hand, somesimple expressions for the gas velocity distribution can be derived from

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experimental measurements. For example, on the basis of the experimental results of Bewlay et al.,[609][617] the decay in the mean axial gas velocity at the spray centerline has been formulated using an exponential function,[156] and the radial distributions of the axial and radial gas velocities specified with power functions.[156] Although these experimental correlations are very simple and easy to use, some parameters in the correlations are specific to the atomizer geometry and operation conditions under which they are derived, and therefore must be determined by experimental measurements.

Grant et al.[619] presented a modeling study of droplet dynamic and thermal histories during spray forming. In spite of the 1-D nature of this modeling, a major advance in the formulation of atomization gas velocity was made by analyzing previous experimental and theoretical results and correlating them into a simple, non-apparatus-specific form. In addition, a detailed solidification model was proposed that involves three thermal regions (i.e., nucleation and recalescence, segregated solidification, and eutectic solidification) and hence is applicable to rapid solidification processes of eutectic alloys.

Using the Lagrangian approach, Liu et al. developed both 1-D[154] and 2-D[156] flow and heat transfer models and rapid solidification kinetics models, including a reaction kinetics model for reactive atomization and deposition, to study droplet-gas interactions in the spray. The formulations are suitable for both Newtonian and non-Newtonian conditions. The models and the related numerical methods as well as the computer codes developed[154][156] were tested and validated for Ni3Al intermetallic compound and Ta-W alloy droplets sprayed using nitrogenoxygen mixture gases, argon and nitrogen, respectively. These models and codes can be used to calculate the momentum, heat and mass transfer in the spray and to predict the spatial distributions of droplet velocity, temperature, cooling rate, solid fraction, and spray enthalpy, prior to deposition. Some microstructural characteristics in semisolid or fully solidified droplets/particles (for example, secondary dendrite arm spacing, SDAS) can be determined on the basis of the thermal histories predicted by the codes.

In some spray forming processes, the atomization gas exits the atomizer nozzle(s) at sonic or supersonic velocities. The gas may expand

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and accelerate within a short distance and then decay through shocks and diffusion to subsonic speed in a non-uniform “roller-coaster” fashion. The compressibility of the gas may influence the transfer processes between the gas and droplets, particularly in the near-nozzle region. Usually, Henderson’s formulations[575] may be used for calculating the drag coefficients of droplets in subsonic and supersonic gas flows. However, within certain Reynolds number range, the dependence of drag coefficients on the Mach number seems to be negligibly small. Thus, the approximation[615] of the standard drag curve may be used in the numerical calculations of droplet trajectories in the spray. In addition, other effects on the drag coefficients need to be considered in the numerical calculations, such as the effects of local void (air/gas) fraction,[550] droplet oscillation and distortion,[315][572] droplet vaporization,[573] and turbulence.[577][578]

For a given system of metal/alloy and atomization gas, the 2-D velocity distributions of the gas and droplets in the spray can be then calculated using the above-described models, once the initial droplet sizes and velocities are known from the modeling of the atomization stage, as described in the previous subsection. With the uncoupled solution of the gas velocity field in the spray, the simplified Thomas’ 2-D nonlinear differential equations for droplet trajectories may be solved simultaneously using a 4th-order Runge-Kutta algorithm, as detailed in Refs. 154 and 156.

During the flight of droplets in the spray, the forced convective and radiative heat exchanges with the atomization gas lead to a rapid heat extraction from the droplets. A droplet undergoing cooling and phase change may experience three states: (a) fully liquid, (b) semisolid, and (c) fully solid. If the Biot number of a droplet in all three states is smaller than 0.1, the lumped parameter model[156] can be used for the calculation of droplet temperature. Otherwise, the distributed parameter model[154] should be used.

In the lumped parameter model, the transient temperature ofa single droplet during flight in a high speed atomization gas is calculated using the modified Newton’s law of cooling,[156] considering the frictional heat produced by the violent gas-droplet interactions due

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to the compressibility of the gas. In thedistributed parameter model, the Fourier equation is used to describe the transient temperature variation within a droplet. Thus, boundary conditions at the droplet surface and center as well as moving interfaces within the droplet are to be specified in addition to the initial droplet temperature.

In a supersonic gas flow, the convective heat transfer coefficient is not only a function of the Reynolds and Prandtl numbers, but also depends on the droplet surface temperature and the Mach number (compressibility of gas).[154][156] However, the effects of the surface temperature and the Mach number may be substantially eliminated if all properties are evaluated at a film temperature defined in Ref. 623. Thus, the convective heat transfer coefficient may still be estimated using the experimental correlation proposed by Ranz and Marshall[505] with appropriate modifications to account for various effects such as turbulence,[587] droplet oscillation and distortion,[585] and droplet vaporization and mass transfer.[555] It has been demonstrated[156] that using the modified Newton’s law of cooling and evaluating the heat transfer coefficient at the film temperature allow numerical calculations of droplet cooling and solidification histories in both subsonic and supersonic gas flows in the spray.

For the purpose of analysis, the thermal history of a metal droplet in the spray can be divided into six regions:

(1)rapid cooling in fully liquid state,

(2)undercooling,

(3)nucleation,

(4)recalescence,

(5)post-recalescence solidification,

(6)cooling in fully solid state.

For an alloy droplet, the post-recalescence solidification involves segregated solidification and eutectic solidification.[619] Droplet cooling in the region (1), (2) and (6) can be calculated directly with the abovedescribed heat transfer model. The nucleation temperature (the achievable undercooling) and the solid fraction evolution during recalescence and post-recalescence solidification need to be determined additionally on the basis of the rapid solidification kinetics.[154][156]

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In spray forming, a large undercooling may generally be attained prior to the onset of nucleation. The most likely solidification mechanism is dendrite growth. The upper bound of the degree of undercooling may be estimated using Hirth’s formulation[624] for homogeneous nucleation within a droplet, coupled with the expression for the homogeneous nucleation rate[156] derived on the basis of the classical nucleation theory.[625] In view of the high probability of heterogeneous nucleation associated with the conditions common in spray forming processes, the undercooling calculated assuming homogeneous nucleation needs to be modified properly by means of theoretical analyses or experimental measurements.[156][620]

Following nucleation, the release of the latent heat of fusion generally occurs at a rate that is substantially higher than that of convective and radiative heat dissipation at the droplet surface, leading to an overall temperature rise in the droplet (lumped parameter model) or a temperature rise at the solid-liquid interface (distributed parameter model). This phenomenon is termed recalescence. The extent of the temperature rise during recalescence depends primarily on the undercooling and cooling rate. The solid-liquid interface velocity during recalescence is also a function of the undercooling and cooling rate. Different expressions for the interface velocity have been used in previous studies, including:

(1)Linear and exponential laws for planar continuous growth at small and large undercoolings,

respectively,[155][619][626]

(2)Ivantsov equation for growth of paraboloid dendrites,[497]

(3)Power law for dendrite growth.[627]

In view of the flexibility of power laws with different power indices to describe different growth mechanisms, the power law may be used to estimate the growth velocity of dendrites into the undercooled droplet and the solid fraction evolution during recalescence.[155][626] The limit of the droplet temperature rise during recalescence is the liquidus temperature. Satisfying this criterion, the maximum solid fraction during recalescence may be calculated on the basis of an

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energy balance.[156] As recalescence proceeds, the interface velocity and hence the release rate of the latent heat of fusion decrease. When the latter equals the rate of heat extraction at the droplet surface, or the solid fraction during recalescence attains the maximum, recalescence terminates. Following recalescence, the interface velocity decreases to a relatively steady value. Further solidification of the droplet in the mushy (semi-solid) zone, referred to as post-recalescence solidification, is dictated by the Scheil equation.[619][628] After all the latent heat of fusion is removed, the droplet cooling enters into the region (6).

With the above-described heat transfer model and rapid solidification kinetic model, along with the related process parameters and thermophysical properties of atomization gases (Tables 2.6 and 2.7) and metals/alloys (Tables 2.8, 2.9, 2.10 and 2.11), the 2-D distributions of transient droplet temperatures, cooling rates, achievable undercoolings, and solid fractions in the spray can be calculated, once the initial droplet sizes, temperatures, and velocities are established by the modeling of the atomization stage, as discussed in the previous subsection. For the implementation of the heat transfer model and the rapid solidification kinetic model, finite difference methods or finite element methods may be used. To characterize the entire size distribution of droplets, some specific

droplet sizes (for example, D0.16, D0.5, and D0.84 ) are to be considered in the calculations of the 2-D motion, cooling and solidification histories.

For the numerical calculations, the gas temperature may be assumed to be constant due to its large mass flow rates under some operation conditions. However, modeling studies using coupled solution approach[576][620] have showed that thermal coupling may lead to a hot core of the spray, a reduced gas density in the core, and thermal expansion of the spray to satisfy continuity. The gas temperature exhibits a gradient across the spray that affects the momentum and heat transfer between the gas and droplets. Due partly to this gas temperature distribution, the droplets in the center region of the spray may still remain in fully liquid state while those at the edge of the spray may be already completely solidified at certain axial distance. The gas temperature distribution may also influence the radial distributions of droplet sizes, velocities, volume fractions, and mass fluxes in the spray.

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The numerical modeling of the spray stage[154][156] revealed some features of gas flows that are in qualitative agreement with the LDA-measured gas velocities.[609][617] Fig. 5.6 shows the calculated velocities of gas and droplets in the spray. In this figure and the following figures, u is the axial velocity, umax is the axial gas velocity at the starting point of the calculations, r is the radial distance from the spray centerline, z is the axial distance from the starting point, andzmax is the maximum axial distance from the starting point in the calculations. The axial gas velocity decays exponentially along the spray axis as the spray spreads. The radial profiles of axial gas velocities exhibit a shape akin to a Gaussian probability distribution. The axial droplet velocities increase initially along the axial direction and attain rapidly their maximum values. With increasing axial distance, the radial profiles of the axial droplet velocities become wider and approach the profiles of gas velocities. The calculated droplet trajectories show a good qualitative agreement with the high-speed video imaging of the actual spray, as illustrated in Fig. 5.7.

The calculated droplet temperatures and solid fractions in the spray are depicted in Fig. 5.8 and Fig. 5.9, respectively. In these figures, T is the droplet temperature, T 0 is the initial droplet temperature, and fs is the solid fraction of droplet. At any axial distance, the droplet velocity, temperature, cooling rate and solidification rate all exhibit a maximum at the spray axis, and decrease to a minimum at the periphery of the spray, except for the radial locations where solidification occurs. For a given droplet diameter, the achievable undercooling is smaller, the flight time required to reach a given axial distance is longer, and the secondary dendrite arm spacing formed during post-recalescence solidification is larger in the periphery region of the spray than elsewhere in the radial direction. Hence, the droplets in the periphery region of the spray solidify within a shorter axial distance relative to those at the spray axis due partly to the longer flight time in the periphery. These modeling results are consistent with those obtained from microstructural characterization studies[629] in which thick prior-droplet- boundaries (more porosity and less droplet deformation) were observed in the fringe region of spray-deposited materials. In addition, the calculated radial distributions of droplet velocity, cooling rate and SDAS

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Figure 5.7. Comparison of numerical modeling to high-speed video imaging of droplet trajectories in the spray during spray forming. Left: calculated droplet trajectories; Right: high-speed video imaging of an actual spray. (Courtesy of Prof. Dr.-Ing. Klaus Bauckhage at University of Bremen, Germany.)

The numerical modeling of the spray stage[154][156] also provided insight into the effect of droplet size on droplet trajectory and thermal history in the spray (Figs. 5.6, 5.8, and 5.9). With decreasing droplet size, the flight distance required to attain the maximum droplet velocities decreases and the flight time required to reach any given flight distance also decreases. Compared to a large droplet, a small droplet exhibits a large axial velocity, a low temperature, a high cooling rate, and a large undercooling and solid fraction at any radial distance, except for the locations where solidification takes place and the fringe region where the relative velocity between the gas and small droplets is low. In addition, a small droplet has a wider radial distribution at any axial distance. Hence, it may be postulated that coarse droplets comprise the core of the spray whereas the periphery of the spray is populated by fine droplets. These findings are supported

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It should be noted, however, that some of the tendencies described above may become invalid for very small droplets (for example, smaller than 10 µm under conditions in Ref. 156). Such small droplets may require a longer flight time to a given axial distance far from the atomizer due to the high deceleration, and their cooling rates may decrease as a result of the reduced relative velocity and temperature. In addition, the two-way coupling[576] may affect the momentum and heat transfer between atomization gas and droplets so that the droplet behavior may be different from that discussed above, particularly the radial distributions of droplet sizes and velocities.

For a stationary spray without scanning, a Gaussian shaped mass distribution typically develops with an annular-jet or discrete-jet atomizer. The radial mass distribution in the spray can be formulated in terms of mass flux:[632]

Eq. (12)	&	&		r		m
	M	= M 0	exp(−	rc	)
·						·

where M is the mass flux in the spray,M0 is the maximum mass flux at

the spray axis, r is the radial coordinate, rc is the characteristic radius that controls the spread of the spray, and m is the sensitivity index that influences the shape of mass flux profile (m = 1.2 ~ 2.0). More delicate formulation for the mass flux has been developed recently by Uhlenwinkel et al.[631]

The mass flux in the spray scales with liquid metal flow rate. Gas pressure tends to narrow the spray whereas melt superheat tends to flatten the spray.[3] By changing the process parameters and/or manipulating the configuration and/or motion of the spray, the mass distribution profile can be tailored to the desired shape. For example, a linear atomizer produces a relatively uniform mass distribution in the spray. The mass flux distribution in the spray generated with a linear atomizer has been proposed to follow the elliptical form of the Gaussian distribution:[178]

	& &	ì	x		2		y		2	ü
		ï								ï
Eq. (13)	M = M	0 expí– (		)		- (		)		ý
			Bx				By
		ï								ï
		î								þ