Materials Science and Engineering A304–306 (2001) 235–239

A simplified model for gas atomizationG. Vedovato∗, A. Zambon, E. RamousDIMEG, Università di Padova, 35131 Padova (PD), Italy

Abstract

Asimplifiedmodel describing the cooling behaviour and the solidification of undercooled droplets in gas atomization has been developed.The resulting computer code can predict the cooling behaviour both in the liquid and in the solid state for any powder size. The dropletvelocity in the collecting chamber aswell as its temperature, solid fraction and cooling rate can be predicted as a function of the characteristicsof the processed alloy, the superheating of the melt, the atomizing gas, its velocity at the nozzles and the droplet diameter. The modelovercomes the problem of the undetermined value of the undercooling and evaluates the effects of the simplified approach. © 2001 ElsevierScience B.V. All rights reserved.

Keywords: Gas atomization; Modelization; Recalescence; Equilibrium; Nucleation temperature

1. Introduction

The present model can describe the solidification ofdroplets in gas atomization. In this process it is importantto know the thermal transient of the particles. This is notan easy task because:

• there is a range of different transients due to differentsizes of the particles;

• solidification occurs with undercoolings, which are vari-able, unknown and not measurable.

Previous numerical models [1–7] were concerned mainlywith the recalescence: undercooling was considered a pa-rameter or it was predicted using heterogeneous nucleationtheories. In a strict sense those are valid only for massivenucleation. In a molten batch there are a number of differenttypes of nucleant particles, each one with a different nucle-ation temperature. In conventional solidification (such as incastings and ingots) undercoolings are generally small andconstant because they are originated only by the particleswith the highest nucleant potency. In gas atomization thoseparticles can no longer spread solidification to the wholemolten bulk, because this is fragmented and the nucleantparticles are scattered in a large number of small droplets.In each droplet the nucleation is originated by the particlewith the highest nucleant potency that are not necessarilythe same: thus undercooling is variable. There are modelsand data that confirm this hypothesis [8–11]. Furthermore,there is experimental evidence that there are both single (one

∗ Corresponding author.

for each droplet) and multiple (several events in a singledroplet) nucleation sites.On the other hand, it can be hard to model the variation of

the nucleation temperature with just a single parameter. Toovercome these problems a model was developed that canpreview the thermal history considering two extreme cases:

• solidification without undercooling;• cooling without nucleation (in the liquid phase).

2. Model formulation

2.1. Motion of a spherical droplet

To describe the motion of a droplet in the atomizer cham-ber, the following hypotheses are made:

1. droplets have a spherical shape which is rigid and con-stant (contractions due to cooling and the phase changesare neglected);

2. droplets form immediately at the nozzle outlet (thespheroidization times were calculated for definite dia-meters to be very short with respect to the total solidifi-cation time);

3. interactions with other droplets are neglected;4. the velocity field of gas flow is constant;5. droplets are supposed to move on a straight path on theflow centreline, therefore:

• gas fluctuations due to turbulence are neglected;• gas turbulence, that could alter particles trajectory, isneglected.

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236 G. Vedovato et al. /Materials Science and Engineering A304–306 (2001) 235–239

Under such hypotheses, the motion of a spherical droplet(in a one-dimensional gas flow) can be described by thefollowing equation:

ρdVddud(t)dt

= Vd(ρd − ρg)g − 12ρgAdCD(z)|ud(t)

−ug(z)|(ud(t) − ug(z)) (1)

where ud, ρd, Vd and Ad are the speed, density, volume andcross-sectional area of a droplet, respectively, ug is the axialvelocity of the gas, ρg the density of the gas, g the accel-eration due to gravity, t the time. CD is the drag coefficientwhich can be approximated with an accuracy better than 7%by the following empirical equation [5]:

CD = 0.28+ 6Re

+ 21√Re

(2)

where Re ≡ ρgDd|ud(t)−μg(z)|/μg is the droplet Reynoldsnumber, Dd the droplet diameter, μg the dynamic viscosityof the gas. Eq. (2) is applicable over the range of 0.1 <

Re < 4000. Analytical relations describing the velocityfield of gas flow are not available. Instead an equation validfor a turbulent circular free jet emerging from an orificeinto a resting fluid is used. It describes the velocity de-cay profile along the centreline of the incompressible freejet [5,12,13]:

ug(z)

u0=

[1+

(z

ξ

)20]−0.05(3)

where, u0 the gas velocity at nozzle outlet, z the flow axis dis-tance from nozzle, ξ the gas velocity decay constant whosevalue is ξ = α

√Ag, where α is an empirical constant that

was estimated as 7.414 [14] for a turbulent free jet and Agthe nozzle throat-area.

2.2. Cooling and solidification

The typical cooling behaviour of an atomized droplet (foran eutectic alloy) is shown in Fig. 1. If the enthalpy of adroplet is H, the sum of the internal energy (function of thetemperature) and the latent heat 1 (a function of the liquidfraction), its value during flight is given by ∂H(t)/∂t = q,where q is the heat flux. The radiation contribution can beneglected.During cooling, the temperature gradient inside the par-

ticle can be assumed null (because its Biot number (Bi =hd/λ) is smaller than 0.1). Thus, the thermal behaviour of anatomized particle can be described by a simple Newtonianformulation:

Vd

(cmρd

∂Td(t)

∂t− L

∂fs(Td(t))

∂t

)= hSd(Td(t) − Tg) (4)

where Vd and Sd are the droplet volume and surface area, Lis the latent heat per unit mass, Td the droplet temperature,

1 The pressure contribution is not relevant and is neglected.

Fig. 1. Schematic behaviour of a molten particle: (1) liquid phase cooling,(2) recalescence, (3) solidification, (4) eutectic solidification, (5) solidphase cooling.

Tg the gas temperature, t the time, fs the solid fraction andcm = cl− (cs− cs)fs is the specific heat of the mushy alloy(with cl, cs heat capacities of the liquid and solid alloy), hthe convective heat transfer coefficient.The gas temperature is assumed to be constant and equal

to 20◦C. If the solid fraction is a function of the temperatureonly, Eq. (4) becomes

Vd

(cmρd

∂Td(t)

∂t− L

dfs(t)dTd

∂Td(t)

∂t

)= hSd(Td(t) − Tg)

(5)

This can be rewritten as

1τ0

(Td(t) − Tg) =(1− L

cmρd

dfs(t)dTd

)∂Td(t)

∂t(6)

where τ 0 is a time coefficient defined as 1/τ0 = hSd/cmρdVd.

2.2.1. Heat coefficientThe convective heat transfer coefficient h is given by the

Ranz–Marshall relation [15]:

h = λg

Dd(2.0+ 0.6

√Re 3

√Pr) (7)

where λg is the gas thermal conductivity, Pr = cgμg/k thegas Prandtl number, cg the gas specific heat (per unit mass).

2.3. Solidification and undercooling

After the temperature of a liquid particle reaches theliquidus temperature, there will be a certain degree of un-dercooling before a nucleation event occurs. Nevertheless,the ideal case of a solidification without undercooling willbe considered at first.

G. Vedovato et al. /Materials Science and Engineering A304–306 (2001) 235–239 237

2.3.1. Solidification with no undercoolingThe solid fraction behaviour can be calculated making the

following assumptions:

• complete solute mixing in the liquid phase (there is nosolute gradient in the liquid phase);

• kinetic and capillarity effects are neglected.Furthermore, two different extreme assumptions can beconsidered:(a) Full diffusion in the solid phase. The solid fraction

behaviour is given by the “lever rule”:

fs = 11− k0

(TL − T

Tm − T

)(8a)

k0 is the equilibrium distribution coefficient, Tm is themelting temperature of the pure solvent, TL is the liquidustemperature. It is interesting to point out that this would bean equilibrium transformation.(b) No diffusion in the solid (Ds = 0). Under this hypoth-

esis, the solid fraction behaviour is described by the Schiel’sequation [5]:

fs = 1−(

Tm − T

Tm − TL

)1/(k0−1)(8b)

Of course, this is no longer an equilibrium transformation.The temperature will be (in each instant) lower than in case(a) because a higher quantity of enthalpy would increasethe energy of the resulting metastable structure (“structuralenthalpy”). Eqs. (8a) and (8b) can be introduced in Eq. (6).

2.3.2. Solidification with undercoolingIn a real transformation, a droplet will cool in the liquid

phase until a nucleation event is obtained. Then the tem-perature Td of the droplet quickly rises due to the latentheat release (recalescence phase). It is not easy to quantifythe temperature rising, because recalescence is not an equi-librium transformation. We will evaluate this value withconsideration to the total enthalpy.We made the assumption that the droplet geometry (and

therefore the motion) does not change: the heat transmissioncoefficient h (Eq. (7)) is always the same, independent of thecooling path. In both cases (with or without undercooling),the heat loss by convection during flight is derived fromNewton’s law: q = hSd(Td(t) − Tg); the terms h, S and Tgare the same, the temperature Td is similar, so the heat lossis very similar.As a consequence, the undercooled droplet will start so-

lidification with an enthalpy just slightly higher than it wouldhave had if it begun to solidify without undercooling. Thedifference is rather small as will be seen in Section 4.The recalescence will stop in the temperature interval

between the two lines of full diffusion and no diffusion (seeFig. 2). Also the temperature difference between the twolines is small. We point out that, due to the high solidifi-cation rate it is more realistic to consider that a complete

Fig. 2. Different theoretical cooling paths of an atomized particle.

segregation occurs. Thus the recalescence will stop at thetemperature given by Schiel’s equation.To conclude, a solidification with a generic undercooling

can be studied by calculating just two extreme cases: so-lidification without undercooling (and full segregation) anda solidification always in the liquid phase. The only inde-terminate factor is the nucleation temperature that can bedetermined in a second time.The possible nucleation of different phases must be con-

sidered (just like in Fe–Ni alloys). We believe that in thiscase, the model could be used with more important con-nection to experimental data. Anyway, in this case the finaltemperature will be lower than in the case of simple seg-regation because a higher quantity of enthalpy will go toincrease the resulting metastable structure.The model will work with pure metals and alloys with-

out nucleation of metastable phases (such as Ni–Cu). Animportant limit is given by the hypercooling (or “critical nu-cleation temperature”): when the undercooling exceeds thislimit, all the liquid fraction solidifies during recalescence.The hypercooling is given by Thyp = H/Cl.We point out that according to the model just developed:

• The recalescence arrest temperature is a function of thesolid fraction corresponding to a certain enthalpy. It doesnot depend on parameters such as the heat coefficient,time, nucleation rate and density, etc.

• The criterion just exposed is valid even if the recalescencestep is not adiabatic: in any case, it will stop when the“Schiel’s line” is reached (see Fig. 2).

3. Model assessment

The model previously described was solved by meansof a computer code, where Eqs. (1) and (6) were solvedsimultaneously by means of finite differences schemes. Itis possible to change the following simulation parameters:processed alloy (all the characteristics), atomizing gas (all

238 G. Vedovato et al. /Materials Science and Engineering A304–306 (2001) 235–239

Table 1Thermophysical properties of nitrogen

Specific heat (or heat capacity) 1039 J kg/KThermal conductivity 2.6× 10−2 W/mKViscosity 1.78× 10−5 N s/mDensity 1.16 kg/m3

the characteristics), gas velocity at nozzle outlet, dropletdiameter, superheating of the molten alloy. The main pa-rameters that the program can predict (as a function of thetime or the axial position) are: droplet temperature, solidfraction, gas velocity, droplet velocity, Reynolds number,heat coefficient, cooling rate. The characteristics of the at-omizing gas are shown in Table 1. To test the model, theAl–4.5wt.% Cu alloy was used (Table 2), because its proper-ties are well known, and it has already been studied in otherpapers.

4. Results and discussion

In Fig. 3, the results obtained by the computing of the twoextreme cases of thermal behaviour (no undercooling andno nucleation) are traced. A generic solidification behaviourof an undercooled particle can be traced by means of thesetwo cases. Initially a particle cools according to the lowerline (no nucleation): when a generic recalescence occurs thetemperature rises until the limit given by the Schiel’s law(see Section 2.3.2). This applies to any possible recalescencebehaviour, instantaneous or not.It is possible to see the maximum error involved, assuming

that the heat flux is invariant with undercooling. Thus, when

Fig. 3. Cooling behaviours of Al–4.5wt.% Cu droplet. Just the two extreme cases (no undercooling and no nucleation) were computed. Some possibleexamples of recalescence are drawn (dotted lines). Diameter 50�m, superheating: 40K, gas initial velocity: 260m/s. (1) Instantaneous recalescence (andadiabatic), (2) non-instantaneous recalescence, (3) nucleation event under hypercooling limit: all the liquid fraction solidifies during recalescence. Themaximum error made considering the enthalpy independent of undercooling is shown.

Table 2Thermophysical properties of Al–4.5wt.% Cu alloy

Specific heat of liquid alloy 982 J kg/KSpecific heat of solid alloy 900 J kg/KDensity 2800 kg/m3Latent heat per unit mass 348 000 J/kgThermal conductivity 164W/mKEquilibrium solute distribution coefficient 0.14Slope of liquidus −3.4Melting temperature of pure solvent 934KLiquidus temperature 919KSolidus (or eutectic) temperature 821KHypercooling 354.37K

solidification following path A is completed, the differencebetween values of curve A and B should be exactly equal toThyp. A certain discrepancy has instead to be expected dueto the lower heat flux experienced by a particle solidifyingaccording to path B. In Fig. 3 this discrepancy is about 30K,in the present example this is the maximum possible error.The hyperundercooling (Thyp = H/Cl) limit is traced.Whenthe undercooling exceeds this limit, all the liquid fractionsolidifies during recalescence.

5. Summary and conclusions

In order to predict the solidification behaviour of particlesin gas atomization, a mathematical model was developed.A new theoretical approach to determine the arrest recales-cence condition is proposed: when a nucleation event occursin an undercooled alloy, the temperature rises until a valuegiven by the Schiel’s equation.

G. Vedovato et al. /Materials Science and Engineering A304–306 (2001) 235–239 239

Thus, the solidification of an undercooled particle can bedescribed by computing two extreme cooling paths:

1. solidification without undercooling;2. cooling occurring entirely in the liquid phase (with nonucleation event).

A particle will initially cool through the second path.When a nucleation event occurs the temperature will in-crease until the first path is reached. The solidification willbe completed through the first path (see Fig. 3). We pointout that this approach can be used even if the recales-cence phase is neither adiabatic nor instantaneous. In thisway it is no longer necessary to determine the nucleationtemperature exactly. Therefore the model can be useful inindustrial applications, in the designing of a gas atomizeror to evaluate whether an existing atomizer can be used toprocess a specific alloy. In this case an approximate un-dercooling range can be fixed. In an experimental field itcan be used to study the solidification evolution. With theaid of experimental data it allows evaluation of an existingundercooling extent and prediction of the influence of theprocess parameters on the microstructure.

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