Modeling of droplet-gas interactions in spray atomization of Ta-2.5W alloy

Materials Science and Engineering

A191 (1995) 171-184

Modeling of droplet-gas interactions in spray atomization of Ta-2.5 W alloy Huimin Liu”, Roger H. Rangel”, Enrique J. Laverniab aDepartment of Mechanical and Aerospace Engineering, University of California, Irvine, CA 92717, USA hDepartmen t of Chemical Engineering and Materials Science, University of California, Irvine, CA 92717, USA Received 23 November 1993; in revised form 28 February 1994

Abstract In the present paper, the droplet-gas interactions that are present during spray atomization of a Ta-2.5W alloy using N? gas are numerically investigated. A simple two-dimensional (2D) flow model and a lumped parameter formulation based on the modified Newton’s law of cooling are developed to simulate the flow and heat transfer phenomena, including rapid solidification of droplets in the spray cone. The 2D distribution of droplet velocity, temperature, cooling rate and solid fraction is calculated. The microstructural characteristics of solidified particles and as-deposited materials are discussed briefly. The effect of droplet size on the 2D distribution of flow, thermal and solidification histories is also addressed. The numerical results demonstrate that, 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 cone, except for the locations where solidification occurs. The droplets in the periphery region solidify within a shorter flight distance relative to those at the spray axis owing to longer flight time in the periphery. At any axial distance, a small droplet exhibits a wider radical distribution. Hence, coarse droplets constitute the core whereas the periphery of the spray cone is populated by fine droplets. Accordingly, the microstructure of spray deposited materials is predicted to be fine in the edges of the deposits as a result of high cooling rates associated with small droplets. These results are in qualitative agreement with available findings. Keywords: Tantalum; Tungsten; Alloys; Droplet-gas

interactions;

Spray atomization

1. Introduction During the past decade, spray atomization and deposition has attracted considerable interest as a viable process for manufacturing structural materials of a variety of geometries with benefits associated with rapid solidification, such as fine grained microstructures, increased solid solubility, non-equilibrium phases and small sized precipitates [ 11.The microstructure and mechanical properties of as-deposited materials are critically dependent on the thermal and solidification histories of atomized droplets. For example, variations in microstructure at different locations within an as-deposited material have been noted in earlier experiments, as reported in Ref. [2], which suggested the importance of the spatial distribution of thermal and solidification histories of the droplets in the spray cone. To elucidate the microstructural 0921-5093/95/$9.50 0 1995 - Elsevier Science S.A. All rights reserved WI~0921-5093(94)09631-6

evolution of the droplets, it is necessary to analyze systematically the two-dimensional (2D) fluid flow, momentum and heat transfer, as well as rapid solidification phenomena, which occur simultaneously during droplet-gas interactions. Inspection of the available literature shows that numerical treatments of the momentum, heat and mass transfer in atomization and deposition processes, as well as rapid solidification phenomena, have been reported by a number of investigators [2-141. In limited studies [7,13], 2D experimental measurements of atomization gas and droplet velocities have been conducted. A few complicated 2D eulerian models [3,14], based on the continuum medium theory, have been used to calculate the momentum and thermal interactions between the atomization gas and droplets in the spray cone. In most of these studies, however, numerical treatments are confined to one-dimension, and only the droplet trajec-

172

H. Liu et ul.

1

Materials Science and Engineering Al91 (199.5) 171-184

tory and thermal history along the center-line of the spray cone are addressed. More recently, Grant et al. [ 151 presented a modeling study of droplet dynamic and thermal histories during spray forming. In spite of the one-dimensional (1D) nature of this modeling, a major advance in the formulation of atomization gas velocity has been made by analyzing previous experimental and theroetical results and correlating them into a simple, non-apparatus-specific form. In addition, the solidification process has been treated in detail by considering three thermal regions, i.e. nucleation and recalescence, segregated solidification, and eutectic solidification, which is hence applicable to the rapid solidification process of eutectic alloys. In the present paper, we introduce a simple 2D lagrangian model and apply the model to investigate the momentum and thermal interactions that occur between N, atomization gas and Ta-2.5 W (in weight per cent) alloy droplets. The rapid solidification phenomena that are present during the droplet-gas interactions will also be considered by combining the 2D model with a phase change kinetic model. A fourthorder Runge-Kutta algorithm is employed for the simultaneous solution of the 2D non-linear differential equations of motion of droplets. Particular emphasis is placed on elucidating the spatial distribution of droplet velocity, temperature and rapid solidification histories, as well as the microstructural characteristics of solidified particles in the spray cone and the as-deposited materials. Such a mathematical modeling is highly desirable, because of the large expense and difficulties associated with the melting and spray processing of the material considered in the present numerical investigation. The selection of this alloy composition was promoted by recent interest in this material for heat exchangers and other welded-tube applications as a result of its unique combination of high melting temperature, high ductility, high density, high formability and ultra corrosion resistance at temperatures below 150 “C [16].

2. Model formulation The droplet dispersion that is present during atomization and deposition comprises a distribution of droplets of differing sizes. The momentum and heat transfer processes between the atomization gas and droplets may be treated using either an eulerian or a langrangian approach. We employ the langrangian approach because of its relative simplicity in numerical calculations. In the 2D langrangian model that will be introduced below, we consider first the 2D motion, cooling and solidification histories of a single droplet. Then, calculations are carried out for some specific droplet

diameters determined by earlier experimental measurements to characterize the entire size distribution of droplets. 2. I. Two-dimensional flow

model

The spatial motion of a droplet during atomization and deposition not only determines its residence time in the spray cone, but also critically influences its cooling and solidification by changing the surface interaction conditions. Therefore, knowledge of the velocity fields of both the gas and droplets is a precondition of the numerical calculation of droplet temperature and solidification histories. The 2D motion equations of a single droplet may be derived on the basis of a 2D force balance on a single sphere in a high speed, axisymmetric gas flow, and may be written in a langrangian frame as [ 17,181

in the axial direction dud

31p,

dt

4d,Q,

uriug-

ud

) Cdrag

(2)

in the radial direction where it has been assumed that (a) the concentration of droplets in the atomization gas is sufficiently dilute, so that droplet-droplet interactions may be safely neglected, (b) the Basset history term and lift force may be ignored compared with the inertia and the standard drag forces, (c) droplets are spherical and their size remains unchanged during flight, so that the correlation for the drag coefficient of a solid sphere is applicable, and (d) the flow of the atomization gas is axisymmetric, and hence droplets neither rotate nor move in the angular direction of the spray cone so that the motion of droplets may be considered to be 2D (in the axial and radial directions of the spray cone). A description of the relevant variables is given in nomenclature at the end of the paper. The gas velocity distribution that is necessary for solution of Eqs. (1) and (2) can be determined using data measured by Bewlay and Cantor [7,13]. Based their experiments, which were conducted with atomizer similar to that used in our experiments, decay in the mean axial gas velocity at the center-line the spray cone satisfies an exponential relationship the following type [2]:

the the the on an the of of

H. Liu et al. Table 1 Parameters

Materials Science and Engineering A191 (1995) 171-184 Table 2 Physical properties

[2,20,2 l]

used in computations

Initial axial velocity of gas Initial axial velocity of droplet Geometrical parameter in Eq. (3) Geometrical parameter in Eq. (3) Geometical parameter in Eqs. (4) and (5) Geometrical parameter in Eqs. (4) and (5) Gas temperature Initial droplet temperature Liquidus temperature Solvent melting temperature Solidus temperature Equilibrium partition coefficient Effective molecular diameter Self-diffusivity in melt Solid-liquid interfacial energy Molar volume of melt Emissivity of surface Stephen-Boltzmann constant Boltzmann’s constant Gas constant Atomic diameter Avogadro’s constant Specific heat ratio of gas Mechanical equivalent of heat

/

u,,,(ms-‘) udll (m s- ‘) z, (m)

520 2 0.02

z,(m)

0.17

D(m)

0.004

C

0.268

$(K) Tdo(K) T, (K) T,, (K) T, (K) k,

220 3750 3306 3293 3299 2

d, (m) D,, (mz s-l) a,,, (J mm’)

2.94 X lo-“’ 3.335 X lo-’ 0.322

V, (m” mol-‘) E o(W m-‘K-“)

1.086 X 10m5 0.5 5.677 X lo-”

x(J K-‘) R(J K-’ mol-I) a(m) Njatoms mol-‘)

138 X lo-?” 8:319 2.94 X lo- ‘(I 6.02 X 1O23 1.4 0.102

Y(mkgJ-I)

cpd(J kg-’ Km’) Pd (kg m-‘! AH,,, (J kg-‘)

of alloy used in computations

2 -2,

i

Ta-2.5wt.%W(s!

244 14862 171400

234 15 000

It should be indicated that, under our experimental conditions, the gas exits the atomizer nozzle at supersonic velocity [2]. The gas expands and accelerates within the distance Zi and then decays through shocks and diffusion to subsonic speed in a non-uniform “roller-coaster” fashion. Hence, the compressibility of the gas may significantly influence the transfer processes between the gas and droplets. However, the dependence of the drag coefficient C,,,, on the Mach number seems to be negligibly small in-our Reynolds’ number regime according to Schlichting [ 191. Hence, Cdral:is expressed here only as a function of Re using an improved approximation [3] for the standard drag curve: c O
C drag

=(

24/Re0.6”h,

1
0.5,

40O
3.66 X lOPa ReO,jZx, 3 X 10”
1

z,- z,

The gas exit velocity us0 and the constants Zi and z, as listed in Table 1, are related to atomizer geometry and atomization parameters and have been determined and discussed elsewhere [2]. On the basis of our experimental observations and the experimental results of Bewlay and Cantor [7,13], the radial distribution of the axial and radial gas velocities may be formulated as f2 (r z) L_[l_(,,:i.j’ ~s

(4)

u,(r,z)=~,(rTg)L

(5)

@A

D/C+z

[20,22]

Ta-2.5wt.%W(l)

0.18,

u&O,z) =ug,,exp -___

173

under the axisymmetric assumption. In Eqs. (4) and (5), the constants D and C are related to atomizer geometry and atomization parameters and have been determined experimentally, as listed in Table 1.

105

(6)

2 X 10h

2x lO”
The 2D velocity distributions of the droplets and gas in the spray cone can be then calculated using Eqs. (l)-(6), once the droplet size and the properties of the droplets and gas have been established. Eqs. (1) and (2), together with Eqs. (3)-( 6) and the parameters listed in Tables 1-3, were solved using a fourth-order Runge-Kutta algorithm. 2.2. Lumpedparameterformulation In atomization and deposition of a Ta-2.5 W alloy with N, atomization gas, a droplet undergoing cooling and phase change may experience three states: (a) fully liquid, (b) mushy, and (c) fully solid. In all three states, the Bi number calculated under our experimental conditions is smaller than 0.1. Thus, the lumped parameter formulation used in previous studies [ 2,5,11,12] can be employed for the droplet temperature calculation. During the flight of droplets in the spray cone, the forced convective and radiative heat exchange with the atomization gas leads to a rapid heat extraction from the droplets. On the contrary, the frictional heat produced by the violent interactions between gas and droplets reduces the heat extraction rate. This effect

H. Liu et al.

174 Table 3 Physical properties [231

of N, atomization

c,,(Jkg-‘K-l)

cpg=1021.290+0.135T’-

/

Materials Science and Engineering A191 (1995) 171-184

gas used in computations

1.794x

10h/T’?

[24]. However, the effects of the surface temperature and the Mach number may be substantially eliminated if all properties are evaluated at a film temperature T’ defined by [24] I

,I

K,(W m-l K-‘)

$=

1+0.032Ma2+0.58

g pg(kgme’)

Pg=

WQ,,,

2.401 x lo-? 0.8 1.250 1.658 x 10m5 118 0.770 302.396 0.893 296.893

n ,ogo (kg m-“) pgo (kg m-’ s-‘) C

r, U,(ms-‘) Ma R(m? se2 Km’)

becomes important in a supersonic atomization gas where the compressibility of the gas is significant [24]. Hence, the transient temperature of a single droplet during flight in the high speed atomization gas is calculated using the modified Newton’s law of cooling [24] dT,_ dt

1

(9) 1

Thus, h may still be estimated using the experimental correlation proposed by Ranz and Marshall [25]:

T’

h=:

Kg0W m-’ K-‘)

$( g

6

(2 + 0.6 Re”* P?),

l
(10)

It should be indicated that a discrepancy between modeling and experimental results has been found and reported in Ref. [2] when modeling was conducted using Newton’s law of cooling and the convective heat transfer coefficient was calculated directly using the Ranz and Marshall correlation. In that particular study [2], the discrepancy was eliminated by modifying the heat transfer coefficient on the basis of the individual experimental data, hence limiting generalization of the method and results. In the present work, however, a good agreement between the modeling and experimental results can be ensured through employing the modified Newton’s law of cooling and calculating the heat transfer coefficient at the film temperature. Hence, such treatment may obtain more wide application in the numerical simulation of droplet cooling and solidification histories during supersonic atomization.

&dcpd 2

%I Td - Tg - r, ~ wc,,

+ m( Td4- q4)

1

2.3. Rapid solidification of droplets (7)

with fully liquid

dfs

-AH m

dTd

mushy fully solid

where the term ~,u~~/2gJc,, in Eq. (7) represents the effect of the frictional heat on the droplet temperature. The recovery factor r, as a function of the Mach number is approximated according to experimental results in Ref. [24]. In a supersonic gas flow, the convective heat transfer coefficient h not only is a function of the Reynolds number and the Prandtl number, but also depends on the droplet surface temperature and the Mach number

For the purpose of analysis, the thermal history of atomized droplets in the spray cone can be divided into four regions: (a) rapid cooling in the fully liquid state, (b) nucleation and recalescence, (c) post-recalescence solidification, and (d) cooling in the fully solid state. The thermal histories in regions (a) and (d) can be described directly using Eqs. (7) and (8). The nucleation temperature (hence the achievable undercooling) and the solid fraction evolution during recalescence and post-recalescence solidification must be determined additionally. In atomization and deposition, a large undercooling may generally be attained prior to the onset of nucleation. An upper bound of the degree of undercooling may be estimated using Hirth’s formulation [26] for homogeneous nucleation within a droplet: 0.01Z$rd3 AT -i =l

(11)

H. Liu et al.

The homogeneous using [81

/

Materials Science and Engineering Al 91 (1995) 171-184

nucleation rate I may be calculated

V,’ 7;21Jm’ ;KT~AT2 AHm2

(12)

It should be indicated that in Eq. (12), the dependence of the free energy change on undercooling has been formulated according to the work of Turnbull and Fisher [27] in order to maintain the problem tractable, although an improved correlation [28] has recently been reported. In view of the high probability of heterogeneous nucleation associated with the conditions that are present in atomization and deposition processes, the undercooling calculated with Eqs. (11) and (12) must be modified by a factor depending on droplet size. This factor has been derived in Ref. [2] based on microstructural observations of the atomized powders. In the present work, we estimate the degree of undercooling for heterogeneous nucleation using the method suggested in Ref. [ 21. 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, i.e. recalescence. The extent of the temperature rise during recalescence depends basically on undercooling and cooling rate, and can be described by Eq. (7), ignoring the surface heat extraction [ 151:

dT, AH, dL -=-dt cpl dt

(13)

The solid-liquid interface velocity during recalescence is also a function of undercooling and cooling rate. Different expressions for the interface velocity have been employed in previous studies [6,9,12,15,29], including the linear and exponential laws for planar continuous growth at small and large undercoolings respectively [6,9,12,15,29] and the Ivantsov equation for growth of paraboloid dendrites [5], as well as the power law for dendrite growth [30]. In view of the ability of power laws with different power indices to describe different growth mechanisms, we employ the power law to estimate the growth velocity of dendrites into undercooled melt. Accordingly, the solid fraction evolution during recalescence is calculated using the following expression [ 121, ignoring the errors induced by higher order d t: dk dt

6K,AT” d

(14)

where K, properties, and should we estimate

175

and m are related to the relevant droplet cooling conditions and growth mechanisms be experimentally determined. In this study, K, using [6,9]

(15) and select m as unity on the basis of the analyses in Ref. [30]. Although it appears that there is no difference between the interface velocity expression in Eq. ( 14) (for m = 1) and the linear law, it should be noted that the linear law is valid only for planar continuous growth at small undercoolings. Our experimental data [2] suggest that the undercoolings experienced by droplets during spray atomization are large, and the most likely growth mechanism is dendrite growth. Hence, we refer to the interface velocity expression in Eq. (14) as a power law, as defined in Ref. [30]. The limit of the droplet temperature rise during recalescence is the liquidus temperature. The maximum solid fraction s during recalescence may be calculated using the following relationship [2]: s=(7;-T+&-

(16)

m

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 s, whichever occurs first, recalescence terminates. Following recalescence, the interface velocity decreases to a relatively steady value. Further solidification of the droplet in the mushy zone, referred to as post-recalescence solidification, is dictated by the Scheil equation [ 15,3 11:

dfs_ dT,

l--S (k-

l)(Tm-

T)

(17)

Using Eqs. (7)-( 15) and ( 17), together with the related parameters given in Tables l-3, the 2D distributions of transient droplet temperature, cooling rate, solid fraction, and achievable undercooling in the spray cone can be calculated, once the droplet size, velocity and trajectory have been established. Eqs. (7), (13), (14) and (17) were solved using a finite difference method with a time step of 10-s s. During the calculations, the gas temperature is assumed to be constant owing to the large mass flow rate of the gas under the present conditions. All computations were completed on a DEC 3000/400 AXP workstation.

H. Liu et al.

176

1

Materials Science and Engineering A191 (1995) 171-184

3. Results and discussion

The numerical model described above is used to compute 2D velocity, temperature, cooling and solidification histories of Ta-2.5 W droplets in N, atomization gas. Five specific droplet diameters, 10, 30, 60, 100 and 200 pm, are selected to characterize the entire size distribution of droplets. Moreover, it is assumed that molten Ta-2.5 W at a superheat temperature of 3750 K and an initial velocity of 2 m SC’ is atomized by relatively cool (220 K) N, gas. The physical properties of Ta-2.5 W and Nz gas used in the computations, together with other relevant parameters, are summarized in Tables l-3. In the following sections, we present and discuss the numerical results. 3.1. Velocity evolution and two-dimensional distributions of atomization gas and droplets in spray cone The velocity vectors of the atomization gas in the spray cone are summarized in Fig. 1. It can be seen that the axial gas velocity on the axis line decays rapidly with increasing distance in the axial direction. At the same time, the gas spreads and mixes with the surrounding environment, leading to an increase in the gas flow diameter. The radial profiles of axial gas velocities

exhibit a shape that is akin to a gaussian probability distribution. The gas flow is predominantly in the axial direction as a result of the axisymmetric assumption. These results are in qualitative agreement with the features of the laser Doppler anenometry (LDA) measurements of the gas velocities reported in Refs. [7,131. Fig. 2 shows the velocity vectors of the droplets of 60 ,um diameter in the spray cone. Initially, both the axial and radial droplet velocities increase along the axial direction and attain rapidly their maximum values. At the same time, the radial profiles of axial droplet velocities approach rapidly a shape that is analogous to that of the axial gas velocity at the axial distance of approximately 0.2 m. With increasing axial distance, the radial profiles become wider. Moreover, at any axial distance, the droplet velocity at the spray axis is larger than at the periphery of the spray cone. Hence, the flight time required to reach a given axial distance is shorter at the spray axis than at the periphery of the spray cone. The computed profiles are in with the deposit contours qualitative agreement obtained in related experiments [ 5,3 21. Fig. 3 shows the axial velocity evolution of the atomization gas and droplets of different diameters along the axis line of the spray cone. The gas velocity remains constant within the distance zi and then decays expo-

-0.1

-0.05

Radial

Di”,,,(In)

Fig. 1. Velocity field of the atomization

0.05

gas in the spray cone.

0.05

xM5

0.

Radial Dikncc (m) Fig. 2. Velocity field of the droplets spray cone.

of 60 pm diameter

in the

/

H. Liu et al.

177

Materials Science and Engineering Al 91 (1995) 171-184

--=

r = 0 (mm)

d

~

-

z x

.Z

-

-

d d d d

10

I = = =

(pm) 30 (pm) 60 (pm) 100 (pm) 200 (pm)

‘,, -;--.

260

d

4

1’0 1’5 2’0 Radial Distance (mm)

0

0

100

200

300

Axial Distance

400

500

30

600

(mm)

z = 60 (mm)

520 Fig. 3. Axial velocity evolution of the atomization lets along the axis line of the spray cone.

25

gas and drop-

__-d

-

-

0

6

520

- -

d = 200 (P) Gas

18 24 12 Radial Distance (mm)

-

lb)

d = 30 (pun) = 60 (P) d = 100 (m)

-----

nentially in the axial direction. Each droplet is accelerated to its maximum velocity at the flight distance where the relative velocity between the droplet and gas is zero. Beyond this point, the droplet is decelerated. However, because of the large density of the material considered (Table 2), the deceleration of larger droplets (for example, d= 60, 100 and 200 pm) is small, leading to a nearly constant velocity during further flight. The flight distance required to attain the maximum velocity decreases with increasing droplet size. A small droplet generally has a large acceleration and a high velocity. Hence the flight time that is required for any given flight distance decreases with decreasing droplet size. However, there is an exception with the droplets of 10 pm diameter, as evident from the figure. These droplets follow closely the gas flow owing to their small inertia. Consequently, the flight time is shorter for these droplets in the region close to the atomizer and turns to longer in the region far from the atomizer. For example, the flight time to a distance of 600 mm at the spray axis is 5.1 ms, 4.4 ms, 5.5 ms, 6.6 ms and 8.8 ms for 10 pm, 30 pm, 60 pm, 100 pm and 200 pm droplets respectively. Similar behavior has also been reported in earlier studies [ 5,151. Fig. 4 shows the radial distributions of the axial velocities of the atomization gas and droplets of different diameters in the spray cone. Clearly, small droplets exhibit larger axial velocities at any radial distance. Small droplets also have a wider radial distribution at any axial distance (for example, z = 30,60 and 90 mm). This suggests that a larger population of small droplets impinge on the surface in the periphery region of the sprayed deposit. This suggestion is verified by an experimental study of particle size distributions [33] which demonstrated that the oversprayed particles collected during atomization and deposition exhibit a

I d = 10 (m)

30

2 = 90 (mm)

,

-_-d

I d = 10 (P)

-d

= 30 (W)

= 6s (~0 _____ d - 100 tW

---Gas

0

(c)

/

6

1 ”

b 1,

7 7 ‘I’

d = 200 (~0

r

I’

I

”

12

18 24 Radial Distance (mm)

Fig. 4. Axial velocity distributions of the atomization droplets in the radial direction of the spray cone.

7

30 gas and

small median size than the average size of powders generated and acquired during atomization. The experimental observation of small median particle diameters and a narrow size distribution of oversprayed powders indicates that the coarse droplets generally form the core whereas the periphery of the spray cone is populated by fine droplets [33]. In another analysis of the spray deposition process [5], the experimental results also revealed that the mass-median droplet

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Materials Science and Engineering A191 (1995) 171-184

diameter decreases along the radial direction from the spray axis. These phenomena may be attributed to the fact that small droplets have small inertia and hence can be easily dragged and accelerated by the atomization gas in both the axial and radial directions. The good agreement between the numerically predicted and the experimentally determined tendency supports the model formulation described herein. 3.2. Temperature, cooling and solidification histories of droplets in spray cone Figs. 5, 6 and 7 show the temperature, cooling and solidification histories respectively of droplets of different diameters along the axis line of the spray cone. From Fig. 5 it can be seen that the droplet temperature rapidly decreases to the nucleation temperature within a short flight distance of approximately 1 mm, 13 mm, 41 mm, 63 mm and 121 mm for 10 pm, 30 pm, 60 pm, 100 pm and 200 pm droplets respectively. The achievable undercooling is calculated to be 699 K, 633 K, 476 K, 241 K and 44 K for 10 pm, 30 pm, 60 pm, 100 pm and 200 pm droplets respectively. The corresponding values of A T/T, are 0.21, 0.19, 0.14, 0.07 and 0.01 respectively, which are within the range observed experimentally for a number of materials [34]. Although the amount of the achievable undercooling was not measured for tantalum, it was believed to be high [34]. From Fig. 6 it can be seen that the cooling rate experienced by each droplet attains a maximum value initially (corresponding to the negative side in Fig. 6) on the order of 105-lo6 K s-l for the droplets larger

than 30 ,um, and 10’ K s-l in magnitude for 10 pm droplets. The cooling rate then decreases continuously along the axial direction until the onset of solidification. Following nucleation, the latent heat of fusion is released at a rate that is substantially higher (positive side in Fig. 6) than the convective heat extraction rate at the droplet surface because of the large undercooling, leading to recalescence. During recalescence, the droplet temperature rises to the equilibrium liquidus temperature (Fig. 5) and the solid fraction increases rapidly (Fig. 7). For example, a dominant amount of solid forms during recalescence in the 10 pm and 30 pm droplets (f, = 0.99 and 0.90 respectively), while the solid forms during recalescence in the 10 pm and 30 pm droplets (f, = 0.99 and 0.90 respectively), while the soild fraction in the 60 pm, 100 pm and 200 ,um droplets increases to 0.68, 0.34 and 0.06 respectively during recalescence. Apparently, the solid formed during recalescence is critically dependent on undercool-

4oJI

s

r = 0 (mm)

-- = I[,, ,,,,,,,,;,,,,,,,,, d

-------- d =

-40

-

--------

-

-

..

-----

d = 10 (pm) d = 30 (km) d = 60 (pm) d = 100 (pm) d = 200

100

Fig. 6. Cooling spray cone.

200 300 Axial Distance

histories

of droplets

(pmj

400 (mm)

500

600

along the axis line of the

(pm)

r = 0 (mm) 1.0 ,,,

0

60

100 (km) ,, = z,,,, (km)

-30 I

0

-

t

II

r = 0 (mm)

I

,.

c

?,,,,,,,,,,,,,,,,,,,,,,,,,,,,,C 0

Fig. 5. Temperature spray cone.

100

200 300 Axial Distance

histories of droplets

400 (mm)

500

600

along the axis line of the

0 Fig. 7. Solidification the spray cone.

100

200 300 400 Axial Distance (mm)

histories

of droplets

500

600

along the axis line of

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/

Materials Science and Engineering A191 (1995) 171-184

ing. The supercooling may be expected if the droplet diameter is smaller than 10 pm. At the end of recalescence, the cooling rate of each droplet changes from positive (heating) to negative (cooling) (Fig. 6) and then exhibits a small, progressive increase during post-recalescence solidification occurring at a cooling rate of approximately 103-lo5 K s-r in magnitude for the droplets larger than 30 pm, and 10h K s-l in magnitude for 10 pm droplets (Fig. 6). During post-recalescence solidification, the temperature of each droplet decreases slowly from the liquidus to the solidus temperature (Fig. 5) and the solid fraction increases progressively (Fig. 7). At the end of postrecalescence solidification, there is a sharp increase in cooling rate due to the termination of the release of the latent heat of fusion (Fig. 6). The cooling rate then turns to decrease with decreasing temperature and velocity difference between the atomization gas and droplets. Solidification is completed at a flight distance of approximately 2 mm, 18 mm, 70 mm, 138 mm and 305 mm, corresponding to a total solidification time of approximately 0.04 ms, 0.25 ms, 0.72 ms, 1.30 ms and 2.71 ms for lOpm, 30 pm, 60 pm, 100 pm and 200 pm droplets respectively (Fig. 7). At an axial distance of 100 mm, the droplet temperature is approximately 290, 1797 and 3034 K for 10, 30 and 60 pm droplets, and the temperature of 100 pm and 200 pm droplet remains close to the solidus and liquidus temperatures respectively (Fig. 5). At an axial distance of 600 mm, the droplet temperature is calculated to be 220 K, 324 K, 1286 K, 2187 K and 2940 K for 10 pm, 30 pm, 60 pm, 100 ym and 200 pm droplets respectively. The effect of the droplet size on the temperature, cooling and solidification histories is evident from Figs. 5, 6 and 7. Overall, small droplets exhibit high cooling rate, large undercooling, short solidification time and low temperature. These phenomena result from the high convective heat transfer rate between small droplets and the atomization gas. As the flight distance increases, the cooling rate of small droplets decreases rapidly because of the decreasing relative velocity and temperature.

0

10 Radial

5

(a)

_

. ..___ _c‘“.“,.

3040-‘*-

-

*.., \

-d

d = 10 @I) = 30 (cun)

---:=;t,@&, _____

’

d = 200 (w)

\ \

’ \ \

760 0

/,/,,,,,,,,,,,,,,,,,,,,,,,,,,,~

0

5

b)

3800

15 Distance

“__._r._‘-‘““.,

-

25

30

2280-

‘. \

l\

d -

-

--...

. . . .

h;

20 (mm)

2 = 90 (mm)

3040:-____

E” g 1520-

10 Radial

,< .---Pr*

-d

10 Cm)

= NJ (rnun)

. ---d=60Wmun) _____ d - ~0 Oun) d = 200tm) *. .“‘. L .\

\

\

\

\ \ \

760:

\

0 0 (c)

\

\

J

;

30

\

1520 -

2 B $

25

(mm)

-

.

-\

2280 -,

8

20

15 Distance

z = 60 (mm)

3800

&

179

,,,1,1,,,,,,,,,,,,,,,,,,,1,,, 5 10 15 Radial

Fig. 8. Temperature distributions tion of the spray cone.

Distance

20

25

30

(mm)

of droplets

in the radial direc-

3.3, Two-dimensional distributions of temperature, cooling rate and solidfraction of droplets in spray cone Figs. 8, 9 and 10 show the distributions of the temperature, cooling rate and solid fraction of droplets of different diameters in the radial direction of the spray cone. As demonstrated in Figs. 8 and 9, at most axial distances (where droplets are in a single-phase state: fully liquid or fully solid), the droplet temperature and cooling rate attain their maximum values at the spray axis and decrease to minimum values at the periphery

of the spray cone. There are some exceptions, for example, at z = 30 mm (Fig. 8(a)). At this axial distance, the 60 and 100 pm droplets experience solidification at some radial distance between 2 and 8 mm, and solidification is completed at the periphery of the spray cone, while at the spray axis solidification has not begun yet (Fig. 10(a)). Therefore, the temperature maxima of the 60 and 100 pm droplets are not at the

H. Liu et al.

180

/

Materials Science and Engineering Al 91 (1995) 171-184 z = 30 (mm)

z 5 30 (mm)

5

1.0

k

0

c 2

k

-5

-

d -

--

___--

ia)

-40I

10 ()uD)

d = 30 t&W d = 60 (bud d = 100 Tim) d = ZOO (&m)

---

t

0 Radial

Dlstnnce

(mm)

10

5

ia)

p

2

-

Z III

d =

10 (w) d = 30 (W

--

0.6-

I’ I’

0.4-

= 60 (wua) _____ d = 100 (PI@

0

( ,,,,I,

I,,,,

5

10 Radial

15

20

Distance

25

0.0

30

(mm)

d = 10 (F) 4-d=30(& -a-d-60@) __*_ d I 100 (m) d = 200 (wn)

,I’ -

1 I II,

1,1,,,,1,,

a’

,‘, ,

I I, I I I I,

10

15

Radial

(bi

,’

,

5

I

20

Distance

I, I I I I 25

30

(mm)

z = 90 (mm)

1.0

k

_ -

..

, ,=I,, 5

i,,

0

2 = 90 (mm)

: --t

d = 200 (@SIB)

,, /,

i

:$

0.2-

---_d

:

,’

E

5

30

t

8 P

tb)

25

(mm)

t 0.8-

-40

20

Distance

z = 60 (mm)

z = 60 (mm)

5,

15

Radial

,

I’

,’

0’

1

,Q’

I

‘J

2 z

s

Lil

-20

3

F -25 -= g -30 u

-

-

---_d

-d

d = 10 @ID) = 30 (W

a m

--e d = lo(~) 4-d-30(p) -+-d=6O()un) --+d - 100 (iAm) d - 200 (IUI))

0.2-

= 60 (w) d = 100 (run) d = 200 (P)

___--

-35

i

0.4-

.. .

0.0 ii I I 2 1 =s 0 ( I / 1 ( 0 I 1 1 ’ I ( 10 15 20 0 5

(ci

Radial

Fig. 9. Cooling rate distributions tion of the spray cone.

Distance

(mm)

of droplets

in the radial direc-

spray axis, but instead at the radial locations where solidification takes place. From the same reason, the temperature of the 60 pm droplet is higher at the spray axis than that of the 100 pm droplet (Fig. 8(b)). This tendency is also reflected in the distributions of cooling rates. It can be seen that the cooling rate of a droplet is relatively low at the radial location where post-recalescence solidification is occurring (compare Fig. 9 with Fig. IO), and even positive (heating) at the radial loca-

fci

Radial

Distance

Fig. 10. Solid fraction distributions direction of the spray cone.

’I * 25

_ _ -

30

(mm)

of droplets

in the radial

tion where recalescence is taking place (Fig. 9(b)). In contrast to the radial distributions of the temperature and cooling rate, the radial distributions of the solid fraction exhibit a distinct tendency, i.e. a maximum at the periphery and a minimum at the spray axis (Fig. 10). This implies that at a given axial distance, the droplets in the periphery will solidify earlier than those at the spray axis. These phenomena may be rationalized as follows.

H. Liu et al.

1 MaterialsScience and Engineering A191 (1995) 171-184

As illustrated in Fig. 4, the velocity difference between droplets and the atomization gas is much larger at the spray axis than at the periphery of the spray cone, leading to a much higher convective heat transfer rate and hence the much larger cooling rate. In addition, the relatively low droplet velocities at the periphery of the spray cone correspond to relatively long flight time to a given axial distance, as discussed in the previous section. The long flight time of droplets at the periphery of the spray cone causes the lower droplet temperature and the higher solid fraction at the periphery for a given axial distance. Experimental support for these numerical results may be found in an earlier microstructural investigation of spray deposited materials at different radial positions [32] in which thick prior droplet boundaries (more porosity and less droplet deformation) were observed in the fringe region of the deposits. Moreover, our calculations demonstrate that, for a given droplet diameter, the achievable undercooling decreases and the solidification time increases with increasing radial distance. These results may be explained similarly on the basis of the droplet velocity and cooling rate distributions in the radial direction. The results summarized in Figs. 8, 9 and 10 also show that with increasing axial distance, for example from z = 30, 60-90 mm, the radial distributions of the droplet temperature, cooling rate and solid fraction become wider and the difference between the spray axia and the periphery of the spray cone decreases. In addition, the effect of the droplet size on the radial distributions is also apparent from Figs. 8, 9 and 10. A small droplet has a low temperature, a high cooling rate and a large solid fraction at any radial distance, the exception being at the fringe of the spray cone (Fig. 9), where the cooling rate of a small droplet is low owing to the small relative velocity between the droplet and the atomization gas (see Fig. 4). The larger the droplet, the smaller the difference in the droplet temperature and cooling rate between the spray axis and the fringe of the spray cone, and hence the more even the distributions are. The only exception is for the 10 pm droplets, as discussed above in the velocity distribution.

3.4. Microstructural characteristics of particles and sprayed materials

In gas atomization, the solidification process is typified by the formation of a fine dendritic microstructure in atomized powders. The secondary dendrite arm spacing SDAS depends critically on the cooling rate during post-recalescence solidification [35]. Refinement of dendritic structure by increasing the rate of heat extraction during solidification has become the

181

r = 0 (mm)

c 2

70-

“0 3 L 3

SO-

1

30-

c” ” 8

V

-x-

Cooling Rate

-

SDAS s % E -1

IO-

-10 ,IIII,I/I/,I,II,III,,11,I,,III(II,, 0 0

30

60 90 120 Droplet Diameter

150 (Km)

180

210

Fig. 11. SDAS and cooling rate during post-recalescence solidification as a function of droplet diameter.

most important advantage of atomization and deposition processes. As mentioned above, post-recalescence solidification takes place at a cooling rate of 103-lo6 K s- l in magnitude, depending on droplet size and processing conditions. In order to determine quantitatively the effect of the droplet size on SDAS, the cooling rates of droplets during post-recalescence solidification are calculated and summarized in Fig. 11 as a function of droplet diameter. The corresponding values of SDAS are also plotted in Fig. 11. SDAS is estimated using the following relationship [ 3 51: SDAS = 39.8 T - IJJ

(18)

As expected, the cooling rate decreases drastically with increasing droplet diameter (from 8.8 X 10h, 2.8 X 105, 5.3 x 104, 1.7 x lo4 to 5.5 X lo3 K s-* when increasing the diameter from 10, 30, 60, 100 to 200 ym). Correspondingly, SDAS increases from 0.33, 0.92, 1.53, 2.13 to 3.00 pm. Since more small droplets impact on the fringe region of the sprayed deposit (as discussed in Section 3.1) and a small droplet exhibits a high cooling rate and a fine SDAS (as shown in Fig. 1 l), it may be predicted that the microstructure is fine in the fringe region of the spray deposited material. This tendency is in good, qualitative agreement with the findings of the microstructural characterization studies of spray deposited materials [32,33] in which a distribution of the microstructural characteristics in the radial direction of the spray cone was found and finer grain size was observed in the edges of the deposits.

4. Conclusions A simple 2D lagrangian flow model and a lumped parameter formulation based on the modified Newton’s law of cooling are developed to investigate

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Materials Science and Engineering Al 91 (I 995) 171-184

the interactions between N, atomization gas and Ta-2.5 W alloy droplets during atomization and deposition processing. The rapid solidification phenomena accompanying droplet-gas interactions are addressed by combining the 2D model with a phase change kinetic model. A fourth-order Runge-Kutta algorithm is applied for the solution of the 2D non-linear differential equations of motion of droplets. The 2D distributions of droplet velocity, temperature and rapid solidification histories are calculated and discussed in detail, while the microstructural characteristics of solidified particles in the spray cone and the as-deposited materials are discussed briefly. The effect of the droplet size on the 2D flow, thermal and solidification histories is also addressed. From the numerical results, some primary concluding remarks may be summarized as follows. (1) The axial gas velocity decays exponentially along the spray axis, while the gas flow diameter increases in the axial direction. The radial profiles of axial gas velocities exhibit a shape akin to a gaussian probability distribution. The gas flow is predominantly in the axial direction as a result of the axisymmetric assumption. These results are in qualitative agreement with the features of the LDA measurements of the gas velocities reported in Refs. [7,13]. (2) The axial and radial 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 rapidly the profiles of gas velocities. The predicted droplet velocity profiles exhibit a shape that is similar to that of the deposit contours obtained in experimental studies [5,32]. (3) 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 cone, except for the radial locations where solidification occurs. Accordingly, 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 of the droplets is larger in the periphery region of the spray cone than elsewhere in the radial direction. Hence, the droplets in the periphery region of the spray cone solidify within a shorter flight distance relative to those at the spray axis because of the longer flight time in the periphery. These results are consistent with those obtained from microstructural studies 1321 in which thick prior droplet

boundaries (more porosity and less droplet deformation) were observed in the fringe region of spray deposited materials. (4) With decreasing droplet size, the flight distance required to attain the maximum droplet velocities decreases and the flight time required for any given flight distance also decreases. Compared with a large droplet, a small droplet exhibits a large axial velocity, a low temperature, a high cooling rate and a large 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. At the same time, a small droplet has a wider radial distribution at any axial distance. Hence, it may be expected that coarse droplets constitute the core whereas the periphery of the spray cone is populated by fine droplets. This suggestion is verified by the experimental analyses of particle size distributions which demonstrated that the oversprayed particles collected during atomization and deposition exhibit a smaller median size than the average size of powders generated and acquired during atomization [33], and the mass-median droplet diameter decreases along the radial direction from the spray axis [5]. Finally, it should be noted that some of the tendencies stated above may become invalid for the droplets smaller than 10 pm. Such small droplets may require a longer flight time to a given flight distance far from the atomizer because of the high deceleration, and their cooling rates may decrease as a result of the reduced relative velocity and temperature. (5) The calculated radial distributions of droplet trajectory, cooling rate and SDAS are in qualitative agreement with the findings of the microstructural characterization studies of spray deposited materials [32,33] in which a distribution of the microstructural characteristics in the radial direction was found and finer grain size was observed in the edges of the deposits.

Acknowledgments The authors wish to acknowledge the Army Research Office (Grant DAALG3-92-G-018 1) for financial support and encouragement. This research was supported in part by the University of California, Irvine, through on allocation of computer resources. The authors would also like to thank Mr. Weidong Cai for collecting data of physical properties of the related materials from literature.

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1

Materials Science and Engineering A191 (I 9%) 171-184

References E.J. Lavemia, J.D. Ayers and T.S. Srivatsan, lnt. Mater. Rev., 37(1)(1992) l-44. [21 X. Liang and E.J. Lavernia, Mater. Sci. Eng. A, 161 (2) (1993) 221-235. [31 H. Liu, Ph.D. Thesis, University of Bremen, Bremen, February 1990. D. Apelian and A. Lawley, [41 P. Mathur, S. Annavarapu, Mater.Sci.Eng.A,142(2)(1991)261-276. L-51P. Mathur, D. Apelian and A. Lawley, Acta Metall., 37 (1989) 429-443. [a C.G. Levi and R. Mehrabian, Metall. Trans. A, 23 (1982) 221-234. [71 B.P. Bewlay and B. Cantor, Mater. Sci. Eng. A, 118 (1989) 207-222. [81 Y. Wu and E.J. Lavernia, Metall. Trans. A, 23 (1992) 2923-2937. [91 T.W. Clyne, Metall. Trans. B, 15 (1984) 369-381. [lOI E.J. Lavernia, T.S. Srivatsan and R.H. Rangel, Atm. Sprays, 2 (1992) 253-274. E.J. Lavernia, G.M. Trapaga, J. [Ill E. Gutierrez-Miravete, Szekely and N.J. Grant, Metall. Trans. A, 20 (1989) 71-85. I121 E.J. Lavernia, E.M. Gutierrez, J. Szekely and N.J. Grant, Int. J. RapidSolid., 4(1988) 89-124. [I31 B.P. Bewlay and B. Cantor, Metall. Trans. B, 21 (1990) 899-912. MA, [I41 G.M. Trapaga, Sc.D. Thesis, MIT, Cambridge, September 1990. [I51 P.S. Grant, B. Cantor and L. Katgerman, Acta Metall., 41 (11)(1993)3097-3118. P. Kumar, C.A. Michaluk and H.D. [I61 SM. Cardonne, Schwaltz, Adv. Mater. Processes, 142 (3) (1992) 16-20. [I71 G. Rudinger, Fundamentals of Gas-Particle Flow, Elsevier, Amsterdam, 1980. 1181 S.L. Soo, Multiphase Fluid Dynamics, Gower, Aldershot, 1990. [I91 H. Schlichting, Boundary Layer Theory, McGraw-Hill, New York, 1955. [201 E.A. Brandes, Smithells Metals Reference Book, Butterworths, London, 6th edn., 1983. [211 C. Kittel, Introduction to Solid State Physics, Wiley, New York, 6th edn., 1991. [=I M.V. George, A Handbook of Chemistry and Physics, Van Nostrand Reinhold, New York, 2nd edn., 1970. P31 D. Liu, Fundamentals of Thermal Energy Engineering, Metallurgy Industry, Beijing, 1980. [241 H.A. Johnson and M.W. Rubesin, Trans. ASME, 71 (5) (1949) 447-456. [251 W.E. Ranz and W.R. Marshall, Chem. Eng. Prog., 48 (3) (1952) 141-146,173-180. J.P. Hirth, Metall. Trans. A, 9 (1978) 40 l-404. ;;67; D. Turnbull and J.C. Fisher, .I. Chem. Phys., 17 (1) (1949) 71-73. Acta Metall., 32 (1) [281 K.S. Dubey and P. Ramachandrarao, (1984)91-96. P91 G.-X. Wang and E.F. Matthys, Int. J. Heat Mass Transfer, 35 (1992) 141-153. [301 X. Zhang and A. Atrens, Int. J. Rapid Solidif, 7 (1992) 83-107. I311 T.W. Clyne and W. Kurz, Metall. Trans. A, 12 (1981) 965-971. [321 X. Liang, J.C. Earthman and E.J. Lavernia, Acta Metall., 40 (11) (1992) 3003-3016.

[ll

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[33] S.N. Ojha, A.K. Tripathi and S.N. Singh, Powder Metall. Znt., 25 (2) (1993) 65-69. [34] M.C. Flemings and Y. Shiohara, Mater. Sci. Eng., 65 (1984) 157-170. [35] P.A. Joly and R. Mehrabian, J. Muter. Sci., 9 (1974) 1446-1455.

Appendix A: Nomenclature a C ‘pd,

‘pg

Cpb

Cps

C C drag d d,

D ;m drag &T

h AH,

I J k Kg

atomic diameter (m) constant in equation for ,LL~ in Table 3 thermal capacity of droplet and atomization gas respectively (J kg- ’K- ‘) thermal capacity of liquid and solid alloy respectively (J kg- ’K- ‘) constant related to local gas flow width Fd,,g/[+&U,2~( d/2)2], drag coefficient droplet diameter (m) effective molecular diameter (m) initial half-width of gas flow (m) self-difisivity in melt (m2 s - ’) solid fraction of droplet drag force gravity acceleration (m s-?) convective heat transfer coefficient (W rn-: K-l) latent heat of fusion (J kg- ’) nucleation rate (mm3 s- ‘) mechanical equivalent of heat (m kg J- ’) equilibrium partition coefficient thermal conductivity of atomization gas (W ,-I K-l 1

Kg,, Km ii Ma :: Pr Y ; Re L4S t T’ iTd,

TdO

Tg.

thermal conductivity of atomization gas at 0 “C -1 K-1 ) (Wm solid-liquid interface mobility (m s- ’K- ‘) power index in Eq. ( 14) molar mass of melt (kg mol- ’) fig/v,, Mach number power index in equation for Kg in Table 3 Avogadro’s constant (atoms mol- ’) ,LL~ c,,,/ K,, Prandtl number radial coordinate (m) recovery factor gas constant (J K- ’molt ’) dU,p,lpu,, Reynolds number solid fraction of droplet during recalescence secondary dendrite arm spacing (pm) time (s) film temperature (K) cooling rate (K s- ‘) temperature of droplet and atomization gas respectively (K) initial temperature of droplet (K)

H. Liu et al.

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Materials Science and Engineering AI 91 (1995) I71 -184

liquidus and solidus temperature of alloy respectively (K) solvent melting temperature (K) nucleation temperature (K) T, - T,, undercooling free stream velocity defined originally in Ref. [24] and taken in the present computations as relative velocity between droplet and atomizationgas (m s-r) dr/dt, radial velocity of droplet and atomization gas respectively (m s - ’) relative speed [b,-vd)2+bgu,)2]1’2, between gas and droplet (m s _ ’) dz/dt, axial velocity of droplet and atomization gas respectively (m s- ‘) and atomization vdO, vgO initial axial velocity of droplet gas respectively (m s l) average gas velocity along axis line of spray gg cone (m s-l)

sound speed of gas at 220 K (m s- ‘) molar volume of melt (m3 mol- ’) axial coordinate, constants related to the atomizer geometry and atomization parameters respectively (m) specific heat ratio of gas emissivity of surface Boltzmann’s constant (J K- ’) kinematic viscosity of atomization gas (kg m- ’ (yRT)“*,

s-1)

kinematic viscosity of atomization gas at 0 “C (kg m-l s-l) density of droplet and atomization gas respectively (kg m 3, density of atomization gas at 0 “C (kg m- 3, Stephen-Boltzmann constant (W mm2 K-‘) solid-liquid inter-facial energy (J m-*)