(Received May 28, 2010; in revised form March 16, 2011)
The work is devoted to investigation of thermophysical processes occurring at granulation of lead-containing aluminum alloys. The researchers proposed the mathematical model of heat transfer that served a basis for calculation of cooling of the alloy drops with different dimensions. With the use of laboratory ex-periment, the boundary conditions of simulation have been specified, and the quality of the obtained granules evaluated.
Antifriction lead-containing alloys allow reducing friction coefficient and hence increasing endurance and operational reliability of the manufactured mechanisms ele-ments. These alloys are comparatively light, solid, and pliant that serves to apply them as structural materials.
Generalization of the published data on the methods of molding and ligature pro-duction from Al-Pb alloys has shown that at atmospheric pressure and low cooling ve-locity (not exceeding 60 °С/s) at hardening, lead segregation on density is observed. The most suitable here is granulation process that enables obtaining dispersed heteroge-neous structure with even distribution of inclusions in high-alloys of aluminum with components (Sn, Pb, Cd, Bi) that are easily melted, practically insoluble in aluminum and drastically changing in their density. Development of rational technology of alloy granulation is bound with determination of the influence of a number of factors among which there are first of all velocity of granule cooling, their composition, and di-mensions.
The work [1] includes recommendation on the technology of ligature manufactur-ing from aluminum alloys with lead content of 15 % (mass) in granules with the diame-ter of 4÷6 mm and inclusions’ dimensions in ligature not exceeding 30 μm. It is shown therein that the lesser granules with high specific surface actively form hydrated oxide films polluting alloys at alloying. Crystallization of the alloy drops with larger size re-sults in inclusions of lead with dimensions exceeding 30 μm that may lead at the use of granules as ligatures to production of low-quality alloy.
To obtain high-concentrated alloys with preset composition and structure they shall be overheated by 100÷150 ºС over the liquidus temperature that provides complete solu-tion and further high-velocity crystallization of lead in aluminum alloy. Thus, an important
task is determination of cooling velocity at which homogeneous solution aliquation does not take place.
By now the problem of determination of the detailed temperature field of the crystallizing drop of high-concentrated aluminum alloys with limited solubility of the alloying component during its free motion in the cooling medium has not been approached to in practice. The only known are the analytical dependences determining crystallization time of such material alloys and providing approximate estimate of the particle temperature averaged over volume and hence of its phase state [2].
To carry out the detailed analysis of the Al-Pb alloy droplet we used the calculation cylindrical area in axisymmetrical coordinate system (Fig. 1).
The calculation grid is presented in Fig. 2. To improve the solution convergence special thickening was arranged in the vicinity of the spherical droplet surface on the grid. It was assumed that high gradients of temperature, velocity, and pressure should be observed in this particular area.
The mathematical model is a system of differential equations of continuity, momentum, and energy conservation. The energy equation includes the terms re-sponsible for internal heat generation at phase transfer in the droplet melt during crystallization and boiling of water. At consideration of turbulence, instantaneous values of hydrodynamic characteristics are written as an amount of statistical aver-age values and pulsation components. Thus, velocities in the equations of motion are averaged after Reynolds [3]:
( ) ( ) 0,div vt
ρ ρ∂ + =∂
(1)
( ) ( ) ( ' '),v div vv p v vt
ρ ρ τ ρ∂ + = −∇ + ∇ −∂
(2)
( ) ( ) ( ) ,hh div vh T St
ρ ρ λ∂ + = ∇ ∇ +∂
(3)
2.
3ji k
ijj i k
vv v
x x xτ μ δ
⎛ ⎞∂∂ ∂= + −⎜ ⎟⎜ ⎟∂ ∂ ∂⎝ ⎠
(4)
Fig. 1. Calculation area of alloy droplet motion in aqueous medium.
1 ⎯ cooling medium, 2 ⎯ droplet.
Fig. 2. The calculation grid of the area occupied by water-vapor mixture.
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Here ρ is the density, kg/m3; v is the velocity, m/s; p is the pressure, Pa; μ is the co-
efficient of dynamic viscosity, Pa·s; v′ is the pulsation component of velocity, m/s;
h is the enthalpy, J/kg; λ is the coefficient of heat conductivity, W/(m⋅K); hS is the
source term responsible for energy inflow (outflow) during phase transition and radiation
from the surface of the droplet melt, W/m3; τ is the tensor of viscous stress, Pa; ijδ is
the Kronecher symbol, 0ijδ = ( ),i j≠ 1.iiδ =
For turbulence simulation, the Boussinesq hypothesis relating Reynolds stresses
( )v vρ ′ ′ with averaged velocity gradient was used. This hypothesis used in k ε− turbu-
lence models is applied for closing equations (1)−(3).
The transfer equations for kinetic energy of turbulence k (m2/s2) and its dissipation
ε (m2/s3) take the following form:
( ) ( ) ,t
k
k div vk k Pt
μρ ρ μ ρεσ
⎛ ⎞⎛ ⎞∂ + = ∇ + ∇ + −⎜ ⎟⎜ ⎟⎜ ⎟∂ ⎝ ⎠⎝ ⎠ (5)
2
1 2( ) ( ) ,tdiv v C P Ct k kε
μ ε ερε ρ ε μ ε ρσ
⎛ ⎞⎛ ⎞∂ + = ∇ + ∇ + −⎜ ⎟⎜ ⎟⎜ ⎟∂ ⎝ ⎠⎝ ⎠ (6)
where P is the velocity of turbulence generation, m2/s3; tμ is the turbulent viscosity, Pa·s.
Empirical constants according to the standard k ε− model have the following values: 1,kσ = 1.3,εσ = C1 = 1.44, C2 = 1.92 [4].
On the surface of the melt droplet, the boundary condition of the third type was used with consideration of the radiant component of heat transfer determined according to Stefan — Boltzman law. The coefficient of convective heat emission was determined at the solution of system (1)−(6). On the lower surface of the cylinder, the velocity of water motion assumed equal to the one of the melt droplet was preset. Due to the lack of reliable data on the dynamics of spherical particles motion in the presence of vapor layer and intense heat transfer, this dependence was obtained experimentally.
At the outlet the following boundary condition was set:
0.zv z∂ ∂ = (7)
Over the cylinder perimeter, the condition of non-permeation and slipping was set:
0, 0,r zv τ= = (8)
where rv is the radial velocity component, m/s, zτ is the tangential component of stress
tensor, Pa. Kinetic energy and velocity of its dissipation at the inlet had zero values. In the equations (1)−(6), temperature dependence of thermophysical parameters of
the media were considered. So vapor formation during the droplet cooling was consid-ered in Stefan problem approximation, where the boundary vapor/water was determined at the solution of motion and heat transfer equations: at the temperature over 100 ºС, the thermophysical parameters of vapor were taken and lower ⎯ the ones of water. The problem was solved in two-dimensional axisymmetrical statement with the use of numerical method of control volume. In publications, as a rule, there are dependences of the drag coefficient to sphere motion on Reynolds number; the formation of vapor and intense heat transfer are not taken into account [5].
A.P. Skuratov and A.A. Pianykh
124
Consideration of internal sources of heat emission at phase transition in water at vaporization and in the volume of a droplet at crystallization included in the value ,hS is
performed by introduction of effective coefficient of heat capacity. Besides, it is taken into account that the particle has homogeneous chemical structure over its entire volume
sol
liq
( ) .T
h pT
S c T dT L= +∫� (9)
In the model, the melting heat L was not considered separately but was included in the dependence of heat capacity on temperature (effective heat capacity) in the form
eff ( ) ( ) ( ),pc T c T c T= + � (10)
sol
liq
( ) ,T
pT
L c T dT= ∫ � (11)
where ( )pc T� is the additional member responsible for internal heat emission.
The authors studied the process of granulation of melt droplets of Al ⎯ 15 % Pb (on mass) with dimensions 4.5; 6.0 and 7.5 mm. It was considered that the system Al-Pb on the phase diagram is characterized by a wide area of immiscibility in liquid state [6], where high velocity of alloy cooling shall be provided. For the melt of the given compo-sition, this area shall be in the range between the temperature of two liquids immisci-bility equal approximately to 990 ºС, and the temperature of monotectical transformation that is lower than the one of pure Al melting by 1.5 ºС (658.5 ºС).
Initial values of water and droplet temperature were 15 and 1100 ºС, respectively. Temperature dependence of thermophysical parameters of alloy and cooling medium was taken according to [7, 8]. During simulation of alloy droplet cooling initially its temperature field was determined in air medium up to the moment of its contact with
Fig. 3. Temperature field (°С) of air medium and alloy droplet with 6.0-mm diameter at the moment of time 0.1 s.
а ⎯ air mediа, b ⎯ granule section.
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water surface. At calculation of droplet motion, velocity influence of the force of droplet friction with air was not considered — the velocity was determined from kinematic equ-ations of material point motion being affected only by gravity. As further calculations showed the temperature of the melt drop for the time of its free motion in air decreases not more than by 10 ºС.
As an example, in Figs. 3 and 4 there are calculated temperature fields of the alloy droplet with 6-mm diameter and ambient medium during its motion.
As was noted, experimental studies were performed with the view of determination of the velocity of alloy droplets motion in water and evaluation of correctness of granulation mode selected as a result of numerical simu-lation.
The laboratory unit for granulation of the Al-Pb alloy consisted of induction furnace and water bath (see Fig. 5). In the basis of the fur-nace crucible, there was an orifice where the al-loy droplet was formed. The distance from the orifice to water surface in the bath was 6 cm. Along with the drop motion in the air and further the processes of cooling, crystallization, and granule formation occurred in the water bath.
Fig. 4. Temperature field (°С) of vapor-water medium and melt droplet with 6.0-mm diameter at the moment of time 0.5 s.
а ⎯ vapor-water mixture, b ⎯ granule section.
Fig. 5. Laboratory unit for granulation.
A.P. Skuratov and A.A. Pianykh
126
In the experiments, the temperature of the media participating in heat transfer was
(in ºС): water ⎯ 15, air ⎯ 20, alloy in induction furnace ⎯ 1100.
Insignificant influence of droplets diameter in the studied range (4.5÷7.5 mm) on their motion was experimentally determined. Therefore, in calculations we used the av-
eraged dependence of instantaneous velocity of the melt droplets motion on time ( )v t
(Fig. 6). Temperature changes in the volume of droplets of different size during their mo-
tion in vapor-water medium obtained as a result of numerical experiment are shown in
Fig. 7. Drastic temperatures drop in the range 640÷700 ºС is explained by the change of thermophysical parameters of aluminum at crystallization in this area [6]. It was ascer-tained that for the droplets with 4.5, 6.0, and 7.5-mm diameter, the cooling velocities average for all period of motion were, respectively, 680, 393, and 325 ºС/s. A significant difference of maximal velocity of alloy droplet cooling from its average values was
Fig. 6. Dynamics of changing averaged velocity of the alloy droplets with 4.5÷7.5-mm diameter in water.
Fig. 7. Dependence of the velocity of alloy droplets coolig in water on their dimensions and motion
time. 1, 3, 5 ⎯ minimal temperature of droplets with 4.5, 6.0 and 7.5-mm diameter, respectively; 2, 4, 6 ⎯ maximal temperature of droplets with 4.5, 6.0 and 7.5-mm diameter, respectively.
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found out. So in the initial period of motion (0÷0.2 s) for the drops with the studied sizes, the cooling velocities are 1200, 800, and 600 ºС/s, respectively. On the basis of numerical investigation we have, besides, determined the height of water bath necessary for full hardening of granules of different sizes.
Results of microscopic analysis performed jointly with V.G. Babkin and A.I. Che-repanov have shown that at cooling of the alloy droplets of 4.5-mm diameter with the average velocity 680 ºС/s (in the temperature range 1100÷650 ºС), the size of lead inclusions in the granules does not exceed 30 μm. The particles of the lead phase are prominent in the background of aluminum matrix and their distribution over the section field is rather uniform (Figs. 8 and 9). For such conditions of granulation, the minimal height of water capacity is 0.7 m.
References
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2. V.I. Dobatkin and V.I. Elagin, Granulated Aluminum Alloys, Metallurgy, Moscow, 1981. 3. S.V. Patankar, Numerical Heat Transfer and Fluid Flow, Hemisphere Publ. Corp., New York, 1980. 4. B.E. Launder and D.B. Spalding, Lectures in Mathematical Models of Turbulence, Academic Press,
London, England, 1972, P. 157−162. 5. S.S. Kutateladze, Heat Transfer and Hydrodynamic Resistance, Reference Book, Energoatomizdat,
Moscow, 1990. 6. Diagrams of Binary Metal Systems, Reference Book, in 3 Vols. by N.P. Lyakishev (Ed.), Mashinostroyenie,
Moscow, 1996. 7. V.E. Zinoviev, Thermophysical Properties of Metals at High Temperatures, Reference Book, Metallurgiya,
