3948
Ind. Eng. Chem. Res. 2004, 43, 3948-3956
Material and Energy Flow in a Metal Evaporation System with Moving Boundaries Matthew A. McClelland* and Kenneth W. Westerberg† Lawrence Livermore National Laboratory, P.O. Box 808, L-282, Livermore, California 94550
Finite-element calculations are performed for the transient material and energy flow in a system to evaporate pure titanium. A 60-kW electron beam is used to heat the end of a 10.16-cm-diameter cylindrical rod, which is fed vertically through a water-cooled crucible. Vapor emanates from a liquid pool in which flow is driven strongly by buoyancy and capillary forces. At high evaporation rates, the vapor exerts strong shear and normal forces on the liquid-vapor interface. The MELT finite-element code is used to calculate transient, axisymmetric flow and temperature fields, along with liquid-solid and liquid-vapor interface locations. The influence of the vapor on the liquid top surface is treated using boundary conditions with parameters determined from Monte Carlo results. The upper and lower interfaces of the liquid pool are tracked using a mesh structured with rotating spines. The finite-element results show a characteristic response time of 20 ms to variations in electron beam power. Introduction Electron-beam technology plays an important role in the processing of refractory metals. High energy fluxes generate the temperatures needed for melting and highrate evaporation. In addition, processing is performed in an atmosphere that is relatively free of contaminants. These features are beneficial in processes such as physical vapor deposition (PVD), welding, refining, and casting.1,2 An improved understanding of mass, momentum, and energy transport would yield improvements in process throughput, control, and economics. One electron-beam process of interest involves the fabrication of metal matrix composites (MMCs) for use in high-performance aircraft components (see Figure 1). In this vacuum PVD process, an electron beam is swept over the top end of a Ti-6V-4Al rod using a magnetic field. Metal evaporates from the top surface of the resulting liquid pool and deposits on moving ceramic fibers. The level of the pool is kept constant by advancing the rod in the vertical direction. In subsequent processing steps, the fibers are arranged and consolidated for later use in part fabrication. Goals for this PVD process include rapid startup, high-rate deposition, composition control, and high vapor utilization. We are interested in the transient response of the melt from the sweeping of the electron beam. It is often desirable to use a single electron gun to provide the power for multiple evaporation sources. Transients resulting from the movement of the electron beam from one melt surface position to another and between evaporation melts can create variations in the deposition profile. A detailed model of material and energy transport would help to select the optimum electron beam power, sweep rate, and sweep pattern for vapor deposition. The behavior of this evaporation system is complex as the locations of the liquid-vapor and solid-liquid * To whom correspondence should be addressed. Tel.: (925) 422-5420. Fax: (925) 424-3281. E-mail:
[email protected]. † Dedicated to the memory of Kenneth W. Westerberg (1965-2001).
Figure 1. Electron-beam deposition of vapor for the fabrication of metal matrix composites.
Figure 2. Axisymmetric finite-element model for evaporation of vapor from the top of an advancing feed rod.
interfaces are influenced by material and energy flow (see Figure 2). The electron beam passes through the vapor stream to the top surface of the liquid pool. A portion of the incident electron beam is lost with the formation of “skip” electrons. A small fraction of the absorbed energy provides the heat necessary to evaporate the metal, and the remaining energy is transported
10.1021/ie030731i CCC: $27.50 © 2004 American Chemical Society Published on Web 06/12/2004
Ind. Eng. Chem. Res., Vol. 43, No. 14, 2004 3949
by convection and conduction to the cold crucible wall. The top surface of the pool deforms at the electron-beam impact site from the recoil of the departing vapor. The lower boundary of the pool is a melting front. Flow in the pool is driven by density gradients (buoyancy effect), surface tension gradients (Marangoni effect), and surface shear forces generated by the expanding vapor plume. The flow intensity is high because of the low viscosity of the liquid metal and the relatively large driving forces. Numerical models have been developed for a variety of ideal and applied materials processing systems with similar features. The behavior of turbulence models was recently explored in a three-dimensional flow system with a deformable free surface3 and in Czochralski crystal growth systems.4,5 Carey et al.6 examined parallel computing approaches for Marangoni flows with thermal and species transport. Extensive attention has been given to liquid bridge flow systems,7-12 floatingzone systems,13 and other systems14 relevant to microgravity environments. Chen et al.15 developed a numerical model for a two-dimensional surface tension flow with two free surfaces, and Kim and Na16 studied a gas tungsten arc welding system with liquid-vapor and solid-liquid interfaces. Bojarevics et al.17 used a psuedospectral method to model axisymmetric electromagnetic casting system with a free surface and solidification front. Cairncross et al.18 and Baer et al.19 developed a finite-element method for free surface flows in three dimensions with wetting lines. Early numerical models of electron-beam evaporation systems have been reviewed by Westerberg et al.20 Guilbaud21 studied the steady-state fluid flow and free surface deformation for the evaporation of cerium from a two-dimensional trough. The numerical strategy involved the iteration between liquid flow and surface evaporation calculations. Westerberg et al.20 used the MELT finite-element code to analyze the steady-state evaporation of aluminum from a disk confined in a water-cooled crucible. Pivoting spines were employed in the finite-element method to track the moving liquidvapor and liquid-solid interfaces along with the trijunction. Karcher et al.22 investigated the electromagnetic control of convective heat transfer for the improvement of evaporation efficiency. McClelland et al.23 and Westerberg et al.24 made material and energy flow measurements for the evaporation of titanium alloys using the evaporation system of Figure 1. MELT finite-element predictions for the evaporation rate were higher than the measured values, but showed the same linear dependence with respect to electron-beam power. In addition, the calculated depths of the pools were in good agreement with the measured values. Improvements in surface evaporation models were made with direct simulation Monte Carlo (DSMC) methods. Fleche et al.25 calculated the coupled vapor and liquid flow fields for the steady, axisymmetric evaporation of cerium. They obtained similar results in a second set of calculations in which surface boundary conditions replaced the DSMC results. They also determined that the contribution of vapor shear stresses is quite important at high evaporation rates. Braun et al.26 performed DSMC calculations for a vapor flow field of the type shown in Figure 1. They also noted the important effects of backscattered vapor on evaporation rates and interfacial stress.
In recent work,27,28 we compared model and experimental results for the electron-beam evaporation system of Figure 2. An analysis was performed for the steadystate evaporation of pure titanium. The finite-element approach of Westerberg et al.20 was employed with improvements to the surface evaporation models. The DSMC results of Braun et al.26 were used to specify parameters in these models, including a new model for vapor shear. The finite-element results provided a good representation of measured evaporation rates, heat flows, and lower pool boundaries made using the methods and apparatus initially described by McClelland et al.23 and Westerberg et al.24 In this paper, we explore the transient response of a similar titanium evaporation system to help with the understanding of electron-beam sweeping. In particular, we present finite-element calculations for the abrupt shutoff of electron-beam power to the evaporation melt. Model Equations A continuum model is employed for the transient, twodimensional liquid and energy flow in the metal ingot. Boundary conditions are applied at the free surface of the pool to account for the influence of the vapor phase. The parameters in these boundary equations are calculated from direct simulation Monte Carlo (DSMC) results as described elsewhere.28 Field Equations. The flow of metal in the pool is governed by the time-dependent continuity and momentum equations for a nearly incompressible Newtonian liquid
∇‚u ) 0
(1)
F0(∂u/∂t + u‚∇u) ) -∇p + µ∇2u + F0g[1 - β(T - T0)] (2) in which u is the velocity, p is the pressure, and T is the temperature. The reference density F0, viscosity µ, and volumetric coefficient expansion β are taken to be uniform in the liquid pool. The Boussinesq approximation is employed in which the liquid density is varied linearly about the melting point value in the body force term. The density is taken to be constant and uniform in a given phase. The time-dependent energy equation includes the effects of convection and conduction
F0Cp(∂T/∂t + u‚∇T) ) k∇2T
(3)
The heat capacity at constant pressure, Cp, and thermal conductivity, k, are taken to be uniform within a given phase. In the solid phase, the upward velocity of the metal rod is assumed to be small, and the term with u is neglected. The Reynolds and Peclet numbers provide measures of the flow intensity and thermal convection, respectively.
Re )
F0umaxdpool µ
Pe )
umaxdpool R
The characteristic length in the above expression is the depth of the liquid pool at the symmetry axis, dpool. The maximum liquid speed, umax, is selected as the characteristic velocity. Both of these quantities are results of simulations.
3950 Ind. Eng. Chem. Res., Vol. 43, No. 14, 2004
Mass Inventory and Flow Boundary Conditions. For the calculation of steady-state solutions for use in initial conditions, it is assumed that the rod feed rate is adjusted so that the average height of the liquid level is maintained at a specified value h0
∫0Rhr dr
h0 ) 2
R2
(4)
In addition, the effects of overflow onto the top of the crucible are neglected. For the calculation of transient solutions, this constraint is relaxed. For the specification of flow conditions at the pool boundaries, it is assumed that the evaporation rate and associated rod advance rate are small and have a negligible kinematic effect on the flow field. Thus, there is no flow through any of the boundaries of the pool. At the free surface, the kinematic condition is applied
n‚dxlv/dt ) n‚u
at x ) xlv
(5)
Here, n is the outward pointing surface normal vector, and xlv is the location of the free interface. As described elsewhere,20,28 this is a distinguishing condition that determines the mesh motion at the free surface. In addition, the tangential velocity is zero at the solidliquid interfaces
t‚u ) 0
at x ) xsl
(6)
The shear stress vanishes at the symmetry axis
n‚τ‚t ) 0
at r ) 0
(7)
We apply the following force balance at the liquid-vapor interface
0 ) n‚(pδ + τ) + σ
(
)
dt dt1 + + ds ds1 dσ dT t - τvt - nπv ) 0 (8) dT ds
The pressure and viscous contributions for the liquid are included in the first term of eq 8. The second term is the normal stress resulting from surface tension and interfacial curvature. In this term, there are contributions in the plane and normal to the r-z calculation domain. In the r-z plane, t is the surface tangent vector, and s is the arc length along the interface in the r direction. The quantities t1 and s1 are used in a plane that includes the surface normal vector and a vector normal to the r-z plane. The third term includes the Marangoni effect in which a stress is generated by temperature gradients in the surface tension. The quantities σ and dσ/dT are taken to be constant at their melting-point values. The final two terms account for the shear and normal stresses generated by the departing vapor. They are discussed in further detail below. Because eq 8 includes surface tension, two end-point conditions are required. At the symmetry axis, the interface is horizontal
n ) δz
at r ) 0
(9)
It is assumed that the liquid wets the crucible to the lip, giving a fixed contact line at the crucible lip (see Figure 2).
h ) hR
at r ) R
(10)
Thermal Boundary Conditions. At the liquidvapor interface, an energy balance accounts for the power provided by the electron beam and the energy losses due to evaporation and thermal radiation
n‚q ) -(n‚δz)qb + qv + qr
(11)
The coefficient of the first term on the right-hand-side of eq 11 accounts for the influence of surface deformation on the flux of electron-beam power arriving at the liquid metal surface. The electron beam is swept in a circular pattern over the top surface of the melt pool. It is assumed that the beam is swept sufficiently fast that there are no variations in the field variables along the sweep path of the beam. The absorbed energy flux is taken to follow a Gaussian profile in the r direction with width σb
qb )
γQb exp{-[(r - rb)/2σb]2} 2π
∫0Rexp{-[(r - rb)/2σb]2}r dr
(12)
This profile is centered about the sweep circle that has a radius of rb. The fraction of electron-beam energy absorbed by the pool, γ, is taken to be uniform. The term in the denominator provides the normalization to obtain the specified total power Qb. An expression for the evaporative heat flux qv is given below, and the heat flux resulting from radiant heat exchange to the cold surroundings is given by
qr ) (T)σSB(T 4 - T∞4)
(13)
in which is the total hemispherical emissivity and T∞ is the temperature of the surroundings. There is assumed to be no exchange of radiant energy from liquidvapor surfaces that face each other as a result of surface deformation. For a pure metal, the temperature at the solid-liquid interface is the melting-point value.
T ) Tmp
at x ) xsl
(14)
This is a distinguishing condition that determines the mesh motion at the solid-liquid interface.20,28 An energy balance at the solid-liquid interface gives
Fs∆Hf(dxsl/dt) ) -n‚(k∇T)|l + n‚(k∇T)|s
(15)
The normal heat flux vanishes at the axis of symmetry
n·q ) 0
at r ) 0
(16)
Newton’s law of cooling is employed to describe the heat losses by contact from the pool to the water-cooled crucible wall with temperature Tw.
n·q ) hl(T - Tw) at x ) xlw
(17)
The heat-transfer coefficient, hl, is taken to be constant at this interface. For the solid ingot side, there are
Ind. Eng. Chem. Res., Vol. 43, No. 14, 2004 3951
contributions due to crucible contact and thermal radiation
n‚q ) T - T1 h + sσSB(T 4 - T∞4) for T1 e T e Tmp Tmp - T1 l ssσSB(T 4 - T∞4) for T e T1
{
(18) The thermal contact term increases linearly from zero to the liquid value as the temperature increases from T1 to the melting-point value. Below a temperature of T1, the contact term is taken to be negligible, and the only contribution is from thermal radiation. The solid hemispherical emissivity is taken to be constant along the side of the ingot. It is noted that, in the physical system, the contribution due to thermal radiation vanishes as contact improves moving up the side of the solid ingot toward the melting line. The neglect of this effect in eq 18 is believed to be minor because contact heat transfer dominates near the melting line. The heattransfer coefficient for the feed platform is taken to be constant
n·q ) hp(T - Tw)
at z ) 0
(19)
Vapor-Phase Boundary Conditions. We account for the vapor phase by applying boundary conditions for mass, momentum, and energy at the liquid-vapor interface. Each of these expressions includes a Langmuir term for the evaporation of an ideal monatomic gas in the absence of backscatter.29 A second term accounts for the influence of backscatter. The mass evaporation flux is given by
M x2πRT
nv ) (1 - ξn)pvap
(20)
Here, pvap is the equilibrium vapor pressure, and ξn is the backscatter fraction, the fraction of vapor that returns to the liquid-vapor interface as a result of backscatter. Note that this evaporation flux is assumed to have a negligible influence on the liquid flow field, but is incorporated directly in the surface energy balance given below. The normal stress, πv, generated by the thrust of the departing vapor is given by
πv )
pvap (1 + ξπ) 2
(21)
The quantity ξπ is the fractional increase in normal stress due to the backscattered atoms. The shear stress generated by the expanding vapor plume is represented by
τv ) ξτσb
dπv ds
(22)
in which ξτ is a coefficient, πv is given by eq 21, and the Gaussian beam width σb is taken to be a characteristic dimension of the trench region. In this model, the expanding vapor plume generates a surface shear stress in proportion to the surface gradient in normal stress. A similar expression was employed by Fleche et al.25 The evaporative energy flux, qv, includes contributions
Figure 3. Mesh deformation for the electron beam evaporation of metal from a melt pool. Pivoting spines pass through the liquid pool and the solid feed rod below.
from the liquid-vapor phase change and kinetic energy
[
qv ) nv ∆Hvap +
2RT (1 + ξq) M
]
(23)
The coefficient ξq is the fractional increase in the kinetic term resulting from backscatter. The coefficients ξi, i ) n, π, τ, q, are calculated from DSMC calculations as described by McClelland et al.28 Numerical Method The field and boundary equations for the solid and liquid phases are discretized using a Galerkin finiteelement method. We use a mesh with two sets of rotating spines as illustrated in Figure 3. Each spine in the first set (nos. 1-3) begins at a common anchor point above and to the left of the pool. These spines enter the liquid along the axis of symmetry, bend at the solid-liquid interface, and proceed to base points along the lower and right boundaries of the solid region. The first and last spines in this set (nos. 1 and 3, respectively) pass through the axis of symmetry at the junctions with the solid-liquid and liquid-vapor interfaces, respectively. Each spine in the second set (nos. 4 and 5) begins at an anchor point above the top of the
3952 Ind. Eng. Chem. Res., Vol. 43, No. 14, 2004 Table 1. Model Parameters for Electron Beam and Ingot Geometry parameter electron-beam power electron-beam sweep circle radius electron-beam spot radius rod radius rod length liquid level heat-transfer coefficient, liquid heat-transfer coefficient, lower solid cooling water temperature temperature of surroundings
Qb rb σb R h0 h0 hl hL Tw T∞
Table 2. Physical Properties and Parameters for Titanium Metal
value
units
property
59.2 0.0251 0.0044 0.0508 0.254 0.254 6000 100 25 25
kW m m m m m W/m2‚K W/m2‚K °C °C
melting point molecular weight liquid properties viscosity viscosity ratio density volumetric expansion thermal conductivity heat capacity thermal transport factor surface tension
ingot. They pass through the liquid-vapor interface, bend at the solid-liquid interface, and end at anchor points along the outside radius of the ingot. The last spine (no. 5) in this set is vertical and passes through the contact line and liquid-solid interface at the crucible wall. As in the earlier studies, the interfacial mesh points move with the interfaces, whereas the interior points move along the spines. This mesh strategy helps with the tracking of nearly vertical liquid boundaries encountered in deep pools. The use of vertical spines leads to excessive mesh deformation. The diagonal rotating spines of this study provide for less mesh deformation as the spines are more orthogonal to the solid-liquid interface. The baseline case in this study had 21 655 unknowns. As in earlier studies,20,23,24 the MELT computer code was used to solve the steady-state locations of the two interfaces simultaneously with the flow and temperature fields. A fully coupled approach was employed with the Newton-Raphson method. The time integration was performed with the trapezoid rule using error control. To minimize spurious transients, a backward Euler method was used for the first few steps of the calculation. The steady-state results were found by integrating from one steady-state point to another point using the false-transient method.30 The backward Euler method was used with large time steps and error tolerances to damp strong transients and progress rapidly through the parameter space to the point of interest. This approach was particularly effective for traversing regions of the parameter space for which steady-state solutions were not available using traditional parameter continuation methods. Finite-Element Results for Transient Evaporation Operating Conditions. Finite-element simulations were performed for the transient response of a titanium melt ingot after the electron-beam power had been shut off. The simulations were based on a Ti-6Al-4V evaporation experiment (no. 7) performed in the Advanced Development Coater (ADC) at 3M. The ADC has a topmounted electron-beam gun that was operated at 59.2 kW in this experiment (see Table 1). Magnets were used to sweep the electron beam in a ring pattern with diameter 0.0251 m over the top of the feed rod. The spot size was 0.88 cm, and the sweep frequency was 600 Hz. A 4-in. (10.16-cm) -diameter rod of the Ti alloy evaporation metal was advanced with a water-cooled pusher through a water-cooled crucible (see Figure 2). The ingot is taken to have a length of 0.254 m at the time that the power is shut off. The initial average liquid level is taken to be the same as the length of the ingot and to have the same elevation as the crucible lip. The liquid
value 1667 0.0479
°C kg/g‚mol
µ µcomp/µmeas λ β k Cp σ σ dσ/dT
3.200 × 10-3 15 4130 5.570 × 10-5 31.0 8.842 × 102 1.125 1.65 -2.400 × 10-4
kg/m-s
p1 E ∆Hv a0 a1 γ
2.317 × 1011 -4.355 × 105 9.092 × 106 a0+a1T 0.1810 2.533 × 10-5 0.347
N/m2 J/g‚mol J/kg
ks s
31.0 0.4
W/m‚K
ξn ξπ ξτ ξq
0.200 0.123 0.0288 0
vapor pressure heat of vaporization thermal emissivity skip fraction solid properties thermal conductivity thermal emissivity vapor backscatter parameters mass normal thrust vapor shear kinetic energy
units
Tmp M
kg/m3 K-1 W/m‚K J/kg‚K N/m N/m‚K
K-1
metal is assumed to wet to the crucible lip, which is a fixed contact line. At these conditions, the nominal evaporation rate is 2 kg/h (0.5 g/s). The heat-transfer coefficient for the liquid in contact with the crucible is 6000 W/m2‚K, and that for the pusher is 100 W/m2‚K. These values are in the range of values reported for liquid and solid metals in contact with a cold surface.31 The heat-transfer coefficient for the solid contact with the crucible decreases linearly with temperature from the liquid value at the melting point to zero at 300 °C below the melting point (see eq 18). Properties and Parameters. For the modeling of these experiments with the MELT code, the Ti-6Al-4V metal was treated as pure titanium because Al and V are minor constituents and vanadium is similar in many respects to titanium. Most physical properties are assigned their melting-point values for the respective phases (see Table 2). One exception is the viscosity, which is a factor of 15 higher than the physical value to numerically accommodate the strong convection encountered in this system. The use of this high viscosity provides converged steady-state solutions with good mesh resolution. Even with this increased viscosity, the flows are still quite intense, with Reynolds and Peclet numbers having magnitudes on the scale of 1000. In an attempt to compensate for the reduced thermal transport resulting from the artificially increased viscosity, the convection and conduction terms in the energy equation are increased by a factor of σ ) 1.125. This approach allows simulations to be performed without the need for excessively fine meshes or turbulent flow models. The skip fraction of γ ) 0.347 is greater than the ideal value of 0.22 for an electron beam impinging on a Ti ingot at a 90° angle.1 This higher value for the skip fraction was estimated from an overall heat balance for this experiment. There is evidence that the incident angle was considerably less than 90°.
Ind. Eng. Chem. Res., Vol. 43, No. 14, 2004 3953
Figure 4. Stream function contours in the liquid pool (left half of field) and temperature contours (right half of field) before (t ) 0) and after (t > 0) the shutoff of electron beam power to a titanium evaporation melt. (a) Steady-state results with electron-beam power on. There is surface flow in two directions from the hot beam impact area to the cold crucible wall and the axis of symmetry. Locations b-d are surface locations for the variables shown in Figure 5. (b) Electron-beam power is off. The trench fills in with liquid and surface velocities have decreased. (c, d) Wave motion has created a bulge in the beam impact area. (e) The trench has partially reformed from wave action. (f) The bulge in the hot zone has reappeared, and the melt surface has cooled significantly.
The parameters ξn, ξπ, and ξτ in boundary eqs 20-22 were determined from Monte Carlo results for Ti in a similar evaporation system.26-28 The backscattering of vapor results in 20% of the vapor returning to the surface and a 12% increase in vapor thrust. The influence of the vapor shear parameter ξτ is discussed below. Flow and Temperature Fields. The predicted flow and temperature fields are shown in Figure 4a-f for the shutoff of the electron-beam power to the Ti ingot. At steady state, the flow field shows two counterrotating cells in which the surface liquid flows strongly from the hot beam area toward the cold wall and a cooler region near the symmetry axis (see Figure 4a). Buoyancy, Marangoni, and vapor shear forces all have the same sign and contribute significantly to the flow. The warm low-density liquid rises below the beam and sinks near the cold wall and interior region. Surface tension gradients drive liquid from the hot zone to the colder surface regions. The surface tension is lower in the hot zone and higher near the crucible wall. The expanding vapor plume imparts a surface shear stress, dragging
surface liquid away from the hot zone. The thrust from the departing vapor creates a depression in the liquidvapor interface. The downward flow near the symmetry axis creates a deep pool. Hot liquid from the electronbeam impact area is convected strongly toward the bottom of the pool, driving the melting line in the downward direction. The flow is quite strong, with Re ) 634 and Pe ) 867, and is in the transition region between laminar and turbulent flow. It is noted that if the model viscosity is replaced by the 15 times smaller physical viscosity, Re ) 9510, and the flow is turbulent. A heat balance provides a measure of evaporation efficiency and numerical accuracy (see Table 3). The flow and temperature fields are well resolved as the incident electron-beam power differs from the total heat removed by 1%. The heat balance also shows that 49% of the energy is lost from the top of the ingot. Only 8% of the energy is consumed in the evaporation of titanium, and less than 1% of the energy flows to the pusher. Flow and temperature fields are shown in Figure 4b4f at several times after the power is shut off at t ) 0.
3954 Ind. Eng. Chem. Res., Vol. 43, No. 14, 2004
Figure 5. (a) Vapor rate, (b) temperature, (c) free surface position, and (d) velocity after the shutdown of an electron beam. Locations for temperature, free surface position, and velocity are shown in Figure 4a. Table 3. Energy as a Fraction of Incident Electron-Beam Power (%) for Steady-State Evaporation skip evaporation thermal radiation, top crucible pusher total
34.7 8.1 6.2 51.4 0.6 101.0
At t ) 44.0 ms, liquid has moved to fill in the trench in the electron-beam impact area. The maximum surface temperature has decreased by 160 °C, and the hottest surface liquid has moved from the hot zone to the symmetry axis. The cooling surface liquid has significantly decelerated. At t ) 57.5 ms, wave motion has created a bulge in the original beam impact area and a depression near the symmetry axis. The continuing wave action drives the liquid at the symmetry axis upward at t ) 77.8 ms to give a partially reformed depression at t ) 112 ms. At t ) 169 ms, there is once again a bulge in the original hot zone. Time Traces for Surface Variables. Four top surface variables are plotted versus time after the electron-beam power shutoff in Figure5a-d. The evaporation rate decreases by 10% in 0.4 ms and by a factor of e in 20 ms, indicating rapid surface cooling (see Figure 5a). Although, evaporation, thermal radiation, convection, and conduction all contribute, it is likely that evaporation is the strongest factor during the early stages of cooling. These results are important in assessing the effects of sweeping an electron beam over the surfaces of melt pools. The sweeping of electron beams provides for considerable flexibility in forming patterns on one or several melt pools. If an electron beam of 120 kW were being used to heat two melt pools of the type studied here, the residence time on each pool could be no greater than 0.4 ms to keep the evaporative fluctuations below 10%. Thus, the electron beam sweep system would have to be swept at rate of ν ) 1/(2 × 0.0004 s) ) 1250 Hz. This sweep rate is achievable given that commercial systems can provide sweep rates as large as 10 kHz.
The temperature at a node (point b in Figure 4a) in the electron-beam impact area is shown in Figure 5b. The temperature drops by over 300 °C in approximately 100 ms. The small oscillations in temperature are likely the result of horizontal motion in the selected node. This horizontal motion is due to the angular motion of the spines resulting from vertical motion in the free surface. The height of the free surface at the symmetry axis (point c in Figure 4a) is shown in Figure 5c. This height is referenced to the elevation of the contact line at the crucible wall. Gravity waves are seen with an amplitude of 6 mm and a period of approximately 130 ms. These waves result from the rapid decay in vapor pressure in the electron-beam impact area. The radial component of velocity (point d in Figure 4a) is shown in Figure 5d versus time. This velocity decreases from nearly 20 cm/s to between 5 and 10 cm/ s. The oscillations are attributed to wave motion and the associated horizontal motion of the selected node resulting from mesh deformation. The decrease in hotzone temperature reduces the Marangoni driving force and the surface velocity. Also, the associated decrease in pressure results in a decrease in the shear force generated by the vapor plume. Thus, the surface velocity responds very quickly to changes in the electron beam power. There is the question of whether the sweeping of the beam induces any flow in the sweep direction. At the 600-Hz sweep frequency for the experiment, the sweep velocity at the free surface is 9500 cm/s, which is much faster than the 20 cm/s velocity calculated above for the surface fluid. Although the beam sweep generates some surface variation in the temperature and evaporation rate, it does not drive flow in the sweep direction. Conclusions A finite-element simulation is given for the transient response of a titanium evaporation melt to the shutoff of electron-beam power. The model includes the liquid pool, the solid feed rod, and the effects of the vapor
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plume at the pool surface. The flow in the pool is driven strongly by temperature-induced Marangoni and buoyancy forces. The flow is driven in the same direction by the drag of the vapor plume on the free surface of the pool. The position of the free surface is also influenced by surface tension, the thrust of the departing vapor, and gravity waves. The lower boundary of the pool is a melting surface. The upper and lower interfaces of the liquid pool are tracked using a mesh structured with rotating spines. Before shutoff, a 60-kW electron beam is swept in a circular pattern on the top of a 10.16-cm-diameter feed rod. The thrust of the departing vapor significantly depresses the liquid-vapor interface. The flow field includes two counter-rotating cells and is in the transition region between laminar and turbulent flow. After the electron beam power is reduced to zero, gravity waves propagate through the melt pool with a period of 130 ms, and the vapor rate decreases by 10% in a fraction of a millisecond. Work is in progress to compare model predictions with measurements of the decay in evaporation rate. These results suggest that, for this evaporation system, the electron beam needs to be swept at a rate of greater than 1 kHz to keep vapor rate fluctuations less than 10%. These high sweep rates can be achieved with commercial electron-beam systems. Acknowledgment Jonathan Storer and Chris Shelton of 3M Corp. are acknowledged for their many helpful discussions concerning the Advanced Development Coater for the fabrication of Ti64 metal matrix composites. This work was partially support by DARPA through the vaporphase manufacturing initiative (administered by Steve Wax). This work was performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under Contract W-7405-Eng-48. Nomenclature Cp ) mass heat capacity at constant pressure (E/MT) dpool ) depth of pool (L) E = unit of energy (ML2/t2) g ) gravitational vector (L/t2) h ) local elevation of the liquid-vapor interface (L) hl ) liquid-crucible heat-transfer coefficient (E/L2Tt) hp ) solid-pusher heat-transfer coefficient (E/L2Tt) hR ) elevation of the crucible lip (L) h0 ) average elevation of the liquid-vapor interface (L) k ) thermal conductivity (E/LTt) L = unit of length (L) M = unit of mass (M) n ) outward-pointing surface normal vector p ) liquid pressure (M/Lt2) pvap ) vapor pressure (M/Lt2) q ) heat flux (E/L2t) Qb ) incident electron-beam power (E/t) qb ) incident energy flux from the electron beam (E/L2t) qr ) energy flux loss due to thermal radiation (E/L2t) qv ) energy flux loss due to evaporation (E/L2t) R ) crucible radius (L) R ) gas constant (E/mol‚T) r ) radial coordinate (L) rb ) radius of circle pattern swept by electron beam (L) s ) surface coordinate in the radial direction (L) s1 ) surface coordinate in the azimuthal direction (L) T ) temperature (T) T = unit of temperature (T)
Tl ) lower temperature of the contact region (T) Tmp ) melting-point temperature (T) Tw ) temperature of the cooling water (T) T0 ) reference temperature (T) T∞ ) temperature of surroundings (T) t ) surface tangent vector in the radial direction t1 ) surface tangent vector in the azimuthal direction t ) time (t) t = unit of time (t) u ) velocity vector (L/t) umax ) maximum magnitude of the liquid velocity (L/t) x ) position vector (L) xsl ) location of the solid-liquid interface (L) xlv ) location of the liquid-vapor interface (L) z ) axial coordinate (L) R ) thermal diffusivity (L2/t) β ) thermal coefficient of volumetric expansion coefficient (1/T) γ ) energy absorption fraction for electron beam ∆Hf ) heat of fusion (E/M) δ ) unit tensor δz ) unit vector in the z direction ) emissivity of the liquid s ) emissivity of the solid µ ) liquid viscosity (M/Lt) Fs ) solid density (M/L3) F0 ) liquid density (M/L3) ξn ) fractional decrease in mass evaporation flux due to backscattered vapor ξq ) fractional increase in evaporative heat flux due to backscattered vapor ξπ ) fractional increase in surface normal stress due to backscattered vapor ξτ ) coefficient for surface shear stress generated by the vapor flow field πv ) magnitude of the normal stress exerted on the liquidvapor interface by the departing vapor (M/Lt2) σ ) surface tension (M/Lt) σSB ) Stephan-Boltzmann constant (E/L2T 4t) σb ) Gaussian half-width for the swept electron beam spot (L) τ ) viscous stress tensor (M/Lt2) τv ) magnitude of the shear stress exerted on the liquidvapor interface by the vapor flow field (M/Lt2)
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Received for review September 15, 2003 Revised manuscript received April 5, 2004 Accepted April 7, 2004 IE030731I
