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Analysis of the steady state molten pool obtained by heating a substrate with an electron beam
Acta metall, mater. Vol. 39, No. 5, pp. 725-733, 1991 Printed in Great Britain. All rights reserved
0956-7151/91 $3.00 + 0.00 Copyright © 1991 Pergamon Press pie
ANALYSIS OF THE STEADY STATE MOLTEN POOL OBTAINED BY HEATING A SUBSTRATE WITH A N ELECTRON BEAM B. B A S U I, J. A. S E K H A R 2, R. J. S C H A E F E R 3 and R. M E H R A B I A N 4 ~Tata Research Design Development Centre, Pune 411001, India, 2Department of Materials Science and Engineering, University of Cincinnati, Cincinnati, OH 45221-0012, 3National Institute of Standards and Technology, Galthersberg, Md and 4Carnegie Mellon University, Pittsburgh, PA 15213, U.S.A. (Received 28 April 1990; in revised form 25 September 1990) Abstract--Surface melting and solidification with high powered beams can be used for enhancing surface properties. The dimensions of the molten zone define the extent of the modified properties and are critical parameters which must be predicted during process design. The flow field in the molten pool has been reported to be one of the key factors which controls the dimensions of the surface layer. However, the calculation of this is only possible through complex numerical schemes and there is a need to look for simple analytical expressions which may be adequate. One approach for this search involves the precise determination of the steady state stationary profiles and then developing a method for extending these values to include the effect of beam motion for predicting the pool dimensions during processing. In this paper, a study of the flow field and its effect on the depth and width of the steady state pool is presented, based on numerical and analytical methods. To validate the predictions, an experimental study is carried out using surface melting of AI-4.5 wt%Cu alloy with an electron beam. The pool shapes are presented through optical micrographs and the depth and width of the pool is measured from these micrographs. The experiments are then simulated using a numerical model which includes fluid flow. The flow field is analyzed using streamline plots and the predicted pool shapes are compared with the micrographs. Further, the results are compared to an analytical method based on pure conduction and the pool depth and width are predicted when the liquid thermal conductivity is modified. The numerical and analytical predictions of the pool depth and width are found to be in good agreement with the experimental measurements (obtained from steady state stationary pools and from dimensions inferred on extrapolating moving beam measurements to zero velocity). The reasons for the success of the analytical model is discussed with reference to the two-dimensional flow fields and vortices predicted by the numerical model. Rtsumt--Pour amdliorer les proprittts de surface, on pent utiliser la fusion et la solidification superficielle ~i raide de faisceaux d'61ectrons de haute puissance. Les dimensions de la zone fondue dtfinissent l'ttendue des proprittts modifites et sont des paramttres critiques qui doivent 6tre prtvus pendant le processus d'61aboration. Le champ d'tcoulement dans le liquide est considtr6 comme un des facteurs cits qui contr61ent les dimensions de la couche superficielle. Cependant, cette estimation n'est possible que par des calculs num&iques complexes et il est ntcessaire de chercher des expressions analytiques simples qui peuvent convenir. L'une des approches de cette recherche, c'est la dttermination prtcise des profils en rtgime permanent stationnaire suivie du dtveloppement d'une mtthode pour adapter ces valenrs afin d'inclure l'effet du mouvement du faisceau pour prtvoir les dimensions de la zone liquide pendant le traitement. Dans cet article, on prtsente une 6tude du champ d'tcoulement et son effet sur la profondeur et la largeur de la zone fondue en rtgime permanent; cette 6tude est baste sur des mtthodes numtriques et analytiques. Pour valider les prtvisions, une 6tude exptrimentales est effectute en utilisant la fusion superficielle de l'alliage A1-4,5% en poids Cu par un faisceau d'61eetrons. Les formes des zones fondue sont prtsenttes sur des micrographies optiques et leur profondeur et leur largeur sont mesurtes ;i partir de ces micrographies. Des exptriences sont ensuite simultes en utilisant un modtle num6rique qui inclut l'tcoulement de fluide. Le champ d'tcoulement est analyst en utilisant des courbes de profil et les formes de zones fondues prtvues sont compartes aux micrographies. De plus, les rtsultats sont comparts ~ une mtthode analytique baste sur la conduction pure et la profondeur et la largeur de la zone fondue sont prtvues lorsque la conductivit6 thermique liquide est modifite. Les prtvisions numtriques et analytiques de la profondeur et de rtpaisseur de la zone fondue sont en bon accord avec les mesures exptrimentales (obtenues ~i partir de zones fondues en rtgime permanent stationnaire et pour des dimensions dtduites, par extrapolation ~i la vitesse nulle, de mesures de dtplacement du falsceau). Les raisons du succts du modtle analytique sont discuttes en faisant rtftrence aux champs d'&:oulement bidimensionnels et aux tourbillons pr6vus par le moddle num&ique. Zupmmenfauug---Oberfl~chenschmelzen und -erstarren durch hochenergetisehe Strahlen kann zur Verbesserung der Obert~cheneigenschaften ausgenutzt werden. Die Malk des geschrnolzenen lkreiches bestimmen das Ausmal3 der g~nderten Figenschaften und sind dither kritische Parameter, diue w~hrend der Prozelkntwicklung vorhergesagt werden miissen, lkrichtet wurde, dab das Flietfeld in der Schmelze einer der wichtigsten Faktoren bei der KontroUe der Dimension der Oberfiichenschicht ist. Allerdings ist die entsprechende lkrecfinung nur mit komplexen numerischen Anstitzen m6glich; man sollte also nach anwendbaren einfacheren analytischen Ausd~cken suchen. Fine M6giichkeit hiefffir besteht in der genauen Bestimmung der station~ren Profile, diese mit einer zu entwickelnden Methode zu erweitern und AM 39/~-A
725
726
BASU et al.: MOLTEN POOL CAUSED BY ELECTRON BEAM HEATING so den EinfluB der Strahlbewegung auf die Vorhersage der Abmessungen des Schmelzbereiches zu ermitteln. In dieser Arbeit wird das FlieBfeld und sein EinfluB auf Tiefe und Breite der Schmelze im station/iren Fall auf der Grundlage numerischer und analytischer Methoden bestimmt. Zur Prfifung werden Experimente zum Aufschmelzen der Oberflfiche der Legierung AI-4,5 Gew.-%Cu wit einem Elektronenstrahl durchgeffihrt. Die Forme der Schmelzbereiche wird mit optischen Aufnahmen dargestellt, Tiefe und Breite werden an den Aufnahmen ausgemessen. Dann werden die Experimente mit einem numerischen Modell unter EinschluB yon FlieBen der Schmelze simuliert. Das FlieBfeld wird mit Str6mungsdiagrammen analysiert, die Form der Schmelzbereiche wird mit den Aufnahmen verglichen. Au~rdern werden die Ergebnisse mit einer analytischen Methode verglichen, die auf reiner Leitung aufbaut; Tiefe und Breite des Schmelzbereicheswerden vorausgesagt ffir .~nderungen in der W~irmeleitung der Schmelze. Die numerischen und analytischen Voraussagen stirnmen mit dem experimentellen Ergebnissen (erhalten aus stationfiren Schmelzbcreichenund aus Dimensionen, die sich bci Extrapolation der Strahlgeschwindigkeit zu Null ergeben) gut fiberein. Die Grfinde ffir den Erfolg des analytischen Modells werden im Hinblick auf die zweidimensionalen FlieBfelder und die Wirbel, die yore numerischen Modell vorausgesagt werden, diskutiert.
1. INTRODUCTION Electron and laser beams are often used for surface processing involving melting and resolidification. On account of the small time of heating and subsequent rapid solidification after melting, surfaces can be modified to display enhanced properties, such as better resistance to the wear and corrosion. A very shallow heat affected zone is another advantage that high heat intensity beam processing may offer. The accurate prediction of the melt depth and width during electron and laser beam surface melting and solidification is therefore important, for the selection of correct process parameters. Anthony and Cline [1] first proposed a closed form analytical solution with a finite area source to calculate melt depths based on a pure conduction model. Phase change was not considered. There are now various publications available based on similar conduction numerical models [2-6] which relax many of the assumptions made in the earlier model [1]. While the first three models [2-4] include phase change in a pure metal in one, two and three-dimensions, respectively, Sekhar et al. [5] proposed a two-dimensional model including alloy solidification. In an earlier paper, Sekhar et al. [7] compared the predicted and experimental pool shapes seen during electron beam melting. The experimental results deviated from the measured value. A higher depth and smaller width prediction was made by the conduction model than was observed. The deviation was attributed to the effect of the fluid flow present in the molten pool. It has now been conclusively established that the fluid flow always occurs and plays an important role in electron and laser beam processing [7-9]. Chan et al. [8] first proposed a comprehensive numerical model for laser melting including the effect of flow field. They however did not compare the predicted and measured melt depth and width. Their study mainly focused on the fundamental analysis of the flow field and its effect on the heat transfer mechanism. Subsequently however Chan et al. [10] compared the predicted and measured values and reported adequate agreement. Kou and Wang [11] and Rappaz [12] have also reported good to adequate
agreement between predicted and measured pool shapes based on a three-dimensional numerical model. Rappaz et al. [12] have further succesfully incorporated microstructure models into the calculations. For a good estimate of the melt depth and width during electron and laser beam melting, the fluid flow is therefore an important consideration. However, a three-dimensional fluid flow numerical calculation is complex and cumbersome and an analytical technique or an intelligent semi-analytical technique should be developed if possible. One approach for this search involves the precise determination of the steady state stationary profiles and then developing techniques for extrapolating these values to include the effect of beam motion. In this paper, the effect of the flow field on the steady state melt depth and width in an A1-4.5 wt% Cu alloy is first studied using a numerical model including fluid flow and then comparing the results with experimental measurements obtained after electron beam melting. Next, a previously suggested analytical procedure [13] is employed for the same melt depth and width prediction with a new modification; namely where the thermal conductivity of the liquid alloy is altered. Finally all the predictions are compared with the experimental measurements. We show below that the prediction of the steady state melt dimensions (i.e. the dimensions of the pool with no relative motion between the beam and the substrate) is calculated with relative ease and work is recommended to obtain proper extrapolations from this starting point of the melt dimensions with finite velocity of the beam. 2. THE EXPERIMENTAL PROCEDURE The experiments were conducted on a 10kW electron beam machine [13]. Al-4,5%Cu alloys were prepared by diluting pure aluminum with a master alloy of eutectic composition, AI-33%Cu. The homogeneity of the samples was checked to a high degree of accuracy by wet chemical and electron microprobe analysis. Experiments were performed in a steady state condition [7] by keeping the stationary beam on for longer times ( > 5s) and, also, in the quasi-steady
BASU et al.: MOLTEN POOL CAUSED BY ELECTRON BEAM HEATING state condition where the samples were traversed at various scanning velocities. Scanningspeeds up to 0.066 m/s were obtained using a linear translating assembly and a rotating assembly was used to attain a translational speed up to 0.66 m/s. Beam profile and spot size were measured by scanning the electron beam over an array of 4 mm thick tungsten wires placed approximately 2000#m apart. The induced current was recorded on an oscilloscope to indicate the position and dimensions of the beam. The same height was maintained for all the experiments to ensure the same beam profile. The beam was noted to be of a top hat shape. Further details of the experiment are recorded in Ref. [13].
2. I. Steady state conditions for stationary beam For the stationary beam experiments, the incident powers were chosen in such a way that the steady (a)
(b)
727
1.0 1.2 0.8
0.7 "~
~
0.5
0.8 "~ '~
0.4
0.6
0.2
I
I
I
I
I
1
I
0,4
8+
8+
8+
8+
8+
o
o
o
o
8 .
q
,~
~
~.
qo
Fig. 2. Dimensionless melt depth (z/a) and width (r/a) as a function of qa for the steady state molten pool. state melt profiles could be maintained for a long time [7]. Steady state conditions were obtained when the absorbed power on the specimen was exactly offset by the conduction of heat into the semi-infinite substrate. A limiting condition to steady state was obtained when the vaporization rate from the melt became significant in the 10 -5 torr vacuum chamber. For AI-4.5%Cu alloy, surface melting was observed at a minimum qa (q and a are the input heat flux and spot radius respectively) of the order of 1.05 x 105 W/m. The steady state profiles could be maintained up to qa of 2.5 × 105 W/m. Figure 1 shows the micrographs of typical steady state profiles for various qa's. Figure 2 shows a plot of dimensionless melt depth and width as a function of qa. The experimental conditions for this plot are given in Table 1.
Quasi-steady state conditions for moving beam
22
Experiments were also performed by moving the samples at different power levels and velocities. The following conditions were commonly employed during each experiment:
(c)
(a) The samples were polished with 600 grit paper to maintain the same absorptivity of the solid before melting starts, (b) experiments were performed at the same height where the spot diameter and spatial energy distribution were measured, (c) after each pass, the samples were allowed to cool to the room temperature before making any subsequent passes, (d) The samples were thick enough to permit the assumption of a semi-infinite substrate ( ~ 2 0 mm thick). The time required to reach the quasi-steady Table I. Depth and width of the steady state molten pool with a stationary electron beam of radius 0.5 mm Incident
Fig. 1. Micrographs of steady-state molten pool with an electron beam of 0.5 mm radius for (a) qa = 1.22.105 W/m, (b) qa = 1.98.105 W/m and (c) qa = 2.5- 105 W/re.
Melt depth
Melt width
flux q
qa
No.
(10s W/m 2)
(10 s W/m)
(z/a)
(r/a)
I 2 3 4 5
2.43 3.97 4.11 4.68 5.00
1.22 1.98 2.05 2.34 2.50
0.292 0.402 0.460 0.960 0.938
0.476 0.940 1.000 1.250 I,154
728
BASU et al.: MOLTEN POOL CAUSED BY ELECTRON BEAM HEATING
state varied with the speed and power of the beam and was largest for the highest power level with slowest speed. Figures 3 and 4 show the variation of pool dimensions with processing conditions and micrographs of typical pool shapes respectively.
(a)
...-
3. THE NUMERICAL MODEL Electron beam melting is a thermo-fluid dynamic process and is, therefore, governed by the laws of conservation of energy, momentum and mass. The physical definition of the electron beam melting problem based on which the numerical scheme is developed is shown in Fig. 5. The fluid flow in the molten pool is assumed to be driven by the surface tension gradient at the free surface [8]. The numerical model is based on the following assumptions: (1) There is symmetry in the angular direction and the problem is solved in the ( r - z ) plane only. (2) The surface of the melt is flat. (3) The buoyancy force is of a negligible order of magnitude due to the small dimension of the pool []41. (4) Although electro-magnetic force is present, it is also negligible compared to the surface tension gradient [15]. (5) The properties, except for the surface tension, are not the function of temperature. For the purpose of computation, the average properties are considered. (6) A critical Reynolds number for transition to turbulent flow for thermocapillary flow is not precisely defined in a cavity. Because of this and accumulated experimental evidence (on stationary
S ~e
=*
•
20.143r a m .
£o)
Fig. 4. Micrographs of the quasi-steady state molten pool with a moving electron beam of 0.5 mm radius and 0.012 m/s scanning speed (U) with (a) qa = 2.8" 105W/m and (b) qa = 4.2.105 W/m. steady state flows [13]), the flow is assumed to be laminar. The governing equations of energy, momentum and mass, when normalized in the following way dp = ( H * - H * t ) / A H a ;
O~
2.0 F
1.8 1,6 "~ 1.4 0 x
i
X/o Z/o Ext rapolotion
e---
L
~
e'.,
~ ~e
0.8 ~ - ~ _ ~ -
- Tm)/AH;
r = r*/a
A1--4.5% Cu Homogenized (I s • 7.44x 105m2/s units of qo: (W/m)
1.2 x . ' ~
-<
0 = C*p(T
v = v*/UR; k = k*/k*; ELectron beom
~ - ~ ~-e 2.80x 105 •
2"05X105 r(V r )
• I ~" t"r~"-.,....,~ L2.05"~:5 ~ ....
°21 0
2"34 x 10'~ 1.22x 105
I
I
I
0,1
0.2
0.3
Ua/2a s
Fig 3 Variation of the dimensionlesspool depth (z/a) and w i d t h (x / a ) w i t h dimensionless scanning speed ( Ua /2= ) and heat flux per unit length qa W/M. The extrapolation of the moving beam results to obtain the stationary steady-state values is also shown.
zcv~l Fig. 5, The geometrical representation of electron beam melting used for the numerical model.
BASU et al.: MOLTEN POOL CAUSED BY ELECTRON BEAM HEATING Cp ~
729
A pproximo t.eO solid / Liquid interface
Cp/Cp~ * *
R o e URa/v;
t
~.,
Ma = URa /~t ;
IIIllllll
',7,',11111 ',~;;lll I
Ste = C*~(Tm - TD/AH~,
111111111 illlllllll
Bf = qaCr~/k* AH~I
IIIlillll IllJlllll
UR = (da /dT" AH~O/C*" # )
p =p*/pv~ take the following form for the problem of electron beam melting with the assumptions detailed above.
3.I. Energy equation div (v~) = (I/Ma) grad (k div 0).
(I)
3.2. r-Momentum equation
Fig. 6. The non-uniform grid distribution and the approximation of the solid-liquid interface.
div (v - v,) = - (@/6r) + (I/R,) div (grad v,) -- v,/r 2.
(2)
3.3. z-Momentum equation
div (v - v:) = - ( @ / 6 z ) + ( I / R , ) div (grad v:). (3) 3.4. Continuity equation
div (v) = O.
(4)
The corresponding boundary conditions are as follows: at r = 0, 60/6r = 6v:/6r = v, = 0; 0 < z < oo
(5)
at r = oo, 0 = - Ste; 0 < z < oo
(6)
a t z =O, k 6 0 / 6 z = - B f ; 0 < r < 1 =0;
l
0
4.1. Experimental
(8)
It should be noted that the energy equation is written in the enthalpy form to include the latent heat absorption during melting using the enthalpytemperature relationship. The normalized enthalpytemperature relationship is as follows. In the solid phase, 0 =¢;
~ <0.0
(9)
In a cell containing the liquid and solid (interface ceil), 0=0;
0<~<1
(10)
In the liquid, 0 =(4, -
l)C~/C~;
4. RESULTS
(7)
6vr/hz = -- 60/6r, v: = O; 0 < r < rmax
atz=~,0=-Ste;
turn and the continuity equations. The flow field is calculated in the liquid domain only by tracking the solid/liquid interface by the "Switch Off" method I17]. The algebraic equations resulting from the finite difference equations are solved by the Gauss-Seidel iterative method with (1764) grids of non-uniform distribution used for the calculation. Very fine grids are used below the beam and coarse grids are used away from the beam. This distribution of grids allows for the incorporation of fine grid spacings at the surface melt pool boundary [18]. A representative grid distribution is shown in Fig. 6 with the approximated interface.
4, > i . 0 .
(ll)
The governing equations (1)-(4), along with the boundary conditions [equations (5)-(8)], and the enthalpy-temperature relationship [equations (9)-(11)] define the complete problem of electron beam melting. These equations are solved by the finite difference method using a control volume formulation [16]. The SIMPLE algorithm [16] is used to solve the momen-
Figure 3 shows the change of the dimensionless melt width and depth with the dimensionless speed (Ua/2rt) for various values of qa. The depth and width for a particular power level and various speeds are extrapolated to zero speed, i.e. stationary beam. The extrapolation was carded out based on the shapes of adjacent curves. Because of this, the extrapolated values are expected to be within ,-,20% of the true value. From Table 2 it can be seen that these experimental values closely match with the experimental values of the stationary beam shown in Fig. 2. At higher qa values and low velocities (i.e. qa = 6.4 x 105W/m and U =0.1 m/s) vaporization occurs resulting in deep penetration and, hence, steady state cannot be attained. Figure 4 shows the Table 2. Depth and width of the steady state molten pool with a stationaryelectronbeamof radius 0.5 mm obtained by extrapolating
quasi-steadymoving beam results
Incident
Melt
Melt
(I05 W/m)
depth (z/a)
width (r/a)
1.56 2.81
0.371 1.050
4.20
--
0.817 1.200 1.450
flux q
qa
NO.
(lOS W/m 2)
l 2
3.12 5.62
3
8.40
730
BASU et al.: MOLTENPOOL CAUSED BY ELECTRON BEAM HEATING (C)
q = 5.0 xl0 e W/m 2 A I , - Cu
AL-Cu
f
0.7
No. Volue
1.2
i
104
i
5=0.50
I
6= 1,00 7 = 1.75 8= 2.50
Q905
(d)
q = 5 . 6 x 108 W / m 2 AL-Cu
(b}
1.3
q • 3 , 9 7 x 10 a W / m z AL-Cu
/S
!
1.o
i
4
;
I= 1 . 0 0
I
2=o.~'5
0.455
3=0.25 4=0.15 5=OAO 6:0.55
1 500 2 = 2.50 ]=1..,.)0 4=0.50 5=0.50
I j
6 = 1.00 7=1.75
1.005
8=2,75
Fig. 7. Streamline and pool shape of the steady-state molten pool with a stationary electron beam of 0.5 m m radius for different qa's (a) q ffi 2.43. I08 W / m 2, (b) q = 3.97-10s W / m 2, (c) q = 5.0. l 0 s W / m 2, (d) q = 5 . 6 . 1 0 s W / m 2. The values of the parameters given along the z and y axis refer to dimensionless
melt depth and width, respectively. The other numbers given along with the streamlines refer to the dimensionless stream function.
micrographs for different qa and scan speeds for the moving beam quasi-steady state conditions. 4.2. Simulation
The experiments are numerically simulated to predict the pool width and depth. The flow field for various power levels is also studied using streamline contours. The property values, along with the model parameters (i.e. the dimensionless numbers as specified before) used for the simulation are presented in Table 3. The results in Table 3 are given as a function of Bf although in the subsequent figures the results are presented against the parameter qa in preference to the boundary heating factor Bf. Although all our modelling work has been performed
Table 3. Property values of AI-4.5%Cu and process parameters q (108 W/m 2) Br 2.43 3.12 3.97 4.10 5.00 5.60
2.815 3.615 4.599 4.750 5.793 6.490
AHe = 3.95 × 10s J/kg; k? = 100.8W/mK; k* = 180.6W/InK;
Tm = 821.0 K;
p = 2700.0 kg/m3; C~ = 882 J/kgK;
C~ = 924 J/kg K; ~ = 10-3 kg/ms; de/dT = - 0.35 x 10 -3 kg/sC. Process parameters: a =0.5ram; M a r 1851.0; 1~ = 20196.0
731
BASU et al.: MOLTEN POOL CAUSED BY ELECTRON BEAM HEATING with this factor and the boundary conditions have been similarly written in this form e.g. equation (7), we have been conscious that regardless of the generality of the modelling and analytical results the detailed experimental work has been carried out by us only on one alloy. Thus multiplying qa by C~/k*AH~ only scales the results without adding any real generality to the comparison of experimental and analytical results. Consequently and in order to remain consistent with the previous results in Refs [1-11] we have preferred to present the data with qa instead of Bf. In addition, it is important to recognize that qa has been used in the literature in preference to Bf because it represents a pure processing parameter (i.e. it does not contain material constants). Figures 7 (a-d) show the streamlines in the steady state molten pool for various qa's: 1.215 × 105, 1.985 x 105, 2.5 x l05 and 2.8 × 105 W/m. With the increasing values of qa, it can be seen that there is a dramatic change in the nature of the flow pattern. The flow field for qa = 1.215 x 105 W/m consists of a single cell which will be called the primary vortex [Fig. 7(a)]. At qa = 1.985 × 105 W/m, the flow field shows existence of another vortex at the bottom of the primary one. This will be referred to as the secondary vortex [Fig. 7(b)]. For the subsequent higher qa's, the secondary cells occupy almost the whole molten pool region with a small primary cell at the top. The primary vortex enhances the heat transfer radially outwards at the top and reduces it near the line of symmetry where the flow is upwards (see Fig. 7). Hence, the primary vortex produces a pool of larger width and smaller depth (i.e. a shallow pool) in the presence of strong convection. Due to the low Prandtl number of the A1-Cu alloy (Pr = 0.01 which means thermal diffusion is 100 times more than the viscous diffusion), the contribution of the fluid flow to the net heat transfer is small. As a result, the primary vortex fails to produce a shallow pool and a deep pool results due to the thermal diffusion. Once a deep pool forms, there are two vortices due to the inability of the fluid particles with the high momentum (i.e. at the top of the pool) to take a sharp turn near the solid-liquid interface and follow a path
2.0 1.8 1.6 1.4 1.2 "~I.0N
o
o
0.8
mericoL 0.6
/
o
- -
/~a 0.4
o Exherirnent.oL
8 o / P"
J
0.2
II 1
0
I 2
Theoretical
A 1 - 4 5 % Cu Homogenized
I I 3 4 qo xl0"~ Wm -1
I 5
L 6
I 7
Fig. 9. Comparison of the numerically and theoretically predicted steady state melt depths with the experimental measurements. along the interface. This pattern is similar to the flow field in a cavity-like surface tension driven flow in rectangular cavities [14]. For a lower qa, the flow field consists of a single vortex due to the small size of the pool. The secondary vortex exists for the higher qa [Fig. 7(a)]. i.e. larger molten pool, and its strength increases with qa [Fig. 7(b), (c) and (d)]. 5. ANALYTICAL CALCULATION Consider a cartisian coordinate system fixed to the center of the heat source, with the y direction being the beam travelling direction and x and z directions 2.0 1.8 1.6 1.4 1.2
Et::mt r o n
~
y
1.0 0.8
f r
0.6
o
Experlmento t
--
TheoreUcot
A 1 - 4 . 5 % Cu
Homogenized
0.4 0.2
0
I
I
t
I
I
I
t
I
2
3
4
5
6
7
8
9
q o x 10-5 Wm "1
Fig. 8. The coordinates for the analytical approach.
Fig. I0. Comparison of the numerically and theoretically predicted steady state melt widths with the experimental measurements.
732
BASU et al.: MOLTEN POOL CAUSED BY ELECTRON BEAM HEATING
being the radial and axial distances along the substrate, respectively (see Fig. 8). Carslaw and Jaeger [19] have presented analytical expressions to determine the time dependent temperature profiles for a stationary point source. Sekhar [13] modified these equations to include the effect of an area source and also to account for the latent heat absorption due to the phase change. At long times (i.e. steady state) these are given as:
k*[T(0, z)
- To + (AH~/C~)]
qa = (z/a)[{1 + ( a 2 / z 2 ) } Ill - 1} (12)
k*[T(r, O) - To + (AH,,/C~, )] qa = (2/n) E(r/a)
for (r/a) < 1
= (2/n)[E(a/r) -- {1 - (a2/r2)}K(a/r)
for (r/a) > 1.
(13)
In equations (12) and (13), K and E are the elliptic integrals of the first and second kind, respectively. The latent heat absorption has been accounted for by an equivalent sensible heat absorption using a pseudotemperature, i.e. CpT = A H ~ . The melt depth and width can, now, be determined from equations (12) and (13), respectively. Substituting T(0, z) by Tm in equation (12), the depth is calculated by solving equation (12). Similarly, the melt width can be determined from equation (13). Sekhar [13] has provided the solution of equation (12) and (13) in graphical form and it is used for the present calculation. For the prediction of the depth and width, a higher liquid thermal conductivity is used in equations (12) and (13) apart from the Cm and AH~ as given in Table 3. The average of the solid and liquid thermal conductivities is used for the calculation, i.e. k* for calculation = (k? + k*)/2 = (100.8 + 180.6)/2 = 140.7 W/mK. The selection of higher thermal conductivity for both depth and width prediction is validated by the nature of the flow field as discussed below. 5.1. Comparison o f the pool depth and width
Figures 9 and 10 show the experimentally observed melt depth and width along with that predicted by the numerical model and the calculation procedure described above. The values predicted by the calculation procedure are referred to as theoretical values in the figures, and in the following discussion. It can be seen that the theoretical predictions are in very good agreement with the numerical as well as the experimental ones (see Figs 9 and 10). As discussed in the previous section, the theoretical prediction is based on a pure conduction calculation with an increased thermal conductivity of the liquid AI-4.5%Cu, i.e. 140.7 W/mK instead of 100.8W/inK. It has been recorded before [8-10] that it may not be adequate to include the effect of
recirculating fluid flow simply with an enhanced thermal conductivity as attempted by Hsu et al. [1]. The reason for the correct analytical predictions even with a pure conduction solution (with a higher liquid thermal conductivity) lies in the nature of the flow pattern as illustrated by the numerical model. It can be seen from Fig. 7 that the flow is radially outward at the top of the pool due to the primary vortex and is axially downward near the line of symmetry due to the presence of a large secondary vortex. This effect of fluid flow, namely of enhancing heat transfer in both directions is, thus, precisely incorporated through a high liquid thermal conductivity. As a result, the theoretical predictions using an enhanced thermal conductivity match so well with the experimental as well as the numerical ones. It should be noted here that this calculation procedure may fail to produce realistic depth and width for the cases where the primary vortex dominates the liquid pool with a small and weak secondary vortex at the bottom; this kind of flow pattern exists for example in a molten pool of steel [8] because of the higher Prandtl number ( = 0.078). For such cases, the theoretical prediction of the depth using higher thermal conductivity would be higher than observed, on account of the reduction of net heat transfer along the line of symmetry resulting from the strong upward flow of the primary vortex. The possible differences between e.b and laser [20] in terms of depth of penetration of the beam has evidently not affected the solution. The model assumes that heat is deposited at the surface whereas the experiments are with an electron beam. The reason for this lack in difference is possibly because the depths of the melts that we are considering are much larger than the effective e.b penetration (typically in the order of 10/am) where heat is released. In addition as pointed out in Ref. [15] previous results have indicated that e.m.f, forces are much smaller than the forces due to surface temperature gradients.
6. CONCLUSIONS Several different methods for predicting the stationary beam melt dimensions are explored in this paper. Using a numerical model, the pool shape and flow field are predicted for several experiments. The flow fields show a dramatic change from low to high power levels of the beam. The molten pool consists of a primary vortex at low power levels of the beam. With an increase in the power level, a secondary vortex forms which occupies most of the molten pool. This secondary vortex is present at all subsequent higher power values (i.e. qa = 2.05 x 10~W/m and above). Using an analytical approach with an enhanced liquid thermal conductivity, the melt depth and width are calculated for all the cases and are found to be in good agreement with the experiments and also with
BASU et al.:
MOLTEN POOL CAUSED BY ELECTRON BEAM HEATING
733
the predictions by the numerical model. T h e reasons for this agreement are explored a n d are n o t e d to be o n a c c o u n t of the presence o f the large secondary vortex in the liquid pool which ensures a higher net heat t r a n s f e r a l o n g the line o f symmetry.
19. H. S. Carlslaw and J. C. Jaeger, Conduction o f Heat in Solids, p. 270. Oxford Univ. Press (1978). 20. W. M. Stecn, Laser Surface Cladding, in Principles of Solidification and Materials Processing (edited by R. Trivedi, J. A. Sekhar and J. Mazumdar), Vol. 1, pp. 163-178. Oxford and IBH, New Delhi (1989).
REFERENCES
APPENDIX
1. T. R. Anthony and H. E. Cline, J. appl. Phys. 48, 3888 (1977). 2. S. C. Hsu, S. Chakraborty and R. Mehrabian, Metall. Trans. 9B, 221 (1978). 3. S. C. Hsu, S. Kou and R. Mehrabian, Metall. Trans. liB, 24 (1980). 4. S. Kou, S. C. Hsu and R. Mehrabian, MetalL Trans. 12B, 33 (1981). 5. J. A. Sekhar, S. Kou and R. Mehrabian, Metall. Trans. 14A, 1169 (1984). 6. R. Mehrabian, S. Kou, S. C. Hsu and A. Munitz, in Laser-Solid Interactions and Laser Processing (edited by S. D. Ferris, H. J. Leamy and J. M. Poate). Mater. Res. Soc., Boston, Mass. (1979). 7. J. A. Sekhar, R. Mehrabian and H. Fraser, in Lasers in Metallurgy (edited by K. Mukherjee and J. Mazumdar), pp. 207-219. Metall. Soc. A.I.M.E., Philadelphia, Pa (1981). 8. C. Chart, J. Mazumdar and M. Chen, Metall. Trans. 15A, 2175 (1984). 9. B. Basu and J. Srinivasan, Int. J. Heat Mass Transf 31, 2331 (1988). 10. C. Chart, J. Mazumdar and M. Cben, Fifth Int. Congr. on Lasers and Electro-optics, San Fransisco, Calif. (1985). I 1. S. Kou and Y. H. Wang, Metall. Trans. 17A, 2265 (1986). 12. M. Rappaz, B. Carrupt, M. Zimmerman and W. Kurz, Helvetica Acta 60, 924 (1987). 13. K. V. Rama Rao and J. A. Sekhar, Acta metall. 3, 81 (1981); see also J. A. Sekhar, P h . D . thesis, Univ. of Illinois, Urbana-Champaign (1982). 14. J. Srinivasan and B. Basu, Int. J. Heat Mass Transf 29, 563 (1986). 15. S. Kou and D. K. Sun, Metall. Trans. 16A, 203 (1985). 16. S. V. Patankar, Numerical Heat Transfer and Fluid Flow. McGraw Hill, New York (1980). 17. V. R. Voller, M. Cross and N. C. Markatose, Int. J. Num. Meth. Engng 24, 271 (1987). 18. A. Zebib, G. M. Honist and E. Meiburg, Phys Fluids 28, 3467 (1985).
Nomenclature a = the spot radius of the electron beam Bf = boundary heating factor qaC~ k* AH,, = specific heat, J/kg K elliptic integral of the second kind H = enthalpy, J/kg AH~= latent heat of fusion, J/kg = thermal conductivity, W/ink K = elliptic integral of the first kind Ma = Marangoni number =
URa/a
p = pressure, N/m 2 q ffi input heat flux from the electron beam, W/m 2 r = radial space co-ordinate, m rm~ = width of the liquid pool R¢ = surface tension Reynolds number =
URa/v
Ste = Stefan number = Ch(T m -- T,)/AH., T temperature, K Tm= melting point, K TO= initial temperature, K UR = the reference surface tension velocity, m/s U = the scanning speed, m/s v, = radial velocity vz = axial velocity x, y = space co-ordinates z = axial space co-ordinates Greek symbols ~, = thermal diffusivity, m2/s dynamic viscosity, Nslm 2 V kinematic viscosity, m2/s p = density, kg/m 2 0 = nondimensional temperature = nondimensional enthalpy a = surface tension, N/m Subscripts 1 = liquid s = solid sat = saturated state = ambient Superscripts * = dimensional value
0956-7151/91 $3.00 + 0.00 Copyright © 1991 Pergamon Press pie
ANALYSIS OF THE STEADY STATE MOLTEN POOL OBTAINED BY HEATING A SUBSTRATE WITH A N ELECTRON BEAM B. B A S U I, J. A. S E K H A R 2, R. J. S C H A E F E R 3 and R. M E H R A B I A N 4 ~Tata Research Design Development Centre, Pune 411001, India, 2Department of Materials Science and Engineering, University of Cincinnati, Cincinnati, OH 45221-0012, 3National Institute of Standards and Technology, Galthersberg, Md and 4Carnegie Mellon University, Pittsburgh, PA 15213, U.S.A. (Received 28 April 1990; in revised form 25 September 1990) Abstract--Surface melting and solidification with high powered beams can be used for enhancing surface properties. The dimensions of the molten zone define the extent of the modified properties and are critical parameters which must be predicted during process design. The flow field in the molten pool has been reported to be one of the key factors which controls the dimensions of the surface layer. However, the calculation of this is only possible through complex numerical schemes and there is a need to look for simple analytical expressions which may be adequate. One approach for this search involves the precise determination of the steady state stationary profiles and then developing a method for extending these values to include the effect of beam motion for predicting the pool dimensions during processing. In this paper, a study of the flow field and its effect on the depth and width of the steady state pool is presented, based on numerical and analytical methods. To validate the predictions, an experimental study is carried out using surface melting of AI-4.5 wt%Cu alloy with an electron beam. The pool shapes are presented through optical micrographs and the depth and width of the pool is measured from these micrographs. The experiments are then simulated using a numerical model which includes fluid flow. The flow field is analyzed using streamline plots and the predicted pool shapes are compared with the micrographs. Further, the results are compared to an analytical method based on pure conduction and the pool depth and width are predicted when the liquid thermal conductivity is modified. The numerical and analytical predictions of the pool depth and width are found to be in good agreement with the experimental measurements (obtained from steady state stationary pools and from dimensions inferred on extrapolating moving beam measurements to zero velocity). The reasons for the success of the analytical model is discussed with reference to the two-dimensional flow fields and vortices predicted by the numerical model. Rtsumt--Pour amdliorer les proprittts de surface, on pent utiliser la fusion et la solidification superficielle ~i raide de faisceaux d'61ectrons de haute puissance. Les dimensions de la zone fondue dtfinissent l'ttendue des proprittts modifites et sont des paramttres critiques qui doivent 6tre prtvus pendant le processus d'61aboration. Le champ d'tcoulement dans le liquide est considtr6 comme un des facteurs cits qui contr61ent les dimensions de la couche superficielle. Cependant, cette estimation n'est possible que par des calculs num&iques complexes et il est ntcessaire de chercher des expressions analytiques simples qui peuvent convenir. L'une des approches de cette recherche, c'est la dttermination prtcise des profils en rtgime permanent stationnaire suivie du dtveloppement d'une mtthode pour adapter ces valenrs afin d'inclure l'effet du mouvement du faisceau pour prtvoir les dimensions de la zone liquide pendant le traitement. Dans cet article, on prtsente une 6tude du champ d'tcoulement et son effet sur la profondeur et la largeur de la zone fondue en rtgime permanent; cette 6tude est baste sur des mtthodes numtriques et analytiques. Pour valider les prtvisions, une 6tude exptrimentales est effectute en utilisant la fusion superficielle de l'alliage A1-4,5% en poids Cu par un faisceau d'61eetrons. Les formes des zones fondue sont prtsenttes sur des micrographies optiques et leur profondeur et leur largeur sont mesurtes ;i partir de ces micrographies. Des exptriences sont ensuite simultes en utilisant un modtle num6rique qui inclut l'tcoulement de fluide. Le champ d'tcoulement est analyst en utilisant des courbes de profil et les formes de zones fondues prtvues sont compartes aux micrographies. De plus, les rtsultats sont comparts ~ une mtthode analytique baste sur la conduction pure et la profondeur et la largeur de la zone fondue sont prtvues lorsque la conductivit6 thermique liquide est modifite. Les prtvisions numtriques et analytiques de la profondeur et de rtpaisseur de la zone fondue sont en bon accord avec les mesures exptrimentales (obtenues ~i partir de zones fondues en rtgime permanent stationnaire et pour des dimensions dtduites, par extrapolation ~i la vitesse nulle, de mesures de dtplacement du falsceau). Les raisons du succts du modtle analytique sont discuttes en faisant rtftrence aux champs d'&:oulement bidimensionnels et aux tourbillons pr6vus par le moddle num&ique. Zupmmenfauug---Oberfl~chenschmelzen und -erstarren durch hochenergetisehe Strahlen kann zur Verbesserung der Obert~cheneigenschaften ausgenutzt werden. Die Malk des geschrnolzenen lkreiches bestimmen das Ausmal3 der g~nderten Figenschaften und sind dither kritische Parameter, diue w~hrend der Prozelkntwicklung vorhergesagt werden miissen, lkrichtet wurde, dab das Flietfeld in der Schmelze einer der wichtigsten Faktoren bei der KontroUe der Dimension der Oberfiichenschicht ist. Allerdings ist die entsprechende lkrecfinung nur mit komplexen numerischen Anstitzen m6glich; man sollte also nach anwendbaren einfacheren analytischen Ausd~cken suchen. Fine M6giichkeit hiefffir besteht in der genauen Bestimmung der station~ren Profile, diese mit einer zu entwickelnden Methode zu erweitern und AM 39/~-A
725
726
BASU et al.: MOLTEN POOL CAUSED BY ELECTRON BEAM HEATING so den EinfluB der Strahlbewegung auf die Vorhersage der Abmessungen des Schmelzbereiches zu ermitteln. In dieser Arbeit wird das FlieBfeld und sein EinfluB auf Tiefe und Breite der Schmelze im station/iren Fall auf der Grundlage numerischer und analytischer Methoden bestimmt. Zur Prfifung werden Experimente zum Aufschmelzen der Oberflfiche der Legierung AI-4,5 Gew.-%Cu wit einem Elektronenstrahl durchgeffihrt. Die Forme der Schmelzbereiche wird mit optischen Aufnahmen dargestellt, Tiefe und Breite werden an den Aufnahmen ausgemessen. Dann werden die Experimente mit einem numerischen Modell unter EinschluB yon FlieBen der Schmelze simuliert. Das FlieBfeld wird mit Str6mungsdiagrammen analysiert, die Form der Schmelzbereiche wird mit den Aufnahmen verglichen. Au~rdern werden die Ergebnisse mit einer analytischen Methode verglichen, die auf reiner Leitung aufbaut; Tiefe und Breite des Schmelzbereicheswerden vorausgesagt ffir .~nderungen in der W~irmeleitung der Schmelze. Die numerischen und analytischen Voraussagen stirnmen mit dem experimentellen Ergebnissen (erhalten aus stationfiren Schmelzbcreichenund aus Dimensionen, die sich bci Extrapolation der Strahlgeschwindigkeit zu Null ergeben) gut fiberein. Die Grfinde ffir den Erfolg des analytischen Modells werden im Hinblick auf die zweidimensionalen FlieBfelder und die Wirbel, die yore numerischen Modell vorausgesagt werden, diskutiert.
1. INTRODUCTION Electron and laser beams are often used for surface processing involving melting and resolidification. On account of the small time of heating and subsequent rapid solidification after melting, surfaces can be modified to display enhanced properties, such as better resistance to the wear and corrosion. A very shallow heat affected zone is another advantage that high heat intensity beam processing may offer. The accurate prediction of the melt depth and width during electron and laser beam surface melting and solidification is therefore important, for the selection of correct process parameters. Anthony and Cline [1] first proposed a closed form analytical solution with a finite area source to calculate melt depths based on a pure conduction model. Phase change was not considered. There are now various publications available based on similar conduction numerical models [2-6] which relax many of the assumptions made in the earlier model [1]. While the first three models [2-4] include phase change in a pure metal in one, two and three-dimensions, respectively, Sekhar et al. [5] proposed a two-dimensional model including alloy solidification. In an earlier paper, Sekhar et al. [7] compared the predicted and experimental pool shapes seen during electron beam melting. The experimental results deviated from the measured value. A higher depth and smaller width prediction was made by the conduction model than was observed. The deviation was attributed to the effect of the fluid flow present in the molten pool. It has now been conclusively established that the fluid flow always occurs and plays an important role in electron and laser beam processing [7-9]. Chan et al. [8] first proposed a comprehensive numerical model for laser melting including the effect of flow field. They however did not compare the predicted and measured melt depth and width. Their study mainly focused on the fundamental analysis of the flow field and its effect on the heat transfer mechanism. Subsequently however Chan et al. [10] compared the predicted and measured values and reported adequate agreement. Kou and Wang [11] and Rappaz [12] have also reported good to adequate
agreement between predicted and measured pool shapes based on a three-dimensional numerical model. Rappaz et al. [12] have further succesfully incorporated microstructure models into the calculations. For a good estimate of the melt depth and width during electron and laser beam melting, the fluid flow is therefore an important consideration. However, a three-dimensional fluid flow numerical calculation is complex and cumbersome and an analytical technique or an intelligent semi-analytical technique should be developed if possible. One approach for this search involves the precise determination of the steady state stationary profiles and then developing techniques for extrapolating these values to include the effect of beam motion. In this paper, the effect of the flow field on the steady state melt depth and width in an A1-4.5 wt% Cu alloy is first studied using a numerical model including fluid flow and then comparing the results with experimental measurements obtained after electron beam melting. Next, a previously suggested analytical procedure [13] is employed for the same melt depth and width prediction with a new modification; namely where the thermal conductivity of the liquid alloy is altered. Finally all the predictions are compared with the experimental measurements. We show below that the prediction of the steady state melt dimensions (i.e. the dimensions of the pool with no relative motion between the beam and the substrate) is calculated with relative ease and work is recommended to obtain proper extrapolations from this starting point of the melt dimensions with finite velocity of the beam. 2. THE EXPERIMENTAL PROCEDURE The experiments were conducted on a 10kW electron beam machine [13]. Al-4,5%Cu alloys were prepared by diluting pure aluminum with a master alloy of eutectic composition, AI-33%Cu. The homogeneity of the samples was checked to a high degree of accuracy by wet chemical and electron microprobe analysis. Experiments were performed in a steady state condition [7] by keeping the stationary beam on for longer times ( > 5s) and, also, in the quasi-steady
BASU et al.: MOLTEN POOL CAUSED BY ELECTRON BEAM HEATING state condition where the samples were traversed at various scanning velocities. Scanningspeeds up to 0.066 m/s were obtained using a linear translating assembly and a rotating assembly was used to attain a translational speed up to 0.66 m/s. Beam profile and spot size were measured by scanning the electron beam over an array of 4 mm thick tungsten wires placed approximately 2000#m apart. The induced current was recorded on an oscilloscope to indicate the position and dimensions of the beam. The same height was maintained for all the experiments to ensure the same beam profile. The beam was noted to be of a top hat shape. Further details of the experiment are recorded in Ref. [13].
2. I. Steady state conditions for stationary beam For the stationary beam experiments, the incident powers were chosen in such a way that the steady (a)
(b)
727
1.0 1.2 0.8
0.7 "~
~
0.5
0.8 "~ '~
0.4
0.6
0.2
I
I
I
I
I
1
I
0,4
8+
8+
8+
8+
8+
o
o
o
o
8 .
q
,~
~
~.
qo
Fig. 2. Dimensionless melt depth (z/a) and width (r/a) as a function of qa for the steady state molten pool. state melt profiles could be maintained for a long time [7]. Steady state conditions were obtained when the absorbed power on the specimen was exactly offset by the conduction of heat into the semi-infinite substrate. A limiting condition to steady state was obtained when the vaporization rate from the melt became significant in the 10 -5 torr vacuum chamber. For AI-4.5%Cu alloy, surface melting was observed at a minimum qa (q and a are the input heat flux and spot radius respectively) of the order of 1.05 x 105 W/m. The steady state profiles could be maintained up to qa of 2.5 × 105 W/m. Figure 1 shows the micrographs of typical steady state profiles for various qa's. Figure 2 shows a plot of dimensionless melt depth and width as a function of qa. The experimental conditions for this plot are given in Table 1.
Quasi-steady state conditions for moving beam
22
Experiments were also performed by moving the samples at different power levels and velocities. The following conditions were commonly employed during each experiment:
(c)
(a) The samples were polished with 600 grit paper to maintain the same absorptivity of the solid before melting starts, (b) experiments were performed at the same height where the spot diameter and spatial energy distribution were measured, (c) after each pass, the samples were allowed to cool to the room temperature before making any subsequent passes, (d) The samples were thick enough to permit the assumption of a semi-infinite substrate ( ~ 2 0 mm thick). The time required to reach the quasi-steady Table I. Depth and width of the steady state molten pool with a stationary electron beam of radius 0.5 mm Incident
Fig. 1. Micrographs of steady-state molten pool with an electron beam of 0.5 mm radius for (a) qa = 1.22.105 W/m, (b) qa = 1.98.105 W/m and (c) qa = 2.5- 105 W/re.
Melt depth
Melt width
flux q
qa
No.
(10s W/m 2)
(10 s W/m)
(z/a)
(r/a)
I 2 3 4 5
2.43 3.97 4.11 4.68 5.00
1.22 1.98 2.05 2.34 2.50
0.292 0.402 0.460 0.960 0.938
0.476 0.940 1.000 1.250 I,154
728
BASU et al.: MOLTEN POOL CAUSED BY ELECTRON BEAM HEATING
state varied with the speed and power of the beam and was largest for the highest power level with slowest speed. Figures 3 and 4 show the variation of pool dimensions with processing conditions and micrographs of typical pool shapes respectively.
(a)
...-
3. THE NUMERICAL MODEL Electron beam melting is a thermo-fluid dynamic process and is, therefore, governed by the laws of conservation of energy, momentum and mass. The physical definition of the electron beam melting problem based on which the numerical scheme is developed is shown in Fig. 5. The fluid flow in the molten pool is assumed to be driven by the surface tension gradient at the free surface [8]. The numerical model is based on the following assumptions: (1) There is symmetry in the angular direction and the problem is solved in the ( r - z ) plane only. (2) The surface of the melt is flat. (3) The buoyancy force is of a negligible order of magnitude due to the small dimension of the pool []41. (4) Although electro-magnetic force is present, it is also negligible compared to the surface tension gradient [15]. (5) The properties, except for the surface tension, are not the function of temperature. For the purpose of computation, the average properties are considered. (6) A critical Reynolds number for transition to turbulent flow for thermocapillary flow is not precisely defined in a cavity. Because of this and accumulated experimental evidence (on stationary
S ~e
=*
•
20.143r a m .
£o)
Fig. 4. Micrographs of the quasi-steady state molten pool with a moving electron beam of 0.5 mm radius and 0.012 m/s scanning speed (U) with (a) qa = 2.8" 105W/m and (b) qa = 4.2.105 W/m. steady state flows [13]), the flow is assumed to be laminar. The governing equations of energy, momentum and mass, when normalized in the following way dp = ( H * - H * t ) / A H a ;
O~
2.0 F
1.8 1,6 "~ 1.4 0 x
i
X/o Z/o Ext rapolotion
e---
L
~
e'.,
~ ~e
0.8 ~ - ~ _ ~ -
- Tm)/AH;
r = r*/a
A1--4.5% Cu Homogenized (I s • 7.44x 105m2/s units of qo: (W/m)
1.2 x . ' ~
-<
0 = C*p(T
v = v*/UR; k = k*/k*; ELectron beom
~ - ~ ~-e 2.80x 105 •
2"05X105 r(V r )
• I ~" t"r~"-.,....,~ L2.05"~:5 ~ ....
°21 0
2"34 x 10'~ 1.22x 105
I
I
I
0,1
0.2
0.3
Ua/2a s
Fig 3 Variation of the dimensionlesspool depth (z/a) and w i d t h (x / a ) w i t h dimensionless scanning speed ( Ua /2= ) and heat flux per unit length qa W/M. The extrapolation of the moving beam results to obtain the stationary steady-state values is also shown.
zcv~l Fig. 5, The geometrical representation of electron beam melting used for the numerical model.
BASU et al.: MOLTEN POOL CAUSED BY ELECTRON BEAM HEATING Cp ~
729
A pproximo t.eO solid / Liquid interface
Cp/Cp~ * *
R o e URa/v;
t
~.,
Ma = URa /~t ;
IIIllllll
',7,',11111 ',~;;lll I
Ste = C*~(Tm - TD/AH~,
111111111 illlllllll
Bf = qaCr~/k* AH~I
IIIlillll IllJlllll
UR = (da /dT" AH~O/C*" # )
p =p*/pv~ take the following form for the problem of electron beam melting with the assumptions detailed above.
3.I. Energy equation div (v~) = (I/Ma) grad (k div 0).
(I)
3.2. r-Momentum equation
Fig. 6. The non-uniform grid distribution and the approximation of the solid-liquid interface.
div (v - v,) = - (@/6r) + (I/R,) div (grad v,) -- v,/r 2.
(2)
3.3. z-Momentum equation
div (v - v:) = - ( @ / 6 z ) + ( I / R , ) div (grad v:). (3) 3.4. Continuity equation
div (v) = O.
(4)
The corresponding boundary conditions are as follows: at r = 0, 60/6r = 6v:/6r = v, = 0; 0 < z < oo
(5)
at r = oo, 0 = - Ste; 0 < z < oo
(6)
a t z =O, k 6 0 / 6 z = - B f ; 0 < r < 1 =0;
l
0
4.1. Experimental
(8)
It should be noted that the energy equation is written in the enthalpy form to include the latent heat absorption during melting using the enthalpytemperature relationship. The normalized enthalpytemperature relationship is as follows. In the solid phase, 0 =¢;
~ <0.0
(9)
In a cell containing the liquid and solid (interface ceil), 0=0;
0<~<1
(10)
In the liquid, 0 =(4, -
l)C~/C~;
4. RESULTS
(7)
6vr/hz = -- 60/6r, v: = O; 0 < r < rmax
atz=~,0=-Ste;
turn and the continuity equations. The flow field is calculated in the liquid domain only by tracking the solid/liquid interface by the "Switch Off" method I17]. The algebraic equations resulting from the finite difference equations are solved by the Gauss-Seidel iterative method with (1764) grids of non-uniform distribution used for the calculation. Very fine grids are used below the beam and coarse grids are used away from the beam. This distribution of grids allows for the incorporation of fine grid spacings at the surface melt pool boundary [18]. A representative grid distribution is shown in Fig. 6 with the approximated interface.
4, > i . 0 .
(ll)
The governing equations (1)-(4), along with the boundary conditions [equations (5)-(8)], and the enthalpy-temperature relationship [equations (9)-(11)] define the complete problem of electron beam melting. These equations are solved by the finite difference method using a control volume formulation [16]. The SIMPLE algorithm [16] is used to solve the momen-
Figure 3 shows the change of the dimensionless melt width and depth with the dimensionless speed (Ua/2rt) for various values of qa. The depth and width for a particular power level and various speeds are extrapolated to zero speed, i.e. stationary beam. The extrapolation was carded out based on the shapes of adjacent curves. Because of this, the extrapolated values are expected to be within ,-,20% of the true value. From Table 2 it can be seen that these experimental values closely match with the experimental values of the stationary beam shown in Fig. 2. At higher qa values and low velocities (i.e. qa = 6.4 x 105W/m and U =0.1 m/s) vaporization occurs resulting in deep penetration and, hence, steady state cannot be attained. Figure 4 shows the Table 2. Depth and width of the steady state molten pool with a stationaryelectronbeamof radius 0.5 mm obtained by extrapolating
quasi-steadymoving beam results
Incident
Melt
Melt
(I05 W/m)
depth (z/a)
width (r/a)
1.56 2.81
0.371 1.050
4.20
--
0.817 1.200 1.450
flux q
qa
NO.
(lOS W/m 2)
l 2
3.12 5.62
3
8.40
730
BASU et al.: MOLTENPOOL CAUSED BY ELECTRON BEAM HEATING (C)
q = 5.0 xl0 e W/m 2 A I , - Cu
AL-Cu
f
0.7
No. Volue
1.2
i
104
i
5=0.50
I
6= 1,00 7 = 1.75 8= 2.50
Q905
(d)
q = 5 . 6 x 108 W / m 2 AL-Cu
(b}
1.3
q • 3 , 9 7 x 10 a W / m z AL-Cu
/S
!
1.o
i
4
;
I= 1 . 0 0
I
2=o.~'5
0.455
3=0.25 4=0.15 5=OAO 6:0.55
1 500 2 = 2.50 ]=1..,.)0 4=0.50 5=0.50
I j
6 = 1.00 7=1.75
1.005
8=2,75
Fig. 7. Streamline and pool shape of the steady-state molten pool with a stationary electron beam of 0.5 m m radius for different qa's (a) q ffi 2.43. I08 W / m 2, (b) q = 3.97-10s W / m 2, (c) q = 5.0. l 0 s W / m 2, (d) q = 5 . 6 . 1 0 s W / m 2. The values of the parameters given along the z and y axis refer to dimensionless
melt depth and width, respectively. The other numbers given along with the streamlines refer to the dimensionless stream function.
micrographs for different qa and scan speeds for the moving beam quasi-steady state conditions. 4.2. Simulation
The experiments are numerically simulated to predict the pool width and depth. The flow field for various power levels is also studied using streamline contours. The property values, along with the model parameters (i.e. the dimensionless numbers as specified before) used for the simulation are presented in Table 3. The results in Table 3 are given as a function of Bf although in the subsequent figures the results are presented against the parameter qa in preference to the boundary heating factor Bf. Although all our modelling work has been performed
Table 3. Property values of AI-4.5%Cu and process parameters q (108 W/m 2) Br 2.43 3.12 3.97 4.10 5.00 5.60
2.815 3.615 4.599 4.750 5.793 6.490
AHe = 3.95 × 10s J/kg; k? = 100.8W/mK; k* = 180.6W/InK;
Tm = 821.0 K;
p = 2700.0 kg/m3; C~ = 882 J/kgK;
C~ = 924 J/kg K; ~ = 10-3 kg/ms; de/dT = - 0.35 x 10 -3 kg/sC. Process parameters: a =0.5ram; M a r 1851.0; 1~ = 20196.0
731
BASU et al.: MOLTEN POOL CAUSED BY ELECTRON BEAM HEATING with this factor and the boundary conditions have been similarly written in this form e.g. equation (7), we have been conscious that regardless of the generality of the modelling and analytical results the detailed experimental work has been carried out by us only on one alloy. Thus multiplying qa by C~/k*AH~ only scales the results without adding any real generality to the comparison of experimental and analytical results. Consequently and in order to remain consistent with the previous results in Refs [1-11] we have preferred to present the data with qa instead of Bf. In addition, it is important to recognize that qa has been used in the literature in preference to Bf because it represents a pure processing parameter (i.e. it does not contain material constants). Figures 7 (a-d) show the streamlines in the steady state molten pool for various qa's: 1.215 × 105, 1.985 x 105, 2.5 x l05 and 2.8 × 105 W/m. With the increasing values of qa, it can be seen that there is a dramatic change in the nature of the flow pattern. The flow field for qa = 1.215 x 105 W/m consists of a single cell which will be called the primary vortex [Fig. 7(a)]. At qa = 1.985 × 105 W/m, the flow field shows existence of another vortex at the bottom of the primary one. This will be referred to as the secondary vortex [Fig. 7(b)]. For the subsequent higher qa's, the secondary cells occupy almost the whole molten pool region with a small primary cell at the top. The primary vortex enhances the heat transfer radially outwards at the top and reduces it near the line of symmetry where the flow is upwards (see Fig. 7). Hence, the primary vortex produces a pool of larger width and smaller depth (i.e. a shallow pool) in the presence of strong convection. Due to the low Prandtl number of the A1-Cu alloy (Pr = 0.01 which means thermal diffusion is 100 times more than the viscous diffusion), the contribution of the fluid flow to the net heat transfer is small. As a result, the primary vortex fails to produce a shallow pool and a deep pool results due to the thermal diffusion. Once a deep pool forms, there are two vortices due to the inability of the fluid particles with the high momentum (i.e. at the top of the pool) to take a sharp turn near the solid-liquid interface and follow a path
2.0 1.8 1.6 1.4 1.2 "~I.0N
o
o
0.8
mericoL 0.6
/
o
- -
/~a 0.4
o Exherirnent.oL
8 o / P"
J
0.2
II 1
0
I 2
Theoretical
A 1 - 4 5 % Cu Homogenized
I I 3 4 qo xl0"~ Wm -1
I 5
L 6
I 7
Fig. 9. Comparison of the numerically and theoretically predicted steady state melt depths with the experimental measurements. along the interface. This pattern is similar to the flow field in a cavity-like surface tension driven flow in rectangular cavities [14]. For a lower qa, the flow field consists of a single vortex due to the small size of the pool. The secondary vortex exists for the higher qa [Fig. 7(a)]. i.e. larger molten pool, and its strength increases with qa [Fig. 7(b), (c) and (d)]. 5. ANALYTICAL CALCULATION Consider a cartisian coordinate system fixed to the center of the heat source, with the y direction being the beam travelling direction and x and z directions 2.0 1.8 1.6 1.4 1.2
Et::mt r o n
~
y
1.0 0.8
f r
0.6
o
Experlmento t
--
TheoreUcot
A 1 - 4 . 5 % Cu
Homogenized
0.4 0.2
0
I
I
t
I
I
I
t
I
2
3
4
5
6
7
8
9
q o x 10-5 Wm "1
Fig. 8. The coordinates for the analytical approach.
Fig. I0. Comparison of the numerically and theoretically predicted steady state melt widths with the experimental measurements.
732
BASU et al.: MOLTEN POOL CAUSED BY ELECTRON BEAM HEATING
being the radial and axial distances along the substrate, respectively (see Fig. 8). Carslaw and Jaeger [19] have presented analytical expressions to determine the time dependent temperature profiles for a stationary point source. Sekhar [13] modified these equations to include the effect of an area source and also to account for the latent heat absorption due to the phase change. At long times (i.e. steady state) these are given as:
k*[T(0, z)
- To + (AH~/C~)]
qa = (z/a)[{1 + ( a 2 / z 2 ) } Ill - 1} (12)
k*[T(r, O) - To + (AH,,/C~, )] qa = (2/n) E(r/a)
for (r/a) < 1
= (2/n)[E(a/r) -- {1 - (a2/r2)}K(a/r)
for (r/a) > 1.
(13)
In equations (12) and (13), K and E are the elliptic integrals of the first and second kind, respectively. The latent heat absorption has been accounted for by an equivalent sensible heat absorption using a pseudotemperature, i.e. CpT = A H ~ . The melt depth and width can, now, be determined from equations (12) and (13), respectively. Substituting T(0, z) by Tm in equation (12), the depth is calculated by solving equation (12). Similarly, the melt width can be determined from equation (13). Sekhar [13] has provided the solution of equation (12) and (13) in graphical form and it is used for the present calculation. For the prediction of the depth and width, a higher liquid thermal conductivity is used in equations (12) and (13) apart from the Cm and AH~ as given in Table 3. The average of the solid and liquid thermal conductivities is used for the calculation, i.e. k* for calculation = (k? + k*)/2 = (100.8 + 180.6)/2 = 140.7 W/mK. The selection of higher thermal conductivity for both depth and width prediction is validated by the nature of the flow field as discussed below. 5.1. Comparison o f the pool depth and width
Figures 9 and 10 show the experimentally observed melt depth and width along with that predicted by the numerical model and the calculation procedure described above. The values predicted by the calculation procedure are referred to as theoretical values in the figures, and in the following discussion. It can be seen that the theoretical predictions are in very good agreement with the numerical as well as the experimental ones (see Figs 9 and 10). As discussed in the previous section, the theoretical prediction is based on a pure conduction calculation with an increased thermal conductivity of the liquid AI-4.5%Cu, i.e. 140.7 W/mK instead of 100.8W/inK. It has been recorded before [8-10] that it may not be adequate to include the effect of
recirculating fluid flow simply with an enhanced thermal conductivity as attempted by Hsu et al. [1]. The reason for the correct analytical predictions even with a pure conduction solution (with a higher liquid thermal conductivity) lies in the nature of the flow pattern as illustrated by the numerical model. It can be seen from Fig. 7 that the flow is radially outward at the top of the pool due to the primary vortex and is axially downward near the line of symmetry due to the presence of a large secondary vortex. This effect of fluid flow, namely of enhancing heat transfer in both directions is, thus, precisely incorporated through a high liquid thermal conductivity. As a result, the theoretical predictions using an enhanced thermal conductivity match so well with the experimental as well as the numerical ones. It should be noted here that this calculation procedure may fail to produce realistic depth and width for the cases where the primary vortex dominates the liquid pool with a small and weak secondary vortex at the bottom; this kind of flow pattern exists for example in a molten pool of steel [8] because of the higher Prandtl number ( = 0.078). For such cases, the theoretical prediction of the depth using higher thermal conductivity would be higher than observed, on account of the reduction of net heat transfer along the line of symmetry resulting from the strong upward flow of the primary vortex. The possible differences between e.b and laser [20] in terms of depth of penetration of the beam has evidently not affected the solution. The model assumes that heat is deposited at the surface whereas the experiments are with an electron beam. The reason for this lack in difference is possibly because the depths of the melts that we are considering are much larger than the effective e.b penetration (typically in the order of 10/am) where heat is released. In addition as pointed out in Ref. [15] previous results have indicated that e.m.f, forces are much smaller than the forces due to surface temperature gradients.
6. CONCLUSIONS Several different methods for predicting the stationary beam melt dimensions are explored in this paper. Using a numerical model, the pool shape and flow field are predicted for several experiments. The flow fields show a dramatic change from low to high power levels of the beam. The molten pool consists of a primary vortex at low power levels of the beam. With an increase in the power level, a secondary vortex forms which occupies most of the molten pool. This secondary vortex is present at all subsequent higher power values (i.e. qa = 2.05 x 10~W/m and above). Using an analytical approach with an enhanced liquid thermal conductivity, the melt depth and width are calculated for all the cases and are found to be in good agreement with the experiments and also with
BASU et al.:
MOLTEN POOL CAUSED BY ELECTRON BEAM HEATING
733
the predictions by the numerical model. T h e reasons for this agreement are explored a n d are n o t e d to be o n a c c o u n t of the presence o f the large secondary vortex in the liquid pool which ensures a higher net heat t r a n s f e r a l o n g the line o f symmetry.
19. H. S. Carlslaw and J. C. Jaeger, Conduction o f Heat in Solids, p. 270. Oxford Univ. Press (1978). 20. W. M. Stecn, Laser Surface Cladding, in Principles of Solidification and Materials Processing (edited by R. Trivedi, J. A. Sekhar and J. Mazumdar), Vol. 1, pp. 163-178. Oxford and IBH, New Delhi (1989).
REFERENCES
APPENDIX
1. T. R. Anthony and H. E. Cline, J. appl. Phys. 48, 3888 (1977). 2. S. C. Hsu, S. Chakraborty and R. Mehrabian, Metall. Trans. 9B, 221 (1978). 3. S. C. Hsu, S. Kou and R. Mehrabian, Metall. Trans. liB, 24 (1980). 4. S. Kou, S. C. Hsu and R. Mehrabian, MetalL Trans. 12B, 33 (1981). 5. J. A. Sekhar, S. Kou and R. Mehrabian, Metall. Trans. 14A, 1169 (1984). 6. R. Mehrabian, S. Kou, S. C. Hsu and A. Munitz, in Laser-Solid Interactions and Laser Processing (edited by S. D. Ferris, H. J. Leamy and J. M. Poate). Mater. Res. Soc., Boston, Mass. (1979). 7. J. A. Sekhar, R. Mehrabian and H. Fraser, in Lasers in Metallurgy (edited by K. Mukherjee and J. Mazumdar), pp. 207-219. Metall. Soc. A.I.M.E., Philadelphia, Pa (1981). 8. C. Chart, J. Mazumdar and M. Chen, Metall. Trans. 15A, 2175 (1984). 9. B. Basu and J. Srinivasan, Int. J. Heat Mass Transf 31, 2331 (1988). 10. C. Chart, J. Mazumdar and M. Cben, Fifth Int. Congr. on Lasers and Electro-optics, San Fransisco, Calif. (1985). I 1. S. Kou and Y. H. Wang, Metall. Trans. 17A, 2265 (1986). 12. M. Rappaz, B. Carrupt, M. Zimmerman and W. Kurz, Helvetica Acta 60, 924 (1987). 13. K. V. Rama Rao and J. A. Sekhar, Acta metall. 3, 81 (1981); see also J. A. Sekhar, P h . D . thesis, Univ. of Illinois, Urbana-Champaign (1982). 14. J. Srinivasan and B. Basu, Int. J. Heat Mass Transf 29, 563 (1986). 15. S. Kou and D. K. Sun, Metall. Trans. 16A, 203 (1985). 16. S. V. Patankar, Numerical Heat Transfer and Fluid Flow. McGraw Hill, New York (1980). 17. V. R. Voller, M. Cross and N. C. Markatose, Int. J. Num. Meth. Engng 24, 271 (1987). 18. A. Zebib, G. M. Honist and E. Meiburg, Phys Fluids 28, 3467 (1985).
Nomenclature a = the spot radius of the electron beam Bf = boundary heating factor qaC~ k* AH,, = specific heat, J/kg K elliptic integral of the second kind H = enthalpy, J/kg AH~= latent heat of fusion, J/kg = thermal conductivity, W/ink K = elliptic integral of the first kind Ma = Marangoni number =
URa/a
p = pressure, N/m 2 q ffi input heat flux from the electron beam, W/m 2 r = radial space co-ordinate, m rm~ = width of the liquid pool R¢ = surface tension Reynolds number =
URa/v
Ste = Stefan number = Ch(T m -- T,)/AH., T temperature, K Tm= melting point, K TO= initial temperature, K UR = the reference surface tension velocity, m/s U = the scanning speed, m/s v, = radial velocity vz = axial velocity x, y = space co-ordinates z = axial space co-ordinates Greek symbols ~, = thermal diffusivity, m2/s dynamic viscosity, Nslm 2 V kinematic viscosity, m2/s p = density, kg/m 2 0 = nondimensional temperature = nondimensional enthalpy a = surface tension, N/m Subscripts 1 = liquid s = solid sat = saturated state = ambient Superscripts * = dimensional value