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Surface solidification with a moving heat source: A study of solidification parameters
Acra meraN. Vol. 35, No. 1, pp. 81-87, 1987 Printed in Great Britain. All rights reserved
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SURFACE SOLIDIFICATION WITH A MOVING SOURCE: A STUDY OF SOLIDIFICATION PARAMETERS
OOOI-6160/87 $3.00 + 0.00 1987 Pergamon Journals Ltd
HEAT
K. V. RAMA RAO and J. A. SEKHAR Defence Metallurgical Research Laboratory, P.O. Kanchanbagh, Hyderabad-500 258, India (Received
30 January
1986; in revised form
16 April 1986)
Abstract-The heat flow and solidification parameters, such as the temperature gradient and solid-liquid interface velocity, have been studied for the case of a moving heat source on an aluminium substrate. The problem was developed and solved for both the large initial transient and the final quasi steady state. Results indicate that exceptionally high temperature gradients may exist in the liquid pool. During the initial transient, the temperature gradient at the solid-liquid interface drops from a high value to a low one at a near constant interface velocity. These results are expected to aid in the design of solidification experiments. The predictive capacity of the model was tested by measuring the secondary dendrite arm spacing (SDAS) along the region of the initial transient on the surface of a AlAS% Cu alloy laser treated with a 5 kW laser. Results obtained showed the SDAS increasing to a steady state value only after the melt width had achieved a steady state, thus validating the theoretical conclusions of the model. RCsum~Nous avons itudit les parametres d’ecoulement de chaleur et de solidification (tels que le gradient de temperature et la vitesse de l’interface solide-liquide) dans le cas d’une source de chaleur mobile sur un substrat d’aluminium. Nous avons developpe et resolu le probleme a la fois pour I’etat transitoire initial et l’etat final quasi-stationnaire. Les resultats indiquent que des gradients de temperature exceptionnellement eleves peuvent exister dans la partie liquide. Pendant l’etat transitoire initial, le gradient de temperature a l’interface solid*liquide passe dune valeur &levee a une valeur faible pour une vitesse d’interface a peu pres constante. Nous esperons que ces resultats aideront a concevoir des experiences de solidification. Nous avons test6 l’aptitude a prediction de ce modile en mesurant l’espacement secondaire des branches de dendrites dans le domaine d’etat transitoire initial a la surface d’un alliage A14,5% Cu trait& au laser sous une puissance de 5 kW. Les rtsultats ont montre que cet espacement secondaire augmente jusqu’a une valeur de regime permanent, seulement lorsque la largeur du bain a atteint un &tat stationnaire, ce qui valide les conclusions theoriques du modele.
Zusammenfassung-Die Parameter von WIrmefluB und Erstarrung, wie Temperaturgradient und Geschwindigkeit der fest-fliissigen Grenzfliche, wurden fiir den Fall einer sich iiber ein Aluminium-Substrat bewegenden Wiirmequelle untersucht. Das Problem wurde formuliert und gel&t sowohol fiir die grol3e anfingliche Transiente als such den quasi-stationlren Endzustand. Die Ergebnisse zeigen, dal3 in der Schmelze auBergew6hnlich hohe Temperaturgradienten auftreten konnen. Wahrend der anfanghchen Transienten fillt der Temperaturgradient an der fest-fliissigen Grenztllche von einem hohen auf einen niedrigen Wert bei einer fast konstanten Grenzfllchengeschwindigkeit. Es wird erwartet, dafi diese Ergebnisse die Planung von Erstarrungsexperimenten erleichtern. Die Aussagekraft dieses Modelles wurde mit einer Messung des Abstandes sekundarer Dendritenarme im Bereich der anfanglichen Transienten an der Oberfliiche einer Al4,5% Cu-Legierung, die mit einem Laser (Leistung 5 kW) behandelt worden war, gepriift. Der Armabstand stieg auf einen stationaren Wert nur an, nachdem die Schmelzweite einen stationlren Zustand angenommen hatte, womit die theoretischen Folgerungen des Modells bestltigt werden. 1. INTRODUCTION
solidification rate than what is possible by the quasi stationary continuous moving beam situation [9-131. In this paper we examine the cooling rates and temperature gradient at the solid-liquid interface in a liquid pool (in a metallic substrate) under a high heat intensity source. The substrate is moving in the negative y direction with a velocity (U). We examine both the initial transient and the final quasi steady state. The quasi steady state is first examined as it is dealt with more in the literature. Results obtained show a similarity between studying the heat flow at different velocities at steady state or at different times during the initial transient. The solutions not only will aid in characterizing solidification microstructure but will also aid in studying the surface fluid flow during the melting and resolidification process [6].
Several recent publications have dealt with the heat flow characterization during the passage of, or the stationary switching of a point or an area source on a substrate [l-17]. Several of these calculations have been verified by experimental observations, on alloy substrates [lo, 121. Even though the potential of these intense heat sources has been recognized for causing rapid solidifications and even forming metallic glass, none of the calculations have been aimed at describing clearly the temperature gradients at the solid-liquid interface or the interface velocities. There has emerged, however, a clear understanding that for the same depths of penetration the pulsed stationary source can give rise to greater cooling and 81
82
RAMA
RAO and SEKHAR:
2. MATHEMATICAL
SURFACE
SOLIDIFICATION
FORMULATION
We consider a high intensity beam (top hat profile) of radius a moving on the bounding surface of a semi-infinite solid with a constant velocity U in the y direction as shown in Fig. 1. The unsteady heat conduction equation referred to any orthogonal coordinate system (u,, u2, uJ is written as
WITH A MOVING
HEAT SOURCE
Where Au,, Au,, Au, and At are the step sizes in the three co-ordinate directions and in the time direction respectively. Now following [5,9], the dimensionless nodal enthalpy $ and the dimensionless temperature 0 are defined by:
s PW
-
*=$y
(H - H*)
H*) d?’
AH Sl
=
AH,,
0 = C,(T - T,)/AH, t,band 0 satisfy the following relations. In the solid region $ < 0 and equal to 0 i.e. $ = 0 = C,(T - T,)/AH,, (5) and in the super heated region -p
(1)
$ = 0 + 1 = CJT - T,,,)/AH, + 1. Finally in the solid-liquid
Where the last term on the r.h.s. accounts for the conduction of heat due to motion of the beam. We take the co-ordinates (u, , u,, u3) as the oblate spheroidal co-ordinates as this is the most natural coordinate system for our problem. Now following [S] the finite-difference analogue of (1) is written as
(6)
interface, we have
O<$&l.O
and
0=0.
(7)
In view of these relations, the above finite difference equation (2) is modified to (C~+D,)~~j,k=c,0~-,,j,k+C20~,,j,k +C3@$-,,k+Cd0?j+,,k
Cs Tfl;;~ = C, V- ,,j.k + C2 r+ ,,j,k
+
CSO?j.k-,
+
ch@?j,k+
I
+C3~j-,,k+C4T~j+,.k+CST~j,k-, -C7$~,I,j,k+C8~?i-,j,k -
(C,/CJ
f
c6?j,k+,
+
(CJCJ
+
(C,o/CJ
f
(C12/cp) K..,k+I
fC+
,,j, k -
&+
-DI (&,k -
I,k
ff?
I, j. k
(CdCp)
H?j
-
(C,,/Cp>
H?j,k-
COS’
c’9
ti;-,,k
+
c,,$;+,,k
-
c,,
$$,k-I
+
c,2ti;,k+,
I
H?‘j>‘)
Pa2 +(cosh2 U,i -
- I, k
x (cosh2u,, - sin2u2j)
(2)
UzJ
+ c~ flij,k
(8)
4
where C,,* = Au;’ T (2Au,)-’ tanh uli C3,d= k,k;’
6~;~ T (2Au,)-’ coth uzj
C,,6 = k,k;’
Au;2(cosh2 ulj
(3)
- sin2 u2j)/cosh2 u,~sin* uu & = plJuC,(2Auik,)-’ sinh uli sin uzj sin u3k
P = rate of heat generated/unit
C,,,,, = pUaC,(2Au2k,)-, cash Ulicos Cu.,2
=
uy
volume
uq cash u2j sin Au2 C, = k, cash u,,Au,Au, ‘G
sin ujk
for q = Constant.
pfJaCp(2Au3W-’
(cosh2 #,j - sin2 u2i>cos u,k cash u,, sin uu C,=2
WhereJJ,k is the weight fraction of the element in the liquid state. This is the general difference equation which is valid at all points of the substrate irrespective of the fact whether the point is in the solid or liquid or in the interface. The source term can be conveniently incorporated as a volume heat source [3,5, 11, 171.Here an appropriate value for the source term P is to be substituted, depending on the type of beam used. In general P is given by
Au;2+k2k,Au;2 [ + k, k ; ’ (cosh2 uli - sin’ uZj) cosh2 uli sin2 uy
1
D, = pa2(cosh2 uli - sin2 +)/k, At.
(4)
(9)
For non-constant q, e.g. Gaussian beam profile, the equation is easily modified as shown in Refs [3,5, 111. Now, to solve (8) appropriate boundary conditions are to be prescribed. When time t = 0, we assume the substrate is at a given temperature T, and when t = 00 the temperature reaches a steady state value. The remaining boundary conditions at any time t are exactly similar as given by Kou et al. [5] and thus
RAMA RAO and SEKHAR:
83
SURFACE SOLIDIFICATION WITH A MOVING HEAT SOURCE
their discussion is omitted here. As previously assumed [S] we have considered no surface heat losses.
3. COMPUTATIONAL PROCEDURE The system of equation (8) with the relations (S)-(7) are solved using an iterative procedure, for the given boundary conditions. Initially at time t = 0, the whole substrate is assumed to be at a given temperature T,, . Then using the boundary conditions and the relations (Z+-(7), equation (8) is solved giving new value for $ at the nodal point (i,j, k, WI). Then depending on whether the nodal point is in solid, solid-liquid interface or super heated region, the corresponding value of 0 is calculated. This calculation is continued for the whole substrate which we have assumed to be semi-infinite. Once one set of values for $ and 0 are obtained the procedure is continued for the same time level until the convergence I$%(New)
- Jl$Jold)l
< 10v4
is satisfied. Using the values of $ and 0 at the mth level, the values at the m + 1th time level are obtained exactly in a similar way. The computations are continued until a steady state is reached. Here we have taken the condition to be
Calculations were carried out for a number of parameters and the results are presented in the next section. Thermophysical constants for aluminium and nickel are given in Table Al. Constant averaged thermophysical properties have been assumed.
4. RESULTS (a) Steady and quasi-steady state Referred to a Cartesian coordinate system fixed to the centre of the heat source, they direction is parallel to the movement of the heat source, the z direction is into the substrate and the x direction is the third orthogonal axis as shown in Fig. 1. The general equation for the temperature distribution in a semiinfinite solid can be manipulated [2] to give the steady state radial temperature distribution T(r, 0) (where r could be x or y in the stationery case) as a function of the distance r/a. For the case involving a change of phase this was given as [18].
Fig. 1. Schematic of a high heat intensity source (such as a laser beam) moving at a constant velocity “U” on a substrate. The axis of a co-ordinate system moving with the beam have been labelled. where K and E are the complete elliptic integrals of the first and second kind. Appendix A gives the distribution along the z-axis. In equation (10) thermophysical properties of the solid and liquid are assumed to be equal and independent of temperature. The equation was tested for aluminium and nickel substrates (thermophysical values are given in Table Al). Plots of equation (10) and equation (Al) (given in Appendix A) are shown in Figs 2 and 3. These curves permit the determination of steady state temperature distribution in the liquid zone on the surface and along the r and z axis of any given substrate material as a function of qu. For example, minimum values of qu = 2.3 x IO5W m-’ and qu = 1.4 x lo5 W m-I are deduced from either of the figures for the initiation of melting at the centre of the circular region for aluminium nickel substrates respectively. Similarly, the corresponding maximum melt depths at steady state for the two substrates occurs at z/a = 1.3 and z/u = 0.6 (i.e. when the temperature of the centre of the circular region is steady at its boiling point).
k(T(r, 0) - T0 + AH,&) qa =i =
E(r/a)
z [
x
if
r/a < 1.0
E(a/r) - (1 - a*/r*) K(u/r)
1
if
0.0 L -1.6
I
I
-1.2
-0.8
I -0.4
1
I
I
0
04
08
, 1.2
FRACTIONAL DISTANCE, r/a r/a>1 (10)
Fig. 2. The general steady state temperature distribution on the melted source of a substrate heated by a stationary beam 1181.
1.6
84
RAMA RAO and SEKHAR:
SURFACE SOLIDIFICATION WITH A MOVING HEAT SOURCE
SE I
I
La
x
,
o0
I
0.4
FRACTIONAL
0
2.0
1.6
1.2
DISTANCE,
z/a Fig. 3. The general steady state temperature distribution along the z-axis (depth axis) of a substrate heated by a stationary beam [18]. Table 1. Direction of change in the depth and width of the steady state melt with increase in material constants Material constants
k
WI
TO
c, Melt depth or width
1
1
r
TM
1
7 = Increases; 1 = Decreases.
Table 1 gives the direction of change of melt depth and width with changes in the material parameters at steady state with a moving heat source. The trends remain the same when a moving source quasi steady state is established. However the exact value of the depth and width would now be obtained by multiplying the 1.h.s. of equation (10) by some function of velocity f(2a/Uu) < 1.0 and f(2cc/Uu) decreasing with increasing U. This will account for the additional rate of heat removal on increasing the velocity. 2.8 [
I
2c IOQ
2.4
Y 3 T
2.0
Y 2
2.6
a l
I= L 9 r, = t U x
1.2
0.8
0.4
-2b
-1.6
-0.8
0
0.8
1.0 -
0
I
I
0.11
2.0 -
a \,
16
2L
FRACTIONAL DISTANCE, y/a Fig. 4. Quasi-steady state dimensionless temperature distribution along the y-axis of the moving heat flux showing trends for different substrate materials [18].
1
2
3
4
5
6
(-y/a] Fig. 5. Dimensionless temperature gradient along the (-y/a) direction at z = 0. The respective values of qa and Ua/Za are 6.4 x IO5W/m and 0.9.
Figure 4 plots the dimensionless temperature multiplied by such a function. This plot is for a nickel substrate. Also shown are values for aluminium from Refs [3, 51. In this figure the r/a of Fig. 3 is substituted by y/a as now the solution is no longer symmetrical about the z axis. From Fig. 2, the temperature gradient (slope of the curve) is seen to start at a value of zero at r/a = 0 and progressively increase to a maximum value at r/a = 1, and then decrease again to zero at large values of r/u. This trend is again seen in Fig. 4. Two important observations may now be made. (i) If the interface position is greater than r/a = 1, then there is a peak in the temperature gradient ahead of it. (ii) As the average cooling rate during solidification at the interface is given as G,*. U, then for any steady state value of U the maximum cooling rate should occur when the interface is at the point r/a = 1.0 (as CL*is maximum at this point). Any change in the processing parameter qa which shifts the interface to a position greater than r/a = 1.0, will actually decrease the average cooling rate during solidification, i.e. increasing qa can increase or decrease the cooling rate depending on where the interface is located with respect to the r/a = 1.0 position. The dimensionless temperature gradient G, = aT*/d(y/a) along the y direction has been calfor culated increasing Ua/2u at a qu = 6.4 x lo5 Wm-‘. A typical result for Uu/2u = 0.9 is plotted in Fig. 5. Significantly as the velocity is increased, the peak in the dimensionless temperature gradient also increases. However the real value of the peak gradient corrected for velocity remains unchanged. There is a more interesting peak in the gradient again at the solid-liquid interface and a corresponding minimum in the gradient between the two peaks. This minimum occurs only when the interface is far away from the r/a = 1 and is necessitated physically by the requirement of a concave temperature profile at the interface. A concave tem-
RAMA RAO and SEKHAR:
SURFACE
SOLIDIFICATION
85
WITH A MOVING HEAT SOURCE
0.5
0.b
s
0.3
d d
rI-
0.2
0.1
0
0
100
50
-1
-2
zoo
150
Fig. 6. Dimensioni~s temperature T*(O, 0,O) as a function of dimensionless time F,. The qa and Ua/2ct values respectively are 6.4 x lo5 W/m and 0.3.
perature profile is indicated when the temperature gradient in the liquid increases as the solid-liquid interface is approached from the centre of the beam. The profile is considered on the x-y plane at the z = 0 position. (b) Initial transienl
The initial transient in the mdt presents a more varied combination of temperature gradient and solidification velocity and is now discussed. Figure 6 shows that the T(0, 0,O)temperature reaches its near
1
I
L
50
100
I
KO
Fig. 8. D~ensionless temperature dist~bution along the y axis during the initial transient.
steady state value in dimensionless time given by F, 3 10. The interface position along the x, z and y axis (shown in Fig. 7) however, takes dimensionless time of N 10, 10 and w 20 respectively to achieve near steady state. At a velocity of 0.1 m s-’ and a beam radius of 5 x IOe4m for an aluminium substrate a distance of 5.9 mm is covered in a dimensionless time of F, = 20.0 before steady state is achieved. This is a barge distance and several solidification parameters can be studied before the quasi state is achieved. Several features such as the temperature gradient and interface velocity prior to achieving steady state are discussed below.
I 200
0 1
Ff.l Fig. 7. Dimensionless interface position along the x, y, and z axis as a function of dimensionl~ time F,. The values of qu and Ua/Za are 6.4 x 10s W/m and 0.3. Table 2. Solidification
6 0.335 3.35 33.5 167.5
2
( y/al
FO
a
1
0
Time (9 w’ 10-Z 10-l 0.5
qa=6.4xiO~Wm~‘;U=O.lms-‘.
(-y/a) Fig. 9. Dimensionless tem~rature gradient aiong the (-y/a) direction at z = 0 position.
parameters
during
the initial
transient
G:
Peak G:
Vf
(K me’)
(K me-‘)
(m s-l)
5.4 I.29 5 3.5
x x x x
Id IO6 105 105
3
2
8.3 2.7 2.6 2.7
Y x x x
106 10’ 106 lob
-3 -6 -5
x ia-” x 10. ’ x 10-e 0
v’=(u+ (m s-‘f 0.07 0.1 0. I 0.1
v*)
86
RAMA RAO and SEKHAR:
SURFACE SOLIDIFICATION
The dimensionless temperature for different Fourier numbers along the y axis is shown in Fig. 8. It must be noted that this is a similar curve to that shown in Fig. 4, where the ordinate axis was scaled by multiplication with a velocity term. The ordinate axis in Fig. 8 is additionally scaled with the time term. The inflexion in the peak in temperature gradient is observed even for low Fourier numbers and is plotted in Fig. 9. A concluding table (Table 2) is provided to highlight the significance of studying the heat flow and solidification during the initial transient prior to the quasi steady state being established. The table is given for qa = 6.4 x 10' W m-’ and a Us/2a = 0.3. Initially the velocity of the solid liquid interface (V’) is zero but, very quickly achieves the source velocity. However the gradient ahead of the interface relaxes much more slowly. This allows for a one experiment study [23] of interface destabilization and is a unique situation where the velocity of the interface is fixed but the temperature gradients may be varied. (c) Experimental
uerz$caCion
An available multimode 5 kW Co, laser was employed to verify the trends predicted by the model discussed above. A 1 mm beam diameter was chosen using a 190 mm focal length lens. A 1 kW beam power focused onto a 1 mm spot diameter gives a qa of 5 x 10’ W m-l. Experiments were now conducted on a commercial 2024 alloy (mainly 4.5 wt% Cu in aluminium). Beam absorptivity was improved by coating the alloy substrate with a 10 pm black paint layer applied with a marker pen. The experiment consisted of initially letting the sample move at a velocity of 0.1 m s-l under the beam and then suddenly switching on the beam. This condition simulates the theoretical experiment discussed above. After the laser pass, the top surface of the trail was slightly polished and etched with Kellers reagent. Two measurements were made (i) the steady state trail width =2x, and (ii) the variation of the dendrite arm spacing along the y direction. It was observed that the trail width had reached a steady state after covering a distance of 1 mm from the position at which the interface could first be located. The model discussed above had predicted a steady state value of and qa =6.4 x lO’Wm_’ for a x/a = 1.5 Us/2a = 0.3. The results of the experiment at qa = _5x 10sW m-l (a = 0.6 x 10m3m) and Us/2a = 0.3 show that a steady state value of x/a = 1.0 was qa = beam power With 2 kW reached. 10 x lo5 W m-’ and the steady state value of x/a was measured to be 1.5 when Us/2a = 0.3. Any discrepancy between the predicted and measured value is thought to arise on account of low absorptivity of the beam by the aluminium substrate [ll]. Table 2 shows that the cooling rate Gz. V should decrease by an order of magnitude along the Y/,=0,,=0 position when steady state is approached.
WITH A MOVING HEAT SOURCE
We note that the initial velocity of the solid-liquid interface must start at zero. This will then reach a steady state value quickly while the temperature gradient at the solid-liquid interface GE relaxes much more slowly. Following Fig. 4.18 in Ref. [25], this must reflect in initially an aligned dendritic structure forming, which tends to become equiaxed dendritic as the velocity increases to steady state and then as the gradient relaxes the cooling rate, Gt V must fall and thereby an increase in the size of equiaxed dendrites is expected. This is exactly seen in our experiment with qa = 5 x 1O’W m-’ and Us/2a = 0.3. Initially an aligned cellular/dendritic region is seen with about 1.5 pm spacing. This further becomes equiaxed dendritic with a SDAS value of 1 + 0.5 pm. The SDAS value subsequently increases to 3 + 0.5 pm in about 1.5 mm along the y direction. This change in magnitude when compared to the data of Bower et al. [24] reflects a one order of magnitude change in the average cooling rate during solidification. We have not attempted any further confirmation of the model as the numerical values of thermophysical constants used in the calculation may not adequately represent the actual values for the 2024 alloy. 5. CONCLUSION We have studied the heat flow, melting and solidification of a surface during the passage of a constant heat source over it and have focused on (1) the large initial transient, (2) the temperature gradients and cooling rates in the liquid pool and (3) the implications of our findings on solidification experimentation. We conclude that (i) the results of the study of the temperature distribution during the initial transient at different times is similar to that obtained on studying the temperature profiles during steady state but with varying beam velocities, (ii) the cooling rate Gt.U may increase or decrease with increasing qa for the same velocity depending on the initial position of the interface, (iii) the Gt during the initial transient decrease at near constant interface velocity while approaching steady state conditions and (iv) that exceptionally high temperature gradients may occur in the liquid. Experiments conducted to verify the theoretical predictions of the model indicated that all the predicted order of magnitude cooling rate changes during the initial transient are true. Acknowledgement-The authors thank Dr P. Rama Rae, Director DMRL for his encouragement and permission to publish this paper. REFERENCES 1. D. Rosenthal, Weld. J. 20, 220s (1941). 2. H. S. Carslaw and J. C. Jaeger (Eds), Conduction of Hear in Solids, 2nd edition, p. 270. Oxford University
Press (1978). 3. S. C. Hsu, S. Kou and R. Mehrabian, Metall. Trans. llB, 30 (1980). 4. M. I. Cohen, J. Franklin Znsf. 283, No. 4, 271 (1967).
RAMA RAO and SEKHAR:
SURFACE SOLIDIFICATION
5. S. Kou, S. C. Hsu and R. Mehrabian,
Metafi.
Trans.
lZB, 33 (1981).
6. H. E. Chne and T. R. Anthony, J. appi. Pkys. 48,4906 r1977‘). I. k. L.‘Nissim, A. Litoila, R. B. Good and J. F. Gibbons, J. appl. Phys., 5, 274 (1974). 8. R. Guenot and J. Racinet, Br. Weld. 1. 14, 427 (1967). 9. J. A. Sekhar, S. Kou and R. Mehrabian, Metail. Trans. l4A, 1169 (1983). 10. J. A. Sekhar, R. Mehrabian and H. L. Fraser, Proceedings of AIME Conference on Lasers in Metallurgy;
p. 201(1982). II. J. A. Sekhar and R. Mehrabian. MetaN. Trans. 12B. 411 (1981).
12. J. M. Brown and J. A. Sekhar, Metali. Trans. HiA, 29 (1984).
13. J. A. Sekhar, Proceedings of the Symposium on Amorphous Materials, p. 19. Bhabha Atomic Research Centre (1983). 14. J. M. Hode and J. P. Joly, J. DC Whys. U-343 (1983). IS. A. Grill and J. Pelleg, Metall. Trans. llB, 257 (1980). 16, S. Kou, ~efalI. Trans. lZA, 2025 (1981). I?. S. Kou, Metali. Trans. 13A, 363 (1982). 18. J. A. Sekhar, Ph.D. Thesis, University of Illinois (1982). 19. G. E. Schnieder, A. B. Strong and M. M. Yovanovich,
Gt h H AH,, k P 4 t T T*
r,
TM 4, u2 9 u3
iJ V
V V
V*
Proc. International Symposium on Computer Methods for Partial Differential Equation (edited by R. Vich-
nevetsley), pp, 312-317. AICA, New Brunswick, N.J. (1975). 20. N. Shamsunder and E. M. Sparrow, J. Z-fearTransf 233 (1975).
a
0
21. D. R. Athey, J. Inst. Math. Appl. 13, 333 (1974). 22. W. W. Mullins and R. F. Sekerka, J. appl. Phys. 134, 323 (1963). 23. W. W. Mullins and R. F. Sekerka, .I. appl. Phys. 35,444 (1964). 24. T. F. Bower, H. D. Brody and M. C. Flemings, Trans. A.l.M.E. 236 (1966). 25. W. Kurz and D. J. Fisher, Fundamentals of Solidification, p. 91. Trans. Tech, Switzerland (1984).
APPENDIX a
X*Y,Z
radius of the circular region (m) of the top hat beam profile specific heat, J.kg-‘K-i fraction liquid in any discretized volume in the finite difference scheme (kt/pC,a*) Fourier number non dimensional temperature gradient in the liquid at the solid-liquid interface
II/ P i,.i, k, m
WITH A MOVING HEAT SOURCE
87
temperature gradient in the liquid at any position ahead of soIid-liquid interface, Km-’ scale factor specific enthalpy, J kg-’ heat of fusion, J kg-’ thermal conductivity J m-i s-’ K-’ rate of heat generation per unit volume, Wmm3 absorbed heat flux, W mm2 time, s temperature, K (=k (T - To + AH&) U/2 aq) non dimensional temperature ambient temperature, K melting tem~~ture, K oblate spheroidal co-ordinates velocity of heat source, ms-’ velocity of solid-liquid interface. V’ = U at steady state with respect to an observer fixed to the solid volume, m3 velocity m s-’ velocity of solid-liquid interface for an observer fixed to the heat source (V* = 0 at steady state) half width of resolidified trail on the X-J’ plane. cartesian co-ordinates thermal diffusivity (k fpC,), m* S- ’ dimensionless temperature dimensionless enthalpy variable density kg mm3 nodal point subscripts in u,, u,, u, and t directions respectively.
APPENDIX A The steady state temperature distribution along the z axis is given as [ 181:
Table Al.
Thenumerical values of thermophysical Constants used in this study. Source reference [IS) Al
Density (kg rn-‘) Melting temperature (K) Boiling temperature (K) Thermal conductivity Specific heat (J kg-’ K-‘) Latent heat of fusion (J La-‘)
2545 933 2123 228 1067 3.95 x 105
Ni
8400 1728 3005 74 606 2.99 x IO’
Copyright
0
SURFACE SOLIDIFICATION WITH A MOVING SOURCE: A STUDY OF SOLIDIFICATION PARAMETERS
OOOI-6160/87 $3.00 + 0.00 1987 Pergamon Journals Ltd
HEAT
K. V. RAMA RAO and J. A. SEKHAR Defence Metallurgical Research Laboratory, P.O. Kanchanbagh, Hyderabad-500 258, India (Received
30 January
1986; in revised form
16 April 1986)
Abstract-The heat flow and solidification parameters, such as the temperature gradient and solid-liquid interface velocity, have been studied for the case of a moving heat source on an aluminium substrate. The problem was developed and solved for both the large initial transient and the final quasi steady state. Results indicate that exceptionally high temperature gradients may exist in the liquid pool. During the initial transient, the temperature gradient at the solid-liquid interface drops from a high value to a low one at a near constant interface velocity. These results are expected to aid in the design of solidification experiments. The predictive capacity of the model was tested by measuring the secondary dendrite arm spacing (SDAS) along the region of the initial transient on the surface of a AlAS% Cu alloy laser treated with a 5 kW laser. Results obtained showed the SDAS increasing to a steady state value only after the melt width had achieved a steady state, thus validating the theoretical conclusions of the model. RCsum~Nous avons itudit les parametres d’ecoulement de chaleur et de solidification (tels que le gradient de temperature et la vitesse de l’interface solide-liquide) dans le cas d’une source de chaleur mobile sur un substrat d’aluminium. Nous avons developpe et resolu le probleme a la fois pour I’etat transitoire initial et l’etat final quasi-stationnaire. Les resultats indiquent que des gradients de temperature exceptionnellement eleves peuvent exister dans la partie liquide. Pendant l’etat transitoire initial, le gradient de temperature a l’interface solid*liquide passe dune valeur &levee a une valeur faible pour une vitesse d’interface a peu pres constante. Nous esperons que ces resultats aideront a concevoir des experiences de solidification. Nous avons test6 l’aptitude a prediction de ce modile en mesurant l’espacement secondaire des branches de dendrites dans le domaine d’etat transitoire initial a la surface d’un alliage A14,5% Cu trait& au laser sous une puissance de 5 kW. Les rtsultats ont montre que cet espacement secondaire augmente jusqu’a une valeur de regime permanent, seulement lorsque la largeur du bain a atteint un &tat stationnaire, ce qui valide les conclusions theoriques du modele.
Zusammenfassung-Die Parameter von WIrmefluB und Erstarrung, wie Temperaturgradient und Geschwindigkeit der fest-fliissigen Grenzfliche, wurden fiir den Fall einer sich iiber ein Aluminium-Substrat bewegenden Wiirmequelle untersucht. Das Problem wurde formuliert und gel&t sowohol fiir die grol3e anfingliche Transiente als such den quasi-stationlren Endzustand. Die Ergebnisse zeigen, dal3 in der Schmelze auBergew6hnlich hohe Temperaturgradienten auftreten konnen. Wahrend der anfanghchen Transienten fillt der Temperaturgradient an der fest-fliissigen Grenztllche von einem hohen auf einen niedrigen Wert bei einer fast konstanten Grenzfllchengeschwindigkeit. Es wird erwartet, dafi diese Ergebnisse die Planung von Erstarrungsexperimenten erleichtern. Die Aussagekraft dieses Modelles wurde mit einer Messung des Abstandes sekundarer Dendritenarme im Bereich der anfanglichen Transienten an der Oberfliiche einer Al4,5% Cu-Legierung, die mit einem Laser (Leistung 5 kW) behandelt worden war, gepriift. Der Armabstand stieg auf einen stationaren Wert nur an, nachdem die Schmelzweite einen stationlren Zustand angenommen hatte, womit die theoretischen Folgerungen des Modells bestltigt werden. 1. INTRODUCTION
solidification rate than what is possible by the quasi stationary continuous moving beam situation [9-131. In this paper we examine the cooling rates and temperature gradient at the solid-liquid interface in a liquid pool (in a metallic substrate) under a high heat intensity source. The substrate is moving in the negative y direction with a velocity (U). We examine both the initial transient and the final quasi steady state. The quasi steady state is first examined as it is dealt with more in the literature. Results obtained show a similarity between studying the heat flow at different velocities at steady state or at different times during the initial transient. The solutions not only will aid in characterizing solidification microstructure but will also aid in studying the surface fluid flow during the melting and resolidification process [6].
Several recent publications have dealt with the heat flow characterization during the passage of, or the stationary switching of a point or an area source on a substrate [l-17]. Several of these calculations have been verified by experimental observations, on alloy substrates [lo, 121. Even though the potential of these intense heat sources has been recognized for causing rapid solidifications and even forming metallic glass, none of the calculations have been aimed at describing clearly the temperature gradients at the solid-liquid interface or the interface velocities. There has emerged, however, a clear understanding that for the same depths of penetration the pulsed stationary source can give rise to greater cooling and 81
82
RAMA
RAO and SEKHAR:
2. MATHEMATICAL
SURFACE
SOLIDIFICATION
FORMULATION
We consider a high intensity beam (top hat profile) of radius a moving on the bounding surface of a semi-infinite solid with a constant velocity U in the y direction as shown in Fig. 1. The unsteady heat conduction equation referred to any orthogonal coordinate system (u,, u2, uJ is written as
WITH A MOVING
HEAT SOURCE
Where Au,, Au,, Au, and At are the step sizes in the three co-ordinate directions and in the time direction respectively. Now following [5,9], the dimensionless nodal enthalpy $ and the dimensionless temperature 0 are defined by:
s PW
-
*=$y
(H - H*)
H*) d?’
AH Sl
=
AH,,
0 = C,(T - T,)/AH, t,band 0 satisfy the following relations. In the solid region $ < 0 and equal to 0 i.e. $ = 0 = C,(T - T,)/AH,, (5) and in the super heated region -p
(1)
$ = 0 + 1 = CJT - T,,,)/AH, + 1. Finally in the solid-liquid
Where the last term on the r.h.s. accounts for the conduction of heat due to motion of the beam. We take the co-ordinates (u, , u,, u3) as the oblate spheroidal co-ordinates as this is the most natural coordinate system for our problem. Now following [S] the finite-difference analogue of (1) is written as
(6)
interface, we have
O<$&l.O
and
0=0.
(7)
In view of these relations, the above finite difference equation (2) is modified to (C~+D,)~~j,k=c,0~-,,j,k+C20~,,j,k +C3@$-,,k+Cd0?j+,,k
Cs Tfl;;~ = C, V- ,,j.k + C2 r+ ,,j,k
+
CSO?j.k-,
+
ch@?j,k+
I
+C3~j-,,k+C4T~j+,.k+CST~j,k-, -C7$~,I,j,k+C8~?i-,j,k -
(C,/CJ
f
c6?j,k+,
+
(CJCJ
+
(C,o/CJ
f
(C12/cp) K..,k+I
fC+
,,j, k -
&+
-DI (&,k -
I,k
ff?
I, j. k
(CdCp)
H?j
-
(C,,/Cp>
H?j,k-
COS’
c’9
ti;-,,k
+
c,,$;+,,k
-
c,,
$$,k-I
+
c,2ti;,k+,
I
H?‘j>‘)
Pa2 +(cosh2 U,i -
- I, k
x (cosh2u,, - sin2u2j)
(2)
UzJ
+ c~ flij,k
(8)
4
where C,,* = Au;’ T (2Au,)-’ tanh uli C3,d= k,k;’
6~;~ T (2Au,)-’ coth uzj
C,,6 = k,k;’
Au;2(cosh2 ulj
(3)
- sin2 u2j)/cosh2 u,~sin* uu & = plJuC,(2Auik,)-’ sinh uli sin uzj sin u3k
P = rate of heat generated/unit
C,,,,, = pUaC,(2Au2k,)-, cash Ulicos Cu.,2
=
uy
volume
uq cash u2j sin Au2 C, = k, cash u,,Au,Au, ‘G
sin ujk
for q = Constant.
pfJaCp(2Au3W-’
(cosh2 #,j - sin2 u2i>cos u,k cash u,, sin uu C,=2
WhereJJ,k is the weight fraction of the element in the liquid state. This is the general difference equation which is valid at all points of the substrate irrespective of the fact whether the point is in the solid or liquid or in the interface. The source term can be conveniently incorporated as a volume heat source [3,5, 11, 171.Here an appropriate value for the source term P is to be substituted, depending on the type of beam used. In general P is given by
Au;2+k2k,Au;2 [ + k, k ; ’ (cosh2 uli - sin’ uZj) cosh2 uli sin2 uy
1
D, = pa2(cosh2 uli - sin2 +)/k, At.
(4)
(9)
For non-constant q, e.g. Gaussian beam profile, the equation is easily modified as shown in Refs [3,5, 111. Now, to solve (8) appropriate boundary conditions are to be prescribed. When time t = 0, we assume the substrate is at a given temperature T, and when t = 00 the temperature reaches a steady state value. The remaining boundary conditions at any time t are exactly similar as given by Kou et al. [5] and thus
RAMA RAO and SEKHAR:
83
SURFACE SOLIDIFICATION WITH A MOVING HEAT SOURCE
their discussion is omitted here. As previously assumed [S] we have considered no surface heat losses.
3. COMPUTATIONAL PROCEDURE The system of equation (8) with the relations (S)-(7) are solved using an iterative procedure, for the given boundary conditions. Initially at time t = 0, the whole substrate is assumed to be at a given temperature T,, . Then using the boundary conditions and the relations (Z+-(7), equation (8) is solved giving new value for $ at the nodal point (i,j, k, WI). Then depending on whether the nodal point is in solid, solid-liquid interface or super heated region, the corresponding value of 0 is calculated. This calculation is continued for the whole substrate which we have assumed to be semi-infinite. Once one set of values for $ and 0 are obtained the procedure is continued for the same time level until the convergence I$%(New)
- Jl$Jold)l
< 10v4
is satisfied. Using the values of $ and 0 at the mth level, the values at the m + 1th time level are obtained exactly in a similar way. The computations are continued until a steady state is reached. Here we have taken the condition to be
Calculations were carried out for a number of parameters and the results are presented in the next section. Thermophysical constants for aluminium and nickel are given in Table Al. Constant averaged thermophysical properties have been assumed.
4. RESULTS (a) Steady and quasi-steady state Referred to a Cartesian coordinate system fixed to the centre of the heat source, they direction is parallel to the movement of the heat source, the z direction is into the substrate and the x direction is the third orthogonal axis as shown in Fig. 1. The general equation for the temperature distribution in a semiinfinite solid can be manipulated [2] to give the steady state radial temperature distribution T(r, 0) (where r could be x or y in the stationery case) as a function of the distance r/a. For the case involving a change of phase this was given as [18].
Fig. 1. Schematic of a high heat intensity source (such as a laser beam) moving at a constant velocity “U” on a substrate. The axis of a co-ordinate system moving with the beam have been labelled. where K and E are the complete elliptic integrals of the first and second kind. Appendix A gives the distribution along the z-axis. In equation (10) thermophysical properties of the solid and liquid are assumed to be equal and independent of temperature. The equation was tested for aluminium and nickel substrates (thermophysical values are given in Table Al). Plots of equation (10) and equation (Al) (given in Appendix A) are shown in Figs 2 and 3. These curves permit the determination of steady state temperature distribution in the liquid zone on the surface and along the r and z axis of any given substrate material as a function of qu. For example, minimum values of qu = 2.3 x IO5W m-’ and qu = 1.4 x lo5 W m-I are deduced from either of the figures for the initiation of melting at the centre of the circular region for aluminium nickel substrates respectively. Similarly, the corresponding maximum melt depths at steady state for the two substrates occurs at z/a = 1.3 and z/u = 0.6 (i.e. when the temperature of the centre of the circular region is steady at its boiling point).
k(T(r, 0) - T0 + AH,&) qa =i =
E(r/a)
z [
x
if
r/a < 1.0
E(a/r) - (1 - a*/r*) K(u/r)
1
if
0.0 L -1.6
I
I
-1.2
-0.8
I -0.4
1
I
I
0
04
08
, 1.2
FRACTIONAL DISTANCE, r/a r/a>1 (10)
Fig. 2. The general steady state temperature distribution on the melted source of a substrate heated by a stationary beam 1181.
1.6
84
RAMA RAO and SEKHAR:
SURFACE SOLIDIFICATION WITH A MOVING HEAT SOURCE
SE I
I
La
x
,
o0
I
0.4
FRACTIONAL
0
2.0
1.6
1.2
DISTANCE,
z/a Fig. 3. The general steady state temperature distribution along the z-axis (depth axis) of a substrate heated by a stationary beam [18]. Table 1. Direction of change in the depth and width of the steady state melt with increase in material constants Material constants
k
WI
TO
c, Melt depth or width
1
1
r
TM
1
7 = Increases; 1 = Decreases.
Table 1 gives the direction of change of melt depth and width with changes in the material parameters at steady state with a moving heat source. The trends remain the same when a moving source quasi steady state is established. However the exact value of the depth and width would now be obtained by multiplying the 1.h.s. of equation (10) by some function of velocity f(2a/Uu) < 1.0 and f(2cc/Uu) decreasing with increasing U. This will account for the additional rate of heat removal on increasing the velocity. 2.8 [
I
2c IOQ
2.4
Y 3 T
2.0
Y 2
2.6
a l
I= L 9 r, = t U x
1.2
0.8
0.4
-2b
-1.6
-0.8
0
0.8
1.0 -
0
I
I
0.11
2.0 -
a \,
16
2L
FRACTIONAL DISTANCE, y/a Fig. 4. Quasi-steady state dimensionless temperature distribution along the y-axis of the moving heat flux showing trends for different substrate materials [18].
1
2
3
4
5
6
(-y/a] Fig. 5. Dimensionless temperature gradient along the (-y/a) direction at z = 0. The respective values of qa and Ua/Za are 6.4 x IO5W/m and 0.9.
Figure 4 plots the dimensionless temperature multiplied by such a function. This plot is for a nickel substrate. Also shown are values for aluminium from Refs [3, 51. In this figure the r/a of Fig. 3 is substituted by y/a as now the solution is no longer symmetrical about the z axis. From Fig. 2, the temperature gradient (slope of the curve) is seen to start at a value of zero at r/a = 0 and progressively increase to a maximum value at r/a = 1, and then decrease again to zero at large values of r/u. This trend is again seen in Fig. 4. Two important observations may now be made. (i) If the interface position is greater than r/a = 1, then there is a peak in the temperature gradient ahead of it. (ii) As the average cooling rate during solidification at the interface is given as G,*. U, then for any steady state value of U the maximum cooling rate should occur when the interface is at the point r/a = 1.0 (as CL*is maximum at this point). Any change in the processing parameter qa which shifts the interface to a position greater than r/a = 1.0, will actually decrease the average cooling rate during solidification, i.e. increasing qa can increase or decrease the cooling rate depending on where the interface is located with respect to the r/a = 1.0 position. The dimensionless temperature gradient G, = aT*/d(y/a) along the y direction has been calfor culated increasing Ua/2u at a qu = 6.4 x lo5 Wm-‘. A typical result for Uu/2u = 0.9 is plotted in Fig. 5. Significantly as the velocity is increased, the peak in the dimensionless temperature gradient also increases. However the real value of the peak gradient corrected for velocity remains unchanged. There is a more interesting peak in the gradient again at the solid-liquid interface and a corresponding minimum in the gradient between the two peaks. This minimum occurs only when the interface is far away from the r/a = 1 and is necessitated physically by the requirement of a concave temperature profile at the interface. A concave tem-
RAMA RAO and SEKHAR:
SURFACE
SOLIDIFICATION
85
WITH A MOVING HEAT SOURCE
0.5
0.b
s
0.3
d d
rI-
0.2
0.1
0
0
100
50
-1
-2
zoo
150
Fig. 6. Dimensioni~s temperature T*(O, 0,O) as a function of dimensionless time F,. The qa and Ua/2ct values respectively are 6.4 x lo5 W/m and 0.3.
perature profile is indicated when the temperature gradient in the liquid increases as the solid-liquid interface is approached from the centre of the beam. The profile is considered on the x-y plane at the z = 0 position. (b) Initial transienl
The initial transient in the mdt presents a more varied combination of temperature gradient and solidification velocity and is now discussed. Figure 6 shows that the T(0, 0,O)temperature reaches its near
1
I
L
50
100
I
KO
Fig. 8. D~ensionless temperature dist~bution along the y axis during the initial transient.
steady state value in dimensionless time given by F, 3 10. The interface position along the x, z and y axis (shown in Fig. 7) however, takes dimensionless time of N 10, 10 and w 20 respectively to achieve near steady state. At a velocity of 0.1 m s-’ and a beam radius of 5 x IOe4m for an aluminium substrate a distance of 5.9 mm is covered in a dimensionless time of F, = 20.0 before steady state is achieved. This is a barge distance and several solidification parameters can be studied before the quasi state is achieved. Several features such as the temperature gradient and interface velocity prior to achieving steady state are discussed below.
I 200
0 1
Ff.l Fig. 7. Dimensionless interface position along the x, y, and z axis as a function of dimensionl~ time F,. The values of qu and Ua/Za are 6.4 x 10s W/m and 0.3. Table 2. Solidification
6 0.335 3.35 33.5 167.5
2
( y/al
FO
a
1
0
Time (9 w’ 10-Z 10-l 0.5
qa=6.4xiO~Wm~‘;U=O.lms-‘.
(-y/a) Fig. 9. Dimensionless tem~rature gradient aiong the (-y/a) direction at z = 0 position.
parameters
during
the initial
transient
G:
Peak G:
Vf
(K me’)
(K me-‘)
(m s-l)
5.4 I.29 5 3.5
x x x x
Id IO6 105 105
3
2
8.3 2.7 2.6 2.7
Y x x x
106 10’ 106 lob
-3 -6 -5
x ia-” x 10. ’ x 10-e 0
v’=(u+ (m s-‘f 0.07 0.1 0. I 0.1
v*)
86
RAMA RAO and SEKHAR:
SURFACE SOLIDIFICATION
The dimensionless temperature for different Fourier numbers along the y axis is shown in Fig. 8. It must be noted that this is a similar curve to that shown in Fig. 4, where the ordinate axis was scaled by multiplication with a velocity term. The ordinate axis in Fig. 8 is additionally scaled with the time term. The inflexion in the peak in temperature gradient is observed even for low Fourier numbers and is plotted in Fig. 9. A concluding table (Table 2) is provided to highlight the significance of studying the heat flow and solidification during the initial transient prior to the quasi steady state being established. The table is given for qa = 6.4 x 10' W m-’ and a Us/2a = 0.3. Initially the velocity of the solid liquid interface (V’) is zero but, very quickly achieves the source velocity. However the gradient ahead of the interface relaxes much more slowly. This allows for a one experiment study [23] of interface destabilization and is a unique situation where the velocity of the interface is fixed but the temperature gradients may be varied. (c) Experimental
uerz$caCion
An available multimode 5 kW Co, laser was employed to verify the trends predicted by the model discussed above. A 1 mm beam diameter was chosen using a 190 mm focal length lens. A 1 kW beam power focused onto a 1 mm spot diameter gives a qa of 5 x 10’ W m-l. Experiments were now conducted on a commercial 2024 alloy (mainly 4.5 wt% Cu in aluminium). Beam absorptivity was improved by coating the alloy substrate with a 10 pm black paint layer applied with a marker pen. The experiment consisted of initially letting the sample move at a velocity of 0.1 m s-l under the beam and then suddenly switching on the beam. This condition simulates the theoretical experiment discussed above. After the laser pass, the top surface of the trail was slightly polished and etched with Kellers reagent. Two measurements were made (i) the steady state trail width =2x, and (ii) the variation of the dendrite arm spacing along the y direction. It was observed that the trail width had reached a steady state after covering a distance of 1 mm from the position at which the interface could first be located. The model discussed above had predicted a steady state value of and qa =6.4 x lO’Wm_’ for a x/a = 1.5 Us/2a = 0.3. The results of the experiment at qa = _5x 10sW m-l (a = 0.6 x 10m3m) and Us/2a = 0.3 show that a steady state value of x/a = 1.0 was qa = beam power With 2 kW reached. 10 x lo5 W m-’ and the steady state value of x/a was measured to be 1.5 when Us/2a = 0.3. Any discrepancy between the predicted and measured value is thought to arise on account of low absorptivity of the beam by the aluminium substrate [ll]. Table 2 shows that the cooling rate Gz. V should decrease by an order of magnitude along the Y/,=0,,=0 position when steady state is approached.
WITH A MOVING HEAT SOURCE
We note that the initial velocity of the solid-liquid interface must start at zero. This will then reach a steady state value quickly while the temperature gradient at the solid-liquid interface GE relaxes much more slowly. Following Fig. 4.18 in Ref. [25], this must reflect in initially an aligned dendritic structure forming, which tends to become equiaxed dendritic as the velocity increases to steady state and then as the gradient relaxes the cooling rate, Gt V must fall and thereby an increase in the size of equiaxed dendrites is expected. This is exactly seen in our experiment with qa = 5 x 1O’W m-’ and Us/2a = 0.3. Initially an aligned cellular/dendritic region is seen with about 1.5 pm spacing. This further becomes equiaxed dendritic with a SDAS value of 1 + 0.5 pm. The SDAS value subsequently increases to 3 + 0.5 pm in about 1.5 mm along the y direction. This change in magnitude when compared to the data of Bower et al. [24] reflects a one order of magnitude change in the average cooling rate during solidification. We have not attempted any further confirmation of the model as the numerical values of thermophysical constants used in the calculation may not adequately represent the actual values for the 2024 alloy. 5. CONCLUSION We have studied the heat flow, melting and solidification of a surface during the passage of a constant heat source over it and have focused on (1) the large initial transient, (2) the temperature gradients and cooling rates in the liquid pool and (3) the implications of our findings on solidification experimentation. We conclude that (i) the results of the study of the temperature distribution during the initial transient at different times is similar to that obtained on studying the temperature profiles during steady state but with varying beam velocities, (ii) the cooling rate Gt.U may increase or decrease with increasing qa for the same velocity depending on the initial position of the interface, (iii) the Gt during the initial transient decrease at near constant interface velocity while approaching steady state conditions and (iv) that exceptionally high temperature gradients may occur in the liquid. Experiments conducted to verify the theoretical predictions of the model indicated that all the predicted order of magnitude cooling rate changes during the initial transient are true. Acknowledgement-The authors thank Dr P. Rama Rae, Director DMRL for his encouragement and permission to publish this paper. REFERENCES 1. D. Rosenthal, Weld. J. 20, 220s (1941). 2. H. S. Carslaw and J. C. Jaeger (Eds), Conduction of Hear in Solids, 2nd edition, p. 270. Oxford University
Press (1978). 3. S. C. Hsu, S. Kou and R. Mehrabian, Metall. Trans. llB, 30 (1980). 4. M. I. Cohen, J. Franklin Znsf. 283, No. 4, 271 (1967).
RAMA RAO and SEKHAR:
SURFACE SOLIDIFICATION
5. S. Kou, S. C. Hsu and R. Mehrabian,
Metafi.
Trans.
lZB, 33 (1981).
6. H. E. Chne and T. R. Anthony, J. appi. Pkys. 48,4906 r1977‘). I. k. L.‘Nissim, A. Litoila, R. B. Good and J. F. Gibbons, J. appl. Phys., 5, 274 (1974). 8. R. Guenot and J. Racinet, Br. Weld. 1. 14, 427 (1967). 9. J. A. Sekhar, S. Kou and R. Mehrabian, Metail. Trans. l4A, 1169 (1983). 10. J. A. Sekhar, R. Mehrabian and H. L. Fraser, Proceedings of AIME Conference on Lasers in Metallurgy;
p. 201(1982). II. J. A. Sekhar and R. Mehrabian. MetaN. Trans. 12B. 411 (1981).
12. J. M. Brown and J. A. Sekhar, Metali. Trans. HiA, 29 (1984).
13. J. A. Sekhar, Proceedings of the Symposium on Amorphous Materials, p. 19. Bhabha Atomic Research Centre (1983). 14. J. M. Hode and J. P. Joly, J. DC Whys. U-343 (1983). IS. A. Grill and J. Pelleg, Metall. Trans. llB, 257 (1980). 16, S. Kou, ~efalI. Trans. lZA, 2025 (1981). I?. S. Kou, Metali. Trans. 13A, 363 (1982). 18. J. A. Sekhar, Ph.D. Thesis, University of Illinois (1982). 19. G. E. Schnieder, A. B. Strong and M. M. Yovanovich,
Gt h H AH,, k P 4 t T T*
r,
TM 4, u2 9 u3
iJ V
V V
V*
Proc. International Symposium on Computer Methods for Partial Differential Equation (edited by R. Vich-
nevetsley), pp, 312-317. AICA, New Brunswick, N.J. (1975). 20. N. Shamsunder and E. M. Sparrow, J. Z-fearTransf 233 (1975).
a
0
21. D. R. Athey, J. Inst. Math. Appl. 13, 333 (1974). 22. W. W. Mullins and R. F. Sekerka, J. appl. Phys. 134, 323 (1963). 23. W. W. Mullins and R. F. Sekerka, .I. appl. Phys. 35,444 (1964). 24. T. F. Bower, H. D. Brody and M. C. Flemings, Trans. A.l.M.E. 236 (1966). 25. W. Kurz and D. J. Fisher, Fundamentals of Solidification, p. 91. Trans. Tech, Switzerland (1984).
APPENDIX a
X*Y,Z
radius of the circular region (m) of the top hat beam profile specific heat, J.kg-‘K-i fraction liquid in any discretized volume in the finite difference scheme (kt/pC,a*) Fourier number non dimensional temperature gradient in the liquid at the solid-liquid interface
II/ P i,.i, k, m
WITH A MOVING HEAT SOURCE
87
temperature gradient in the liquid at any position ahead of soIid-liquid interface, Km-’ scale factor specific enthalpy, J kg-’ heat of fusion, J kg-’ thermal conductivity J m-i s-’ K-’ rate of heat generation per unit volume, Wmm3 absorbed heat flux, W mm2 time, s temperature, K (=k (T - To + AH&) U/2 aq) non dimensional temperature ambient temperature, K melting tem~~ture, K oblate spheroidal co-ordinates velocity of heat source, ms-’ velocity of solid-liquid interface. V’ = U at steady state with respect to an observer fixed to the solid volume, m3 velocity m s-’ velocity of solid-liquid interface for an observer fixed to the heat source (V* = 0 at steady state) half width of resolidified trail on the X-J’ plane. cartesian co-ordinates thermal diffusivity (k fpC,), m* S- ’ dimensionless temperature dimensionless enthalpy variable density kg mm3 nodal point subscripts in u,, u,, u, and t directions respectively.
APPENDIX A The steady state temperature distribution along the z axis is given as [ 181:
Table Al.
Thenumerical values of thermophysical Constants used in this study. Source reference [IS) Al
Density (kg rn-‘) Melting temperature (K) Boiling temperature (K) Thermal conductivity Specific heat (J kg-’ K-‘) Latent heat of fusion (J La-‘)
2545 933 2123 228 1067 3.95 x 105
Ni
8400 1728 3005 74 606 2.99 x IO’