Copyright © 2011 Praise Worthy Prize S.r.l. - All rights reserved International Review of Mechanical Engineering, Vol. 5, N. 2 Special Issue on Heat Transfer
260 temperatures of the both sides of the interface as well as
of the products of the heat conductivity and the temperature gradients,
ii areas with a Newton type heat contact namely the product of the thermal transfer coefficient and the
difference between the temperatures of the ingot outer wall and of the inner crucible wall,
iii areas in which the radiation losses predominate. The magnitudes of these areas are not known and
probably change with the time. Due to this the effective coefficient of the heat
transfer α or λ
i
λ together with the chosen total heat contact area namely the height of the contact ring
h define the heat flow to the crucible.
In a similar way an effective heat transfer coefficient α λ
i
λ in the pulleringot interface is used.
IV. Results from Numerical Experiments
IV.1. Thermal Transfer at Melting of Al, Cu, Ti The material characteristics used for the calculations
are given in Table I. The effective heat transfer coefficients on the
boundary areas G
1
, G
2
, G
4
are assumed as follows: α
1
λ
1
λ
Cu
=1.2, α
2
λ
2
λ
Cu
=0.8, α
3
λ
4
λ
Cu
=0.8 .
TABLE I
M
ATERIAL
C
HARACTERISTICS
U
SED
I
N
C
ALCULATIONS
[12] Parameter Al
Cu Ti
λ [WmK] 184.5 at 973 K 318.1at 1280 K
13 at 973 K T
m
[K] 823 1356 1938 C
p
[JgK] 1.087 0.38
0.58 a [m
2
s] 0.63·10
-4
1.13·10
-4
0.534·10
-5
C
p
. ρ T
m
[Jcm
3
] 2415.5 4612
6120
Fig. 5 shows temperature distributions in a cross- section of copper ingot during electron beam melting in
the axis-symmetrical case that is obtained by solving the two-dimensional quasi-steady-state heat model.
The figure presents half of the ingot cross-section. There have no pouring of the liquid metal into the
crucible, the ingot diameter is 60 mm and ingot length is 100 mm. In these cases the beam radius is 20 mm. The
curve marked with 1356 in Fig. 5 corresponds to the crystallization front.
The beam power is chosen under condition that the liquid metal pool is extended to cover the whole top
surface of the ingot. Due to that the direct contact between the heated ingot
and the inner wall of the copper crucible G
2
is negligible and ingot side surface is cooled only by
radiation losses. The beam power value is reduced by the energy of the
back scattered and secondary electrons.
Fig. 5. Temperature contours of a half of the electron beam melted ingot
The important role of the choice of thermal contact conditions is shown in Fig. 6. Some experimental
verifications of these calculations are given in [2].
Fig. 6. The liquid-solid interfaces calculated in copper ingot at beam power 15 kW and a heat contact width 1 mm: curve 1 – ideal thermal
contact in G
2
; curve 2-at effective heat transfer coefficient α=0.78 for
G
2
and G
4
; curve 3- α is chosen as 0.84
The knowledge of the shape of the crystallization front is directly connected with the quality control of the
ingot structure and surface – it is a key factor for producing ingots with a perfect crystal structure. A flat
crystallization front permits forming of dendrite structures, parallel to the block axis as well as the
uniform impurity displacement toward the ingot top surface. Conversely, in the case of an ingot that is deep
in the center and shallow at the periphery of the cast block a non-uniform structure along its radius will be
created. On the other hand, the presence of melted, not solidified segments, below the solidified area at the
Copyright © 2011 Praise Worthy Prize S.r.l. - All rights reserved International Review of Mechanical Engineering, Vol. 5, N. 2 Special Issue on Heat Transfer
261 contact ingot-crucible of type solid-solid is
investigated. In the next moment a solid scull is created there and the contact becomes again solid-solid. As a
result the adjacent segments are melted and for a short time a good heat contact occurs of type liquid-solid,
after that they are solidified and so on. Such non-steady process is responsible for the roughness of the ingot side
walls subject to further machining and in such way the yield of the re-melted and refined metal [3].
Fig. 7 shows the typical temperature distributions for an aluminum ingot with diameter 60 mm, heated by
electron beam with a power P=7.5 kW after the
correction for the reflected electron energy losses. The heated area is a circle of diameter 20 mm on the top
surface. The other computation parameters and the assumed heat contact conditions are given in Table II.
The heat fluxes for the EB melting and refining of Al at three casting velocities are summarized in Table II.
Fig. 7. Temperature contours and liquid solid boundaries 823 K, calculated at beam power 7.5 kW for Al ingot. There L is the ingot
length, R is the cast ingot radius. The cast speed is 3 mmmin
Note that Q means the heat, added by the pouring
metal; P
1
is the total heating power at the top ingot surface G
1
, taking in account the radiation losses and added energy from molten droplets;
P
2
is the thermal losses at the poolcrucible interface G
2
, P
3
is energy losses from rest side ingot walls G
3
and P
4
are the energy , transferred to the puller G
4
. The coefficients
α is the Newton heat contact characteristics.
The heat fluxes for the EB melting and refining of Ti, at the same three casting velocities as in the case of
Al ingot are summarized in Table III together with the assumed heat contact conditions. In this case, according
to the experimental data we had to eliminate the radiation energy losses from the ingots top surface. In the real
process, the convection flow of the liquid metal considerably diminishes the surface temperatures.
It can be noted that unexpected break in the solidified skin around the molten pool, observed in all cases of
computer simulation of the ingotcrucible interface gives a new explanation of the ingot side surface roughness.
This molten part of the side ingot wall with width of ∆h
– Fig. 4 is bellow the top ring, where it was assumed an ideal heat contact, causing its solidification. The change
of the ideal contact rings width does not modify this situation, because there are an self-sufficiency. If the size
of this contact area is extended, the energy losses will be intensified up to unacceptable values in comparison with
experimentally observed ones and a part of the ideal contact area will be solidified, lowering energy flux to
the crucible. In [8] the heat flux through the interface liquid metalwater cooled wall is evaluated to10
4
Wm
2
and for the case of good thermal contact between the solidified ingot wall and a water cooled wall is of order
of 10
3
Wm
2
. A calculation of the thermal field in one half of Ti cast
ingot during EBMR is demonstrated in Fig. 8. The melted metal is marked with dark area.
TABLE II
C
ALCULATED
F
LUXES
A
ND
S
OLIDIFIED
S
KIN
D
IMENSIONS
F
OR
EB M
ELTING
O
F
AL W
ITH
P=7.5 kW
No V mmmin
Q W
α crucible
h mm
α puller
P
1
W P
2
W P
3
W P
4
W R
1
mm h
mm h
L
mm ∆h
mm 1
2 3
3 6
9 161
946 2710
0,88 1
1 3
3 3
0,55 0,55
0,6 7555
8300 10
4
5470 71,4
6050 71,2
6250 61,2
60 80
120 2075
27,1 2000
27,5 2230
21,9 23
24 25
28 29
30 14
12,5 10
1 6
12 TABLE
III C
ALCULATED
F
LUXES
A
ND
S
OLIDIFIED
S
KIN
D
IMENSIONS
F
OR
EB M
ELTING
O
F
T
I
W
ITH
P=10 kW No V
mmmin Q
W α
crucible h
mm α
puller P
1
W P
2
W P
3
W P
4
W R
1
mm h
mm h
L
mm ∆h
mm 1
2 3
3 6
9 486
973 1460
1 1
1 5
7 8
1,7 1,7
1,7 1,04.10
4
1,09.10
4
1,14.10
4
5800 55,3
6300 57,4
8400 73,3
1490 1520
1760 1500
14,3 1550
14,1 1640
14,3 25
25,5 26
30 32
34 9
13 12
6 7
10
Copyright © 2011 Praise Worthy Prize S.r.l. - All rights reserved International Review of Mechanical Engineering, Vol. 5, N. 2 Special Issue on Heat Transfer
262
Fig. 8. The temperature contour lines at EBMR of Ti - input power P=18.75 kW, beam radius r
b
=20 mm, casting velocity v=9 mmmin, contact zone with crucible at the top side of ingot is 8 mm
Another example of EBMR of copper with beam power 45 kW is shown in Fig. 9.
The curve marked with 1356 corresponds to the crystallization front. In these cases the beam radius is
also 20 mm. The critical molten pool volume namely molten pool
occupied the whole upper ingot surface for this ingot dimensions is about 40-50 cm
3
and at the studied case of beam power 45 kW the volume of the molten pool is
about 70 cm
3
.
Fig. 9. Temperature distribution for beam power 45 kW, crystallization speed – 2 mmmin, height of the heat contact interface
ingot-crucible – 5 mm
The appearance of the liquid ring below the upper solidified shell can be seen in Figs. 7, 8, 9.
IV.2. Thermal Transfer at Melting of Refractory Metals The material characteristics used for the calculations
are given in Table IV. Fig. 10 and Fig. 11 show the temperature distributions
in a cross-section of tungsten and molybdenum ingots, respectively, during electron beam melting. These results
in the axis-symmetrical case are obtained by applying the two-dimensional quasi-steady-state heat model eq.
1 and suitable boundary conditions.
TABLE IV
M
ATERIAL
C
HARACTERISTICS
, U
SED
I
N
C
ALCULATIONS
Parameter W Mo
λ [WmK] 101 at 1800 K
83 at1280K T
m
[K] 3650 2896
C
p
[JgK] 0,14 0,25
ρ [gcm
3
] 17,6 10,22
The molten pool is presented as a dark area and the crystallization front is denoted by the corresponding
melting point temperature on these presentations of the calculated temperature fields in one half part of the ingot
Figs. 10, 11. The created molten pool depth is small and usually operator moves the molten pool by
movement of the beam spot center to fill the whole crucible area.
Here the reason for the side wall roughness is different in comparison with the case of EBMR of metals
in the previous paragraph. The radial temperature gradient, created here, is a reason for the segregation
processes of the impurities during EBMR of refractory metals.
Fig. 10. The temperature contour lines at EBMR of W: input power P=200 kW, beam radius is 30 mm, casting velocity v=15 mmmin,
contact zone with crucible is 8 mm
Fig. 11. The temperature contour lines at EBM of Mo: power P=100 kW, casting velocity v=9 mmmin, beam radius r=30 mm, contact zone
with crucible at the top side of ingot is 8 mm
Copyright © 2011 Praise Worthy Prize S.r.l. - All rights reserved International Review of Mechanical Engineering, Vol. 5, N. 2 Special Issue on Heat Transfer
263
V. Statistical Processing
