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
In order to study the influence of the value of the effective coefficient of the heat transfer on the EBMR of
a copper ingot with diameter d = 60 mm and length
L=100 mm a beam power of 38.7 kW, distributed uniformly in the central part of the top ingot surface
d=20 mm and a crystallisation speed 6 mmmin were chosen. The heat transfer coefficient
α
1
λ
1
λ
Cu
= Z
1
, α
2
λ
2
λ
Cu
= Z
2
, α
4
λ
4
λ
Cu
= Z
4
, that are on the boundary areas
G
1
, G
2
and G
4
, are chosen as parameters. The height
h of the heat contact ring ingotcrucible is denoted as
Z
4
. The limits of variation of Z
1
are from 1 to 2, for Z
2
0.4–1, for Z
3
are 0.4–1 and for Z
4
– from 1 to 12 mm. The output values under study are:
P
1
, the beam and pouring
metal input powers diminished taking into account the radiation and secondary electron losses;
P
2
, the heat losses through the heat contact area;
P
3
, the heat losses through the rest side ingot surface and
P
4
, the heat losses through the ingot bottom as well as
V, the volume of the molten pool.
A plan of the experiments is made on the basis of the D-optimality criterion [13]. Computer simulated
experiments were performed and the calculated results are given in Table V. A geometrical presentation of
parameters is given in Fig. 12, where h
is the depth of the liquid pool,
h
1
is the width of the liquid pool, h
L
is the height of the solidified ring adjacent to the crucible
and ∆h is the height of the liquid metal.
The variables Z
1
, Z
2
, Z
3
and Z
4
were coded in the region from
−1 to +1 in order to be made dimensionless and to allow the estimation of statistical regression
models, and will be referred to, respectively, as X
1
, X
2
, X
3
and X
4
. Using MINITAB, full quadratic regression models are estimated for the output values
P
1
, P
2
, P
3
, P
4
and V as follows:
P
1
=36685+838· X
1
+267· X
2
+173· X
3
+942· X
4
−886·X
1 2
+ +355·
X
2 2
+181· X
3 2
−617·X
4 2
−145·X
1
· X
2
−77X
1
· X
3
+ −350·X
1
· X
4
−102·X
2
· X
3
−195·X
2
· X
4
−372·X
3
· X
4
, P
2
= −23003+1509·X
1
−5920·X
2
+2305· X
3
−4407·X
4
+ +2343·
X
1 2
+809· X
2 2
+2599· X
3 2
+2663· X
4 2
+187· X
1
· X
2
+ +788
X
1
· X
3
+587· X
1
· X
4
−294·X
2
· X
3
−78·X
2
· X
4
−372·X
3
· X
4
, P
3
= −110−12·X
1
+524· X
2
+403· X
3
+1221· X
4
−771·X
1 2
+ +61·
X
2 2
+345· X
3 2
−743·X
4 2
−14·X
1
· X
2
−32·X
1
· X
3
+ −102·X
1
· X
4
−209·X
2
· X
3
−474·X
2
· X
4
−341·X
3
· X
4
, P
4
= −12593−2372·X
1
+4479· X
2
−2891·X
3
+2185· X
4
+ −715·X
1 2
−1214·X
2 2
+2079· X
3 2
−1347·X
4 2
−40·X
1
· X
2
+ −681·X
1
· X
3
−189·X
1
· X
4
+620· X
2
· X
3
+739· X
2
· X
4
+903· X
3
· X
4
, V=8836+5701·X
1
−39943·X
2
−33760·X
3
−51434·X
4
+ −77836·X
1 2
+92049· X
2 2
+49563· X
3 2
+19088· X
4 2
+ +8274·
X
1
· X
2
−21196·X
1
· X
3
−9125·X
1
· X
4
+26883· X
2
· X
3
+ −9710·X
2
· X
4
+13298· X
3
· X
4
Some two-dimensional plots of the relations between some of the input values and the factors are presented in
Figures 12, 13, 14. They are given as examples of the considerable
potential of the method to provide a better understanding of the process and hence to optimise the technology and
equipment. Fig. 12 presents a contour plot of the corrected input
power P
1
versus the heat transfer coefficients Z
2
and Z
3
at the ingotcrucible and ingotpuller interfaces,
respectively. In this figure a minimum of the corrected input power
P
1
is seen in the region of lower heat transfer coefficients Z
2
to the crucible and Z
3
to the ingot bottom. Under these conditions maximal radiation losses and
evaporation of the base metal are expected. Therefore, these regimes are more suitable for EB evaporation and
a deposition of coatings but not for EBMR. For small heat transfer coefficients Z
2
and large values of
Z
3
the change of Z
2
is not important in determination of
P
1
. At these conditions the change of Z
3
plays a considerable role.
TABLE V
R
ESULTS
O
BTAINED
F
ROM
T
HE
P
LAN
O
F
N
UMERICAL
E
XPERIMENT
F
OR
C
OPPER
A
T
P=38,8 kW
A
ND
V=6 mmmin
I
N
T
HE
R
EGIONS
F
OR
Z
1
= 1-2,
Z
2
= 0,4-1
, Z
3
= 0,4-1
AND
Z
4
=1-12mm No P
1
W P
2
W P
3
W P
4
W h
mm h
1
mm ∆h
mm h
L
mm V
mm
3
1 2
3 4
5 6
7 8
9
10 11
12 13
14 15
16 32348
36255 34563
36167 37154
37143 37151
35828 35895
35702 36925
36645 35845
36733 35471
37097 -14076
- 31986 -21097
- 23644 - 25839
- 14892 - 30697
- 23163 - 21327
- 10530 - 20215
- 28898 - 31649
- 23993 - 17661
- 24236 - 4623,4
- 30,5 - 907,9
- 1811.6 208,4
196.6 278
- 210 - 10615
- 1666 - 1507
- 214 - 71
- 63,9 - 322
103 - 13649
- 4281,2 - 12584
- 10714 - 11535
- 22452 - 6736
- 12455 - 13508
- 23505 - 15197
- 7532,4 - 4120,5
- 12683 - 17488
- 12967 75
12 36
65 7
9 8
68 51
65 76
47 17
21 45
10 29
14 26
27 10
11 11
16 26
26 19
17 16
15 18
12 73
28 61
38 45
60 62
21
16 2
6 4
30 5
5 14
24 26
211579 5401
94106 182067
1343 2170
1790 131051
137419 181017
197453 88403
1096 12591
78496 3551
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
264
Fig. 12. Contour plot of the input power taking into account the radiation losses: P
1
vs. coefficients of thermal transfer at the ingotcrucible interface Z
2
and at ingotpuller interface Z
3
at transfer coefficient to the molten pool stirring Z
1
=1.5 and the crystallized ring height Z
4
=6.5 mm
For high values of Z
2
and small values of Z
3
the transfer to the crucible
Z
2
plays a critical role. At the same time the heat transfer to the cooled puller
Z
3
is not so important with big values of
Z
2
and Z
3
- the heat transfers to the crucible and to the puller are of equal
importance at the determination of P
1
. In Fig. 13 a plot of the heat flow to the crucible
P
2
versus the factors Z
2
and Z
3
is given. One can see that for small values of the heat transfer coefficient in the contact zone ingotcrucible
Z
2
the heat flow to the crucible is defined by both heat transfer coefficients
Z
2
and Z
3
. For high heat transfer Z
2
the heat flow to the cooled crucible P
2
is defined by Z
3
. A minimal
P
2
is observed for large Z
2
and small Z
3
. At the same time the maximal
P
2
is seen for small Z
2
and high
Z
3
.
Fig. 13. Contour plot showing the dependence of the radial heat flow towards the crucible P
2
on the coefficients of thermal transfer at the ingotcrucible interface Z
2
and at ingotpuller interface Z
3
at transfer coefficient to the molten pool stirring Z
1
=1.5 and the crystallized ring height Z
4
=6.5 mm
Fig. 14 shows that a maximal volume of the molten pool is reached for lower values of
Z
2
and Z
4
. Here the role of
Z
2
is definitive. The role of the coefficient of the heat transfer
Z
2
is negligible for medium values of Z
2
and a wide contact zone
Z
4
. Under these conditions the volume of the molten pool is minimal. For low values of
Z
4
narrow heat contact to the crucible and high values of the heat transfer
Z
2
an optimal Z
4
exists. At high values of the
Z
2
and narrow contact zone Z
4
as well as the small values of
Z
2
and small values of Z
4
the volume of the molten pool is influenced by both factors
Z
2
and Z
4
.
Fig. 14. Contour plot showing the changes of the molten pool volume V at variation of coefficients of thermal transfer at the ingotcrucible
interface Z
2
and the crystallized ring height Z
4
at Z
1
=1.5 and a coefficient of thermal transfer ingotpuller interface Z
3
=0.7
VI. Conclusion
