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
238 In that a way a second value of angle to maximum
point φ
2
could be used in next calculation. In that a way through the sequential approximations an enough exact
value of dimensionless distance R to the point with
maximal temperature, as well as value of corresponding φ, correlating with the chosen distance y
could be found. Then from eq. 9 is calculated the curve
θ
m
= θ
m
[ Y=yv2a] given on Fig. 6.
In the case of EBW of thin plates using shown on Fig. 5 bended edges of joining plates and assuming the
maximal temperature T
m
equal to T
melting
one can use eq. 9 and curve
θ
m
vy2a and at ξ=0 will find a relation for
beam current calculation of EBW of thin plates of thickness
h: I
a
=
2 2
m a
e m
h T T
v U
h a
πλ η θ
− ⎛
⎞ ⋅ ⋅ ⎜
⎟ ⎝
⎠ 12
Usually T
could be neglected, but eq.12 open possibility to calculate the cases of EBW using pre-
heated samples as example welding on second or third rotation.
Some experimentally confirmed curves for thin plates of stainless steel, copper, nickel and kowar are shown on
Figs. 7, and thermo-physical parameters used at calculations are shown in Table I.
Note, that the curve numbers are as they are in the Table I.
TABLE I
T
ERMO
-P
HYSICAL
P
ARAMETERS
O
F
W
ELDED
M
ATERIALS
No Material T
melting
[
o
C ] λ
[Wcm deg] a
[cm
2
s]
e
1. 2.
3. 4.
5. Copper
Nickel Kowar
St. steel 1X18H9T
Steel 08K
П 1083
1453 1450
1350 1350
3,85 0,83
0,18 0,194
0,51 1,13
0,2 0,03
0,462 1,24
0,79 0,8
0,81 0,93
0,93
In the case of EB but welding of thin plates with gap between bended welded edges a similar simple equation
could be created electron beam in the gap is assumed to be lost and weld cross section is a bigger semi sphere:
2 1
2 2
m a
e m
h T T
a I
v U
h a
ξ πλ
η θ
− +
= ⎛
⎞ ⋅ ⋅ ⎜
⎟ ⎝
⎠
13
This equation is in good agreement with experimental results [4], [5].
In the case of EBW of thin details with a glassmetal joints near to the weld place a increase of the beam
current could be not available and pre-heating of welded edges could be more practical. On base of eq.13
tolerances of detail production could be more clear also.
This calculations for an tilted Decart coordinate system, where x is situated on sample surface and
coincides with the sample movement, y is the sample
depth and z is electron beam scanning direction, the
depth of melted material at EB surface thermal treatment were successfully applied too[6].
Fig. 7a. Dependencies of Electron beam current at but welding on plates thickness U
a
=23 kV;v=1 cms; ξ=0
Fig. 7b. Electron beam current vs. welding speedU
a
=23 kV; h = 1 mm;
ξ=0. The curve numbers are as in the Table I
III. Temperature Fields at Deep
Penetrated EBW and Discussion on the Use of that Calculation to evaluate the
Weld Geometry
In the general case of EB welding with a powerful deep penetrating beam mean power density of which is
more than mentioned critical value the energy flux density that is absorbed on keyhole walls is a complex
function of the coordinates. In order to simplify the problem of non-known and non-steady distribution of the
real heat source a quasi steady state heat source can be assumed.
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
239 The solution of thermal balance on heating a sample
with a deep penetrating electron beam, if the crater depth is h, we could also use as approximation a linear moving
thermal source with a constant intensity P, assuming no
phase changes in the sample during heat transfer, at known thermal conductivity coefficient-
λ, and assuming T
m
– the difference between the maximum of the temperature cycle in a point and the room temperature, as
were written in 9 is:
1
2 2
2 2
2
m m
hT r v
K P
a r v
r v r v
exp K
K a
a a
πλ θ
⋅ ⎛
⎞ =
= ⋅
⎜ ⎟
⎝ ⎠
⎡ ⋅ ⋅
⋅ ⎤ ⎛
⎞ ⎛
⎞ ⎛
⎞ ⋅
⋅ ⎜
⎟ ⎜
⎟ ⎜
⎟ ⎢
⎥ ⎝
⎠ ⎝
⎠ ⎝
⎠ ⎣
⎦
14
where r is the radius vector, moving together with the
heat source, K
and K
1
are the modified Bessel functions of second kind, respectively of order zero and of order
one; ν is the welding speed.
In the case of EBW at disregarding the keyhole in the melting pool, the same model and solution of heating
theory, using the electron beam characteristics, formulae and graphs for the evaluation of the weld geometry
parameters at electron beam welding with deep penetrating beam were presented as an approximation
[8]-[10].
Other possibilities for evaluation the weld geometry in case of deep penetrating powerful beam are models
[11],[12] utilizing the ideas for heating by moving heat source that content a sum of linear and point heat sources
as a cylindrical or conical steady, continuously operating heat sources. As were appointed, more practical, instead
eq. 9 is find by iterations [3],[9],[10] the graphical correlation between dimensionless maximal temperature
θ
m
and dimensionless distance, measured perpendicularly to heat source movement direction
y ν2a shown on Fig. 6. Note, that a another important
characteristic of all fusion welding processes is thermal efficiency
η
t
defined as a ratio between the energy P
f
absorbed and spent for heating of the metal of the volume of the weld up to melting temperature usually
including the fusion heat, and total beam energy converted in the thermal energy
P: η
t
= P
f
P = v·F
w
·S P 15
where v is the welding speed, F
w
is the cross - section area of the melted zone,
S is the heat content per unit volume of the material of work-piece during heating
from room temperature up to fusion temperature T
m
S = C
p
.T
m
+ H
f
, being
C
p
the mean specific heat for the temperature range between the room and fusion
temperatures. H
f
is the heat of fusion. The thermal
efficiency value accounts for losses due to the following processes and mechanisms:
i thermal conductivity towards cold sample regions, ii super-heating of the weld metal above melting
temperature, iii heat transfer by vapor-gas flow leaving the
welding crater, iv radiate dissipation of heat from weld surface.
Fig. 8 gives the transformed plot of θ
m
b ν2a that is
shown for a concrete case of stainless steel - namely curve
νbPh. On that figure with three strait lines the 100, 50 and 20 thermal efficiencies of the EB
welding are presented. The theoretical limit of that efficiency is 48,4 [3] seen as near to the straight line
part of the theoretical curve νbPh. The experimental
data of two extensive studies of the geometry characteristics of EB welds there are partially presented
the fulfilled 140 experiments [9] are denoted by points. The values of experimental data cover ranges of
Ph 1.33-10 kWcm and of
b ν 0.1- 0.75 cm
2
s and are in an agreement with the theoretical curve, shown by
continuous curve, that could be accepted as a first approximation for calculation of heat processes at EBW
at known weld characteristics b and h and EBW
parameters P,v .
Fig. 8. Comparison between experimental and theoretical data for thermal efficiency
The discrepancy of the experimental points and theoretical curve are due the assumptions:
i Use a steady state instead a non-stationary heat source. Note that idea for model using a non-stationary
variable or oscillating intensity of the EB heat source in work-piece were discussed in [7],[13].
An experimental confirmation of this idea were found by direct measurements of the temperatures in the points
placed at distances y = 0,1 cm, y = 0,15 cm y = 0,2 cm, y = 0,25 cm, and
y = 0,3 cm respectively from the line of the movement of heat source beam using the WW-
Re thermocouples. The beam was 60 kV, 50 mA and 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
240 welding speed was 1 cm s
-1
. The measured dependencies of the sample
Tt are shown in Fig. 9. Non - monotonous character of temperature changes
in the vicinity of the maximum of the first curve in the initial stages of the quenching of the material were
observed. In this case the thermocouple is fixed just outside the weld but near the fast moved liquid - solid
boundary. Analysis of great number of such curves showed that they all contain a component representing
periodical temperature changes of a frequency of 3,5 - 5 Hz. Therefore, the real heat source operating within the
welding bath has variable components too. The lowest frequency component is responsible for non-stationary
changes of the temperature cycles.
Fig. 9. The measured dependencies of the sample temperature on time
Avail from the welding pool curves for bigger distances y in Fig. 9 owing to metal capacity, the heat
waves originating by the variable component of the non - stationary thermal source attenuate and observed thermal
cycles are similar to the cycles, generating by a moving heat source of constant intensity.
ii Approximation of disregarding keyhole the real distances to any point in the sample are less, due to
distances between the beam axis and the keyhole walls; the heat source is more distributed on keyhole walls than
in central region of a line heat source.
In [13] paradoxically the reconstructed through calculations heat sources intensities by the experimental
dimensions of the melted and heat affected zones prove to be different. That can be explained due to the lower
distances between these zones and the cylindrical keyhole walls, where heat is absorbed. The phase
transitions in solid state and the turbulent flows as the variations in the shape of crater in the melt can be
additional reasons for that discrepancy as well as for difference between the theoretical curve b
νPh and experimental weld geometry characteristics, presented by
points. The same reasons lead to increased thermal efficiency
of the EB welding higher than theoretical limit 0.485. The problem is that geometry parameters of the weld
are unknown and prognostication of these characteristics using chosen electron beam power P and welding speed
v is desired. From the theory-equation 9 or by curve θ
m
vy2a extended by a lot of experimental data to get regions of allowable values of main EBW parameters
vb and
Ph one could evaluate only very rough regions for expected geometry of the beam as this were done in [9]
and is shown for Stainless steel on Fig. 10 and Fig. 11 .
Fig. 10. Ranges of the weld depths as dependence on the beam power, calculated for welding speeds 0,2-15 cms : 1- h
max
at Ph = 2 kWmm and 2- h
min
at Ph=10 kWmm
Fig. 11. Ranges of the weld width as dependence of welding speed or of reciprocal value 1v: 1- b
max
at vb=0,75 cm
2
s; 2- b
min
at vb=0,15 cm
2
s
A second example of the possibility to use theoretical function
b νPh for prognosis of EB weld characteristics
is shown on Fig. 12 - Fig. 14 [8]. On Fig. 12 and Fig. 13, respectively for steel and copper are shown the
correlations between the beam power P at two weld
widths the solid line is for b=1mm and dashed line is for
b=2mm and EB weld depth h given near the continuous and dashed curves can be seen. The inclined straight
lines on that figure show the different welding speeds.
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
241 There
v
1
=0.5cms; v
2
=1cms; v
3
=1,5 cms; v
4
=2cms and v
5
=2.5cms. On abscise in Fig. 12 are given the energy per one
unit of the weld length Wl=P
v. It can be seen that typical welding parameter - energy per one unit of the
weld length Wl=Pv, wide used in conventional welding
processes - for the EB welding is not sufficient characteristic of the process. If the weld width is changed
at constant Wl - the beam penetration depth is changed
too, but the weld depth can be less or more dependently on the welding velocity. On Fig. 14 is shown a
comparison between weld parameters in steel solid lines and copper dashed lines. The thermal parameters
are given in Table II.
TABLE II T
ERMO
-P
HYSICAL
P
ARAMETERS
O
F
W
ELDED
M
ATERIALS
No Material T
melting
[K] λ
[Wcm deg] a
[cm
2
s] 1.
2. Steel
Copper 1808
1356 0,51
3,89 0,15
1,12
Fig. 12. Contour graph hP,Wl; atv
1
=0.5cms; v
2
=1cms; v
3
=1.5cms; v
4
=2cms; v
5
=5cms; Continuous line ─ presents b=1mm; dashed line - -
is for b=2mm. The material is steel
Fig. 13. Contour graph hP,Wl; at v
1
=0.5cms; v
2
=1cms; v
3
=1.5cms; v
4
=2cms; v
5
=5cms; Continuous line ─ presents b=1mm; dashed line - -
is calculated for b=2mm. The material is copper
The results of these calculations using steady state models namely moving linear heat source can be used
for practical use for rough initial technology parameter choice. One can apply this model at admission of known
value of the width or the depth of the weld as well as at prognosis of the pair width and depth of the weld at
calculating its values on the basis of known welding and material characteristics. But all they have a big
disadvantage to not taking in the account the position of focus relatively sample surface or the beam focusing
current changes and the variations of the distance gun- sample. The beam parameters radial and angular
distributions or the respective more practically values - the beam focusing current and the distance between the
treated sample and the gun at invariable geometry and electrical parameters of the gun, generating electron
beam are not included too. These parameters are included in more exact determination of the weld
characteristics on base of statistical models of EBW [14]- [20] where using the multi-response surface
methodology it is found polynomial regressions or simulation of EBW processes using neural networks
[21],[22].
Fig. 14. A comparison between the depth at EBW welding of steelsolid lines and copperdashed lines at b=2mm for various powers P and
welding speeds v
1
-v
5
, as they are in Fig.12 and Fig.13
IV. Conclusion