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
236 relationship between the coordinates of the point where
the temperature rise is concerned along the X- or x- axis at any time t.
The abscise of the moved and absolute coordinate systems is given by X = x - vt.
There v is the velocity of the heat source movement and moving with the heat source coordinate system is
oriented in such a way, that
x
v v
v = =
. Substituting this in Eq. 1, and taking in the account
transfer of the heat by the sample material, the general differential equation of heat conduction in a moving
coordinate system can be obtained as:
2 2
2 2
2 2
1 v
a t
x X
y z
θ θ
θ θ
θ
∂ ∂
∂ ∂
∂ ⎛
⎞ +
+ =
− ⎜
⎟ ∂
∂ ∂
∂ ∂
⎝ ⎠
2 Equation 2, even for 2D heat flow at chosen
condition of infinite in z direction heat source, e.g., along the X-axis and y-axis, involves three variables, namely,
X, y, and t. Hence, solution of this equation by the separation of
variables or other similar techniques would not be feasible.
From the heat dispersion calculations based on 2, assuming a quasi - stationary temperature distribution
θ
∂ t
∂ =
equation 2 can be solved and one is able to obtain approximately relation between the weld
parameters and the geometry characteristics of the weld, as well to explain many process features.
In the case of EB welding or surface thermal treatment of a semi-infinite sample with electron beam,
characterized with mean power density on work-piece surface less than a critical power density of order of 10
6
Wcm
2
a hemi-spherical fusion zone can be obtained due to near to point heating source Fig. 1.
At that critical power density the shape of cross section of the weld is sharply going to a deep and narrow
cross section and heat affected zone become narrow. That cross section is shown in Fig. 2.
The depth of penetration is 78 mm and the beam power is 15 kW, welding speed is 1 cms, the beam is
focused 60 mm below the sample surface. The explanation of the deep penetration of the intense
electron beams into the treated material is connected with the generating of a key-hole crater or plasma cavity
within the liquid metal welding pool through which the energy beam entering in the heated samples see Fig. 3.
Fig. 1. Cross-section of EBW at average power density less than 10
6
Wcm
2
Fig. 2. Cross-section of EBW at average power density bigger than 5.10
6
Wcm
2
Fig. 3. Schema of a longitudinal cross section of a EB weld during deep penetration of the electron beam
II. Application of Thermal Field
Evaluation on Weld Geometry Parameters at EBW of Thin Plates
EBW of thin plates using plain butt welding or edge fusion in butt welding with bended edges are typical
design of the joints Fig. 4 and Fig. 5. The welding is realized without use of additional material welding
wire by fusion of plates edges.
Fig. 4. Plain butt welding of thin plates
Fig. 5. Butt welding with bended edges of plates
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
237 In the case of plate thickness h and gap between plates
ξ to get semispherical shape for joining, shown on Fig. 5 the depth in the plates d could be evaluated by the
geometry relation:
d =
1 4
2 h
h
π ξ
⎛ ⎞
+ ⎜
⎟ ⎝
⎠
3 At EBW of thin plates with welding speeds of order
of 0,5 ─ 3 cmsec the depth of the welding bath is small
in comparison with extension of thermal field, as well as the superheating of molten metal is small compared to
the melting temperature T
surface
-T
melt
T
melt
. Due to that, neglecting the phase transformation heat
and using constant thermo-physical parameters, near to that of solid sample, the calculation could be fulfilled by
approximated solution of the equation 2 for distances of order or bigger of h.
Note that in that case nevertheless that real thermal source could be a point thermal source, due to heat
reflection from the back surface of the plate at distances of order of h a 2D temperature field take place.
Therefore in all cases of heating of thin plates a line source can be used.
At the case of welding with continuous operating electron beam the moving line heat source have intensity:
a a
e
U I
P h
h η
⋅ ⋅ =
4 where
U
a
is accelerating voltage measured in kV; I
a
is the beam current in mA;
e
is the electron efficiency of the electron irradiating process.
Also let it be assumed that temperature of sample is T=T
at r →∞ , and the thermal flow at r→0 is equal to
intensity of the heat source 4 the temperature of quasi- state temperature field, moved together with the moving
heat source is:
2 2
2 P
v v
T exp
x K
r T
h a
a
πλ
⎛ ⎞
⎛ ⎞
= −
⋅ +
⎜ ⎟
⎜ ⎟
⎝ ⎠
⎝ ⎠
5 where
K is modified Bessel function of imaginary
argument of second kind, of order zero or Mc Donald’s function.
There one could use dimensionless Peclet numbers: X
= Xv2a; Y = yv2a; R
= rv2a 6
There a is the temperature diffusivity, a =
λC·γ and the dimensionless relative temperature rise is:
2 h
T T
P
πλ θ
= −
7 Then relation 5 could be written as:
exp X
K R
exp R
cos K
R
θ ϕ
∗ ∗
∗ ∗
= −
= =
− ⋅ ⋅
8 where
φ is the angular coordinate in more practical here cylindrical coordinate system, moved with the heat
source. The maximal temperature, that could be reach in the
time for a material point, distanced to y from the moving
heat source direction axes x one could get through maximal temperature
θ
m
for Y
=y ·
v2a:
1 m
K R
K R
exp R K
R
θ
∗ ∗
∗ ∗
⎛ ⎞
⎜ ⎟
= ⋅
⎜ ⎟
⎝ ⎠
9
There:
2 v
R y
a sin
ϕ
∗
= ⋅
10 where:
2 1
1 K
R sin
K R
ϕ
∗ ∗
⎡ ⎤
⎢ ⎥
= − ⎢ ⎥
⎣ ⎦
11 It is more convenient to use a graphical dependence
θ
m
[ hv2a]. For calculation one value of
φ
1
is calculated from eq. 10 at chosen y
and this value is included in eq. 11.
Fig. 6. Tilted diagram of the curve θ
m
= θ
m
vy2a. In the normal view here the parameters on the abscise and the ordinate are X=Ph
λ
T and Y=vy
2a
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
