High Temperatures ^ High Pressures, 2003/2004, volume 35/36, pages 117 ^ 126
DOI:10.1068/htjr086
Heat flux determination in the gas ^ tungsten-arc welding
process by using a three-dimensional model in inverse
heat conduction problem
Sandro M M Lima e Silva, Louriel O Vilarinho, Amërico Scotti, Tiong H Ong,
Gilmar Guimara¬es
Heat and Mass Transfer and Fluid Dynamics Laboratory, and Laboratory for Welding Process
Development, School of Mechanical Engineering, Federal University of Uberlaªndia, Campus Santa
Moªnica, Uberlaªndia, MG, Brazil; email: metrevel@mecanica.ufu.br
Presented at the 16th European Conference on Thermophysical Properties, Imperial College, London,
England, 1 ^ 4 September 2002
Abstract. Heat input measurement during the welding process is a highly complex task, primarily because the welding arc is a non-uniform heat source. To solve this problem, various analytical
and numerical approaches have been proposed. They can be divided into two categories, the direct
and the inverse problems of heat transfer. The problem is considered direct when all the boundary
conditions are given for the outside surface of the domain. In an inverse problem, information
concerning one or more boundary conditions is unknown. Thus, an inverse problem requires the
knowledge of temperature at a fixed point inside the domain, in order to provide the temperature
profile at the surface. Here, a methodology is proposed to calculate the heat flux delivered to the
workpiece during the welding process. The inverse-problem technique is based on the conjugated
gradient method with an adjoint problem. The proposed model is based on three-dimensional
heat transfer with spatial and temporal heat source variation. To assess the proposed technique,
different welding conditions were used in the gas ^ tungsten-arc welding process. The arc heat input
was estimated by temperature measurement at the surface at the rear of the weld, with ten
thermocouples equally spaced along the middle line of the plate.
1 Introduction
One of the most widely used welding processes is the gas ^ tungsten-arc (GTA) process,
used for welding of stainless steels and non-ferrous materials. In this process, a tungsten
electrode is shielded by a flow of inert gas such as argon (generally used), helium, or a
mixture of the two. The intensity of the heat flux and the temperature gradients in the
workpiece are extremely important for the study of the welding process. The analysis of
thermal behaviour of the physical phenomena that take place in the process is crucial
for evaluating, for example the width and depth of weld penetration, changes of the microstructure in the base metal and the residual stresses produced by the welding process.
Not all the electrical power necessary for producing the arc (calculated from the
voltage and current) is absorbed by the workpiece. The difference is due to heat losses to
the atmosphere by convection or radiation during the process and by the Joule effect
in the electrode. The inverse heat conduction problem represents an alternative way of
arriving at the heat flux that enters the workpiece. This procedure is justified by the
difficulties in carrying out measurements in the thermally affected zone in the weld
area. The inverse technique can be used to derive the heat flux to the weld face of the
workpiece by temperature measurement at the face opposite to the weld bead.
Inverse heat conduction problems have been used recently in the studies of welding
processes. Katz and Rubinsky (1984) used a finite-element method with one-dimensional
treatment, while Hsu et al (1986), also using the finite-element method, proposed a twodimensional model. Here, the temperatures measured by thermocouple sensors in the
solid region are used to calculate the position of the solid ^ liquid interface and
the temperature field in the solid region of the workpiece. For this, a Newton ^ Raphson
S M M Lima e Silva, L O Vilarinho, A Scotti, T H Ong, G Guimara¬es
118
interpolation is used, with the assumption of a stationary welding process. Gonc°alves
et al (2002) used a quasi-stationary analytic model (Rosenthal 1935) to derive the direct
problem equation and applied the simulated annealing method to obtain the heat flux
input. It should be noted, however, that in the quasi-steady two-dimensional model the
influence of heat diffusion through the plate thickness is neglected and the moving heat
source is considered constant during the whole welding process. Although these considerations are justified by practical results in GTA welding of thin plates, they represent a
limitation for other processes when welding thicker plates.
In this work, we are determining the heat flux and the temperature field during a
GTA process using a three-dimensional transient thermal model. An inverse technique
based on the conjugated gradient optimisation method with an adjoint equation is used.
The temperature is measured at the face at the rear of the weld. The moving heat source
is taken as boundary condition with time and position dependence. The welded plate is a
sample of austenitic stainless steel AISI 304 fitted with ten thermocouples at the opposite
face of the plate. The use of the measured temperatures together with the inverse technique
allows us to obtain the heat flux to the front surface of the plate and consequently the
rate of heating necessary for welding, in addition to the temperature field in the plate.
2 Theoretical fundamentals
2.1 The direct problem
The thermal problem in GTA welding can be represented by figure 1. The temperature
field can be obtained through the solution of the heat diffusion equation, considering
as thermal excitation a heat source moving in direction y with the remaining surfaces
subject to convective heat losses.
h3 , T1
c
Welding speed
h6 , T1
z
zh
y
x
z0
h2 , T1
a
yh
h6 , T1
y0
b
h4 , T1
Unknown heat flux, q … y; z; t†
h5 , T1
Figure 1. Three-dimensional thermal
welding problem.
The thermal model is obtained by numerical solution of the transient three-dimensional
heat diffusion equation considering as constant the thermal properties of the plate initially
at temperature T0 . The equation can then be written down as
q2 T…x, y, z, t† q2 T…x, y, z, t† q2 T…x, y, z, t† 1 qT…x, y, z, t†
‡
‡
ˆ
qx 2
qy 2
qz 2
a
qt
(1)
in the region R (0 5 x 5 a, 0 5 y 5 b, 0 5 z 5 c) at t 4 0, with the boundary conditions:
ÿk
qT…0, y, z, t†
ˆ q…y, z, t† at S1 …y0 4 y 4 y1 , z0 4 z 4 z1 † ,
qx
(2)
ÿk
qT…0, y, z, t†
ˆ h3 ‰T1 ÿ T…0, y, z, t†Š at S2 …y, z 2 S j…y , z† 2
= S1 † ,
qx
(3)
Heat flux in gas ^ tungsten-arc welding
119
ÿk
qT…a, y, z, t†
ˆ h5 ‰T…a, y, z, t† ÿ T1 Š ,
qx
(4)
ÿk
qT…x, 0, z, t†
ˆ h6 ‰T1 ÿ T…x, 0, z, t†Š ,
qy
(5)
ÿk
qT…x, b, z, t†
ˆ h2 ‰T…x, b, z, t† ÿ T1 Š ,
qy
(6)
ÿk
qT…x, y, 0, t†
ˆ h4 ‰T1 ÿ T…x, y, 0, t†Š ,
qz
(7)
ÿk
qT…x, y, c, t†
ˆ h1 ‰T…x, y, c, t† ÿ T1 Š ,
qz
(8)
and the initial condition
T…x, y, z, 0† ˆ T0 .
(9)
Here S is the front surface of the plate, S1 (zh , x, yh ) the area subjected to the moving
heat source (y0 5 y 5 y0 ‡ yh and z0 5 z 5 z0 ‡ zh ), and S2 the rear face of the plate
not exposed to the heat source but experiencing heat loss by convection. In the above
equations, a is the thermal diffusivity, k the thermal conductivity, and hi (i ˆ 1 to 6) are
the heat transfer coefficients at the faces of the plate. T1 is the ambient temperature,
while the moving heat source is represented by q(y, z, t) whose intensity is allowed to
vary with surface position and the duration of welding.
If the value of the heat flux, q(y, z, t), is known, equations (1) ^ (9) represent the
direct problem related to the inverse problem studied. To obtain the solution, the finite
volume method is used with a grid of 73671650 volumes.
2.2 The inverse problem
The inverse problem technique is based on the conjugated gradient method with the
adjoint problem. To estimate the heat flux, temperatures are measured at the rear surface
of the test plate. This method uses two auxiliary problems, known as the sensitivity
problem and the adjoint problem, that are solved in order to compute a search step size
b k and a gradient HSq . Here, Sq is an ordinary least-squares function
t
2
‰Y…t† ÿ T…xi ; t†Š dt ,
(10)
Sq ˆ
f
tˆ0
where Y…t† is the measured temperature, T(xi , t) is the estimated temperature at a single
measurement location, xi , and tf the measurement duration (Ozisik and Orlande 2000).
The heat flux can then be estimated through a computational algorithm that includes
an iterative procedure for the solution of the direct problem, the inverse problem, the
sensitivity problem, the adjoint problem, and the gradient equation. The discrepancy
principle is used here as the stopping criterion (Ozisik and Orlande 2000).
The experimental data were processed by a computational algorithm developed
specifically for inverse heat flux estimation in cutting processes. Derivation of the heat
diffusion equation, the discretisation, and the inverse algorithm procedure can be found
in detail in Lima et al (2000) and Lima (2001). In addition, thermocouples are attached
to the rear face, away from the region where the weld is formed, in order to validate
the results. It means that, once the heat flux is obtained, the three-dimensional direct
model can be used to calculate any temperature in the insert, including at the positions
of the thermocouples. The agreement between the calculated and measured temperature
at the rear face generates confidence in the estimated temperature field and heat flux.
Although the inverse technique can be used to estimate the 2-D component heat flux q(y, z, t),
the unknown heat flux can only be recovered if a great number of thermocouples are
S M M Lima e Silva, L O Vilarinho, A Scotti, T H Ong, G Guimara¬es
120
used in the y and z directions. In this work, keeping in mind that the behaviour of the
GTA torch has little spatial change and the main interest is in its average value,
q(y, z, t) is assumed a priori as a q(y, t) heat flux. This means that the heat flux
obtained here represents an average over the bead width (constant in z direction) as in
figure 1. The bead width is obtained for each experiment individually. This procedure is
discussed in the next section.
3 Experimental procedure
Figure 2 represents the welding rig used in the experimental procedure. The GTA torch,
representing a point heat source, moves at a specific speed along a straight path, with
the use of a totally automated X ^ Y coordinate table. To avoid test-plate dimension
Voltage and
current acquisition
Coordinated
table control
Power supply
Thermocouples
Welding plate
Multimeter HP75000
Multimeter
control
Shielding
gas
Coordinated
table
Figure 2. Schematic representation of the experimental rig.
To power
supply
GTA torch
Test plate
Ground cable
Support
To DAS
Figure 3. Schematic diagram showing
the test-plate support.
Heat flux in gas ^ tungsten-arc welding
121
1
2
3
4
5
6
7
8
9
10
25
50
25
interference, the test plate was held suspended in air by four pointed cylindrical bars, so
that only very small contact area existed (figure 3).
Figure 4 shows how the ten type K thermocouples were located at the rear side of
the test plate, ie at the face opposite to the bead (heat source). They were equally spaced
from each other (16.7 mm), while the first was 25 mm from the test-plate edge.
25
16.7
16.7
16.7
16.7
16.7 16.7 16.7 16.7
16.7
25
Figure 4. Thermocouple positions in test plate, at the rear of the weld (all dimensions are in millimetres).
Capacitor discharge was used to fix the ten type K thermocouples to the rear plate
surface. The plate was of AISI 304 steel with dimensions of 200 mm650 mm64 mm.
A detailed description of the procedure can be found in Vilarinho (2001). Collection
and storage of the data from the thermocouples was done with a microcomputer-based
data-acquisition system (HP 75000 B E1326B), DAS for short. The DAS, under a software control, sampled (multiplexed) each thermocouple signal at intervals of 0.38 s
(totaling 1028 points for each thermocouple).
A set of experiments with different welding conditions, such as current (40, 70, and
100 A), arc length (2, 3, and 4 mm), gas (Ar and Ar ‡ 25% He) and electrode angle
(308, 608, and 708) was carried out. The electrode used was AWS EWTh-2, tungsten
doped with 2% of thorium.
In order to reduce the number of experiments whilst keeping the results reliable,
a statistical planning method Taguchi (Robust Project) with L9 matrix was used.
4 Analysis of results
Nine experiments were performed. The experimental conditions are listed in table 1.
Four factors at three levels [electrode angle, length of the arc (LA), current, and shielding
gas] were used to obtain the optimal experimental matrix.
Table 1. Test conditions.
Test
Electrode angle=8
Arc length=mm
Current=A
Shielding gas
Bead width=mm
A01
A02
A03
A04
A05
A06
A07
A08
A09
30
30
30
60
60
60
90
90
90
2
3
4
2
3
4
2
3
4
40
70
100
70
100
40
100
40
70
Ar
Ar ‡ 25%
Ar ‡ 25%
Ar ‡ 25%
Ar
Ar ‡ 25%
Ar ‡ 25%
Ar ‡ 25%
Ar
2
3
4
3
4
2
4
2
3
He
He
He
He
He
He
The width of the weld for each experimental condition was recorded with an image
analyser, Neophot 21. The weld width determination is shown in figure 5 for the test
A01 (sample 01). Figure 6 presents the voltage signs and current during the welding
process for this test.
The values of the thermal properties of AISI 304 (considered constant) were obtained
from the literature (Incropera and DeWitt 1996): thermal conductivity was taken as equal
to 14.9 W mKÿ1 and thermal diffusivity as equal to 3:95610ÿ6 m2 sÿ1. The coefficient of
S M M Lima e Silva, L O Vilarinho, A Scotti, T H Ong, G Guimara¬es
122
Dark-etching
region
Bead width
Weld pool
2 mm
(a)
(b)
Figure 5. Weld bead: (a) measured dimensions, (b) view from above.
Voltage=V
15
Current=A
100
80
60
40
20
0
2000
10
2010
2020
2030
Time=ms
2040
2050
5
0
2000
2010
2020
2030
Time=ms
2040
2050
Figure 6. Welding voltage and current.
heat transfer by convection was also taken as constant and equal to 15 W mÿ2 Kÿ1 for
all the surfaces. It should be noted that this value represents an average value obtained
from empirical correlation (Incropera and DeWitt 1996) for specific geometry and the
temperature.
A mesh of 73671650 volumes in the respective directions of y, z, and x has been
constructed so as to coincide with the position of each thermocouple and its respective
control volume. Figure 7 shows the measured temperatures at the rear surface of the
test plate A02 (sample 02). The time for all tests (table 1) was about 25 s. In figure 7 only
the first 149 points are shown.
Thermocouple
Experimental temperature=8C
400
350
1
3
5
7
9
2
4
6
8
10
300
250
200
150
100
50
0
Figure 7. Temperature evolution at the surface
at the rear of the weld ötest A02.
0
10
20
30
Time=s
40
50
60
Figure 8 shows the heat input rate estimated at the surface calculated from the
experimental temperatures shown in figure 7. The estimated heat input rate for the most
severe welding condition (test A03) is shown in figure 9.
It is seen in figure 7 that, although the maximum temperatures fluctuate around an
average temperature, the dispersion is significant. This dispersion at the rear side is
responsible for a deviation in the estimated power of the order of 35 W (test A03).
An investigation of this behaviour can indicate, in the future, procedures for optimising
welding processes and obtaining more stable effective power.
Heat flux in gas ^ tungsten-arc welding
123
800
500
Estimated heat input rate=W
Estimated heat input rate=W
600
400
300
200
100
0
0
10
20
30
40
Time=s
50
700
600
500
400
300
200
100
0
60
0
10
20
30
40
Time=s
50
60
Figure 8. Estimated heat input rate in test A02. Figure 9. Estimated heat input rate in severe
conditions (test A03).
Figure 10 shows the heat flux field at the welding face in test A03, at the instants of
7.98 s, 10.64 s, 14.06 s, and 17.86 s. One can see the effect of the speed of the moving
source as well as the stability of the estimated heat flux. One can also see the symmetry
effect related to the direction of the moving source.
The behaviour of the estimated components of the heat flux can be better seen in
figure 11 for four specific positions, those of thermocouples 2, 4, 7, and 9. The effect of
the heat source moving in the y direction can also be noticed.
The comparison between the estimated and experimental temperatures at three
positions for the severe welding condition is shown in figure 12. There is a reasonable
agreement of the experimental and estimated temperatures with deviations of 9%, 3.3%,
and 4.4% of the maximum values. Several factors can be responsible for these small
discrepancies, for example, the assumptions of constant values for the thermal properties
and the assumed coefficient of heat transfer by convection in the theoretical model.
z=mm
30
(a)
Heat flux=W mÿ2
27
9597966
8958101
8318237
7678372
7038508
6398644
5758779
5118915
4479051
3839186
3199322
2559457
1919593
1279729
639864
24
21
(b)
z=mm
30
27
24
21
(c)
50
100 150
y=mm
200
50
(d)
100 150
y=mm
200
Figure 10. Heat flux field in welding at the following instants: (a) 7.98 s, (b) 10.64 s, (c) 14.06 s,
and (d) 17.86 s.
S M M Lima e Silva, L O Vilarinho, A Scotti, T H Ong, G Guimara¬es
124
Heat flux= 10ÿ6 W mÿ2
12
10
8
6
4
2
0
ÿ2
ÿ4
(a) position 2
(b) position 4
Heat flux=10ÿ6 W mÿ2
12
10
8
6
4
2
0
ÿ2
ÿ4
0
10
20
(c) position 7
30 40
Time=s
50
60
0
10
20 30 40
Time=s
50
60
(d) position 9
Figure 11. Estimated heat flux at four different thermocouple positions.
Thermocouple
450
2
400
5
8
experimental
model
350
T=8C
300
250
200
150
100
Figure 12. Comparison of calculated and measured
temperatures at the surface of the rear of the weld for
thermocouples 2, 5, and 8.
50
0
0
10
20
30
40
Time=s
50
60
Besides, effects such as phase change and heat losses by radiation have not been taken
into account. A theoretical model that takes account of all these aspects is being
developed.
All the same, the results obtained here are promising. They show that the experimental
methodology is adequate and that it is necessary to optimise the thermal model.
It is also seen that the use of a discrete model (space and time) for the moving heat
source is appropriate in analysing a thermal phenomenon. This represents, in fact,
an advance over the three-dimensional model of Rosenthal which is based on a quasistationary state and constant heat source. An a priori assumption of constant power
does not allow a real investigation of the effective power delivered to the plate.
Heat flux in gas ^ tungsten-arc welding
125
5 Statistical planning method
The statistical planning method, Taguchi, was made through a L9 matrix in order to reduce
the number of experiments. The method uses the average heat input rate for each experimental condition (table 1). This average heating rate is used to obtain the thermal efficiency
Zˆ
q
,
VI
(12)
where V is the voltage, I is the current, and q the average heat input rate or effective
power. Table 2 shows for each test, the average voltage and current, Vm and Im respectively, the calculated total power, Pt , the estimated heat input rate, q, and the calculated
thermal efficiency, Z.
Table 2. Heat input rate and thermal efficiency for each test.
Test
Electrode Arc length
angle=8
/mm
Shielding
gas
A01
A02
A03
A04
A05
A06
A07
A08
A09
30
30
30
60
60
60
90
90
90
Ar
Ar ‡ 25%
Ar ‡ 25%
Ar ‡ 25%
Ar
Ar ‡ 25%
Ar ‡ 25%
Ar ‡ 25%
Ar
2
3
4
2
3
4
2
3
4
He
He
He
He
He
He
Im =A
Vm =V Total
q (average heat
power=W input rate)=W
Z=%
41
71
102
71
101
40
101
41
71
8.2
9.8
10.8
9.0
9.6
10.6
8.5
10.4
9.7
83.4
74.2
69.5
76.1
59.1
68.6
66.6
86.9
72.0
336
696
1102
639
970
424
859
426
689
280
516
766
486
573
291
572
371
496
With the thermal efficiency, Z, for each test (table 2), the larger-the-better function
of the Robust Planning of Taguchi was used to analyse the results. In the analysis, the
variable z was defined to relate thermal efficiency through the following equation
z ˆ 8:6859 ln Z ÿ 3610ÿ6 .
(13)
Figure 13 shows the variation of z for the four factors studied. It can be seen that
maximum efficiency should be obtained when the electrode angle is 308, the arc length
is 2 mm, the current is 40 A, and the gas used is Ar ‡ 25% He. This combination was
then executed in test A10. The results obtained are presented in table 3. It is seen that
the efficiency found in test A10, of 86.2%, is very close to the maximum thermal
efficiency, 86.9%, obtained in previous tests (table 2). This procedure eliminates the need
for a larger number of experiments.
38.0
37.8
37.6
37.4
z
37.2
37.0
36.8
36.6
36.4
36.2
36.0
30 60 90
Electrode
angle=8
2
3
4
Arc length
=mm
40 70 100 Ar Ar ‡ 25%He
Current=A Shielding
gas
Figure 13. Statistical Taguchi results.
S M M Lima e Silva, L O Vilarinho, A Scotti, T H Ong, G Guimara¬es
126
Table 3. Results of test A10.
Test
Electrode Arc length
angle=8
/mm
Shielding
gas
Im =A
Vm = V Ptotal = W
q (average heat
input rate)=W
Z=%
A10
30
Ar ‡ 25% He
41
9.2
323
86.2
2
375
6 Conclusions
The use of the inverse heat conduction problem techniques to determine temperature
fields, heat input rate, and thermal efficiency in the GTA welding process is presented.
The heat rate at the weld surface has been estimated by using the conjugated gradient
method with the adjoint equation. The experimental technique is seen to be adequate in
spite of not accounting for the effects of phase change and heat losses by radiation. It is
seen that the values found for the thermal efficiency indicate the potential of the method
as an alternative to techniques based on calorimeters.
Acknowledgements. We would like to thank Fapemig and CNPq Government Agencies for financial
support of this work. The technical support of Frederico R S Lima was invaluable.
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ß 2003 a Pion publication