Civil Engineering Reference
In-Depth Information
Welding Parameters For Data Set 2:
Type: bead in groove with flat top surface
Weld speed
V
B
: 3.81
3
10
m/s
Average energy input
Q
/
D
: 2.159
10
3
kJ/m
Bead depth below surface: 9.5 mm
Bead width at surface: 15 mm
Plate thickness: 0.0254 m
Maximum length of trailing solidification boundary at surface: 14.5 mm
Power input (Q/D)
V
B
: 8.226
10
3
4
10
Grid spacing:
m
Time interval for passage of discrete volume elements past a given point:
l
2
2
10
t
l
/
V
B
2.625
s
Appendix B
Derivation of Weighted Finite-Sum Representation
of Pseudo Steady State Based on Rosenthal Solution
According to the Rosenthal solution of Eq. (6.1) for thick plate boundary conditions, the change in
temperature at a position
p
due to an energy source at position
p
1 whose power is
q
p
1
, distance from
p
is
R
and speed relative to the workpiece is
V
B
is
q
p
1
V
B
2
T
p
----------------exp
----------
(
Rx
)
,
(6.B1)
2
k
p
R
p
where
k
p
p
-------------------------------.
(6.B2)
T
()
C
p
T
()
For a uniform discretization of the solution domain corresponding to a node separation of
l
, it follows
that
3
T
()
C
p
T
()
(
T
p
T
p
1
)
()
l
q
p
1
--------------------------------------------------------------------------,
(6.B3)
t
l
/
V
B
. Next, we note that for a discrete partitioning of the temperature field,
a change in temperature at position
p
is related to the changes in temperature of its nearest neighbor
positions by the expression
R
l
,
x
l
and
y
6
1
T
p
--------
i
T
i
,
(6.B4)
6
p
i
1
where
T
i
T
i
T
p
. Equating Eq. (6.B1) and Eq. (6.B2), it follows that
6
(
T
p
T
p
1
)
V
B
l
V
B
l
1
--------
------------------------------------------
------------
i
T
i
(
T
p
)
exp
(6.B5)
6
p
2
p
p
i
1
