Environmental Engineering Reference
In-Depth Information
2
O(ʵ
)
e
b
ʸ
0
(
b
ʸ
2
d
ˆ
0
d
ʳ
n
n
b
ʸ
1
d
ˆ
0
d
ʳ
n
−
1
d
ˆ
1
d
ʳ
2
d
d
ʳ
b
ʸ
1
)
d
ˆ
2
d
ʳ
+
+
+
2
2
d
ˆ
0
d
ʳ
n
−
2
d
ˆ
1
d
ʳ
n
2
−
n
−
=
,
0
(20)
2
e
b
ʸ
0
b
ʸ
1
d
ˆ
0
d
ʳ
n
+
1
d
ˆ
0
d
ʳ
n
d
ˆ
1
d
ʳ
d
2
ʸ
0
d
ʳ
+
+
(
n
+
1
)
=
0
.
(21)
2
The above equations are subjected to the following boundary conditions:
ʳ
=
0
;
ˆ
0
=
0
;
ʸ
0
=
0
;
ˆ
1
=
0
;
ʸ
1
=
0
;
ˆ
2
=
0
;
ʸ
2
=
0
ˉ
R
p
(
1
−
ʴ)
ʳ
=
1
;
ˆ
0
=
;
ʸ
0
=
1
;
ˆ
1
=
0
;
ʸ
1
=
0
;
ˆ
1
=
0
;
ʸ
1
=
0
(22)
U
1
3
,
2
,
3
,
4
,1.Thevalueof
The solution of Eqs. (
16
)-(
21
) was carried out for
n
=
n
1 represents a Newtonian regime as a way to validate the solution. Furthermore
a recursive solution for whatever value of 0
=
<
≤
1(Eqs.(
23
)-(
28
)) is obtained.
Figure
2
shows a comparison between the analytical solution using power law model
and the numerical solution using Perzyna's model, employing data given in Table
2
n
n
exp
−
b
ʸ
0
1
n
+
n
n
1
1
+
n
n
ˆ
0
=
(
c
0
ʳ
+
c
1
)
−
c
,
(23)
1
c
0
(
n
+
1
)
3
A
6
ʸ
0
=
ʳ
−
ʳ
+
ʳ,
(24)
Ta b l e 2
Material's
properties and data used in
the calculations
Property/parameter
Va l u e
Pin radius, mm
3.95
Shoulder radius, mm
9.50
Characteristic length H, mm
5.75
Welding speed U, mm
0.31
Workpiece material
AISI 1018
AISI 1018 density, kg m
−
3
7860.00
Pressure plunge, MPa
65.90
Conductivity, W m
−
1
K
−
1
30.00
T
0
temperature, K
1242.00
T
1
temperature, K
1552.00
m
0
,Pas
n
10
6
3.5
×
n
1
/
3, 1
/
2, 2
/
3, 3
/
4, 1
.
0
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