Civil Engineering Reference
In-Depth Information
kT
i
m
+
1
x
2
2
k
x
2
2
2
k
a
2
w
L
die
ρ
A
1
c
+
htw
ρ
A
1
2
c
T
i
+
1
m
=
T
i
m
+
t
T
i
m
−
+
ρ
c
ρ
c
(6.8)
+
2
k
a
2
wT
die
L
die
ρ
A
1
c
+
htwT
∞
ρ
A
1
2
c
+
e
gen,clamp
ρ
c
For the nodes in the testing region,
kT
i
m
−
1
2
k
x
2
ρ
c
+
2
hA
22
ρ
A
11
xc
T
i
+
1
m
=
T
i
m
+
t
T
i
m
x
2
ρ
c
−
(6.9)
+
kT
i
m
+
1
x
2
ρ
c
+
2
hA
22
T
∞
ρ
A
11
xc
+
e
gen,test
ρ
c
As the electrical current will be passing through the die and heat will be trans-
ferred from the sheet metal to the die, the die temperature will also change as a
function of time. To conserve the 1D nature of this analysis, the die is considered
to be a lumped mass with a uniform temperature. This is an accurate assumption
as the Biot number is less than 0.1 for the die geometry and heat transfer proper-
ties [
3
].
Thus, the power balance for the dies is given by
T
(6.11)Tiavg,mg clamp
−
T
die
2
k
a
2
A
3
T
∞
−
T
die
hA
s
,die
+
L
die
+
e
gen,die
V
Ti+1die
= ρ
a
2
V
die
c
a
2
T
i
+
1
(6.10)
Ti+1die
−
T
die
t
where
A
s
,die
is the die surface area,
T
die
is the present die temperature,
A
3
is the full
conduction area between the sheet and the die,
T
Tiavg, mg clamp
is the average temper-
ature at the clamping region for the sheet,
V
die
is the volume of the die,
ρ
a
2
is the
density of the die material (A2 Steel) which is a function of temperature,
c
a
2
is the
heat capacity of the die material, and
T
i
+
1
Ti+1die
−
T
die
is the temperature change of the
die from the present time to the future time.
Additionally, the average temperature at the clamping region for the sheet is
defined by
b
T
(6.11)Tiavg,mg clamp
=
1
b
T
i
m
,clamp region,
b
(6.11)
1
where
b
is the number of nodes in the clamping region.
Using an explicit solution approach, Eq. (
6.10
) can be written as follows:
