Civil Engineering Reference
In-Depth Information
where
E
part
is the rate of change of the energy content in the part analyzed (
Q
),
E
g
is the rate of heat generation due to the electric resistive heating (
Q
e
), and
E
out
is
the rate of energy leaving the part due to conduction into the dies in the contact
zone (
Q
cond
), convection (
Q
conv
) and radiation from the surface (
Q
rad
). To clarify,
E
g
has only one contributor (
Q
e
), whereas
E
out
has three contributors (
Q
cond
,
Q
conv
, and
Q
rad
) that are the three forms of heat transfer. After developing each component, the
heat equation can be written in one dimension as Eq. (
4.3
):
ρ
V
v
C
p
∂
T
∂
t
=−
A
s
[
h
(
T
−
T
∞
)]−
2
kA
c
∂
T
T
4
−
T
surr
∂
x
−
A
s
εσ
SB
+ η(
1
− ξ )
VI
(4.3)
where
ρ
is the density of the material,
V
v
is the volume of the part,
C
p
is the
specific heat of the material,
T
is the temperature,
t
is time,
A
s
is the lateral sur-
face area of the part,
h
is the convection heat transfer coefficient,
T
∞
=
T
surr
is
the surrounding temperature,
k
is thermal conductivity for the die material,
A
c
is
the cross-sectional area,
x
is coordinate,
ε
is radiative emissivity for the part, and
σ
SB
is the Stefan-Boltzmann constant. The input from electricity was taken from
Eq. (
4.4
).
P
e
=
P
heat
+
P
def
=
η(
1
−
ξ )
VI
+
ηξ
VI
(4.4)
where
P
e
is the electric power (I
·
V),
η
is the efficiency, and
ξ
is the EEC.
P
heat
represents the amount of electric power that will dissipate into heat through resis-
tive heating of the workpiece.
P
def
is the electrical power component that will aid
the plastic deformation.
The changes in geometry are calculated assuming compression at constant
speed, as well as volume constancy, as given by Eq. (
4.5
):
∂
h
∂
t
=−
u
;
∂
D
∂
t
=
V
v
π
u
h
√
h
(4.5)
where
D
and h are instantaneous dimensions of the workpiece, and
u
is the com-
pression speed. The second equation was determined from the volume constancy
condition
(
V
v
=
π
D
2
h
4
)
.
The change in temperature solely due to the portion of electrical power that did
not contribute toward plastic deformation can be shown in the integral in Eq. (
4.6
).
t
f
T
(4.6)Tf
−
T
o
=
η
VI
mC
p
(
1
−
ξ )
d
t
(4.6)
0
where
m
is the mass of the workpiece. The EEC profile with respect to time can be
shown in Eq. (
4.7
).
ξ
=
ξ
0
∗
t
b
(4.7)
