Environmental Engineering Reference
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and the energy equation:
DT
Dt
=∇·
K
∇
T
+
=
:∇
ʽ,
(3)
where
denotes the velocity, pressure, thermal conductivity, density
and the stress tensor, respectively.
As a first step in addressing the problem we assume that the screw is stationary
and the barrel is rotatory, it is also assumed that the screw can be unrolled.
ʽ
,
p
,
K
,
ˁ
and
=
•
The flow is time independent, fully developed and only occurs in a longitudinal
direction (z axis).
•
=
An incompressible fluid with constant thermal conductivity (
k
constant) and
non-Newtonian (power law model) is considered.
•
Constant pressure gradient along the channel.
•
The viscosity depends on the shear rate and on the temperate.
Because the PLA is a polymer in which their behavior is shear rate dependent
(non-Newtonian), the power law model is used as a constitutive model coupled with
the energy equation.
n
−
1
ʷ( ʳ)
=
m
(
T
) ʳ
,
(4)
where
m
0
e
−
a
(
T
−
T
0
)
,
m
(
T
)
=
(5)
is a constant with units
T
.
It is preferable to work in dimensionless terms, so defining the following dimen-
sionless variables:
a
=
H
,ˆ
=
ʽ
z
y
ʾ
=
V
,ʸ
=
a
(
T
−
T
0
),
(6)
the velocity profile is
ˆ
=−
[
(
e
ʸ
]
1
/
n
ʾ
+
)
,
G
M
1
(7)
subjected with the following boundary conditions
ˆ
(ʾ
=
0
)
=
0
,
ˆ
(ʾ
=
1
)
=
1
,
(8)
where
1
=
p
L
H
V
=
,
G
H
,
(9)
1
/
n
ʷ
0
where
M
1 is an integration constant.
In the same way the temperature profile is:
ʸ
+
β
[
(
e
ʸ
]
1
/
n
G
ʾ
+
M
1
)
(
G
ʾ
+
M
1
)
=
0
,
(10)
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