Environmental Engineering Reference
In-Depth Information
transient processes. The perfusion term can be written in a dimensionless form when
compared with a diffusive time as follows
(L
2
W
b
)/
(L
2
Q)/ (kT
c
)
˃
=
ʱ
ˆ
=
,justas
is the diffusive flux of energy, and
T
c
the dimensionless blood temperature.
A thermal Deborah number is defined to characterize the fluidity of materials under
specific flow conditions,
De
T
=
ʱʻ/
ʸ
a
=
T
a
/
L
2
; based on the premise that given enough time
even a solid-like material will flow. Flow characteristics are not inherent properties
of the material itself, but relative properties which depend on two fundamental dif-
ferent characteristic times. By substituting the above mentioned terms, the following
dimensionless differential equation for temperature distributions is obtained.
∂
2
2
2
De
T
∂
ʸ
∂˄
2
+
∂ʸ
ʸ
∂ʵ
+
∂
ʸ
ʸʷ
=
+
˃
[
ʸ
a
−
ʸ
]={∅
m
+∅
r
}
(3)
2
2
∂˄
The proposed dimensionless solution for the Eq. (
3
) is given by
e
−
˃˄
ʸ(ʾ,ʷ,˄)
=
ʸ
0
(ʾ, ʷ)
+
ʸ
t
(ʾ,ʷ,˄)
(4)
where
the transient tempera-
ture. By using the Eq. (
4
)inEq.(
3
), a differential equationwith relaxation is obtained:
ʸ
0
(ʾ, ʷ)
is the steady state temperature and
ʸ
t
(ʾ,ʷ,˄)
2
2
2
D
e
t
∂
ʸ
˄
∂˄
D
e
t
]
∂ʸ
˄
∂˄
2
De
t
=
∂
ʸ
∂ʵ
+
∂
ʸ
˄
∂ʷ
+∅
r
e
˃˄
}
,
+[
1
−
2
˃
+
˃
(5)
2
2
2
where
.
The corresponding dimensionless boundary conditions for
ʸ
t
=
ʸ
0
(ʵ,ʷ,˄)
and
∅
r
=∅
r
(ʵ,ʷ,˄)
˄>
0
are:
d
ʸ
t
(ʾ,
1
,˄)
e
˃˄
,
B
.
C
.
1
:
ʷ
=
1
;−
={ˆ
s
+
Bi
0
[
ʸ
f
−
ʸ
0
s
]}
(5a)
d
ʷ
B
.
C
.
2
:
ʷ
=
0
;
ʸ
t
(ʷ,
0
,˄)
=
0
,
(5b)
d
ʸ
˄
d
B
.
C
.
3
:
ʾ
=
0
;
=
0
,
(5c)
ʾ
d
ʸ
˄
d
B
.
C
.
4
:
ʾ
=
1
;
=
0
.
(5d)
ʾ
Where
Bi
0
is the Biot number in steady state, which compares the internal resis-
tance of the energy transfer with the resistance at the border, in this case, with the
surface of the tissue;
ʸ
0
is the steady state temperature,
ʸ
˄
the transient temperature,
ʸ
0
ʵ
=
ʸ
0
(ʾ
=
ʸ
f
the dimensionless air temperature.
The last parameter of Eq. (
5a
) may be altered by a sudden change or a small pertur-
bation in the atmosphere, which occurs when the surface of the tissue is subjected
to an air stream, creating a stochastic value. Equation (
6
) may be used to determine
this perturbation:
0
)
the surface temperature, and
ʸ
f
=
ʸ
0
s
+∈
(
t
)
;
where
∈
(
t
)
=
ʻ
T
(
0
.
5
−
˃
i
),
(6)
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