Environmental Engineering Reference
In-Depth Information
Fig. 1
a
Representation of the system of study and
b
heat transfer problem
that blood and tissue properties are not altered by time or position, and also, blood
perfusion will remain constant. It will be assumed a thermal equilibrium between
the arterial blood temperature and the interstitial tissue core, as it remains constant,
T
a
=
T
c
. The surface temperature
T
s
will be determined by the transfer processes
within the tissue and the thermal exchange with the environment, as shown in Fig.
1
.
For transient processes or very small scales, it will be necessary to take into account
the delay between the temperature gradient and the energy flux. Accordingly, the
classical Fourier equation is modified by the Maxwell-Cattaneo Joseph and Preziosi
(
1989
,
1990
), Chandrasekharaiah (
1998
), Ostoja-Starzewski (
2007
) model given by:
)
+
ʻ
∂
(
,
)
q
x
t
(
,
+
ʻ)
=
(
,
=−
∇
(
,
),
q
x
t
q
x
t
k
T
x
t
(2)
∂
t
C
2
denotes the relaxation time of the tissue,
where
the thermal diffusivity,
C
the thermal velocity of propagation in the medium,
k
is the thermal conductivity
and
q
is the heat flux. The relaxation time represents the time lag required to establish
steady heat conduction in a volume element once a temperature gradient has been
imposed across it Chandrasekharaiah (
1998
). The partial time derivative added by
Cattaneo in the constitutive relationship between the heat flux and the temperature
succeeded in resolving the main shortcoming of the Fourier model, rendering the
heat-conduction equation to a damped hyperbolic equation.
ʻ
=
ʱ
/
ʱ
2.1 Dimensionless Equation
A dimensionless form of the heat transfer model (Eq.
1
) can be obtained by using the
following dimensionless variables:
y
L
T
T
C
x
ʸ
=
for temperature;
ʾ
=
L
and
ʷ
=
for
˄
=
L
2
t for time, defined
as the Fourier number which compares the conduction energy with the storage in
position, which are bounded in a range from 0 to 1; and
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