Biomedical Engineering Reference
In-Depth Information
where
Q
,
m
f
,
U,
and
P
are the heat flow from the warm end to the cold
end, the mass flow rate of the hot (or cold) fluid, the overall heat transfer
coecient, and the wetted perimeter, respectively. The group (
m
f
c
pf
)
2
/
(
UP
)
plays the same role as
Ak
ef f
in the one-dimensional insulation system. Upon
noting that
m
f
=
Aρ
f
ε
a
a
and
P
=
Aa
f
, Bejan's equation (1.57) may be
translated in the present bioheat transfer problem as
u
(1
d
+(1
ε
)
k
s
+
ρ
f
c
p
f
ε
a
a
2
s
d
dx
u
T
−
−
ε
)
S
m
= 0
(1.58)
a
f
U
dx
In these countercurrent heat transfer models, namely, Bejan's and Wienbaum
and Jiji's, the perfusion heat sources are ignored. Thus, in what follows, we
shall attempt to reduce the present set of governing equations to a single equa-
tion for the tissue temperature variations, without neglecting these perfusion
heat source terms.
When the blood flow is strong enough to neglect the macroscopic diffusion,
the energy equations (1.39) and (1.42) for arterial and venous blood flows for
the one-dimensional steady state reduce to
d
dx
ε
a
a
a
=
a
s
)
ρ
f
c
p
f
ω
a
a
ρ
f
c
p
f
u
T
−
a
f
h
f
(
T
−
T
−
T
(1.59)
d
dx
ε
v
v
v
=
v
s
)+
ρ
f
c
p
f
ω
a
v
ρ
f
c
p
f
u
T
−
a
f
h
f
(
T
−
T
T
(1.60)
where the interfacial heat transfer coecients are assumed to be the same
as in the case of Chato. However, the foregoing equations are different from
Chato's equations (1.50) and (1.51), since we do take account of the heat
transfer between the bloods and tissue. Upon noting the continuity relation-
ship
ε
a
a
=
v
as given by the continuity equations (1.37) and (1.40)
u
−
ε
v
u
with
ω
a
=
−
ω
v
, we subtract equation (1.60) from (1.59) to obtain
d
dx
ε
a
a
(
a
+
v
)=
a
v
)
a
+
v
)
(1.61a)
ρ
f
c
p
f
ω
a
(
ρ
f
c
p
f
u
T
T
−
a
f
h
f
(
T
−
T
−
T
T
or
a
d
dx
(
a
+
v
)=
a
v
)
ρ
f
c
p
f
ε
a
u
T
T
−
a
f
h
f
(
T
−
T
(1.61b)
a
)
/dx
=
ω
a
.
Weinbaum and Jiji (1985) proposed that the mean tissue temperature around
an artery-vein pair can be approximated as
as we note the continuity relationship, namely,
d
(
ε
a
u
−
a
+
v
s
=
T
T
T
(1.62)
2
Following their approximation, we obtain
a
s
2
ρ
f
c
p
f
ε
a
u
d
T
a
v
=
T
−
T
−
(1.63)
a
f
h
f
dx
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