Biomedical Engineering Reference
In-Depth Information
value of
k
for the fingertips, whereby abundant microvessels exist, is set to be 5.0
×
10
−
13
m
2
, while
13
m
for the other parts of the hand (Nield and Bejan 1998).
Considering the continuity equation and the momentum equation, an equation for pressure in
porous media can be obtained that is expressed as
it is set to be
1.0
×
10
−
µ
µ
=−
∇=∇⋅ −
2
P
V
∇⋅
V
= 0
(23.14)
Darcy
Darcy
k
k
The dimensionless form of Equations 23.13 and 23.14 can be written as
∇=0
2*
P
(23.15)
V
*
=−
Da
⋅
Re
⋅ ∇
P
*
(23.16)
Darcy
where
Da
is the Darcy number such that
Da = k / D
2
and
Re
is the Reynolds number such that
ρµ
Re
=
UD
/
.
23.4.3.2 Heat Transfer Modeling based on Pennes equation
In order to obtain the temperature of living tissue, energy equations for the blood and tissue phase
are needed, and this may cause the simulation to become complex. As an alternative, Pennes equa-
tion is used to investigate heat transfer in living tissues, which is expressed as
∂
∂
T
t
ρ
c
=∇ ++
λ
2
TQ
ωρ
cT T
(
−
)
(23.17)
mb
bb
b
where
r
indicates the tissue density,
c
is the specific heat of the tissue,
r
b
and
c
b
represent the den-
sity and specific heat of blood, respectively,
T
b
is the temperature of blood that perfuses the tissue,
which is assumed to be constant,
l
is the thermal conductivity of the tissue,
Q
m
is the heat produc-
tion per unit volume, and
w
b
indicates the blood perfusion rate.
If the relationship between Darcy's velocity and the blood perfusion rate is known, the local
blood perfusion can be obtained. Thus, coupling Darcy's equation and Pennes equation is an ele-
gant method to describe the non-uniformity of blood perfusion. It is assumed that the diameter
and length of the microvessels are the same in different parts of the tissue, thus, the ratio of the
microvessel blood flow and microvessel volume can be written as
QV vL
b
/
=
/
(23.18)
b
b
where
v
b
is the blood velocity and
L
is the length of the microvessel. Since the microvascular volume
and Darcy's velocity are written as
VV
b
ϕ
t
and
V
=
ϕ
v
b
, where
j
is the porosity of the tissue,
Darcy
the blood perfusion can be expressed as
ω
=
QV
/
=
ϕ
QV
/
=
ϕ
vLVL
/
=
/
(23.19)
b
b
t
b
b
b
Darcy
Substituting Darcy's velocity for the blood perfusion rate, the dimensionless form of the energy
equation can be expressed as
∂
∂
T
t
*
1
1
=∇+
2*
T
Pe
QV
*
+
βγ
*
(
TT
*
−
*
)
(23.20)
m
Darcy
b
e
*
where
Pe
is the Peclet number such that
Pe UD
= ,
b
is the ratio of the heat capacity of blood and
tissue, and
g
is the ratio of the diameter to the length of the microvessel.
/
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