Biomedical Engineering Reference
In-Depth Information
equation (1.11) for the solid matrix into the corresponding volume averaged
equations as follows:
For the blood phase:
f
T
∂x j
f
ερ f c p f
+ ρ f c p f
u j
T
∂t
εk f
ερ f c p f u j T f
f
∂x j
T
+ k f
V
=
Tn j dA
∂x j
A int
1
V
k f ∂T
1
V
+
∂x j n j dA
ρ f c p f u j Tn j dA
(1.19)
A int
A int
For the solid matrix phase:
(1
Tn j dA
s
s
ε ) ρ s c s
T
∂x j
ε ) k s
T
k s
V
(1
=
∂t
∂x j
A int
k f ∂T
1
V
1
V
∂x j n j dA +
ρ f c p f u j Tn j dA +(1
ε ) S m
(1.20)
A int
A int
s is the intrinsic average of the solid matrix temperature. Note that
the dispersion heat flux ρ f c pf u j T = ερ f c pf u j T f appears in the volume
averaged energy equation (1.19) for the blood phase, which may well be mod-
eled under the gradient diffusion hypothesis:
where
T
f
ερ f c pf u j T f = εk dis kj
T
(1.21)
∂x k
A number of expressions have been proposed for the thermal dispersion
thermal conductivity k dis kj . Nakayama et al. (2006) obtained a transport equa-
tion for the dispersion heat flux vector, which naturally reduces to the forego-
ing gradient diffusion form. For a bundle of vessels of radius R , they obtained
the following expression for the predominant axial component of k dis kj :
ρ f c pf
2
f R
f R
1
48
u
ρ f c pf
u
k dis xx =
k f
< 1
(capillary blood vessels)
k f
k f
(1.22a)
k dis xx =2 . 55 ρ f c pf
7 / 8
f R
f R
u
ρ f c pf
u
Pr 1 / 8 k f
> 1
k f
k f
(large arteries and veins)
(1.22b)
To close the foregoing macroscopic energy equations (1.19) and (1.20),
the terms associated with the surface integral, describing the interfacial heat
 
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