Environmental Engineering Reference
In-Depth Information
By rearranging Eq. (
5.57
) the energy equation can now be expressed as:
o
T
o
M
o
H
o
M
o
c
H
o
Þ l
0
T
o
t
þ l
0
T
o
t
þ
c
H
~
ð
v
r
T
ð
~
v
r
H
Þ
T
T
ð
5
:
58
Þ
H
H
h
i
¼
viscous
:
r~
¼
r
_
q
þ
v
eld-
dependent effective thermal conductivity (see some examples in Fang et al. [
98
,
99
],
Reinecke et al. [
100
], Parekh and Lee [
101
], and Nkurikiyimfura et al. [
102
,
103
]),
we de
Since in a magnetic
fl
uid, one may experience the anisotropy and the
ne the heat
fl
ux vector to be [
91
]:
_
q
¼
j r
½
T
ð
5
:
59
Þ
where
j
represents the second-order tensor, i.e. the thermal conductivity tensor.
If the magnetic material is isotropic, then Eq. (
5.59
) takes a conventional form:
q ¼
kr
T
ð
5
:
60
Þ
For the isotropic effective thermal conductivity, the energy equation Eq. (
5.58
)
takes the following form:
o
T
o
M
o
H
o
M
o
c
H
o
þ l
0
T
o
t
þ l
0
T
o
t
þ
c
H
~
v
r
T
~
v
r
H
T
T
ð
5
:
61
Þ
H
H
h
i
Þþ
¼
viscous
:
r~
¼
r kr
ð
T
v
5.3.1 A Short Note on the Effective Thermal Conductivity
The thermal conductivity is a well-de
ned physical property for monophase
fl
uids.
In the case of a suspension of solid particles and the carrier or base
uid, the effective
thermal conductivity becomes a model quantity. The basic approaches apply thermal
resistivity networks in analogy to the calculation methods for electric circuits [
104
].
Since Maxwell [
105
], who
fl
rst investigated the effective thermal conductivity,
many different models have been developed, as shown in Table
5.4
.
The effective thermal conductivity for the magnetic
ned by use of
different models, which have been developed for suspensions, solid composites as well
as nano
fl
uid can be de
uids. In most of the correlations, the effective thermal conductivity is
expressed as a function of the volume fraction of the solid phase, and by taking into
account the thermal conductivities of the solid phase and the liquid phase, respectively:
fl
k
¼
f
ðk
liquid
; k
solid
; /
V
Þ
ð
5
:
62
Þ
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