Biomedical Engineering Reference
In-Depth Information
where
is the specific internal energy density for the mixture and
r
is the heat
supply density for the mixture given by
e
X
N
a¼
1
r
ðaÞ
e
ðaÞ
;
X
N
a¼
1
r
ðaÞ
1
r
1
r
e ¼
r
¼
r
ðaÞ
;
(10.41)
^
^
e
e
ð
t
Þ
and where the sum of the energy supplies
is denoted by
,
ðaÞ
n
o
X
N
^
^
ðaÞ
þ e
ðaÞ
^
^
e
ð
t
Þ¼
e
ðaÞ
ð
t
Þ
e
ð
t
Þ:
(10.42)
a¼
1
The key results of this section are the statements of the conservation of mass,
momentum, and energy for each constituent and the summation of these component
forms to yield statements of these conservation principles for the mixture. The
kinematic identities (
10.29
) and (
10.34
) form the other important result; its deriva-
tion is a suggested problem below.
Problems
10.5.1 Show that the constituent form of mass balance (
10.18
) summed over all the
constituents will produce the continuum statement (3.6).
10.5.2 Derive the formula relating the sum of the density-weighted, constituent-
specific, time derivatives to the time derivative following the selected
component, (
10.29
). In the course of this derivation you will likely employ
(
10.11
), (
10.12
), and (
10.28
).
10.5.3 Derive the formula (
10.34
) from (
10.29
) by setting
ˆ
ðaÞ
equal to
v
ðaÞ
.
10.6 A Statement of Irreversibility in Mixture Processes
Many physical quantities can be considered as influencing the specific internal
energy density
e
of a material object. (Recall that the specific internal energy density
e
was introduced in the section on the conservation of energy, Sect. 3.5). These
include, for example, the specific volume, the components of a tensor measuring
deformation or strain, the densities, or the concentrations of the constituents of the
mixture, and so on. In the mixtures of interest here the set of parameters
characterizing the thermodynamic substate of a particle
X
of the mixture, which
actually represents an RVE, will be the infinitesimal strain tensor
E
given by (2.52)
and each of the densities of the constituents,
r
(
a
)
. The notation {
E
,
r
(
a
)
} is introduced
for these parameters. The set of parameters {
E
,
r
(
a
)
} is said to characterize the
thermodynamic substate of a particle
X
in an object (i.e. a thermodynamic system).
Knowledge of the thermodynamic substate {
E
,
r
(
a
)
} does not, however,
completely characterize the thermodynamic properties of a thermodynamic system
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