Biomedical Engineering Reference
In-Depth Information
First we will develop the conservation of mass principle for fluid mechanics. The basis
for this principle is that mass can neither be created nor destroyed within the volume/
system of interest. If the inflow mass flow rate is not balanced by the outflow mass flow
rate, then there will be a change in volume or density within the volume of interest. If the
inflow mass flow rate exceeds the outflow mass flow rate, mass will accumulate within the
system. If the reverse scenario is true, mass will be removed from the system. Under normal
conditions (e.g., not under strenuous activity), the blood volume within the heart remains
constant from beat to beat. Stated in other words, the mass ejected from the aorta and the
pulmonary arteries is recovered from the superior vena cava, the inferior vena cava, and
the pulmonary veins. However, you can imagine a case where the residual fluid mass
within the heart decreases from beat to beat. If you are experiencing severe blood loss due
to a laceration, early on the heart would continue to eject the normal amount of blood, but
the venous return would not be equal to this ejection volume. Therefore, the blood volume
in the heart would decrease. No matter what the case is regarding mass changes within the
volume of interest, mass must be conserved within the system of interest.
Before we move forward into the derivation of the conservation of mass of a system, we
will derive a general relationship for the time rate of change of a system property as a
function of the same property per unit mass of the volume (inherent property). This is
sometimes referred to as the Reynolds Transport Theorem (RTT) formulation. For mass
balance, the system property is mass and the inherent property is 1 (i.e., mass divided by
mass). For balance of linear momentum, the system property is momentum (
-
) and the
inherent property is velocity (
-
) (i.e., momentum divided by mass). For energy balance,
the system property is energy,
E
(or entropy,
S
) and the inherent property is energy per
unit mass,
e
(or entropy per unit mass,
s
). The system and volume of interest used in this
derivation will be a cube, but this same analysis technique can be applied to any geometry
(
Figure 3.8
).
We will also assume that the shape remains the same, but this analysis holds
for deformation as well. The system and volume have been chosen so that there is a region
that overlaps at some later time (area 2). Mass from area 1 enters the volume of interest
during
t
.
The following derivation will relate the time rate of change of any system property (
W
)
to its inherent property (
w
).
W
and
w
are arbitrary properties that are only used for for-
mula derivation (RTT). This formulation starts by using the formula of a derivative:
Δ
t
and mass from area 3 exits the volume of interest during
Δ
dW
dt
5
W
j
t
1
Δt
2
W
j
t
lim
Δ
Δ
t
t
0
-
FIGURE 3.8
t
t
+
Δ
t
System and volume of interest used to derive
the formula for conservation laws. The system of interest is
shown by the gray shaded cube, and the volume of interest is
the dashed cube. To use this formulation, one would need to
know the change in time between the two states shown in this
figure.
System
12
3
Volume
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