Biomedical Engineering Reference
In-Depth Information
out of the volume. However, the volume flow rate can be calculated at any one location at
any time within the system of interest. Its definition would be
ð
d
-
-
Q
ð
3
:
28
Þ
5
U
area
The volume flow rate divided by area is defined as the average velocity at a particular
section of interest:
ð
Q
A
5
1
A
d
-
-
v
avg
5
ð
3
:
29
Þ
U
area
From the special cases that we have discussed, as well as the general formula, we can
now use the conservation of mass to solve various fluid mechanics problems.
Example
Determine the velocity of blood at cross-section 4 of the aortic arch schematized in
Figure 3.9
.
Assume that the diameter of the blood vessel is 3 cm, 1.5 cm, 0.8 cm, 1.1 cm, and 2.7 cm at cross
sections 1, 2, 3, 4, and 5, respectively. Branches 2, 3, and 4 make a 75
,85
, and a 70
angle with
the horizontal direction, respectively. The velocity is 120 cm/s, 85 cm/s, 65 cm/s, and 105 cm/s
at 1, 2, 3, and 5, respectively. There is inflow at 1 and outflow at all of the remaining locations.
Assume steady-flow at this particular instant in time and that the volume of interest is non-
deformable.
Solution
Figure 3.10
highlights the given geometric constraints in this problem. The gray dashed box on
this figure represents one of the possible choices for the volume of interest. We will also make the
assumption that blood density does not change and has a value of 1050 kg/m
3
.
FIGURE 3.9
Schematic of the aortic arch.
3
2
4
1
5
FIGURE 3.10
3
Figure associated with the in-text example.
2
4
85°
75°
70°
Y
1
5
X
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