Biomedical Engineering Reference
In-Depth Information
Substituting
Equation 3.33
and converting the mass integral into a volume integral,
Equation 3.38
becomes
ð
ð
ð
5
@
@
-
b
-
s
d
-
-
r
-
xyz
-
xyz
-
xyz
1
2
ρ
dV
ρ
dV
1
ρ
U
ð
3
:
39
Þ
t
V
V
area
To account for the acceleration of an inertial body, relative to the inertial reference
frame, the conservation of linear momentum formulation requires one extra term. When
the system is not accelerating relative to the inertial frame,
-
r
is zero and
Equation 3.39
simplifies to
Equation 3.33
. To apply
Equation 3.39
to a system, it is required that there are
two coordinate systems defined at the beginning of the problem; one is inertial coordinate
system (
XYZ
), and the other stays with the moving control volume (
xyz
). This formula is
valid for one instant in time, similar to
Equation 3.33
. However, it is possible in particular
situations that the mass (i.e.,
-
b
; ρ
) and the acceleration (
-
r
) are functions of time. This
equation can easily be adapted for that scenario. Also,
Equation 3.39
is a vector equation,
with all velocity components related to the non-inertial reference frame (
xyz
). Written in
component form,
Equation 3.39
becomes
ð
ð
ð
5
@
@
-
bx
1
-
sx
2
d
-
-
xyz
U
-
rx
ρ
dV
u
xyz
ρ
dV
u
xyz
ρ
1
t
area
V
V
ð
ð
ð
5
@
@
-
by
-
sy
d
-
-
ry
-
xyz
ρ
dV
v
xyz
ρ
dV
v
xyz
ρ
1
2
1
U
ð
3
:
40
Þ
t
V
V
area
ð
ð
ð
5
@
@
-
bz
1
-
sz
2
d
-
-
rz
ρ
-
xyz
U
dV
w
xyz
ρ
dV
w
xyz
ρ
1
t
V
V
area
Example
One of the first implantable mechanical heart valves was designed as a ball within a cage that
acted as a check valve. Using the conservation of momentum (with acceleration), we will model
the acceleration of the ball after it is hit by a jet of blood being ejected from the heart
(
Figure 3.15
). The ball has a turning angle of 45
and a mass of 25 g. Blood is ejected from the
heart at a velocity of 150 cm/s, through an opening with a diameter of 27 mm. Determine the
velocity of the ball at 0.5 sec. Neglect any resistance to motion (except mass).
FIGURE 3.15
Acceleration of a ball and cage mechanical heart valve
for the in-text problem.
45
↓
Y
A, v
inflow
v
ball
m
X
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