Biomedical Engineering Reference
In-Depth Information
The pressure difference between the input and the output is a function of the density,
ρ
(kg/m
3
)
, and the acceleration due to gravity,
g
(m/s
2
),
P
=
ρ
gh
(4.48)
Rewriting equation (4.47) in terms of the pressure gives
A
ρ
g
dP
dt
q
1
−
q
2
=
(4.49)
Where the hydraulic capacitance,
C
, is defined as
A
ρ
g
C
=
(4.50)
Equation (4.49) can then be written as a differential equation in terms of the hydraulic
capacitance
C
dP
dt
q
1
−
q
2
=
(4.51)
or in terms of an integral as
1
C
P
=
(
q
1
−
q
2
)
dt
(4.52)
Finally, hydraulic inertiance is equivalent to electrical inductance. Consider the cylin-
drical section of fluid shown in Figure 4-12. The difference between the two forces can
be described in terms of the difference between the two pressures
(
P
1
−
P
2
)
across the
cylinder
F
1
−
F
2
=
P
1
A
−
P
2
A
=
(
P
1
−
P
2
)
A
(4.53)
The net force,
F
(N), that causes the mass of fluid to accelerate at
a
(m/s
2
)is
F
=
ma
(4.54)
Therefore,
(
P
1
−
P
2
)
A
=
ma
(4.55)
Equation (4.55) can be rewritten in terms of the rate of change of velocity
(
P
1
−
P
2
)
A
=
m
d
dt
(4.56)
FIGURE 4-12
Forces applied to a
cylinder of fluid.
