Biomedical Engineering Reference
In-Depth Information
Fig. 8.9 Internal friction in the liquid or gas
Condition 1. The pressure difference,
D
p
, and the viscosity,
m
, of the liquid or
gas is constant. The pressure force on the piston is:
S
2
S
2
F
A
¼ D
p
A
A
A
¼ D
p
B
A
B
¼
F
B
;
(8.56)
where
A
is the area of the cylinder, and
p
is the pressure difference on the piston.
The velocity
V
of the piston could be limited by the internal friction of the liquid or
gas in the channel that feeds them to the cylinder (Fig.
8.9
). To estimate the velocity
limit, it is necessary to calculate the flow rate,
Q
, using the equation:
D
D
4
128
m
p
Q
¼
L
ð
p
1
p
2
Þ;
(8.57)
where
D
and
L
are the diameter and the length of the channel,
p
1
and
p
2
are the inlet
pressure and the exit pressure of the channel, and
m
is the viscosity. From equation
(8.57) we have:
p
D
A
p
S
4
D
B
S
3
Q
A
¼
L
A
ð
p
1
p
2
Þ¼
L
B
ð
p
1
p
2
Þ¼
Q
B
(8.58)
128
m
A
128
m
B
S
From formula (8.58) we conclude that the flow rate decreases with the cube of
the device size. In this case, the velocity of channel flow can be calculated as:
Q
A
4
S
3
Q
B
4
V
A
¼
D
A
¼
D
B
¼
S
V
B
:
(8.59)
p
p
S
2
Equation (8.59) is equivalent to the condition
t
=
const
.
Condition 2. We mentioned that a more advanced condition is
V
=
const
. In order
to obtain this condition, it is necessary to decrease the viscosity of the liquid or gas
in device
B
in accordance with the equation:
m
A
¼
S
m
B
:
(8.60)
