move, the volume V can be considered as the volume of the rigid pressure vessel.
According to the relationship between the rate of change of gas weight entering the
pressure chamber and the rate of pressure rise (the change of gas state is an
isothermal process), it can be written that,
d DW
ð
Þ
dt
¼
V 0
RT
dp
dt
ð8:28Þ
If the change of gas in cylinder V 0 is an adiabatic process, then
d DW
ð
Þ
dt
¼
V 0
kRT
dp
dt
or
G ¼
V 0
kRT
dp
dt
ð8:29Þ
where
DW Mass of the gas entering cylinder;
V 0 When the piston is in the initial position (or when the piston is in a certain
residence position), the volume surrounded by the inner diameter of cylinder
and the piston. The value of V 0 remains unchanged before the piston starts,
which can be understood as the volume of the rigid chamber;
k
Gas state change index, for atmosphere k ¼ 1:4;
R
Gas constant;
p
In-cylinder gas pressure, the initial pressure is p0;
T
Absolute temperature of gas in cylinder;
t
Time.
There may be two different situations for the flow into the working chamber V 0
of the cylinder, i.e., sonic flow or subsonic flow.
For the sound adiabatic flow p 0 =p s 0:5283 (when k ¼ 1:4), the maximum
weight flow into the cylinder is achieved.
G max ¼ c 0 A 0
p s
ffiffiffiffi ffi
T s
p
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi
kg
R
2
k þ 1
k þ 1
ð
Þ = kÀ1
ð
Þ
s
ð8:30Þ
For subsonic adiabatic flow p 0 =p s [ 0:5283, the gas mass flow into the cylinder
is
G ¼ c 0 A 0
p s
ffiffiffiffi ffi
T s
p
p
p s
1=k
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2kg
k þ 1
ð
ÞR
1 À
p
p s
kÀ1
ð
Þ=k
"
#
v
u
u
t
ð8:31Þ
8.2 Structure and Characteristics of Actuators
29