where
n Variation index, n % 1:2 for compressed air.
If the flow process of the air in the intake passage and the change of the gas state
in the cylinder are all polytropic process, then there is
c 0 A 0
p s
ffiffiffiffi ffi
T s
p
T s
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ng
R
2
n þ 1
n þ 1
ð
Þ= nÀ1
ð
Þ
s
dt ¼
V 0
nRT 0
p
p 0
1Àn
ð
Þ=n
dp
ð8:33’Þ
or
p
1Àn
ð
Þ=n dp ¼
nRT 0
V 0
p
1Àn
ð
Þ=n
0
G max
ð
Þ n dt
ð8:34’Þ
The gas variation index n is constant. The pressure change in the working
chamber of the cylinder can be obtained by integrating the above formula.
Z p
p 0
p
1Àn
ð
Þ=n dp ¼
Z t
0
nRT 0
V 0
p
1Àn
ð
Þ=n
0
G max
ð
Þ n dt
Initial conditions: when t ¼ 0, the in-cylinder pressure p ¼ p 0 % p a (ambient
atmospheric pressure). So it is obtained
p
1=n
À p
1=n
a ¼
nRT 0
V 0
p
1Àn
ð
Þ=n
a
G max
ð
Þ n t
ð8:35Þ
Because of the critical state of the maximum weight flow [Reference Eq. (8.30)]
G max
ð
Þ n ¼ c 0 A 0
p s
T s R
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi
ngRT s
2
n þ 1
n þ 1
ð
Þ = nÀ1
ð
Þ
s
So the pressure change in the working chamber of cylinder is calculated by
Eq. (8.35) as
p ¼ p a
RT 0
V 0 p a
c 0 A 0
p s
T s R
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi
ngRT s
2
n þ 1
n þ 1
ð
Þ = nÀ1
ð
Þ
s
2
4
3
5
n
ð8:36Þ
The above formula describes the variation of the pressure p on the piston (i.e., in
the working chamber of the cylinder) with time t during the charging process. It
shows that with the increase of intake time, the pressure in the working chamber of
cylinder increases; when the p value reaches the starting pressure p a of the piston,
the piston begins to move. The gas pressure p in the working chamber of cylinder
8.2 Structure and Characteristics of Actuators
31
n Variation index, n % 1:2 for compressed air.
If the flow process of the air in the intake passage and the change of the gas state
in the cylinder are all polytropic process, then there is
c 0 A 0
p s
ffiffiffiffi ffi
T s
p
T s
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ng
R
2
n þ 1
n þ 1
ð
Þ= nÀ1
ð
Þ
s
dt ¼
V 0
nRT 0
p
p 0
1Àn
ð
Þ=n
dp
ð8:33’Þ
or
p
1Àn
ð
Þ=n dp ¼
nRT 0
V 0
p
1Àn
ð
Þ=n
0
G max
ð
Þ n dt
ð8:34’Þ
The gas variation index n is constant. The pressure change in the working
chamber of the cylinder can be obtained by integrating the above formula.
Z p
p 0
p
1Àn
ð
Þ=n dp ¼
Z t
0
nRT 0
V 0
p
1Àn
ð
Þ=n
0
G max
ð
Þ n dt
Initial conditions: when t ¼ 0, the in-cylinder pressure p ¼ p 0 % p a (ambient
atmospheric pressure). So it is obtained
p
1=n
À p
1=n
a ¼
nRT 0
V 0
p
1Àn
ð
Þ=n
a
G max
ð
Þ n t
ð8:35Þ
Because of the critical state of the maximum weight flow [Reference Eq. (8.30)]
G max
ð
Þ n ¼ c 0 A 0
p s
T s R
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi
ngRT s
2
n þ 1
n þ 1
ð
Þ = nÀ1
ð
Þ
s
So the pressure change in the working chamber of cylinder is calculated by
Eq. (8.35) as
p ¼ p a
RT 0
V 0 p a
c 0 A 0
p s
T s R
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi
ngRT s
2
n þ 1
n þ 1
ð
Þ = nÀ1
ð
Þ
s
2
4
3
5
n
ð8:36Þ
The above formula describes the variation of the pressure p on the piston (i.e., in
the working chamber of the cylinder) with time t during the charging process. It
shows that with the increase of intake time, the pressure in the working chamber of
cylinder increases; when the p value reaches the starting pressure p a of the piston,
the piston begins to move. The gas pressure p in the working chamber of cylinder
8.2 Structure and Characteristics of Actuators
31