*Corr. Author’s Address: Inner Mongolia University of Science & Technology, Department of Mechanical Engineering, Baotou 014010, China, 380264335@qq.com 119
Strojniški vestnik - Journal of Mechanical Engineering 69(2023)3-4, 119-134 Received for review: 2022-07-01
© 2023 The Authors. CC BY 4.0 Int. Licensee: SV-JME Received revised form: 2022-11-24
DOI:10.5545/sv-jme.2022.266 Original Scientific Paper Accepted for publication: 2023-02-14
Influence Analysis and Performance Optimization  
of a Pneumatic Actuator Exhaust Utilization System
Yu, Q. – Li, F. – Tan, X.
Qihui Y u
1,2,3
 –·Fengqi Li
1,
* – Xin Tan
1
1 
Inner Mongolia University of Science & Technology, Department of Mechanical Engineering, China 
2 
Beijing Key Laboratory, Pneumatic and Thermodynamic Energy Storage and Supply, China  
3 
University of Nottingham, Department of Mechanical, Materials and Manufacturing Engineering, UK
Due to the loss of exhaust energy, the compressed air energy utilization efficiency in pneumatics system is low. To realize the recovery and 
utilization of cylinder compressed air, a new type of compressed air utilization system is proposed, and the composition and working principle 
of the system are introduced. According to the working process of the system, the mathematical model of the system is established by using 
equations such as energy equation and gas state equation. The model is simulated by MATLAB/Simulink tools, and the correctness of the 
mathematical model is verified by experiments. The mathematical models are converted into dimensionless models, and the main parameters 
affecting the system's operating characteristics are obtained through model simulation analysis. The influencing parameters are optimized by 
using the analytic hierarchy process and the grey correlation analysis method, and the exhaust utilization efficiency of the system is 34.7 % 
under the optimal parameter combination. To further utilize the compressed air expansion, the opening and closing time of the solenoid valve 
was controlled to study the energy-saving effect of the system. The study found that when the initial pressure of the compressed air supply 
tank was set to 0.5 MPa, 0.6 MPa, and 0.7 MPa, the maximum energy saving efficiency was 23.25 %, 24.99 %, and 26.12 %, respectively. 
When the volume of the compressed air supply tank is set to 0.8 L, 1 L, and 1.2 L, the maximum energy-saving efficiency is 30.01 %, 24.99 %, 
and 21.51 %, respectively. This paper provides a new technical scheme for compressed air recovery and reuse, as well as a theoretical basis 
for the control of subsequent compressed air utilization systems.
Keywords: pneumatic system, exhaust utilization, analysis, characteristic, energy saving
Highlights
• 	 An air exhaust utilization loop is proposed, which can recycle the air exhaust to realize energy savings. 
• 	 The speed of piston movement and the efficiency of exhaust gas utilization determine the performance of the system. 
• 	 The multi-objective parameters are optimized to keep the system stable and good performance. 
• 	 Controlling the on-off time of the solenoid valve can further improve the efficiency of gas utilization.
0  INTRODUCTION
The pneumatic system has the characteristics of low-
cost, high-strength ratio, convenient maintenance, 
cleanliness and environmental protection, and high 
safety [1] to [4]. However, compressed air is expensive 
to produce and is one of the most expensive energy 
carriers [5]. Research has found that the actual energy 
efficiency of some pneumatic systems is low, or even 
80 % of the energy will be wasted in the form of heat 
or other energy in the worst case, which increases the 
application cost of pneumatic systems [6]. Developing 
and researching the energy savings of pneumatic 
systems are urgent tasks.
In recent years, some researchers have changed 
and optimized the structure of the pneumatic circuit 
to improve its energy efficiency. Du et al. [7] proposed 
a bridge-type energy-saving loop that fully utilizes 
the expansion energy to do work, and significantly 
improved the system energy efficiency by 50 % to 
70 % through algorithm optimization control. After 
that, Du et al. [8] used a value to connect the two 
chambers of the cylinder to improve the bridge circuit 
and optimized the control through the finite element 
configuration method and the interior point method, 
which could achieve 55 % to 87 % of air energy saving 
and the cylinder runs more smoothly. Yang et al. [9] 
and [10] proposed a novel energy recovery booster 
valve (BVER) and studied its energy efficiency by 
introducing the concept of aerodynamics. Based on 
the thermodynamics of metamorphic compressed air, 
the energy efficiency evaluation and pressure response 
model of BVER is proposed, which increases the 
boost ratio by 15 % to 20 %, and the efficiency can 
be increased by 5 % to 10 % depending on the supply 
pressure. Endler et al. [11] added a quick-acting on-off 
valve to the circuit to reduce the impact of external 
load on compressed air consumption and proposed 
a compressed air energy-saving decision-making 
algorithm based on differential pressure and control 
signals. The research results show that compressed air 
consumption can be reduced by 47 % to 54 %. Shi et 
al. [12] and Shi and Cai [13] proposed an expansion 
energy booster, which cuts off the air supply before 
the piston moves to the end of the stroke and uses 
the air expansion to drive the remaining stroke to 

Strojniški vestnik - Journal of Mechanical Engineering 69(2023)3-4, 119-134
120 Yu, Q. – Li, F. – Tan, X.
achieve energy savings. The SMC company proposes 
a product that applies the air-saving circuit. It 
cooperates with the cylinder through the valve and 
uses the compressed air in the exhaust chamber of 
the cylinder as the power source when the piston rod 
retracts to achieve the effect of energy saving, which 
reduces the air consumption by 45 % compared with 
the general circuit [14]. For the horizontal installation 
of the cylinder, Beater [15] adds a pressure-reducing 
valve to the circuit to reduce the consumption of 
compressed air by controlling the air supply pressure 
and shows that the use of this circuit can save 25 % of 
compressed air. Doll et al. [16] and Doll and Sawodny 
[17] took full advantage of the expansion energy by 
controlling five reversing valves, and the studies show 
that the circuit can save 85 % of the air under optimal 
conditions. Some scholars make full use of exhaust 
energy in the form of energy conversion. Cummins 
recovers energy and stores strain energy with a rubber 
inner bladder [18]. Luo et al. [19] adopt a closed-loop 
coordinated control strategy to convert exhaust energy 
into electrical energy for reuse by utilizing scroll 
expansion technology. 
There are two ways to improve compressed 
air energy efficiency: changing the structure of the 
optimized pneumatic circuit and converting the 
energy of the compressed air. However, the former 
has a narrow application range and is difficult to 
control, and the latter has higher requirements on the 
conversion device and larger energy loss. In most 
conventional pneumatic circuits, the energy contained 
in the compressed air is rarely used efficiently. The 
method of recovering and utilizing compressed air is 
simple and easy to control and is an effective way to 
improve energy efficiency.
In some circuits for compressed air recovery 
and utilization, the two processes of recovery and 
utilization cannot be applied to the same scene, 
resulting in wasted compressed air energy. In view of 
the above problems, this paper proposes a pneumatic 
actuator exhaust utilization circuit and introduces 
the system composition and working principle. 
Secondly, the mathematical model of the system is 
established and an experimental platform is built, and 
the model is verified with experiments. Afterward, 
through a dimensionless model, the main parameters 
affecting the performance of the system are studied 
and optimized. Finally, by controlling the opening 
and closing time of the solenoid valve, the further 
utilization of the compressed air expansion energy 
is studied, which lays a good theoretical foundation 
for the next step to efficiently control the pneumatic 
system.
1  SYSTEM COMPOSITION AND WORKING PRINCIPLE
Fig. 1 shows the circuit diagram of the compressed 
air utilization system. It is mainly composed of the 
1  the compressed air power source
2  pressure reducing valve
3, 4, 19, 20  pressure transducer
5, 7, 10, 12, 17, 22, 23  solenoid directional valve
6, 11  silencer
8, 9  compressed air tank
13, 14  check valve
15, 18  one-way throttle valve
16  cylinder; and 
21  speed displacement sensor
Fig. 1.  Exhaust utilization circuit

Strojniški vestnik - Journal of Mechanical Engineering 69(2023)3-4, 119-134
121 Influence Analysis and Performance Optimization of a Pneumatic Actuator Exhaust Utilization System 
compressed air source, cylinder, compressed air tank, 
directional valve, one-way valve, etc. The rod-less 
chamber of the cylinder is named chamber a, the 
rod chamber is named as chamber b, the compressed 
air tank connected with the chamber a is a recycle 
compressed air tank (tank1), and the compressed air 
tank connected with the chamber b is a compressed 
air supply tank (tank2). The pressure of chamber 
a, chamber b, tank1 and tank2 are named by p
a
, p
b
, 
p
tank1
, p
tank2
. To study the influence of the relationship 
between the pressure in the compressed air tank 
and the cylinder chamber on the performance of the 
system, define the pressure ratio λ
1
 = p
a 
/p
tank1
 as 
the recovery switching criterion, and λ
2
 = p
tank2 
/p
b
 
as the supply switching criterion. The principle of a 
pneumatic actuator exhaust utilization system is as 
follows.
When the piston rod is extended, the compressed 
air energy source 1 supplies compressed air to 
chamber a, the released compressed air is adjusted 
by valve 2, and flows into cylinder chamber a after 
passing through valve 7, value 17 and value 18, and 
drives the cylinder to the right, the compressed air 
in the chamber b is discharged into the environment 
through valve 15 and valve 12.
When the piston rod is retracted, tank2 is used as 
an energy source to supply compressed air to chamber 
b through valve 14, valve 10, value 12, and valve 15 
when p
tank2 
/p
b
 > λ
2
. If p
tank2 
/p
b
 < λ
2
, the compressed 
air energy source 1 is adjusted by valve 2, and the 
compressed air is supplied to chamber b through valve 
5 and t valve 15. If p
a 
/p
tank1
 > λ
1
, the compressed air 
in chamber a flows into the recovery compressed 
air tank through valve 18, valve 17, and valve 13. If  
p
a 
/p
tank1
 < λ
1
, Then the compressed air in chamber a 
is discharged into the environment through valve 18, 
valve 17, and value 7.
When the recovery process is completed, but 
the energy is not enough to drive the movement of 
the piston rod, compressed air energy source 1 is 
used to pressurize the compressed air tank through 
electromagnetic reversing valve 21 and value 22 to 
meet the requirements. 
2  EXPERIMENTAL VERIFICATION OF MATHEMATICAL 
MODELS
2.1  Mathematical Models
To simplify the calculation, the following assumptions 
are proposed: (1) The compressed air in the loop 
follows the ideal compressed air law; (2) The 
compressed air source is a stable pressure source, 
there is no leakage in the system, and the effective 
area of the inlet and exhaust ports is the same; (3) The 
compressed air flows into a stable one-dimensional 
state; (4) The supply compressed air temperature 
is the same as the atmospheric temperature; (5) The 
initial pressure of tank2 is a fixed value.
2.1.1  Energy Equation
Chambers a and b do not admit compressed air at the 
same time; according to [20], the energy equation of 
chambers a or b can be described as follows
 
:
Cm
dT
dt
Sh CG TT RGTp Au
vi
i
ii vi ei ii ii
     , (1)
 Cm
dT
dt
Sh TT RGTp Au
vi
i
ii ei ii ii
   , (2)
where, C
v
 is the constant volume specific heat,  
[J/(kg·K)], m
 
is compressed air mass, [kg], T is the 
temperature, [K], T
e
 is the atmospheric temperature, 
[K], S
 
is the heat exchange area, [m
2
], h is the heat 
exchange coefficient, [W/(m
2
K)], R is the compressed 
air constant, [J/(kg·K)], G is the mass flow rate of the 
compressed air, [kg/s], p is the pressure, [MPa], A is 
the area, [m
2
], and u is the speed of piston movement, 
[m/s]. Subscript I represents a or b.
The energy equation of tank can be described as 
follows [21]:
dT
dt
RT
CpV
GCTG CT hS TT
jj
vjj
jp uj vj jj ej
 





, (3)
where, V is the volume, [m
3
]. Subscript j represents 
tank1 or tank2.
2.1.2  Mass Flow Equation
The mass flow equations of chambers a, b, tank1 and 
tank2 can be expressed as follows:
 G
ApD
T
p
p
b
ApB
T
pp
ij
eu
u
d
u
eu
u
du
,
,



                      
        
p
p
b
d
u








, (4)
where
 (,), pp
p
p
p
p
du
d
u
k
d
u
k
k














 21
 (5)
 B
K
Rk


2
1 ()
, (6)

Strojniški vestnik - Journal of Mechanical Engineering 69(2023)3-4, 119-134
122 Yu, Q. – Li, F. – Tan, X.
 D
k
k
Rk
k








22
1
1
+1
-1
()
, (7)
where, p
d
 and p
u
 are the downstream and upstream 
pressures, respectively, [MPa], A
e 
is the effective 
area of the intake and exhaust ports, [m
2
], T
u
 is the 
upstream temperature, [K], k is the specific heat ratio, 
and b is the critical pressure ratio.
2.1.3  Dynamic Equation
The force diagram of the piston is shown in Fig. 2. 
The equation of motion of the piston can be obtained 
according to Newton's second law. 
Fig. 2.  Movement direction and force diagram of piston
The motion equation of the piston is shown in Eq. 
(8):
d
d
,
2
2
1
0
x
t
M
pA pA pA FxnLn
xn Ln
ak ab kb er fa b
ab

  






() ,
,
 (8)
where, M is the mass of load, [kg], L is the cylinder 
stroke, [m], F
f
 is the friction, [N], A
ki
 is the effective 
area of the piston, [m
2
], p
e
 is atmospheric pressure, 
[MPa], A
r
 is the effective area of the piston rod, [m
2
]; 
n is the dead zone displacement, [m].
The friction force of the pneumatic system 
includes many factors. The composite model proposed 
by Canudas in 1995 can comprehensively reflect the 
friction characteristics of the contact surface of the 
pneumatic system under lubrication conditions
 
[22] 
and [23]. In this paper, the composite model described 
by Eq. (9) is used as the mathematical model of the 
friction force of the cylinder.
F
uFFF eu uu
uu u
f
cs c
uu
e
e
s

 













 







() sgn( )
(/ )
,, (9)
where, α is the cylinder viscosity coefficient,  
[N/(m/s)], F
c
 is the coulomb friction force, [N], F
s
 
is the maximum static friction force, [N], μ is the 
dynamic friction factor; u
s 
is the Stribeck velocity, 
[m/s], δ is an arbitrary index (0.5 to 2), and u
e
 is the 
critical speed, [m/s].
2.1.4  Equation of Compressed Air State
According to the pressure changes of the two chambers 
of the cylinder, the state equations of chambers a and 
b can be derived as:
 
d
d
d
d
p
tV
RTGp Au
pV
T
T
t
i
i
ii ii
ii
i
i
 






1
. (10)
The equation of the state of the compressed air in 
the compressed air tank is expressed as follows [24]:
 
d
d
p
t
R
CV
GCTh STT
j
vj
jp uj hj ej
 





, (11)
where S
h
 is the surface area of the inner wall of the 
tank chamber, [m
2
].
2.1.5  Basic Parameters
The basic parameters are shown in Table 1, and the 
above mathematical model is constructed using 
MATLAB/Simulink. B is the value of process quantity 
in the simulation process.
Table 1.  Basic parameters of the system
Parameter Value Parameter Value Parameter Value
M, [kg] 5 V
tank1
, [L] 1 B 50
L, [mm] 200 V
tank2
, [L] 1 F
c
, [N] 8
D, [mm] 63 C
v
, [J/(kg·K)] 718 F
s
, [N] 50
d, [mm] 20 C
p
, [J/(kg·K)] 1005 T
e
, [K] 293
p
s
, [Pa] 600000 R, [J/(kg·K)] 287 P
0
, [Pa] 600000
p
e
, [Pa] 101300 b 0.5283
2.2  Experimental Verification
In order to verify the correctness of the mathematical 
model, an experimental platform for exhaust 
utilization was built. The compressed air preparation 
unit consists of a regulator, a filter, and a lubrication 
control. Its main function is to reduce the pressure 
to a fixed value. Four pressure sensors are used to 
monitor the pressure in chamber a, chamber b, tank1, 
and tank2. Displacement sensors are used to monitor 
piston displacement. A force sensor is used to measure 
the output force of the pneumatic cylinder; an NI 
data acquisition card and computer are used to obtain 
experimental data.

Strojniški vestnik - Journal of Mechanical Engineering 69(2023)3-4, 119-134
123 Influence Analysis and Performance Optimization of a Pneumatic Actuator Exhaust Utilization System 
The pressure sensor test range is 0 MPa to 1 MPa, 
and the force sensor test range is 3 N to 3000 N. The 
measurement accuracy of pressure, displacement, 
and force are, respectively 1 %, ±5 μm, and ±0.05 %. 
Therefore, the maximum absolute errors of pressure, 
displacement and force are 0.01 MPa, ±5 μm, and 
±1.5 N, respectively.
The compressed air source should be turned on, 
and the pressure in tank2 set to a fixed value through 
the regulator. The system uses LabVIEW to control 
and save the data and realize the return stroke of the 
cylinder by controlling the four solenoid valves. 
Fig. 3 is the experimental schematic diagram 
of the system, and the experimental platform is 
shown in Fig. 4. Using the experimental platform, 
the experimental research on the compressed air 
utilization system is carried out. The compressed air 
source pressure is set to 0.6 MPa, the load mass is set 
to 5 kg. Fig. 5 shows the relationship between some 
parameters and time in the system.
The performance of the system will be shown 
by some parameters. Fig. 5 includes the piston 
displacement, velocity, chamber a, chamber b, and the 
changes of two tanks in simulation and experiment, 
which directly shows the performance of the system. 
It can be seen from Fig. 5 that the experimental 
results and the various parameter curves in the 
simulation have the same trend of change, the pressure 
in chamber b rises first, and then decreases. As the 
pressure in chamber a decreases, the pressure in 
tank1 first rises and then remains unchanged, and the 
pressure in tank2 decreases first, remains unchanged 
in the middle, and finally decreases again. 
Fig. 4.  Experimental platform
There are certain errors in the experimental and 
simulated values, but the maximum error is within 10 
%. The reasons for the error in the analysis may be the 
estimation of parameters such as friction force in the 
simulation, the instability of the compressed air source 
pressure, the existence of compressed air leakage in 
1 Compressed air energy source 
2 pressure reducing valve 
3, 4, 19, 20 pressure sensor 
5, 7, 10, 12, 17 solenoid reversing valve 
6, 11 silencer  
8, 9 compressed air tank 
13, 14 check valve 
15, 18 one-way throttle valve 
16 cylinder, and 
21 speed displacement sensor
Fig. 3.  Schematic diagram of pneumatic system experiment

Strojniški vestnik - Journal of Mechanical Engineering 69(2023)3-4, 119-134
124 Yu, Q. – Li, F. – Tan, X.
the pneumatic circuit, and the expansion energy of 
the compressed air in the cylinder is ignored. On the 
basis of previous studies [25], it can be known that the 
main cause of the error is the instability of gas source 
pressure. The purpose of this experiment is to explore 
the influence law of system performance through 
the general change trend of the curve. Therefore, the 
correctness of the model is verified by the comparison 
of the curve trend between simulation and experiment.
3  SIMULATION ANALYSIS OF DIMENSIONLESS MODELS
To understand the performance changes of the 
system more objectively, it is necessary to use the 
dimensionless method to eliminate physical units and 
study the characteristics of the system. To facilitate 
the research, the assumptions made in Section 2 also 
apply to this section. The reference values and the 
dimensionless variables are shown in Table 2. The 
basic mathematical model can be made dimensionless 
as described below.
Table 2.  Reference values and dimensionless variables
Variable Reference value Dimensionless variable
P p
s
p
*
= p/p
s
T T
p
 = V
a
 /(A
e
DR) t
*
 = t/T
p
G G
max
 = p
s
 A
e
 D/T
e
G
*
 = G/G
max
T T
e
T
*
 = T/T
e
X L x
*
 = x/L
A A
ka
A
*
 = A/A
ka
V V
a
V
*
 = V/V
a
M m
max
 = p
s
 V
a
 /(R θ
0
) m
*
 = m/m
max
3.1  Dimensionless Models
3.1.1  Dimensionless Energy Equation
When the compressed air is charged into chamber a or 
chamber b, its dimensionless energy equation can be 
expressed as follows: 
 
m
T
t
SK
S
GT
kG pAu
i
ii ab
ai
ii ii
*
*
*
*
max
*
**
** **
d
d









 
1
1
 
. (12)
When the compressed air is discharged from 
chamber a or chamber b, its dimensionless energy 
equation can be expressed as follows:
 
m
dT
dt
SK
S
T
kp uG T
i
ii aa
i
ii ii
*
*
*
*
max
*
*
** **
.


  

1
1 (13)
The dimensionless energy equation of the 
compressed air tank can be expressed as follows:
 
d
d
T
t
T
p
kTGK T
jj
j
jj aj j
*
*
*
*
** *
, 







1 (14)
a)
b)
c)
Fig. 5.  Comparison of experimental and simulation curves;  
a) displacement and velocity curve of piston, b) pressure curve of 
chamber a and b, and c) pressure curve of tank1 and tank2

Strojniški vestnik - Journal of Mechanical Engineering 69(2023)3-4, 119-134
125 Influence Analysis and Performance Optimization of a Pneumatic Actuator Exhaust Utilization System 
where, K
a
 is the Kagawa coefficient [25], which 
represents the degree of heat transfer and is defined by 
Eqs. (15) and (16).
 K
T
T
ai
p
hi
= , (15)
 K
T
T
aj
p
hj
= , (16)
 T
Cm
Sh
hi
v
maxi
= , (17)
 T
Cm
Sh
hj
vj
jj
= , (18)
 SA LA
maxk ak a
 22  , (19)
where, T
h
 is the thermal equilibrium time constant; T
p
 
is the reference time constant; S
max
 is the maximum 
heat exchange area of the two chambers of the 
cylinder, [m
2
].
The dimensionless maximum heat exchange area 
of chamber a and b is expressed as follows:
 S
S
A
L
A
max
* max
ka ka
  22

. (21)
The dimensionless heat exchange area of the 
tank1 and tank2 is as follows:
 S
S
A
ALA
A
j
j
ka
jjj
ka
*
. 
 22 
 (22)
3.1.2  Dimensionless Mass Flow Equation
The dimensionless mass flow equation of chamber a 
or b is expressed as:
 G
pA
T
p
p
b
pB A
DT
p
p
p
p
i
ue
u
d
u
ue
u
d
u
k
d
u
*
**
*
*
*
**
*
*
*
*
*














2
 










 k
k
d
u
p
p
b
1
*
*
. (23)
The dimensionless mass flow equation of tank1 is 
expressed as:
 G
pb
p tank
*
tank
*
ta
1
1
                                   



1
1
n nk
*
tank
*
b
b
pb
1
1
          
















1
. (24)
The dimensionless mass flow equation of tank2 is 
expressed as:
G
p
T
p
tank
*
tank
*
tank
*
tank
2
2
2
2
                              

* *
tank
*
tank
*
tank
*
tank
*
b
p
T
pb
b
pb



















2
2
2
2
   1
1
 
. (25)
3.1.3  Dimensionless Motion Equation
The dimensionless motion equation of the piston is 
described as:
dx
dt
T
Ap pp AFxL
f
kb baer f
2
2
2
1
0
0
*
*
*
** ** **
()
() ,









   
                                                    xL 



0,
 



,(26)
where, F
*
f
 is an estimated value that can be calculated 
by Eq. (27):
 F
Fu
FC uu
f
s
c
*
**
** **
, 







                  
       
0
0
 (27)
where, C
*
 is the dimensionless viscous friction 
coefficient which can be described as follows:
 F
F
pA
s
s
sk a
*
, = (28)
 F
F
pA
c
c
sk a
*
, = (29)
 C
Cu
pA
sk a
*
, =
0
 (30)
where, u
0
 is the reference speed, [m/s]; p
s
 is the 
compressed air source pressure, [MPa].
The dimensionless parameter T
*
f
 is defined by 
Eq. (32), and T
*
f
  corresponds to the J parameter used 
when selecting the cylinder.
 
T
T
T
f
f
p
*
, =
 (31)
 T
ML
pA
f
sk a
= . (32)
The mathematical relationship between J 
parameter and T
*
f
 is shown below: 
 J
T
f









1
2
*
. (33)

Strojniški vestnik - Journal of Mechanical Engineering 69(2023)3-4, 119-134
126 Yu, Q. – Li, F. – Tan, X.
a is discharged into the atmosphere, the pressure 
in tank1 remains unchanged. The pressure in tank2 
first decreases, then remains unchanged, and then 
decreases. This is because the system does not meet 
the air supply criterion, and the air supply from 
tank2 is switched to the air supply. Fig. 7a shows 
the working condition of the recovery part and the 
pressure in tank2 remains unchanged. Fig. 7b shows 
the specific working conditions of the gas supply part.
Table 3.  Dimensionless parameter reference value
Dimensionless parameters Symbol
Reference 
value
Dimensionless natural period T
*
f
0.0663
Dimensionless static friction F
*
s
0.0267
Dimensionless viscous friction 
coefficient
C
*
0.0153
Dimensionless piston area of chamber b A
*
kb
0.8992
Dimensionless Coulomb friction F
*
c
0.0043
Effective area of dimensionless 
compressed air inlet
A
*
e2
1
Kagawa coefficient of chamber a K
aa
0.0563
Kagawa coefficient of chamber b K
ab
0.0626
Dimensionless volume of tank1 V
*
tank1
1.604
Dimensionless volume of tank2 V
*
tank2
1.604
Recovery switching criterion λ
1
1
Criteria for compressed air supply 
switching
λ
2
1
Dimensionless initial supply pressure of 
tank2
β
*
1
Dimensionless air source pressure β
*
s
1
4  PERFORMANCE RESEARCH AND PARAMETER 
OPTIMIZATION
To obtain main influence parameters on efficiency 
and speed, further research on its speed and efficiency 
characteristics is essential; it should select the 
parameters with the high degree of impact, optimize 
the main performance of the system, and obtain the 
parameter combination under the optimal conditions.
4.1  Performance Analysis
Piston velocity and exhaust utilization efficiency 
reflect the main performance of exhaust utilization 
system, so the parameters affecting piston velocity 
and exhaust utilization efficiency are studied as 
dimensionless.
3.1.4  Dimensionless Equation of State
According to the pressure changes of chamber a or 
b, dimensionless equation of state is expressed as 
follows:
 
dp
dt
GT pAu
V
p
T
dT
dt
ii ii ii
i
i
i
i
*
*
** ** *
*
*
*
*
*
. 

 (34)
The equation of tank1 is expressed as follows:
 
dp
dt
kG KT
tank
*
tank
*
at ank
* 1
11 *
.  

3
1 (35)
The equation of tank2 is expressed as follows:
     
dp
dt
kG pT KT
tank
*
tank
*
tank tank at ank
* 2
22 22 *
**
.  

4
1 (36)
3.2  Dimensionless Simulation Analysis
To explore the relationship between the dimensionless 
initial pressure of tank2 and the utilization efficiency 
of compressed air, the ratio of the initial pressure p
0
 
of tank2 to the compressed air source pressure p
s
 is 
defined as the dimensionless initial supply pressure of 
tank2, which is expressed as β
*
 = p
0
/p
s
. Table 3 is the 
reference value of dimensionless parameters.
Based on the simulation program, the variation 
trend of each dimensionless physical quantity can be 
obtained by substituting parameters.
Fig. 6 shows the trend of dimensionless 
parameters over time. It can be seen from Fig. 6a 
that the piston does not move for an initial period of 
time. This is because the driving force of the driving 
chamber during this period is less than the sum of 
other resistances such as compressed air resistance 
and friction in the chamber a, and the piston is 
stationary. It can be seen from Fig. 6b that the pressure 
in chamber b first rises and then falls, because the 
friction force changes from static friction force to 
kinetic friction force after the piston moves, and the 
reduction of friction force accelerates the movement 
of the piston. At this time, the volume of chamber b 
increases, and the intake air volume is less than the 
increase in the volume of chamber b, the compressed 
air expands, and the pressure decreases. The pressure 
in chamber b does not rise again because the air 
supply time is short. 
It can be seen from Fig. 6c that the pressure in 
tank1 first increases and then remains unchanged. 
This is because the system does not meet the 
recovery criterion, the compressed air in chamber 

Strojniški vestnik - Journal of Mechanical Engineering 69(2023)3-4, 119-134
127 Influence Analysis and Performance Optimization of a Pneumatic Actuator Exhaust Utilization System 
4.1.1  Dimensionless Speed Analysis
To facilitate the study of the influence of various 
dimensionless parameters on the speed characteristics 
of the piston, the dimensionless speed of the piston is 
defined as:
 u
ut
t
t
*
*
. 

d
0
 (37)
The rate of change of dimensionless speed 
refers to the ratio of a parameter to the sum of 
the dimensionless average rate of change of all 
parameters. The sum of the dimensionless average 
speed change rate can be described as follows:
uu uuuu
u
totalc gT cg Fc gC cg Ac gF cg
Ac
fs kb c
e
  

 

 
**** *
*
2
g gK cg Kc gV cg Vc g
cg cg cg
uuuu
uuuu
aatt


 
 
12 12
12
**
 
s
cg  . (38)
It can be seen from Fig. 8 that u
*
 is mainly 
affected by T
*
f
, A
*
e2
, A
*
kb
, β
*
, and the rate of change of 
these parameters are respectively 23.57 %, 23.48 %, 
36.42 %, and 7.87 %.
a) 
b) 
c) 
Fig. 6.  Simulation curve; a) displacement and velocity curve of 
piston, b) pressure curve of chamber a and b, c) pressure curve of 
tank1 and tank1
a)
b)
Fig. 7.  Change in compressed air mass; a) compressed air mass 
changes in the recovery part, b) compressed air mass changes in 
the supply part 

Strojniški vestnik - Journal of Mechanical Engineering 69(2023)3-4, 119-134
128 Yu, Q. – Li, F. – Tan, X.
4.1.2  Dimensionless Efficiency Analysis
At the initial moment of compressed air reuse, the 
pressure in the exhaust chamber of the cylinder can 
be approximated as the supply pressure. Because 
the temperature change in the pipeline is difficult to 
measure and the compressed air tank is large when 
the system is working, the change in the compressed 
air flow process is not considered, and only the 
enthalpy of the compressed air is considered. For the 
calculation of air volume and pressurized compressed 
air energy in standard state, please refer to the relevant 
literature [26] and [27].
The efficiency of compressed air utilization can 
be calculated by:
  
EE
E
cs
c
**
*
-
%, 100 (39)
where, E
*
c
 is the dimensionless energy at the 
beginning of the return trip, E
*
s
 is the dimensionless 
energy at the end of the return trip.
Through the model established by the above 
process, in the research process, the value of a single 
parameter is changed to obtain the influence rate 
of the change of each parameter on the compressed 
air utilization efficiency. In order to evaluate the 
characteristics of compressed air reuse efficiency more 
objectively, the dimensionless average efficiency is 
defined as follows:
 



dt
t
t
0
. (40)
The change rate of dimensionless efficiency 
refers to the ratio of a parameter to the sum of the 
dimensionless average efficiency change rates of all 
parameters. The sum of the dimensionless average 
efficiency change rate can be described as follows:
 

totalc gT cg Fc gC cg Ac gF cg
Ac
fs kb c
e
  

  
+
**** *
*
2
g gK cg Kc gV cg Vc g
cg cg cg
aatt


 
 



12 12
12
**
 
s
cg 
. (41)
Fig. 9 describes the change rate of the 
dimensionless efficiency of each parameter.
It can be seen from Fig. 9 that the compressed 
air utilization efficiency is mainly affected by V
*
tank2
, 
A
*
kb
, λ
2
, T
*
f
, and A
*
e2
. The change rates of these 
parameters were 25.84 %, 24.1 %, 18.84 %, 12.1 %, 
and 11.49 %. 
4.2  Parameter Optimization
Optimizing the main parameters that affect the 
performance of the system can further improve 
the effect of energy saving. It can be seen from the 
above research that the dimensionless piston speed 
and compressed air utilization efficiency are mainly 
affected by T
*
f
, A
*
e2
, A
*
kb
, λ
2
, β
*
 and V
*
tank2
. The 
orthogonal experiments should be designed in the 
optimization process, and the grey relational analysis 
method used to optimize the key parameters that affect 
the performance of the system.
The overall optimization objective function can 
be defined as follows:
 fx uV () ,    
12 3 tank2
 (42)
where, α
1
 is the speed weighting factor, α
2
 is the 
reused compressed air tank volume weighting factor, 
and α
3 
is the reuse efficiency weighting factor.
Fig. 9.  The rate of change of the dimensionless efficiency  
of each parameter
Fig. 8.  Rate of change of speed

Strojniški vestnik - Journal of Mechanical Engineering 69(2023)3-4, 119-134
129 Influence Analysis and Performance Optimization of a Pneumatic Actuator Exhaust Utilization System 
There are three weighting factors in the objective 
function, and the importance of the three factors is 
different. The analytic hierarchy process can transform 
multi-objective and multi-criteria problems into multi-
level single-objective problems, and the results are 
simpler and clearer. Here we choose the analytic 
hierarchy process to obtain the weighting factor. By 
dividing the importance of the factors, the weighting 
factors of the specific objective function as shown in 
Table 4 are obtained by calculation.
Therefore, the weighting factor of the objective 
function is α = ( α
1
, α
2
, α
3
)
T
 
= (0.5, 0.2, 0.3)
T
.
Table 4.  Objective function weighting factors
Parameter u V
tank2
η

mn
m



mn
m
mn
m n

 
u 1 5/2 5/3 31/6 0.5
V
tank2
2/5 1 2/3 31/15 0.2
η 3/5 3/2 1 31/10 0.3

mn
m n
 
31/3 1
Table 5.  Parameter boundary value
Parameter
Boundary 
value 
T
f
  
[s]
A
e2
 
[mm
2
]
A
kb
 
[mm
2
]
λ
2
β
V
tank2
 
[L]
Lower 
boundary 
0.0103 50.27 2626.37 0.8 0.4 0.8
Upper 
boundary
0.0326 176.72 3004.15 1.2 0.8 1.2
The most representative and smallest combination 
of all combinations of experimental factors is 
selected by orthogonal experiment method, and part 
of the experimental results is used to characterize all 
experimental conditions. When the value of V
tank2
 
is set within 0.2 L from the initial value, the energy 
in tank2 can meet the system requirements, and the 
compressed air recovery and utilization process will 
not be complicated. Values of λ
2
 and β other than 0.2 
from the initial value will have a greater impact on the 
speed of the piston, so λ
2
 and β should be taken within 
0.2 of the initial value. The upper and lower boundary 
values of T
f
, A
e2
, and A
kb
 are calculated according 
to the values of other parameters. Table 5 shows the 
upper and lower boundary values of each parameter.
Five values are selected for each of the six 
parameters, and the L
25
(5
6
) orthogonal table is used to 
obtain the combination of parameters. The parameters’ 
values are shown in Table 6.
Table 6.  Parameters combinations
No.
Parameter
T
f
  
[s]
A
e2
 
[mm
2
]
A
kb
 
[mm
2
]
λ
2
β
V
tank2
 
[L]
1 0.0103 50.27 2626.37 0.8 0.4 0.8
2 0.0103 95.00 2410.39 0.9 0.5 0.9
3 0.0103 113.10 2803.12 1 0.6 1
4 0.0103 153.94 2916.18 1.1 0.7 1.1
5 0.0103 176.72 3004.15 1.2 0.8 1.2
6 0.0146 50.27 2410.39 1 0.7 1.1
7 0.0146 95.00 2803.12 1.1 0.8 1.2
8 0.0146 113.10 2916.18 1.2 0.4 0.8
9 0.0146 153.94 3004.15 0.8 0.5 0.9
10 0.0146 176.72 2626.37 0.9 0.6 1
11 0.0207 50.27 2803.12 1.2 0.5 0.9
12 0.0207 95.00 2916.18 0.8 0.6 1
13 0.0207 113.10 3004.15 0.9 0.7 1.1
14 0.0207 153.94 2626.37 1 0.8 1.2
15 0.0207 176.72 2410.39 1.1 0.4 0.8
16 0.0292 50.27 2916.18 0.9 0.8 1.2
17 0.0292 95.00 3004.15 1 0.4 0.8
18 0.0292 113.10 2626.37 1.1 0.5 0.9
19 0.0292 153.94 2410.39 1.2 0.6 1
20 0.0292 176.72 2803.12 0.8 0.7 1.1
21 0.0326 50.27 3004.15 1.2 0.6 1
22 0.0326 95.00 2626.37 1.1 0.7 1.1
23 0.0326 113.10 2410.39 0.8 0.8 1.2
24 0.0326 153.94 2803.12 0.9 0.4 0.8
25 0.0326 176.72 2916.18 1 0.5 0.9
The parameters’ combinations are substituted 
into the simulation model to obtain the value of each 
objective function, and the results are combined and 
calculated by the grey correlation degree method, 
and the value of the correlation degree is used to 
evaluate the pros and cons of these combinations, so 
as to obtain the optimal result. The corresponding 
parameter combinations are shown in Table 7. The 
optimal result is brought into the model simulation, 
and the single exhaust utilization efficiency of the 
system is 34.7 %.
Table 7.  Optimized parameters
Parameter
T
f
 [s] A
e2
 [mm
2
] A
kb
 [mm
2
] λ
2
β V
tank2
 [L]
0.0292 176.72 2803.12 0.8 0.7 1.1
5  RESEARCH ON ENERGY-SAVING UTILIZATION  
OF EXPANSION ENERGY
In the above research process, only the transmission 
energy of the compressed air is used, while the 

Strojniški vestnik - Journal of Mechanical Engineering 69(2023)3-4, 119-134
130 Yu, Q. – Li, F. – Tan, X.
expansion energy of the compressed air is hardly used, 
which results in a waste of compressed air energy. To 
make use of the expansion of the compressed air, the 
opening and closing sequence of the solenoid valve is 
changed to control the air supply time to the chamber 
b, so that the air is sufficiently expanded to push the 
piston back. During the return process, the energy 
change of the pressurized compressed air in tank2 and 
the speed of the piston are studied.
5.1 Solenoid Valve Working Status
Fig. 10 shows the value opening and closing sequence 
diagrams of the four solenoid valves under the basic 
operating conditions, where u1, u2, u3, and u4 are 
solenoid valve 7, solenoid valve 17, solenoid valve 
12, and solenoid valve 5, respectively.
The 0 state in Fig. 10 indicates that the solenoid 
valve is de-energized and closed, and the 1 state 
indicates that the solenoid valve is energized and 
opened. In the actual working process, the solenoid 
valve does not act instantaneously when the solenoid 
valve is disconnected and closed; there is a 10 ms 
response time. 
5.2  System Performance Analysis of Variable Initial 
Pressure of tank2
The volume of tank2 is 1 L, and the initial pressure of 
tank2 is 0.5 MPa, 0.6 MPa, and 0.7 MPa, respectively, 
and cut off the compressed air supply of tank2 to the 
chamber b at 0.06 s, 0.07 s, 0.08 s, 0.09 s and 0.1 s, 
respectively. The expansion of the compressed air in 
the chamber b can be utilized in the subsequent time. 
Fig. 11 shows the working conditions of the solenoid 
valve and the change of the pressure in tank2 in 
different working conditions.
It can be seen from Fig. 12 that when the initial 
pressure of tank2 is constant, the energy saving 
efficiency decreases with the increase of the time of 
cutting off tank2, and when the time of cutting off 
tank2 is fixed, the energy saving efficiency decreases 
with the increase of the initial pressure of tank2. 
However, when the time of cutting off the tank2 is 
0.06 s, the energy saving efficiency increases with 
the increase of the initial pressure, and the maximum 
is 26.12 %. The reason may be that when the initial 
pressure of the tank2 is large, the time of cutting off 
the tank2 has a great influence on the energy saving 
efficiency.
It can be seen from Fig. 13 that when the initial 
pressure of tank2 is constant, the average piston 
velocity increases with the increase of the time of 
cutting off tank2, and when the time of cutting off 
tank2 is fixed, the average velocity of piston increases 
with the increase of the initial pressure of tank2. In the 
working process of the system, it is necessary to avoid 
cutting off the tank2 too early to affect the normal 
operation of the system.
When the volume of tank2 is 1 L, in order to 
improve the energy saving efficiency of the system 
and not make the piston speed too low affect the 
operation of the system, it is suggested that the time 
of cutting off tank2 is 0.06 s to 0.08 s and the initial 
pressure of tank2 is 0.6 MPa to 0.7 MPa.
     
     
Fig. 10.  Working condition of solenoid valve

Strojniški vestnik - Journal of Mechanical Engineering 69(2023)3-4, 119-134
131 Influence Analysis and Performance Optimization of a Pneumatic Actuator Exhaust Utilization System 
5.3  System Performance Analysis of Variable Volumes of 
tank2
The initial pressure of tank2 is 0.6 MPa, the volume 
of tank2 is 0.8 L, 1 L, and 1.2 L, and cut off the 
compressed air supply of tank2 to the chamber b at 
0.06 s, 0.07 s, 0.08 s, 0.09 s and 0.1 s, respectively. 
The expansion of the compressed air in the chamber b 
can be utilized in the subsequent time. Fig. 14 shows 
the working conditions of the solenoid valve and the 
change of the pressure in tank2 in different working 
conditions.
It can be seen from Fig. 15 that when the volume 
of tank2 is constant, the energy saving efficiency 
decreases with the increase of the time of cutting off 
tank2, and when the time of cutting off tank2 is fixed, 
                 
                 
                 
Fig. 11.  The working condition of the solenoid valve and the initial pressure of tank2 is: a) 0.5 MPa, b) 0.6 MPa, and c) 0.7 MPa

Strojniški vestnik - Journal of Mechanical Engineering 69(2023)3-4, 119-134
132 Yu, Q. – Li, F. – Tan, X.
Energy efficiency [%]
Fig. 12.  Energy efficiency
Average piston speed [%]
Fig. 13.  Piston average speed
the average velocity of the piston decreases with the 
increase of the volume of tank2.
It can be seen from Fig. 16 that when the volume 
of tank2 is constant, the average piston velocity 
increases with the increase of the time of cutting 
off tank2, and when the time of cutting off tank2 is 
fixed, the average velocity of piston increases with the 
increase of tank2 volume. This is because the increase 
in the volume of tank2 leads to a decrease in pressure 
change, resulting in an increase in driving force.
When the initial pressure of tank2 is 0.6 MPa, in 
order to improve the energy saving efficiency of the 
system and not make the piston speed too low affect 
the operation of the system, it is suggested that the 
time of cutting off tank2 is 0.07 s to 0.08 s and the 
volume of tank2 is 0.8 L to 1 L.
The difference in the compressed air supply time 
will make the expansion of the compressed air in the 
chamber b can be utilized to different degrees, but 
the piston speed will also be affected. The air supply 
time should be controlled to avoid the influence of the 
piston movement speed being too low on the system 
work.
6  CONCLUSION
To use the compressed air, this paper proposes a 
pneumatic actuator exhaust utilization system to return 
the compressed air to the compressed air tank and 
supply it under suitable conditions. The conclusions 
of this study are as follows:
1.  A new exhaust utilization circuit is designed to 
recover and utilize compressed air.
2.  The simulation results are in good agreement 
with the experimental results, which verifies 
the accuracy of the mathematical model of the 
system.
3.  The operating characteristics of the system are 
mainly affected by the dimensionless parameters, 
such as natural period, effective piston area, 
chamber b area, initial of tank2, volume of 
tank2, and the criterion of compressed air supply 
switching.
4.  Some parameters that have a greater impact 
on system characteristics are optimized. At 
the optimal parameters, the exhaust utilization 
efficiency of the system is 34.7 %.
5.  Expansion energy can be further utilized by 
controlling the air supply time. By controlling the 
air supply time, when the initial pressure of tank2 
is 0.5 MPa, 0.6 MPa, and 0.7 MPa, the maximum 
energy-saving efficiency is 23.25 %, 24.99 %, 
and 26.12 %. When the volume of tank2 is 0.8 
L, 1 L, and 1.2 L, the maximum energy-saving 
efficiency is 30.01 %, 24.99 %, and 21.51 %.
Overall, the compressed air utilization system 
can recover and utilize compressed air well. During 
the working process, the system parameters should be 
selected and optimized according to the specific work 
requirements. This paper provides a new technical 
solution for compressed air recovery and utilization 
and a theoretical basis for subsequent improvement of 
system control.

Strojniški vestnik - Journal of Mechanical Engineering 69(2023)3-4, 119-134
133 Influence Analysis and Performance Optimization of a Pneumatic Actuator Exhaust Utilization System 
7  ACKNOWLEDGEMENTS
The research work presented in this paper is 
financially supported by a Grant (52065054) from the 
National Natural Science Foundation of China and 
the Beijing Outstanding Young Scientists Program 
(BJJWZYJH01201910006021).
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a)                 
b)                 
c)                 
Fig. 14.  The working condition of the solenoid valve and the pressure change in tank2 when tank2 volume is: a) 0.8 L, b) 1 L, and c) 1.2 L

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