Abstract

A new pneumatic cylinder assembly is proposed as an alternative to classical cylinders which are well known for their poor energetic efficiency. The new system comprises an added expansion volume which permits to recover the energy content of a filled cylinder by a real thermodynamic expansion instead of simply releasing the filled air to the atmosphere. The energetic performance of the new system is evaluated and compared with the performance of an equivalent single cylinder producing the same mechanical work. The paper explains the operation principle and properties through numeric simulation and presents a small experimental prototype.

1 Introduction

In the context of the world situation of global climate change, many reflections, developments, and new industrial strategies have appeared from several decades and reach today a real state of technical and economic maturity. On the side of energy supply, the two dominant alternative technologies of photovoltaics and wind have shown significant augmentation in volume, even if they still represent only a small part of the global energy consumption. Transportation of goods and persons follows the line of the “all electric” systems and tries on their side to reduce the amount of CO2 emissions. In a general way, all domains of energy production and use for and by humans are affected and will have to undergo significant changes in the coming decades. As alternatives to the Internal Combustion Engine (ICE) propulsion systems, fuel-cells or battery-powered electric drives are currently appearing, reducing the green-house gas emissions by significantly improving the system efficiency. In many industrial processes, manufacturing infrastructures, or final use of energy, a huge potential of energy savings exist and is still today marginally considered.

In important sectors of industrial activities, the pneumatic technology has been used for a long term, based on its simplicity, reliability, and low costs. Mainly pneumatic cylinders but also semi-rotary actuators are used, fed from a central reservoir and controlled by distribution valves. For filling the compressed air reservoirs, compressors of different types are used, the energy efficiency of which obviously intervenes in the overall chain and must be analyzed separately.

The energetic properties of pneumatic systems and actuators have been often discussed and several recommendations have been published [18].

In this paper, the energy efficiency of pneumatic actuators is discussed, as the ratio of the produced mechanical work to the transferred enthalpy from the air reservoir. Then, an original solution is presented which permits to nearly double the energetic efficiency of the pneumatic to mechanical transformation compared to classical actuators.

2 The Poor Efficiency of the Classical Pneumatic Actuators

In a classical pneumatic actuator, the mechanical work is obtained from the displacement of the piston under constant pressure. At the end of the stroke, the pressure in the fully deployed cylinder is released to the atmosphere by opening the exhaust valve, allowing the free return of the piston. This corresponds to renounce to the pneumatic energy content of the pressured air inside the cylinder. The pneumatic energy content of the deployed and filled cylinder can be illustrated by the W2d surface in the diagram of Fig. 1 (volume under the expansion curve, right of V2).

Fig. 1
Structure and properties of a classical pneumatic system
Fig. 1
Structure and properties of a classical pneumatic system
Close modal
This internal energy can be calculated as the energy obtained from an isothermal expansion of that volume of air from a pressure P2 down to the atmospheric pressure Pa (Eq. (1)) [9]
W2d=P2V2(lnP2Pa1+PaP2)
(1)
On the other side, the mechanical work produced by the piston’s displacement while the pressure is maintained constant through the PRV valve at a value P2 is calculated through
W2=(P2Pa)(V2V1)
(2)

The work W2 corresponds to the rectangular surface between the P2 and Pa pressure levels in Fig. 1.

Then, the energetic efficiency of the pneumatic cylinder can be defined as
ηpneum_act=W2W2+W2d
(3)

The denominator of Eq. (3) represents the injected enthalpy into the system while the numerator corresponds to the amount of effectively converted energy. The efficiency thus depends on the quantity of air which is released to the atmosphere and depends on the operating pressure of the cylinder. For the common pneumatic industrial devices, the efficiency takes low values between 30% and 40%. Such properties have initiated different attempts of increasing the performance as for example the variation of the supply pressure [1012].

3 Recovering the Energy Content of the Cylinders by Additional Expansion

For the recovery by the expansion of the energy content of a cylinder, two different principles are possible. First, the constant pressure displacement and the expansion of the air occur in the same volume or in the same cylinder. Figure 2(a) illustrates this principle where the V1 volume corresponds to the displaced volume under constant pressure as a part only of the total volume of the cylinder and covered by a part of the piston’s stroke. After reaching this value of volume, the intake valve (Vint) is closed while the piston continues to move. The covered volume during this next part of the stroke is V2 and is the difference to the full volume of the cylinder. While the injected mass of air remains constant due to the closed intake valve, the variation of the volume from V1 up to V1 + V2 causes a real thermodynamic expansion of a ratio equal to V1/(V1 + V2). The curves in the figure illustrate simultaneously the evolution of the volume and pressure.

Fig. 2
Adding an expansion volume (a) in the same cylinder and (b) in an additional cylinder
Fig. 2
Adding an expansion volume (a) in the same cylinder and (b) in an additional cylinder
Close modal

An example of such a process among many others is given by the principle of the so-called Truglia motor [13].

For the second principle, the constant pressure displacement work is realized in a first cylinder over its full stroke where the piston position varies from zero to Xmax (Fig. 2(b)). The expansion of the air is realized during the return stroke through a transfer of the accumulated mass of air into a second cylinder of a larger volume. During this transfer stroke, the volume of the first cylinder varies from V1 to zero, and the volume of the second larger one varies from zero to V2, resulting into an expansion of the ratio of V1/V2 The variation of the volumes and the resultant variation of the pressures are illustrated by the curves of Fig. 2(b). The control signals of the intake valve (Vint), the transfer valve (Vtr), and the exhaust valve (Vexh) are also represented.

Two examples of this second principle are given first by the Motor Development International (MDI) motor of the compressed air car [14] where the sequence of variation of the volumes of the cylinders is somewhat different. The second example is a compressed air-driven generator using coupled semi-rotary actuators [15,16].

This second principle will be described more in detail in Sec. 4.

4 A New Cylinder Assembly for an Increased Energetic Efficiency

The principle of recovering the thermodynamic content of the pressurized air is applied to linear cylinders, where in addition to the simple displacement work produced in a conventional cylinder, the air is additionally expanded in a supplementary pneumatic chamber system, allowing to recover a significant part of the injected enthalpy.

Figure 3 shows one of the possible arrangements of the new assembly, where the displacement work is produced by a central cylinder and where the expansion of the air is done within three identic peripheral cylinders mechanically coupled to the central one. In such an arrangement, the volumetric ratio of the expansion is equal to 1 over 3.

Fig. 3
One of the possible arrangements of the proposed cylinder assembly
Fig. 3
One of the possible arrangements of the proposed cylinder assembly
Close modal

The system illustrated in Fig. 3 is studied, and its performance is compared with a single cylinder producing the same mechanical work. Figure 4(a) shows a front view and a side view of the proposed system while Fig. 4(b) shows a conventional single cylinder. One can see that the two systems have identic mechanical output interfaces (right sides of Figs. 4(a) and 4(b)).

Fig. 4
Proposed system: (a) front and side view and (b) single cylinder with a compatible interface
Fig. 4
Proposed system: (a) front and side view and (b) single cylinder with a compatible interface
Close modal

The cylinder assembly with one central and three peripheral cylinders as represented in Fig. 4 is only one of the possibilities of coupling cylinders for realizing the additional expansion work. A simpler solution using only a central filling and two lateral expansion cylinders is represented in Fig. 5 and will be described in Sec. 5.2. Further arrangements are described in a patent application [17].

Fig. 5
Simpler configuration of coupled cylinder assemblies
Fig. 5
Simpler configuration of coupled cylinder assemblies
Close modal

4.1 Operating Principle and Control.

The proposed system is studied in its execution as an assembly of individual cylinders as represented in Figs. 3 and 4. The chambers of the pull and push sides of the central cylinder are fed from the air reservoir under constant pressure. Then, the air is transferred into the chambers of the push and pull sides (opposite sides) of the three other cylinders running in parallel. As represented in the diagram of Fig. 6(a), two intake valves Vina and Vinb are feeding the chambers of the first cylinder. The transfer of the air from the first cylinder to the three others is controlled through the two transfer valves Vtra and Vtrb. Finally, after the expansion, the air at the final expansion pressure is released to the external atmosphere through two exhaust valves Vexha, Vexhb. The sequences of operation are defined according to the diagram of Fig. 6(c). This diagram indicates the states (intake, transfer, or exhaust) of the different volumes V1a, V1b, V2a, V2b over one and a half cycle of operation.

Fig. 6
Model and control of the system
Fig. 6
Model and control of the system
Close modal

The simulation of the system is considering a simplified model where the three parallel running cylinders are represented by one cylinder of three-times larger volume (Fig. 6(b)).

4.2 The Pressure Variation During the Expansion.

The expansion of the air is supposed to be of the adiabatic type. In reality, the timely defined evolution of the pistons and volumes allows a non-negligible heat transfer from the walls of the cylinders to the air in expansion and will have a better result on the final efficiency. However, considering an adiabatic expansion corresponds to consider a kind of worst case. A comparison of the efforts produced in adiabatic and in isothermal conditions by a similar system is made in Ref. [16]. The resulting pressure P2 in the chambers V2a (first half cycle) and V2b (second half cycle) and of the respective feeding chambers V1b and V1a takes the value of
P2=Pin(V1maxV1a,b+V2b,a)γwithγ=1.4
(4)

In Eq. (4), the term in the parenthesis represents the variation of the volumetric ratio of the expansion. V1max is the constant volume which has been filled during the previous stroke and is the initial value of the variable air containing volume during the expansion. The denominator (V1a,b + V2b,a) represents the changing active volume during the expansion. This term varies from V1max up to V2max.

During the left-to-right stroke, the volume V1a varies from zero to V1max and is filled with air under constant pressure. During the same stroke, the volume V1b varies from V1max down to zero while the volume V2a varies from zero to V2max. The mechanical force produced corresponds to the sum of a constant force issued from the left side of the small cylinder, a variable force on the opposite side of the same cylinder, and a variable expansion force produced by the left side of the larger piston. The opposite side of the second cylinder is connected to the atmospheric pressure through an open exhaust valve.

At the end of the left-to-right motion, the expansion volume ratio is equal to 1/3, and the corresponding pressure ratio becomes
Pin/P2=0.21accordingrel.(4).(5)
(5)

The produced mechanical work by the proposed system must be evaluated as a contribution of four main components namely the forces generated by the intake pressure on the surfaces of the first piston and the forces generated by the expansion pressure on the surfaces of the first and second one. The effect of the atmospheric pressure during the exhaust phase of the second piston is also considered. An analytic calculation of the sum of these four components is not realistic, due to the fact that the expansion pressure depends on the position of the pistons, which further is the result of the double integration of the global force (force to speed, speed to position). For the evaluation of the mechanical performance of the system, but also for the estimation of all internal variables, the way of simulation is chosen.

4.3 Simulation of the Proposed System.

The new cylinder assembly is modeled according to the schematic representation of Fig. 6(b). This model is composed of a first cylinder assuming the function of the intake and the production of mechanical work at constant pressure. A second equivalent cylinder is used for the modeling of the expansion function and the related expansion work. From the sum of the produced components of the forces, the speed V of the output rod is calculated by integration of the inertial phenomenon. From the speed, the position X is calculated through a further integration (Fig. 7).

Fig. 7
Functional diagram of the simulation
Fig. 7
Functional diagram of the simulation
Close modal

The four components of the produced force, namely, the a- and b-side contributions of both cylinders are calculated in dependency of the pressure in the chambers. These pressures depend on the variable volumes of each chamber which are estimated in dependency of the position of the output rod and the intake air pressure Pin. In the functional diagram of Fig. 7, the output power is also calculated as also the produced mechanical work.

In the used model, the friction is neglected, the focus of this study being set on the increase of performance by adding the expansion. In the comparison with a classical cylinder, the friction is neglected in both compared systems. The influence of the friction of a pneumatic actuator is considered in Ref. [15]. A more detailed description of the friction can be found in Ref. [18].

4.4 Simulation Results.

According to the set of parameters given in Table 1, the internal and external variables of the system are calculated. The simulation results are given for a complete round-trip cycle of the cylinders. This cycle can be divided into four sectors as represented in Fig. 8. The sector A corresponds to the motion of the rod from left to right, B illustrates the stop at the right end of the stroke, C corresponds to the return stroke from right to the left, and D illustrates the stop at the left end of the stroke. The different sectors are related to the position of the mobile part. This position is given as a time function in Fig. 11.

Fig. 8
Evolution of the pressures
Fig. 8
Evolution of the pressures
Close modal
Table 1

Parameters of the single cylinder of the 1 + 3 assembly

Parameters of the single cylinder
Diameter0.012 (m)
Stroke0.020 (m)
Operating pressure5 bar
Parameters of the single cylinder
Diameter0.012 (m)
Stroke0.020 (m)
Operating pressure5 bar

Figure 8 shows the evolution of the pressure in the chambers V1a and V1b. In reality, the simulated curves represent the pressure in the exchange lines between the first and the second equivalent cylinders, downstream of the intake valves. The upper values of the curves correspond to the intake pressure (5 bar in the simulated case), and the lower values show the value of the pressure after the expansion (1/3)1.4 * 5 bar = 1.07 bar, see Eq. (4). The translation time of the rod is 0.08 s, while the stop time is 0.42 s.

The pressures indicated in Fig. 8, together with the values of the pressures in the intake, exchange, and exhaust ways produce a global acceleration force as represented in Fig. 9.

Fig. 9
Forces produced in the cylinders
Fig. 9
Forces produced in the cylinders
Close modal

The curve F1 in Fig. 9 represents the force of the new system with expansion. The two other curves represent the forces of a single cylinder without expansion. These curves will be used for the evaluation of the reduction of the air consumption. Their role and significance will be explained in Sec. 5.1.

The force exerted on the mobile part of the parallel-connected piston rods is producing an acceleration and brings this part on a determined velocity. The corresponding curves are shown in Fig. 10. The velocity (speed) S1 corresponds to the force F1 of Fig. 9, the speed S2 to the force F2, and the speed S3 to the force F3.

Fig. 10
Velocity of the mobile part
Fig. 10
Velocity of the mobile part
Close modal

From the curves of the velocity (Fig. 10), the position reached by the mobile part is also simulated. The curves can be seen in Fig. 11. The curve P1 corresponds to the cylinder assembly with expansion. The mobile equipment starts at position zero and reaches the end of the stroke at the value of 0.020 m. The sequences of the movement (A, B, C, D) are the same as explained in Fig. 8.

Fig. 11
Position of the mobile equipment
Fig. 11
Position of the mobile equipment
Close modal

The simulation further calculates the mechanical power transmitted to the mobile part. This variable corresponds to the product of the speed by the accelerating force. The curve of the power is represented in Fig. 12.

Fig. 12
Power dissipated by the mobile equipment
Fig. 12
Power dissipated by the mobile equipment
Close modal

Then, from the transferred power, the produced mechanical work is calculated by a simple integration. The curve is given in Fig. 13.

Fig. 13
Energy (work) transferred to the mobile equipment
Fig. 13
Energy (work) transferred to the mobile equipment
Close modal

5 Efficiency of the New Assembly

In a general case of the addition of an expansion chamber to a classical cylinder, the efficiency can be defined by the ratio of the produced mechanical work to the injected enthalpy to the assembly. The total produced work is the sum of the constant pressure displacement work and the work produced by expansion from the injected pressure down to the lower pressure level reached after the expansion Plow.

Plow is defined by the volumetric expansion ratio
Plow=Pin(V1V2)γ
(5)
If the internal energy content of the injected air is converted into mechanical work depending on the lower pressure Plow, the energy efficiency can be calculated as
η=WoutHin=WoutPinΔV+U=PinΔV+WexpPinΔV+U=PinΔV+PinΔV(lnPinPlow1+PlowPin)PinΔV+PinΔV(lnPinPa1+PaPin)
(6)

From this expression, one can see that the efficiency does not depend on the volume of air injected and expanded, but only on the pressure level of the injected air and on the lower level of the pressure after expansion.

In Table 2, the different efficiencies obtained with an additional expansion chamber are represented in dependency of the lower value of the expansion pressure Plow. The input pressure is 10 bar. The same efficiencies are represented for an input pressure of 5 bar in Table 3.

Table 2

Efficiency in dependency of the expansion pressure Plow. Pin = 10 bar

Pin = 10 bar
Plow (bar)η
100.374(No expansion)
90.376
80.384
70.398
60.420
50.455
40.506
30.584
20.711
11.000(Expansion down to atmospheric)
Pin = 10 bar
Plow (bar)η
100.374(No expansion)
90.376
80.384
70.398
60.420
50.455
40.506
30.584
20.711
11.000(Expansion down to atmospheric)
Table 3

Efficiency in dependency of the expansion pressure Plow. Pin = 5 bar

Pin = 5 bar
Plow (bar)η
50.552(No expansion)
40.565
30.614
20.727
11.000(Expansion down to atmospheric)
Pin = 5 bar
Plow (bar)η
50.552(No expansion)
40.565
30.614
20.727
11.000(Expansion down to atmospheric)
For the simulated example of the new assembly according to the parameters given in Table 1 (Sec. 4.4), the efficiency is calculated on the base of the produced mechanical work and the injected enthalpy into the system. From Fig. 13, the produced work by the moving part during the first stroke is
Wout=0.5*Emax=0.5*2.915J
(7)
The energy efficiency is calculated as in Eq. (3)
ηconv=WoutHin=WoutU+PinΔV
(8)

U is the thermodynamic energy content of the injected air under pressure and is calculated as the energy needed for the compression into a volume V1 of the equivalent mass of air from the atmospheric pressure to the value of Pin, V1 being the filled volume of the first actuator.

V1 = 2.262*10−6 m3
U=Ecomp=PinV1(lnPinPatm1+PatmPin)=5105N/m20.00000226m3(ln5bar1bar1+1bar5bar)=0.91J
(9)
PinΔV=5105N/m20.00000226m3=1.13J
(10)
Finally, and according to Eq. (7) the efficiency becomes
ηconv=WoutHin=0.52.91J1.13J+0.91J=0.713
(11)

5.1 Comparison of Performance.

The value of efficiency given through Eq. (11) characterizes the efficiency of the new proposed system. It is now interesting to compare this value with the efficiency of a classical cylinder operated without expansion and producing the same mechanical work. A first attempt is to simulate the dynamics of the same load driven by a cylinder producing the average value of the force of the new system. The value of the intake pressure and the maximum length of the stroke is the same.

From Figs. 9 (F2), 10 (S2), and 11 (P2), the simulation shows that this force (average) does not produce the same mechanical work. This force is produced by a cylinder in which diameter (0.01665 m) is calculated as a function of the value of the constant average force to be produced under the same pressure (87.09 N) applied for the same duration as the traveling time measured for the new system (0.08 s). The curve P2 in Fig. 11 shows that the maximum length (0.020 m) of the stroke is not reached. From Fig. 10, one can observe that the force F2 produces the same maximum velocity as the new system (see S1 and S2 in Fig. 10).

By a successive iteration process, it is possible to determine the amplitude of a constant force (superior to F2) that produces the same displacement of the load within the same time as for the new system. The value of this force is 104.0621 N (see F3 in Fig. 9, S3 in Fig. 10, and P3 in Fig. 11).

From the value of F3, the diameter of the equivalent single cylinder can be calculated as
deq=4Fπp=4104.06Nπ4105N/m2=0.0182m
(12)

The factor of 4 for the calculation of an equivalent pressure (4 * 105) in Eq. (11) takes into account the action of the atmospheric pressure on the rear side of the piston (4 = 5 – 1).

The intake volume of the single cylinder becomes
V1s=πd24l=π(0.0182m)240.02m=5.2106m3
(13)
The energy content of this intake volume is calculated as given by Eq. (11)
U=Ecomp=PinV1s(lnPinPatm1+PatmPin)=5105N/m20.0000052m3(ln5bar1bar1+1bar5bar)=2.1J
(14)
The associated displacement work is
PinΔV=5105N/m20.0000052m3=2.6J
(15)
The efficiency of the single cylinder becomes
ηconv=WoutHin=0.52.91J2.6J+2.1J=0.309
(16)
The reduction of consumption of air is defined as the ratio of the two intake volumes
kred=V1V1s=2.26106m35.20106m3=0.43
(17)

This factor illustrates the main benefit of adding an expansion chamber to a smaller cylinder compared to a single one producing the same mechanical work. It is evident that the nature of the driven load, the resistant force, and the total inertia play an important role and must be considered from case to case.

6 Experimental Setup

An experimental setup has been realized according to the arrangement initially presented in Sec. 3 (Fig. 3). Four cylinders are connected mechanically at their stator side as at the side of the moving piston rods. The central cylinder has the role of the intake and production of constant pressure displacement work. The schematic representation of the cylinders and control system is shown in Fig. 6. Figure 14 shows the realized system without control valves.

Fig. 14
Views of the first realized setup
Fig. 14
Views of the first realized setup
Close modal

6.1 The Parasitic Effect of the Dead Volumes.

The distribution of the air through the inlet-, transfer- and exhaust valves and through the connecting tubes between the different cylinders affects the system with dead volumes. These volumes were not considered in the ideal simulation and calculation of the performance and the theoretic value of efficiency.

In the recording of the pressures of Fig. 15, the change-over from constant pressure displacement work and stop at the left end (A + B with reference to Fig. 8) to expansion work (C) is affected by an abrupt variation of the pressure at the instant of the opening of the transfer valve. This transient phenomenon can be explained through the simplified model represented in Fig. 16. In this model, there are two dead volumes represented, namely V1p which is a model of the volumes of the tubing between the intake valve, the chamber of the first cylinder V1, and the transfer valve Vtra.

Fig. 15
Abrupt variation of the pressure
Fig. 15
Abrupt variation of the pressure
Close modal
Fig. 16
Simplified model of the system with dead volumes
Fig. 16
Simplified model of the system with dead volumes
Close modal

A second dead volume V2p represents the tubing between the transfer valve Vtra and the chamber of the second cylinder, V2. In reality, this second cylinder is composed of three individual small cylinders, what makes the connections more complex.

At the instant of the change-over where the transient occurs, the pistons have reached their end-of-stroke position at the left side. The intake valve is closed. The intake pressure has been established inside of the fully deployed first cylinder and in the dead volume V1p. Then, the transfer valve opens in order to start the filling of the volume of the second cylinder V2 when the pistons move in the opposite direction. At the instant of the opening of the transfer valve, the downstream volume is not equal to zero as if the piston has not moved, but the air under pressure in V1 and V1p inrushes into the dead volume V2p. As a result, the initial pressure of the expansion stroke is not equal to P1 but is changed to P1′ due to a pre-expansion. The value of P1 is governed by a relation of an adiabatic expansion
P1=P1(V1+V1pV1+V1p+V2p)γ
(18)
The volume of the chamber of the first cylinder V1 is 2.26 cm3, while the dead volume V1p is 1.47 cm3 The dead volume V2p is 1.66 cm3. As a consequence, after the opening of the transfer valve Vtr the initial pressure level in the V1 + V2 volume for expansion is
P1=5bar(2.26cm3+1.47cm32.26cm3+1.47cm3+1.66cm3)1.4=0.5975bar=2.98bar
(19)

The negative effect of the pre-expansion is particularly significant in the case of the realized demonstrator, due to the small volume of the cylinders versus the value of the dead volumes. Additionally, the assumption of an adiabatic pre-expansion should also be questioned.

6.2 A System With Greater Volumes.

With the goal to reduce the effect of the dead volumes, a second demonstrator has been realized. The structure of this system corresponds to the schematic representation of Fig. 5, namely, an arrangement of a central and two lateral cylinders. The dimensions of the cylinders of this new system are given in Table 4.

Table 4

Parameters of the cylinders of the1 + 2 assembly

Cylinder typeBore size (mm)Pressure area (push) (mm2)Pressure area (pull) (mm2)Stroke (mm)
PB12-1001211393.4100
PB16-10016201181100
Cylinder typeBore size (mm)Pressure area (push) (mm2)Pressure area (pull) (mm2)Stroke (mm)
PB12-1001211393.4100
PB16-10016201181100

The realized device is shown in Fig. 17 without and with the control valves.

Fig. 17
Second system realized with two lateral cylinders: (a) cylinders and (b) control valves
Fig. 17
Second system realized with two lateral cylinders: (a) cylinders and (b) control valves
Close modal

6.3 Mechanical Forces and Energetic Properties of the 100 mm Linear Cylinder Assembly.

In Sec. 3, linear cylinder assemblies are presented where the principle of the “added expansion” has been applied. The numeric simulations have shown the dynamic behavior of a specific assembly using a set of 1 + 3 identical cylinders with short strokes (20 mm). The related experimental investigations have shown the noninsignificant effect of the dead volumes. Then, a second assembly with 1 + 2 cylinders and a longer stroke (100 mm) has been realized where the effect of the dead volumes was supposed to be smaller.

In this section, the quasi-static behavior of the new assembly is analyzed as well as the energetic properties. The comparison is made with a single cylinder operated in a classical way, without expansion, which consumes the same amount of compressed air as the new assembly.

6.3.1 Quasi-Static Behavior of the New Assembly.

In opposition to the dynamic simulation of the 1 + 3 cylinders system (Sec. 3.4), the simulation of the 1 + 2 cylinder assembly is done as a system where the piston’s movement is imposed by an external source. This allows to represent the value of the generated forces in dependency of the piston’s position and to better understand the proposed method of the added and imbricated expansion.

The simulated curves represent the pressure (Fig. 18) and the forces (Figs. 19 and 20) exerted on the sides of both pistons, from the previously calculated pressure. This pressure Pexp is calculated as a function of the position of the pistons, as a function of the ratio of the volumes of both cylinders, according to the rule of an adiabatic expansion

Pexp=Pin_air(V1maxV1a+V2b)γwithγ=1.4
(20)
Fig. 18
Pressure by expansion
Fig. 18
Pressure by expansion
Close modal
Fig. 19
Forces exerted on the small piston (left-to-right stroke)
Fig. 19
Forces exerted on the small piston (left-to-right stroke)
Close modal
Fig. 20
Global forces of the two pistons and total force of the new assembly
Fig. 20
Global forces of the two pistons and total force of the new assembly
Close modal

For the simulated curves of Figs. 1821, the horizontal axis represents the time and is related to a total cycle time of 1 s. The total cycle comprises one left-to-right stroke at a constant speed and one return stroke at the same speed.

Fig. 21
Total force with average and force of the small cylinder alone (without expansion)
Fig. 21
Total force with average and force of the small cylinder alone (without expansion)
Close modal

The pressure and force curves illustrate only the first half cycle.

The input pressure is chosen at 10 bar.

According to Eq. (20a) and for one 12 mm and two 16 mm cylinders, the pressure after the expansion is
Plow=Pin_air(V1maxV2max)γ=10105N/m2(0.012220.0162)1.4=1.69105N/m2

The 1 + 2 cylinder system has a 12 mm cylinder used as the filling device, and two 16 mm cylinders are the expansion device. With a stroke of 100 mm, the volumes of the first and second cylinders become

V1max=(0.012m)2π40.1m=0.0000113m3=11.3cm3
(21)
V2max=2(0.016m)2π40.1m=0.0000402m3=40.2cm3
(22)
The volumetric ratio is then
Rvol=V1maxV2max=0.281=13.55
(23)
During the movement of the pistons from the left to the right, the volume V1a varies from V1max to zero, and the volume V2b varies from zero to V2max. At the end of the stroke, the volumetric ratio is V1max/V2max. With a choice of 1/3.55 for this ratio, the pressure ratio becomes
PexpPin_air=(13.55)1.4=0.169
(24)
For the diagram in Fig. 18, an input pressure of 10 bar was chosen. The curves in Fig. 19 show the evolution of the forces on the first piston. The upper constant value curve corresponds to the constant force related to the intake at constant pressure. The value of this force is calculated through
F1a=Pin_airA1
(25)
where A1 is the active surface of the first (small) piston.
The decreasing curve corresponds to the force on the opposite side of this piston, due to the pressure during the expansion. This force is given by
F1b=PexpA1
(26)
Finally, the rising curve in Fig. 19 corresponds to the global force produced by the first cylinder, namely,
F1glob=F1aF1b
(27)
In Fig. 20, the force contributions of the two cylinders are represented. The lower decreasing curve is the global force exerted by the second piston F2a. This force takes the value of
F2glob=F2aF2b=PexpA2PatmA2
(28)

The second term in Eq. (27) illustrates the fact that on the opposite side of this piston the exhaust valve is open and the atmospheric pressure must be taken into account.

In the same figure (Fig. 20), the global force contribution of the first cylinder is represented again (rising curve), as the total force produced by the new assembly (upper decreasing curve). This total force is
Ftot=F1glob+F2glob
(29)
Finally, Fig. 21 shows the comparison of the performance of the new system with the performance of the single (small) cylinder operated alone, without expansion. The air consumption of both systems is the same. The decreasing curve shows the force produced by the new system together with its average value. The average value of the produced force is
Fave=1T/20T/2Ftot=185N
(30)
The force produced by the single cylinder without expansion (lower constant value) is given by
Fsgle=(Pin_airPatm)A1=9e5N/m20.000113m2=101.7N
(31)
and takes into account that the opposite side of the piston receives the atmospheric pressure. A1 is the area of the small piston (12 mm).
On the base of this comparison, the new cylinder assembly produces an average force equal to
Fave=185N101.7NFsgle=182%Fsgle
(32)

The factor of 182% illustrates the benefit of an additional expansion chamber in a different way in the sense of increasing the mechanical performance for a given air consumption in comparison to the reduction factor of the air consumption as was presented in Sec. 4.1.

Another advantage of the new system is that it can accelerate the mass of a given application with an important force at the beginning of the motion. The maximum force of the new system being
Fmax=(Pin_airPatm)A2=9e5N/m20.000402m2=361.9N
(33)

6.3.2 The Effect of the Dead Volumes of the 100 mm System.

An experience is made for the 1 + 2 100 mm system regarding the effect of the dead volumes. The pressures in the cylinders of this system are given in Fig. 22. The upper curve gives the pressure P1 in the first cylinder (V1, Fig. 16) and the lower curve gives the pressure in the second cylinder (V2, Fig. 16) after the opening of the transfer valve. The first abrupt fall from P1 to P1′ is due to the filling of the dead volume V2p, the decrease from P1′ down to P2′ is caused by the displacement of the pistons in the return stroke (expansion).

Fig. 22
Pressures in the cylinders of the second system
Fig. 22
Pressures in the cylinders of the second system
Close modal

7 Conclusions

A new assembly of pneumatic cylinders has been described as an alternative to classical cylinders which are well known for their poor energetic efficiency. The new proposed system comprises an added expansion volume which permits to recover the energy content of a filled cylinder by a real thermodynamic expansion instead of simply releasing the filled air to the atmosphere. The combination of the alternated effects of the expansion on each piston results in a higher resulting force with reduced variation. The energetic performance of the new system has been evaluated and compared with the performance of an equivalent single cylinder producing the same mechanical work. The theoretical result in terms of reduction of the consumed amount of air of the new proposed system is 43% of the amount of air consumed by a classical cylinder. Differently formulated, the new principle generates for the same amount of air consumed as a classical cylinder a higher effort in which an average value is 182% of that produced by the single cylinder without expansion. The paper has explained the operation principle and properties through numeric simulation. Two experimental small systems have been realized. A specific problem of the effect of dead volumes has been recognized which has a non-negligible effect on small cylinder assemblies.

Conflict of Interest

There are no conflicts of interest. This article does not include research in which human participants were involved. Informed consent not applicable. This article does not include any research in which animal participants were involved.

Data Availability Statement

The authors attest that all data for this study are included in the paper.

Nomenclature

d =

diameter of the cylinder

F =

force

U =

internal energy of the injected air under pressure

kred =

factor of reduction of consumption of air

Fave =

average value of the force

Fmax =

maximum value of the force

Ftot =

total force

F1a =

force on the left side of the first cylinder

F1b =

force on the right side of the first cylinder

F2a =

force on the left side of the second cylinder

Hin =

enthalpy injected in the cylinder

Pin =

pressure of the air at the input of the cylinder

Rvol =

volumetric ratio of the expansion

Woutn =

mechanical work produced by the assembly

ηconv =

efficiency of the conversion

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