Towards the Isothermal Gas Compression—A Novel Finned Piston-Cylinder with Increased Efficiency

: In this paper, a novel concept of a finned piston system is presented and analyzed in which the compression heat is continuously extracted from the compression chamber. The resulting compression characteristic moves in the direction of an isothermal process, reducing the temperature of the compressed fluid in the compression chamber and reducing the necessary mechanical work required to carry out the process. The finned piston concept consists in an integrated heat exchanger inside of the chamber that is constituted of imbricated flat fins placed on the stator part and on the mobile piston. The internal heat exchange on the surface is strongly increased in comparison with a classical piston/cylinder. The energetic performance of the new system is evaluated with the help of simulation. The pressures, forces, and temperature of the compressed gas are simulated as well as the mechanical work needed. The different curves are compared with the system’s adiabatic and isothermal characteristics.


Introduction
In the context of climate change and reducing green gas emissions, many efforts are being made to establish new alternative sources.From photovoltaic plants to wind energy farms, high investments are being made towards replacing fossil-based solutions.As a possibility to bridge or compensate for weather or seasonal fluctuations, long-term storage technologies will be based on energy conversion into chemical carriers as such as hydrogen, synthetic gases, or fuels.In such energy chains, gas compression stages will be used intensively and will have to be further developed to show higher energy efficiencies than the industrial solutions known today.
Not only for new energy applications but also in many other industrial processes, compression machines are used today as objects where energy efficiency issues have not been at the forefront of their technical developments.In a more general context, improvements in the energetic performance of compression machines and other machines or processes will be part of the solutions of the energy transition.[1][2][3].
In this paper, an original solution for the realization of compression stages is presented and analyzed.The new concept, called the finned piston, is based on the principle of continuously exporting the heat produced by the compression mechanism itself in order to reduce the amount of mechanical work needed for the elevation of the pressure following the reduction in the volume in the compression chamber.The concept of the finned piston has a heat exchanger integrated in the compression chamber.The new geometry proposed in this paper is following a previous study proposed by Mahbod Heidary in his PhD thesis [4][5][6].The use of imbricated cylindric fins in the stator part of the cylinder as well as on the sliding piston significantly increases the convective heat exchange on the surface in comparison to a classic piston/cylinder assembly and allows a compression process with a reduced elevation of the temperature inside the compression chamber [7][8][9][10].
The principle of the finned piston is applied on a so-called gas booster, used in many applications, with low power and high pressure.Whereas the original gas boosters use pneumatic actuators, well-known for their very low efficiency, a first important step has been achieved by replacing these with a variable-speed electric motor completed by an active compensator for the low-frequency power fluctuations.Figure 1 shows the evolution of the gas booster from its original version with a pneumatic motor (a) to the electromechanically moved versions.Two solutions are proposed, the first using a conventional crankshaft and piston rod (b), and the second using a so-called Scotch Yoke mechanism (c) [11,12].The next step for an increased efficiency is an improvement in the thermodynamic mechanism of the compression itself, which attempts to reach a characteristic of isothermal compression.
The principle of the finned piston is applied on a so-called gas booster, applications, with low power and high pressure.Whereas the original ga pneumatic actuators, well-known for their very low efficiency, a first impo been achieved by replacing these with a variable-speed electric motor com active compensator for the low-frequency power fluctuations.Figure 1 sho tion of the gas booster from its original version with a pneumatic motor (a) mechanically moved versions.Two solutions are proposed, the first using a crankshaft and piston rod (b), and the second using a so-called Scotch Yo (c) [11,12].The next step for an increased efficiency is an improvement in namic mechanism of the compression itself, which attempts to reach a ch isothermal compression.

The Compression Cylinder Based on a Finned Piston
In comparison with a classical cylinder/piston assembly, a so-called concept is built of imbricated metallic fins, with a first part at the stator side one on the side of the sliding piston.The original system analyzed in [4][5][6] us cylindric fins as represented at the bottom of Figure 2. One can see that suc ture uses nearly double the space of a classical cylinder/piston system (to when the volume occupied by the cooling fins is identical to the volume o sion chamber.The gas is compressed in the spaces between the stator fins, of the mobile part play the role of small pistons.The intake and exhaust through the radial perforations from fin to fin, and via the outlet (O1 in Fig other side (piston side), the gas is compressed in the spaces between the fin part where the stator fins play the role of pistons.The intake and exhaust p that side by radial perforations from fin to fin at the rear side of the piston fin a second outlet perforation placed at the top end (right side) of the stator fin 2).Anti-return valves in the intake and exhaust paths should be placed jus fices O1 and O2 in order to avoid excessive dead volumes.The gases of the then be mixed by a dedicated collector.
The system analyzed in this paper is built of imbricated flat fins, as r the middle of the figure.The advantage of using flat fins instead of cylindr explained in the next section.

The Compression Cylinder Based on a Finned Piston
In comparison with a classical cylinder/piston assembly, a so-called finned piston concept is built of imbricated metallic fins, with a first part at the stator side and a second one on the side of the sliding piston.The original system analyzed in [4][5][6] used imbricated cylindric fins as represented at the bottom of Figure 2. One can see that such an architecture uses nearly double the space of a classical cylinder/piston system (top of Figure 2) when the volume occupied by the cooling fins is identical to the volume of the compression chamber.The gas is compressed in the spaces between the stator fins, where the fins of the mobile part play the role of small pistons.The intake and exhaust path is given through the radial perforations from fin to fin, and via the outlet (O1 in Figure 2).On the other side (piston side), the gas is compressed in the spaces between the fins of the mobile part where the stator fins play the role of pistons.The intake and exhaust path is given on that side by radial perforations from fin to fin at the rear side of the piston fins and through a second outlet perforation placed at the top end (right side) of the stator fins (O2 in Figure 2).Anti-return valves in the intake and exhaust paths should be placed just after the orifices O1 and O2 in order to avoid excessive dead volumes.The gases of the two sides can then be mixed by a dedicated collector.
The system analyzed in this paper is built of imbricated flat fins, as represented in the middle of the figure.The advantage of using flat fins instead of cylindric ones will be explained in the next section.

The Heat Exchange Surface of the Classic, Flat and Cylindric Fin Systems
The heat exchange surface of the different systems comprises the external surface of the global compression cylinder, the end and front surfaces of the stator and piston, and also the effective side surface of each fin.A comparison of the three systems is given on the basis of a specific system, with the parameters given in Table 1.

Cyl Internal Diameter
Fin Thickness Active Length of the Cylinder Stroke D = 27.5 mm h = 2.5 mm L = 100 mm 90 mm

Heat Exchange Surface of the Classical Cylinder
The convection surface of the classical cylinder-piston assembly is equal to the sum of the extreme surfaces and the external cylinder surface, with tot classic cylinder without fins S = + = (3) This surface varies depending on the piston's position between 9826 mm 2 and 1187 mm 2 .

Heat Exchange Surface of the Flat Fin Cylinder
This surface is calculated according to the representation in Figure 3.

The Heat Exchange Surface of the Classic, Flat and Cylindric Fin Systems
The heat exchange surface of the different systems comprises the external surface of the global compression cylinder, the end and front surfaces of the stator and piston, and also the effective side surface of each fin.A comparison of the three systems is given on the basis of a specific system, with the parameters given in Table 1.The convection surface of the classical cylinder-piston assembly is equal to the sum of the extreme surfaces and the external cylinder surface, with and This surface varies depending on the piston's position between 9826 mm 2 and 1187 mm 2 .

Heat Exchange Surface of the Flat Fin Cylinder
This surface is calculated according to the representation in Figure 3.The surface of a fin side is given by where a is the fin's width and l is the stroke of the piston in the cylinder: and the fin's width can be calculated as The 11 fins totalize a surface of 46,145 mm 2 .
To this surface, the front and rear surface must be added, namely given by (1), as well as the surface of the external cylinder The details of the calculation are given in Appendix A.

Heat Exchange Surface of the Cylindric Fin Cylinder
The convection surface of the cylindric fins system is calculated in Appendix The external surface is equal to The heat exchange surfaces of the three solutions (classical, flat fins, and c fins) are represented in Figure 4, and are dependant on the position of the piston.The surface of a fin side is given by where a is the fin's width and l is the stroke of the piston in the cylinder: the angle α is and the fin's width can be calculated as The 11 fins totalize a surface of 46,145 mm 2 .To this surface, the front and rear surface must be added, namely S extrem = 1187 mm 2 given by (1), as well as the surface of the external cylinder S ext_cyl = 8639 mm 2 from (2) Finally, the total convection surface of the flat fin piston-cylinder assembly is The heat exchange surfaces of the three solutions (classical, flat fins, and cylindric fins) are represented in Figure 4, and are dependant on the position of the piston.
Figure 4 shows an important property of the classical cylinder-piston system, where the heat exchange surface reaches its maximum value at the BDC (bottom dead centre) position (corresponding to the 0 and 1 value of the dimensionless piston travel time of the cycle represented in the figure).The minimum value of the surface is given at the TDC (top dead centre) position (corresponding to the 0.5 value of the dimensionless piston travel time in the figure).Figure 4 shows an important property of the classical cylinder-piston system the heat exchange surface reaches its maximum value at the BDC (bottom dead position (corresponding to the 0 and 1 value of the dimensionless piston travel tim cycle represented in the figure).The minimum value of the surface is given at t (top dead centre) position (corresponding to the 0.5 value of the dimensionles travel time in the figure).
For the finned pistons (flat fins and cylindric fins), the heat transfer surface constant and independent of the position of the piston since there is a gap betw fins, and this allows the heat transfer to occur on the entire fin surface (Figure 4).H the effective thermal resistance for the heat transfer from the compressed gas to rounding area will be calculated in Section 3.1.2and is dependant on the piston's p

Two Models for the Simulation of the Finned Piston
Two different methods can be applied for the evaluation of the performanc compression cylinder.The first method is based on a common expression of the v in the pressure resulting from the variation in the volume of the chamber: with γ = 1.4 in the adiabatic mode and γ = 1 in the isothermal.
For an implementation of this expression, the volumetric ratio must be know it is relatively easy to define as a time function for representing the time variatio pressure.The polytropic factor γ indirectly represents the quality of the thermal flo the compressed gas to the surrounding area but does not allow us to make a rel the thermal model based on classic parameters like the thermal capacity of the com gas, nor the thermal resistance (which is easy to calculate on the basis of the kno rameters of the geometry of the finned system).For the evaluation of the temper the gas, another expression can be used: For the finned pistons (flat fins and cylindric fins), the heat transfer surface remains constant and independent of the position of the piston since there is a gap between the fins, and this allows the heat transfer to occur on the entire fin surface (Figure 4).However, the effective thermal resistance for the heat transfer from the compressed gas to the surrounding area will be calculated in Section 3.1.2and is dependant on the piston's position.

Two Models for the Simulation of the Finned Piston
Two different methods can be applied for the evaluation of the performance of the compression cylinder.The first method is based on a common expression of the variation in the pressure resulting from the variation in the volume of the chamber: with γ = 1.4 in the adiabatic mode and γ = 1 in the isothermal.
For an implementation of this expression, the volumetric ratio must be known, and it is relatively easy to define as a time function for representing the time variation in the pressure.The polytropic factor γ indirectly represents the quality of the thermal flow from the compressed gas to the surrounding area but does not allow us to make a relation to the thermal model based on classic parameters like the thermal capacity of the compressed gas, nor the thermal resistance (which is easy to calculate on the basis of the known parameters of the geometry of the finned system).For the evaluation of the temperature of the gas, another expression can be used: Again, in this case, the polytropic factor must be known, and it cannot be calculated from the parameters of a thermal model.With these formulas, the evolution of the pressure and of the temperature can be drawn, depending on the variation in the volume for the simple cases of adiabatic or isothermal compressions.
A second method which is based on the general expression P • V = m • R • T can be used and can be implemented in the form of the structural diagram of Figure 5.For this model, the temperature of the gas can be introduced as an output variable of a thermal model.Such a model is simply established on the basis of two known parameters, namely the thermal capacity of the gas and a thermal resistance which defines the thermal flow to the external world.The input variable of the thermal model corresponds to the thermal power transmitted to the gas and is calculated from the piston's velocity and the piston's force.The mechanical power Pow exerted by the piston on the gas is supposed to be identical (friction is neglected) to the transmitted thermal power Pth, as long as the exhaust valve of the cylinder is closed and no fluid in transferred out of the cylinder.The complete system model is represented in Figure 6, where the equivalent scheme for the thermal model with capacitor Cth and resistance Rth is transformed in a structural diagram with the integration block (1/s).The output variable of the thermal model corresponds to the temperature of the gas and, in the equivalent scheme, corresponds to the voltage across the thermal capacitor Cth.The auxiliary (binary) variable X represents the opening/closing of the exhaust valve.

Parameters of the Thermal Model of the Finned Piston System
The energetic performance of the finned piston system will be evaluated with the help of a simulation of one compression stroke.First, the evolution of the temperature of the gas will be simulated and, second, the necessary mechanical work needed for the compression will be calculated.These results will be obtained from the system model For this model, the temperature of the gas can be introduced as an output variable of a thermal model.Such a model is simply established on the basis of two known parameters, namely the thermal capacity of the gas and a thermal resistance which defines the thermal flow to the external world.The input variable of the thermal model corresponds to the thermal power transmitted to the gas and is calculated from the piston's velocity and the piston's force.The mechanical power Pow exerted by the piston on the gas is supposed to be identical (friction is neglected) to the transmitted thermal power P th , as long as the exhaust valve of the cylinder is closed and no fluid in transferred out of the cylinder.The complete system model is represented in Figure 6, where the equivalent scheme for the thermal model with capacitor C th and resistance R th is transformed in a structural diagram with the integration block (1/s).The output variable of the thermal model corresponds to the temperature of the gas and, in the equivalent scheme, corresponds to the voltage across the thermal capacitor C th .The auxiliary (binary) variable X represents the opening/closing of the exhaust valve.For this model, the temperature of the gas can be introduced as an output variable of a thermal model.Such a model is simply established on the basis of two known parameters, namely the thermal capacity of the gas and a thermal resistance which defines the thermal flow to the external world.The input variable of the thermal model corresponds to the thermal power transmitted to the gas and is calculated from the piston's velocity and the piston's force.The mechanical power Pow exerted by the piston on the gas is supposed to be identical (friction is neglected) to the transmitted thermal power Pth, as long as the exhaust valve of the cylinder is closed and no fluid in transferred out of the cylinder.The complete system model is represented in Figure 6, where the equivalent scheme for the thermal model with capacitor Cth and resistance Rth is transformed in a structural diagram with the integration block (1/s).The output variable of the thermal model corresponds to the temperature of the gas and, in the equivalent scheme, corresponds to the voltage across the thermal capacitor Cth.The auxiliary (binary) variable X represents the opening/closing of the exhaust valve.

Parameters of the Thermal Model of the Finned Piston System
The energetic performance of the finned piston system will be evaluated with the help of a simulation of one compression stroke.First, the evolution of the temperature of the gas will be simulated and, second, the necessary mechanical work needed for the compression will be calculated.These results will be obtained from the system model

Parameters of the Thermal Model of the Finned Piston System
The energetic performance of the finned piston system will be evaluated with the help of a simulation of one compression stroke.First, the evolution of the temperature of the gas will be simulated and, second, the necessary mechanical work needed for the compression will be calculated.These results will be obtained from the system model represented in Figure 6, where the parameters of the thermal model will play a significant role.The two parameters C th and R th will be calculated for the example that is defined in Table 1.They are based on the mass of the gas that is compressed and on the variable thermal resistance of the exchange between the fins and within the interspace between fins, as well as being dependent on the piston's position.

Thermal Capacitance
The thermal capacitance C th is expressed as J/K and is calculated through Equation ( 12) with the value of the mass of the gas m which is calculated from the filling conditions of the cylinder at a pressure of 15 bar and at ambient temperature This results in:

. Thermal Resistance
The thermal resistance which defines the heat transfer from the heated gas to the surrounding area results from the parallel connection of two resistances, R th1 and R th2 .R th1 corresponds to the transfer in the space between fins (dominant when piston is deployed, BDC).R th2 corresponds to the gaps between the fins (dominant when the piston is retracted, TDC).The two resistances are represented in Figure 7.
represented in Figure 6, where the parameters of the thermal model will play a significant role.The two parameters Cth and Rth will be calculated for the example that is defined in Table 1.They are based on the mass of the gas that is compressed and on the variable thermal resistance of the exchange between the fins and within the interspace between fins, as well as being dependent on the piston's position.

Thermal Capacitance
The thermal capacitance Cth is expressed as / J K and is calculated through Equa- tion.( 12) with the value of the mass of the gas m which is calculated from the filling conditions of the cylinder at a pressure of 15 bar and at ambient temperature This results in: The thermal resistance which defines the heat transfer from the heated gas to the surrounding area results from the parallel connection of two resistances, Rth1 and Rth2.Rth1 corresponds to the transfer in the space between fins (dominant when piston is deployed, BDC).Rth2 corresponds to the gaps between the fins (dominant when the piston is retracted, TDC).The two resistances are represented in Figure 7.The two resistances vary based on the function of the position and are calculated as: A represents the value of the total heat exchange surface, namely the sum of the surface of the fins, added to the surface of the piston's front and back, and also the surface of The two resistances vary based on the function of the position and are calculated as: A represents the value of the total heat exchange surface, namely the sum of the surface of the fins, added to the surface of the piston's front and back, and also the surface of the outer wall.The factor α changes from zero to one depending on whether the piston is deployed or retracted.
H 1 and H 2 are the thermal transfer coefficients expressed in W/(m 2 •K) where k a is the thermal conduction coefficient of the gas (air in the simulated case) expressed in W/(m•K).D 1 and D 2 are the distances between the fin walls, as represented in Figure 7.
Finally, the value of the global R th is obtained by the calculation of the parallel connection of R th1 and R th2 , according to R th is represented in Figure 8 as a function of the displacement of the piston in the cylinder.
H1 and H2 are the thermal transfer coefficients expressed in W/(m 2 •K) where ka is the thermal conduction coefficient of the gas (air in the simulated case) expressed in W/(m•K).D1 and D2 are the distances between the fin walls, as represented in Figure 7.
Finally, the value of the global Rth is obtained by the calculation of the parallel connection of Rth1 and Rth2, according to Rth is represented in Figure 8 as a function of the displacement of the piston in the cylinder

Simulation Conditions
A finned piston-cylinder with flat fins as represented in the middle of Figure 2 is simulated in the time domain with the parameters given in Table 1.The simulation covers one half-period of the cycle and corresponds to the compression stroke with a constant velocity of the piston.The simulation time is 0.5 s.The corresponding variation in the volume is represented in Figure 9.

Simulation Conditions
A finned piston-cylinder with flat fins as represented in the middle of Figure 2 is simulated in the time domain with the parameters given in Table 1.The simulation covers one half-period of the cycle and corresponds to the compression stroke with a constant velocity of the piston.The simulation time is 0.5 s.The corresponding variation in the volume is represented in Figure 9. Figure 10 gives, first, a comparison of the two models used for the calculation of the pressure (Equation ( 9) and Figure 5).The comparison is made for an isothermal compression.In this comparison, the elevation of the pressure goes up to the limit imposed by the volume reduction limit.For the following simulations, the pressure is limited to a value of 160 bar, which corresponds to the opening of the exhaust valve when the pressure of the gas in the cylinder equals the pressure in the receiving reservoir.The initial value of the pressure (intake) is 15 bar.The curves of the two models are completely superimposed.The limitation of the volume at the TDC (top dead centre) is imposed intentionally to avoid the infinite value of the ratio V comp_max /V compr_var in the simulation.
Figure 10 gives, first, a comparison of the two models used for the calculation of the pressure (Equation ( 9) and Figure 5).The comparison is made for an isothermal compression.In this comparison, the elevation of the pressure goes up to the limit imposed by the volume reduction limit.For the following simulations, the pressure is limited to a value of 160 bar, which corresponds to the opening of the exhaust valve when the pressure of the gas in the cylinder equals the pressure in the receiving reservoir.The initial value of the pressure (intake) is 15 bar.The curves of the two models are completely superimposed.
The limitation of the volume at the TDC (top dead centre) is imposed intentionally to avoid the infinite value of the ratio in the simulation.
Figure 10 gives, first, a comparison of the two models used for the calculation of the pressure (Equation ( 9) and Figure 5).The comparison is made for an isothermal compression.In this comparison, the elevation of the pressure goes up to the limit imposed by the volume reduction limit.For the following simulations, the pressure is limited to a value of 160 bar, which corresponds to the opening of the exhaust valve when the pressure of the gas in the cylinder equals the pressure in the receiving reservoir.The initial value of the pressure (intake) is 15 bar.The curves of the two models are completely superimposed.

Pressures, Forces and Temperatures
In Figure 11, three curves for the elevation of the pressure are given.First, the pressure elevation in the adiabatic mode is shown, where the whole compression heat is maintained inside of the cylinder (blue The initial condition is a value of 15 bar (15 × 10 5 N/m 2 ).When the pressure reaches the value of the external reservoir (160 bar), an antireturn valve opens automatically and the remaining portion of the stroke occurs under this pressure value when the compressed gas is transferred.The second curve (yellow curve) corresponds to a model of the compression according to the diagram of Figure 6.The temperature of the gas is calculated with the help of a thermal model using the parameters of the thermal capacity of the gas and the variable value of the thermal resistance, which were calculated in Section 3.1.2.The curve shows a typically slower rise of the pressure due to the continuous evacuation of a part of the produced heat.The third curve (red curve) corresponds to an isothermal compression.

Pressures, Forces and Temperatures
In Figure 11, three curves for the elevation of the pressure are given.First, the pressure elevation in the adiabatic mode is shown, where the whole compression heat is maintained inside of the cylinder (blue curve).The initial condition is a value of 15 bar (15 × 10 5 N/m 2 ).When the pressure reaches the value of the external reservoir (160 bar), an anti-return valve opens automatically and the remaining portion of the stroke occurs under this pressure value when the compressed gas is transferred.The second curve (yellow curve) corresponds to a model of the compression according to the diagram of Figure 6.The temperature of the gas is calculated with the help of a thermal model using the parameters of the thermal capacity of the gas and the variable value of the thermal resistance, which were calculated in Section 3.1.2.The curve shows a typically slower rise of the pressure due to the continuous evacuation of a part of the produced heat.The third curve (red curve) corresponds to an isothermal compression.Figure 12 shows the variation of the force exerted by the piston on the gas.Again, the three conditions are represented, adiabatic (blue), finned piston (yellow) and isothermal (red).Figure 12 shows the variation of the force exerted by the piston on the gas.Again, the three conditions are represented, adiabatic (blue), finned piston (yellow) and isothermal (red).Figure 12 shows the variation of the force exerted by the piston on the gas.Again, the three conditions are represented, adiabatic (blue), finned piston (yellow) and isothermal (red).In the simulation diagram of Figure 6, the mechanical power (Pow) and the power transmitted to the gas (Pth) are simulated.The evolution of these two variables is represented in Figures 13 and 14.The mechanical powers are calculated from the mechanical force and the velocity of the piston.After the opening of the exhaust valve, the power exerted by the piston serves to exhaust the gas and no more energy is transmitted to the gas in the cylinder.The opening of the valve is activated in the simulation diagram by the binary variable X.In the simulation diagram of Figure 6, the mechanical power (Pow) and the power transmitted to the gas (P th ) are simulated.The evolution of these two variables is represented in Figures 13 and 14.The mechanical powers are calculated from the mechanical force and the velocity of the piston.After the opening of the exhaust valve, the power exerted by the piston serves to exhaust the gas and no more energy is transmitted to the gas in the cylinder.The opening of the valve is activated in the simulation diagram by the binary variable X.  Figure 15 represents the principal result of this study, namely the curves of the elevation of the temperature in the compression cylinder.The yellow curve shows the performance of the proposed system using the finned piston technology.The blue and the red curves represent the temperatures of the gas in the conditions of adiabatic and isothermal compression, respectively.The performance of the integrated heat exchanger (the finned system) is particularly visible through the strong reduction in the elevation of the  Figure 15 represents the principal result of this study, namely the curves of the elevation of the temperature in the compression cylinder.The yellow curve shows the performance of the proposed system using the finned piston technology.The blue and the red curves represent the temperatures of the gas in the conditions of adiabatic and isothermal compression, respectively.The performance of the integrated heat exchanger (the finned system) is particularly visible through the strong reduction in the elevation of the temperature (around 100 K instead of nearly 250 K in adiabatic conditions).
Power mec adia (blue, w fins (yellow), iso (red) [W] Power to the gas adia (blue) w fins (yellow), iso (red) [W] Figure 15 represents the principal result of this study, namely the curves of the elevation of the temperature in the compression cylinder.The yellow curve shows the performance of the proposed system using the finned piston technology.The blue and the red curves represent the temperatures of the gas in the conditions of adiabatic and isothermal compression, respectively.The performance of the integrated heat exchanger (the finned system) is particularly visible through the strong reduction in the elevation of the temperature (around 100 K instead of nearly 250 K in adiabatic conditions).Figure 15 represents the principal result of this study, namely the curves of the elevation of the temperature in the compression cylinder.The yellow curve shows the performance of the proposed system using the finned piston technology.The blue and the red curves represent the temperatures of the gas in the conditions of adiabatic and isothermal compression, respectively.The performance of the integrated heat exchanger (the finned system) is particularly visible through the strong reduction in the elevation of the temperature (around 100 K instead of nearly 250 K in adiabatic conditions).

Effect of a Reduced Speed
Reducing the speed of the piston has the effect that the evacuation of the compression heat is more performant due to the fact the thermal time-constants in the thermal model (C th and R th ) are not changed.The result of this is that the rise in the temperature in the cylinder is reduced.Figure 16 shows, again, the evolution of the temperature when the speed of the piston is reduced from 0.18 m/s to 0.09 m/s (Figure 16a) and 0.045 m/s, respectively (Figure 16b).This means that the 90 mm stroke is covered in 1 and 2 s (instead of 0.5 s).The value of the temperature in adiabatic conditions is unchanged as well in isothermal conditions.For the finned piston system (yellow curve), the elevation of the temperature is one half of the elevation with the original speed (0.18 m/s).With a speed of 0.045 m/s, the elevation is reduced by another 50%.

Effect of a Reduced Speed
Reducing the speed of the piston has the effect that the evacuation of the compression heat is more performant due to the fact the thermal time-constants in the thermal model (Cth and Rth) are not changed.The result of this is that the rise in the temperature in the cylinder is reduced.Figure 16 shows, again, the evolution of the temperature when the speed of the piston is reduced from 0.18 m/s to 0.09 m/s (Figure 16a) and 0.045 m/s, respectively (Figure 16b).This means that the 90 mm stroke is covered in 1 and 2 s (instead of 0.5 s).The value of the temperature in adiabatic conditions is unchanged as well in isothermal conditions.For the finned piston system (yellow curve), the elevation of the temperature is one half of the elevation with the original speed (0.18 m/s).With a speed of 0.045 m/s, the elevation is reduced by another 50%.

Energetic Considerations
Figure 17 shows the performance of the finned piston system in terms of energy.The three curves correspond to the integration of the mechanical power needed to move the piston.It is effectively the energy consumed by the different compression tasks (adiabatic in blue, finned in yellow, and isothermal in red).The reduction in the needed energy from 350 to 300 joules corresponds to a saving of 14%.

Energetic Considerations
Figure 17 shows the performance of the finned piston system in terms of energy.The three curves correspond to the integration of the mechanical power needed to move the piston.It is effectively the energy consumed by the different compression tasks (adiabatic in blue, finned in yellow, and isothermal in red).The reduction in the needed energy from 350 to 300 joules corresponds to a saving of 14%.

Energetic Considerations
Figure 17 shows the performance of the finned piston system in terms of energy.T three curves correspond to the integration of the mechanical power needed to move t piston.It is effectively the energy consumed by the different compression tasks (adiaba in blue, finned in yellow, and isothermal in red).The reduction in the needed energy fro 350 to 300 joules corresponds to a saving of 14%.In Figure 18, the energy transferred from the piston to the gas is represented.This energy comprises two parts, namely the energy transferred to the gas itself (gas internal energy) and a part which is evacuated by the fins.
Gases 2024, 4, 4 13 of In Figure 18, the energy transferred from the piston to the gas is represented.Th energy comprises two parts, namely the energy transferred to the gas itself (gas intern energy) and a part which is evacuated by the fins.

Effect of a Reduced Speed
The mechanical work needed for the compression up to 160 bar is also reduced whe the compression speed is reduced.Figure 19 shows the same variables as in Figure 17 b for when the speed is reduced to 0.09 m/s (Figure 19a) and 0.045 m/s (Figure 19b), respe tively.The final value of the work of the finned piston at t = 1 s is 289 J for a piston spee of 0.09 m/s.The reduction from 350 J to 289 J represents a reduction of 17.4%.For a spee of 0.045 m/s (Figure 19b), the curve of the work of the finned solution is even closer to th curve of the isothermal compression (final value: 279 J).

Effect of a Reduced Speed
The mechanical work needed for the compression up to 160 bar is also reduced when the compression speed is reduced.Figure 19 shows the same variables as in Figure 17 but for when the speed is reduced to 0.09 m/s (Figure 19a) and 0.045 m/s (Figure 19b), respectively.The final value of the work of the finned piston at t = 1 s is 289 J for a piston speed of 0.09 m/s.The reduction from 350 J to 289 J represents a reduction of 17.4%.For a speed of 0.045 m/s (Figure 19b), the curve of the work of the finned solution is even closer to the curve of the isothermal compression (final value: 279 J).
the compression speed is reduced.Figure 19 shows the same variables as in Figure 17 but for when the speed is reduced to 0.09 m/s (Figure 19a) and 0.045 m/s (Figure 19b), respectively.The final value of the work of the finned piston at t = 1 s is 289 J for a piston speed of 0.09 m/s.The reduction from 350 J to 289 J represents a reduction of 17.4%.For a speed of 0.045 m/s (Figure 19b), the curve of the work of the finned solution is even closer to the curve of the isothermal compression (final value: 279 J).

The Limit of the Model of the Finned Piston Systems
The energetic performance of the finned piston concept has been evaluated by simulation on the basis of a dedicated model which includes a thermal model for the cooling effect of the finned system.The gain in the necessary mechanical work for the compression has been evaluated in this first study.Additionally, the elevation of the temperature and the mechanical work for compression has been simulated for a reduced speed of the  The energetic performance of the finned piston concept has been evaluated by simulation on the basis of a dedicated model which includes a thermal model for the cooling effect of the finned system.The gain in the necessary mechanical work for the compression has been evaluated in this first study.Additionally, the elevation of the temperature and the mechanical work for compression has been simulated for a reduced speed of the piston.The verified performance of the proposed system with the simple model used in this study shows good potential for the performance increase of a real system.However, the performance should be calculated more accurately, including, for instance, the friction of the mechanical parts, especially of the sealing elements.Particularly for high pressures, the influence of the seals can be of importance.
All the simulated phenomena and, in the foreground, the elevation of the temperature are based on the assumption that the temperature of the fins and of the whole cylinder body is constant and identical to the ambient temperature.In reality, the temperature of the fins will be dependent on the quality of their own cooling.In consequence, the model should be completed by a part dedicated to the cooling of the stator fins and of the mobile fins.Such a model should include the resistance of the thermal conduction of the element between the compression fins and the fins of the stator and mobile cooling system.Also, the thermal capacitance of the additional elements should be included.

A Finned Compressor with Cooling of the Stator and of the Mobile Finned Piston
In Figure 20, a complete compressor with the finned technology is represented.The compressor consists of two cylinders using the finned technology.The two pistons are connected to a common crankshaft via two connecting rods, and they work with a 180 • phase-shift.This phase-shift allows us to get the same compression sequence of the two lateral cylinders of a classical gas booster, shown in Figure 1.
The cooling of the stator part does not represent a big challenge due to its non-moving character.However, concerning the sliding finned pistons, the transfer of the heat is of a different difficulty.The heat transfer functional elements are given by a couple of sliding horizontal fins, moving inside a cylinder-prolongation section, and are cooled by a transverse air flow.The air flow is produced by a dedicated fan.
The sliding fins play two different roles, the first being to cool the piston at its rear side, and the second being to transmit the compression force from the crankshaft to the mobile piston.
The proposed system in Figure 19 should be modelled accurately for an evaluation of its performance.A finite element representation of the thermodynamic phenomena will certainly produce the best results.
of a different difficulty.The heat transfer functional elements are given by a couple of sliding horizontal fins, moving inside a cylinder-prolongation section, and are cooled by a transverse air flow.The air flow is produced by a dedicated fan.
The sliding fins play two different roles, the first being to cool the piston at its rear side, and the second being to transmit the compression force from the crankshaft to the mobile piston.

Conclusions
A new finned piston system has been analyzed for its energetic performance and for its reduction in the elevation of the temperature.The aim of this study is to develop a cylinder/piston system for which the compression characteristic tends to move in the direction of an isothermal compression.The technique used is to integrate a heat exchange device inside of the compression volume, allowing a permanent evacuation of the produced heat during the compression process.The heat exchange device consists of imbricated fins sliding with the piston movement against the fins of the stator part.The heat transfer area of a system with flat fins is compared to a system with cylindric fins, showing a superior performance of the former system.This study illustrates a simplified system with a linear movement of the piston.The thermodynamic phenomena are represented with simple thermodynamic relations, where ideal adiabatic conditions are simulated as well as ideal isothermal phenomena.The new proposed system with fins is evaluated using an integrated thermal model of the heat exchange and shows a good performance and significant reduction in the temperature in the compression cylinder.The simulations are made for different values of the compression speed.The reduction in the compression speed shows a clear tendency of performance increase, namely reduction in the needed mechanical effort together with a reduction in the temperature of the compressed gas.

Figure 1 .
Figure 1.Gas boosters driven by pneumatic motor (a), and by electric motor (b,c).

Figure 1 .
Figure 1.Gas boosters driven by pneumatic motor (a), and by electric motor (b,c).

Figure 2 .
Figure 2. Classic piston and finned pistons with flat and cylindric fins.

Figure 2 .
Figure 2. Classic piston and finned pistons with flat and cylindric fins.

Figure 3 .
Figure 3. Parameters of the flat fin cylinder.
total convection surface of the flat fin piston-cylinder assembly i

Figure 3 .
Figure 3. Parameters of the flat fin cylinder.

Figure 4 .
Figure 4. Heat transfer areas of the different cylinders.

Figure 4 .
Figure 4. Heat transfer areas of the different cylinders.

Figure 5 .
Figure 5. Thermal model for the pressure.

Figure 6 .
Figure 6.The complete model for the compression cylinder with the thermal part.

Figure 5 .
Figure 5. Thermal model for the pressure.

Figure 5 .
Figure 5. Thermal model for the pressure.

Figure 6 .
Figure 6.The complete model for the compression cylinder with the thermal part.

Figure 6 .
Figure 6.The complete model for the compression cylinder with the thermal part.

Figure 7 .
Figure 7. Definition of the thermal resistances R th1 and R th2 .

Figure 8 .
Figure 8. Thermal resistance as a function of the piston's displacement.

Figure 8 .
Figure 8. Thermal resistance as a function of the piston's displacement.

Figure 9 .
Figure 9. Variation in the volume of the compession chamber.The limitation of the volume at the TDC (top dead centre) is imposed intentionally to avoid the infinite value of the ratio _ max _ var comp compr V V

Figure 9 .
Figure 9. Variation in the volume of the compession chamber.

Figure 10 .
Figure 10.Comparison of the models.

Figure 10 .
Figure 10.Comparison of the models.

Figure 14 .
Figure 14.Power transmitted to the gas.

PoFigure 15 .
Figure 15.Temperatures of the gas during compression.

Figure 18 .
Figure 18.Energy transferred from the piston to the gas in the cylinder.Adiabatic (blue), finned (yellow), and isothermal (red).

Figure 18 .
Figure 18.Energy transferred from the piston to the gas in the cylinder.

5 .
Discussion and Future Development 5.1.The Limit of the Model of the Finned Piston Systems

Figure 20 .
Figure 20.Compressor with two cylinders (180° phase-shift) and cooling of the fins.Figure 20.Compressor with two cylinders (180 • phase-shift) and cooling of the fins.

Figure 20 .
Figure 20.Compressor with two cylinders (180° phase-shift) and cooling of the fins.Figure 20.Compressor with two cylinders (180 • phase-shift) and cooling of the fins.

Table 1 .
Dimensions of the system.

Table 1 .
Dimensions of the system.