Patent Publication Number: US-8117842-B2

Title: Systems and methods for compressed-gas energy storage using coupled cylinder assemblies

Description:
RELATED APPLICATIONS 
     This application is a continuation of U.S. patent application Ser. No. 12/938,853, filed on Nov. 3, 2010, which claims the benefit of and priority to U.S. Provisional Patent Application No. 61/257,583, filed Nov. 3, 2009; U.S. Provisional Patent Application No. 61/287,938, filed Dec. 18, 2009; U.S. Provisional Patent Application No. 61/310,070, filed Mar. 3, 2010; and U.S. Provisional Patent Application No. 61/375,398, filed Aug. 20, 2010, the entire disclosure of each of which is hereby incorporated herein by reference. 
    
    
     STATEMENT REGARDING FEDERALLY SPONSORED RESEARCH 
     This invention was made with government support under IIP-0810590 and IIP-0923633 awarded by the NSF. The government has certain rights in the invention. 
    
    
     FIELD OF THE INVENTION 
     In various embodiments, the present invention relates to pneumatics, power generation, and energy storage, and more particularly, to compressed-gas energy-storage systems and methods using pneumatic cylinders. 
     BACKGROUND 
     Storing energy in the form of compressed gas has a long history and components tend to be well tested, reliable, and have long lifetimes. The general principle of compressed-gas or compressed-air energy storage (CAES) is that generated energy (e.g., electric energy) is used to compress gas (e.g., air), thus converting the original energy to pressure potential energy; this potential energy is later recovered in a useful form (e.g., converted back to electricity) via gas expansion coupled to an appropriate mechanism. Advantages of compressed-gas energy storage include low specific-energy costs, long lifetime, low maintenance, reasonable energy density, and good reliability. 
     If a body of gas is at the same temperature as its environment, and expansion occurs slowly relative to the rate of heat exchange between the gas and its environment, then the gas will remain at approximately constant temperature as it expands. This process is termed “isothermal expansion. Isothermal expansion of a quantity of gas stored at a given temperature recovers approximately three times more work than would “adiabatic expansion, that is, expansion where no heat is exchanged between the gas and its environment, because the expansion happens rapidly or in an insulated chamber. Gas may also be compressed isothermally or adiabatically. 
     An ideally isothermal energy-storage cycle of compression, storage, and expansion would have 100% thermodynamic efficiency. An ideally adiabatic energy-storage cycle would also have 100% thermodynamic efficiency, but there are many practical disadvantages to the adiabatic approach. These include the production of higher temperature and pressure extremes within the system, heat loss during the storage period, and inability to exploit environmental (e.g., cogenerative) heat sources and sinks during expansion and compression, respectively. In an isothermal system, the cost of adding a heat-exchange system is traded against resolving the difficulties of the adiabatic approach. In either case, mechanical energy from expanding gas must usually be converted to electrical energy before use. 
     An efficient and novel design for storing energy in the form of compressed gas utilizing near isothermal gas compression and expansion has been shown and described in U.S. patent application Ser. Nos. 12/421,057 (the &#39;057 application) and 12/639,703 (the &#39;703 application), the disclosures of which are hereby incorporated herein by reference in their entireties. The &#39;057 and &#39;703 applications disclose systems and methods for expanding gas isothermally in staged hydraulic/pneumatic cylinders and intensifiers over a large pressure range in order to generate electrical energy when required. Mechanical energy from the expanding gas is used to drive a hydraulic pump/motor subsystem that produces electricity. Systems and methods for hydraulic-pneumatic pressure intensification that may be employed in systems and methods such as those disclosed in the &#39;057 and &#39;703 applications are shown and described in U.S. patent application Ser. No. 12/879,595 (the &#39;595 application), the disclosure of which is hereby incorporated herein by reference in its entirety. 
     The ability of such systems to either store energy (i.e., use energy to compress gas into a storage reservoir) or produce energy (i.e., expand gas from a storage reservoir to release energy) will be apparent to any person reasonably familiar with the principles of electrical and pneumatic machines. 
     Various embodiments described in the &#39;057 application involve several energy conversion stages: during compression, electrical energy is converted to rotary motion in an electric motor, then converted to hydraulic fluid flow in a hydraulic pump, then converted to linear motion of a piston in a hydraulic-pneumatic cylinder assembly, then converted to mechanical potential energy in the form of compressed gas. Conversely, during retrieval of energy from storage by gas expansion, the potential energy of pressurized gas is converted to linear motion of a piston in a hydraulic-pneumatic cylinder assembly, then converted to hydraulic fluid flow through a hydraulic motor to produce rotary mechanical motion, then converted to electricity using a rotary electric generator. 
     However, such energy storage and recovery systems would be more directly applicable to a wide variety of applications if they converted the work done by the linear piston motion directly into electrical energy or into rotary motion via mechanical means (or vice versa). In such ways, the overall efficiency and cost-effectiveness of the compressed air system may be increased. 
     SUMMARY 
     Embodiments of the present invention obviate the need for a hydraulic subsystem by converting the reciprocal motion of energy storage and recovery cylinders into electrical energy via alternative means. In some embodiments, the invention combines a compressed-gas energy storage system with a linear-generator system for the generation of electricity from reciprocal motion to increase system efficiency and cost-effectiveness. The same arrangement of devices can be used to convert electric energy to potential energy in compressed gas, with similar gains in efficiency and cost-effectiveness. 
     Another alternative, utilized in various embodiments, to the use of hydraulic fluid to transmit force between the motor/generator and the gas undergoing compression or expansion is the mechanical transmission of the force. In particular, the linear motion of the cylinder piston or pistons may be coupled to a crankshaft or other means of conversion to rotary motion. The crankshaft may in turn be coupled to, e.g., a gear box or a continuously variable transmission (CVT) that drives the shaft of an electric motor/generator at a rotational speed higher than that of the crankshaft. The continuously variable transmission, within its operable range of effective gear ratios, allows the motor/generator to be operated at constant speed regardless of crankshaft speed. The motor/generator operating point can be chosen for optimal efficiency; constant output power is also desirable. Multiple pistons may be coupled to a single crankshaft, which may be advantageous for purposes of shaft balancing. 
     In addition, energy storage and generation systems in accordance with embodiments of the invention may include a heat-transfer subsystem for expediting heat transfer in one or more compartments of the cylinder assembly. In one embodiment, the heat-transfer subsystem includes a fluid circulator and a heat-transfer fluid reservoir as described in the &#39;703 application. The fluid circulator pumps a heat-transfer fluid into the first compartment and/or the second compartment of the pneumatic cylinder. The heat-transfer subsystem may also include a spray mechanism, disposed in the first compartment and/or the second compartment, for introducing the heat-transfer fluid. In various embodiments, the spray mechanism is a spray head and/or a spray rod. 
     Gas undergoing expansion tends to cool, while gas undergoing compression tends to heat. To maximize efficiency (i.e., the fraction of elastic potential energy in the compressed gas that is converted to work, or vice versa), gas expansion and compression should be as near isothermal (i.e., constant-temperature) as possible. Several ways of approximating isothermal expansion and compression may be employed. 
     First, as described in the &#39;703 application, droplets of a liquid (e.g., water) may be sprayed into a chamber of the pneumatic cylinder in which gas is presently undergoing compression (or expansion) in order to transfer heat to or from the gas. As the liquid droplets exchange heat with the gas around them, the temperature of the gas is raised or lowered; the temperature of the droplets is also raised or lowered. The liquid is evacuated from the cylinder through a suitable mechanism. The heat-exchange spray droplets may be introduced through a spray head (in, e.g., a vertical cylinder), through a spray rod arranged coaxially with the cylinder piston (in, e.g., a horizontal cylinder), or by any other mechanism that permits formation of a liquid spay within the cylinder. Droplets may be used to either warm gas undergoing expansion or to cool gas undergoing compression. An isothermal process may be approximated via judicious selection of this heat-exchange rate. 
     Furthermore, as described in U.S. Pat. No. 7,802,426 (the &#39;426 patent), the disclosure of which is hereby incorporated by reference herein in its entirety, gas undergoing either compression or expansion may be directed, continuously or in installments, through a heat-exchange subsystem external to the cylinder. The heat-exchange subsystem either rejects heat to the environment (to cool gas undergoing compression) or absorbs heat from the environment (to warm gas undergoing expansion). Again, an isothermal process may be approximated via judicious selection of this heat-exchange rate. 
     As mentioned above, some embodiments of the present invention utilize a linear motor/generator as an alternative to the conventional rotary motor/generator. Like a rotary motor/generator, a linear motor/generator, when operated as a generator, converts mechanical power to electrical power by exploiting Faraday&#39;s law of induction: that is, the magnetic flux through a closed circuit is made to change by moving a magnet, thus inducing an electromotive force (EMF) in the circuit. The same device may also be operated as a motor. 
     There are several forms of linear motor/generator, but for simplicity, the discussion herein mainly pertains to the permanent-magnet tubular type. In some applications tubular linear generators have advantages over flat topologies, including smaller leakage, smaller coils with concomitant lower conductor loss and higher force-to-weight ratio. For brevity, only operation in generator mode is described herein. The ability of such a machine to operate as either a motor or generator will be apparent to any person reasonably familiar with the principles of electrical machines. 
     In a typical tubular linear motor/generator, permanent radially-magnetized magnets, sometimes alternated with iron core rings, are affixed to a shaft. The permanent magnets have alternating magnetization. This armature, composed of shaft and magnets, is termed a translator or mover and moves axially through a tubular winding or stator. Its function is analogous to that of a rotor in a conventional generator. Moving the translator through the stator in either direction produces a pulse of alternating EMF in the stator coil. The tubular linear generator thus produces electricity from a source of reciprocating motion. Moreover, such generators offer the translation of such mechanical motion into electrical energy with high efficiency, since they obviate the need for gear boxes or other mechanisms to convert reciprocal into rotary motion. Since a linear generator produces a series of pulses of alternating current (AC) power with significant harmonics, power electronics are typically used to condition the output of such a generator before it is fed to the power grid. However, such power electronics require less maintenance and are less prone to failure than the mechanical linear-to-rotary conversion systems which would otherwise be required. Operated as a motor, such a tubular linear motor/generator produces reciprocating motion from an appropriate electrical excitation. 
     In a compressed-gas energy storage system, gas is stored at high pressure (e.g., approximately 3000 pounds per square inch gauge (psig)). This gas is expanded into a chamber containing a piston or other mechanism that separates the gas on one side of the chamber from the other, preventing gas movement from one chamber to the other while allowing the transfer of force/pressure from one chamber to the next. This arrangement of chambers and piston (or other mechanism) is herein termed a “pneumatic cylinder or “cylinder. The term “cylinder is not, however, limited to vessels that are cylindrical in shape (i.e., having a circular cross-section); rather, a cylinder merely defines a sealed volume and may have a cross-section of any arbitrary shape that may or may not vary through the volume. The shaft of the cylinder may be attached to a mechanical load, e.g., the translator of a linear generator. In the simplest arrangement, the cylinder shaft and translator are in line (i.e., aligned on a common axis). In some embodiments, the shaft of the cylinder is coupled to a transmission mechanism for converting a reciprocal motion of the shaft into a rotary motion, and a motor/generator is coupled to the transmission mechanism. In some embodiments, the transmission mechanism includes a crankshaft and a gear box. In other embodiments, the transmission mechanism includes a crankshaft and a CVT. A CVT is a transmission that can move smoothly through a continuum of effective gear ratios over some finite range. 
     In the type of compressed-gas storage system described in the &#39;057 application, reciprocal motion is produced during recovery of energy from storage by expansion of gas in pneumatic cylinders. In various embodiments, this reciprocal motion is converted to rotary motion by first using the expanding gas to drive a pneumatic/hydraulic intensifier; the hydraulic fluid pressurized by the intensifier drives a hydraulic rotary motor/generator to produce electricity. (The system is run in reverse to convert electric energy into potential energy in compressed gas.) By mechanically coupling linear generators to pneumatic cylinders, the hydraulic system may be omitted, typically with increased efficiency and reliability. Conversely, a linear motor/generator may be operated as a motor in order to compress gas in pneumatic cylinders for storage in a reservoir. In this mode of operation, the device converts electrical energy to mechanical energy rather than the reverse. The potential advantages of using a linear electrical machine may thus accrue to both the storage and recovery operations of a compressed-gas energy storage system. 
     In various embodiments, the compression and expansion occurs in multiple stages, using low- and high-pressure cylinders. For example, in expansion, high-pressure gas is expanded in a high-pressure cylinder from a maximum pressure (e.g., approximately 3,000 psig) to some mid-pressure (e.g. approximately 300 psig); then this mid-pressure gas is further expanded further (e.g., approximately 300 psig to approximately 30 psig) in a separate low-pressure cylinder. Thus, a high-pressure cylinder may handle a maximum pressure up to approximately a factor of ten greater than that of a low-pressure cylinder. Furthermore, the ratio of maximum to minimum pressure handled by a high-pressure cylinder may be approximately equal to ten (or even greater), and/or may be approximately equal to such a ratio of the low-pressure cylinder. The minimum pressure handled by a high-pressure cylinder may be approximately equal to the maximum pressure handled by a low-pressure cylinder. 
     The two stages may be tied to a common shaft and driven by a single linear motor/generator (or may be coupled to a common crankshaft, as detailed below). When each piston reaches the limit of its range of motion (e.g., reaches the end of the low-pressure side of the chamber), valves or other mechanisms may be adjusted to direct gas to the appropriate chambers. In double-acting devices of this type, there is no withdrawal stroke or unpowered stroke: the stroke is powered in both directions. 
     Since a tubular linear generator is inherently double-acting (i.e., generates power regardless of which way the translator moves), the resulting system generates electrical power at all times other than when the piston is hesitating between strokes. Specifically, the output of the linear generator may be a series of pulses of AC power, separated by brief intervals of zero power output during which the mechanism reverses its stroke direction. Power electronics may be employed with short-term energy storage devices such as ultracapacitors to condition this waveform to produce power acceptable for the grid. Multiple units operating out-of-phase may also be used to minimize the need for short-term energy storage during the transition periods of individual generators. 
     Use of a CVT enables the motor/generator to be operated at constant torque and speed over a range of crankshaft rotational velocities. The resulting system generates electrical power continuously and at a fixed output level as long as pressurized air is available from the reservoir. As mentioned above, power electronics and short-term energy storage devices such as ultracapacitors may, if needed, condition the waveform produced by the motor/generator to produce power acceptable for the grid. 
     In various embodiments, the system also includes a source of compressed gas and a control-valve arrangement for selectively connecting the source of compressed gas to an input of the first compartment (or “chamber) of the pneumatic cylinder assembly and an input of the second compartment of the pneumatic cylinder assembly. The system may also include a second pneumatic cylinder assembly having a first compartment and a second compartment separated by a piston slidably disposed within the cylinder and a shaft coupled to the piston and extending through at least one of the first compartment and the second compartment of the second cylinder and beyond an end cap of the second cylinder and coupled to a transmission mechanism. The second pneumatic cylinder assembly may be fluidly coupled to the first pneumatic cylinder assembly. For example, the pneumatic cylinder assemblies may be coupled in series. Additionally, one of the pneumatic cylinder assemblies may be a high-pressure cylinder and the other pneumatic cylinder assembly may be a low-pressure cylinder. The low-pressure cylinder assembly may be volumetrically larger, e.g., may have an interior volume at least 50% larger, than the high-pressure cylinder assembly. 
     A further opportunity for increased efficiency arises from the fact that as gas in the high-pressure storage vessel is exhausted, its pressure decreases. Thus, in order to extract as much energy as possible from a given quantity of stored gas, the electricity-producing side of such an energy-storage system must operate over a wide range of input pressures, i.e., from the reservoir&#39;s high-pressure limit (e.g., approximately 3,000 psig) to as close to atmospheric pressure as possible. At lower pressure, gas expanding in a cylinder exerts a smaller force on its piston and thus on the translator of the linear generator (or to the rotor of the generator) to which it is coupled. For a fixed piston speed, this generally results in reduced power output. 
     In preferred embodiments, however, power output is substantially constant. Constant power may be maintained with decreased force by increasing piston linear speed. Piston speed may be regulated, for example, by using power electronics to adjust the electrical load on a linear generator so that translator velocity is increased (with correspondingly higher voltage and lower current induced in the stator) as the pressure of the gas in the high-pressure storage vessel decreases. At lower gas-reservoir pressures, in such an arrangement, the pulses of AC power produced by the linear generator will be shorter in duration and higher in frequency, requiring suitable adjustments in the power electronics to continue producing grid-suitable power. 
     With variable linear motor/generator speed, efficiency gains may be realized by using variable-pitch windings and/or a switched-reluctance linear generator. In a switched-reluctance generator, the mover (i.e., translator or rotor) contains no permanent magnets; rather, magnetic fields are induced in the mover by windings in the stator which are controlled electronically. The position of the mover is either measured or calculated, and excitement of the stator windings is electronically adjusted in real time to produce the desired torque (or traction) for any given mover position and velocity. 
     Substantially constant power may also be achieved by mechanical linkages which vary the torque for a given force. Other techniques include piston speed regulation by using power electronics to adjust the electrical load on the motor/generator so that crankshaft velocity is increased, which for a fixed torque will increase power. For such arrangements using power electronics, the center frequency and harmonics of the AC waveform produced by the motor/generator typically change, which may require suitable adjustments in the power electronics to continue producing grid-suitable power. 
     Use of a CVT to couple a crankshaft to a motor/generator is yet another way to achieve approximately constant power output in accordance with embodiments of the invention. Generally, there are two challenges to the maintenance of constant output power. First is the discrete piston stroke. As a quantity of gas is expanded in a cylinder during the course of a single stroke, its pressure decreases; to maintain constant power output from the cylinder as the force acting on its piston decreases, the piston&#39;s linear velocity is continually increased throughout the stroke. This increases the crankshaft angular velocity proportionately throughout the stroke. To maintain constant angular velocity and constant power at the input shaft of the motor/generator throughout the stroke, the effective gear ratio of the CVT is adjusted continuously to offset increasing crankshaft speed. 
     Second, pressure in the main gas store decreases as the store is exhausted. As this occurs, the piston velocity at all points along the stroke is typically increased to deliver constant power. Crankshaft angular velocity is therefore also typically increased at all times. 
     Under these illustrative conditions, the effective gear ratio of the CVT that produces substantially constant output power, plotted as a function of time, has the approximate form of a periodic sawtooth (corresponding to CVT adjustment during each discrete stroke) superimposed on a ramp (corresponding to CVT adjustment compensating for exhaustion of the gas store.) 
     With either a linear or rotary motor/generator, the range of forces (and thus of speeds) is generally minimized in order to achieve maximize efficiency. In lieu of more complicated linkages, for a given operating pressure range (e.g., from approximately 3,000 psig to approximately 30 psig), the range of forces (torques) seen at the motor/generator may be reduced through the addition of multiple cylinder stages arranged, e.g., in series. That is, as gas from the high-pressure reservoir is expanded in one chamber of an initial, high-pressure cylinder, gas from the other chamber is directed to the expansion chamber of a second, lower-pressure cylinder. Gas from the lower-pressure chamber of this second cylinder may either be vented to the environment or directed to the expansion chamber of a third cylinder operating at still lower pressure, and so on. An arrangement using two cylinder assemblies is shown and described; however, the principle may be extended to more than two cylinders to suit a particular application. 
     For example, a narrower force range over a given range of reservoir pressures is achieved by having a first, high-pressure cylinder operating between approximately 3,000 psig and approximately 300 psig and a second, larger-volume, low-pressure cylinder operating between approximately 300 psig and approximately 30 psig. The range of pressures (and thus of force) is reduced as the square root, from 100:1 to 10:1, compared to the range that would be realized in a single cylinder operating between approximately 3,000 psig and approximately 30 psig. The square-root relationship between the two-cylinder pressure range and the single-cylinder pressure range can be demonstrated as follows. 
     A given pressure range R 1  from high pressure P H  to low pressure P L , namely R 1 =P H /P L , is subdivided into two pressure ranges of equal magnitude R 2 . The first range is from P H  down to some intermediate pressure P I  and the second is from P I  down to P L . Thus, R 2 =P H /P I =P I /P L . From this identity of ratios, P I =(P H P L ) 1/2 . Substituting for P I  in R 2 =P H /P I , we obtain R 2 =P H /(P H P L ) 1/2 =(P H P L ) 1/2 =R 1   1/2 . It may be similarly shown that with appropriate cylinder sizing, the addition of a third cylinder/stage reduces the operating pressure range as the cube root, and so forth. In general (and as also set forth in the &#39;595 application), N appropriately sized cylinders reduce an original (i.e., single-cylinder) operating pressure range R 1  to R 1   1/N . Any group of N cylinders staged in this manner, where N≧2, is herein termed a cylinder group. 
     In various embodiments, the shafts of two or more double-acting cylinders are connected either to separate linear motor/generators or to a single linear motor/generator, either in line or in parallel. If they are connected in line, their common shaft may be arranged in line with the translator of a linear motor/generator. If they are connected in parallel, their separate shafts may be linked to a transmission (e.g., rigid beam) that is orthogonal to the shafts and to the translator of the motor/generator. Another portion of the beam may be attached to the translator of a linear generator that is aligned in parallel with the two cylinders. The synchronized reciprocal motion of the two double-acting cylinders may thus be transmitted to the linear generator. 
     In other embodiments of the invention, two or more cylinder groups, which may be identical, may be coupled to a common crankshaft. A crosshead arrangement may be used for coupling each of the N pneumatic cylinder shafts in each cylinder group to the common crankshaft. The crankshaft may be coupled to an electric motor/generator either directly or via a gear box. If the crankshaft is coupled directly to an electric motor/generator, the crankshaft and motor/generator may turn at very low speed (very low revolutions per minute, RPM), e.g., 25-30 RPM, as determined by the cycle speed of the cylinders. 
     Any multiple-cylinder implementation of this invention such as that described above may be co-implemented with any of the heat-transfer mechanisms described earlier. 
     All of the mechanisms described herein for converting potential energy in compressed gas to electrical energy, including the heat-exchange mechanisms and power electronics described, can, if appropriately designed, be operated in reverse to store electrical energy as potential energy in a compressed gas. Since this will be apparent to any person reasonably familiar with the principles of electrical machines, power electronics, pneumatics, and the principles of thermodynamics, the operation of these mechanisms to store energy rather than to recover it from storage will not be described. Such operation is, however, contemplated and within the scope of the invention and may be straightforwardly realized without undue experimentation. 
     In one aspect, embodiments of the invention feature an energy storage and generation system including or consisting essentially of a first pneumatic cylinder assembly, a motor/generator outside the first cylinder assembly, and a transmission mechanism coupled to the first cylinder assembly and the motor/generator. The first pneumatic cylinder assembly typically has first and second compartments separated by a piston, and the piston is typically coupled to the transmission mechanism. The transmission mechanism converts reciprocal motion of the piston into rotary motion of the motor/generator and/or converts rotary motion of the motor/generator into reciprocal motion of the piston. 
     Embodiments of the invention may include one or more of the following, in any of a variety of combinations. The system may include a shaft having a first end coupled to the piston and a second end coupled to the transmission mechanism. The second end of the shaft may be coupled to the transmission mechanism by a crosshead linkage. The piston may be slidably disposed within the cylinder. The system may include a container for compressed gas and an arrangement for selectively permitting fluid communication of the container for compressed gas with the first and/or second compartments of the pneumatic cylinder assembly. A second pneumatic cylinder assembly, which may include first and second compartments separated by a piston, may be coupled to the transmission mechanism and/or fluidly coupled to the first pneumatic cylinder assembly. The first and second pneumatic cylinder assemblies may be coupled in series. The first pneumatic cylinder assembly may be a high-pressure cylinder and the second pneumatic cylinder assembly may be a low-pressure cylinder. The second pneumatic cylinder assembly may be volumetrically larger (e.g., have a volume larger by at least 50%) than the first pneumatic cylinder assembly. The second pneumatic cylinder assembly may include a second shaft having a first end coupled to the piston and a second end coupled to the transmission mechanism. The second end of the second shaft may be coupled to the transmission mechanism by a crosshead linkage. 
     The transmission mechanism may include or consist essentially of, e.g., a crankshaft, a crankshaft and a gear box, or a crankshaft and a continuously variable transmission. The system may include a heat-transfer subsystem for expediting heat transfer in the first and/or second compartment of the first pneumatic cylinder assembly. The heat-transfer subsystem may include a fluid circulator for pumping a heat-transfer fluid into the first and/or second compartment of the first pneumatic cylinder assembly. One or more mechanisms for introducing the heat-transfer fluid (e.g., a spray head and/or a spray rod) may be disposed in the first and/or second compartment of the first pneumatic cylinder assembly. The transmission mechanism may vary torque for a given force exerted thereon, and/or the system may include power electronics for adjusting the load on the motor/generator. 
     In another aspect, embodiments of the invention feature an energy storage and generation system including or consisting essentially of a plurality of groups of pneumatic cylinder assemblies, a motor/generator outside the plurality of groups of pneumatic cylinder assemblies, and a transmission mechanism coupled to each of the cylinder assemblies and to the motor/generator. The transmission mechanism converts reciprocal motion into rotary motion of the motor/generator and/or converts rotary motion of the motor/generator into reciprocal motion. Each group of assemblies includes at least first and second pneumatic cylinder assemblies that are out of phase with respect to each other, and the first pneumatic cylinder assemblies of at least two of the groups are out of phase with respect to each other. Each pneumatic cylinder assembly may include a shaft having a first end coupled to a piston slidably disposed within the cylinder assembly and a second end coupled to the transmission mechanism (e.g., by a crosshead linkage). 
     Embodiments of the invention may include one or more of the following features in any of a variety of combinations. The transmission mechanism may include or consist essentially of a crankshaft, a crankshaft and a gear box, or a crankshaft and a continuously variable transmission. The system may include a heat-transfer subsystem for expediting heat transfer in the first and/or second compartment of each pneumatic cylinder assembly. The heat-transfer subsystem may include a fluid circulator for pumping a heat-transfer fluid into the first and/or second compartment of each pneumatic cylinder assembly. One or more mechanisms for introducing the heat-transfer fluid (e.g., a spray head and/or a spray rod) may be disposed in the first and/or second compartment of each pneumatic cylinder assembly. 
     In yet another aspect, embodiments of the invention feature a method for energy storage and recovery including expanding and/or compressing a gas via reciprocal motion, the reciprocal motion arising from or being converted into rotary motion, and exchanging heat with the gas during the expansion and/or compression in order to maintain the gas at a substantially constant temperature. The reciprocal motion may arise from or be converted into rotary motion of a motor/generator, thereby consuming or generating electricity. The reciprocal motion may arise from or be converted into rotary motion by a transmission mechanism, e.g., a crankshaft, a crankshaft and a gear box, or a crankshaft and a continuously variable transmission. 
     In a further aspect, embodiments of the invention feature an energy storage and generation system including or consisting essentially of a first pneumatic cylinder assembly coupled to a linear motor/generator. The first pneumatic cylinder assembly may include or consist essentially of first and second compartments separated by a piston. The piston may be slidably disposed within the cylinder assembly. The linear motor/generator directly converts reciprocal motion of the piston into electricity and/or directly converts electricity into reciprocal motion of the piston. The system may include a shaft having a first send coupled to the piston and a second end coupled to the mobile translator of the linear motor/generator. The shaft and the linear motor/generator may be aligned on a common axis. 
     Embodiments of the invention may include one or more of the following features in any of a variety of combinations. The system may include a second pneumatic cylinder assembly that includes or consists essentially of first and second compartments and a piston. The piston may be slidably disposed within the cylinder assembly. The piston may separate the compartments and/or may be coupled to the linear generator. The second pneumatic cylinder assembly may be connected in series pneumatically and in parallel mechanically with the first pneumatic cylinder assembly. The second pneumatic cylinder assembly may be connected in series pneumatically and in series mechanically with the first pneumatic cylinder assembly. 
     The system may include a heat-transfer subsystem for expediting heat transfer in the first and/or second compartment of the first pneumatic cylinder assembly. The heat-transfer subsystem may include a fluid circulator for pumping a heat-transfer fluid into the first and/or second compartment of the first pneumatic cylinder assembly. One or more mechanisms for introducing the heat-transfer fluid (e.g., a spray head and/or a spray rod) may be disposed in the first and/or second compartment of the first pneumatic cylinder assembly. The system may include a mechanism for increasing the speed of the piston as the pressure in the first and/or second compartment decreases. The mechanism may include or consist essentially of power electronics for adjusting the load on the linear motor/generator. The linear motor/generator may have variable-pitch windings. The linear motor/generator may be a switched-reluctance linear motor/generator. 
     These and other objects, along with advantages and features of the invention, will become more apparent through reference to the following description, the accompanying drawings, and the claims. Furthermore, it is to be understood that the features of the various embodiments described herein are not mutually exclusive and can exist in various combinations and permutations. Herein, the terms “liquid and “water interchangeably connote any mostly or substantially incompressible liquid, the terms “gas and “air are used interchangeably, and the term “fluid may refer to a liquid or a gas unless otherwise indicated. As used herein, the term “substantially means ±10%, and, in some embodiments, ±5%. A “valve is any mechanism or component for controlling fluid communication between fluid paths or reservoirs, or for selectively permitting control or venting. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       In the drawings, like reference characters generally refer to the same parts throughout the different views. Also, the drawings are not necessarily to scale, emphasis instead generally being placed upon illustrating the principles of the invention. In the following description, various embodiments of the present invention are described with reference to the following drawings, in which: 
         FIG. 1  is a schematic cross-sectional diagram showing the use of pressurized stored gas to operate a double-acting pneumatic cylinder and a linear motor/generator to produce electricity or stored pressurized gas according to various embodiments of the invention; 
         FIG. 2  depicts the mechanism of  FIG. 1  in a different phase of operation (i.e., with the high- and low-pressure sides of the piston reversed and the direction of shaft motion reversed); 
         FIG. 3  depicts the arrangement of  FIG. 1  modified to introduce liquid sprays into the two compartments of the cylinder, in accordance with various embodiments of the invention; 
         FIG. 4  depicts the mechanism of  FIG. 3  in a different phase of operation (i.e., with the high- and low-pressure sides of the piston reversed and the direction of shaft motion reversed); 
         FIG. 5  depicts the mechanism of  FIG. 1  modified by the addition of an external heat exchanger in communication with both compartments of the cylinder, where the contents of either compartment may be circulated through the heat exchanger to transfer heat to or from the gas as it expands or compresses, enabling substantially isothermal expansion or compression of the gas, in accordance with various embodiments of the invention; 
         FIG. 6  depicts the mechanism of  FIG. 1  modified by the addition of a second pneumatic cylinder operating at a lower pressure than the first, in accordance with various embodiments of the invention; 
         FIG. 7  depicts the mechanism of  FIG. 6  in a different phase of operation (i.e., with the high- and low-pressure sides of the pistons reversed and the direction of shaft motion reversed); 
         FIG. 8  depicts the mechanism of  FIG. 1  modified by the addition a second pneumatic cylinder operating at lower pressure, in accordance with various embodiments of the invention; 
         FIG. 9  depicts the mechanism of  FIG. 8  in a different phase of operation (i.e., with the high- and low-pressure sides of the pistons reversed and the direction of shaft motion reversed); 
         FIG. 10  is a schematic diagram of a system and related method for substantially isothermal compression and expansion of a gas for energy storage using one or more pneumatic cylinders in accordance with various embodiments of the invention; 
         FIG. 11  is a schematic diagram of the system of  FIG. 10  in a different phase of operation; 
         FIG. 12  is a schematic diagram of a system and related method for coupling a cylinder shaft to a crankshaft; and 
         FIGS. 13A and 13B  are schematic diagrams of systems in accordance with various embodiments of the invention, in which multiple cylinder groups are coupled to a single crankshaft. 
     
    
    
     DETAILED DESCRIPTION 
       FIG. 1  illustrates the use of pressurized stored gas to operate a double-acting pneumatic cylinder and linear motor/generator to produce electricity according to a first illustrative embodiment of the invention. If the linear motor/generator is operated as a motor rather than as a generator, the identical mechanism employs electricity to produce pressurized stored gas.  FIG. 1  shows the mechanism being operated to produce electricity from stored pressurized gas. 
     The illustrated energy storage and recovery system  100  includes a pneumatic cylinder  105  divided into two compartments  110  and  115  by a piston (or other mechanism)  120 . The cylinder  105 , which is shown in a vertical orientation in  FIG. 1  but may be arbitrarily oriented, has one or more gas circulation ports  125  (only one is explicitly labeled), which are connected via piping  130  to a compressed-gas reservoir  135  and a vent  140 . Note that as used herein the terms “pipe, “piping and the like refer to one or more conduits capable of carrying gas or liquid between two points. Thus, the singular term should be understood to extend to a plurality of parallel conduits where appropriate. 
     The piping  130  connecting the compressed-gas reservoir  135  to compartments  110 ,  115  of the cylinder  105  passes through valves  145 ,  150 . Compartments  110 ,  115  of the cylinder  105  are connected to vent  140  through valves  155 ,  160 . A shaft  165  coupled to the piston  120  is coupled to one end of a translator  170  of a linear electric motor/generator  175 . 
     System  100  is shown in two operating states, namely (a) valves  145  and  160  open and valves  150  and  155  closed (shown in  FIG. 1 ), and (b) valves  145  and  160  closed and valves  150  and  155  open (shown in  FIG. 2 ). In state (a), high-pressure gas flows from the high-pressure reservoir  135  through valve  145  into compartment  115  (where it is represented by a gray tone in  FIG. 1 ). Lower-pressure gas is vented from the other compartment  110  via valve  160  and vent  140 . The result of the net force exerted on the piston  120  by the pressure difference between the two compartments  110 ,  115  is the linear movement of piston  120 , piston shaft  165 , and translator  170  in the direction indicated by the arrow  180 , causing an EMF to be induced in the stator of the linear motor/generator  175 . Power electronics are typically connected to the motor/generator  175 , and may be software-controlled. Such power electronics are conventional and not shown in  FIG. 1  or in subsequent figures. 
       FIG. 2  shows system  100  in a second operating state, the above-described state (b) in which valves  150  and  155  are open and valves  145  and  160  are closed. In this state, gas flows from the high-pressure reservoir  135  through valve  150  into compartment  110 . Lower-pressure gas is vented from the other compartment  115  via valve  155  and vent  140 . The result is the linear movement of piston  120 , piston shaft  165 , and translator  170  in the direction indicated by the arrow  200 , causing an EMF to be induced in the stator of the linear motor/generator  175 . 
       FIG. 3  illustrates the addition of expedited heat transfer by a liquid spray as described in, e.g., the &#39;703 application. In this illustrative embodiment, a spray of droplets of liquid (indicated by arrows  300 ) is introduced into either compartment (or both compartments) of the cylinder  105  through perforated spray heads  310 ,  320 ,  330 , and  340 . The arrangement of spray heads shown is illustrative only; any suitable number and disposition of spray heads inside the cylinder  105  may be employed. Liquid may be conveyed to spray heads  310  and  320  on the piston  120  by a center-drilled channel  350  in the piston shaft  165 , and may be conveyed to spray heads  330  and  340  by appropriate piping (not shown). Liquid flow to the spray heads is typically controlled by an appropriate valve system (not shown). 
       FIG. 3  depicts system  100  in the first of the two above-described operating states, where valves  145  and  160  are open and valves  150  and  155  are closed. In this state, gas flows from the high-pressure reservoir  135  through valve  145  into compartment  115 . Liquid at a temperature higher than that of the expanding gas is sprayed into compartment  115  from spray heads  330 ,  340 , and heat flows from the droplets to the gas. With suitable liquid temperature and flow rate, this arrangement enables substantially isothermal expansion of the gas in compartment  115 . 
     Lower-pressure gas is vented from the other compartment  110  via valve  160  and vent  140 , resulting in the linear movement of piston  120 , piston shaft  165 , and translator  170  in the downward direction (arrow  180 ). Since the expansion of the gas in compartment  115  is substantially isothermal, more mechanical work is performed on the piston  120  by the expanding gas and more electric energy is produced by the linear motor/generator  175  than would be produced by adiabatic expansion in system  100  of a like quantity of gas. 
       FIG. 4  shows the illustrative embodiment of  FIG. 3  in a second operating state, where valves  150  and  155  are open and valves  145  and  160  are closed. In this state, gas flows from the high-pressure reservoir  135  through valve  150  into compartment  110 . Liquid at a temperature higher than that of the expanding gas is sprayed (indicated by arrows  400 ) into compartment  110  from spray heads  310  and  320 , and heat flows from the droplets to the gas. With suitable liquid temperature and flow rate, this arrangement enables the substantially isothermal expansion of the gas in compartment  110 . Lower-pressure gas is vented from the other compartment  110  via valve  155  and vent  140 . The result is the linear movement of piston  120 , piston shaft  165 , and translator  170  in the upward direction (arrow  200 ), generating electricity. 
     System  100  may be operated in reverse, in which case the linear motor/generator  175  operates as an electric motor. The droplet spray mechanism is used to cool gas undergoing compression (achieving substantially isothermal compression) for delivery to the storage reservoir rather than to warm gas undergoing expansion from the reservoir. System  100  may thus operate as a full-cycle energy storage system with high efficiency. 
     Additionally, the spray-head-based heat transfer illustrated in  FIGS. 3 and 4  for vertically oriented cylinders may be replaced or augmented with a spray-rod heat transfer scheme for arbitrarily oriented cylinders as described in the &#39;703 application. 
       FIG. 5  is a schematic of system  100  with the addition of expedited heat transfer by a heat-exchange subsystem that includes an external heat exchanger  500  connected by piping through valves  510 ,  520  to chamber  115  of the cylinder  105  and by piping through valves  530 ,  540  to chamber  110  of the cylinder  105 . A circulator  550 , which is preferably capable of pumping gas at high pressure (e.g., approximately 3,000 psi), drives gas through one side of the heat exchanger  500 , either continuously or in installments. An external system, not shown, drives a fluid  560  (e.g., air, water, or another fluid) from an independent source through the other side of the heat exchanger. 
     The heat-exchange subsystem, which may include heat exchanger  500 , circulator  550 , and associated piping, valves, and ports, transfers gas from either chamber  110 ,  115  (or both chambers) of the cylinder  105  through the heat exchanger  500 . The subsystem has two operating states, either (a) valves  145 ,  160 ,  510 , and  520  closed and valves  150 ,  155 ,  530 , and  540  open, or (b) valves  145 ,  160 ,  510 ,  520  open and valves  150 ,  155 ,  530 , and  540  closed.  FIG. 5  depicts state (a), in which high-pressure gas is conveyed from the reservoir  135  to chamber  110  of the cylinder  105 ; meanwhile, low-pressure gas is exhausted from chamber  115  via valve  155  to the vent  140 . High-pressure gas is also circulated from chamber  110  through valve  530 , circulator  550 , heat exchanger  500 , and valve  540  (in that order) back to chamber  110 . Simultaneously, fluid  560  warmer than the gas flowing through the heat exchanger is circulated through the other side of the heat exchanger  500 . With suitable temperature and flow rate of fluid  560  through the external side of the heat exchanger  500  and suitable flow rate of high-pressure gas through the cylinder side of the heat exchanger  500 , this arrangement enables the substantially isothermal expansion of the gas in compartment  110 . 
     In  FIG. 5 , the piston shaft  165  and linear motor/generator translator  170  are moving in the direction shown by the arrow  570 . It should be clear that, like the illustrative embodiment shown in  FIG. 1 , the embodiment shown in  FIG. 5  has a second operating state (not shown), defined by the second of the two above-described valve arrangements (“state (b) above), in which the direction of piston/translator motion is reversed. Moreover, this identical mechanism may clearly be operated in reverse—in that mode (not shown), the linear motor/generator  175  operates as an electric motor and the heat exchanger  500  cools gas undergoing compression (achieving substantially isothermal compression) for delivery to the storage reservoir  135  rather than warming gas undergoing expansion. Thus, system  100  may operate as a full-cycle energy storage system with high efficiency. 
       FIG. 6  depicts a system  600  that includes a second pneumatic cylinder  600  operating at a pressure lower than that of the first cylinder  105 . Both cylinders  105 ,  600  are, in this embodiment, double-acting. They are connected in series (pneumatically) and in line (mechanically). Pressurized gas from the reservoir  135  drives the piston  120  of the double-acting high-pressure cylinder  105 . Series attachment of the two cylinders directs gas from the lower-pressure compartment of the high-pressure cylinder  105  to the higher-pressure compartment of the low-pressure cylinder  600 . In the operating state depicted in  FIG. 6 , gas from the lower-pressure side  610  of the low-pressure cylinder  600  exits through vent  140 . Through their common piston shaft  620 ,  165 , the two cylinders act jointly to move the translator  170  of the linear motor/generator  175 . This arrangement reduces the range of pressures over which the cylinders jointly operate, as described above. 
     System  600  is shown in two operating states, (a) valves  150 ,  630 , and  640  closed and valves  145 ,  650 , and  660  open (depicted in  FIG. 6 ), and (b) valves  150 ,  630 , and  640  open and valves  145 ,  650 , and  660  closed (depicted in  FIG. 7 ).  FIG. 6  depicts state (a), in which gas flows from the high-pressure reservoir  135  through valve  145  into compartment  115  of the high-pressure cylinder  105 . Intermediate-pressure gas (indicated by the stippled areas in the figure) is directed from compartment  110  of the high-pressure cylinder  105  by piping through valve  650  to compartment  670  of the low-pressure cylinder  600 . The force of this intermediate-pressure gas on the piston  680  acts in the same direction (i.e., in the direction indicated by the arrow  690 ) as that of the high-pressure gas in compartment  115  of the high-pressure cylinder  105 . The cylinders thus act jointly to move their common piston shaft  620 ,  165  and the translator  170  of the linear motor/generator  175  in the direction indicated by arrow  690 , generating electricity during the stroke. Low-pressure gas is vented from the low-pressure cylinder  600  through the vent  140  via valve  660 . 
       FIG. 7  shows the second operating state (b) of system  600 . Valves  150 ,  630 , and  640  are open and valves  145 ,  650 , and  660  are closed. In this state, gas flows from the high-pressure reservoir  135  through valve  150  into compartment  110  of the high-pressure cylinder  105 . Intermediate-pressure gas is directed from the other compartment  115  of the high-pressure cylinder  105  by piping through valve  630  to compartment  610  of the low-pressure cylinder  600 . The force of this intermediate-pressure gas on the piston  680  acts in the same direction (i.e., in direction indicated by the arrow  700 ) as that of the high-pressure gas in compartment  110  of the high-pressure cylinder  105 . The cylinders thus act jointly to move the common piston shaft  620 ,  165  and the translator  170  of the linear motor/generator  175  in the direction indicated by arrow  700 , generating electricity during the stroke, which is in the direction opposite to that shown in  FIG. 6 . Low-pressure gas is vented from the low-pressure cylinder  600  through the vent  140  via valve  640 . 
     The spray arrangement for heat exchange shown in  FIGS. 3 and 4  or, alternatively (or in addition to), the external heat-exchanger arrangement shown in  FIG. 5  (or another heat-exchange mechanism) may be straightforwardly adapted to the system  600  of  FIGS. 6 and 7 , enabling substantially isothermal expansion of the gas in the high-pressure reservoir  135 . Moreover, system  600  may be operated as a compressor (not shown) rather than as a generator. Finally, the principle of adding cylinders operating at progressively lower pressures in series (pneumatic) and in line (mechanically) may involve three or more cylinders rather than merely two cylinders as shown in the illustrative embodiment of  FIGS. 6 and 7 . 
       FIG. 8  depicts an energy storage and recovery system  800  with a second pneumatic cylinder  805  operating at a lower pressure than the first cylinder  105 . Both cylinders  105 ,  805  are double-acting. They are attached in series (pneumatically) and in parallel (mechanically). Pressurized gas from the reservoir  135  drives the piston  120  of the double-acting high-pressure cylinder  105 . Series pneumatic attachment of the two cylinders is as detailed above with reference to  FIGS. 6 and 7 . Gas from the lower-pressure side of the low-pressure cylinder  805  is directed to vent  140 . Through a common beam  810  coupled to the piston shafts  165 ,  815  of the cylinders, the cylinders act jointly to move the translator  170  of the linear motor/generator  175 . This arrangement reduces the operating range of cylinder pressures as compared to a similar arrangement employing only one cylinder. 
     System  800  is shown in two operating states, (a) valves  150 ,  820 , and  825  closed and valves  145 ,  830 , and  835  open (shown in  FIG. 8 ), and (b) valves  150 ,  820 , and  825  open and valves  145 ,  830  and  835  closed (shown in  FIG. 9 ).  FIG. 8  depicts state (a), in which gas flows from the high-pressure reservoir  135  through valve  145  into compartment  115  of the high-pressure cylinder  105 . Intermediate-pressure gas (depicted by stippled areas) is directed from the other compartment  110  of the high-pressure cylinder  105  by piping through valve  830  to compartment  840  of the low-pressure cylinder  805 . The force of this intermediate-pressure gas on the piston  845  acts in the same direction (i.e., in direction indicated by the arrow  850 ) as the high-pressure gas in compartment  115  of the high-pressure cylinder  105 . The cylinders thus act jointly to move the common beam  810  and the translator  170  of the linear motor/generator  175  in the direction indicated by arrow  850 , generating electricity during the stroke. Low-pressure gas is vented from the low-pressure cylinder  805  through the vent  140  via valve  835 . 
       FIG. 9  shows the second operating state (b) of system  800 , i.e., valves  150 ,  820 , and  825  are open and valves  145 ,  830  and  835  are closed. In this state, gas flows from the high-pressure reservoir  135  through valve  150  into compartment  110  of the high-pressure cylinder  105 . Intermediate-pressure gas is directed from compartment  115  of the high-pressure cylinder  105  by piping through valve  820  to compartment  855  of the low-pressure cylinder  805 . The force of this intermediate-pressure gas on the piston  845  acts in the same direction (i.e., in direction indicated by the arrow  900 ) as that exerted on piston  120  by the high-pressure gas in compartment  110  of the high-pressure cylinder  105 . The cylinders thus act jointly to move the common beam  810  and the translator  170  of the linear motor/generator  175  in the direction indicated, generating electricity during the stroke, which is in the direction opposite to that of the operating state shown in  FIG. 8 . Low-pressure gas is vented from the low-pressure cylinder  805  through the vent  140  via valve  825 . 
     The spray arrangement for heat exchange shown in  FIGS. 3 and 4  or, alternatively or in combination, the external heat-exchanger arrangement shown in  FIG. 5  may be straightforwardly adapted to the pneumatic cylinders of system  800 , enabling substantially isothermal expansion of the gas in the high-pressure reservoir  135 . Moreover, this exemplary embodiment may be operated as a compressor (not shown) rather than a generator (shown). Finally, the principle of adding cylinders operating at progressively lower pressures in series (pneumatic) and in parallel (mechanically) may be extended to three or more cylinders. 
       FIG. 10  is a schematic diagram of a system  1000  for achieving substantially isothermal compression and expansion of a gas for energy storage and recovery using a pair of pneumatic cylinders (shown in partial cross-section) with integrated heat exchange. In this illustrative embodiment, the reciprocal motion of the cylinders is converted to rotary motion via mechanical means. Depicted are a pair of double-acting pneumatic cylinders with appropriate valving and mechanical linkages; however, any number of single- or double-acting pneumatic cylinders, or any number of groups of single- or double-acting pneumatic cylinders, where each group contains two or more cylinders, may be employed in such a system. Likewise, a wrist-pin connecting-rod type crankshaft arrangement is depicted in  FIG. 10 , but other mechanical means for converting reciprocal motion to rotary motion are contemplated and considered within the scope of the invention. 
     In various embodiments, the system  1000  includes a first pneumatic cylinder  1002  divided into two compartments  1004 ,  1006  by a piston  1008 . The cylinder  1002 , which is shown in a vertical orientation in this illustrative embodiment, has one or more ports  1010  (only one is explicitly labeled) that are connected via piping  1012  to a compressed-gas reservoir  1014 . 
     The system  1000  as shown in  FIG. 10  includes a second pneumatic cylinder  1016  operating at a lower pressure than the first cylinder  1002 . The second pneumatic cylinder  1016  is divided into two compartments  1018 ,  1020  by a piston  1022  and includes one or more ports  1010  (only one is explicitly labeled). Both cylinders  1002 ,  1016  are double-acting in this illustrative embodiment. They are attached in series (pneumatically); thus, after expansion in one compartment of the high-pressure cylinder  1002 , the mid-pressure gas (depicted by stippled areas) is directed for further expansion to a compartment of the low-pressure cylinder  1016 . 
     In the state of operation depicted in  FIG. 10 , pressurized gas (e.g., approximately 3,000 psig) from the reservoir  1014  passes through a valve  1024  and drives the piston  1008  of the double-acting high-pressure cylinder  1002  in the downward direction as shown by the arrow  1026   a . Gas that has already expanded to a mid-pressure (e.g., approximately 250 psig) in the lower chamber  1004  of the high-pressure cylinder  1002  is directed through a valve  1028  to the lower chamber  1018  of the larger volume low-pressure cylinder  1016 , where it is further expanded. This gas exerts an upward force on the piston  1022  with resulting upward motion of the piston  1022  and shaft  1040  as indicated by the arrow  1026   b . Gas within the upper chamber  1020  of cylinder  1016  has already been expanded to atmospheric pressure and is vented to the atmosphere through valve  1030  and vent  1032 . The function of this two-cylinder arrangement is to reduce the range of pressures and forces over which each cylinder operates, as described earlier. 
     The piston shaft  1034  of the high-pressure cylinder  1002  is connected by a hinged connecting rod  1036  or other suitable linkage to a crankshaft  1038 . The piston shaft  1040  of the low-pressure cylinder  1016  is connected by a hinged connecting rod  1042  or other suitable linkage to the same crankshaft  1038 . The motion of the piston shafts  1034 ,  1040  is shown as rectilinear, whereas the linkages  1036 ,  1042  have partial rotational freedom orthogonal to the axis of the crankshaft  1038 . 
     In the state of operation shown in  FIG. 10 , the piston shaft  1034  and linkage  1036  are drawing the crank  1044  in a downward direction (as indicated by arrow  1026   a ) while the piston shaft  1040  and linkage  1042  are pushing the crank  1046  in an upward direction (as indicated by arrow  1026   b ). The two cylinders  1002 ,  1016  thus act jointly to rotate the crankshaft  1038 . In  FIG. 10 , the crankshaft  1038  is shown driving an optional transmission mechanism  1048  whose output shaft  1050  rotates at a higher rate than the crankshaft  1038 . Transmission mechanism  1048  may be, e.g., a gear box or a CVT (as shown in  FIG. 10 ). The output shaft  1050  of transmission mechanism  1048  drives an electric motor/generator  1055  that generates electricity. In some embodiments, crankshaft  1038  is directly connected to and drives motor/generator  1055 . 
     Power electronics may be connected to the motor/generator  1055  (and may be software-controlled), thus providing control over air expansion and/or compression rates. These power electronics are not shown, but are well-known to a person of ordinary skill in the art. 
     In the embodiment of the invention depicted in  FIG. 10 , liquid sprays may be introduced into any of the compartments of the cylinders  1002 ,  1016 . In both cylinders  1002 ,  1016 , the liquid spray enables expedited heat transfer to the gas being expanded (or compressed) in the cylinder (as detailed above). Sprays  1070 ,  1075  of droplets of liquid may be introduced into the compartments of the high-pressure cylinder  1002  through perforated spray heads  1060 ,  1065 . The liquid spray in chamber  1006  of cylinder  1002  is indicated by dashed lines  1070 , and the liquid spray in chamber  1004  of cylinder  1002  is indicated by dashed lines  1075 . Water (or other appropriate heat-transfer fluid) is conveyed to the spray heads  1060  by appropriate piping (not shown). Fluid may be conveyed to spray head  1065  on the piston  1008  by various methods; in one embodiment, the fluid is conveyed through a center-drilled channel (not shown) in the piston rod  1034 , as described in U.S. patent application Ser. No. 12/690,513 (the &#39;513 application), the disclosure of which is hereby incorporated by reference herein in its entirety. Liquid flow to both sets of spray heads is typically controlled by an appropriate valve arrangement (not shown). Liquid may be removed from the cylinders through suitable ports (not shown). 
     The heat-transfer liquid sprays  1070 ,  1075  warm the high-pressure gas as it expands, enabling substantially isothermal expansion of the gas. If gas is being compressed, the sprays cool the gas, enabling substantially isothermal compression. A liquid spray may be introduced by similar means into the compartments of the low-pressure cylinder  1016  through perforated spray heads  1080 ,  1085 . Liquid spray in chamber  1018  of cylinder  1016  is indicated by dashed lines  1090 . 
     In the operating state shown in  FIG. 10 , liquid spray transfers heat to (or from) the gas undergoing expansion (or compression) in chambers  1004 ,  1006 , and  1018 , enabling a substantially isothermal process. Spray may be introduced in chamber  1020 , but this is not shown as little or no expansion is occurring in that compartment during venting. The arrangement of spray heads shown in  FIG. 10  is illustrative only, as any number and disposition of spray heads and/or spray rods inside the cylinders  1002 ,  1016  are contemplated as embodiments of the present invention. 
       FIG. 11  depicts system  1000  in a second operating state, in which the piston shafts  1034 ,  1040  of the two pneumatic cylinders  1002 ,  1016  have directions of motion opposite to those shown in  FIG. 10 , and the crankshaft  1038  continues to rotate in the same sense as in  FIG. 10 . In  FIG. 11 , valves  1024 ,  1028 , and  1030  are closed and valves  1100 ,  1105 , and  1110  are open. Gas flows from the high-pressure reservoir  1014  through valve  1100  into compartment  1004  of the high-pressure cylinder  1002 , where it applies an upward force on piston  1008 . Mid-pressure gas in chamber  1006  of the high-pressure cylinder  1002  is directed through valve  1105  to the upper chamber  1020  of the low-pressure cylinder  1016 , where it is further expanded. The expanding gas exerts a downward force on the piston  1022  with resulting motion of the piston  1022  and shaft  1040  as indicated by the arrow  1026   b . Gas within the lower chamber  1018  of cylinder  1016  is already expanded to approximately atmospheric pressure and is being vented to the atmosphere through valve  1110  and vent  1032 . In  FIG. 11 , gas expanding in chambers  1004 ,  1006  and  1020  exchanges heat with liquid sprays  1115 ,  1125 , and  1120  (depicted as dashed lines) to keep the gas at approximately constant temperature. 
     The spray-head heat-transfer arrangement shown in  FIGS. 10 and 11  for vertically oriented cylinders may be replaced or augmented with a spray-rod heat-transfer scheme for arbitrarily oriented cylinders (as mentioned above). Additionally, the systems shown may be implemented with an external gas heat exchanger instead of (or in addition to) liquid sprays, as described in the &#39;235 application. An external gas heat exchanger also enables expedited heat transfer to or from the gas being expanded (or compressed) in the cylinders. With an external heat exchanger, the cylinders may be arbitrarily oriented. 
     In all operating states, the two cylinders  1002 ,  1016  in  FIGS. 10 and 11  are preferably 180° out of phase. For example, whenever the piston  1008  of the high-pressure cylinder  1002  has reached its uppermost point of motion, the piston  1022  of the low-pressure cylinder  1016  has reached its nethermost point of motion. Similarly, whenever the piston  1022  of the low-pressure cylinder  1016  has reached its uppermost point of motion, the piston  1008  of the high-pressure cylinder  1002  has reached its nethermost point of motion. Further, when the two pistons  1008 ,  1022  are at the midpoints of their respective strokes, they are moving in opposite directions. This constant phase relationship is maintained by the attachment of the piston rods  1034 ,  1040  to the two cranks  1044 ,  1046 , which are affixed to the crankshaft  1038  so that they lie in a single plane on opposite sides of the crankshaft  1038  (i.e., they are physically 180° apart). At the moment depicted in  FIG. 10 , the plane in which the two cranks  1044 ,  1046  lie is coincident with the plane of the figure. 
     Reference is now made to  FIG. 12 , which is a schematic depiction of a single pneumatic cylinder assembly  1200  and a mechanical linkage that may be used to connect the rod or shaft  1210  of the cylinder assembly to a crankshaft  1220 . Two orthogonal views of the linkage and piston are shown in partial cross section in  FIG. 12 . In this illustrative embodiment, the linkage includes a crosshead  1230  mounted on the end of the rod  1210 . The crosshead  1230  is slidably disposed within a distance piece  1240  that constrains the lateral motion of the crosshead  1230 . The distance piece  1240  may also fix the distance between the top of the cylinder  1200  and a housing (not depicted) of the crankshaft  1220 . 
     A connecting pin  1250  is mounted on the crosshead  1230  and is free to rotate around its own long axis. A connecting rod  1260  is attached to the connecting pin  1250 . The other end of the connecting rod  1260  is attached to a collar-and-pin linkage  1270  mounted on a crank  1280  affixed to the crankshaft  1220 . A collar-and-pin linkage  1270  is illustrated in  FIG. 12 , but other mechanisms for attaching the connecting rod  1260  to the crank  1280  are contemplated within embodiments of the invention. Moreover, either or both ends of the crankshaft  1220  may be extended to attach to further cranks (not shown) interacting with other cylinders or may be linked to a gear box (or other transmission mechanism such as a CVT), motor/generator, flywheel, brake, or other device(s). 
     The linkage between cylinder rod  1210  and crankshaft  1220  depicted in  FIG. 12  is herein termed a “crosshead linkage, which transforms substantially rectilinear mechanical force acting along the cylinder rod  1210  into torque or rotational force acting on the crankshaft  1220 . Forces transmitted by the connecting rod  1260  and not acting along the axis of the cylinder rod  1210  (e.g., lateral forces) act on the connecting pin  1250 , crosshead  1230 , and distance piece  1240 , but not on the cylinder rod  1210 . Thus, advantageously, any gaskets or seals (not depicted) through which the cylinder rod  1210  slides while passing into cylinder  1200  are subject to reduced stress, enabling the use of less durable gaskets or seals, increasing the lifespan of the employed gaskets or seals, or both. 
       FIGS. 13A and 13B  are schematics of a system  1300  for substantially isothermal compression and expansion of a gas for energy storage and recovery using multiple pairs  1310  of pneumatic cylinders with integrated heat exchange. Storage of compressed air, venting of low-pressure air, and other components of the system  1300  are not depicted in  FIGS. 13A and 13B , but are consistent with the descriptions of similar systems herein. Each rectangle in  FIGS. 13A and 13B  labeled PAIR  1 , PAIR  2 , etc. represents a pair of pneumatic cylinders (with appropriate valving and linkages, not explicitly depicted) similar to the pair of cylinders depicted in  FIG. 10 . Each cylinder pair  1310  is a pair of fluidly linked pneumatic cylinders communicating with a common crankshaft  1320  by a mechanism that may resemble those shown in  FIG. 10  or  FIG. 12  (or may have some other form). The crankshaft  1320  may communicate (with or without an intervening transmission mechanism) with an electric motor/generator  1330  that may thus generate electricity. 
     In various embodiments, within each of the cylinder pairs  1310  shown in  FIGS. 13A and 13B , the high-pressure cylinder (not explicitly depicted) and the low-pressure cylinder (not explicitly depicted) are 180° out of phase with each other, as depicted and described for the two cylinders  1002 ,  1016  in  FIG. 10 . For simplicity, the phase of each cylinder pair  1310  is identified herein with the phase of its high-pressure cylinder. In the embodiment depicted in  FIG. 13A , which includes six cylinder pairs  1310 , the phase of PAIR  1  is arbitrarily denoted 0°. The phase of PAIR  2  is 120°, the phase of PAIR  3  is 240°, the phase of PAIR  4  is 360° (equivalent to 0°), the phase of PAIR  5  is 120°, and the phase of PAIR  6  is 240°. There are thus three sets of cylinder pairs that are in phase, namely PAIR  1  and PAIR  4 )(0°), PAIR  2  and PAIR  5  (120°), and PAIR  3  and PAIR  6 ) (240°). These phase relationships are set and maintained by the affixation to the crankshaft  1320  at appropriate angles of the cranks (not explicitly depicted) linked to each of the cylinders in the system  1300 . 
     In the embodiment depicted in  FIG. 13B , which includes four cylinder pairs  1310 , the phase of PAIR  1  is also denoted 0°. The phase of PAIR  2  is then 270°, the phase of PAIR  3  is 90°, and the phase of PAIR  4  is 180°. As in  FIG. 13A , these phase relationships are set and maintained by the affixation to the crankshaft  1320  at appropriate angles of the cranks linked to each of the cylinders in the system  1300 . 
     Linking an even number of cylinder pairs  1310  to a single crankshaft  1320  advantageously balances the forces acting on the crankshaft: unbalanced forces generally tend to either require more durable parts or shorten component lifetimes. An advantage of specifying the phase differences between the cylinder pairs  1310  as shown in  FIGS. 13A and 13B  is minimization of fluctuations in total force applied to the crankshaft  1320 . Each cylinder pair  1310  applies a force varying between zero and some maximum value (e.g., approximately 330,000 lb) during the course of a single stroke. The sum of all the torques applied by the multiple cylinder pairs  1310  to the crankshaft  1320  as arranged in  FIGS. 13A and 13B  varies by less than the torque applied by a single cylinder pair  1310 , both absolutely and as a fraction of maximum torque, and is typically never zero. 
     Generally, the systems described herein may be operated in both an expansion mode and in the reverse compression mode as part of a full-cycle energy storage system with high efficiency. For example, the systems may be operated as both compressor and expander, storing electricity in the form of the potential energy of compressed gas and producing electricity from the potential energy of compressed gas. Alternatively, the systems may be operated independently as compressors or expanders. 
     In addition, the systems described above, and/or other embodiments employing liquid-spray heat exchange or external gas heat exchange (as detailed above), may draw or deliver thermal energy via their heat-exchange mechanisms to external systems (not shown) for purposes of cogeneration, as described in the &#39;513 application. 
     The terms and expressions employed herein are used as terms of description and not of limitation, and there is no intention, in the use of such terms and expressions, of excluding any equivalents of the features shown and described or portions thereof, but it is recognized that various modifications are possible within the scope of the invention claimed.