Patent Publication Number: US-2013227948-A1

Title: Method and system for converting thermal power, delivered from a variable temperature heat source, into mechanical power

Description:
FIELD OF THE INVENTION 
     The present invention relates to a method and a system for converting thermal power delivered from a variable temperature heat source into mechanical power, according to the preambles of claims  1  and  16  respectively. 
     The present invention relates particularly, but without limitation, to a method and a system for converting thermal power delivered from a heat source with temperature varying in a range between a higher temperature and a lower temperature, such as flue gases of a biogas-fuelled endothermic engine, into mechanical power, using a closed thermodynamic cycle operating between the higher temperature of the heat source and a temperature substantially equal to ambient temperature. 
     It should be first noted that the term ambient temperature is intended herein as the temperature of the site in which the closed cycle is installed and operated. 
     BACKGROUND ART 
     Methods and systems are known in the art for converting the thermal power delivered from a heat source into mechanical power using a closed thermodynamic cycle. 
     For example, a widely used technology in large plants (power from a few tens to hundreds MW) consists in using a production cycle comprising two cascade power subcycles, one of which subcycles is an open gas cycle known as Brayton and the other is a closed steam cycle known as Rankine; the two subcycles form together a cycle that can recover the thermal power delivered from the first subcycle to generate further mechanical power. 
     If the plant is of medium size (power from a few hundreds to thousands of kW) Organic Rankine Cycles (ORC) are more cost effective and efficient. 
     If the plant is of small size (e.g. power lower than 200 kW), combined cycles that use steam or organic Rankine subcycles are not economically advantageous, due to their low efficiency and high costs. 
     One example of the above cycles is described in the patent U.S. Pat. No. 6.523.347 B1 (hereinafter referred to for simplicity as U.S. &#39;347), in which a mechanical power production cycle is made up of two cascade power subcycles. Particularly, the topping subcycle is a recuperative Brayton cycle in either open (see diagram of  FIG. 1  of the U.S. &#39;347 patent) or closed (see diagram of  FIG. 2  of the U.S. &#39;347 patent) configuration, whereas the bottoming subcycle is a cycle composed of three processes (see diagram of  FIG. 5  of the U.S. &#39;347 patent). The bottoming subcycle consists of a compression step, which is defined as polytropic, in which a working gas fluid, such as helium or hydrogen, mixed with a finely atomized non-freezable lubricating liquid, is compressed; a step of isobaric heating of the working fluid, after separation of liquid from gas and a step of adiabatic expansion of the gas. The working fluid heating step of the bottoming subcycle is carried out by cooling the fluid that enters the compressor of the topping subcycle. 
     It shall be noted that the bottoming subcycle as described in U.S. &#39;347 is a power cycle, as it receives thermal power (through the heat exchanger  17  of  FIGS. 1 and 2  of the U.S. &#39;347 patent) and generates mechanical power (mechanical shaft  50  of  FIGS. 1 and 2  of the U.S. &#39;347 patent). 
     Nevertheless, it should be stressed that it seems to disagree with the second law of thermodynamics as enunciated by Kelvin (stating that “a cyclic transformation whose only final result is to transform heat extracted from a source into work is impossibile”). The bottoming subcycle includes a gas and liquid compression step, whereupon both the gas and the liquid are at a temperature higher than that at the start of the compression (as is commonly found in lubricated positive-displacement compressors, in which oil needs cooling after compression). Failing this, i.e. if no heat exchange occurred between the liquid and the gas during compression, the gas would not be subjected to polytropic compression (segment  31  of the thermodynamic diagram T-s of  FIG. 5 ) but a common isentropic compression (a vertical segment in the T-s). Therefore, the liquid separated from the gas will be at a higher temperature than the gas with which it is finely mixed. 
     As a result, after compression the separated liquid should be cooled by a process that is not contemplated in the patent. Furthermore, such cooling process should be carried out with a lower-temperature reservoir, which would also allow the bottoming subcycle to comply with Kelvin statement. 
     To prove this, and from an entropic perspective (or according to the second law), it should be noted that energy transfer as work interaction does not involve any entropy transfer, whereas energy transfer as heat interaction involves entropy exchange. Therefore, the introduction of thermal power into the bottoming subcycle (heat exchanger  17  of the U.S. &#39;347 patent) also introduces entropy into the subcycle. However, the delivery of mechanical power (shaft  50  of the U.S. &#39;347 patent) does not involve entropy transfer outside. Therefore, the bottoming subcycle is an entropy accumulating cycle, which characteristic contrasts with the cycle concept. In practice, in a machine operating according to the bottoming subcycle as described in the U.S. &#39;347 patent the liquid (in the tank  19 ) would be continuously heated, which would affect operation. 
     SUMMARY OF THE PRESENT INVENTION 
     In view of the above prior art, the object of the present invention is to obviate the above mentioned prior art drawbacks. 
     According to the present invention, this object is fulfilled by a method for converting thermal power delivered from a variable temperature heat source to mechanical power which uses a closed thermodynamic cycle, as defined in claim  1 . 
     According to the present invention, this object is fulfilled by a system for converting thermal power delivered from a variable temperature heat source to mechanical power, as defined in claim  16 . 
     The present invention affords a more efficient thermal power recovery as compared with prior art cycles. 
     Also, the present invention provides a particularly efficient system for small-sized plants (e.g. with a power of less than 200 kW). 
     Furthermore, the present invention also affords considerable savings in terms of system implementation times and costs, as it allows the use of readily available devices, with little or no changes, to implement the closed thermodynamic cycle. 
    
    
     
       BRIEF DESCRIPTION OF THE FIGURES 
       The characteristics and advantages of the invention will appear from the following detailed description of one practical embodiment, which is illustrated without limitation in the annexed drawings, in which: 
         FIG. 1  shows a temperature entropy T-s diagram of the steps that a first fluid undergoes in a first embodiment of the closed thermodynamic cycle of the present invention; 
         FIG. 2  shows a temperature entropy T-s diagram corresponding to that of  FIG. 1 , of the steps that a second fluid undergoes in the closed thermodynamic cycle of the present invention; 
         FIG. 3  shows another temperature entropy T-s diagram of the steps that a first fluid undergoes in a second embodiment of the closed thermodynamic cycle of the present invention; 
         FIG. 4  shows a temperature entropy T-s diagram corresponding to that of  FIG. 3 , of the steps that a second fluid undergoes in the thermodynamic cycle of the present invention; 
         FIG. 5  shows a further temperature entropy T-s diagram of the steps that a first fluid undergoes in a third embodiment of the closed thermodynamic cycle of the present invention; 
         FIG. 6  shows a temperature entropy T-s diagram corresponding to that of  FIG. 5 , of the steps that a second fluid undergoes in the thermodynamic cycle of the present invention; 
         FIG. 7  shows a further temperature entropy T-s diagram of the steps that a first fluid undergoes in a fourth embodiment of the closed thermodynamic cycle of the present invention; 
         FIG. 8  shows a temperature entropy T-s diagram corresponding to that of  FIG. 7 , of the steps that a second fluid undergoes in the thermodynamic cycle of the present invention; 
         FIG. 9  shows a diagram of the system that implements the fourth embodiment of the cycle, for converting thermal power delivered from a variable temperature heat source into mechanical power, according to the present invention. 
     
    
    
     DETAILED DISCLOSURE OF THE INVENTION 
     A better understanding of the advantages of the invention as disclosed hereinafter would be given by the assessment of the efficiency and useful work values of a few reference cycles operating between a higher temperature Thigh and a lower temperature Tlow to convert thermal power delivered from a variable temperature heat source into mechanical power, where the cold reservoir is at ambient temperature Tamb. Particularly, we will assume that the temperature Tamb is lower than or equal to the temperature Tlow and that the latter is lower than or equal to the temperature Thigh. 
     The following table shows the efficiency and useful work values of three reference cycles, i.e. Lorentz, Carnot and Brayton cycles. 
     
       
         
           
               
               
               
             
               
                   
               
               
                 Cycle 
                 efficiency η 
                 Useful work w 
               
               
                   
               
             
            
               
                 Lorentz 
                 
                   
                     
                       
                         
                           η 
                           LRZ 
                         
                         = 
                         
                           1 
                           - 
                           
                             Tamb 
                             
                               
                                 Thigh 
                                 - 
                                 Tlow 
                               
                               
                                 
                                   ln 
                                    
                                   
                                       
                                   
                                    
                                   Thigh 
                                 
                                 - 
                                 
                                   ln 
                                    
                                   
                                       
                                   
                                    
                                   Tlow 
                                 
                               
                             
                           
                         
                       
                     
                   
                 
                 w LRZ  = η LRZ  c p  (Thigh − Tlow) 
               
               
                   
               
               
                 Carnot 
                 η CRT  = 1 − Tamb/Tcrt 
                 w CRT  = η CRT  c p  (Thigh − Tcrt) 
               
               
                   
                 Tcrt = max({square root over (TambThigh)}; Tlow) 
                   
               
               
                 Brayton 
                 η BRT  = 1 − Tamb/Tbrt 
                 w BRT  = η BRT  c p  (Thigh − Tbrt) 
               
               
                   
                 Tbrt = max({square root over (TambThigh)}; Tlow) 
               
               
                   
               
            
           
         
       
     
     Where “w” designates the useful work and “cp” designates the specific heat at a constant pressure of the working fluid. 
     Assuming the above, and with reference to the accompanying figures, a method is disclosed for converting a thermal power delivered from a variable temperature heat source, into mechanical power. The closed thermodynamic cycle  1  operates between a higher temperature Thigh and a temperature substantially equal to ambient temperature Tamb, where the higher temperature Thigh is much higher than ambient temperature Tamb. 
     It shall be noted that the maximum temperature of the heat source may also be higher than one thousand degrees Celsius but, due to technological and cost requirements of the usable devices, the higher temperature Thigh will fall in a range from 400 to 800° C., whereas ambient temperature Tamb will be the temperature of the site at which the cycle  1  operates. 
     For example, the heat source may be the flue gases of an endothermic engine, e.g. fueled with biogas (in which case the temperature Thigh will range from 400 to 500° C.), or concentrated solar radiation or thermodynamic solar power (in which case the temperature Thigh will range from 600 to 800° C.), or the flue gases from external combustion of biomasses. 
     The maximum temperature of the heat source may also coincide with the higher temperature Thigh at which the closed thermodynamic cycle  1  operates. 
     Ambient temperature Tamb for Italy is typically set to about 15° C. 
     The closed thermodynamic cycle  1  comprises an adiabatic compression process (line  2 ′- 5  of  FIG. 1 , line  2 ′- 4  of  FIG. 3 , line  3 - 5  of  FIG. 5  and line  3 - 4  of  FIG. 7 ) which is adapted to change the temperature of one phase of the two-phase mixture from ambient temperature Tamb to a lower temperature Tlow and to change the specific entropy value of one phase of the two-phase mixture from a first specific entropy value s 1 , s 3  (s 1  for the shapes of  FIGS. 1 and 3  and s 3  for the shapes of  FIGS. 5 and 7 , respectively) to a second specific entropy value s 2 . 
     It shall be noted that ambient temperature Tamb for the cycle  1  acts as a cold reservoir, whereas the lower temperature Tlow is a free parameter of the closed cycle  1 , as described in greater detail below, and anyway the lower temperature Tlow is preferably a temperature value ranging from 80 to 120° C. 
     Particularly, the adiabatic compression process (line  2 ′- 5  of  FIG. 1 , line  2 ′- 4  of  FIG. 3 , line  3 - 5  of  FIG. 5  and line  3 - 4  of  FIG. 7 ) is such that the second specific entropy value s 2  of one phase of the two-phase mixture is lower than the first specific entropy value s 1 , s 3  (s 1  for the shapes of  FIGS. 1 and 3  and s 3  for the shapes of  FIGS. 5 and 7 , respectively) and that the ambient temperature value Tamb is lower than the lower temperature value Tlow. 
     It shall be further noted that the higher temperature Thigh is higher than the lower temperature value Tlow. 
     It should be first underlined that the cycle exposed to low temperature as disclosed in the U.S. &#39;347 document and the closed thermodynamic cycle  1  differ to a substantial extent, in that the cycle as described in U.S. &#39;347 only uses one reservoir, whereas the closed thermodynamic cycle  1  uses two reservoirs. Particularly, the U.S. &#39;347 cycle uses ambient air as a reservoir in the open configuration, and the working fluid of the topping cycle in the closed version, which are both used for introducing thermal power in the bottoming cycle. 
     The closed thermodynamic cycle  1  of the present invention uses a hot reservoir (such as the above mentioned flue gases from the endothermic engine fueled with biogases or else), to introduce thermal power into the cycle, and the environment for rejection of thermal power from the cycle. 
     Therefore, besides the number of reservoirs, the cycle of U.S. &#39;347 and the closed thermodynamic cycle  1  also differ in the way they use the environment, i.e. in the former case for introduction of thermal power into the cycle, in the latter case for rejection of thermal power from the cycle. 
     Furthermore, since the U.S. &#39;347 cycle provides a cryogenic system other than the environment, a considerable increase of costs and construction complexity is involved, as compared with the closed cycle  1 . 
     The cycle  1  of the present invention is a closed cycle, which affords the freedom of selecting the two-phase mixture and the pressurization level for cycle operation. 
     Therefore, the adiabatic compression process can change the specific entropy value of one phase of the two-phase mixture from a first specific entropy value s 1 , s 3  (s 1  for the shapes of  FIGS. 1 and 3  respectively and s 3  for the shapes of  FIGS. 5 and 7  respectively) to a second specific entropy value s 2 . 
     For this purpose, the two-phase mixture is preferably obtained by a mixing process, in which: 
     one phase of the mixture is a first working fluid and 
     a second phase of the mixture is a second working fluid. 
     Particularly, the first working fluid is mixed with the second working fluid in a ratio indicatively ranging from 0.1 to 1000 times, preferably 10 times, the kilograms of the second working fluid per one kilogram of the first working fluid. 
     Preferably, the first working fluid is a non soluble gas, whereas the second working fluid is a non volatile liquid. 
     Therefore, in a preferred embodiment, the two-phase mixture ratio is 10 kg of non-volatile liquid per kg of non soluble gas, 
     The peculiar characteristics of the two working fluids, i.e. non-volatility and non-solubility are important characteristics, because when the two fluids are mixed, they retain their purity even upon mutual contact, which means that they only have thermal interactions any chemical or physical interaction being prevented. 
     Therefore, once they are mixed, they form two distinct phases that can be separated by appropriate separating means, as described above with reference to the system. 
     For simplicity, the two working fluids will be referred to hereinbelow as gases and liquids, their “non-properties” being implied. 
     The above mentioned ratio limits the increase of gas temperature, and hence the specific volume of the gas and the specific work of the gas, as the liquid acts as a thermally inert material during adiabatic compression of the two-phase mixture. 
     Thus, adiabatic compression of the two-phase mixture may be considered as a gas-liquid compression process designed to increase the two-phase mixture temperature from Tamb to Tlow, and to reduce the specific entropy of the gas phase of the mixture from the value s 1 , s 3  to the value s 2 . 
     After adiabatic compression of the gas-liquid mixture (i.e. the two-phase mixture), that starts from ambient temperature Tamb, and considering that the gas and the liquid only have thermal interactions during adiabatic compression, an intermediate situations is obtained, in which the temperature of the two-phase mixture is not constant, as it would be with liquid only, but changes less than with gas only. 
     Particularly, the specific entropy of the gas phase of the two-phase mixture will decrease, whereas the specific entropy of the liquid phase of the two-phase mixture will increase, not necessarily to the same extent. 
     In extensive terms, the entropy of the gas phase of the two-phase mixture will decrease, whereas the entropy of the liquid phase of the two-phase mixture will increase to the same extent, whereby the entropy of the two-phase mixture will remain constant. 
     It shall be noted that a limited temperature change in the two-phase mixture has an advantageous effect on gas compression, as the specific volume of gas increases less with an increase of pressure, although this implies liquid compression. 
     Generally, both the liquid and the gas should be non-toxic, non-corrosive, non-combustible, environment-friendly and inexpensive. 
     Particularly concerning the gas, the latter may be selected from simple-structure molecules (small number of atoms), or complex-structure molecules (large number of atoms), or light gases (low molar mass) or heavy gases (high molar mass). 
     The gas may be pure or a mixture of other pure gases. 
     Preferably, monotomic gases (also known as noble gases) seem to be more appropriate, due to their favorable combination of the temperature, pressure and specific volume ratios in adiabatic processes (compression and expansion), i.e. the peculiar steps of the whole cycle  1 . 
     More in detail, with the same ratio between the final temperature and the initial temperature in an adiabatic reversible process, monoatomic gases are less exposed than all the others to pressure and specific volume changes, which is a very advantageous characteristic for the cycle and operation of compressors or expanders of the systems, as described below. 
     Preferably, the non-volatile liquid is selected from the group comprising vegetable, mineral or synthetic lubricating oils. 
     For example, the use of a lubricating oil for endothermic engines is preferred, as it has a low evaporation pressure, a relatively high density and a medium heat capacity. 
     Among monoatomic gases, Argon is preferred, as it seems to best reflect the above listed general characteristics. 
     Particularly, it shall be noted that the ambient temperature value Tamb must be lower than the lower temperature Tlow and that the second specific entropy value s 2  of one phase of the mixture must be lower than the first specific entropy value s 1 , s 3 . 
     It shall be also noted that, assuming adiabatic compression of gas only, temperature would change from the value Tamb to a temperature value equal to or lower than the higher temperature value Thigh, whereas adiabatic compression of liquid only would result in constant temperature. In both cases, i.e. adiabatic compression of gas only or liquid only, the entropy value would remain constant. 
     The gas and the liquid should evolve through equal temperatures, which requires perfect heat exchange between the two phases, i.e. the gas phase and the liquid phase; this condition may be obtained, for instance, by spraying the liquid into the gas. 
     In other words, the mixing step will consist in spraying the fluid and putting it in contact with the gas, to form the two-phase mixture to be later adiabatically compressed. 
     The temperature Tlow is a function, among other parameters, of the ratio of the mass throughputs of the liquid and the gas, i.e. it is a function of the ratio whereby the kilograms of fluid indicatively range from 0.1 to 1000 times, preferably 10 times per each kilogram of gas. As this ratio increases, the temperature Tlow at the end of adiabatic compression of the two-phase mixture decreases. Therefore, an appropriate change of the ratio between the mass throughputs of the liquid and the gas will provide the desired end-of-compression temperature. 
     Concerning the pressurization level of the closed cycle  1 , a pressure, for instance the minimum pressure (e.g. the isobars indicated in the Figures as P 2  or P 3  in the various embodiments), may be adjusted to change the mass flow rate of the gas, and hence the powers exchanged therefrom, while other parameters, such as geometric dimensions and flow velocities, remain the same. 
     Although this minimum pressure is free, it should be preferably higher than high vacuum, to avoid problems to the structure and seals of the devices to be used in the system as described below, and not higher than a few times ambient pressure, to avoid an excessively high value of maximum pressure. 
     Various embodiments of the steps of the cycle  1  will be now described. 
     Triangle 
     Also referring to  FIGS. 1 and 2 , these show a first possible implementation of the steps of the cycle  1 , and particularly the progress of temperature T relative to specific entropy s for the gas ( FIG. 1 ) and the liquid ( FIG. 2 ). 
     It shall be noted that before adiabatic compression of the two-phase mixture (in  FIG. 1 , the line from  2 ′ to  5 ), the closed cycle  1  includes a mixing process, for mixing the first working fluid (i.e. the gas) with the second working fluid (i.e. the liquid). 
     After adiabatic compression of the process mixture (in  FIG. 1 , the line from  2 ′ to  5 ), the cycle  1  includes a separation step, to separate the gas from the fluid. 
     Advantageously, after the separation step (point  5  of the T-s diagram of  FIG. 1 ), the thermodynamic cycle  1  involves isobaric heating of gas only, at pressure P 1 , because once the liquid is separated from the gas it is treated separately; the isobaric heating process (in  FIG. 1 , the line from  5  to  1 ) starts from the lower temperature Tlow and reaches the higher temperature Thigh. 
     In the first embodiment, prior to adiabatic compression of the two-phase mixture (in  FIG. 1 , the line from  2 ′ to  5 ) the thermodynamic cycle  1  involves adiabatic expansion of gas only from the higher temperature Thigh and the first pressure P 1  (point  1  of the T-s diagram of  FIG. 1 ), to reach ambient temperature Tamb and a second pressure P 2  (point  2 ′ of the T-s diagram of  FIG. 1 ). 
     As for the liquid, before being mixed with the gas (point  2 ′ of the T-s diagram of  FIG. 1 ), it is cooled to Tamb temperature. 
     Furthermore, depending on the pressure required to spray the liquid into the gas, the liquid may be possibly changed from the pressure P 1  existing at the end of adiabatic compression of the two-phase mixture (point  5  of the T-s diagram of  FIG. 1 ) to an appropriate level, e.g. ranging from a few tens of bars to about one hundred bars. 
     Thus, it shall be noted that the only process of the cycle  1  that involves the two-phase mixture is adiabatic compression (in  FIG. 1 , the line from  2 ′ to  5 ), whereas the other steps involve the gas only or the liquid only. 
     In other words, the thermodynamic process as shown in  FIG. 1  defines a sort of triangle, whose sides identify the following processes: 
     from  1  to  2 ′: adiabatic expansion of gas from the higher temperature Thigh and the first pressure P 1  to ambient temperature Tamb and a second pressure P 2 ; 
     mixing (point  2 ) of the gas and the fluid in a preferred ratio, such as 10 kilograms of liquid per 1 kilogram of gas; 
     from  2 ′ to  5 : adiabatic compression of the two-phase gas-liquid mixture to change the temperature of the mixture from ambient temperature Tamb to the lower temperature Tlow and to change the specific entropy value “s” of the gas phase of the mixture from the first entropy value s 1  to the second entropy value s 2 ; 
     separation of the two-phase mixture (point  5 ); 
     from  5  to  1 : isobaric heating of gas only, at pressure P 1 , from the lower temperature Tlow to the higher temperature Thigh. 
     Referring now to  FIG. 2 , it can be noted that, during adiabatic compression of the mixture, the liquid increases its specific entropy from the value sliq 1  to the value sliq 2 , and temperature increases from ambient temperature Tamb to the lower temperature value Tlow (points  2 ′ to  5  of  FIG. 2 ). 
     Therefore, it can be appreciated that the temperature of the two-phase mixture is not constant as it would be with adiabatic compression of liquid only, but changes less than with gas only. Furthermore, the specific entropy of the gas decreases (from s 1  to s 2 , where s 1  is higher than s 2 ), whereas the specific entropy of the liquid increases, not necessarily to the same extent (from sliq 1  to sliq 2 , where sliq 2  is higher than sliq 1 ), but the total entropy of the two-phase mixture remains constant. 
     Quadrangle 
     In view of increasing the figures of merit of the closed thermodynamic cycle  1 ,  FIGS. 3 and 4  show a second embodiment of the processes that form the cycle  1 . 
     In addition to what has been described with reference to  FIGS. 1 and 2 , in this second embodiment adiabatic compression of the gas-liquid mixture may end at an end of two-phase compression temperature T 2  (point  4  of the T-s diagram of  FIG. 3 ), in which this temperature T 2  is lower than the lower temperature value Tlow (point  5  of the T-s diagram of  FIG. 3 ). 
     It shall be noted that the end of two-phase compression temperature T 2  ranges from Tamb to Tlow. 
     The end of two-phase compression temperature T 2  reached by the gas-liquid mixture lies on an isobar P 4 . 
     The cycle  1  comprises adiabatic compression of gas only from the end of two-phase compression temperature T 2  and the fourth pressure P 4  to the lower temperature Tlow. 
     Concerning the liquid, it shall be noted that in this second embodiment, depending on the pressure required to spray the liquid into the gas (point  2 ′ of the T-s diagram of  FIG. 3 ), the liquid may be possibly changed from the pressure P 4  existing at the end of adiabatic compression of the two-phase mixture (point  4  of the T-s diagram of  FIG. 3  or  FIG. 4 ) to an appropriate level, e.g. ranging from a few tens of bars to about one hundred bars. 
     In other words, the thermodynamic cycle as shown in  FIG. 3  defines a sort of quadrangle, whose sides identify the following processes: 
     from  1  to  2 ′: adiabatic expansion of gas from the higher temperature Thigh and the first pressure P 1  to ambient temperature Tamb and a second pressure P 2 ; 
     mixing (point  2 ′) of the gas and the fluid in a preferred ratio, such as 10 kilograms of liquid per 1 kilogram of gas; 
     step from  2 ′ to  4 : adiabatic compression of the two-phase gas-liquid mixture to change the temperature of the mixture from ambient temperature Tamb to the end of two-phase compression temperature T 2  and to change the specific entropy value “s” of the gas phase of the mixture from the first entropy value s 1  to the second entropy value s 2 ; 
     separation of the two-phase mixture (point  4 ); 
     from  4  to  5 : adiabatic compression of gas from the end of two-phase compression temperature T 2  and the fourth pressure P 4  to the lower temperature Tlow. 
     from  5  to  1 : isobaric heating of gas only, at pressure P 1 , from the lower temperature Tlow to the higher temperature Thigh. 
     Referring now to  FIG. 4 , it can be noted that, during adiabatic compression of the two-phase mixture, the liquid increases its specific entropy from the value sliq 1  to the value sliq 3 , and temperature increases from ambient temperature Tamb to the end of two-phase compression temperature T 2  (points  2 ′ to  4  of  FIG. 4 ). 
     Also in this embodiment, it can be appreciated that the temperature of the mixture is not constant as it would be with adiabatic compression of one liquid only, but changes less than with adiabatic compression of one gas only. Furthermore, during such adiabatic compression step (points to  2 ′ to  4  of  FIG. 3 ) the specific entropy of the gas decreases (from s 1  to s 2 , where the specific entropy value s 1  is higher than s 2 ), whereas the specific entropy of the liquid increases, not necessarily to the same extent (from sliq 1  to sliq 3 , where sliq 3  is higher than sliq 1 ), but the total entropy of the two-phase mixture remains constant. 
     Referring to  FIGS. 5 and 6 , a third embodiment of the processes of the cycle  1  is shown. In addition to what has been described above with reference to the previous figures, in this third embodiment, adiabatic expansion of the gas (line from  1 - 2 ′ of  FIG. 1  or  FIG. 3 ) ends at an end of expansion temperature T 1  (point  2  of the T-s diagram of  FIG. 5 ), which is higher than ambient temperature Tamb. 
     The end of expansion temperature T 1  reached by the gas lies on an isobar P 3  and, along such isobar, the thermodynamic cycle comprises an isobaric cooling process for cooling the gas from the end of expansion temperature T 1  to ambient temperature Tamb (point  3  of the T-s diagram of  FIG. 5 ). 
     Due to such isobaric cooling step (line from  2  to  3  of  FIG. 5 ), the specific entropy value during adiabatic compression of the gas (i.e. the gas phase of the two-phase mixture) changes from a specific entropy value s 1  to the specific entropy value s 3 , where the value s 3  is lower than the specific entropy value s 1 . 
     Concerning the liquid, it shall be noted that in this third embodiment, depending on the pressure required to spray the liquid into the gas (point  3 ′ of the T-s diagram of  FIG. 5 ), the liquid may be possibly changed from the pressure P 1  existing at the end of adiabatic compression of the two-phase mixture (point  5  of the T-s diagram of  FIG. 5 ) to an appropriate level, e.g. ranging from a few tens of bars to about one hundred bars. 
     It shall be noted that the end of expansion temperature T 1  has a lower limit represented by Tamb and an upper limit represented by the temperature at which the point  3  (end of isobaric cooling) has a specific entropy value s 3  coinciding with the specific entropy value s 2 . 
     In other words, the thermodynamic cycle as shown in  FIG. 5  defines a sort of quadrangle, whose sides identify the following processes: 
     from  1  to  2 : adiabatic expansion of gas from the higher temperature Thigh at the first pressure P 1  to the end of expansion temperature T 1  at the pressure P 3 ; 
     from  2  to  3 : isobaric cooling of gas along the isobar P 3  from the temperature T 1  to the temperature Tamb; 
     mixing (point  3  of  FIG. 5 ) of the gas and the fluid in a preferred ratio, such as 10 kilograms of liquid per 1 kilogram of gas; 
     from  3  to  5 : adiabatic compression of the two-phase gas-liquid mixture to change the temperature of the mixture from ambient temperature Tamb to the lower temperature Tlow and to change the specific entropy value “s” of the gas phase of the two-phase mixture from the first entropy value s 3  to the second entropy value s 2 ; 
     separation of the two-phase mixture. (point  5  of  FIG. 5 ); 
     from  5  to  1 : isobaric heating of gas only, at pressure P 1 , from the lower temperature Tlow to the higher temperature Thigh. 
     Referring now to  FIG. 6 , it can be noted that, during adiabatic compression of the two-phase mixture, the fluid increases its specific entropy from the value sliq 1  to the value sliq 2 , and temperature increases from ambient temperature Tamb to the lower temperature value Tlow (points  3  to  5  of  FIG. 6 ). 
     Also in this embodiment, it can be appreciated that the temperature of the two-phase mixture is not constant as it would be with adiabatic compression of liquid only, but changes less than with adiabatic compression of gas only. Furthermore, during the adiabatic compression step (points to  3  to  5  of  FIG. 5 ) the specific entropy of the gas decreases (from s 3  to s 2 , where the s 3  is higher than s 2 ), whereas the specific entropy of the liquid increases, not necessarily to the same extent (from sliq 1  to sliq 2 , where sliq 2  is higher than sliq 1 ), but the total entropy of the two-phase mixture remains constant. 
     It should be understood that the embodiments as shown in  FIGS. 3 and 5  (quadrangular) provide advantages over the embodiment as shown in  FIG. 1  (triangle) in terms of achievable useful work, because the area enclosed by the closed cycle  1  in the T-s plane of  FIG. 3  or  FIG. 5  is larger than the area defined by the closed cycle  1  of  FIG. 1 . 
     This is because the area enclosed by the cycle is equal to the useful work and the specific work increases with such area. Therefore, the passage from the so-called Triangle embodiment to the so-called Quadrangle embodiment affords an increase of the useful work of the cycle with the same power being introduced into the isobaric cooling  5 - 1 . 
     In other words, the passage from the so-called Triangle embodiment to the so-called Quadrangle embodiment affords an increase of the useful work of the cycle while maintaining the energy introduced into the cycle unchanged. 
     Pentagon 
     In order to further increase the figures of merit of the closed thermodynamic cycle  1 . The previous figures show a fourth embodiment of the steps that form the cycle  1 . 
     In addition to the description referred to  FIGS. 3 and 5 , what can be appreciated in this fourth embodiment is the result of the combination of the cycles of  FIG. 3  and  FIG. 5 . Thus, we can note that: 
     the process of adiabatic compression of the gas-liquid mixture ends at the end of two-phase compression temperature T 2  (point  4  of the T-s diagram of  FIG. 7 ) in which such second temperature T 2  is lower than the lower temperature Tlow and 
     the process of adiabatic expansion of the gas (line  1 - 2  of  FIG. 7 ) ends at the end of expansion temperature T 1  (point  2  of the T-s diagram of  FIG. 7 ), which is higher than the ambient temperature Tamb. 
     It shall be noted that the end of expansion temperature T 1  may be equal to or lower or higher than, i.e. independent of the end of two-phase compression temperature T 2 . 
     The end of two-phase compression temperature T 2  ranges from Tamb to Tlow. 
     The end of expansion temperature T 1  has a lower limit represented by Tamb and an upper limit represented by the temperature at which the point  3  (end of isobaric cooling) has a specific entropy value s 3  coinciding with the specific entropy value s 2 . 
     The end of two-phase compression temperature T 2  reached by the gas-liquid mixture lies on the isobar P 4 , whereas the end of expansion temperature T 1  reached by the gas only upon expansion, lies on the isobar P 3 . 
     The cycle  1  comprises a process of adiabatic compression of gas from the end of two-phase compression temperature T 2  and the fourth pressure P 4  to the lower temperature Tlow. 
     Concerning the liquid, it shall be noted that in this fourth embodiment, depending on the pressure required to spray the liquid into the gas (point  3  of the T-s diagram of  FIG. 7 ), the liquid may be possibly changed from the pressure P 4  existing at the end of adiabatic compression of the two-phase mixture (point  4  of the T-s diagram of  FIG. 7 ) to an appropriate level, e.g. ranging from a few tens of bars to about one hundred bars. 
     In other words, the cycle  1  as shown in  FIG. 7  describes a pentagon having the following processes: 
     from  1  to  2 : adiabatic expansion of gas from the higher temperature Thigh at the first pressure P 1  to the end of expansion temperature T 1  at the pressure P 3 ; 
     from  2  to  3 : isobaric cooling of gas along the isobar P 3  from the temperature T 1  to the temperature Tamb; 
     from  3  to  4 : adiabatic compression of the gas-liquid mixture to change the temperature of the mixture from ambient temperature Tamb to the end of two-phase compression temperature T 2  and to change the specific entropy value “s” of the gas phase of the two-phase mixture from the third entropy value s 3  to the second entropy value s 2 ; 
     mixing (point  3 ) of the gas and the fluid in a preferred ratio, such as 10 kilograms of liquid per 1 kilogram of gas; 
     from  4  to  5 : adiabatic compression of gas from the end of two-phase compression temperature T 2  to the lower temperature Tlow and from the pressure P 4  to the pressure P 1 ; 
     separation of the two-phase mixture (point  5 ); 
     from  5  to  1 : isobaric heating of gas, at pressure P 1 , from the lower temperature Tlow to the higher temperature Thigh. 
     Referring now to  FIG. 8 , it can be noted that, during adiabatic compression of the two-phase mixture, the liquid increases its temperature from ambient temperature Tamb to the end of compression temperature T 2  as it is cooling the gas. 
     Also in this embodiment, it can be appreciated that the temperature of the two-phase mixture is not constant as it would be with adiabatic compression of liquid only, but changes less than with adiabatic compression of gas only. Furthermore, during the adiabatic compression step (points to  3  to  4  of  FIG. 7 ) the specific entropy of the gas decreases (from s 3  to s 2 , where the s 3  is higher than s 2 ), whereas the specific entropy of the liquid increases, not necessarily to the same extent (from sliq 1  to sliq 3 , where sliq 3  is higher than sliq 1 ), but the total entropy of the two-phase mixture remains constant. 
     It shall be noted that the passage from the so-called Quadrangle embodiment to the so-called Pentagon embodiment (as shown in  FIG. 7 ) affords an increase of the useful work of the cycle while maintaining the energy introduced into the cycle unchanged. 
     Calculations show that, when compared with the Lorentz cycle, that appears in the table as the one with the maximum attainable useful work, although technologically unfeasible, the thermodynamic cycle  1  of the fourth embodiment (the so-called Pentagon embodiment) can achieve about 90% the useful work that can be obtained with such Lorentz cycle, whereas Carnot and Brayton cycles achieve about 60% Lorentz cycle results. 
     Once the various embodiments of the closed thermodynamic cycle have been disclosed, we can describe the system that employs the cycle  1 , preferably but without limitation the fourth embodiment of the thermodynamic cycle as shown with reference to  FIGS. 7 and 8 . 
     The system can convert the thermal power delivered from the variable temperature heat source (such as the flue gases of a biogas-fueled endothermic engine) into mechanical power. This system operates in a range from the higher temperature Thigh (ranging from 400° C. to 800° C.) to an ambient temperature Tamb that acts as a cold reservoir. 
     For this purpose, also with reference to  FIG. 9 , the system comprises compression means  7 B which are appropriate to adiabatically compress a two-phase mixture for the temperature of the mixture to change from ambient temperature Tamb to a lower temperature Tlow and for the entropy value of one phase of the two-phase mixture to change from the specific entropy value s 3  to the specific entropy value s 2 , where the ambient temperature Tamb is lower than the lower temperature value Tlow and the second specific entropy value s 2  is lower than the specific entropy value s 3 . 
     Preferably, the compression means  7 B are in fluid communication with mixing means  7 A to receive the two-phase mixture at their inlet. 
     These mixing means  7 A are appropriate to mix the first working fluid (gas) and the second working fluid (liquid) in a preferred a ratio of 10 kilograms of the second working fluid per one kilogram of the first working fluid. 
     It shall be noted that, to ensure adiabatic compression of the two-phase mixture, the compression means  7 B compress to an end of two-phase compression temperature T 2  and to a fourth pressure P 4 , in which the end of two-phase compression temperature T 2  is lower than said lower temperature Tlow. 
     Particularly, the compression means  7 B consist of positive-displacement compressors or turbocompressors or equivalent devices, whereas the mixing means  7 A consist of a spraying device. 
     In one embodiment, the compression means  7 B and the mixing means  7 A may be integrated in a single device  7 . 
     The system comprises separator means  8  for separating the gas from the liquid, once adiabatic compression of the two-phase mixture has been accomplished by the compression means  7 B. 
     The separator means are in fluid communication with said compressor means  7 B to receive said compressed two-phase mixture at their inlet, and divide it into the gas and the fluid. 
     Preferably, the separator means  8  consist of coalescence filters. 
     Upstream from said separator means  8 , the system comprises heating means  9  for isobaric heating of the gas at a first pressure P 1  from the lower temperature Tlow to said higher temperature Thigh. 
     It shall be noted that, to increase gas temperature from Tlow to Thigh, the exchanger utilizes, for instance, the heat  16  of the hot source gases (which is shown by a dotted and dashed line in  FIG. 9 ). 
     Preferably, the heating means  9  consist of gas-gas exchangers. 
     The system comprises expansion means  10  for adiabatic expansion of the gas from the higher temperature Thigh and the first pressure P 1  to ambient temperature Tamb and a second pressure P 2 . 
     Preferably, the expansion means  10  expand the gas to a first end of expansion temperature T 1 , which is higher than ambient temperature Tamb and to a third pressure P 3 . 
     Preferably, the expansion means  10  consist of positive displacement expanders or turboexpanders or equivalent devices. 
     Downstream from the separator means  8  and upstream from said heater means  9 , the system comprises second compression means  12  for adiabatic compression of the gas from the second end of two-phase compression temperature T 2  and the fourth pressure P 4  to the lower temperature Tlow and the first pressure P 1 . 
     Downstream from the expansion means  10 , the system comprises cooling means  11  for isobaric cooling of the gas, along the pressure isobar P 3 , from the first end of expansion temperature T 1  to said ambient temperature Tamb. 
     It shall be noted that, in the first embodiment ( FIG. 1 ) and the second embodiment ( FIG. 3 ), the cooling means  11  are unnecessary, as no isobaric cooling is provided. 
     The system comprises cooling means  13  for cooling the liquid when it has been separated by the separator means  8 . 
     Particularly, the cooling means  13  are located downstream from the separator means  8  and upstream from the spraying device  7 A. 
     Preferably, the cooling means  11  and  13  consist of air coolers or equivalent devices. 
     In order to convert the thermal power delivered from the variable temperature heat source (such as the flue gases of a biogas-fueled endothermic engine) into mechanical power, there is provided a drive shaft  14 , operably connected between the compressor means  7 B and the expansion means  10  and possibly a power take-off  15  to allow an alternator (not shown) coupled to such power take-off  15  to generate electric current. 
     All typical changes to a gas cycle (Brayton cycle), such as intercooling, regeneration and postheating, shall be intended to apply to the present method and system. 
     Also, any embodiment of a gas cycle alternative to the present one, such as a single-shaft machine, a separate-shaft machine and a multi-shaft machine shall be intended to apply to the present system. 
     For example, the system may comprise a pump (not shown), interposed between the separator means  8  and the cooling means  13 . Depending on the pressure required in the spraying means  7 A to spray the liquid into the gas, this pump operates to change the liquid from the pressure P 4  existing at the end of adiabatic compression of the mixture to an appropriate level, e.g. ranging from a few tens of bars to about one hundred bars. 
     Those skilled in the art will obviously appreciate that a number of changes and variants may be made to the method and system for converting thermal power delivered from a variable heat source into mechanical power as described hereinbefore, still within the scope of the invention, as defined in the following claims.