Abstract:
Method for dimensioning a solar generation system and the solar generation system obtained, including a solar radiation heat absorber for a Stirling engine. The Stirling engine includes a head and a heat exchanger surrounding the head of the engine, the absorber having a cavity shaped so as to be joined onto the head of the engine and to transfer heat to the heat exchanger. The method includes the step of giving the absorber such a mass as to guarantee stable operation of the Stirling engine during temporary periods of predefined duration wherein the solar radiation is insufficient to guarantee operation of the engine (Pric&lt;Pabs+Ploss).

Description:
CROSS-REFERENCE TO RELATED APPLICATIONS 
       [0001]    This application is a continuation of PCT/IB2012/051758, filed Apr. 11, 2012 which claims the benefit of IT-RM2011A000181, filed Apr. 11, 2011, the contents of each of which are incorporated herein by reference. 
     
    
     FIELD OF THE INVENTION 
       [0002]    The present invention relates to a system for receiving solar radiation reflected by a solar concentrator and to the respective dimensioning method thereof. In particular, the receiving system has the purpose of adapting the distribution of a solar radiation reflected by a primary reflector on to an absorber of a Stirling engine contained in the interior of a secondary reflector. The heat absorber is advantageously dimensioned in respect of both form and weight so as to guarantee the stable operation of the Stirling engine in transient conditions of predefined duration, wherein said solar radiation is insufficient to guarantee the steady-state operation of the engine. 
       BACKGROUND OF THE INVENTION 
       [0003]    The Stirling engine is a closed-circuit, external-combustion thermal engine that uses a gas as a thermodynamic fluid, usually air, nitrogen or helium in the high-yield variants: the latter is shown to be highly versatile by comparison with different heat sources. The Stirling engine is composed of one or more cylinders in which one or more pistons glide following the expansion and contraction of the thermodynamic fluid. Alternatively, the gas flows from a heat exchanger and a cold exchanger after being passed from a heat regenerator: the corresponding harmonic movement of the pistons can produce electrical energy via a mechanical-electrical converter. 
         [0004]    The heat is generally supplied to the heat exchanger, which is positioned at one extremity of the cylinder/s and provides for heating of the internal thermodynamic fluid, which sets the piston/s in motion. 
         [0005]    The heat exchanger is usually formed from a plurality of fins made of a material having good thermal conductivity or from a plurality of small conductors in which the operating fluid flows. 
         [0006]    The regenerator alternatively absorbs and restores heat from/to the operating fluid and increases the transformational efficiency. The cold exchanger, which constitutes the low-temperature source, is usually a compound-tube, crossover flow exchanger, in which the tubes are externally lapped by refrigerating water, while the operating fluid flows within the tubes. 
         [0007]    The fact that the Stirling engine is a combustion engine is advantageous because the heat can be supplied externally from a vast range of fuels, even ones of low calorific value. 
         [0008]    One highly widespread application of the Stirling engine provides for production of the heat source by combusting natural gas. 
         [0009]    Another application provides for the use of the Stirling engine for the production of renewable-source electrical energy, and in particular by means of the uptake and possible accumulation of heat obtained from solar radiation for the purpose of defining said hot spot within the engine. 
         [0010]    To this extent, the use of the Stirling engine is divided between applications in which the heat is given up to a vector fluid and transported thereby to the head of one or more Stirling engines, and applications in which the engine head is directly exposed to the solar radiation. This implies that the engine is aligned along a focusing axis of a reflective mirror, the said primary reflector. The receiving system thus comprises an absorber, directly mounted on the head of the Stirling engine, and a secondary reflector, pre-arranged about the absorber for the purpose of limiting the losses due to radiation emitted by said absorber, which can reach temperatures very close to 500° C. 
         [0011]    Examples of this type of application are given in U.S. Pat. No. 6,735,946, U.S. Pat. No. 7,026,722 and WO2010/045269. 
         [0012]    In WO2010/045269, in particular, there is evidence of problems relating to possible overheating of the engine head, which problems are resolved by rotating the engine about the focusing axis, in such a way as to distribute the radiation over a wider area of the head. 
         [0013]    US2004031517 describes a system in which the absorption head of the Stirling engine is out of focus, in particular, it is at a greater distance from the primary reflector that its own focus. 
         [0014]    Whilst this solution improves the distribution of the heat over the head of the Stirling engine, it does not prove satisfactory in terms of stabilising the operation of the engine. 
         [0015]    It is in fact observed that during transient conditions in which the sky clouds over, the engine can shut down even in the case of transients of low irradiation lasting only a few seconds. 
         [0016]    For the purpose of stabilising this behaviour, the known art tends to oversize the primary reflector to make provision for variability of insolation, allowing slight operation of the engine in conditions of little or average radiation, but in the case of excessive irradiation, the engine tends to overheat and even to sustain damage. Hence there are various systems for preserving the engine from overheating. 
         [0017]    Conversely, by reducing the surface area of the primary reflector, the engine tends to shut down as soon as the solar radiation decreases, even transiently, for example during cloud passage. 
         [0018]    According to another aspect, free-piston Stirling engines of small size are known that are optimised to operate by means of natural gas combustion, guaranteeing optimum efficiency. An example of similar engines is given by WO2005/054654. Because these engines have witnessed a great development in the sphere of gas combustion, this has entailed an inevitable perfecting of said engines, which are especially reliable and efficient. 
       SUMMARY OF THE INVENTION 
       [0019]    The present invention proposes to teach the sizing of a solar generator directly and/or indirectly exposed to solar radiation in such a way as to overcome the above-mentioned limitations of the known art. 
         [0020]    A subject of the present invention is a method for dimensioning a solar generator directly exposed to solar radiation, according to claim  1 . 
         [0021]    According to the present invention, the absorber is dimensioned in such a way as to possess a thermal capacity such as to guarantee the operation of the engine even when the solar radiation is insufficient to power the steady-state operation of the engine. 
         [0022]    A further aim of the present invention is to demonstrate an advantageous juxtaposition of the absorber, relative to the primary reflector of a solar system, in such a way as to overcome the above-mentioned problems of the known art. 
         [0023]    Also a subject of the present invention is an absorber and a solar generator that is dimensioned according to the present invention. 
         [0024]    According to a first variant of the invention, the totality of the solar radiation reflected by the primary reflector impinges upon the absorber directly, that is, without reflections, which absorber is positioned at a first distance interval relative to the focal point of the primary reflector. 
         [0025]    According to a second variant of the invention, a portion of the solar radiation reflected by the primary reflector impinges upon the absorber directly, while a residual portion is reflected by means of the secondary reflector onto the absorber. At that moment, the absorber is positioned at a second distance interval, relative to the focal point of the primary reflector. This second distance interval can at least partially overlap said first distance interval. 
         [0026]    The present invention is especially suitable for the dimensioning of systems suited to the use of Stirling engines of small size, that is, requiring heat flows of less than 50 Watt/cm 2 . 
         [0027]    According to a further aspect of the present invention, a method is described of controlling solar tracking, now made possible by the arrangements related to dimensioning of the absorber and of the generator in general as described herein. 
         [0028]    The claims describe preferred embodiments of the invention, forming an integral part of the present description. 
     
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         [0029]    Further characteristics and advantages of the invention will become more evident in the light of the detailed description of, albeit non-exclusive, embodiments and methods for a system for receiving solar radiation, which is illustrated by way of a non-limiting example with the aid of the combined design tables, wherein: 
           [0030]      FIG. 1  represents a diagrammatic axial cross-section of the receiving system that is the subject of the present invention, 
           [0031]      FIG. 2  represents the characteristic operating curves of a Stirling engine of small size used in the invention, 
           [0032]      FIG. 3  represents an axial cross section of part of the system of  FIG. 1 , 
           [0033]      FIG. 4  represents two variants of the receiving system according to the present invention, 
           [0034]      FIG. 5  represents a schematic construction of an axial cross-section of  FIG. 4 . 
           [0035]      FIG. 6  represents the absorber axial cross-section of one of the variants of  FIG. 4 . 
       
    
    
       [0036]    Identical letters and identical reference numerals in the figures identify identical members or components, even in different variants. 
       DETAILED DESCRIPTION OF THE INVENTION 
       [0037]    With reference to  FIG. 1 , the receiving system comprises a secondary reflector  1  and an absorber  2 . 
         [0038]    The absorber  2  is directly associated with a head  31  of a Stirling engine  3 . 
         [0039]    The head  31  is generally cylindrical, proving suitable for being associated with a paraboloid primary reflector  4 , which defines a focusing axis and a focal distance F for focusing. 
         [0040]    The absorber  2  is tasked with picking up the light radiation and transferring the heat to an annular zone around the head  31  of the engine, alongside a heat exchanger  32 . Therefore the absorber comprises at least one cavity  25  complementary with the head  31  of the engine. 
         [0041]    According to the present invention, the absorber  2  is dimensioned, not only as a solar-energy uptake member and conveyor of thermal energy. According to the present invention, the absorber  2  is dimensioned in such a way as to guarantee stable operation of the Stirling engine, in the event of a reduction in the intensity of minimum reflected radiation necessary for the correct operation of the engine. In particular the latter is dimensioned in such a way as to function as a thermal capacitor, that is, so as to provide the heat necessary for starting the engine and for stabilising the temperature of the annular heat exchanger  32  of the engine head. 
         [0042]    At steady-state operation, the absorber  2  is advantageously capable of compensating for variations in solar radiation captured by the primary reflector, and therefore by the receiving system. 
         [0043]    To determine the thermal capacity of the absorber effectively, it is necessary to determine empirically on a test bench the characteristic absorption and productivity curves of the converter upon variations in the temperature present at the interface between the absorber  2  and the heat exchanger  32  (or operating temperature). 
         [0044]    The thermodynamic converter used in the tests conducted is a free-piston Stirling engine produced by Microgen™ for operation by combusting natural gas, and is suitable for producing both electric power and thermal power by means of direct heating of the cooling water entering the engine. 
         [0045]    With the use of known mathematical techniques for minimising the mean square error between the measured data for electrical power produced Pe, thermal power produced Pt and thermal power absorbed Pabs as a function of the temperature T operating at the interface between absorber  2  and the heat exchanger  32  of the Stirling engine  3  and the interpolating graphs of measured data, are obtained the following curves:
       the electrical production curve Pe(T);   the thermal production curve Pt(T);   the thermal absorption curve Pabs(T).
 
where T indicates the operating temperature of the converter.
       
 
         [0049]      FIG. 2  shows the respective curves for the Microgen™ converter used as an example. In particular, it is shown that
       The Stirling engine necessitates absorption of a thermal power Pstart for start-up and necessitates absorption of a power of Psteady to supply, in the steady state, the maximum electrical and thermal power; normally Pstart≦Psteady;   The temperature Tstart is the temperature present at the interface between the absorber  2  and the heat exchanger  32  when the power absorbed by the Stirling engine is Pstart;   The temperature Tsteady is the temperature present at the interface between the absorber  2  and the heat exchanger  32  when the power absorbed by the Stirling engine is Psteady; normally Tsteady≧Tstart;   At the temperature Tstart, the engine supplies the minimum electrical and thermal power.   At the temperature Tsteady, the engine supplies the maximum electrical and thermal power.       
 
         [0055]    Therefore starting up the engine and maintaining it in operation requires a quantity of heat dQ or a corresponding useful power P u =dQ/dt, in such a way as to:
       Start up the thermodynamic converter and maintain it in operation,   sufficiently heat the heat absorber such that, in the absence of concentrated solar radiation, the amount of heat accumulated to excess, succeeds in maintaining the thermodynamic cycle in operation for a predefined interval of time δt.       
 
         [0058]    Such characterization of the Stirling engine can be provided by the manufacturer of the engine itself, or by predisposing a heater of known thermal power in contact with the engine head, and arranging a temperature sensor at the interface between the engine head and the heater. These operations are known, and in any case are accessible to the person skilled in this sector. 
         [0059]    According to the present invention, a suitable energy balancing equation of the receiving system is the following: 
         [0000]        Pric=Piner+Pabs−Ploss    (e1)
 
         [0000]    where
       Pric is the power entering the secondary reflector,   Piner is the thermal power accumulated by the heat absorber, and   Ploss are the thermal losses of the receiving system.       
 
         [0063]    From this we deduce that the useful power for operating the thermodynamic system is: 
         [0000]        P   u   =Pric−Ploss  that is,  P   u   =Pabs+Piner    
         [0064]    The heat absorber is dimensioned for functioning as a thermal capacitor, that is, it stores the energy necessary for starting up the Stirling engine and supplies the heat necessary for maintaining the Stirling engine in operation when the amount of heat, due to the concentrated solar radiation, is insufficient for the optimum operation of the converter. In particular, Piner is expressed by: 
         [0000]        Piner=C   thermal   *dT/dt    (e2)
 
         [0000]    where T is the temperature of the interface between absorber  2  and heat exchanger  32  and C thermal  is the thermal capacity of the absorber  2 . 
         [0065]    Using the expression (e2) and making P u =Pabs+Piner=0, that is, the solar radiation is insufficient to supply the thermal energy Psteady to the absorber, the following is obtained: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                        
                       T 
                     
                     
                        
                       t 
                     
                   
                   = 
                   
                     - 
                     
                       Pabs 
                       
                         C 
                         thermal 
                       
                     
                   
                 
               
               
                 
                   ( 
                   e3 
                   ) 
                 
               
             
           
         
       
     
         [0066]    In particular, by resolving the non-linear differential equation (e3), the characteristic thermal capacity value C thermal  of the absorber is found to be such that the system thereby remains turned on, for a defined interval of time δt, notwithstanding that the solar radiation may be insufficient to maintain the absorber at the temperature Tsteady. 
         [0067]    According to the present invention it is assumed that the power absorbed by the thermodynamic converter is approximating by a linear curve of the type Pabs=AT, in which case (e3) may be rewritten as follows: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                        
                       T 
                     
                     
                        
                       t 
                     
                   
                   = 
                   
                     - 
                     
                       AT 
                       
                         C 
                         thermal 
                       
                     
                   
                 
               
               
                 
                   ( 
                   e4 
                   ) 
                 
               
             
           
         
       
     
         [0000]    for which the close-form solution is 
         [0000]    
       
         
           
             
               
                 
                   
                     T 
                      
                     
                       ( 
                       t 
                       ) 
                     
                   
                   = 
                   
                     
                       T 
                       0 
                     
                     * 
                     
                        
                       
                         
                           A 
                           
                             C 
                             thermal 
                           
                         
                         * 
                         
                           ( 
                           
                             t 
                             - 
                             
                               t 
                               0 
                             
                           
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   e5 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where T 0  is the temperature at the interface between the heat exchanger  32  and the absorber  2  at the instant t=t 0 , that is T(t 0 )=T 0 , instant in which P u =0. 
         [0068]    In a following instant t 1 =δt+t 0 , (e5) may be rewritten as 
         [0000]    
       
         
           
             
               
                 
                   
                     T 
                      
                     
                       ( 
                       
                         t 
                         1 
                       
                       ) 
                     
                   
                   = 
                   
                     
                       T 
                       0 
                     
                     * 
                     
                        
                       
                         
                           A 
                           
                             C 
                             thermal 
                           
                         
                         * 
                         δ 
                          
                         
                             
                         
                          
                         t 
                       
                     
                   
                 
               
               
                 
                   ( 
                   e6 
                   ) 
                 
               
             
           
         
       
     
         [0000]    and so the temperature present at the interface decreases by ΔT alter a time interval δt: 
         [0000]    
       
         
           
             
               
                 
                   
                     Δ 
                      
                     
                         
                     
                      
                     T 
                   
                   = 
                   
                     
                       
                         T 
                          
                         
                           ( 
                           
                             t 
                             0 
                           
                           ) 
                         
                       
                       - 
                       
                         T 
                          
                         
                           ( 
                           
                             t 
                             1 
                           
                           ) 
                         
                       
                     
                     = 
                     
                       
                         T 
                         0 
                       
                       * 
                       
                         ( 
                         
                           1 
                           - 
                           
                              
                             
                               
                                 A 
                                 
                                   C 
                                   thermal 
                                 
                               
                               * 
                               δ 
                                
                               
                                   
                               
                                
                               t 
                             
                           
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   e7 
                   ) 
                 
               
             
           
         
       
     
         [0069]    It is desirable that, when the temperature at the interface between the absorber  2  and the heat exchanger  32  reaches the temperature T(t 1 ) the engine should still be capable of operating, that is, T(t 1 )≧Tstart, to achieve the aim of guaranteeing stable operation of the solar converter at least for a time interval δt. 
         [0070]    From the formula (e7) the thermal capacity of the absorber is obtained, equal to 
         [0000]    
       
         
           
             
               
                 
                   
                     C 
                     thermal 
                   
                   = 
                   
                     
                       
                         - 
                         A 
                       
                       * 
                       δ 
                        
                       
                           
                       
                        
                       t 
                     
                     
                       ln 
                        
                       
                         ( 
                         
                           1 
                           - 
                           
                             
                               Δ 
                                
                               
                                   
                               
                                
                               T 
                             
                             
                               T 
                               0 
                             
                           
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   e8 
                   ) 
                 
               
             
           
         
       
     
         [0071]    Solely for the purpose of demonstrating that the equations shown lead to concrete results, an example is provided of preliminary functional data to dimensioning of the absorber: 
         [0000]    
       
         
           
             
               δ 
                
               
                   
               
                
               t 
             
             = 
             
               300 
                
               
                   
               
                
               sec 
                
               
                   
               
                
               
                 ( 
                 
                   5 
                    
                   
                       
                   
                    
                   min 
                 
                 ) 
               
             
           
         
       
       
         
           
             Psteady 
             = 
             
               4500 
                
               
                   
               
                
               Watt 
             
           
         
       
       
         
           
             Pstart 
             = 
             
               2550 
                
               
                   
               
                
               Watt 
             
           
         
       
       
         
           
             
               
                 T 
                 0 
               
               = 
               
                 530 
                  
                 ° 
                  
                 
                     
                 
                  
                 
                   C 
                   . 
                 
               
             
             ; 
           
         
       
       
         
           
             
               
                 Δ 
                  
                 
                     
                 
                  
                 T 
               
               = 
               
                 
                   
                     T 
                     0 
                   
                   - 
                   
                     
                       T 
                        
                       
                         ( 
                         
                           t 
                           1 
                         
                         ) 
                       
                     
                      
                     
                         
                     
                      
                     with 
                      
                     
                         
                     
                      
                     
                       T 
                       ( 
                       
                         t 
                         1 
                       
                       ) 
                     
                   
                 
                 = 
                 
                   Tstart 
                   ≅ 
                   
                     250 
                      
                     ° 
                      
                     
                         
                     
                      
                     
                       C 
                       . 
                     
                   
                 
               
             
             ; 
           
         
       
       
         
           
             A 
             = 
             
               
                 
                   Psteady 
                   - 
                   Pstart 
                 
                 
                   Δ 
                    
                   
                       
                   
                    
                   T 
                 
               
               = 
               
                 8.0 
                  
                 
                     
                 
                  
                 
                   Watt 
                   / 
                   ° 
                 
                  
                 
                     
                 
                  
                 
                   C 
                   . 
                 
               
             
           
         
       
     
         [0000]    with A representing the angular coefficient of the interpolating line Pabs, represented in  FIG. 2 , the thermal capacity of the heat absorber is obtained as C thermal ≈3000 J/° C. If, for example, copper is selected as the material of which the absorber  2  is composed, a mass of the absorber m absorber ≈8 Kg is obtained. Thanks to the present invention, a generator may remain stably operating for time intervals longer than 10 s in which the solar radiation is insufficient to guarantee powering of the Stirling engine: Pric&lt;Pabs+Ploss. 
         [0072]    From the experiments performed, the above-mentioned assumption, that is that the power absorbed by the thermodynamic converter is approximating by a linear curve of the type Pabs=AT was found to be verified. 
         [0073]    The secondary reflector  1  has a form, at least internally, tapering towards the head of the engine, defining a Y-axis of development, preferably having cylindrical symmetry. 
         [0074]    The secondary reflector  1  according to the present invention does not have the capacity to absorb radiation, but only to reflect it. 
         [0075]    Said reflector comprises a wall  11 , preferably planar, having a barycentric aperture  12 , through which enters the solar radiation concentrated by the primary reflector  4 . This closing wall  11  is situated at the level of the maximum transverse cross-section of the secondary reflector. 
         [0076]    Said reflector further comprises a second aperture  14  at the level of a minimum transverse cross-section thereof into which the absorber  2  is inserted. 
         [0077]    As will become clearer in what follows, the absorber  2  has a cylindrically symmetrical shape comprising a first base  21  and a second base  24 . In the second base  24 , a cavity is formed into which the head of the Stirling engine  31  is inserted. For this reason the second base defines a circular rim having an internal diameter d 24   —   i  and an external diameter d 24   —   e.    
         [0078]    The diameter of the front surface, defining the first base  21 , may vary according to the project characteristics, which will be better described below, between a diameter d 21  minimum and maximum. This is obviously analogous for the second base  24 . In particular the minimum diameter d 21  of the first base  21  and the internal diameter d 24   —   i  of the second base  24  must be greater than or equal to the diameter dtesta of the Stirling engine head. 
         [0079]    According to a particular aspect of the present invention, the wall  11  is located at the focus distance of reflection of the primary reflector. Therefore, the absorber  2  and accordingly the head of the engine  31  are out of focus, in particular beyond the focal distance F measured from the convex surface of the primary reflector  4 , that is to a distance greater than F on the axis of the primary reflector. 
         [0080]    The solar radiation therefore impinges upon the absorber  2 , becoming distributed at least over the whole of the front surface  21  thereof, resolving the problems of the known art. 
         [0081]    If the aperture  12  were arranged at a distance of less than F from the primary reflector, the aperture  12  would have to be larger to allow all the radiation to enter into the secondary reflector, involving a larger dispersion of radiation reflected by the absorber. 
         [0082]    Beyond the focal point, the radiation irradiated by the reflector  4  has a tendency to widen. To the extent to which the absorber  2  is out of focus, the radiation impinges only upon the absorber  2 , or also the internal surface  13  of the secondary reflector  1 . The latter reflects on to the lateral surfaces of the absorber  2  the portion of the radiation that has not impinged upon the absorber  2  directly. Therefore the absorber is illuminated directly and/or indirectly, after one or more reflections on to the internal surface  13  of the secondary reflector  1 . 
         [0083]    The inlet aperture  12 , preferably circular in shape and of diameter Dfmin≦dric≦DFmax, is preferably positioned at the focal distance F from the primary reflector  4  and has the objective of causing entry of the concentrated power Pric by reducing the thermal losses Ploss. 
         [0084]    DFmin and DFmax represent the maximum diameters of the focal region at the focal distance. These parameters depend on the care with which the primary reflector  4  has been manufactured. 
         [0085]    Advantageously, because the thermal losses are due to convective and radiative losses and are directly proportional to the area of the aperture  12 , by locating the wall  11 , comprising the aperture  12 , at the focal length F, a smallest possible aperture is obtained. 
         [0086]    A further advantage results from the fact that the aperture  12 , having a predefined diameter, allows filtration of excesses of illumination due to manufacturing imperfections of the primary reflector  4 , protecting the engine from overheating. According to a first variant of the invention, it is advantageous to insert thermocouples into apposite slits  23 , parallel with the lateral surfaces of the head of the Stirling engine  31 , for measuring the temperature of the interface between the absorber  2  and the heat exchanger  32  in such a way as to control in a closed loop the solar tracking system in order to obtain better centering of the solar radiation within the aperture  12 . 
         [0087]    A further advantage of inserting the further thermocouples into slits  23  arises from the fact that, since they are not directly exposed to solar radiation, the sensors are not subject to intense thermal stresses, thus increasing its average lifetime. 
         [0088]    The secondary reflector  1  is preferably realised with use of a compound parabolic concentrator (cpc) that is, a rotating solid shape which tapers starting from the wall  11  towards the Stirling engine  3 . In particular, the latter comprises at one end the wall  11  at the other end an aperture  14  wherein the absorber  2  is inserted. The latter, with reference  FIG. 3 , has a concentration ratio C equal to: 
         [0000]        C= 1/sin(θ cpc ) 2 =( D in/ D out) 2    (e9)
 
         [0000]    wherein θcpc, Din and Dout are respectively the angle of acceptance, the diameter of the wall  11 , and the diameter of the aperture  14  for insertion of the absorber  2 . In the absorber  2  is inserted the head  31  of the Stirling engine  3 . The secondary reflector  1  is arranged perpendicularly to a barycentric point at the parabolic surface of the primary reflector  4 , that is axially relative to the Y-axis. 
         [0089]    As mentioned above, the aperture  12  is positioned at a distance F, coincident with the focal length of the parabolic surface of the primary reflector  4 . 
         [0090]    The secondary reflector  1  preferably optimises the transfer of the concentrated solar radiation by reflecting internally and thus by efficiently reconcentrating, the sun rays entering from the aperture  12 , on to the heat absorber  2  positioned in direct contact with the head of the Stirling engine  31 . 
         [0091]    As will become clear below, the absorber  2 , preferably having cylindrical symmetry, comprises at least one preferably annular peripheral part  24 , which is inserted into the aperture  14  into which is inserted the absorber  2 , which aperture is formed in the secondary reflector, opposite the surface  21 . The head  31  of the Stirling engine is inserted into the cavity  25  formed in the absorber. 
         [0092]    Otherwise expressed, the absorber  2  behaves as a plug for the aperture  14  of the secondary reflector  1 , and in the absorber  2  a cavity  25  is formed into which the head  31  of the Stirling engine is inserted. In particular, the absorber  2  behaves as a cap, covering the head of the engine in the portion which, of said head, protrudes into the interior of the secondary reflector; stably connecting the head  31  to the secondary reflector  1 . 
         [0093]    According to the present invention, for dimensioning the secondary reflector θcpc Dout are selected in such a way that the following conditions must be satisfied simultaneously: 
         [0000]    
       
         
           
             
               θ 
                
               
                   
               
                
               refl 
                
               
                   
               
                
               1 
                
               ° 
             
             ≤ 
             
               θ 
                
               
                   
               
                
               cpc 
             
             &lt; 
             
               π 
               2 
             
           
         
       
       
         
           
             i.e. θcpc is greater than the angle of acceptance of the primary reflector θrefl1° and less than π/2, so as to direct all the sun rays reflected by the primary reflector towards the insertion aperture  14 ; 
             Dout≧that the diameter of the annular part  24 . 
           
         
       
     
         [0096]    Din is dependent upon θcpc and Dout according to the relationship Din=Dout/sin(θcpc). 
         [0097]    According to a preferred embodiment, it is necessary for θcpc=θrefl1° and for Dout to be approximately equal to the outer diameter d 24   —   e  and of the annular surface  24  of the absorber. 
         [0098]    Thus, Din=d 24   —   e /sin(θrefl1°). 
         [0099]    The secondary reflector  1  is a three-dimensional object achieved by rotation of the section represented in  FIG. 3  about the Y-axis. 
         [0100]    The angle of acceptance θrefl1° of the primary reflector  4  is obtained from the intersection of the development axis Y with a line R 1  passing through the end of the primary reflector and the centre of the aperture  12  for entry of the radiation reflected by the primary reflector ( 4 ), see  FIG. 3 . 
         [0101]    The angle of acceptance θcpc is obtained from the intersection of the Y-axis with a line R 2  passing through two opposing points relative to the development axis Y, these points being arranged at opposite ends of the rotation solid defining the secondary reflector: see  FIG. 3 . 
         [0102]    The shape of the section is described by the following two parametric equations, expressed as a function of the unknown variables x and y respectively determined perpendicularly and on said Y-axis, with the origin O, as a function of the common angular parameter φ: 
         [0000]    
       
         
           
             
               
                 
                   x 
                   = 
                   
                     
                       
                         2 
                         * 
                         z 
                         * 
                         
                           sin 
                            
                           
                             ( 
                             
                               ϕ 
                               - 
                               
                                 θ 
                                  
                                 
                                     
                                 
                                  
                                 cpc 
                               
                             
                             ) 
                           
                         
                       
                       
                         1 
                         - 
                         
                           cos 
                            
                           
                             ( 
                             ϕ 
                             ) 
                           
                         
                       
                     
                      
                     
                       Dout 
                       2 
                     
                   
                 
               
               
                 
                   ( 
                   e10 
                   ) 
                 
               
             
             
               
                 
                   y 
                   = 
                   
                     
                       2 
                       * 
                       z 
                       * 
                       
                         cos 
                          
                         
                           ( 
                           
                             ϕ 
                             - 
                             
                               θ 
                                
                               
                                   
                               
                                
                               cpc 
                             
                           
                           ) 
                         
                       
                     
                     
                       1 
                       - 
                       
                         cos 
                          
                         
                           ( 
                           ϕ 
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   e11 
                   ) 
                 
               
             
             
               
                 where 
               
               
                 
                     
                 
               
             
             
               
                 
                   z 
                   = 
                   
                     
                       Dout 
                       2 
                     
                     * 
                     
                       ( 
                       
                         1 
                         + 
                         
                           sin 
                            
                           
                             ( 
                             
                               θ 
                               cpc 
                             
                             ) 
                           
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   e12 
                   ) 
                 
               
             
             
               
                 
                   
                     2 
                     * 
                     θ 
                      
                     
                         
                     
                      
                     cpc 
                   
                   ≤ 
                   ϕ 
                   ≤ 
                   
                     
                       π 
                       2 
                     
                     + 
                     
                       θ 
                        
                       
                           
                       
                        
                       cpc 
                     
                   
                 
               
               
                 
                   ( 
                   e13 
                   ) 
                 
               
             
           
         
       
     
         [0103]    The parametric functions x,y and φ are known from the art. The present invention teaches the use thereof in the specific case of a solar generator by introducing the parameter θcpc. These equations are therefore used in calculations, particularly in Matlab™, TracePro™ and Comsol™, for the purpose of simulating and understanding the optical, thermal and mechanical dynamics of the solar generator. 
         [0104]    Having defined the shape of the secondary reflector  1 , its maximum length Lcpc_max is 
         [0000]    
       
         
           
             
               
                 
                   Lcpc_max 
                   = 
                   
                     
                       
                         Dout 
                         + 
                         Din 
                       
                       
                         tan 
                          
                         
                           ( 
                           
                             θ 
                              
                             
                                 
                             
                              
                             cpc 
                           
                           ) 
                         
                       
                     
                     . 
                   
                 
               
               
                 
                   ( 
                   e14 
                   ) 
                 
               
             
           
         
       
     
         [0105]    A secondary reflector  1  having axial development equal to Lcpc_max is represented in  FIG. 4(A) . 
         [0106]    According to the present invention it is possible to limit the length of the axial development of the secondary reflector  1  without diminishing its the optical efficiency: that is, if Din, Dout and θcpc are equal, it is possible to obtain reflectors with lengths Lcpc less than 
         [0000]    
       
         
           
             
               
                 D 
                  
                 
                     
                 
                  
                 out 
               
               + 
               
                 D 
                  
                 
                     
                 
                  
                 in 
               
             
             
               tan 
                
               
                 ( 
                 
                   θ 
                    
                   
                       
                   
                    
                   cpc 
                 
                 ) 
               
             
           
         
       
     
         [0000]    of comparable optical efficiency. 
         [0107]    When the length of the axial development Lcpc of the secondary reflector is reduced, the number of reflections has to be limited, possibly by designing the reflector and the absorber in such a way that there are no reflections. 
         [0108]    In this case the following equation must be verified: 
         [0000]        L&lt;Lcpc≦L+Lh    (e15)
 
         [0000]    where, with reference to  FIG. 5 ,
   L is the axial length of the heat absorber, Lcpc is the axial length of the secondary reflector, and Lh is the difference between the first two:   
 
         [0000]        Lh=Lcpc−L    (e16)
 
         [0110]    So in order for the absorber to be illuminated without reflections the diameter d 21  of the front surface  21 , the following must be satisfied: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       d 
                        
                       
                           
                       
                        
                       21 
                     
                     2 
                   
                   = 
                   
                     Lh 
                     * 
                     
                       tan 
                        
                       
                         ( 
                         
                           θ 
                            
                           
                               
                           
                            
                           refl 
                            
                           
                               
                           
                            
                           1 
                            
                           ° 
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   e17 
                   ) 
                 
               
             
           
         
       
     
         [0111]    Furthermore, because the absorber  2  is inserted into the interior of the secondary reflector  1 , the maximum outer diameter d 21 , which the front surface  21  can assume, is delimited by the internal width of the secondary reflector  1  corresponding to the axial length of the absorber  2 . In particular, using the parametric formula (e11), the axial length of the absorber is expressed as 
         [0000]    
       
         
           
             
               
                 
                   L 
                   = 
                   
                     
                       2 
                       * 
                       z 
                       * 
                       
                         cos 
                          
                         
                           ( 
                           
                             
                               ϕ 
                               L 
                             
                             - 
                             
                               θ 
                                
                               
                                   
                               
                                
                               cpc 
                             
                           
                           ) 
                         
                       
                     
                     
                       1 
                       - 
                       
                         cos 
                          
                         
                           ( 
                           
                             ϕ 
                             L 
                           
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   e18 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where φ L  is the parametric angle φ calculated in correspondence with the height L of the absorber according to the equation (e11). 
         [0112]    The beam of the secondary reflector  1  corresponding to an axial length L of the absorber is then x L =d 21 /2, and using the parametric formulae (e10) and (e11), the maximum possible dimensions of the surface  21  are obtained: 
         [0000]        d 21=2* L *tan(φ L   −θcpc )− D out   (e19)
 
         [0113]    This equation defines the maximum value of Lh, on account of which direct illumination reaches only the absorber, and so substituting (e19) into (e17) we have: 
         [0000]    
       
         
           
             
               
                 
                   Lh 
                   = 
                   
                     
                       
                         L 
                         * 
                         
                           tan 
                            
                           
                             ( 
                             
                               
                                 ϕ 
                                 L 
                               
                               - 
                               
                                 θ 
                                  
                                 
                                     
                                 
                                  
                                 cpc 
                               
                             
                             ) 
                           
                         
                       
                       - 
                       
                         D 
                          
                         
                             
                         
                          
                         
                           out 
                           / 
                           2 
                         
                       
                     
                     
                       tan 
                        
                       
                         ( 
                         
                           θ 
                            
                           
                               
                           
                            
                           refl 
                            
                           
                               
                           
                            
                           1 
                            
                           ° 
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   e20 
                   ) 
                 
               
             
           
         
       
     
         [0114]    Then, substituting (e20) into (e15), two limit values which Lcpc can assume are obtained, for the precise purpose of the absorber being irradiated only on the front surface  21 , as illustrated by  FIG. 4(B) : 
         [0000]    
       
         
           
             
               
                 
                   L 
                   &lt; 
                   
                     L 
                      
                     
                         
                     
                      
                     cpc 
                   
                   ≤ 
                   
                     L 
                     + 
                     
                       
                         
                           L 
                           * 
                           tan 
                            
                           
                             ( 
                             
                               
                                 ϕ 
                                 L 
                               
                               - 
                               
                                 θ 
                                  
                                 
                                     
                                 
                                  
                                 cpc 
                               
                             
                             ) 
                           
                         
                         - 
                         
                           D 
                            
                           
                               
                           
                            
                           
                             out 
                             / 
                             2 
                           
                         
                       
                       
                         tan 
                          
                         
                           ( 
                           
                             θ 
                              
                             
                                 
                             
                              
                             refl 
                              
                             
                                 
                             
                              
                             1 
                              
                             ° 
                           
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   e21 
                   ) 
                 
               
             
           
         
       
     
         [0115]    Accordingly, a limit length Lcpc_lim of the secondary reflector is identified as being equal to: 
         [0000]    
       
         
           
             Lcpc_lim 
             = 
             
               L 
               + 
               
                 
                   
                     L 
                     * 
                     tan 
                      
                     
                       ( 
                       
                         
                           ϕ 
                           L 
                         
                         - 
                         
                           θ 
                            
                           
                               
                           
                            
                           cpc 
                         
                       
                       ) 
                     
                   
                   - 
                   
                     D 
                      
                     
                         
                     
                      
                     
                       out 
                       / 
                       2 
                     
                   
                 
                 
                   tan 
                    
                   
                     ( 
                     
                       θ 
                        
                       
                           
                       
                        
                       refl 
                        
                       
                           
                       
                        
                       1 
                        
                       ° 
                     
                     ) 
                   
                 
               
             
           
         
       
     
         [0116]    Thus, when Lcpc≦Lcpc_lim, the absorber has a shape which tapers towards the Stirling engine. 
         [0117]    On the other hand, when Lcpc exceeds the value Lcpc_lim, the absorber tapers towards the primary reflector. In this case 
         [0000]    
       
         
           
             
               
                 
                   Lcpc_lim 
                   &lt; 
                   Lcpc 
                   ≤ 
                   
                     
                       
                         D 
                          
                         
                             
                         
                          
                         out 
                       
                       + 
                       
                         D 
                          
                         
                             
                         
                          
                         in 
                       
                     
                     
                       tan 
                        
                       
                         ( 
                         
                           θ 
                            
                           
                               
                           
                            
                           cpc 
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   e22 
                   ) 
                 
               
             
           
         
       
     
         [0000]    wherein the irradiation of the heat absorber  2  occurs as follows: a portion of the solar rays reflected by the primary reflector  4  impinge directly upon the absorber  2 , while another portion impinges upon it by means of at least one reflection on the secondary reflector  1 , as shown in  FIG. 4(A) . 
         [0118]    Then, having defined the length Lcpc of the secondary reflector  1 , two different ways of irradiating the absorber  2  may be obtained. This comprises variations in the shape of the absorber  2 , as will be illustrated in what follows. 
         [0119]    Once the length of the axial development Lcpc of the secondary reflector  1  has been selected, the shape of the absorber must be optimised for the purpose of
       optimising the absorption of the radiation on one or more surfaces of the absorber;   transferring the absorbed energy, in the form of heat, from said surfaces to the zone of contact with the heat exchanger of the Stirling engine which, in the example, has an annular shape.       
 
         [0122]    Otherwise it is necessary to guarantee that the thickness of the absorber is not reduced too much in any areas, to enable a sufficient transmission of heat from the uptake surfaces towards the zone in which the heat is given up to the exchanger  32 , in order that the Stirling engine  3  is able to operate correctly. This constraint is in contrast with the need to make the absorber as compact as possible to reduce the thermal losses thereof. 
         [0123]    According to the present invention, following intensive experimentation, the following further design constraints have come to light with regard to the absorber dimensions:
       with reference to  FIG. 1 , it would be desirable for the thickness s of the absorber on the head of the engine to be the minimum possible, so as to minimise the heat conduction pathway to the peripheral part of the heat exchanger ( 32 ) of the engine head. In this regard, it has been observed that this thickness s cannot be small ad libitum, since this would cause localised overheating of the engine head;   with reference to  FIG. 1 , the insertion part of the absorber, destined to come into contact with the peripheral part of the heat exchanger ( 32 ) of the engine head, must be of a sufficient thickness to guarantee low resistance to the passage of the heat flow towards the engine.       
 
         [0126]    This dimensioning is further constrained by the thermal capacity of the absorber, that is, by its mass as calculated above. 
         [0127]    As mentioned above, it is assumed that the minimum diameter d 21  of the first base  21  and the internal diameter d 24   —   i  of the circular rim  24  is greater than or equal to the diameter (testa of the engine head.
   (e19) indicates the possible value of the maximum diameter of the first base  21 , but the outer diameter d 24   —   e  of the second base  24  remains to be ascertained.   
 
         [0129]      FIG. 6  represents an axial section of a variant heat absorber that is the subject of the present invention. 
         [0130]    The general equation of thermal conduction (Fourier&#39;s law) is taken into consideration. The present invention also teaches how to use such an equation and how to select the parameters to be substituted into said equation: 
         [0000]    
       
         
           
             
               
                 
                   W 
                   = 
                   
                     λ 
                     × 
                     
                       
                         ( 
                         
                           
                             T 
                              
                             
                                 
                             
                              
                             irr 
                           
                           - 
                           
                             T 
                              
                             
                                 
                             
                              
                             int 
                           
                         
                         ) 
                       
                       L 
                     
                     × 
                     S 
                   
                 
               
               
                 
                   ( 
                   
                     e 
                      
                     
                         
                     
                      
                     23 
                   
                   ) 
                 
               
             
           
         
       
     
         [0000]    wherein:
       W is the power transmitted from the zone exposed to solar radiation to the zone in direct contact with the heat exchanger of the Stirling;   λ is the thermal conductibility of the material of the absorber, for example copper, and is known;   Tirr is the mean temperature of the surface exposed to solar radiation;   Tint is the mean temperature of the interface between the absorber and the heat exchanger of the engine cylinder;   S represents the area of the circular rim defined by the second base  24  and by the cavity included therein, via which the heat exchanger  32  of the head  31  of the Stirling engine  3  is contacted;   L is the height of the heat absorber;       (e23) is resolved analytically and in a closed form by stipulating the following:
       Tint is equal to Tsteady,   L is at least equal to the distance between the vault of the cylinder  31  and the position of the annular heat exchanger  32  of the Stirling engine, yet can be assimilated to the portion of the absorber which protrudes into the interior of the secondary reflector.   Tirr is set as equal to Tmax, the maximum temperature that the preselected material, for example copper, is able to withstand without sustaining damage.   
       
 
         [0141]    The specific parameters of the engine have the following values, with specific reference to the above-mentioned Microgen™ engine:
       W=4500 Watt, i.e. equal to the steady-state power Psteady   λ=400 W/m*K   L=80 mm,   Tint=823° K (550° C.)   Tmax for copper is 1100° K (827° C.) and       
 
         [0000]    
       
         
           
             S 
             = 
             
               
                 ( 
                 
                   
                     
                       d 
                        
                       
                           
                       
                        
                       24 
                        
                       
                         _e 
                         2 
                       
                     
                     - 
                     
                       d 
                        
                       
                           
                       
                        
                       test 
                        
                       
                           
                       
                        
                       
                         a 
                         2 
                       
                     
                   
                   4 
                 
                 ) 
               
                
               π 
             
           
         
       
     
         [0000]    which is the area of the circular rim S, that is, the exchange surface between the absorber  2  and the heat exchanger  32  of the Stirling engine, for which the diameter dtesta of the head  31  of the Stirling engine  3  is dtesta=116 mm. 
         [0147]    Therefore, by substituting the above-mentioned parameters into the equation (e23), the minimum diameter d 24   —   e  for transmitting the steady-state power to the Stirling engine is calculated as: 
         [0000]    
       
         
           
             
               
                 
                   
                     d 
                      
                     
                         
                     
                      
                     24 
                      
                     _e 
                   
                   = 
                   
                     
                       2 
                       × 
                       
                         
                           
                             
                               W 
                               × 
                               L 
                             
                             
                               
                                 λ 
                                  
                                 
                                   ( 
                                   
                                     
                                       T 
                                        
                                       
                                           
                                       
                                        
                                       max 
                                     
                                     - 
                                     
                                       T 
                                        
                                       
                                           
                                       
                                        
                                       int 
                                     
                                   
                                   ) 
                                 
                               
                               × 
                               π 
                             
                           
                           + 
                           
                             
                               d 
                                
                               
                                   
                               
                                
                               test 
                                
                               
                                   
                               
                                
                               
                                 a 
                                 2 
                               
                             
                             4 
                           
                         
                       
                     
                     ≈ 
                     
                       130 
                        
                       
                           
                       
                        
                       mm 
                     
                   
                 
               
               
                 
                   ( 
                   
                     e 
                      
                     
                         
                     
                      
                     24 
                   
                   ) 
                 
               
             
           
         
       
     
         [0148]    By this means, having selected a thermal capacity using (e8) and then the material constituting the absorber, the mass thereof is obtained and the front surface  21  defining the first base of the absorber is connected to the second base  24  of the absorber, yielding a cylindrically symmetrical solid wherein the first base  21  is planar and has a diameter d 21  and the second base is substantially a circular rim, due to the cavity formed therein for the purpose of inserting the head  31  of the Stirling engine  3 . 
         [0149]    In particular, this cylindrically symmetrical solid tapers from the first base  21  towards the second base  24  when the secondary reflector has axial development included in the limits defined by (e21). This first base  21  assumes a diameter of a value comprised between those obtained by (e17) and by (e19). Having stipulated the mass and the height of the absorber and having selected the diameter d 21  from among those just cited, d 24   —   e  must assume a dimension smaller than d 21  by taking on the shape described above. 
         [0150]    In contrast, this cylindrically symmetrical solid tapers from the second base  24  towards the first base  21  when the absorber has axial development comprised within the limits of (e22); e d 24   —   e  assumes a value greater than or equal to that obtained by mean of (e24). Because the mass and the height L of the absorber are stipulated and the value d 24   —   e,  the diameter d 21  is obtained as a consequence of this limitation of the mass of the absorber. It is precisely this that stipulates that the absorber in this second example tapers from the second base  24  towards the first  21 . 
         [0151]    According to an axial section of the absorber, said absorber may therefore present rectilinear, concave or convex lateral profiles  22  in relation to the above-mentioned connecting operation. 
         [0152]    According to one embodiment, the heat absorber can be covered with a flexible film of material having selective properties capable of increasing the absorption of solar radiation and thus reducing the thermal emissions due to heating of the absorber. The use of this covering therefore enables the radiative losses of the secondary reflector to be appreciably reduced. 
         [0153]    The secondary reflector  1  is preferably internally covered with a reflective surface which may preferably be glass or another reflective material with specular reflectance at least greater than 90% and resistant to temperatures above 200° C. Furthermore the secondary reflector  1  is externally covered with a refractory material so as to minimise convective and radiative thermal losses of the secondary reflector. 
         [0154]    The advantages deriving from application of the present invention are clear:
       With the use of the present invention it is made possible to improve the operation of a solar generator directly exposed to solar radiation, which generator is based on a Stirling engine, the head of which is exposed to the solar radiation with use of a suitable absorber.   The absorber protects the cylinder head, guaranteeing to maintain the temperature of the head within the limits of operation of the engine, the present invention teaches to confer a suitable shape to the absorber depending on the shape of the secondary reflector.   The present invention teaches how to dimension the thermal capacitor of the heat absorber in such a way that it succeeds both in storing the energy necessary for starting the Stirling engine in operation, and in supplying the heat necessary to maintain the Stirling engine in operation, even when, due to the concentrated solar radiation, the amount of heat is not sufficient for optimum operation of said engine;   The present invention also teaches to dimension the secondary reflector in such a way as to optimise the operation thereof,   The present invention also teaches a pre-existing Stirling engine, originally used for production of energy from the combustion of natural gas, teaching to obtain the characteristic curves thereof and to use them according to the purposes of the present invention.   Also presented are an analytical method for dimensioning absorbers, and a practical example of application of the method.