Abstract:
A volumetric absorption solar collector in combination with a conventional flat panel collector. Addition of the volumetric absorption solar collector to the conventional flat panel collector allows the overall solar collection system to reach higher collection temperatures, increase collection efficiency, and reduce ecological footprint.

Description:
CROSS-REFERENCE TO RELATED APPLICATIONS 
       [0001]    This nonprovisional application is a continuation of and claims priority to provisional application No. 61/642,182, entitled “Solar Collection Enhancement by Volumetric Absorption”, filed May 3, 2012 by the same inventors. 
     
    
     BACKGROUND OF THE INVENTION 
       [0002]    1. Field of the Invention 
         [0003]    This invention relates, generally, to solar energy. More particularly, it relates to volumetric absorption solar collectors with increased effectiveness and efficiency. 
         [0004]    2. Description of the Prior Art 
         [0005]    Multiple solar collection technologies exist. However, the conventional art does not make use of volumetric absorption solar collectors (“VASC”), which are semitransparent solids that act as an insulation layer that lowers energy losses to the ambient while allowing for the radiation to reach the desired element. Thus, conventional technologies experience significant energy losses to the ambient, which increases costs, decreases efficiency, and reduces effectiveness. 
         [0006]    Accordingly, what is needed is a volumetric absorption solar collector. However, in view of the art considered as a whole at the time the present invention was made, it was not obvious to those of ordinary skill how the art could be advanced. 
         [0007]    While certain aspects of conventional technologies have been discussed to facilitate disclosure of the invention, Applicants in no way disclaim these technical aspects, and it is contemplated that the claimed invention may encompass one or more of the conventional technical aspects discussed herein. 
         [0008]    The present invention may address one or more of the problems and deficiencies of the prior art discussed above. However, it is contemplated that the invention may prove useful in addressing other problems and deficiencies in a number of technical areas. Therefore, the claimed invention should not necessarily be construed as limited to addressing any of the particular problems or deficiencies discussed herein. 
         [0009]    In this specification, where a document, act or item of knowledge is referred to or discussed, this reference or discussion is not an admission that the document, act or item of knowledge or any combination thereof was at the priority date, publicly available, known to the public, part of common general knowledge, or otherwise constitutes prior art under the applicable statutory provisions; or is known to be relevant to an attempt to solve any problem with which this specification is concerned. 
       SUMMARY OF THE INVENTION 
       [0010]    The long-standing but heretofore unfulfilled need for an improved, more effective and more efficient solar collector is now met by a new, useful and nonobvious invention. 
         [0011]    A significant advantage of the current invention is that it can be used with current solar collection technologies. 
         [0012]    Certain embodiments of the current invention also allow present solar collection technologies to reach higher collection temperatures, increase collection efficiency, and/or reduce ecological footprint. 
         [0013]    Novel and unusual features of the present invention include non-intuitive inclusion of a semitransparent solid layer and compatibility with solar collector fabrication methods. 
         [0014]    These and other important objects, advantages, and features of the invention will become clear as this disclosure proceeds. 
         [0015]    The invention accordingly comprises the features of construction, combination of elements, and arrangement of parts that will be exemplified in the disclosure set forth hereinafter and the scope of the invention will be indicated in the claims. 
     
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS  
         [0016]    For a fuller understanding of the nature and objects of the invention, reference should be made to the following detailed disclosure, taken in connection with the accompanying drawings, in which: 
           [0017]      FIG. 1  depicts temperature values of a VASC mounted on an FPC compared to a VASC individually and FPC individually; 
           [0018]      FIG. 2  depicts a VASC-FPC layout according to an embodiment of the current invention; 
           [0019]      FIG. 3  depicts a graphical illustration of the variation of absorptance ratio with respect to the depth of the VASC; 
           [0020]      FIG. 4  depicts a graphical illustration of the variation of thermal resistance ratio with respect to the depth of the VASC; 
           [0021]      FIG. 5  depicts a graphical illustration comparing the efficiency of a FPC with different VASC-FPC systems; 
           [0022]      FIG. 6  depicts a graphical illustration comparing the efficiency of an advanced FPC with various VASC-FPC systems; and 
           [0023]      FIG. 7  depicts a graphical illustration comparing price levels for different optical glasses. 
       
    
    
     DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENT 
       [0024]    In the following detailed description of the preferred embodiments, reference is made to the accompanying drawings, which form a part thereof, and within which are shown by way of illustration specific embodiments by which the invention may be practiced. It is to be understood that other embodiments may be utilized and structural changes may be made without departing from the scope of the invention. 
         [0025]    It was proven that the volumetric absorption solar collector has the potential to accumulate significant amounts of heat and therefore it can be used for applications such as water heating, space heating and all other applications for which flat plate collectors (“FPCs”) are typically used. In different cases, the analysis showed that a VASC system allows for more heat to be extracted than the FPC and is therefore thermodynamically superior. 
         [0026]    An example application of the VASC is to use it in conjunction with the FPC. In this configuration, a well selected depth of the VASC could be mounted on the absorber surface of the FPC. The radiation received by the volumetric absorption solar collector would be collected and then transferred to the FPC&#39;s absorber surface. This would increase the temperature values that can normally be achieved by using the conventional FPC alone, as depicted in  FIG. 1 . The sun&#39;s radiation hits the surface of the VASC at ambient temperature and increase temperature of the collector at the bottom by “n” times the ambient temperature. The same happens with the FPC, where the temperature is increased by “m” times. When combined, the overall increase of temperature is “m+n”. 
         [0027]    The process is, however, not as straightforward as it might look. In reality, there are different kinds of heat losses, reflection losses, and convection losses, among other types of energy loss. This would make the overall increase of temperature not “m+n” but a fraction of “m+n” that is still greater than “m” and “n” individually. 
       EXAMPLE 
       [0028]    The following analysis is a mathematical study of real case where a VASC is used in conjunction with a FPC and a comparison of the performance of the system with the performance of the FPC individually is made. 
         [0029]    A. Heat Transfer Analysis of a VASC-FPC System 
         [0030]    This configuration takes a conventional FPC and mounts a layer of glass (VASC) on top of the absorber plate. All other factors remain constant. A layout of this system is shown in  FIG. 2 . An incoming solar radiation I C  hits the glazing and a portion of it, I cτs , is transmitted to the VASC. After a series of internal and external reflections, R, I cτs e −μL (1−R) 2  reaches the absorber plate. The quantity I cτs e −μL (1−R) 2 α s  represents the amount of heat absorbed in the plate. The term 1/U c A c  represents the thermal resistance between the VASC and the ambient temperature. 
         [0031]    Applying the conservation of energy principle to the VASC, it was observed 
         [0000]    
       
         
           
             
               
                 
                   
                     k 
                      
                     
                       
                         
                            
                           2 
                         
                          
                         T 
                       
                       
                          
                         
                           x 
                           2 
                         
                       
                     
                   
                   = 
                   
                     
                        
                       
                         ( 
                         
                           
                             l 
                             c 
                           
                            
                           
                             
                               τ 
                               s 
                             
                              
                             
                               ( 
                               
                                 1 
                                 - 
                                 R 
                               
                               ) 
                             
                           
                            
                           
                              
                             
                               
                                 - 
                                 μ 
                               
                                
                               
                                   
                               
                                
                               x 
                             
                           
                         
                         ) 
                       
                     
                     
                        
                       x 
                     
                   
                 
               
               
                 
                   ( 
                   1 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where I c  is the incident radiation hitting the glazing surface, τ s  is the reflection coefficient and R is the reflected radiation. This is similar to Q*=I cτs (1−R). 
         [0032]    Applying the boundary conditions at x=0 and at x=L, 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         - 
                         k 
                       
                        
                       
                         
                            
                           T 
                         
                         
                            
                           x 
                         
                       
                     
                      
                     
                       | 
                       
                         x 
                         = 
                         0 
                       
                     
                   
                   = 
                   
                     - 
                     
                       
                         U 
                         c 
                       
                        
                       
                         ( 
                         
                           T 
                            
                           
                             | 
                             
                               x 
                               = 
                               0 
                             
                           
                            
                           
                             - 
                             
                               T 
                               ∞ 
                             
                           
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   2 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       
                         
                           I 
                           C 
                         
                          
                         
                           
                             τ 
                             s 
                           
                            
                           
                             ( 
                             
                               1 
                               - 
                               R 
                             
                             ) 
                           
                         
                          
                         
                           
                              
                             
                               
                                 - 
                                 μ 
                               
                                
                               
                                   
                               
                                
                               L 
                             
                           
                            
                           
                             ( 
                             
                               1 
                               - 
                               R 
                             
                             ) 
                           
                         
                          
                         
                           α 
                           s 
                         
                       
                       - 
                       
                         k 
                          
                         
                           
                              
                             T 
                           
                           
                              
                             x 
                           
                         
                       
                     
                      
                     
                       | 
                       
                         x 
                         = 
                         L 
                       
                     
                   
                   = 
                   
                     Q 
                     u 
                   
                 
               
               
                 
                   ( 
                   3 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where U c  is the equivalent convection coefficient between the glass material and the ambient, and α s  is the absorptance of the glass. 
         [0033]    The equation can be solved to arrive at the final form of the heat conducted in the absorber plate, which is 
         [0000]        Q=Q   u   +Q′e   −μL (1−α s (1− R ))  (4)
 
         [0000]    where Q is the total heat reaching the bottom of the glass, Q* is the radiation entering the glass material, and Q u  is the heat extracted and ready to be used. 
         [0034]    a. The Efficiency of the System 
         [0035]    For the VASC component of the system, 
         [0000]    
       
         
           
             
               
                 
                   η 
                   = 
                   
                     
                       Q 
                       
                         Q 
                         * 
                       
                     
                     = 
                     
                       
                         
                           Q 
                           u 
                         
                         + 
                         
                           
                             Q 
                             * 
                           
                            
                           
                             
                                
                               
                                 
                                   - 
                                   μ 
                                 
                                  
                                 
                                     
                                 
                                  
                                 L 
                               
                             
                              
                             
                               ( 
                               
                                 1 
                                 - 
                                 
                                   
                                     α 
                                     s 
                                   
                                    
                                   
                                     ( 
                                     
                                       1 
                                       - 
                                       R 
                                     
                                     ) 
                                   
                                 
                               
                               ) 
                             
                           
                         
                       
                       
                         Q 
                         * 
                       
                     
                   
                 
               
               
                 
                   ( 
                   5 
                   ) 
                 
               
             
           
         
       
     
         [0036]    The overall efficiency of the system VASC-FPC is the ratio between the energy extracted from the absorber plate Q u  and the incident radiation that hits the top surface I c . 
         [0000]    
       
         
           
             
               
                 
                   
                     η 
                     
                       vasc 
                       - 
                       fpc 
                     
                   
                   = 
                   
                     
                       
                         Q 
                         u 
                       
                       
                         I 
                         c 
                       
                     
                     = 
                     
                       
                         
                           Q 
                           u 
                         
                         
                           Q 
                           * 
                         
                       
                        
                       
                         
                           τ 
                           s 
                         
                          
                         
                           ( 
                           
                             1 
                             - 
                             R 
                           
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   6 
                   ) 
                 
               
             
           
         
       
     
         [0037]    Combining Equations (5) and (6) with Q*, the overall efficiency of a combined VASC-FPC as a function of the temperature difference can be found to be 
         [0000]    
       
         
           
             
               
                 
                   
                     η 
                     
                       vasc 
                       - 
                       fpc 
                     
                   
                   = 
                   
                     
                       
                         
                           τ 
                           s 
                         
                          
                         
                           ( 
                           
                             1 
                             - 
                             R 
                           
                           ) 
                         
                       
                        
                       
                         [ 
                         
                           
                             
                               A 
                               - 
                               
                                 A 
                                  
                                 
                                     
                                 
                                  
                                 
                                    
                                   ɛ 
                                 
                               
                               + 
                               B 
                             
                             
                               
                                 A 
                                  
                                 
                                     
                                 
                                  
                                 ɛ 
                               
                               + 
                               B 
                             
                           
                           - 
                           
                             
                                
                               
                                 - 
                                 ɛ 
                               
                             
                              
                             
                               ( 
                               
                                 1 
                                 - 
                                 
                                   
                                     α 
                                     s 
                                   
                                    
                                   
                                     ( 
                                     
                                       1 
                                       - 
                                       R 
                                     
                                     ) 
                                   
                                 
                               
                               ) 
                             
                           
                         
                         ] 
                       
                     
                     - 
                     
                       
                         
                           
                             τ 
                             s 
                           
                            
                           
                             ( 
                             
                               1 
                               - 
                               R 
                             
                             ) 
                           
                         
                         
                           
                             ( 
                             
                               
                                 A 
                                  
                                 
                                     
                                 
                                  
                                 ɛ 
                               
                               + 
                               B 
                             
                             ) 
                           
                            
                           
                             T 
                             ∞ 
                           
                         
                       
                        
                       
                         ( 
                         
                           
                             T 
                             B 
                           
                           - 
                           
                             T 
                             ∞ 
                           
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   7 
                   ) 
                 
               
             
           
         
       
     
         [0000]    Equation (7) is a straight line in the ρ−ΔT plane. 
         [0038]    If the effects of reflections are disregarded and the ε=0, Equation (7) is reduced to 
         [0000]    
       
         
           
             
               
                 
                   
                     η 
                     fpc 
                   
                   = 
                   
                     
                       
                         τ 
                         s 
                       
                        
                       
                         α 
                         s 
                       
                     
                     - 
                     
                       
                         
                           U 
                           c 
                         
                         
                           I 
                           c 
                         
                       
                        
                       
                         ( 
                         
                           
                             T 
                             B 
                           
                           - 
                           
                             T 
                             ∞ 
                           
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   8 
                   ) 
                 
               
             
           
         
       
     
         [0000]    which is the normal equation for the efficiency as a function of the temperature difference between the plate and the ambient for a common FPC. 
         [0039]    The efficiency can also be expressed in terms of fixed losses (F L ) and variable losses (V L ) as follows: 
         [0000]      η=1− F   L   −V   L   (9)
 
         [0000]    For a common FPC 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       F 
                       L 
                     
                     = 
                     
                       1 
                       - 
                       
                         
                           τ 
                           s 
                         
                          
                         
                           α 
                           s 
                         
                       
                     
                   
                    
                   
                     
 
                   
                    
                   and 
                 
               
               
                 
                   ( 
                   
                     9 
                      
                     a 
                   
                   ) 
                 
               
             
             
               
                 
                   
                     V 
                     L 
                   
                   = 
                   
                     
                       
                         U 
                         c 
                       
                       
                         I 
                         c 
                       
                     
                      
                     
                       ( 
                       
                         
                           T 
                           B 
                         
                         - 
                         
                           T 
                           ∞ 
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   
                     9 
                      
                     b 
                   
                   ) 
                 
               
             
           
         
       
     
         [0000]    and for the VASC-FPC system 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       F 
                       L 
                     
                     = 
                     
                       1 
                       - 
                       
                         
                           
                             τ 
                             s 
                           
                            
                           
                             ( 
                             
                               1 
                               - 
                               R 
                             
                             ) 
                           
                         
                          
                         
                           [ 
                           
                             
                               
                                 A 
                                 - 
                                 
                                   A 
                                    
                                   
                                       
                                   
                                    
                                   
                                      
                                     ɛ 
                                   
                                 
                                 + 
                                 B 
                               
                               
                                 
                                   A 
                                    
                                   
                                       
                                   
                                    
                                   ɛ 
                                 
                                 + 
                                 B 
                               
                             
                             - 
                             
                               
                                  
                                 
                                   - 
                                   ɛ 
                                 
                               
                                
                               
                                 ( 
                                 
                                   1 
                                   - 
                                   
                                     
                                       α 
                                       s 
                                     
                                      
                                     
                                       ( 
                                       
                                         1 
                                         - 
                                         R 
                                       
                                       ) 
                                     
                                   
                                 
                                 ) 
                               
                             
                           
                           ] 
                         
                       
                     
                   
                    
                   
                     
 
                   
                    
                   and 
                 
               
               
                 
                   ( 
                   
                     9 
                      
                     c 
                   
                   ) 
                 
               
             
             
               
                 
                   
                     V 
                     L 
                   
                   = 
                   
                     
                       
                         
                           τ 
                           s 
                         
                          
                         
                           ( 
                           
                             1 
                             - 
                             R 
                           
                           ) 
                         
                       
                       
                         
                           ( 
                           
                             
                               A 
                                
                               
                                   
                               
                                
                               ɛ 
                             
                             + 
                             B 
                           
                           ) 
                         
                          
                         
                           T 
                           ∞ 
                         
                       
                     
                      
                     
                       ( 
                       
                         
                           T 
                           B 
                         
                         - 
                         
                           T 
                           ∞ 
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   
                     9 
                      
                     d 
                   
                   ) 
                 
               
             
           
         
       
     
         [0040]    Finally, the efficiency can be expressed in terms of the temperature difference between the fluid entering the collector and the ambient temperature, as it is usually done for the flat plate collectors. The FL and VL values become: 
         [0041]    For a common FPC: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       F 
                       L 
                     
                     = 
                     
                       1 
                       - 
                       
                         
                           F 
                           R 
                         
                          
                         
                           τ 
                           s 
                         
                          
                         
                           α 
                           s 
                         
                       
                     
                   
                    
                   
                     
 
                   
                    
                   and 
                 
               
               
                 
                   ( 
                   
                     9 
                      
                     e 
                   
                   ) 
                 
               
             
             
               
                 
                   
                     V 
                     L 
                   
                   = 
                   
                     
                       
                         
                           F 
                           R 
                         
                          
                         
                           B 
                           c 
                         
                       
                       
                         I 
                         c 
                       
                     
                      
                     
                       ( 
                       
                         
                           T 
                           
                             f 
                             , 
                             in 
                           
                         
                         - 
                         
                           T 
                           ∞ 
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   
                     9 
                      
                     f 
                   
                   ) 
                 
               
             
           
         
       
     
         [0042]    For the VASC-FPC system: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       F 
                       L 
                     
                     = 
                     
                       1 
                       - 
                       
                         
                           F 
                           R 
                         
                          
                         
                           
                             
                               τ 
                               s 
                             
                              
                             
                               ( 
                               
                                 1 
                                 - 
                                 R 
                               
                               ) 
                             
                           
                            
                           
                             [ 
                             
                               
                                 
                                   A 
                                   - 
                                   
                                     A 
                                      
                                     
                                         
                                     
                                      
                                     
                                        
                                       ɛ 
                                     
                                   
                                   + 
                                   B 
                                 
                                 
                                   
                                     A 
                                      
                                     
                                         
                                     
                                      
                                     ɛ 
                                   
                                   + 
                                   B 
                                 
                               
                               - 
                               
                                 
                                    
                                   
                                     - 
                                     ɛ 
                                   
                                 
                                  
                                 
                                   ( 
                                   
                                     1 
                                     - 
                                     
                                       
                                         α 
                                         s 
                                       
                                        
                                       
                                         ( 
                                         
                                           1 
                                           - 
                                           R 
                                         
                                         ) 
                                       
                                     
                                   
                                   ) 
                                 
                               
                             
                             ] 
                           
                         
                       
                     
                   
                    
                   
                     
 
                   
                    
                   and 
                 
               
               
                 
                   ( 
                   
                     9 
                      
                     g 
                   
                   ) 
                 
               
             
             
               
                 
                   
                     V 
                     L 
                   
                   = 
                   
                     
                       
                         
                           F 
                           R 
                         
                          
                         
                           
                             τ 
                             s 
                           
                            
                           
                             ( 
                             
                               1 
                               - 
                               R 
                             
                             ) 
                           
                         
                       
                       
                         
                           ( 
                           
                             
                               A 
                                
                               
                                   
                               
                                
                               ɛ 
                             
                             + 
                             B 
                           
                           ) 
                         
                          
                         
                           T 
                           ∞ 
                         
                       
                     
                      
                     
                       ( 
                       
                         
                           T 
                           
                             f 
                             , 
                             in 
                           
                         
                         - 
                         
                           T 
                           ∞ 
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   
                     9 
                      
                     h 
                   
                   ) 
                 
               
             
           
         
       
     
         [0000]    where T ∞  is the temperature of the fluid entering the plate and F R  is the heat removal factor. The equation resulting from Equations (9), (9e) and (9f) 
         [0000]    
       
         
           
             
               
                 
                   η 
                   = 
                   
                     1 
                     - 
                     
                       ( 
                       
                         1 
                         - 
                         
                           
                             F 
                             R 
                           
                            
                           
                             τ 
                             s 
                           
                            
                           
                             α 
                             s 
                           
                         
                       
                       ) 
                     
                     - 
                     
                       
                         
                           
                             F 
                             R 
                           
                            
                           
                             U 
                             c 
                           
                         
                         
                           I 
                           c 
                         
                       
                        
                       
                         ( 
                         
                           
                             T 
                             
                               f 
                               , 
                               in 
                             
                           
                           - 
                           
                             T 
                             ∞ 
                           
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     9 
                      
                     i 
                   
                   ) 
                 
               
             
           
         
       
     
         [0000]    is known as the Hottel-Whillier-Bliss equation, which is the equation for the efficiency of a FPC in terms of fixed losses and variable losses and temperature difference between the fluid and ambient. 
         [0043]    b. Comparison Between a FPC and a VASC-FPC 
         [0044]    The comparison between the performances of the two systems can be based on their efficiencies, but these efficiencies depend on the foxed losses and variable losses. Comparing fixed losses and variable losses can give an idea of how the two systems perform. 
         [0045]    i. Comparing Fixed Losses 
         [0046]    From Equations (9e) and (9g), α s,eq  can be defined as the equivalent absorptance of the VASC-FPC system, therefore 
         [0000]    
       
         
           
             
               
                 
                   
                     α 
                     
                       s 
                       , 
                       eq 
                     
                   
                   = 
                   
                     
                       ( 
                       
                         1 
                         - 
                         R 
                       
                       ) 
                     
                      
                     
                       [ 
                       
                         
                           
                             A 
                             - 
                             
                               A 
                                
                               
                                   
                               
                                
                               
                                  
                                 ɛ 
                               
                             
                             + 
                             B 
                           
                           
                             
                               A 
                                
                               
                                   
                               
                                
                               ɛ 
                             
                             + 
                             B 
                           
                         
                         - 
                         
                           
                              
                             
                               - 
                               ɛ 
                             
                           
                            
                           
                             ( 
                             
                               1 
                               - 
                               
                                 
                                   α 
                                   s 
                                 
                                  
                                 
                                   ( 
                                   
                                     1 
                                     - 
                                     R 
                                   
                                   ) 
                                 
                               
                             
                             ) 
                           
                         
                       
                       ] 
                     
                   
                 
               
               
                 
                   ( 
                   10 
                   ) 
                 
               
             
           
         
       
     
         [0000]    Making substitution and rearranging Equation (10), the equivalent absorptance can also be written as 
         [0000]    
       
         
           
             
               
                 
                   
                     α 
                     
                       s 
                       , 
                       eq 
                     
                   
                   = 
                   
                     
                       
                         
                           α 
                           s 
                         
                          
                         
                           ( 
                           
                             1 
                             - 
                             R 
                           
                           ) 
                         
                       
                        
                       
                          
                         
                           
                             - 
                             μ 
                           
                            
                           
                               
                           
                            
                           L 
                         
                       
                     
                     + 
                     
                       
                         
                           1 
                           - 
                           R 
                         
                         
                           ( 
                           
                             
                               
                                 μ 
                                  
                                 
                                     
                                 
                                  
                                 L 
                               
                               
                                 μ 
                                  
                                 
                                     
                                 
                                  
                                 k 
                               
                             
                             + 
                             
                               1 
                               
                                 U 
                                 c 
                               
                             
                           
                           ) 
                         
                       
                        
                       
                         [ 
                         
                           
                             
                               ( 
                               
                                 
                                   1 
                                   
                                     μ 
                                      
                                     
                                         
                                     
                                      
                                     k 
                                   
                                 
                                 + 
                                 
                                   1 
                                   
                                     U 
                                     c 
                                   
                                 
                               
                               ) 
                             
                              
                             
                               ( 
                               
                                 1 
                                 - 
                                 
                                    
                                   
                                     - 
                                     μL 
                                   
                                 
                               
                               ) 
                             
                           
                           - 
                           
                             
                               L 
                               k 
                             
                              
                             
                                
                               
                                 
                                   - 
                                   μ 
                                 
                                  
                                 
                                     
                                 
                                  
                                 L 
                               
                             
                           
                         
                         ] 
                       
                     
                   
                 
               
               
                 
                   ( 
                   11 
                   ) 
                 
               
             
           
         
       
     
         [0047]    Thus, the ratio, γ, between the equivalent absorptance of the VASC-FPC system, α s,eq , and the absorptance of the common FPC, α s , can be determined: 
         [0000]    
       
         
           
             
               
                 
                   γ 
                   = 
                   
                     
                       
                         α 
                         
                           s 
                           , 
                           eq 
                         
                       
                       
                         α 
                         s 
                       
                     
                     = 
                     
                       
                         
                           ( 
                           
                             1 
                             - 
                             R 
                           
                           ) 
                         
                          
                         
                            
                           
                             
                               - 
                               μ 
                             
                              
                             
                                 
                             
                              
                             L 
                           
                         
                       
                       + 
                       
                         
                           
                             1 
                             - 
                             R 
                           
                           
                             
                               α 
                               s 
                             
                              
                             
                               ( 
                               
                                 
                                   
                                     μ 
                                      
                                     
                                         
                                     
                                      
                                     L 
                                   
                                   
                                     μ 
                                      
                                     
                                         
                                     
                                      
                                     k 
                                   
                                 
                                 + 
                                 
                                   1 
                                   
                                     U 
                                     c 
                                   
                                 
                               
                               ) 
                             
                           
                         
                          
                         
                           ( 
                           
                             
                               
                                 ( 
                                 
                                   
                                     1 
                                     
                                       μ 
                                        
                                       
                                           
                                       
                                        
                                       k 
                                     
                                   
                                   + 
                                   
                                     1 
                                     
                                       U 
                                       c 
                                     
                                   
                                 
                                 ) 
                               
                                
                               
                                 ( 
                                 
                                   1 
                                   - 
                                   
                                      
                                     
                                       - 
                                       μL 
                                     
                                   
                                 
                                 ) 
                               
                             
                             - 
                             
                               
                                 L 
                                 k 
                               
                                
                               
                                  
                                 
                                   
                                     - 
                                     μ 
                                   
                                    
                                   
                                       
                                   
                                    
                                   L 
                                 
                               
                             
                           
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   12 
                   ) 
                 
               
             
           
         
       
     
         [0048]    If γ is greater than 1, then more energy is absorbed by the VAFC-FPC system&#39;s absorber plate than the FPC alone, which increases the amount of heat delivered to the end user, Q u . If this is the case, the VASC-FPC system has a better performance compared to the FPC in terms of the fixed losses. If γ is less than 1, the VASC-FPC performance is poorer compared to FPC alone in terms of the fixed losses. 
         [0049]    i. Comparing Variable Losses 
         [0050]    From equations (9h) and (9f), U c,eq  can be defined as the equivalent U c  for the VASC-FPC system, therefore 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         F 
                         R 
                       
                        
                       
                         U 
                         
                           c 
                           , 
                           eq 
                         
                       
                     
                     
                       I 
                       C 
                     
                   
                   = 
                   
                     
                       
                         F 
                         R 
                       
                        
                       
                         
                           τ 
                           s 
                         
                          
                         
                           ( 
                           
                             1 
                             - 
                             R 
                           
                           ) 
                         
                       
                     
                     
                       
                         ( 
                         
                           
                             A 
                              
                             
                                 
                             
                              
                             ɛ 
                           
                           + 
                           B 
                         
                         ) 
                       
                        
                       
                         T 
                         ∞ 
                       
                     
                   
                 
               
               
                 
                   ( 
                   13 
                   ) 
                 
               
             
           
         
       
     
         [0000]    Rearranging Equation (13) and making substitutions, the equivalent thermal resistance of the overall VASC-FPC system is 
         [0000]    
       
         
           
             
               
                 
                   
                     1 
                     
                       U 
                       
                         c 
                         , 
                         eq 
                       
                     
                   
                   = 
                   
                     
                       L 
                       k 
                     
                     + 
                     
                       1 
                       
                         U 
                         c 
                       
                     
                   
                 
               
               
                 
                   ( 
                   14 
                   ) 
                 
               
             
           
         
       
     
         [0051]    The ratio, ρ, between the equivalent resistance of the combined system and the FPC alone is 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       1 
                       
                         U 
                         
                           c 
                           , 
                           eq 
                         
                       
                     
                     
                       1 
                       
                         U 
                         c 
                       
                     
                   
                   = 
                   
                     ρ 
                     = 
                     
                       1 
                       + 
                       
                         
                           
                             μ 
                              
                             
                                 
                             
                              
                             L 
                           
                           
                             μ 
                              
                             
                                 
                             
                              
                             k 
                           
                         
                          
                         
                           U 
                           c 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   15 
                   ) 
                 
               
             
           
         
       
     
         [0000]    If ρ is greater than 1, the VASC-FPC thermal resistance is higher, which reduces the amount of variable heat losses of the system and increases the overall performance. If ρ is less than 1, the VASC-FPC system is poorer than FPC in terms of variable losses. 
         [0052]      FIGS. 3 and 4  show the variation of the ratios γ ( FIG. 3 ) and ρ ( FIG. 4 ) with respect to the depth of the collector, for different types of glasses (μk) and for fixed U c =7.5 W/m 2 K, R=4%, and α s =95%. 
         [0053]    It can be seen from  FIG. 3  that the ratio γ is always less than 1 for all the given glass materials. This means that the fixed losses are larger for the VASC-FPC system compared to the FPC alone. This puts the combined VASC-FPC system at a competitive disadvantage in comparison to the common FPC. 
         [0054]    In  FIG. 4 , the ratio ρ is always greater than 1, which means that the variable heat losses of the combined VASC-FPC are smaller than the FPC alone. This puts the VASC-FPC system at a competitive advantage in comparison to the normal FPC. 
         [0055]    From  FIGS. 3 and 4 , it can be concluded that the fixed losses are larger for the combined system and the variable losses are lower. This cannot provide a definite insight on which collector has better performance, even though it can be seen that the variable losses are more decreased than fixed losses are increased. This results in the overall performance being greater for the combined system in comparison to the FPC alone. 
         [0056]      FIGS. 5 and 6  show the variation of the collector efficiency as a function of the temperature difference between the entering fluid and ambient. This is an all-inclusive technique of comparing VASC-FPC with the FPC. 
         [0057]    Three different depths of the VASC are used and compared to a common FPC. The following parameters have been used in  FIG. 5  and are typical for a common FPC. The same parameters were used to reproduce an efficiency curve (not shown) for the FPC. The parameters are as follows: 
         [0000]      μk=1 W/m 2 K
 
         [0000]      I c =1000 W/m w   
         [0000]      F R =0.9 
         [0000]      T s =0.92 
         [0000]      U c =7.5 W/m 2 K 
         [0000]      R=4% 
         [0058]    It can be noticed that for temperature differences below 20° C., the effects of fixed losses cause the efficiency of a normal collector to be greater than the efficiency of the combined system, but for higher temperatures, the effects of lower variable losses lead to a higher efficiency of the combined system. For this case in particular, it can be seen that for a temperature difference of 100° C., the efficiency of a FPC with a relatively small depth VASC component is three times higher than the FPC alone. 
         [0059]      FIG. 6  shows the same phenomenon for an advanced FPC (“APC”). Two parameters were changed to improve the performance of the FPC and the combined VASC-FPC. The two changed parameters are as follows: 
         [0000]      F R =0.9 
         [0000]      U c =7.5 W/m 2 K 
         [0000]    All other parameter values were kept. In this case, it can be seen that for the same temperature difference of 100° C., the efficiency can be significantly improved. 
         [0060]    B. Locating Materials 
         [0061]    A borosilicate glass cylinder was used to carry out the experiments. The experimental results matched the theoretical prediction of the VASC model to an appreciable degree. The highest temperature that was measured in the material at steady state conditions was about 40° C. In an effort to improve the performance of the collector, it is commendable to locate other potential materials for this application. The search would focus on materials whose material number, A, is high and that can withstand high temperatures. The material used in the experiment had an A value of approximately 0.58. Higher values of A would provide higher temperatures. Optical glass materials were deemed to be best for this application due to their relatively high values of A. A sample of some of these glass materials is shown in the table below. These are different optical glasses from Scott, as shown in Table 1. Different materials and glasses used can be purchased from HOYA OPTICS. 
         [0000]    
       
         
               
             
               
               
               
               
               
               
             
           
               
                 TABLE 1 
               
             
             
               
                   
               
               
                 Schott optical glass properties. 
               
             
          
           
               
                   
                 p 
                 k 
                 pk 
                   
                   
               
               
                 Glass 
                 (l/m) 
                 (W/mK) 
                 (W/m 2 K) 
                 T max   
                 A 
               
               
                   
               
               
                 F2 
                 1.59 
                 0.78 
                 1.24 
                 434 
                 2.69 
               
               
                 F2HT 
                 1.38 
                 0.78 
                 1.08 
                 434 
                 3.08 
               
               
                 F5 
                 1.57 
                 0.88 
                 1.47 
                 438 
                 2.27 
               
               
                 K10 
                 1.43 
                 1.18 
                 1.60 
                 459 
                 2.08 
               
               
                 LF5 
                 1.47 
                 0.87 
                 1.28 
                 418 
                 2.81 
               
               
                 LLF1 
                 1.30 
                 0.99 
                 1.20 
                 431 
                 2.58 
               
               
                 N-BAK1 
                 1.18 
                 0.80 
                 0.94 
                 592 
                 3.56 
               
               
                 N-BAK2 
                 1.23 
                 0.92 
                 1.13 
                 554 
                 2.95 
               
               
                 N-BAK4 
                 1.88 
                 0.68 
                 1.85 
                 581 
                 2.02 
               
               
                 N-BALF4 
                 2.08 
                 0.86 
                 1.77 
                 574 
                 1.83 
               
               
                 N-BK10 
                 1.11 
                 1.32 
                 1.46 
                 551 
                 3.28 
               
               
                 N-BK7 
                 1.40 
                 1.11 
                 1.56 
                 557 
                 2.14 
               
               
                 N-BK7HT 
                 1.11 
                 1.11 
                 1.24 
                 957 
                 2.69 
               
               
                 N-FK5 
                 1.21 
                 0.93 
                 1.12 
                 466 
                 2.99 
               
               
                 N-FK514 
                 1.18 
                 0.70 
                 0.90 
                 464 
                 3.71 
               
               
                 N-K5 
                 1.44 
                 0.95 
                 1.37 
                 546 
                 2.43 
               
               
                 N-KZFS11 
                 1.80 
                 0.81 
                 1.48 
                 551 
                 2.29 
               
               
                 N-KZFS2 
                 2.24 
                 0.81 
                 1.81 
                 872 
                 1.84 
               
               
                 N-KZFS4 
                 2.17 
                 0.84 
                 1.82 
                 535 
                 1.83 
               
               
                 N-KZFS5 
                 2.20 
                 0.95 
                 2.09 
                 584 
                 1.80 
               
               
                 N-KZFS8 
                 2.34 
                 1.05 
                 2.48 
                 509 
                 1.38 
               
               
                 N-LAF2 
                 3.00 
                 0.67 
                 2.03 
                 653 
                 1.64 
               
               
                 N-LAF21 
                 2.69 
                 0.83 
                 2.23 
                 653 
                 1.49 
               
               
                 N-LAF33 
                 2.52 
                 0.80 
                 2.01 
                 600 
                 1.66 
               
               
                   
               
             
          
         
       
     
         [0062]      FIG. 7  shows the cost of different types of materials. It can be seen that there is a large range of materials with a low μk (high A) that are relatively lowly priced. Therefore, for this application, the best materials are not necessarily the most expensive ones. 
         [0063]    It will thus be seen that the objects set forth above, and those made apparent from the foregoing disclosure, are efficiently attained. Since certain changes may be made in the above construction without departing from the scope of the invention, it is intended that all matters contained in the foregoing disclosure or shown in the accompanying drawings shall be interpreted as illustrative and not in a limiting sense. 
         [0064]    It is also to be understood that the following claims are intended to cover all of the generic and specific features of the invention herein described, and all statements of the scope of the invention that, as a matter of language, might be said to fall therebetween.