Patent Publication Number: US-11644219-B2

Title: Secondary reflectors for solar collectors and methods of making the same

Description:
CROSS-REFERENCE TO RELATED APPLICATIONS 
     This application claims the benefit of U.S. Non-Provisional Patent Application Ser. No. 15/631,522 filed Jun. 23, 2017, which claims the benefit of U.S. Provisional Patent Application Nos. 62/354,595, 62/354,948, and 62/370,086 filed Jun. 24, 2016, Jun. 27, 2016, and Aug. 2, 2016 respectively, the contents of which are incorporated herein by reference in their entirety. 
    
    
     CONTRACTUAL ORIGIN 
     The United States Government has rights in this disclosure under Contract No. DE-AC36-08GO28308 between the United States Department of Energy and the Alliance for Sustainable Energy, LLC, the Manager and Operator of the National Renewable Energy Laboratory. This invention was made with State of California support under California Energy Commission grant number EPC-14-047. The Energy Commission has certain rights to this invention. 
    
    
     BACKGROUND 
     There are two main approaches for collecting solar energy, photovoltaic (PV) and concentrating solar power (CSP). Unlike PV that directly converts solar irradiation into electricity, CSP utilizes reflectors to concentrate light (e.g. sunlight) onto absorber tube(s) at the reflector focal point contained in a receiver. The concentrated light is subsequently absorbed as thermal energy by a heat transfer fluid flowing through the absorber tubes. Finally, a thermodynamic power cycle (e.g. power block) converts the thermal energy contained in the high-temperature heat-transfer fluid into electricity. The heat transfer fluid exiting the receiver may have a temperature between 200° C. and 600° C. Because of the inclusion of the power block, a CSP power plant is typically deployed at the utility-scale for electricity generation and can be more capital-intensive than an equivalent utility-scale PV power plant. However, in addition to electricity generation, CSP systems can generate heat for a variety of applications such as heating, cooling, water desalination, process heat and complimentary heat addition to other energy sectors such as geothermal and fossil-fuel power plants. With the reducing cost of emerging new technologies in the near future, CSP may be very competitive in the energy market. Thus, CSP and PV can be complementary to each other in future renewable-dominating electricity markets, along with other types of renewable energy such as wind, geothermal and biomass. 
     CSP technologies include parabolic trough, linear Fresnel, power tower (also called central receiver) and dish/engine technologies. The global CSP installed capacity has reached over 6 GWe as of 2016. Linear Fresnel systems include a low-profile array of tracking primary reflectors (e.g. mirrors) and a fixed receiver containing at least one absorber tube and an optional secondary reflector. Fresnel technology can be a relatively low-cost technology compared to parabolic trough technology, but historically suffers from low performance metrics. The secondary reflector is a key component used to improve optical performance of a linear Fresnel CSP plant. The shape of a secondary reflector is particularly influential in affecting the solar power captured by the absorber tube(s) and thus the system&#39;s total optical performance. Thus, there remains a need for improved secondary reflectors and methods for designing these improved secondary reflectors. 
     SUMMARY 
     An aspect of the present disclosure is a device that includes a reflecting surface having a length aligned along a first axis (z), where a cross-section of the reflecting surface in a plane perpendicular to the first axis (z) forms a curve comprising a concave section positioned between a first endpoint and a second endpoint, at least a portion of the concave section is accurately approximated by a polynomial equation, an aperture is formed by a straight line connecting the first endpoint to the second endpoint, and the concave section is configured to focus a plurality of beams of light passing through the aperture onto a focal point. 
     In some embodiments of the present disclosure, the focal point may be positioned at least partially within an area enclosed by the concave section and the straight line. In some embodiments of the present disclosure, the focal point may be positioned outside an area enclosed by the concave section and the straight line. 
     In some embodiments of the present disclosure, the plane may contain a second axis (x) that is perpendicular to the first axis (z) and a third axis (y) that is perpendicular to the second axis (x), and the third axis (y) may intersect the second axis (x) to form a dimensionless xy-coordinate system having an origin of (0, 0) at the intersection. The concave section may be positioned in the dimensionless xy-coordinate system, the concave section may include a first portion that starts at the first endpoint positioned at (1, 0) and terminates at an intermediate point at (0, D), and the concave section may include a second portion that starts at the second endpoint positioned at (−1, 0) and terminates at the intermediate point. D may be a dimensionless number between 0.40 and 1.00. 
     In some embodiments of the present disclosure, the second portion may be a symmetrical, mirror-image of the first portion. In some embodiments of the present disclosure, the first portion may have a slope of 1.0 at the intermediate point. In some embodiments of the present disclosure, the first portion may have a slope not equal to 1.0 at the intermediate point. In some embodiments of the present disclosure, the first portion may be accurately described within the xy-coordinate system by the polynomial equation. 
     In some embodiments of the present disclosure, the polynomial equation may be a third-order polynomial equation defined as y=+b′x 2 +c′x+d, a′ may be between −1.0 and −0.10, b′ may be between −0.70 and −0.01, c′ may be between 0.03 and 0.30, d may be between 0.40 and 1.0, and a′, b′, c′, and d may all be dimensionless numbers. In some embodiments of the present disclosure, the first portion may be accurately described as indicated by a regression R 2  value between 0.999 and 1.0 inclusively. 
     In some embodiments of the present disclosure, the device may further include an absorber tube positioned at the focal point. In some embodiments of the present disclosure, the absorber tube may be configured so that a heat-transfer fluid can flow through the absorber tube. 
     An aspect of the present disclosure is a method that includes defining a geometric configuration of a field of primary reflectors and the optics of the field, such that the field can receive and reflect a light; defining a width of an aperture of a secondary reflector configured to receive at least a portion of the light, such that the portion passes through the aperture; and calculating a profile for a reflecting surface of the secondary reflector, where the aperture is positioned between a first endpoint and a second endpoint of the profile, the secondary reflector includes the aperture, and the profile resulting from the calculating, results in the reflecting surface maximizing the portion reflected off of the reflecting surface and directed to a focal point. 
     In some embodiments of the present disclosure, the calculating of the profile may include placing a first point of the profile on the first endpoint; a first calculating, for the first point, of a principal incidence of a fraction of the portion striking the first point; a second calculating, for the first point, of at least one variable that directs the fraction towards the focal point; and repeating the first calculating and the second calculating for n additional points until the profile is defined, including an n th  point of the profile placed on the second endpoint, where n is between 10 and 1000, and at least a portion of the profile is accurately described by a polynomial equation. 
     In some embodiments of the present disclosure, the variable may include at least one of a target reflection and/or a surface normal to the first point and the n th  point. In some embodiments of the present disclosure, n may be determined by defining a step distance between neighboring points. In some embodiments of the present disclosure, n may be between 100 and 400. In some embodiments of the present disclosure, the method may further include, prior to the defining of the width of the aperture, defining a distance between the aperture and the field of primary reflectors. 
     In some embodiments of the present disclosure, the method may further include, repeating the defining of the distance and the calculating m times until a final profile is obtained, where the final profile results in a final reflecting surface that maximizes the portion directed to the focal point, and m is between 1 and 100. In some embodiments of the present disclosure, the field of primary reflectors may include a linear Fresnel primary reflector. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       Exemplary embodiments are illustrated in referenced figures of the drawings. It is intended that the embodiments and figures disclosed herein are to be considered illustrative rather than limiting. 
         FIG.  1    illustrates a concentrating solar power plant utilizing a receiver having a secondary reflector, according to some embodiments of the present disclosure. 
         FIG.  2    illustrates a cross-sectional view of the receiving/reflecting surface of the secondary reflector (without a cusp) shown in  FIG.  1   , where the cross section is relative to the xy-plane, according to some embodiments of the present disclosure. 
         FIG.  3    illustrates a cross-sectional view of a receiving/reflecting surface of a secondary reflector having a cusp, where the cross section is relative to the xy-plane, according to some embodiments of the present disclosure. 
         FIG.  4    illustrates a profile of a receiving/reflecting surface of secondary reflector having a compound parabolic concentrator (CPC) design.  FIG.  4    also shows variables associated with equations (2) through (5) for a polar coordinate system for approximating the surface (e.g. profile) of a CPC design. 
         FIG.  5    illustrates a profile of a receiving/reflecting surface of a secondary reflector having a butterfly design. 
         FIG.  6    illustrates a profile of a receiving/reflecting surface of a secondary reflector having a butterfly design.  FIG.  6    also shows variables associated with equations (6) through (10) for an xy-coordinate system that utilizes sinusoidal functions to approximating the surface of a butterfly design. 
         FIG.  7    illustrates the profile (for one half of a secondary reflector) for a receiving/reflecting surface of a secondary reflector (with a cusp) designed according to methods described herein, where the profile is accurately described by a third-order polynomial equation, according to some embodiments of the present disclosure. 
         FIG.  8    illustrates the profile (for one half of a secondary reflector) for a receiving/reflecting surface of a secondary reflector (without a cusp) designed according to methods described herein, where the profile is accurately described by a third-order polynomial equation, according to some embodiments of the present disclosure. 
         FIG.  9    illustrates the profile (for one half of a secondary reflector) for a receiving/reflecting surface of a secondary reflector (with a cusp) having a CPC design, where the profile is not accurately described by a third-order polynomial equation. 
         FIG.  10    illustrates the profile (for one half of a secondary reflector) for a receiving/reflecting surface of a secondary reflector (with a cusp) having a butterfly design. 
         FIG.  11    illustrates various geometric variables and/or parameters of a CSP plant utilizing Fresnel design features, according to some embodiments of the present disclosure. 
         FIG.  12    illustrates various optical features of a CSP plant utilizing Fresnel design features, according to some embodiments of the present disclosure. 
         FIG.  13    illustrates the incidence of reflected sunlight to secondary reflectors, according to some embodiments of the present disclosure. 
         FIG.  14    illustrates flowcharts summarizing method steps for an adaptive method for designing a profile for the receiving/reflecting surface of a secondary reflector, according to some embodiments of the present disclosure. 
         FIG.  15    illustrates some steps for the method illustrated in  FIG.  14    for determining the optimal points on the receiving/reflecting surface (or profile) of a secondary reflector, according to some embodiments of the present disclosure. 
         FIG.  16    illustrates an example of a profile (e.g. a cross-section of a receiving/reflecting surface) of a secondary reflector obtained using the method illustrated in  FIG.  14   , according to some embodiments of the present disclosure. 
         FIG.  17    illustrates how the acceptance angle of a secondary reflector aperture changes with primary reflector position, according to some embodiments of the present disclosure. 
         FIG.  18    illustrates how the power intensity changes at Point A with primary reflector position because the relative deviation from the nominal tracking direction changes for each primary reflector, according to some embodiments of the present disclosure. 
         FIG.  19    illustrates a step for determining the principal incidence point using the receiver acceptance angle, according to some embodiments of the present disclosure. 
         FIG.  20    illustrates the profiles of receiving/reflecting surfaces of secondary reflectors resulting from the methods described herein for varying receiver heights and receiver sizes, according to some embodiments of the present disclosure. 
         FIG.  21    illustrates an example of a secondary reflector profile calculated by the method described herein and summarized in  FIG.  14   , according to some embodiments of the present disclosure, designed specifically for the Hyperlight Energy primary reflector field. 
         FIG.  22    illustrates what is referred to herein as the relative edge height (Δh), a design parameter, for designing the profile for a secondary reflector, according to some embodiments of the present disclosure. 
         FIG.  23    illustrates the secondary intercept factor of secondary reflector profiles as a function of relative edge height, according to some embodiments of the present disclosure. 
         FIG.  24    illustrates the average surface distance of secondary reflector profiles to absorber centers as a function of relative edge height, according to some embodiments of the present disclosure. 
         FIG.  25    illustrates the secondary intercept factor as a function of incidence angle to characterize the performance of a secondary reflector profile, according to some embodiments of the present disclosure. A secondary-reflector reflectance equal to 0.94 is assumed in the ray-trace simulations. 
         FIG.  26    illustrates the incidence angle modifier as a function of incidence angle for the Hyperlight Energy primary reflector field example and the resultant secondary reflector as designed using the method described herein, according to some embodiments of the present disclosure. The nominal optical efficiency is calculated as 0.699. 
         FIG.  27    illustrates stacked optical losses based on individual error sources for varying incidence angle (transversal), according to some embodiments of the present disclosure. 
         FIG.  28    illustrates stacked optical losses based on individual error sources for varying incidence angle (longitudinal), according to some embodiments of the present disclosure. 
         FIGS.  29 A through  29 D  illustrate profiles for secondary reflectors obtained using methods described herein, according to some embodiments of the present disclosure. 
     
    
    
     REFERENCE NUMBERS 
       100  . . . concentrating solar power plant 
       110  . . . light source 
       115  . . . light 
       120  . . . primary reflector field 
       130  . . . primary reflector 
       135  . . . surface 
       140  . . . receiver 
       150  . . . secondary reflector 
       155  . . . surface 
       160  . . . absorber tube 
       200  . . . aperture 
       400  . . . glass envelope 
       410  . . . profile 
       420  . . . cusp 
       700  . . . extremum 
       1400  . . . method 
       1410  . . . defining the primary reflector field 
       1420  . . . defining the secondary reflector aperture 
       1430  . . . calculating the secondary reflector profile 
       1431  . . . defining the position of starting edge point 
       1432  . . . calculating the principal incidence 
       1433  . . . calculating dependent variables for current point on profile 
       1434  . . . calculating the position of the next point 
       1435  . . . determining whether the profile is complete 
       1440  . . . optimizing the secondary reflector height 
       1450  . . . proposed secondary reflector design 
       1500  . . . point on secondary reflector profile 
       1600  . . . focal point 
     DETAILED DESCRIPTION 
     The present disclosure may address one or more of the problems and deficiencies of the prior art discussed above. However, it is contemplated that some embodiments as disclosed herein may prove useful in addressing other problems and deficiencies in a number of technical areas. Therefore, the embodiments described herein should not necessarily be construed as limited to addressing any of the particular problems or deficiencies discussed herein. 
     The present disclosure relates to adaptive methods for designing secondary reflectors having improved performance characteristics. As described herein, the design methods do not assume any pre-defined analytical form or shape, although it will be shown that the methods described herein result in a generally preferred shape that is accurately described mathematically. This preferred shape is attained through an adaptive process that maximizes the energy collection received by the receiver&#39;s absorber tube. The low performance of existing linear Fresnel technologies comes from the fact that a large number of design parameters exist in a Fresnel system&#39;s design, including geometric and optical parameters for each collector component and no robust numerical method exists that can fully optimize such a system. The methods described herein, and the resultant secondary reflector designs, address these deficiencies. 
       FIG.  1    illustrates a concentrating solar power (CSP) plant  100  utilizing linear Fresnel concepts, according to some embodiments of the present disclosure. This example of a CSP plant  100  includes a primary reflector field  120  configured to receive light  115  (e.g. sunlight) from a light source  110  (e.g. the sun). The primary reflector field  120  includes at least one, and often a plurality of primary reflectors  130  ( FIG.  1    illustrates a total of 18 primary reflectors, generally called out using reference number  130 , with three specifically called out;  130 A,  130 B, and  130 C) that collect and concentrate the light  115  onto a receiver  140  having a secondary reflector  150  and an absorber tube  160 . In some situations, the secondary reflector  140  may be utilized to collect a larger percentage of the light  115  received by the primary reflectors  130 , such that a larger percentage of the light  115  is directed onto the absorber tube  160 , ultimately resulting in a higher power production output than could be achieved by a CSP plant not utilizing a secondary reflector  150 . Referring again to  FIG.  1   , the light source  110  produces light  115 A that strikes a surface  135  of a primary reflector  130 C, which reflects at least a portion of the light  115 B towards the receiver  150 . A portion of the light  115 B reflected by the surface  135  of the primary reflector  130 C may then strike against a surface  155  of the secondary reflector  150 , resulting in a portion of the light  115 C being reflected by the surface  155  and directed towards the absorber tube  160 , where the light  115 C is absorbed, resulting in the transfer of heat to a heat-transfer fluid (not shown) configured to flow through the receiver absorber tube  160 . Referring again to  FIG.  1   , the light  115  can be visualized as individual beams or rays of light that travel in a linear path. 
     As shown in  FIG.  1   , an absorber tube  160  may be in the form of a pipe and/or tube, or any other suitable conduit, with the heat-transfer fluid flowing through the pipe and/or tube. In some embodiments of the present disclosure, an absorber tube  160  may have a length between 0.5 meters and 10 meters long (in the z-axis direction of  FIG.  1   ). An absorber tube  160  may have in outside diameter between 2 cm and 20 cm. Examples of a heat-transfer fluid, which flows through the absorber tube  160  and receives the solar energy may be a liquid (e.g. an oil, water, a molten salt, sodium) a solid (e.g. sand) and/or a gas (e.g. air and/or steam). Similarly, a secondary reflector  150  may have a length between 0.5 meters and 10 meters long. The receiving/reflecting surface  155  may be constructed of a polished metal and/or polymer, or any other sufficiently reflective material that can withstand the high operating temperatures of the system. In some embodiments of the present disclosure, the surface  155  may include a high-temperature resistant glass coating. The thickness of the receiving/reflecting surface  155  may be between 0.5 mm and 10 mm. 
       FIG.  1    illustrates a receiving/reflecting surface  155  for the secondary reflector having a curved cross-section in the xy-plane. For the example shown in  FIG.  1   , any cross-section in the xy-plane, along the z-axis (e.g. length dimension) of the receiving surface  155  of the secondary reflector  150  will result in a similar or identical curve in two-dimensional space.  FIG.  2    illustrates a cross-section (in the xy-plane) for the exemplary surface  155  of the secondary reflector  150  shown in  FIG.  1   , where the cross-section includes a first half of the surface  155 A connected to a second half of the surface  155 B at the apex (or maximum) of secondary reflector. Thus, in some embodiments of the present disclosure, a secondary reflector may be constructed of a first half of the secondary reflector that is substantially the mirror image of a second half of the secondary reflector, having corresponding receiving/reflecting surfaces ( 155 A and  155 B). The two halves of the surface ( 155 A and  155 B) may also be substantially symmetrical. The first half of the surface  155 A may connect with the second half of the surface  155 B at a point (0, D) in space designated by the xy-coordinates, where D refers to a depth dimension (e.g. measured in feet, meters, etc.). In addition, both halves of the surface ( 155 A and  155 B) may be defined by a width dimension in the xy-plane, such that the first half of the surface  155 A may terminate at a point in two-dimensional space at the xy-coordinates of (−W, 0) and where the second half of the surface  155 B may terminate at a point in two-dimensional space at the xy-coordinates of (W, 0), where W is equal to one-half of the total width of the secondary reflector  150  as measured from the reflecting/receiving surface  155 . 
       FIG.  2    also illustrates how an area of the cross-section may be defined, labeled A, where A is bounded by the borders defined by the surfaces ( 155 A and  155 B) and a straight line drawn between the points (−W, 0) and (W, 0). In some embodiments of the present disclosure, the absorber tube may be placed entirely within the area, A, as shown in  FIG.  4   . In other cases, the absorber tube may be positioned partially within the area, A, as shown in  FIG.  16   . In still other examples, the absorber tube may be positioned outside of the area, A. 
     Thus, the receiving/reflecting surface  155  of a secondary reflector  150  constructed of a first half that is symmetrical and the mirror image of a second half may be uniquely defined by a mathematical representation of a curve fitted to the cross-section of the receiving/reflecting surface  155 . This concept is elaborated below. Referring again to  FIG.  2   , the surface  155  of the secondary reflector  150  forms an opening referred to herein as an aperture  200  having a width spanning between the points (−W, 0) and (W, 0) on the x-axis. Thus, the absolute width of the aperture  200  corresponds to 2*W. 
       FIG.  3    illustrates a curve resulting from a cross-section of a first half of a receiving/reflecting surface  155 A for a secondary reflector having a shape referred to as a compound parabolic concentrator (CPC). The second half (not shown) of the receiving/reflecting surface  155  of the secondary reflector  150  having a CPC design is typically a symmetrical, mirror-image of the first half of the surface  155 A. The curve resulting from the cross-section of the first half of the surface  155 A may be approximated (e.g. traced) by a mathematical equation that describes a position for each point of the surface  155 A of the secondary reflector  150  in two-dimensional space. Thus, for a cross-section in the xy-plane, the shape of the secondary reflector receiving/reflecting surface  155 A may be described by a mathematical function generally described as,
 
 y=f ( x )  (1)
 
for an xy-coordinate system where x is the position of a point on the surface  155 A in the x-axis direction (width), and y is the position of the point on the surface  155 A in the y-axis direction (depth). As will be shown below, a characteristic equation that describes the curve approximating the shape of the surface  155 A of the secondary reflector in the xy-plane of a compound parabolic reflector is poorly described by a second-order or third-order polynomial equation, using the xy-coordinate system. A better fit for a CPC secondary reflector shape may be better achieved using a polar coordinate system, as shown in  FIG.  4   .
 
     Referring to  FIG.  4   , the two-dimensional curve resulting from a cross-section of the secondary reflector&#39;s receiving/reflecting surface  155  in the xy-plane, referred to herein as the profile  410  of the surface, may be defined for a CPC secondary reflector using the following equation, 
                     r   ⁡     (   θ   )       =       r   a       cos   ⁡     (     θ   -       tan     -   1       ⁡     (       I   ⁡     (   θ   )       ⁢     /     ⁢     r   a       )         )                 (   2   )               
where r a  is the radius of the absorber tube, θ is the polar coordinate angle, and I(θ) is the tangent distance from the absorber tube to the CPC secondary reflector&#39;s receiving/reflecting surface. The CPC acceptance angle (θ a ) may influence the range of accepted angles for the CPC design by defining the secondary reflector&#39;s aperture  200  width (W) and distance from the absorber tube  160  to a secondary edge (Δ h ). In some embodiments, the involute may touch the absorber tube  160  but a gap may exist between the absorber tube  160  and the CPC cusp  420  due to a glass envelope  400  that may encompass the absorber tube  160 . The involute and parabolic shapes may be adjusted by a factor of θ 0 ; if there is no gap, the value of θ 0  is equal zero. The involute is defined by the bounds
 
             [     0   ,       θ   a     +     π   2         ]         
and illustrated by the line segment  AB  in  FIG.  4   . The macro-focal parabola is bounded by
 
             [         θ   a     +     π   2       ,         3   ⁢   π     2     -     θ   a         ]         
and illustrated by the line segment  BC  in  FIG.  4   . The equations for I(θ) are detailed in equations (3)-(5) as follows:
 
     
       
         
           
             
               
                 
                   
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       FIG.  5    illustrates another exemplary profile resulting from a cross-section of a surface of a secondary reflector in the xy-plane, in this case for a secondary reflector having what is referred to herein as a butterfly design. As with the CPC design, only a first half of the surface&#39;s profile  410 A of the secondary reflector is shown. The second half (not shown) of the profile for a butterfly design is typically a symmetrical, mirror-image of the first half of profile  410 A. Referring to  FIG.  6   , the profile  410  of a butterfly design may also be described as two parabolic wings (two halves  410 A and  410 B) that are used to receiving, reflecting, and focusing light on the absorber tube  160 . The two halves of such a secondary reflector may be fashioned by rotating a parabola by an angle of (θ B ). The parabola&#39;s vertex may be defined by a focal length and the angle of rotation. For the case shown in  FIG.  6   , the focal length is the radius of the absorber tube&#39;s outer layer or glass envelope  400 . The parabola&#39;s focal point is at the center of the absorber tube (0, 0) and the vertex of the parabola is defined by (x v , y v ), as seen in  FIG.  6   . The profile  410  for the first half of the symmetric parabola may be defined by the equation below: 
                     x   =         x   ′     ⁢     cos   ⁡     (     θ   B     )         +       y   ′     ⁢     sin   ⁡     (     θ   B     )             ;     y   =         -     x   ′       ⁢     sin   ⁡     (     θ   B     )         +       y   ′     ⁢     cos   ⁡     (     θ   B     )                     (   6   )                 y   ′     =       y   v     +         (       x   ′     -     x   v       )     2       4   ⁢     r   a                   (   7   )               
where θ B  is defined by the primary field width and the height of the absorber tube  160  above the primary reflectors. Other constants in equation (6) and (7) are defined as follows:
 
     
       
         
           
             
               
                 
                   
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                       ( 
                       
                         
                           - 
                           
                             r 
                             a 
                           
                         
                         ⁢ 
                         
                           cos 
                           ⁡ 
                           
                             ( 
                             
                               θ 
                               B 
                             
                             ) 
                           
                         
                       
                       ) 
                     
                   
                   ; 
                   
                     
                       y 
                       v 
                     
                     = 
                     
                       
                         r 
                         a 
                       
                       ⁢ 
                       
                         sin 
                         ⁡ 
                         
                           ( 
                           
                             θ 
                             B 
                           
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   9 
                   ) 
                 
               
             
             
               
                 
                   
                     x 
                     max 
                   
                   = 
                   
                     
                       
                         cos 
                         ⁡ 
                         
                           ( 
                           
                             θ 
                             B 
                           
                           ) 
                         
                       
                       * 
                       
                         
                           2 
                           ⁢ 
                           
                             r 
                             a 
                           
                         
                       
                     
                     
                       
                         1 
                         - 
                         
                           
                             1 
                             
                               1 
                               + 
                               
                                 w 
                                 
                                   f 
                                   ⁢ 
                                   ield 
                                 
                                 2 
                               
                               + 
                               
                                 h 
                                 2 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   10 
                   ) 
                 
               
             
           
         
       
     
     In Equation (8), w field  refers to the width of the primary reflector field  120  and h to the height of the absorber tube  160  above the primary reflectors  130  (see  FIG.  1   ). 
       FIG.  7    illustrates a profile  410 A of the first half of a receiving/reflecting surface of a secondary reflector designed by the methods described herein. As described above, the profile  410  is obtained from a cross-section of the secondary reflector&#39;s receiving/reflecting surface in the xy-plane. The secondary reflector&#39;s surface profile  410 A shown in  FIG.  7    is referred to herein as a “proposed” design and corresponds generally to secondary reflector having a surface profile that is accurately described by a third-order polynomial. The second half of the surface profile (not shown) resulting from the proposed secondary reflector design is a symmetrical, mirror-image of the first half of the profile  410 A. 
     As used herein, a polynomial equation is described as,
 
 y=f ( x )= ax   n   +bx   n−1   +cx   n−2   + . . . a′x   3   +b′x   2   +c′x+d   (11)
 
where x describes the position of a point on the x-axis of the surface of the secondary reflector in two-dimensional space and the exponents of x are integer values greater than or equal to one and the parameters of x are any real number (a, b, c, a′, b′, c′, and d). As described above, y refers to the position of the corresponding point on the secondary reflector&#39;s receiving/reflecting surface on the y-axis. The largest numerical value of an exponent in equation (11) is referred to herein as the “order” of the polynomial. Thus, a third-order polynomial that fits and approximates the profile of the surface of a secondary reflector may be represented as,
 
 y=f ( x )= a′x   3   +b′x   2   +c′x+d   (12)
 
     The profile  410 A shown in  FIG.  7   , for half of the secondary reflector&#39;s surface, was obtained by normalizing the width (W) of the cross-section of the first half of the secondary reflector to one (1.0), while maintaining the aspect ratio of the secondary reflector. The aspect ratio is referred to herein as the depth (D) of the secondary reflector divided by one-half of the total width (2 W is equal to the total width) of the secondary reflector, as measured from the reflecting/receiving surface. In addition, the leading edge of the secondary reflector, in the y-axis direction (e.g. depth direction) was placed on the x-axis (e.g. y=0). This standardization procedure allows the profiles of different secondary reflector profiles to be compared on an equivalent basis, without the parameters being skewed due to different scales. All of the profiles below, that were fitted mathematically were consistently analyzed using the procedure described in this paragraph. 
     Referring again to  FIG.  7   , a third-order polynomial provides a good mathematical fit of the profile for this embodiment of a secondary reflector (e.g. one embodiment of the proposed design), with an R 2  value of greater than 0.999, with a dimensionless depth (D) value of about 0.799 and a dimensionless width (where W equals one half of the secondary reflector&#39;s total width) value of 1.0. The profile  410  shown in  FIG.  7    includes a “cusp”  420  and an extremum  700 , in this case a maximum. As used herein, an “extremum” refers to a point on a curve having a slope equal to zero or infinity, which corresponds to at least one of a point of discontinuity, a maximum, or a minimum. Thus, a cusp is also an extremum, having a slope of infinity and the extremum on the profile labeled  700  corresponds to a local maximum having a slope of zero. Including a cusp in the surface of a secondary reflector may enable a larger percentage of the light (not shown) to be captured by the secondary reflector. However, inclusion of a cusp in the design of a secondary reflector may also complicate the manufacturing process and increase manufacturing costs. Thus, some embodiments of a secondary reflector may not include a cusp, while others may include at least one cusp. 
       FIG.  8    illustrates another embodiment of the present disclosure, a proposed design for a secondary reflector, without a cusp, where the profile  410  of the surface of the secondary reflector ( 410 A representing a first half of the profile). Instead, the first half of the secondary reflector&#39;s surface meets/joins the second half of the secondary reflector&#39;s surface at a local maximum, i.e. extremum  700 . For this example, this extremum  700  (e.g. a maximum having a slope of zero) occurs at point where the first half of the profile  410 A joins with the second half of the profile (e.g. at the point where x=0 or (0, D)). The second half of the profile (not shown) may be a symmetrical, mirror-image of the first half of the profile  410 A. As shown in  FIG.  8   , a third-order polynomial equation fits the profile  410 A of the secondary reflector&#39;s receiving/reflecting surface very well with an R 2  value of greater than 0.999. 
       FIG.  9    illustrates a profile  410  obtained when fitting with a third-order polynomial equation the cross-section of a CPC secondary reflector&#39;s receiving/reflecting surface lacking a cusp.  FIG.  9    illustrates that the fit is relatively poor (e.g. R 2 &lt;0.97), and the shape of fitted equation does not accurately describe the actual profile of the secondary reflector. For example, the best fitting third-order polynomial produces a cusp  420 , even though the actual CPC cross-section of the secondary reflector&#39;s surface has no cusp. In addition, the half of the profile  410 A shown in  FIG.  8    has two extrema, a local minimum, i.e. extremum, at  700 A, and a local maximum, i.e. extremum, at  700 B. Thus, fitting a third-order polynomial equation to the profile  410 A of a secondary reflector having a CPC design results in a poor fit and equations (2)-(5) provided above result in a better approximation/fit of the shape of secondary reflector&#39;s receiving/reflecting surface. 
       FIG.  10    illustrates a profile  410  obtained when fitting a third-order polynomial equation to the cross-section of the receiving/reflecting surface of a secondary reflector having a butterfly design.  FIG.  10    illustrates that a third-order polynomial equation may accurately fit the first half of the profile  410 A of a secondary reflector having a butterfly design. However, the profile  410 A resulting from the butterfly design differs from the proposed profile of the secondary reflector shown in  FIG.  8   , in that the butterfly design&#39;s profile  410 A includes a cusp and/or a local minimum at the origin (the point where the first half of the profile  410 A joins to the second half of the profile). In addition, a butterfly design is fundamentally different from the designs shown in  FIGS.  7  and  8    in that its maximum depth occurs at a value of x equal to 1.0, or at the coordinates (W, D) instead of (0, D). In addition, at least some of the signs (e.g. + or −) for the parameters obtained by the third-order polynomial equation to fit the butterfly design differ between the parameters used in the third-order polynomial equations used to fit the designs summarized in  FIGS.  7  and  8    (as well as the CPC design). 
     Thus, in summary,  FIGS.  7 - 10    clearly illustrate that secondary reflector profiles obtained from the cross-sections of the receiving/reflecting surfaces in the xy-plane can differ greatly when these profiles are described mathematically using a third-order polynomial equation. Some of these differences are summarized in Table 1 below. Note that all numbers listed in Table 1 are dimensionless. See equation 12 for a reference to the parameters a′, b′, c′, and d. 
     
       
         
           
               
             
               
                 TABLE 1 
               
             
            
               
                   
               
               
                 Secondary Reflector Surface Profile Comparison 
               
            
           
           
               
               
               
               
               
               
               
               
               
               
               
            
               
                 secondary reflector 
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
               
               
                 design type 
                 a′ 
                 b′ 
                 c′ 
                 d 
                 R2 
                 # maxima 
                 # minima 
                 f(x = 0) 
                 f(x = W) 
                 D/W 
               
               
                   
               
            
           
           
               
               
               
               
               
               
               
               
               
               
               
            
               
                 proposed design w/cusp 
                 −0.59 
                 −0.42 
                 0.23 
                 0.79 
                 1.00 
                 1 
                 1 
                 D 
                 0 
                 0.78 
               
               
                 proposed design w/o cusp 
                 −0.78 
                 −0.08 
                 0.04 
                 0.82 
                 1.00 
                 1 
                 0 
                 D 
                 0 
                 0.82 
               
               
                 CPC design w/cusp 
                 −2.25 
                 1.61 
                 −0.15 
                 0.95 
                 0.96 
                 2 
                 1 
                 D 
                 0 
                 0.89 
               
               
                 CPC design w/o cusp 
                 −2.78 
                 2.61 
                 −0.71 
                 1.04 
                 0.97 
                 2 
                 1 
                 D 
                 0 
                 1.00 
               
               
                 butterfly design 
                 0.22 
                 −0.62 
                 0.88 
                 0.00 
                 1.00 
                 0 
                 1 
                 0 
                 D 
                 0.47 
               
               
                   
               
            
           
         
       
     
     An aspect of the present disclosure relates to methods for designing secondary reflectors having characteristics as described above. Referring to  FIGS.  11  and  12   , a linear Fresnel CSP plant  100  may be characterized by various geometric variables and/or parameters and optical elements. With respect to its geometry, and as described above for  FIG.  1   , the CSP plant  100  includes an array of primary reflectors  130  and a receiver  140  having an optional secondary reflector  150 . The primary reflectors  130  track the light source  110  (e.g. the sun) and reflect the light (e.g. the sun&#39;s rays) to the receiver&#39;s absorber tube  160 , either directly or indirectly through the use of a secondary reflector  150 , where the reflected sun&#39;s rays form a concentrated beam. 
     Various parameters in the system optics of a Fresnel collector CSP plant  100  may contribute to optical errors, including the finite-size of the sun&#39;s shape, imperfect primary reflector  120  specularity, primary reflector slope error, primary reflector tracking error, absorber tube position error, and/or errors associated with the secondary reflectors  150 , all of which may contribute to reduced CSP plant energy efficiencies. One effect of these optical errors is a broadening of the reflected beam, relative to the original shape of the sun. For example, if the beam is significantly broadened, the receiver  140  may obstruct at least a portion of the reflected beam lying within the acceptance angle window of the receiver  140 . 
     In some embodiments of the present disclosure, the root mean square (RMS) of the overall beam spread distribution (e.g. beam broadening) is estimated by assuming the sun shape and all system optical errors follow a Gaussian distribution:
 
σ total   2 =σ sun   2 +σ specularity   2 +4×σ slope   2 +4×σ track   2 +4×σ receiver   2   (13)
 
where σ specularity , σ slope , σ track  and σ receiver  are the RMS for mirror specularity, mirror slope error, mirror tracking error, and receiver-related optical error, respectively. For embodiments, where the sun shape and/or some optical error sources are not a Gaussian distribution, the total beam distribution RMS σ total  may be obtained through convolutions of individual distributions. The coefficients in Equation 13 may vary for other solar technologies such as parabolic trough, power tower, and dish/engine.
 
     Based on the geometry of a linear Fresnel CSP plant  100 , each point on a secondary reflector surface has on open line of sight to all of the primary reflector surfaces except the parts which are shaded or obstructed, for example by the receiver itself. Thus, the sun ray&#39;s incidence angular range will typically cover the entire primary reflector surface along its width.  FIG.  13    illustrates the incidence angular range for two arbitrary points. Point A has an unobstructed view of the entire primary reflector field, and as a result, its incidence angular range covers the entire field. However, the incidence angular range of the primary reflector array for Point B is partially obstructed by the absorber tube. By repeating this incidence angular range analysis for the entire surface of the secondary reflector, a central incidence can be readily calculated for each point. The central incidence sun rays for Points A and B are illustrated in the  FIG.  13   . As used herein, “incidence” is defined as the incoming sun ray direction from the sun with respect to the secondary reflector&#39;s aperture. 
     As shown in the example of  FIG.  13   , the incidence angular range for the secondary reflector  150  is much wider than the sun shape and the beam spread resulting from the system optical errors is significant. As a result, a secondary reflector having a CPC design may not provide the most efficient solar energy capture, because a CPC secondary reflector is typically designed based on the assumption that the sun light arrives at the CPC secondary reflector&#39;s aperture within a tight angle of only a few milliradians. Thus, a non-parabolic reflector shape is needed for secondary reflectors incorporated in linear Fresnel CSP systems utilizing both primary reflectors and secondary reflectors. 
     To resolve at least this problem and to provide a more efficient CSP plant, some of the methods described herein begin at a pre-specified edge point and adaptively advance it to a complete profile by determining a secondary reflector&#39;s receiving/reflecting surface profile that is able to reflect the maximum amount of power/energy to the absorber tube. The resultant final optimum secondary reflector profile depends on features of the specific linear Fresnel system&#39;s configuration (e.g. geometry and optics), including parameters such as absorber size, position, collector field size and system optical errors. 
       FIG.  14    illustrates two flowcharts (left and right) that summarize a method  1400  for adaptively designing a profile for the receiving/reflecting surface (profile) of a secondary reflector having features as described above, for a specific Fresnel CSP plant design. However, it should be noted that the method described below can also be applied to other concentrating solar devices including solar thermal collectors or concentrating PV with secondary reflectors/concentrators. In addition, the method described herein can accommodate different configurations of absorbers (e.g. shape, size, number, and/or geometrical arrangement). Details of each step shown in  FIG.  14    are illustrated in more detail in subsequent figures and their corresponding text. Generally, the method  1400  includes the four steps illustrated in the flowchart illustrated on the left side of  FIG.  14   . Such a method  1400  may begin with the defining  1410  of the primary reflector field&#39;s geometry and optics, which will vary depending on the specific CSP plant design details. This defining  1410  provides the CSP system variables and/or metrics necessary to proceed with the defining  1420  of the aperture size for the secondary reflector (e.g. width). With the aperture defined, the method  1400  may then continue through a sequential and iterative process for calculating  1430  the profile of the secondary reflector, based on the various variables and parameters obtained from the upstream defining steps ( 1410  and  1420 ). Finally, with a first profile designed, the method  1400  may repeat steps  1420  and  1430  by repeating their calculations for optimizing  1440  the secondary reflector&#39;s profile as a function of the height of the secondary reflector relative to the primary reflector field, to finally arrive at the proposed secondary reflector design  1450  for the Fresnel CSP power plant being evaluated. 
     The flowchart on the right side of  FIG.  14    summarizes some of the details of the calculating  1430  step of the first flowchart described above, for calculating  1430  the profile of the secondary reflector. The calculating  1430  of the secondary reflector&#39;s profile may begin with the defining  1431  of the vertical position of the starting edge point of the secondary reflector&#39;s profile relative to the primary reflector field. Once this position is determined, the method  1400  may proceed with the calculating  1432  of the principal incidence of the incoming light relative to this starting point, which in turn, enables the calculating  1433  of various dependent variables (e.g. target reflection direction, surface normal, surface tangent, etc.), which are all described in more detail below. Once the calculating  1433  is completed, the method  1400  has sufficiently defined the system such that the method  1400  may proceed with the calculating  1434  of the position of the next point on the secondary reflector&#39;s profile. The calculating  1444  may be achieved by utilizing a predefined step-distance. Steps  1431  through  1434  may then be repeated until the entire profile is calculated for the secondary reflector. Thus, the method  1400  may include a determining step  1435 , which evaluates whether the profile has been completed across the entire aperture (e.g. width) of the secondary reflector and whether or not steps  1431  through  1434  need to be repeated. The final outcome is the proposed secondary reflector design  1450  for the Fresnel CSP power plant being evaluated. Each of the steps described generally above, will now be described in more detail 
     Defining the secondary reflector aperture: For a given linear Fresnel design, the secondary reflector aperture is assumed to be capable of intercepting up to 95%, 96%, 97%, 98%, 99%, or up to greater than 99% of the reflected power from the outer-most mirror n. Thus, the acceptance angle for primary reflector  120   n  may be defined by:
 
β n   pr =2·σ total   (14)
 
     The secondary reflector aperture may be defined by:
 
 a   2nd ≅2 ·l   n ·sin(β n   pr )  (15)
 
     The underlying assumptions are: 1) the primary reflector has a parabolic shape; 2) the secondary reflector&#39;s aperture width is small compared with the absorber height relative to the primary reflector field (in the y-axis direction; see  FIG.  1   ). However, these assumptions are for illustrative purposes and may vary for different linear Fresnel designs. 
     Calculating the secondary reflector profile: The secondary reflector aperture (a 2nd ) is determined by collector optimization. As an adaptive method, initially the secondary reflector surface may be determined by the aperture width and, in the absence of other mechanical design limitations (e.g. wind load analysis), the starting point may be placed at the same height (in y-axis direction; see  FIG.  1   ) as the absorber tube&#39;s center because this results in the maximum absorber acceptance angle (given the same secondary reflector aperture). 
     As illustrated in  FIG.  15   , initially the principal incidence may be calculated first (see step  1432  in  FIG.  14   ). The principal incidence may be calculated so that the maximum amount of sun ray power is reflected to the absorber tube  160  at a starting point  1500 A on the secondary reflector&#39;s profile. Strictly speaking, the principal incidence may be determined by performing detailed ray-trace simulation. In a simplified approach, the principal incidence may be defined as the central incidence accounting for the absorber tube blockage (by using the tangent vector to the absorber tube), as illustrated in  FIG.  13   . As shown in  FIG.  14   , the method  1400  may proceed with the calculating  1433  of various intermediate variables. The target reflection direction is calculated by connecting the starting point with the absorber tube&#39;s center using a straight line. Once the principal incidence and the target reflection direction are identified, the surface normal to the starting point  1500 A may be calculated by bisecting the principal incidence and the target reflection. The adaptive method  1400  (see  FIG.  14   ) continues by calculating  1434  the next point  1500 B along the secondary reflector&#39;s surface (or profile). This point may be determined by specifying a step distance (depending on the surface accuracy requirement) along the secondary reflector&#39;s surface tangent vector. Then the principal incidence, the reflection, the surface normal, and the surface tangent at the second surface point  1500 B may be determined by repeating the process for the next point (not shown) on the secondary reflector&#39;s surface/profile. An optional iterative process may be utilized to further refine the point ( 1500 B) determined above (by specifying a step distance) by calculating the angle between the surface normal at the first two surface points ( 1500 A and  1500 B) and adjusting the surface tangent by half of the calculated angle so that the angular difference between the surface normal and the surface tangent are the same for the two neighboring surface points ( 1500 A and  1500 B). 
     The process described above is then repeated for as many subsequent surface points as necessary to determine the remaining surface, resulting in a complete reflector shape (see  FIG.  14   ).  FIG.  16    illustrates an example of a profile  410  for a secondary reflector, including a first half of the profile  410 A and a second half of the profile  410 B, using the adaptive method described above.  FIG.  16    illustrates how a secondary reflector  150  may be configured to focus the light  115  received from the primary reflectors (not shown) onto a focal point  1600  (e.g. the center point of the circular cross-section of the absorbing tube  160 ). 
     Calculating the principal incidence using the accumulated power intensity: The principal incidence direction for any point on the optimum secondary reflector surface may be determined so that the maximum amount of power will be reflected to the absorber tube. The acceptance angle of the absorber tube a ab  is typically much less than the incidence angular range for a point on the surface of the secondary reflector. Thus, the central incidence may not be a good approximate for the principal incidence. Thus, as described herein, some embodiments of the present disclosure may use a more accurate but simpler approach for calculating the principal incidence at any point on the surface of the secondary reflector. First, as illustrated in  FIG.  17   , the acceptance angle of the primary reflector may be smaller for the primary reflectors located closer to the primary reflector field&#39;s center than those of primary reflectors located farther away, which may be considered the deviation angel of Point A from the primary reflector tracking direction. Referring again to  FIG.  15   , it can be seen that,
 
β center &lt;β side .  (16)
 
     This implies that the power intensity for incidence ray i is more than the power intensity for incidence ray j. Thus, for the point A on the secondary reflector&#39;s surface, the power intensity varies with the position of the primary reflector. As shown in  FIG.  18   , power intensity decreases from the left outer-most primary reflector to the center primary reflector and then increases toward the right outer-most primary reflector. More specifically, the power intensity of each ray is a function of the deviation angle, which is a function of primary reflector position. Assuming the reflected beam spread follows a Gaussian distribution, the power intensity may be obtained:
 
 P ( x )= g (σ total ,β pr ( x ))  (17)
 
where g is a Gaussian function with a standard deviation of σ total  and defines the intensity at a deviation angle of β pr (x). As illustrated in  FIG.  18   , β pr (x) may be calculated as a function of the primary reflector position.
 
       FIG.  19    illustrates the distribution of the power intensity of the incidence rays as a function of mirror position. As expected, the power intensity may decrease first and then increase with increasing distance to the mirror array center. However, the power intensity for mirrors located on the right side of  FIG.  19    is higher than the power intensity of mirrors located on the left side because the elongated optical path to the mirrors located on the right leads to a smaller deviation angle and thus higher power intensity. As illustrated in  FIG.  19   , an absorber acceptance window may be formed using two vertical lines. The mirror distance between two vertical lines corresponds to the absorber acceptance angular size for point A. The acceptance window moves from left to right and the accumulated power intensity is calculated for this acceptance window as a function of mirror position. At the principal mirror position, the acceptance window has the maximum accumulated power intensity. If the absorber is large enough, the acceptance angle window will be larger than the entire mirror array and the central incidence would be equal to the principal incidence. 
     This process may be repeated for any other points on the surface of the secondary reflector. The view blockage of the absorber tube may be accounted for geometrically.  FIG.  20    provides three examples of optimized surface shapes (profiles) for secondary reflectors, as a function of the relative height of the secondary reflector and absorber tube size. For all cases, starting from the left edge point on the surface of the secondary reflector, the principal incidence comes from the opposite side of primary reflectors first (e.g. referring to  FIG.  16    starting at the left side of the surface for the secondary reflector, the principal incidence comes from primary reflectors located on the right side), and then shifts to the primary reflectors located on the same side because of the absorber tube blockage. For the profile shown on the right side of  FIG.  20   , the central flat part of the profile is the result of a minimum distance set between the absorber tube and the secondary reflector because of the existence of a glass envelope for situations where a vacuum absorber tuber is utilized. 
     Optimizing the secondary reflector height: When the vertical position of the starting edge point on the surface of the secondary reflector is not fully determined, the relative height of the secondary reflector may not be determined. A practical criterion to determine the optimum secondary reflector height is the average distance between the surface of the secondary reflector and absorber tube. The shorter the average distance, the larger the average absorber tube acceptance angle, which results in more energy intercepted by the absorber tube from reflected light from the secondary reflector. 
     Example: Optimization of a secondary reflector profile for the Hyperlight Energy linear Fresnel technology. The geometrical attributes of such a system are summarized in Tables 2 and 3 below. 
     
       
         
           
               
             
               
                 TABLE 2 
               
               
                   
               
               
                 Geometrical specifications of the Hyperlight Energy reflector assembly. 
               
               
                   
               
             
            
               
                   
               
            
           
           
               
               
            
               
                 Total collector width 
                 12.8 
               
               
                 Total collector effective width (W field , m) 
                 12.5 
               
               
                 Measured from secondary reflector edges at the two 
                   
               
               
                 ends 
                   
               
               
                 Net secondary reflector aperture width (W primary , m) 
                 10.5 
               
               
                 No. of basin along width 
                 2 
               
               
                 Basin width (m) 
                 6.1 
               
               
                 Basin length (m) 
                 15.7 
               
               
                 Basin spacing along width (m) 
                 1.0 
               
               
                 Basin spacing along length (m) 
                 0.67 
               
               
                 No. of primary reflectors per basin 
                 50 
               
               
                 Primary reflector width (m) 
                 0.105 
               
               
                 Primary reflector spacing (m) 
                 0.115 
               
               
                 Position of inner most primary reflector 
                 0.15 
               
               
                 relative to the inner edge of secondary 
                   
               
               
                 refelctor(m) 
                   
               
               
                 Position of outer most reflector relative to the 
                 5.9 
               
               
                 inner edge (m) 
                   
               
               
                 Receiver assembly 
                   
               
               
                 Absorber tube height (m) 
                 7 
               
               
                 Absorber tube diameter (m) 
                 0.09 
               
               
                   
               
            
           
         
       
     
     
       
         
           
               
             
               
                 TABLE 3 
               
               
                   
               
               
                 Optical properties of the Hyperlight Energy reflector assembly. 
               
               
                   
               
             
            
               
                   
               
            
           
           
               
               
               
            
               
                   
                 Primary reflector 
                   
               
               
                   
                 Reflectance of primary reflector 
                 0.94 
               
               
                   
                 RMS of specularity (Gaussian, 
                 1.5 
               
               
                   
                 mrad) 
                   
               
               
                   
                 RMS of slope error and tracking 
                 3 
               
               
                   
                 error(Gaussian, mrad) 
                   
               
               
                   
                 Absorber tube 
                   
               
               
                   
                 Transmittance of glass envelope 
                 0.97 
               
               
                   
                 Absorptance of surface coating 
                 0.96 
               
               
                   
                 Position error (vertical, m) 
                 [0, 30] 
               
               
                   
                 Secondary reflector 
                   
               
               
                   
                 Reflectance of primary reflector 
                 0.94 
               
               
                   
                 Specularity (Gaussian, mrad) 
                 1.5 
               
               
                   
                 Slope error (Gaussian, mrad) 
                 2 
               
               
                   
                 Position error(vertical, m) 
                 [0, 20] 
               
               
                   
                   
               
            
           
         
       
     
     Defining the secondary reflector aperture for the Hyperlight system: It was assumed that the secondary reflector aperture needs to capture a certain amount of reflected solar power from the outer-most reflector. By using the beam spread RMS estimation of Equation (13) and ignoring receiver position error, the total reflector beam RMS may be obtained by,
 
σ total−reflector   2 =σ sun   2 +σ specularity +4×σ slope+track   2 =2.8 2 +1.5 2 +4×3.0 2   (18)
 
     In this example, the sun shape was assumed to have a circumsolar ratio (CSR) of 10%, which has a RMS of about 2.8 mrad; the mirror specularity, slope error and tracking error are listed in Table 3 above. σ total−reflector  is the RMS of the beam spread after accounting for all reflector errors. It was assumed that the beam spread after secondary reflector reflection follows a Gaussian distribution. Thus, the required acceptance angle to capture 95% of reflected power under normal incidence may be defined as:
 
β n   pr =2×σ total−reflector =13.6 m  rad  (19)
 
where β n   pr  is illustrated in  FIG.  12   .
 
     The required secondary reflector aperture to capture 95% of reflected power at the outer edge of the outmost primary reflector at the normal incidence may be determined by, 
                         a     2   ⁢   nd       2     ≅         l   n     ·     sin   ⁡     (     β   n   pr     )         +         w   n     2     ·     cos   ⁡     (     τ   n     )             =             [       7   2     +       (       12.5   2     -     0.105   2       )     2       ]       ×     sin   ⁡     (       0   .   0     ⁢   1   ⁢   3   ⁢   6     )         +       0.105   2     ×     cos   ⁡     (       0   .   3     ⁢   6     )           ≅     0.176   ⁢           ⁢   m               (   20   )               
where l n  is the distance between the n-th primary reflector center and the absorber tube center; w n  is the n-th primary reflector width; τ n  is the tracking angle of the n-th primary reflector. The addition of the second term in Equation 20 is the result of a flat primary reflector. Thus,
 
 a   2nd ≈0.35 m   (21)
 
     Selection of the absorber tube: The geometric concentration ratio of a linear Fresnel collector may be defined by, 
     
       
         
           
             
               
                 
                   c 
                   = 
                   
                     
                       W 
                       
                         p 
                         ⁢ 
                         r 
                         ⁢ 
                         i 
                         ⁢ 
                         m 
                         ⁢ 
                         a 
                         ⁢ 
                         r 
                         ⁢ 
                         y 
                       
                     
                     
                       D 
                       
                         a 
                         ⁢ 
                         b 
                         ⁢ 
                         s 
                         ⁢ 
                         o 
                         ⁢ 
                         r 
                         ⁢ 
                         b 
                         ⁢ 
                         e 
                         ⁢ 
                         r 
                       
                     
                   
                 
               
               
                 
                   ( 
                   22 
                   ) 
                 
               
             
           
         
       
     
     The higher the concentration, the less the absorber tube surface area per primary reflector area, which leads to less heat loss and increases the system thermal efficiency. With such a high concentration ratio, the secondary reflector may attain relatively high temperatures resulting in limited secondary reflector design options and increased costs. In this example, a vacuum absorber was selected. For vacuum absorbers in the market, the largest available absorber tube diameters for parabolic trough collectors is about 0.09 m. Due to the elongated optical path for a linear Fresnel system compared to a parabolic trough, a vacuum absorber tube diameter of 0.09 m was selected. As a result, the geometric concentration ratio was determined to be about 117 for the Hyperlight example described herein. 
     Calculating the secondary reflector profile: The method described above and summarized in  FIG.  14    was then used to design a proposed secondary reflector profile for the Hyperlight Energy collector configuration, as described above. The resultant optimum vertical position of the edge point on the surface of the secondary reflector was determined to be about 0.05 m lower than the absorber center, as shown in  FIG.  21   . 
     The performance of a proposed secondary reflector can be characterized by the following metric, the secondary intercept factor: 
     
       
         
           
             
               
                 
                   
                     
                       γ 
                       
                         2 
                         ⁢ 
                         nd 
                       
                     
                     ⁡ 
                     
                       ( 
                       
                         
                           θ 
                           
                             ⊥ 
                             , 
                           
                         
                         ⁢ 
                         
                           θ 
                            
                         
                       
                       ) 
                     
                   
                   = 
                   
                     
                       
                         Power 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         intercepted 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         by 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         absorber 
                       
                       
                         
                             
                         
                         ⁢ 
                         
                           
                             
                               
                                 Power 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 intercepted 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 by 
                               
                             
                           
                           
                             
                               
                                 secondary 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 reflector 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 aperture 
                               
                             
                           
                         
                         ⁢ 
                         
                             
                         
                       
                     
                     ≅ 
                     
                       
                         
                           γ 
                           
                             2 
                             ⁢ 
                             nd 
                           
                           t 
                         
                         ⁡ 
                         
                           ( 
                           
                             θ 
                             ⊥ 
                           
                           ) 
                         
                       
                       · 
                       
                         
                           γ 
                           
                             2 
                             ⁢ 
                             nd 
                           
                           l 
                         
                         ⁡ 
                         
                           ( 
                           
                             θ 
                              
                           
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   23 
                   ) 
                 
               
             
           
         
       
     
     For a selected absorber tube, the higher the intercept factor, the better the efficiency of the secondary reflector performance. Here, θ ⊥  and θ ∥  are the transversal angle and longitudinal incidence angle, respectively. The secondary intercept factor may be decomposed as two individual secondary intercept factors (γ 2nd   t  and ↓ 2nd   l ) along transversal and longitudinal directions for mathematical simplification. 
     For the Hyperlight Energy collector configuration, the absorber tube position may be fixed at 7 meters and the secondary reflector may have one additional design parameter—the relative edge height, which is defined as the relative vertical distance between the absorber tube center and the secondary edge point, as shown in  FIG.  22   . To calculate the impact of the relative edge height to the secondary intercept factor, a ray-trace software—SolTrace—was employed to perform optical simulation of the Hyperlight Energy collector. SolTrace is a software tool that the National Renewable Energy Laboratory (NREL) has developed to model and analyze the optical performance of CSP collectors. It uses the Monte-Carlo algorithm to simulate the sun shape and can efficiently account for effects of various system optical errors. A SolTrace model was developed to accommodate the Hyperlight Energy collector configuration (geometry and optics) and simulate the optical performance of derived secondary-reflector profiles.  FIG.  23    shows the secondary intercept factor at normal incidence with varying relative edge height of secondary reflectors. It can be seen that the secondary intercept factor becomes highest at the relative edge height of 0.05 m, but the secondary intercept factor is higher than 90% in most cases. To understand the rationale behind this fact, the average surface distance to the absorber tube center is also plotted as a function of relative edge height (see  FIG.  24   ). The maximum secondary intercept factor corresponds to the minimum average distance. The shorter the average distance, the larger the absorber tube acceptance angle to the secondary reflector. 
     After the relative edge height was optimized, the final secondary reflector profile was determined as well. Its secondary intercept factor at different sun positions was then calculated using Soltrace. As shown in  FIG.  25   , the secondary intercept factor of the optimal secondary-reflector profile was plotted as a function of both transversal and longitudinal incidence angles. Note that the secondary reflector reflectance was assumed to be 0.94. From the plot, the secondary intercept factor stayed between 0.91 and 0.92 for the entire range of transversal incidence; and with the increasing longitudinal incidence angle, the secondary intercept factor only started to drop significantly at incidence angles higher than 70°. It is evident that the derived secondary reflector profile is able to remain at a consistently high performance level for the majority of sun positions during a calendar year and exhibit its design robustness with varying incidence angles. 
     Projected collector performance: The collector optical efficiency can be defined as:
 
η(θ)=ρ·τ·α·γ 1st (θ ⊥ ,θ ∥ )·γ 2nd (θ ⊥ ,θ ∥ )=η o   ·IAM (θ ⊥ ,θ ∥ )≈η o   ·IAM   t (θ ⊥ )· IAM   l (θ ∥ )  (24)
 
     Here, ρ is the primary mirror reflectance, τ is the absorber tube glass-envelope transmittance, and α is the average absorber tube coating absorptance. η o  is the nominal optical efficiency and IAM is the incidence angle modifier, which can be approximated by decomposing IAM into two incidence angle modifiers (IAM t  and IAM l ) along transversal and longitudinal directions. γ 1st  is the primary intercept factor accounting for the cosine loss, shading and blocking effect, and intercept accuracy of primary reflector array with respect to the secondary-reflector aperture: 
     
       
         
           
             
               
                 
                   
                     
                       γ 
                       
                         1 
                         ⁢ 
                         s 
                         ⁢ 
                         t 
                       
                     
                     ⁡ 
                     
                       ( 
                       
                         
                           θ 
                           ⊥ 
                         
                         , 
                         
                           θ 
                            
                         
                       
                       ) 
                     
                   
                   = 
                   
                     
                       
                         
                           
                             Power 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             intercepted 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             by 
                           
                         
                       
                       
                         
                           
                             secondary 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             reflector 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             aperture 
                           
                         
                       
                     
                     
                       
                         
                           
                             Power 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             intercepted 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             by 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             the 
                           
                         
                       
                       
                         
                           
                             entire 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             primary 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             reflector 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             aperture 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   25 
                   ) 
                 
               
             
           
         
       
     
     By employing SolTrace, the optical performance of the Hyperlight Energy primary reflector field with the optimal secondary profile was calculated. The resultant incidence angle modifiers are plotted in  FIG.  26   . The nominal optical efficiency at normal incidence is 0.699 from the SolTrace simulations. 
     The optical loss of a linear Fresnel collector comes from four sources: (a) Loss due to component optical properties. This is the effect of primary reflector reflectance, absorber tube absorptance, and absorber-envelope transmittance; (b) Secondary reflector intercept loss. This accounts for the secondary reflector reflectance and the fact that some reflected light by the secondary reflector will miss the absorber tube; (c) Primary intercept loss excluding cosine loss. This accounts for shading and blocking of primary reflector field and the fact that some reflected light by the primary reflector will miss the secondary reflector aperture; and (d) Primary intercept loss—cosine loss. This accounts for the cosine loss of the primary reflector field. 
     The relative contribution of each individual optical loss source is plotted for varying transversal and longitudinal incidence angles in  FIGS.  27  and  28   , respectively. For an increase in both incidence angles, the optical loss due to component optical properties and secondary intercept loss remain roughly constant and this again illustrates the robustness of the secondary reflector surface design (e.g. profile). The cosine loss of the primary reflectors increases with increasing transversal or longitudinal incidence angle, but this is due to the nature of the linear Fresnel configuration and cannot be significantly reduced any further. The primary reflector intercept loss, excluding cosine loss, has a substantial contribution to the collector performance as shown in the  FIGS.  27  and  28   . Particularly for the increasing transversal incidence angle, the portion of optical loss starts to dominate the overall collector optical loss. This portion of primary intercept loss comes primarily from shading and blocking and largely depends on the collector geometrical configuration such as primary reflector spacing and the receiver height. With increasing primary spacing, the shading and blocking will decrease, but more reflected sun rays will miss the receiver due to the increasing optical path. This indicates that there exists an optimal primary spacing and a receiver height that will produce the highest collector performance. At the same time, the determination of a set of optimal collector geometrical parameters will also require accounting for associated economics of the overall linear Fresnel collector system depending on specific applications. The stacked bar plots with different optical-loss sources will shed light on future collector optimization. 
     The optimum secondary reflector shape derived from the method described above depends on a number of collector geometrical parameters. Therefore, a sensitivity analysis was completed where various input variables to the method were varied, including the width of secondary aperture, the secondary height, the absorber size and the absorber height relative to the secondary aperture. The resultant proposed secondary reflector profile for each case is summarized in Table 4 below; e.g. the best fit parameters for the best-fitting third-order polynomial equations (see equation 12 above).  FIGS.  29 A through  29 B  shows the plots obtained for each secondary reflector profile for each test case listed in Table 4. Each of the secondary reflector profiles evaluated had a first half and a second half that was a symmetrical, mirror-image of the first half. As shown in the figures, only the first half of each profile was evaluated. In addition, as described above, the point corresponding to the where the first half and second half of the profile join was placed on the y-axis (x=0), and the outer most point, at one-half the width of the profile, was placed on the x-axis (y=0). Next, all of the data points describing the first half of the profile were normalized and converted to dimensionless numbers by dividing each value of x and y by the width of half of the profile. As a result, the width of the first half of the profile was set to 1.0, positioned at (W, 0), for each of the cases analyzed with the point where the two halves join positioned at (0, D). This procedure insures that a meaningful comparison can be made between differing shaped secondary reflector profiles, even if the dimensionalized shapes (e.g. measure in units of length; feet, meters, etc.) vary significantly due to, for example, different power plant scales. 
     
       
         
           
               
             
               
                 TABLE 4 
               
             
            
               
                   
               
               
                 Sensitivity Test - Third-Order Polynomial Equation Parameters. 
               
            
           
           
               
               
               
               
               
               
               
               
               
               
            
               
                 # 
                 case # 
                 cusp? 
                 a′ 
                 b′ 
                 c′ 
                 d 
                 R2 
                 D 
                 W 
               
               
                   
               
               
                  1 
                 FIG. 7 
                 yes 
                 −0.5889 
                 −0.4195 
                 0.2260 
                 0.7899 
                 0.9998 
                 0.7830 
                 1 
               
               
                  2 
                 FIG. 8 
                 no 
                 −0.7768 
                 −0.0758 
                 0.0378 
                 0.8191 
                 0.9998 
                 0.8210 
                 1 
               
               
                  3 
                 30 
                 yes 
                 −0.3551 
                 −0.3689 
                 0.2223 
                 0.5064 
                 0.9997 
                 0.5392 
                 1 
               
               
                  4 
                 35 
                 yes 
                 −0.3554 
                 −0.4034 
                 0.2110 
                 0.5460 
                 0.9999 
                 0.5684 
                 1 
               
               
                  5 
                 40a 
                 yes 
                 −0.4507 
                 −0.3502 
                 0.2143 
                 0.5872 
                 1.0000 
                 0.6132 
                 1 
               
               
                  6 
                 40b 
                 yes 
                 −0.2600 
                 −0.5967 
                 0.2333 
                 0.6257 
                 1.0000 
                 0.6471 
                 1 
               
               
                  7 
                 40c 
                 yes 
                 −0.3222 
                 −0.4806 
                 0.2279 
                 0.5742 
                 0.9999 
                 0.5955 
                 1 
               
               
                  8 
                 40d 
                 yes 
                 −0.5084 
                 −0.4038 
                 0.1926 
                 0.7262 
                 0.9999 
                 0.7449 
                 1 
               
               
                  9 
                 40e 
                 yes 
                 −0.8999 
                 −0.0655 
                 0.0714 
                 0.9144 
                 0.9996 
                 0.9281 
                 1 
               
               
                 10 
                 40f 
                 yes 
                 −0.3638 
                 −0.3753 
                 0.2061 
                 0.5314 
                 0.9998 
                 0.5575 
                 1 
               
               
                 11 
                 40g 
                 yes 
                 −0.3160 
                 −0.4010 
                 0.2203 
                 0.4969 
                 0.9997 
                 0.5281 
                 1 
               
               
                 12 
                 40h 
                 yes 
                 −0.2223 
                 −0.5719 
                 0.2328 
                 0.5607 
                 0.9999 
                 0.5798 
                 1 
               
               
                 13 
                 40i 
                 yes 
                 −0.1467 
                 −0.6347 
                 0.2350 
                 0.5464 
                 0.9999 
                 0.5667 
                 1 
               
               
                   
                   
                 min 
                 −0.8999 
                 −0.6347 
                 0.0378 
                 0.4969 
                 0.9996 
                 0.5281 
                 1 
               
               
                   
                   
                 avg 
                 −0.4282 
                 −0.3959 
                 0.1947 
                 0.6327 
                 0.9998 
                 0.6517 
                 1 
               
               
                   
                   
                 max 
                 −0.1467 
                 −0.0655 
                 0.2350 
                 0.9144 
                 1.0000 
                 0.9281 
                 1 
               
               
                   
               
            
           
         
       
     
     The foregoing discussion and examples have been presented for purposes of illustration and description. The foregoing is not intended to limit the aspects, embodiments, or configurations to the form or forms disclosed herein. In the foregoing Detailed Description for example, various features of the aspects, embodiments, or configurations are grouped together in one or more embodiments, configurations, or aspects for the purpose of streamlining the disclosure. The features of the aspects, embodiments, or configurations, may be combined in alternate aspects, embodiments, or configurations other than those discussed above. This method of disclosure is not to be interpreted as reflecting an intention that the aspects, embodiments, or configurations require more features than are expressly recited in each claim. Rather, as the following claims reflect, inventive aspects lie in less than all features of a single foregoing disclosed embodiment, configuration, or aspect. While certain aspects of conventional technology have been discussed to facilitate disclosure of some embodiments of the present invention, the 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. Thus, the following claims are hereby incorporated into this Detailed Description, with each claim standing on its own as a separate aspect, embodiment, or configuration.