Patent Publication Number: US-9842954-B1

Title: Radiation collection apparatus with flexible stationary mirror

Description:
RELATED APPLICATIONS 
     This application claims priority based upon the Provisional Patent Application No. 61/862,993, filed Aug. 7, 2013. 
    
    
     TECHNICAL FIELD 
     The present invention relates to a radiation collection apparatus including a stationary but flexible mirror for optimally concentrating solar radiation on a movable radiation absorption element. 
     BACKGROUND OF THE INVENTION 
     A well-known way of capturing and converting solar radiation into a more useful form of energy such as electricity is to use a radiation focusing device (typically in the form of one or more mirrors) to concentrate and focus incident solar radiation on a radiation absorption element (typically a vessel containing a working fluid or an array of photovoltaic cells). As the angle of incidence of the solar radiation changes throughout the course of a day, the relative positions of the radiation focusing device and the absorption element are adjusted to keep the concentrated solar radiation focused on the radiation absorption element. Intuitively, the most practical and cost-effective mechanizations pair a fixed radiation focusing device with an absorption element that moves with respect to the radiation reflector. Representative examples of this approach are disclosed, for example, in the U.S. Pat. Nos. 3,868,823 and 4,071,017 to Russell, Jr. et al., the U.S. Pat. No. 3,994,435 to Barr, the U.S. Pat. No. 4,318,394 to Alexander, and the U.S. Patent Publication No. 2011/0168160 to Martinez Moll et al. 
     The radiation focusing device may comprise a series of flat mirrors, as disclosed by Russell Jr. et al., but more commonly comprises one or more curved (parabolic, spherical or cylindrical) mirrors, as disclosed by Barr, Alexander, and Martinez Moll et al. As disclosed in each of the foregoing references, the mirrors can be configured as a linearly extending trough so that the reflected solar radiation is focused above the mirrors on a linearly extending line or region that moves in a circular/cylindrical path as the angle of incidence of the solar radiation changes. Such a configuration is particularly advantageous in terms of mechanical simplicity because the radiation absorption element can simply be pivoted about a fixed point aligned with the center of the circular/cylindrical path of focus. But in practice, the disclosed reflector configurations allow significant divergence of the reflected solar energy as the solar angle of incidence strays from an optimal angle. As a result, the energy focused on the radiation absorption element is less concentrated, resulting in less efficient energy conversion. Accordingly, many such systems include a provision for adjusting the curvature of the radiation focusing device as a function of the solar angle of incidence to improve the concentration of solar energy focused on the radiation absorption element. See the aforementioned patent to Alexander, for example. And in mechanizations including a movable radiation reflector, see for example, the U.S. Pat. No. 4,056,309 to Harbison et al. and the U.S. Patent Publication No. 2012/0285440 to Kosaka et al. 
     SUMMARY OF THE INVENTION 
     The present invention is directed to an improved solar radiation collection apparatus including a stationary radiation focusing device and a movable radiation receiving element, where the radiation focusing device is a curved mirror optimally configured to concentrate the reflected solar energy in a linear circle of focus aligned with the central axis of the mirror. 
     Key to the invention is the recognition that solar energy (from any angle) incident on any given point of the mirror&#39;s surface is naturally focused onto a circle of focus that intersects the mirror at that point; and that points symmetrically disposed about the central axis of the mirror perfectly reflect onto the same circle of focus. If the radiation receiving element is constrained to follow the circle of focus associated with a given point(s), solar radiation incident on other regions of the mirror&#39;s surface will be less focused on the radiation receiving element. The various embodiments of this invention take advantage of this phenomenon in an apparatus that includes a radiation receiving element constrained to follow a circle of focus associated with a given point (or points) on the mirror&#39;s surface, a mirror support structure that holds fixed the surface of the mirror in a region about the given point(s), and an adjustment mechanism coupled to the mirror at locations removed from the given point(s) for flexing the other regions of the mirror in a manner to compensate for focusing error so that solar radiation incident on such other regions is more nearly focused on the radiation receiving element. 
     In one embodiment, the radiation receiving element is constrained to follow a circle of focus corresponding to the center of the mirror, which circle of focus is tangent to the surface of the mirror at its center. In this case, the mirror support structure holds fixed the center of the mirror, and the adjustment mechanism is coupled to the ends of the mirror for flexing the non-central portions of the mirror to compensate for focusing errors associated with solar energy incident on those portions of the mirror&#39;s surface. 
     In another embodiment, the radiation receiving element is constrained to follow a circle of focus corresponding to the ends of the mirror, which circle of focus intersects the ends of the mirror. In this case, the support structure holds fixed the ends of the mirror, the mirror is split at its center, and the adjustment mechanism is coupled to the inboard end of each mirror segment to compensate for focusing errors associated with solar energy incident on all but the ends of the mirror&#39;s surface. 
     In a further embodiment, the radiation receiving element is constrained to follow a circle of focus corresponding to a pair of points midway between the center and ends of the mirror, which circle of focus intersects the midpoints of the mirror. In this case, the support structure holds fixed the midpoints of the mirror, the mirror is split at its center, and the adjustment mechanism is coupled both the ends of the mirror and to the inboard end of each mirror segment to compensate for focusing errors associated with solar energy incident on all but the midpoints of the mirror&#39;s surface. 
     Due to the intrinsically optimal nature of these configurations, the reflected solar energy remains more concentrated with off-center angles of solar incidence, so that less corrective adjustment of the mirror curvature is needed. In variations on these embodiments, the radiation receiving element may be a secondary mirror that re-focuses and concentrates the reflected solar radiation on an absorption element. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG. 1  is a diagram of a bowl-shaped mirror. 
         FIGS. 2, 2A and 2B  are diagrams of a trough-shaped mirror. 
         FIG. 2C  is a diagram showing various circles of focus associated with different points on a trough shaped mirror. 
         FIG. 3  is a representation of a solar radiation collection apparatus according to a first embodiment of this invention. 
         FIG. 4  is a diagrammatic representation of a solar radiation collection apparatus according to a second embodiment of this invention. 
         FIG. 5  is an isometric representation of the solar radiation collection apparatus of  FIG. 4 . 
         FIG. 6  is a representation of a solar radiation collection apparatus according to a third embodiment of this invention. 
         FIG. 7  is a representation of a solar radiation collection apparatus according to a fourth embodiment of this invention. 
         FIG. 8  is a representation of a solar radiation collection apparatus according to a fifth embodiment of this invention. 
         FIG. 9  is a representation of a solar radiation collection apparatus according to a sixth embodiment of this invention. 
     
    
    
     DESCRIPTION OF THE PREFERRED EMBODIMENT 
     The solar radiation collection apparatus of the present invention is described in the context of a solar energy conversion apparatus in which solar radiation reflected from a stationary radiation focusing device (mirror) is focused on a radiation receiving element. In the illustrated embodiments, the absorption element is depicted as a tubular vessel through which is circulated the working fluid of a turbine-based electricity generator plant. Alternately, the absorption element could be implemented with an array of photovoltaic cells for converting the solar radiation into electricity directly. And in some applications, of course, the collected solar energy may be used for its heat without other energy conversion mechanisms. Hence the novelty of the present invention does not reside in the energy conversion aspect per se, but in the configuration and structure of the stationary radiation reflector element, and its coordinated control relative to the position of a movable radiation receiving element. 
     As illustrated in  FIGS. 1, 2 and 2A-2C , the radiation focusing device is a slightly-curved mirror  10 ; it may be bowl-shaped (i.e.,  3 D) as illustrated in  FIG. 1 , or trough-shaped (i.e.,  2 D), as illustrated in  FIGS. 2 and 2A-2C . At the center of a bowl-shaped mirror  10 , designated in  FIG. 1  by the point O, the curvature or bend of the mirror can be decomposed into two orthogonal perpendicular axes A and B. Each of the axes A and B is contained in a corresponding plane perpendicular to the mirror  10  at point O; for example, PA designates a plane containing the axis A. The trough-shaped mirrors of  FIGS. 2 and 2A-2C  are represented by the mirror section lying in plane PA. 
     Referring to  FIG. 2A , when the solar radiation incident on the center O of the mirror  10  has an angle of incidence a with respect to the normal vector V, and the reflected radiation will intrinsically focus at a point ƒ(α) on the plane PA. Since the curvature of the mirror is slight, the function ƒ draws out a circle C in the plane PA as the incidence angle α varies. The circle of focus C has radius about one-quarter that of radius of curvature of the mirror  10 ; accordingly, its center F is located about one-quarter radius of curvature from the mirror  10 . Since the mirror  10  of  FIG. 2  is trough-shaped, the reflected radiation will be focused on a line perpendicular to the plane PA and passing through the point ƒ(α), and the radiation absorption element D can be in the shape of a linearly extending bar or tube centered about the point ƒ(α). Advantageously, the absorption element D can be mounted on a pivot arm E that is pivoted at the center F of the circular focus path C. As the incident angle of solar radiation changes, the pivot arm E is pivoted about point F so that the reflected radiation remains focused on the radiation absorption element D. For each incremental change α′ of the solar incidence angle α, the pivot arm E is rotated an angle of 2α′. A control system (not shown) pivots arm E about point F in response the sensed solar radiation angle of incidence, or in response to look-up table data as a function of time and date. 
       FIG. 2A  also illustrates that solar radiation incident on other regions of the mirror&#39;s surface is not naturally focused on the same circle of focus C. In other words, there is one natural circle of focus for any given point on the mirror&#39;s surface, and since the absorption element D follows that circle of focus, solar radiation incident on other points of the mirror will focus less perfectly on the absorption element D. And because the objective is to focus all reflected solar energy on the absorption element D, the radiation reflected from such other points is considered to have a focusing error. 
     As mentioned above, the present invention springs from the recognition that solar energy incident on any given point of a slightly curved mirror is perfectly reflected onto a circle of focus that intersects the mirror at that point, and points equidistant from the center of the mirror  10  perfectly reflect onto the same circle of focus. In all cases, the circle of focus is centered about the mirror&#39;s central axis N—that is, an axis normal to the mirror  10  and passing through its center O.  FIG. 2A  depicts a circle of focus C for the center point O of the mirror  10 ; this circle of focus intersects the mirror  10  at point O, and hence, is tangent to the surface of mirror  10 .  FIGS. 2 and 2B  depict a circle of focus C for the symmetrically displaced points P 1  and P 2 ; this circle of focus intersects the mirror  10  at points P 1  and P 2 . These are but two examples, as there are different circles of focus for any given pair of symmetrically displaced points. 
     In general, the circle of focus for any given point on the mirror&#39;s surface may be defined in terms of the mirror&#39;s radius of curvature R and the arc angle β of the point with respect to the center point O. This is graphically illustrated in  FIG. 2B , where the depicted mirror  10  has a radius of curvature R, and point G represents the center of curvature of the mirror  10 —that is, a point on the central axis N at a distance of R from the mirror&#39;s surface. The arc angle β is the angle referenced to point G, subtending between the central axis N and a given point (P 1  or P 2  in the illustrated example) on the surface of the mirror. Of course, for the central point O, arc angle β is zero. The circle of focus C may be characterized by its radius r and the height h of its center with respect to the mirror&#39;s center point O. The values of r and h are given in terms of radius R and arc angle β as follows: 
     
       
         
           
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       FIG. 2C  shows three different circles of focus C for a given radius of curvature R, and different values of arc angle β. If the mirror  10  has a total arc angle of 24°, the value of arc angle β for the end points of the mirror  10  will be 12°, the value of arc angle β for midpoints of the mirror  10  will be 6°, and of course, the arc angle β is zero for the center point O. Solar radiation incident on center point O will be perfectly focused on the circle of focus C β=0 ; solar radiation incident on the midpoints P 1  and P 2  will be perfectly focused on the circle of focus C β=6 ; and solar radiation incident on the mirror endpoints P 3  and P 4  will be perfectly focused on the circle of focus C β=12 . Among other things,  FIG. 2C  shows that the focusing error associated with solar radiation approximately normal to the mirror  10  is only slight because the various circles of focus nearly coincide, but that the focusing error increases dramatically as the angle of incidence increases beyond that point. And since an absorption element D can be constrained to follow but one circle of focus, it is particularly important to compensate for focusing errors when the solar radiation incidence angle is off-center. While it is generally knows to adjust the curvature of a mirror as a function of the solar angle of incidence to improve the concentration of solar energy focused on a radiation absorption element, the present invention provides an improved solar collection apparatus including an radiation receiving element constrained to follow a circle of focus corresponding to a specified point(s) on the mirror, a mirror support structure that holds fixed the surface of the mirror at the specified point(s), and an adjustment mechanism coupled to the mirror at locations removed from the given point for flexing the mirror in a manner to compensate for focusing error associated with solar energy incident on other points of the mirror&#39;s surface. 
       FIG. 3  depicts a first embodiment in which the radiation reflector is implemented with a single slightly bent trough-shaped mirror  10 , and a radiation receiving absorption element  32  that allows a circular path of focus  20  corresponding to the center of mirror  10 . The mirror  10  is supported by a mirror support structure including a base  16 , sets of upper and lower crossbeams  12  and  14 , and pair of support beams  18   a ,  18   b . The lateral margins of mirror  10  are sandwiched between upper and lower crossbeams  12  and  14  extending parallel to the mirror&#39;s bent axis, and the longitudinal margins of mirror  10  are affixed to support beams  18   a ,  18   b  extending parallel to the mirror&#39;s flat axis. The crossbeams  12 ,  14  vary in thickness (profile) as shown, and their combined thickness varies with distance from the center of the mirror  10  to impart a variable stiffness to the mirror  10  along its bent axis, while the support beams  18  reduce drooping of the mirror surface along its flat axis. The bottom crossbeams  14  are fastened at their centers to the base  16 , to thereby hold fixed the center of the mirror  10 , while the crossbeams  12 ,  14  permit controlled flexure of the non-central regions of the mirror  10 . 
     Solar radiation incident on the center of the mirror  10  will naturally reflect onto a circular focus path  20  tangent to the surface of mirror  10 . The radiation absorption element  32  is constrained to follow the circular focus path  20  by a series of pivot arm assemblies  24  distributed along the flat axis of the mirror  10 . The side view of  FIG. 3  depicts one such pivot arm assembly  24 ; it includes a vertical support column  22  mounted on the base  16 , a support shaft  26  rotatably mounted on the support column  22  at the center of the circular focus path  20 , a pivot arm  28  mounted on the support shaft  26  for rotation therewith, and a cam element  30  also mounted on the support shaft  26  for rotation therewith. The length of the pivot arm  28  is such that it extends somewhat beyond the circular focus path  20 , and the absorption element  32  is supported by the pivot arm  28  at the circular focus path  20 . 
     As the direction of the incident radiation changes, the support shaft  26  is rotated as required to maintain the radiation absorption element  32  in alignment with the focus of the reflected radiation. The rotation of support shaft  26 , and hence pivot arm  28 , will be approximately twice the change in the mean direction of incident radiation, as indicated in the graphical representation of  FIG. 2A . Rotation of the support shaft  26  can be achieved using a motor (not shown), in a manner given by common practice; this involves either preprogramming a fixed path using a computer and the knowledge of the radiation direction as a function of time, or using a feedback mechanism designed to maximize incident radiation on the radiation absorption element  32 . 
     The cam element  30  and cables  34   a ,  34   b  are used to nonlinearly flex the mirror  10  as the support shaft  26  is rotated to move the radiation absorption element  32 . The cam element  30  is rigidly mounted on support shaft  26  proximal to the support column  22 , and the cables  34   a ,  34   b  are oppositely wrapped about the cam element  30 , as depicted in  FIG. 3 . The loose (unwrapped) end of cable  34   a  is fastened to the longitudinal margin of mirror  10  adjacent the support beam  18   a , and the loose (unwrapped) end of cable  34   b  is fastened to the longitudinal margin of mirror  10  adjacent the support beam  18   b . When the support shaft  26  rotates to adjust the location of the radiation absorption element  32  on the circular focus path  20 , the cables  34   a  and  34   b  respectively wrap and unwrap on the cam element  30 , causing the mirror  10  to flex or bend about its bent axis. The amount of flexure of the mirror  10  is a function of the shape of the cam element  30 , the profile of the crossbeams  12  and  14 , and the degree of rotation of the support shaft  26 . 
     The curvature at the center of the mirror  10  is perfect for focusing light along the circular focus path  20 , as this is how the circular focus path  20  was defined. With increasing distance from the center of mirror  10 , however, the surface of the mirror must be increasingly flexed in order to concentrate the reflected radiation on the radiation absorption element  32  as the solar incidence angle changes. And since the amount of mirror flexure depends upon the shape of the cam element  30 , the profiles of the crossbeams  12 ,  14 , and the degree of rotation of the central axle  26 , the concentration of reflected radiation incident on the radiation absorption element  32  can be optimized by suitably shaping the profiles of the crossbeams  12 ,  14  and the cam element  30 . 
     To model the effect of the crossbeams  12  and  14 , it can be assumed that the cable  34   a  applies a force F to the longitudinal margin of the mirror  10 , and that the mirror&#39;s curvature at a distance D from its center changes by a factor kD determined by the combined thickness T of crossbeams  12  and  14 . This provides a linear correction, and matches the fact that no correction is needed at the center of mirror  10  (i.e., at D=0). To create such a linear correction, the crossbeams  12 ,  14  will have a combined thickness T(x) given by:
 
 T ( x )= K [( d   0   −d )/ d]   1/3  
 
where K is any constant, d is the (positive) distance from the center of the mirror  10  along its bent axis, and d 0  is the distance from the center of the mirror  10  to the point where the force F is applied by the cable  34   a . This thickness T(x) goes to zero at the edge of the mirror  10 , and becomes very large for small values of d. In practice, near d=0, T(x) has a large enough value to negate any significant change in curvature at the center of the mirror  10 . The thicknesses of the crossbeams  12 ,  14  are thus restricted. The original curvature of the crossbeams  12 ,  14 , and thus the mirror  10 , with no force applied to cables  34   a ,  34   b  can still be chosen, as can the shape of the cam element  30 . This can be done by first restricting the crossbeam shape to require a mirror shape that focuses light perfectly (which will be the correct subsection of a parabola) at some specific solar incidence angle. Given this restriction, one can numerically fit the shape of the cam element  30 , and thus the distance of cable pulled, as a function of solar incidence angle to minimize the overall error. The overall error can be defined as the maximum deviation of the mirror  10  from one that focuses perfectly for any allowed incidence angle.
 
       FIGS. 4-5  depict a second embodiment in which the radiation reflector is implemented with a split trough-shaped mirror  10 , and the absorption element  32  follows a circular path of focus  20  corresponding to the endpoints of mirror  10 . The endpoints of the mirror  10  are held fixed, and the mirror  10  is split in the middle to form two mirror segments  10 ′ and  10 ″ so that portions of the mirror away from the endpoints can be flexed to compensate for focusing errors. Each mirror segment  10 ′,  10 ″ is attached along its bent axis to upper and lower crossbeams  12  and  14 . The bottom crossbeam  14  is connected to the base  16  near the outboard edge of each mirror  10 ′,  10 ″. As in the embodiment of  FIG. 3 , the absorption element  32  is supported by a series of pivot arm assemblies  24 , each comprising a vertical support column  22  mounted on the base  16 , a support shaft  26  rotatably mounted on the support column  22  at the center of the circular focus path  20 , a pivot arm  28  mounted on the support shaft  26  for rotation therewith, and a cam element  30  also mounted on the support shaft  26  for rotation therewith. 
     In the embodiment of  FIGS. 4-5 , the cables  34   a  and  34   b  are attached to the center, or inboard, ends of the mirror segments  10 ′,  10 ″. The bending of the mirror segments  10 ′,  10 ″ is determined by the profile (and hence, the stiffness) of the crossbeams  12  and  14 , the shape of the cam element  30 , and the amount the pivot arm  28  rotates. The thickness/profile of the crossbeams  12  and  14  and the shape of the cam element  30  are chosen to optimize solar concentration on the radiation absorption element  32  as the angle of incident radiation varies through a chosen range. This optimization can be done as in the first embodiment. In this case, the optimal thickness of the crossbeams  12 ,  14  is constant, as illustrated in  FIGS. 4-5 . 
       FIG. 6  depicts a third embodiment in which the radiation reflector is implemented with a single slightly bent trough-shaped mirror  10 , as in the embodiment of  FIG. 3 . However, the third embodiment utilizes a trough-shaped secondary reflector  40 . The mirror  10  is supported by crossbeams  12 ,  14  and a base  16 , with the crossbeams  12 ,  14  being fastened to the base  16  at their midpoints. Each of the distributed pivot arm assemblies  24  includes a vertical support column  22  mounted on the base  16 , a support shaft  26  rotatably mounted on the support column  22  at the center of the circular focus path  20 , a pivot arm  28  mounted on the support shaft  26  for rotation therewith, and a cam element  30 . The pivot arm  28  supports the radiation absorption element  32  at the circular path of focus  20  as before, and the trough-shaped secondary reflector  40  is rigidly supported on a secondary support shaft  46 . The secondary support shaft  46  is rotatably mounted on pivot arm  28  near its outboard end, outboard of the radiation absorption element  32 , and is coupled to the support shaft  26  by a cable or chain  44  so that rotation of the support shaft  26  produces corresponding rotation of the secondary support shaft  46 , and hence, secondary reflector  40 . But the radius of the secondary support shaft  46  is twice that of the support shaft  26  so that the secondary reflector  40  rotates only half as much as the pivot arm  28 . This optimizes light incident on the secondary reflector  40  and radiation absorption element  32 . 
     Bending of the mirror  10  is determined by the shape and variable profile of the crossbeams  12   14 , the length of the cables  34   a ,  34   b , and the rotary position of support shaft  26 . These are optimized to maximize the radiation incident on the secondary reflector  40 . The secondary reflector  40  is a compound parabolic mirror, and re-focuses the reflected radiation from mirror  10  onto the radiation absorption element  32 . 
       FIG. 7  depicts a fourth embodiment in which the radiation reflector is implemented with a split trough-shaped mirror as in the embodiment of  FIGS. 4-5 , and a radiation receiving element  52  constrained to follow a circular path of focus  20  corresponding to the midpoints of mirror  10 . The midpoints of the mirror  10  are held fixed by the base  16 , and the mirror  10  is split in the middle to form two mirror segments  10 ′ and  10 ″ so that portions of the mirror away from its midpoints can be flexed to compensate for focusing errors. In this case however, the adjustment cables are attached both to the endpoints of the mirror segments  10 ′,  10 ″ and to the inboard ends of the mirror segments  10 ′ and  10 ″. The radiation receiving element  52  is a re-focusing trough-type mirror that cooperates with a tertiary reflector  54  to permit the radiation absorption element to be fixed instead of movable. In this embodiment, the support shaft  26  is rigidly attached to the support column  22 , and also serves as the radiation absorption element. Since the support shaft/absorption element  26  is fixed, the pivot arm  28  is rotatably mounted on the support shaft  26 , and the electric drive motor (not shown) rotates the pivot arm  28  instead of the support shaft/absorption element  26 . A pair of tapered crossbeams  12   a ,  12   b  and  14   a ,  14   b  sandwich lateral margins of the thin slightly bent mirror segments  10 ′,  10 ″ to impart a variable stiffness to the mirror segments  10 ′,  10 ″ along their bent axes, and each lower crossbeam  14   a ,  14   b  rigidly mounts to the base  16  approximately at its mid-point. 
     Bending of the mirror segments  10 ′,  10 ″ in relation to rotation of the pivot arm  28  is achieved with first and second pairs of cables  34   a ,  34   b  and  50   a ,  50   b . The cables  34   a ,  34   b  are attached to the inboard ends of the mirror segments  10 ′,  10 ″ as in the embodiment of  FIGS. 4-5 , and are oppositely wrapped around a cam element  30  rigidly fastened to the inboard end of pivot arm  28 . The cables  50   a ,  50   b  are fastened to the longitudinal margins of mirror segments  10 ′,  10 ″ in a manner analogous to the embodiment of  FIG. 3 , and oppositely wrapped around the cam element  30 , or a secondary cam element also fastened to the inboard end of pivot arm  28 . 
     A secondary support shaft  46  is rotatably mounted on the outboard end of the pivot arm  28  substantially at the circular path of focus  20 , and the secondary reflector  52  is rigidly fastened to the secondary support shaft  46 . A cable or chain  44  couples the support shaft/absorption element  26  to the secondary support shaft  46 , so that when the motor rotates the pivot arm  28  with respect to the base  16 , the secondary reflector  52  also rotates with respect to the base  16 . But the radius of the secondary support shaft  46  is twice that of the support shaft/absorption element  26  so that the secondary reflector  52  rotates only half as much as the pivot arm  28 . The secondary reflector  52  is a hyperbolic mirror, and re-focuses the radiation gathered from the mirror segments  10 ′,  10 ″ on the tertiary reflector  54 , which is rigidly fastened to the inboard end of the pivot arm  28 . The tertiary reflector  54  is a compound parabolic reflector, and re-focuses light the radiation gathered from the secondary reflector  52  on the support shaft/absorption element  26 . The shape of the cam element(s)  30  and the crossbeams  12  and  14  are chosen to maximize the radiation incident on the support shaft/absorption element  26  for a given range of angles of mean incident radiation. This maximization can be done in a manner similar to that described in the other embodiments. 
       FIG. 8  depicts a fifth embodiment that features primary and secondary mirrors  10  and  58  to concentrate radiation along two different axes and achieve higher concentration of radiation on a smaller radiation absorption element  56 . The first axis coincides with the bent axis of the primary mirror  10 , and the second axis coincides with the bent axis of the secondary mirror  58 . Incident radiation is concentrated along the first axis by the primary mirror  10 , and then along the second axis by the secondary mirror  58 . 
     The flat axis of primary mirror  10  is flanked by pair of pivot arm assemblies  24 , each such pivot arm assembly  24  including a vertical support column  22  mounted on a base  16 , a central support shaft  26  rotatably mounted on the vertical support column at its upper end, and a pivot arm  28  rigidly fastened to the support shaft  26  for rotation therewith. The secondary mirror  58  is fastened to the outboard ends of the pivot arms  28  to receive radiation reflected from the primary mirror  10 , and a motor  64  mounted on one of the vertical support columns  22  rotates the support shaft  26  and thus the secondary mirror  58  with respect to the base  16  to keep the secondary mirror  58  as close as possible to the focus point of radiation reflected from the primary mirror  10 . 
     A secondary mount  60  is provided at the midpoint of the secondary mirror  58  for supporting the radiation absorption element  56 . A pair of secondary support arms  62  fastened to the secondary mounting plate  60  extend inward toward the support shaft  26 , and rotatably support a secondary support shaft  68 . The radiation absorption element  56  is supported between a pair of secondary pivot arms  70 , which in turn, are rigidly fastened to the secondary support shaft  68  so that rotation of the secondary support shaft  68  with respect to secondary support arms  62  produces corresponding rotation of the radiation absorption element  56  with respect to the secondary mirror  58 . A second electric motor  74  fastened to one of the secondary support arms  62  is provided to rotate the secondary support shaft  68 , and hence the radiation absorption element  56 , with respect to the secondary mirror  58 . 
     As in the embodiments of  FIGS. 3 and 7 , a set of cables  34   a ,  34   b  at each pivot arm assembly  24  couple the longitudinal margins of the primary mirror  10  to the cam elements (not shown) on the support shaft  26  for flexing the primary mirror  10  in relation to rotation of the support shaft  26 . In addition, the primary mirror  10  may be equipped with crossbeams as in previous embodiments. 
     The bending of primary mirror  10  is coupled to rotation of the support shaft  26 , and hence secondary mirror  58 , by the motor  64 , which in turn is coordinated to match the incident radiation. As before, the speed that the support shaft  26  turns is twice the speed at which the direction of the incident radiation, projected to the first axis, moves. This required motion of the support shaft  26  is achieved in a manner given by common practice, as mentioned in previous embodiments. 
     For sufficiently focused radiation with mean angle within some predetermined range, the incident radiation should all or mostly fall on the secondary mirror  58 . The secondary mirror  58  is curved in a similar manner, though normally at a different scale, to the primary mirror  10 . In this embodiment, for ease of construction the secondary mirror  58  is rigidly mounted to the pivot arm assemblies  24 , but the secondary mirror  58  could alternatively be adaptively bent in a manner similar to the primary mirror  10 . 
       FIG. 9  depicts a sixth embodiment that features a non-linear concentrator design, where the slightly-curved mirror  76  is dish-shaped instead of trough-shaped. The reflected radiation along any given axis of the mirror  76  has a circular path of focus that includes the surface of (i.e., is tangent to) the mirror  76 . The reflected radiation is concentrated where the various circular paths of focus coincide, and a radiation absorption element  88  is located at that point. The structure illustrated in  FIG. 9  and described below provides a way of supporting the radiation absorption element  88 , adjusting its position based on changes in the incident angle of solar radiation, and flexing the mirror  76  accordingly to concentrate the reflected radiation on the radiation absorption element  88  as the solar incidence angle varies from the ideal angle. 
     The radiation absorption element  88  is supported on the outboard end of a rod  86 . The inboard end of rod  86  is supported by a hub  80 , and the hub  80  is rigidly supported over the center of mirror  76  by a distributed array of support arms  78 . The coupling between hub  80  and the inboard end of rod  86  is a ball-and-socket joint  82 , and a two-axis motor controller (not shown) mounted on hub  80  controls the angular orientation of the rod  86  in relation to the incident angle of solar radiation to maintain the radiation absorption element  88  at the point of maximally concentrated reflected radiation; any conventional two-axis tracker can be used for this purpose. 
     Curvature adjustment of mirror  76  is achieved with a cable linkage coupling the rod  86  to the margins of the mirror  76 . A set of four arms  90   a ,  90   b ,  90   c ,  90   d  fastened to the rod  86  extend perpendicular to the rod  86  at intervals of 90 degrees, and a set of four cables  92   a ,  92   b ,  92   c ,  92   d  passing through openings provided in the hub  80  couple the margin of the mirror  76  to the outboard ends of the arms  90   a ,  90   b ,  90   c ,  90   d . The openings in the hub  80  and the points of cable attachment are aligned with the arms  90   a ,  90   b ,  90   c ,  90   d  so that movement of a given arm  90   a ,  90   b ,  90   c  or  90   d  produces a corresponding movement of the cable  92   a ,  92   b ,  92   c ,  92   d  fastened to the opposite side of the mirror  76 . For example, if a given tilting of the rod  86  produces upward movement of the outboard end of arm  90   a , the cable  92   a  attached to the opposite side of the mirror  76  is pulled upward to increase curvature in that quadrant of the mirror  76 ; of course, the opposite effect would simultaneously occur on the opposite side of the mirror  76 . Thus, the cables  92   a ,  92   b ,  92   c ,  92   d  bend the mirror  76  based on the location of the cable attachment to the mirror  76 , the support hub  80  and the cable attachment to an arm  90   a ,  90   b ,  90   c ,  90   d , as well as the possibly variable rigidity of the mirror  76 . These are all chosen to maximize the concentration of radiation on the radiation absorption element  88  for a given range of incident radiation angles. 
     While the present invention has been described with respect to the illustrated embodiment, it is recognized that numerous modifications and variations in addition to those mentioned herein will occur to those skilled in the art. Accordingly, it is intended that the invention not be limited to the disclosed embodiment, but that it have the full scope permitted by the language of the following claims.