Patent Application: US-201113095115-A

Abstract:
parabolic mirror . the mirror includes a flexible material with a reflective surface and a rear surface . a flexible band is in contact with the rear surface of the flexible material . the bending stiffness of the band as a function of distance along its length is selected so that the band and the flexible material in contact therewith assume a parabolic shape when ends of the band are moved toward one another . in a preferred embodiment , the bending stiffness of the band is achieved by controlling the second moment of area of the band along its length . the second moment of area may be adjusted by altering the width of the band along its length or by altering the thickness of the band along its length , or a combination of the two .

Description:
the approach presented herein for designing and fabricating precision parabolic mirrors as shown in fig6 a consists of a thin , flat , very flexible metal sheet 14 with a highly reflective surface 16 and a “ backbone ” band 18 attached to its rear surface . the figure of the “ backbone ” band 18 is optimized to form the sheet 14 into a precision parabola when the two ends of the band 18 are pulled toward each other by a predetermined amount . this result can be achieved using a simple spacer rod or an active position control system when high precision requires real - time adjustment . an analytical model is used to optimize the band &# 39 ; s shape after it is deformed so that it is parabolic . the band 18 is cut from a flat plate with a stiffness that is substantially higher than the mirror sheet 14 . as discussed below , the elastic properties of the band 18 can also be tuned to account for the mirror plate &# 39 ; s stiffness . it is also shown herein that the band 18 profile can be determined numerically using finite element analysis ( fea ) combined with a numerical optimization method . these numerical results agree well with the analytical solutions . rather than optimizing the band stiffness by varying its width , its thickness , ( s ), can also be optimized to achieve the desired shape , see fig6 ( b ). in some designs it may be desirable to vary both the band &# 39 ; s thickness t ( s ) and width b ( s ) on the initial flat band . in general , varying the thickness , t ( s ), would be a more costly manufacture than a uniform thickness hand . however the thickness , as a function of length , t ( s ), can be manufactured more simply by using a multi - layer band that approximates the variable thickness solution . moreover , the bands can also be optimized by punching holes on uniform width bands in approximately continuous patterns . however , this could create stress concentration problems in areas near the holes . the backbone - band concept &# 39 ; s validity is demonstrated herein by finite element analysis and by laboratory experiments . in the experiments , mirror bands of various profiles were fabricated and tested in the laboratory using a collimated light source ( that emulates direct sunlight ) and outdoors in natural sunlight . our studies suggest that this concept would permit essentially mirror elements to be easily fabricated and efficiently packaged and shipped to field sites and then assembled into the parabolic mirrors for mirror solar collectors with potentially substantial cost reductions over current technologies . here a model based on euler - bernoulli beam theory of a flat band that will form a desired parabolic shape by moving its two ends toward each other to a given distance , l , is presented , see fig6 ( a ). it is assumed that by proper selection of the bending stiffness ei ( s ) of the band as a function of the distance , s , along its length a parabolic shape results when the band is deformed , where i ( s ) is the second moment of area of the band and e ( s ) is the modulus of elasticity of the band material . the thickness t ( s ) is much smaller than the length s of the band , so while the deflection is large ( rotation and displacement ), the shear stresses are small and hence euler - bernoulli beam equations can be used . the final distance l ( parabolic chord length ) between the two band ends is specified , and the rim angle of the desired parabola is given as θ , see fig6 ( c ). the end deflection is achieved by the application of forces , f , during assembly and held in place by spacer rods , or an active control system . if the focal length of the parabolic mirror is f , then the desired shape of the deformed band is given by the well - known relationship , see fig6 ( c ): considering the energy efficiency of the mirror , a shallow parabola is selected , hence d ≦ f . the angle θ of the parabola is given by : where u is a dummy integration variable along the longitudinal direction of the beam . based on the above assumptions , euler - bernoulli beam theory applies , and the deflection of the beam are governed by [ 17 ]: where m ( s ) is the bending moment on the band , φ ( s ) is the rotation of band surface normal , and κ ( s ) is the curvature of the final band shape , see fig7 . with the thickness t ( s ) and width b ( s ) varying with length s , the second moment of area i ( s ) for a rectangular cross section is given by : as shown in fig6 ( c ), the bending moment in the band can be calculated as a function of x as : thus , the bending moment along the band length s is governed by : m ( s )= f ·( h + d − f · ln 2 ( s / 2 f +√{ square root over ( s 2 / 4 f 2 + 1 ))}) ( 11 ) it is well - known that loading a band with collinear external forces does not result in a parabolic shape . however , it is possible to shape the band &# 39 ; s cross section to form a parabola shape when its ends are pulled together by horizontal threes . in this process , it will be assumed that both the thickness and the bending stiffness of the thin mirror sheet are much smaller than the corresponding quantities of the band . in these cases , the shape can be tuned to a parabola by varying the band &# 39 ; s thickness t ( s ), its width b ( s ) or both as a function of s , see fig6 ( b ) ( c ). more general situations with non - negligible mirror sheet stiffness and / or bending stiffness can be considered by applying the finite element optimization method described later in this patent application . in a first case , t ( s ) changes and the width h ( s ) is assumed to be a constant h , as shown in fig8 . thus , the thickness t ( s ) as a function of the width b and the second moment of area the band is : for a thick band , large shear stresses could result and produce non - negligible errors . moreover , there might be an error induced by the difference of the curvature of the neutral line and the curvature of the upper surface . these errors are of second order and neglected in the present context . also varying the thickness on the band is difficult and expensive to fabricate . a varying thickness can be approximated by constructing the band from layers , see fig9 . this laminating approach is probably not economically viable compared to the method discussed below . a more cost - effective way to vary the area moment of inertia of the band is to vary its width as a function of s , b ( s ), with the band &# 39 ; s thickness , t , held constant , see fig1 . in this case , the band width is : after substituting equations ( 7 ), ( 8 ) and ( 11 ) into equation ( 14 ), the ideal band width is obtained as the explicit solution : such a design would be much easier to manufacture than a varying thickness design . clearly it is possible to combine the above two approaches by varying both band thickness and width . this might be done when other design constrains need to be met . the bands can also be optimized by punching holes on uniform width and thickness bands in approximately continuous patterns . however , the holes will produce a stress concentration problem . in addition , it is clear that similar results can be achieved by varying the material properties as a function of s , though this does present some significant manufacturing challenges . an analysis of mirror performance for a mirror made according to the invention will now be presented . for this analysis , it is assumed that the mirrors are actively tracking the sun . in this case , the sunlight will be parallel to the axis of the parabola . the objective is to calculate the distance of the reflections of the rays from the focal point where the absorber tube will be mounted . the focal error , ε , is defined as the distance from the focal point to a reflection ray , see fig1 . this error determines the diameter of the absorber tube for the or to insure that all the solar energy intersects the absorber tube . other metrics can be developed such as the percent of the energy that falls on a given absorber tube . the discussion of these metrics is beyond the scope of this disclosure . assuming small variations from the ideal parabolic profile , the focal error can be determined as follows ( see fig1 ). for an arbitrary ray at horizontal position x , assume that the position error of the actual deformed shape is δz , and that the angular error of the surface normal is δφ . taking z as the vertical coordinate of the ideal parabola and x , y as the running coordinates of the reflection ray , one obtains : as it can be seen , the focal error is positive when the reflected ray passes below the focal point and negative when it passes above the focal point . the maximal focal error ε max is defined as the maximum of the absolute values of the focal errors for all rays entering the mirror &# 39 ; s aperture . the performance of solar concentrators is often expressed in terms of their ability to concentrate collimated light , called concentration ratio , c , as a function of the chord length l and the focal diameter d f , 100 % of light entering the mirror to reach the absorber tube . here , for a given chord length , l , the maximum focal error , ε max , is chosen as a power precision performance metric . the analytical euler - bernoulli beam model shows the feasibility of the band - shaping approach for relatively simple cases . a more general approach , suitable also for the treatment of more involved cases ( e . g . non - negligible bending stiffness of mirror sheet ), is to perform a numerical shape optimization procedure based on finite element analysis ( fea ), as discussed below . the objective of the optimization is to minimize the maximum focal error by varying i ( s ): in order to find the optimal profile i ( s ), we describe it via a finite fourier series expansion : where only even terms need to be regarded as the function i ( s ) is symmetric with respect to s . b ( s )=[ 1 cos ( π s / l ) cos ( 2 πs / l ) . . . cos ( nπs / l ] t ( 22 ) will minimize the maximal focal error ε max obtained after performing the corresponding fea computation and evaluating the focal errors from the resulting bent band . this task corresponds to an unconstrained optimization problem with design variables a and cost function ε max , for which several well - known solution schemes exist . we chose here to apply an exact newton search in which at each optimization step the jacobian is computed by repeated evaluations of the fea analysis for small variations of each of the coefficients in a and the corresponding next estimate of a ( i ) is computed such that the linear approximation of the maximal focal error vanishes . a case study will now be presented . in this case study , a parabolic band based on varying width is presented . the optimization is obtained using both the analytical formulation set out above and the finite element based numerical optimization method just described . in this case , the rim angle θ is taken as 180 °. hence d is equal to f and l is equal to 4f . using the given parameters and equation ( 15 ), the band width as a function of s is : with the parameters in table 1 , the ideal analytical shape shown in fig1 is obtained . a finite element model of the analytically shaped band was developed and implemented in adina [ 18 - 20 ]. fig1 shows the boundary conditions and the force and moment loading of the fea analysis . the band is modeled as a shell bending problem . as shown in fig1 , u 1 , u 2 and u 3 are the translations about x , y and z axes , θ 1 and θ 2 are the rotations about x and v axes , the sign “√” means the degree of freedom is active and “−” means it is fixed . boundary conditions are shown at points b and c . the rotation about z axis is fixed for the whole model , in the model , it is assumed that the deformation is large and that strains are small , and that no plastic deformation occurs . the horizontal force , f , and the moment m 0 , which is equal to fh , are divided into two halves and applied as concentrated forces at the two end nodes . the loads were incrementally increased to the final value in 8 steps . the figure also shows the deflection and the stress distribution . the maximum equivalent mises stress is 348 . 52 mpa ( 50536 psi shown in fig1 ), which is below the yield stress of 1050 mpa for the chosen material ( spring steel 38si6 ). to evaluate the precision of the result , ray tracing using the fea deformed shape of the mirror was carried out , see fig1 . assuming collimated rays entering the mirror along the axis of the parabola , the reflected rays are traced based on the normal rotations φ ( s ) and displacements [ x ( s ) z ( s )] from the fea results . the focal error is calculated using equation ( 19 ). the resulting maximum error , ε max , for the analytically shaped band was 1 . 85 mm . this means the diameter of the absorber tube , d f , should be at least 3 . 70 mm if 100 % of the energy is to be absorbed . the fea results show that the band based on the analytical formulation is not a perfect parabola . a fea optimized band was calculated using the shape optimization method discussed above . as initial guess , a rectangular band with width , b , 76 . 2 mm ( 3 . 0 inches ) and thickness , t , 0 . 7937 mm ( 1 / 32 inch ) was employed . the optimization procedure converged after 9 iterations with a termination condition of 10 − 4 for the magnitude of the increment δa of the design parameter vector . fig1 shows the band width h ( s ) as a function of band length s for the optimized fea and the analytical optimized results . it can be seen that the numerical fea approach converges to a similar shape as the analytical approach . the ray tracing for the fea optimized band is shown in fig1 . the maximum focal error is 0 . 38 mm , approximately a factor of five smaller than the idealized analytical result . in order to assess the improvement of solar energy collection properties of the shape - optimized band and a simple rectangular hand , a fea analysis of a rectangular band was carried out . the results shows that the maximal focal error of the optimized band is a factor of 10 smaller than that of the rectangular band , see fig1 . the results of the previous optimization were validated experimentally . the experimental system consists of two main components : a flexible mirror with varying - width backbone band and a collimated light source consisting of a parabolic dish with an led light source at its focal point and an absorber located on the mirror &# 39 ; s focal line , see fig2 two locking blocks are used to construct the mirror &# 39 ; s chord length , l , to its desired value . the concentration absorber was made from a semitransparent sparent white plastic plate with the dimensions 1 . 5 × 26 inches . the fea optimized band was cut from a piece of 0 . 7937 in ( 1 / 32 inch ) spring steel sheet using a water jet cutter with tolerance ± 0 . 0254 mm (± 1 / 1000 inch ). fig2 shows the backbone band with optimized width and a simple rectangular band . fig2 ( a ) shows the band mirror concentrating sunlight . a wire is used to fix the chord length , l , and a black plastic absorber was placed at the focal line of the band . the width of the focal area is less than 3 mm for 100 % energy to be collected . the plastic absorber was quickly burnt by the concentrated light . the burn mark is shown in fig2 ( b ). the width of burn is less than 2 mm . the concentration ratio of the optimized band , c , is about 154 . 8 under sunlight . the result is much higher than those achieved by most current industrial parabolic mirror solar concentrators . for comparison , the non - optimized rectangular band ( see fig2 ) had about 5 mm focal width with only about 90 % energy collected . it was not possible to measure the focal width of 100 % collection as the image was outside of the measurement limits . the focal width of the optimized band is 4 . 6 mm measured in the laboratory for 100 % of the rays collected . and the rectangular band focal width is 10 . 3 mm with about 90 % rays collected . the parabolic shape of the deformed hand was measured in two ways , an edge finder on a cnc milling machine and an optical method . however , since the band was thin and thus highly compliant , the edge finder induced deformation errors that made the measurements unfit for focal error determination . thus , the optical method , in which no physical contact s made with the band , was further pursued . in this method , a photograph of the band on the vertical direction was taken and converted into a monochrome image ( black and white ). the threshold figure yields a high contrast black and white digital image , see fig2 ( a ). this image was then fitted with a high degree polynomial function and thus yielded a shape that closely matched the predicted contour , see fig2 ( b ). as before , the shape was used as the ray tracing algorithm , see fig2 . the focal error was obtained , see fig2 . note that any measured rigid body rotations and translations of the mirror shape in fig2 due to calibration issues have been eliminated from the results shown . the maximum focal error is small , 0 . 72 mm , compared with 6 . 41 mm of the rectangular band . in this disclosure , the design and manufacture of a simple and low cost precision 365 parabolic mirror solar concentrator with an optimized profile backbone band is presented . the band is optimally shaped so that it forms a parabola when its ends are pulled together to a known distance . it could be fabricated and shipped flat , and onsite its ends would be pulled together to distance by a wire , or rod , or actively controlled with a simple control system . varying width of the band as a function of its length appears to be the most cost - 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