Glass bending method and apparatus

TECHNICAL FIELD

p-0002Embodiments of the subject matter described herein relate generally to forming of curved glass sheets. More particularly, embodiments of the subject matter relate to sag-bending deformation of glass sheets.

BACKGROUND

p-0003Glass mirrors, including those with a parabolic shape, are useful for solar applications. In some solar applications, parabolic-shaped mirrors can be used for solar concentrator systems. A solar concentrator system is one where sunlight is reflected with increased, concentrated intensity on a receiving unit. Because of the optical effects associated with parabolic-shaped mirrors, such shapes are useful for focusing concentrated sunlight.

p-0004Forming parabolic mirrors can be accomplished through sag-bending techniques. There are challenges associated with forming parabolic-shaped glass mirrors with sag bending. For example, it can be challenging to form a parabolic-shaped glass sheet because the curve deviation from a flat sheet of glass increases as the length of the glass sheet increases. Thus, longer mirrors have portions that are deformed a small amount and a greater amount. The dissimilarity in deformation amounts can cause increased difficulty in accurately controlling the shape of the glass sheet during deformation.

p-0005Solar concentrator systems can be sensitive to minor variations in operating conditions, such as mirror shape, which can affect the location of concentrated sunlight on the receiving unit, the shape of the concentrated sunlight reflected area, and other aspects of the system, all of which contribute to the efficiency and power output of the system. Accordingly, solar concentrator systems benefit from components made to very high precision.

p-0006Thus, crafting parabolic mirrors with minimized defects or deviations from an ideal shape provides a benefit to a solar concentrator system.

BRIEF DESCRIPTION OF THE DRAWINGS

p-0007A more complete understanding of the subject matter may be derived by referring to the detailed description and claims when considered in conjunction with the following figures, wherein like reference numbers refer to similar elements throughout the figures.

p-0008<figref idrefs="DRAWINGS">FIG. 1</figref> is an illustration of a parabolic glass mold;

p-0009<figref idrefs="DRAWINGS">FIG. 2</figref> is a side view of the parabolic glass mold of <figref idrefs="DRAWINGS">FIG. 1</figref>;

p-0010<figref idrefs="DRAWINGS">FIG. 3</figref> is a top view of the parabolic glass mold of <figref idrefs="DRAWINGS">FIG. 2</figref>;

p-0011<figref idrefs="DRAWINGS">FIG. 4</figref> is an illustration of a parabola;

p-0012<figref idrefs="DRAWINGS">FIG. 5</figref> is an illustration of another parabola with a designated section;

p-0013<figref idrefs="DRAWINGS">FIG. 6</figref> is an illustration of the designated section of <figref idrefs="DRAWINGS">FIG. 5</figref> and transformed section;

p-0014<figref idrefs="DRAWINGS">FIG. 7</figref> is an illustration of an embodiment of a linear-square root composite shape;

p-0015<figref idrefs="DRAWINGS">FIG. 8</figref> is another illustration of an embodiment of a linear-square root composite shape;

p-0016<figref idrefs="DRAWINGS">FIG. 9</figref> is yet another illustration of an embodiment of a linear-square root composite shape;

p-0017<figref idrefs="DRAWINGS">FIG. 10</figref> is an embodiment of a sag-bending glass mold having a linear-square root composite shape;

p-0018<figref idrefs="DRAWINGS">FIG. 11</figref> is a side view of the sag-bending glass mold of <figref idrefs="DRAWINGS">FIG. 10</figref>;

p-0019<figref idrefs="DRAWINGS">FIG. 12</figref> is a top view of the sag-bending glass mold of <figref idrefs="DRAWINGS">FIG. 11</figref>;

p-0020<figref idrefs="DRAWINGS">FIG. 13</figref> is a side view of a sag-bending glass mold having a linear-square root composite shape with associated glass sheet;

p-0021<figref idrefs="DRAWINGS">FIG. 14</figref> is a side view of the sag-bending glass mold of <figref idrefs="DRAWINGS">FIG. 13</figref> forming the glass sheet into a partial-parabolic shape;

p-0022<figref idrefs="DRAWINGS">FIG. 15A</figref> is a side view of the sag-bending glass mold of <figref idrefs="DRAWINGS">FIG. 14</figref> with the shaped glass sheet removed;

p-0023<figref idrefs="DRAWINGS">FIG. 15B</figref> is a side view of the sag-bending glass mold of <figref idrefs="DRAWINGS">FIG. 15A</figref> with the shaped glass sheet separated;

p-0024<figref idrefs="DRAWINGS">FIG. 16</figref> is a side view of embodiments of two sag-bending molds with adjacent glass sheets;

p-0025<figref idrefs="DRAWINGS">FIG. 17</figref> is a side view of the embodiments of <figref idrefs="DRAWINGS">FIG. 16</figref> after sag-bending the adjacent glass sheets;

p-0026<figref idrefs="DRAWINGS">FIG. 18</figref> is another embodiment of a sag-bending glass mold having a linear-square root composite shape;

p-0027<figref idrefs="DRAWINGS">FIGS. 19 and 20</figref> are different embodiments of sag-bending glass molds having joined linear-square root composite shapes; and

p-0028<figref idrefs="DRAWINGS">FIG. 21</figref> is an illustration of a method for forming a partial parabolic glass sheet.

DETAILED DESCRIPTION

p-0029The following detailed description is merely illustrative in nature and is not intended to limit the embodiments of the subject matter or the application and uses of such embodiments. As used herein, the word “exemplary” means “serving as an example, instance, or illustration.” Any implementation described herein as exemplary is not necessarily to be construed as preferred or advantageous over other implementations. Furthermore, there is no intention to be bound by any expressed or implied theory presented in the preceding technical field, background, brief summary or the following detailed description.

p-0030“Inhibit”—As used herein, inhibit is used to describe a reducing or minimizing effect. When a component or feature is described as inhibiting an action, motion, or condition it may completely prevent the result or outcome or future state completely. Additionally, “inhibit” can also refer to a reduction or lessening of the outcome, performance, and/or effect which might otherwise occur. Accordingly, when a component, element, or feature is referred to as inhibiting a result or state, it need not completely prevent or eliminate the result or state.

p-0031In addition, certain terminology may also be used in the following description for the purpose of reference only, and thus are not intended to be limiting. For example, terms such as “upper”, “lower”, “above”, and “below” refer to directions in the drawings to which reference is made. Terms such as “front”, “back”, “rear”, “side”, “outboard”, and “inboard” describe the orientation and/or location of portions of the component within a consistent but arbitrary frame of reference which is made clear by reference to the text and the associated drawings describing the component under discussion. Such terminology may include the words specifically mentioned above, derivatives thereof, and words of similar import. Similarly, the terms “first”, “second”, and other such numerical terms referring to structures do not imply a sequence or order unless clearly indicated by the context.

p-0032A partial parabolic shape, that is, a portion or segment of a parabola, is a desirable for use in some embodiments of solar concentrator applications. When used herein the glass referred to can be used for creating a mirror, as would be used in a reflector element of a solar concentrator.

p-0033A glass sheet can be processed to create a mirror sheet through the addition of a reflective layer to an exposed surface of the glass sheet, or by embedding a reflective layer between glass sheets. Such processing can include the metallization of a surface of the glass, including deposition of successive metal layers, including tin, silver, copper, and other metals, to produce a reflective layer. In some mirrors, the reflectivity is uni-directional.

p-0034Regardless, when described herein the glass sheets described can, in some embodiments, be later processed to produce mirrors, including mirrors appropriate for concentrating photovoltaic or concentrating solar thermal applications. In certain embodiments, the glass sheets can comprise one or more reflective surfaces before the bending techniques described herein are applied. Thus, while reference is sometimes made to embodiments for bending glass sheets, it should be understood that bending the glass sheet to produce a curved glass sheet can also describe a process for bending a glass sheet with a reflective surface to produce a curved mirror. Additional process steps, such as laminating or polishing may also be used to produce a completed mirror from a curved glass sheet without deviating from the advances described herein.

p-0035Common industry practice is to form a desired curved shape by sag-bending a flat sheet of glass (or other material) either to form an entire parabola, which includes the shaping of wasted segments of the parabola, or to form the partial parabolic shape in its original orientation as extending from an imaginary origin of the parabolic shape. The former approach is expensive in that it wastes glass. The latter approach is challenging to mold from a flat sheet of glass because the parabolic shape deviates an increasing amount from a flat shape the longer the parabola extends. Forming the curved shape from a flat glass sheet requires differing deformation amounts and therefore different heating amounts, all of which are challenging and costly.

p-0036Additionally, concentrator glass sheets can be wider than they are long, sometimes having an aspect ratio of 5:1 or greater. This aspect ratio introduces additional challenges because high-aspect ratio glass sheets do not easily sag at the differing amounts required by a partial parabolic shape. It is preferable to have a lower aspect ratio for more uniform sag-bending.

p-0037To reduce cost and simplify manufacture, the inventors have discovered that it is possible to form a partial parabolic shape from a non-parabolic-shaped mold. Additionally, the non-parabolic shape can have one side with a flat edge. Multiple non-parabolic shapes can be molded by joining the flat edges, thereby decreasing the aspect ratio of the molded glass. Further, this can be accomplished with little to no waste glass, and while simultaneously increasing manufacturing throughput. The advantageous end result is that partial parabolic curved glass sections can be formed with lower cost, greater simplicity, and at a faster rate, than previously possible.

p-0038<figref idrefs="DRAWINGS">FIG. 1</figref> illustrates a typical mold <b>100</b> for forming a parabolic glass sheet as used in the art. The mold <b>100</b> has an upper surface <b>102</b> which can be solid or distributed across several longitudinal or lateral rib portions. The mold <b>100</b> has a cross-sectional shape as shown in <figref idrefs="DRAWINGS">FIG. 2</figref>. With reference to both <figref idrefs="DRAWINGS">FIGS. 1 and 2</figref>, the mold <b>100</b> can be considered to have an origin <b>104</b> and an upper point <b>106</b>. The parabolic shape of the upper surface <b>102</b> is used to form the shape of a curved glass segment when flat glass is heated to sag bend onto it. Typical molds, such as mold <b>100</b>, are constructed to form a single parabolic shape for use in molding a single glass sheet.

p-0039As can be seen in <figref idrefs="DRAWINGS">FIG. 2</figref>, two portions of the curved upper surface <b>102</b> are identified as Δ<b>1</b> and Δ<b>2</b>. Δ<b>1</b> indicates an area near the imaginary origin <b>104</b>, whereas Δ<b>2</b> indicates a region further from the origin <b>104</b>. As can be seen, and is true for parabolic shapes, the rate of change of the curved upper surface <b>102</b> from a flat vertical shape is significantly higher in the region of Δ<b>2</b> than in Δ<b>1</b>. When sag-bending a flat glass sheet, the temperature to which the glass sheet must be heated increases corresponding to the amount of deflection from the flat shape that the glass will be curving. Thus, a shallow curved shape can be formed at a lower temperature than a deeply-curved shape. Additionally, it is desirable to have as little variance between different portions of the curved shape's depth as possible to promote uniform heating and uniform bending. Finally, it is desirable to form as shallow a curve as possible to reduce localized deformation in the glass sheet beyond sheet bending.

p-0040It should be noted that the scale and proportion of all figures is for descriptive purposes only, and should not be considered for actual measurements. Additionally, exaggerations for clarity may be used when necessary. For example, the rates of curvature of regions near Δ<b>1</b> and Δ<b>2</b> may in fact be much less than illustrated, but shown thus for descriptive purposes. Notwithstanding such alterations, the manufacturing difficulties described may still be present even for smaller differences in rates of curvature.

p-0041When sag-bending a flat glass sheet onto the upper surface <b>102</b>, the glass sheet will need a greater amount of heat in the region near Δ<b>2</b> than near Δ<b>1</b>. Additionally, because of the slope of curvature of the region near Δ<b>2</b>, some localized deformation in the glass may occur in addition to bending from a flat shape to match the curve of the upper surface <b>102</b>. These considerations increase manufacturing complexity and cost.

p-0042<figref idrefs="DRAWINGS">FIG. 3</figref> illustrates a top view of the mold <b>100</b>. This view permits display of the relative width w and length l. The ratio of w to l forms the aspect ratio. For solar concentrator applications, the ratio, as can be seen, although not to scale, can be as high as 3:1 or greater. This high aspect ratio additionally complicates manufacture.

p-0043<figref idrefs="DRAWINGS">FIG. 4</figref> illustrates a parabola P. A parabola is a conic section and in a plane is the locus of points that are equidistant between a point focus and a line directrix. In the illustration of <figref idrefs="DRAWINGS">FIG. 4</figref>, the parabola P is formed in relation to the focus f and directrix d. Coordinate axes x and y are provided for reference. Although the focus f is used for illustrative purposes, it should not be confused with the focus of the finished mirror or curved glass used in solar concentrating applications, whether thermal or photovoltaic, or otherwise benefitting from the glass produced. Additionally, as used throughout, a parabola refers to such a shape as illustrated here, and parabolic is used to describe shapes having properties associated with the shape.

p-0044A parabola formed with symmetry about the y-axis, such as parabola P, can be expressed by the mathematical formula:

p-0045<br /><i>P</i>(<i>x</i>)=<i>ax</i>²<i>+bx+c; </i>

p-0046where a, b, and c represent constants. Such a parabola is useful for solar concentrator applications for certain values of the constants. As mentioned above, however, for practical reasons, it is a partial parabolic shape that can also be used for solar concentrator applications. One such shape that can be used is:

p-0047<br /><i>P</i>₁(<i>x</i>)=0.001192<i>x</i>²+0.109046544<i>x </i>

p-0048<figref idrefs="DRAWINGS">FIG. 5</figref> illustrates a portion <b>202</b> of parabola P between points A₀and B₀. Point A₀corresponds to the point of the parabola P a distance A from the origin and with a height from the origin of P(A). Thus, (A, P(A)) is the point A₀. Similarly, point B₀is the point of the parabola P that is a distance B from the origin and has a height of P(B).

p-0049Portion <b>202</b> represents a partial parabolic shape that can be used in a solar concentrator application. It can be inefficient, however, to form the entire parabola P from (0, P(0)) to (B, P(B)), subsequently discarding the portion of parabola P between (0, P(0)) and (A, P(A)), to obtain portion <b>202</b> for use in a solar concentrator. This is the approach typically used by mold <b>100</b>, such that the origin <b>104</b> corresponds to the point (0, P(0)) and the upper point <b>106</b> corresponds to the point (B, P(B)). For one such portion <b>202</b> corresponding to the exemplary parabola P₁(x), the portion can be between 45.791 and 483.241 on the x axis.

p-0050<figref idrefs="DRAWINGS">FIG. 6</figref> illustrates the discovery by the inventors that portion <b>202</b> can be transformed into portion <b>212</b> for simplification of manufacture at reduced cost and increased throughput. The inventors have discovered that portion <b>202</b> can be translated such that point (A, P(A)) is located at the origin. Thus, as shown in <figref idrefs="DRAWINGS">FIG. 6</figref>, point A₀has been translated to point A₁located at (0, 0). Point B₀is altered by translation to point B₁. By itself, this translation reduces the waste associated with forming portion <b>202</b>. The slope of the portion <b>202</b> as it approaches the origin (0, 0) does not approach a horizontal, or zero slope. Put another way, the tangent of P(x) does not approach zero as P(x) approaches the origin from the positive x direction. This can make manufacture challenging.

p-0051The inventors have further discovered that P(x) and corresponding portion <b>202</b> can be rotated downward by a negative angle θ to produce Q(x) and corresponding portion <b>212</b>. The negative angle θ is defined as the angle necessary to rotate portion <b>202</b> such that the slope of <b>202</b> approaches zero as Q(x) approaches the origin from the positive x direction. The value of θ varies as the parabola P(x) varies, but can be anywhere from 0.01 to 90 degrees, whether measured negatively or positively.

p-0052After rotating P(x) and portion <b>202</b> to achieve curve Q(x) and portion <b>212</b>, the point A₁is still located at the origin and indicated by point A₂. Point B₁has been moved by the rotation of portion <b>202</b> to be located at new point B₂. Portion <b>212</b> can be described by the curve Q(x). Whereas P(x) was described by the parabolic formula above for symmetry about the y axis, Q(x) is no longer a parabolic curve, but instead can be described by the formula:

p-0053<br /><i>Q</i>(<i>x</i>)=<i>dx</i>+((√(<i>ex+f</i>))/<i>g</i>)+<i>h; </i>

p-0054where d, e,f,g, and h are constants. The shape of the curve of Q(x) is referred to as a linear square root composite shape. The values of a, b, and c associated with parabola P(x) determine the value of negative angle θ and both determine the values of constants d, e, f, g, and h. Accordingly, the exact shape of curve Q(x) will be determined by the shape of parabola P(x) desired for the embodiment. The calculation of x′ and Q(x′), where x′ corresponds to the x coordinate of a point on the curve (x, P(x)), and Q(x′) corresponds to the y coordinate, is performed by the use of a rotation matrix [R] on P(x) such that:

p-0055<maths id="MATH-US-00001" num="00001"><math overflow="scroll"><mrow><mrow><mo>[</mo><mtable><mtr><mtd><mstyle><mspace width="0.3em" height="0.3ex" /></mstyle></mtd></mtr><mtr><mtd><msup><mi>x</mi><mi>′</mi></msup></mtd></mtr><mtr><mtd><mrow><mi>Q</mi><mo></mo><mrow><mo>(</mo><msup><mi>x</mi><mi>′</mi></msup><mo>)</mo></mrow></mrow></mtd></mtr><mtr><mtd><mstyle><mspace width="0.3em" height="0.3ex" /></mstyle></mtd></mtr></mtable><mo>]</mo></mrow><mo>=</mo><mrow><mrow><mo>[</mo><mtable><mtr><mtd><mstyle><mspace width="0.3em" height="0.3ex" /></mstyle></mtd><mtd><mstyle><mspace width="0.3em" height="0.3ex" /></mstyle></mtd></mtr><mtr><mtd><mrow><mi>cos</mi><mo></mo><mstyle><mspace width="0.3em" height="0.3ex" /></mstyle><mo></mo><mi>θ</mi></mrow></mtd><mtd><mrow><mrow><mo>-</mo><mi>sin</mi></mrow><mo></mo><mstyle><mspace width="0.3em" height="0.3ex" /></mstyle><mo></mo><mi>θ</mi></mrow></mtd></mtr><mtr><mtd><mrow><mi>sin</mi><mo></mo><mstyle><mspace width="0.3em" height="0.3ex" /></mstyle><mo></mo><mi>θ</mi></mrow></mtd><mtd><mrow><mi>cos</mi><mo></mo><mstyle><mspace width="0.3em" height="0.3ex" /></mstyle><mo></mo><mi>θ</mi></mrow></mtd></mtr><mtr><mtd><mstyle><mspace width="0.3em" height="0.3ex" /></mstyle></mtd><mtd><mstyle><mspace width="0.3em" height="0.3ex" /></mstyle></mtd></mtr></mtable><mo>]</mo></mrow><mo></mo><mrow><mo>[</mo><mtable><mtr><mtd><mstyle><mspace width="0.3em" height="0.3ex" /></mstyle></mtd></mtr><mtr><mtd><mi>x</mi></mtd></mtr><mtr><mtd><mrow><mi>P</mi><mo></mo><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow></mtd></mtr><mtr><mtd><mstyle><mspace width="0.3em" height="0.3ex" /></mstyle></mtd></mtr></mtable><mo>]</mo></mrow></mrow></mrow></math></maths>

p-0056To produce a Q₁(x) that would correspond to P₁(x), the following constants are present:

p-0057<tables id="TABLE-US-00001" num="00001"><table frame="none" colsep="0" rowsep="0"><tgroup align="left" colsep="0" rowsep="0" cols="3"><colspec colname="offset" colwidth="49pt" align="left" /><colspec colname="1" colwidth="63pt" align="left" /><colspec colname="2" colwidth="105pt" align="left" /><thead><row><entry /><entry namest="offset" nameend="2" align="center" rowsep="1" /></row></thead><tbody valign="top"><row><entry /><entry>d</entry><entry>= 9.170396</entry></row><row><entry /><entry>e</entry><entry>= 0.00052</entry></row><row><entry /><entry>f</entry><entry>= 1.011891</entry></row><row><entry /><entry>g</entry><entry>= 0.0000280154</entry></row><row><entry /><entry>h</entry><entry>= 35906.3</entry></row><row><entry /><entry namest="offset" nameend="2" align="center" rowsep="1" /></row></tbody></tgroup></table></tables><br /> Such constants correspond to the fact that P₁(x) must be rotated by a negative angle θ approximately equal to 6.22° to achieve the linear square root composite curve Q₁(x). Put another way, P₁(x) must be rotated approximately 6.22° clockwise, toward the positive x axis, to achieve curve Q₁(x). The values of a, b, and c associated with parabola P(x), along with the end points of interest, such as examples A₀and B₀, necessarily determine the value of negative angle θ and both necessarily determine the values of constants d, e, g, and h. Accordingly, the exact shape of curve Q(x) will be determined by the shape of parabola P(x) and the selected endpoints desired for the embodiment. Therefore, for every portion of a parabola P(x), such as portion <b>202</b>, there will exist exactly one transformed portion of a linear square root composite shaped curve, such as portion <b>212</b>, of Q(x). Portion <b>212</b> corresponds exactly to the curvature of portion <b>202</b>, except that is has been translated and rotated from the original illustration in <figref idrefs="DRAWINGS">FIG. 5</figref>.

p-0058Thus, a sag-bending mold which is used to fabricate portion <b>212</b>, having the curved shape of portion <b>212</b>, will produce a curved portion of glass which matches the curve of portion <b>202</b>. The portion <b>212</b> is advantageously simpler, faster, and less complex to manufacture than portion <b>202</b> or parabola P(x), greatly reducing cost. Accordingly, the non-parabolic shape of the linear square root composite curved shape can be used to create, by molding for example, a partial parabolic curve shape that previously could only be formed by using a parabolic shaped mold. A non-parabolic surface on a mold can therefore be used to produce a partial parabolic shaped glass surface after sag-bending molding. All references to parabolic and non-parabolic shapes are made with respect to the original defined coordinate references in which the parabola is first described.

p-0059<figref idrefs="DRAWINGS">FIG. 7</figref> illustrates that further improvement can be possible by taking advantage of the zero-slope at endpoint A₂by rotating the portion <b>212</b> around the y axis to obtain portion <b>222</b>. Portions <b>212</b> and <b>222</b> have overlapping endpoints A₂and A₃at the origin. Endpoint B₂of portion <b>212</b> is at the same place as calculated above and illustrated with respect to <figref idrefs="DRAWINGS">FIG. 6</figref>. Endpoint B₃of portion <b>222</b> has the same height as B₂while extending in the negative x direction an equal distance from the origin as B₂is in the positive x direction, as illustrated. Thus, portions <b>212</b> and <b>222</b> can meet smoothly at the origin with a zero slope and form a double-wide joined portion. Each half, if divided at the y axis forms a partial parabolic shape of the type illustrated in <figref idrefs="DRAWINGS">FIG. 5</figref> and desired for solar concentrator systems.

p-0060<figref idrefs="DRAWINGS">FIG. 8</figref> illustrates an inversion pairing of portion <b>212</b>. As can be seen, the portion <b>212</b> has been further rotated such that endpoint B₂is now positioned at the origin and portion <b>212</b> has been oriented such that the curve, which is the inverse or Q⁻¹(x) of Q(x), still has a zero slope, or horizontal tangent, at the origin as Q⁻¹(x) approaches the origin from the positive x direction. Accordingly, endpoint A₂now is positioned at the upper right end of the portion <b>212</b>. As previously stated, it should be noted that while portion <b>212</b> appears to have a shallower upward slope in <figref idrefs="DRAWINGS">FIG. 7</figref> than in <figref idrefs="DRAWINGS">FIG. 8</figref>, and the inversion of curve Q(x) in <figref idrefs="DRAWINGS">FIG. 8</figref> has produced a sharper upward slope, in actual practice, the reverse may be true, and all illustrated elements are shown for descriptive purposes. Accordingly, such features may not be to scale for purposes of clarity of description.

p-0061Rotating the inverted portion <b>212</b> of <figref idrefs="DRAWINGS">FIG. 8</figref> produces portion <b>222</b> which has a similar positioning of end points A₃and B₃as mirrored across the y axis. Depending on the shape of curve Q(x), it may be more advantageous to construct a mold having an upper surface with the shape of joined portions <b>212</b> and <b>222</b> resembling <figref idrefs="DRAWINGS">FIG. 7</figref> or the inverted embodiment illustrated in <figref idrefs="DRAWINGS">FIG. 8</figref>.

p-0062<figref idrefs="DRAWINGS">FIG. 9</figref> illustrates yet another embodiment of a variation of <figref idrefs="DRAWINGS">FIG. 7</figref> where portions <b>212</b> and <b>222</b> have been rotated about the x axis. Thus, while the endpoints A₂and A₃remain at the origin, the curves extend in the negative y direction as they extend outward from the origin. Such a rotation can also be performed on the embodiment shown in <figref idrefs="DRAWINGS">FIG. 8</figref>, producing downward curves of the illustrated inverted portions <b>212</b>, <b>222</b> of <figref idrefs="DRAWINGS">FIG. 8</figref>. In all instances where portion <b>222</b> is formed by rotating portion <b>212</b> about an axis or line, portion <b>222</b> is said to be the mirror image of portion <b>212</b>.

p-0063Any of the shapes of <figref idrefs="DRAWINGS">FIGS. 6-9</figref>, or other variations thereof, can be used to form a mold for sag-bending a flat glass sheet into the partial parabolic shape desired. Several of the shapes will produce multiple partial parabolic curved sections or segments, such as the shape illustrated in <figref idrefs="DRAWINGS">FIG. 7</figref> or <b>8</b>, which can then be separated, divided, or cut into the desired part for use.

p-0064<figref idrefs="DRAWINGS">FIG. 10</figref> illustrates an improved mold <b>300</b> for use in forming partial parabolic shaped glass sections formed by sag-bending. The mold <b>300</b> can be a permiter-support mold with a plurality of ribs <b>308</b> forming an upper surface <b>302</b>. In certain embodiments, the mold <b>300</b> can be formed as a solid component with a single upper surface, while in other embodiments, the upper surface can be distributed across multiple components, such as internal longitudinal or lateral ribs. The mold <b>300</b> of <figref idrefs="DRAWINGS">FIG. 10</figref> illustrates longitudinal ribs <b>308</b> which extend between the two outer edges <b>306</b> of the mold <b>300</b>. The ribs <b>308</b> can be any thickness or width as desired for operation of the mold <b>300</b>.

p-0065With additional reference to <figref idrefs="DRAWINGS">FIG. 11</figref>, the mold <b>300</b> is shown from the side, illustrating the cross-section of the upper surface <b>302</b>, which has a variable height. It should be noted that mold <b>300</b> has been simplified for descriptive purposes. Some embodiments of the mold <b>300</b> can include ports and inlets and outlets for gas venting and vacuum, as well as clips, gutters, heating features, and any other appropriate components for use in a sag-bending mold, including a perimeter-supported sag-bending mold.

p-0066The upper surface <b>302</b> has outer edges <b>306</b>. The mold <b>300</b> can have an upper surface <b>302</b> with a linear square root composite curved shape, in accordance with the shapes and their variations as described above. Thus, the mold <b>300</b> can have a shape similar to that illustrated in <figref idrefs="DRAWINGS">FIG. 7</figref>, where points A₂and A₃would lie at the midpoint of the upper surface <b>302</b>, approximating the origin of <figref idrefs="DRAWINGS">FIG. 7</figref>, and points B₂and B₃would correspond to the points of the upper surface <b>302</b> at the outer edges <b>306</b>. The upper surface <b>302</b> can also be reduced or increased in scale as desired, such for a negative or positive mold. Such a surface as the upper surface <b>302</b> can be continuous or spread across any ribs <b>308</b> of the mold <b>300</b>.

p-0067<figref idrefs="DRAWINGS">FIG. 11</figref> additionally includes regions marked <b>43</b> and <b>44</b>, similar to the respective marked regions Al and <b>42</b> of <figref idrefs="DRAWINGS">FIG. 2</figref>, described above. Each indicator references the same portion of the upper surface of the respective mold <b>100</b>, <b>300</b>. Because of the rotation of the linear square root composite shape, it should be noted that <b>43</b>, which references a position similar to that referenced by Δ<b>1</b> has a shallower curve than Δ<b>1</b>. Δ<b>4</b> references a position similar to that of Δ<b>2</b>. Similarly, and again because of the rotation by negative angle θ of the curved mold <b>300</b> as compared to the parabolic mold <b>100</b>, Δ<b>4</b> is shallower than Δ<b>2</b>. Thus the improved curve of mold <b>300</b> can reduce local deformation during sag bending as compared to a parabolic mold <b>100</b>. Additionally, the reduced disparity between Δ<b>3</b> and Δ<b>4</b>, as compared to between Δ<b>1</b> and Δ<b>2</b> correspondingly reduces manufacturing complexity and, hence, cost.

p-0068<figref idrefs="DRAWINGS">FIG. 12</figref> illustrates a further advantage of joining linear square root composite shapes to form a mold, such as mold <b>300</b>. As compared to the mold <b>100</b> of <figref idrefs="DRAWINGS">FIG. 3</figref>, it can be seen that by joining multiple curved sections, the length/of the mold <b>300</b> is increased to at least double that of mold <b>100</b> in <figref idrefs="DRAWINGS">FIG. 3</figref>. Accordingly, for a constant width w, it can be seen that the aspect ratio decreases from that of <figref idrefs="DRAWINGS">FIG. 3</figref>, which was about 3:1 to about 1.5:1, or, where the width w is varied, to any other aspect ratio of 2:1, 1.6:1, 1.15:1, or even less than one. For example, with reference to <figref idrefs="DRAWINGS">FIGS. 18-20</figref>, it can be seen that molds with aspect rations much less than 1 can be formed.

p-0069<figref idrefs="DRAWINGS">FIGS. 13-17</figref> illustrates a mold <b>400</b> for sag-bending glass to a particular shape. Unless otherwise indicated, the numerical indicators in <figref idrefs="DRAWINGS">FIGS. 13-17</figref> designate components and features similar to those in <figref idrefs="DRAWINGS">FIGS. 10-12</figref>, except that they have been incremented by <b>100</b>. The mold <b>400</b> can be a perimeter-supported mold for sag-bending planar or flat glass sheets, such as sheet <b>450</b>. Sheet <b>450</b> can have a midpoint <b>452</b>, dividing the sheet <b>450</b> into first side <b>454</b> and second side <b>456</b>. The sheet <b>450</b> can have a lower surface <b>460</b>.

p-0070During preparation for sag-bending the sheet <b>450</b>, the sheet <b>450</b> can be positioned on or above the upper surface <b>402</b> of the mold <b>400</b>, as depicted in <figref idrefs="DRAWINGS">FIG. 13</figref>. In <figref idrefs="DRAWINGS">FIG. 14</figref>, the sheet <b>450</b> has been replaced with curved sheet <b>470</b>, formed by heating the sheet <b>450</b> until it reaches a temperature sufficient to sag the glass mirror sheet to deform downwards toward the upper surface <b>402</b> of the mold <b>400</b>. The mold <b>400</b> can then support the lower surface <b>460</b> of the sheet with its upper surface <b>402</b>, molding the glass sheet <b>450</b> to the desired shape as curved sheet <b>470</b>. In certain embodiments, a downward force can be imparted to sheet <b>450</b> to cause it to curve to follow upper surface <b>402</b>. Such a force can be caused by a press from above the sheet <b>450</b> directed downward and forcing the sheet <b>450</b> into the mold <b>400</b>. Alternatively, the mold <b>400</b> can incorporate one or more vacuum ports which draw the sheet <b>450</b> downward toward and onto the upper surface <b>402</b> of the mold <b>400</b>.

p-0071After shaping the sheet <b>450</b> into the curved sheet <b>470</b>, the curved sheet <b>470</b> can be separated from the mold <b>400</b>, as shown in <figref idrefs="DRAWINGS">FIG. 15A</figref>. In certain embodiments, the curved sheet <b>470</b> can be separated, cut, or divided at its midpoint <b>472</b> as shown in <figref idrefs="DRAWINGS">FIG. 15B</figref>. As shown, the curved sheet <b>470</b> is thus divided into first section <b>474</b> and second section <b>476</b>, each of which can have a partial parabolic shape, despite the linear square root composite curved shape of the upper surface <b>402</b> against which it is formed. In certain embodiments, the mold <b>400</b> can include a flat portion between partial parabolic shapes, such as near the middle of the mold <b>400</b> to provide material between the two sections for kerf, although such a flat portion is desirably minimized to inhibit waste material.

p-0072With additional reference again to <figref idrefs="DRAWINGS">FIG. 12</figref>, it should be understood that the increase in length/is greater than that caused simply by doubling the curved upper surface. The rotation of the curved portion <b>212</b> of <figref idrefs="DRAWINGS">FIG. 6</figref> which, as can be seen in <figref idrefs="DRAWINGS">FIG. 6</figref> extends further along the x axis than the parabolic portion <b>202</b>, causes portion <b>212</b> to overlie the x axis to a greater extent than portion <b>202</b>. Although both portions produce the same partial parabolic shape from the same area of glass sheet, the non-parabolic curve of mold <b>300</b> does so with less vertical bending, increasing accuracy and manufacturability.

p-0073<figref idrefs="DRAWINGS">FIG. 16</figref> further illustrates the advantage to aspect ratio caused by the use of a linear square root composite curve shape on a sag-bending mold.

p-0074As can be seen in the cross-sections illustrated in <figref idrefs="DRAWINGS">FIGS. 16 and 17</figref>, a mold <b>500</b>, which is a double-sided parabolic glass mold, has a greater vertical deformation distance and lesser horizontal deformation range. By contrast, another embodiment of a mold <b>530</b>, which is a double-sided linear square root composite curve shape similar to those shapes illustrated in <figref idrefs="DRAWINGS">FIGS. 7 and 8</figref>, has a greater horizontal deformation range and lesser vertical deformation distance. Both molds <b>500</b>, <b>530</b>, however, form similarly long glass sheets <b>510</b> and <b>540</b> respectively. As can be seen with reference to <figref idrefs="DRAWINGS">FIG. 17</figref>, the glass sheets <b>510</b> and <b>540</b> have been formed into curved sheets <b>512</b>, <b>542</b> respectively. As shown above, if each glass sheet <b>512</b>, <b>542</b> is separated at its midpoint, and waste region <b>514</b> removed, they will form substantially identical partial parabolic shapes.

p-0075The glass sheet <b>542</b> formed by mold <b>530</b>, if separated at or near the midpoint, however, is formed by a greatly advantageous process. Because the mold <b>530</b> uses a linear square root composite shape for its molding surface, the aspect ratio is superior to the parabolic shape of mold <b>500</b>, permitting for easier sagging at a lower temperature. Additionally, because there is less vertical deformation when mold <b>530</b> is used, localized deformation effects are inhibited or eliminated, thereby increasing glass quality and ultimately solar system performance. Further, waste region <b>514</b> of curved sheet <b>512</b> is the region between partial parabolic portions, as can be understood by reference to <figref idrefs="DRAWINGS">FIG. 5</figref> and the space between the origin and point A₀in the figure. The waste region <b>514</b> extends on both sides of the mold <b>500</b>, causing waste glass to be cut from the curved sheet <b>512</b>. Thus, while glass sheets <b>510</b>, <b>540</b> appear to be of equal length, sheet <b>510</b> is in fact slightly longer to account for the waste region <b>514</b>. The reduced waste enabled by using a linear square root composite shaped sag-bending mold further contributes to its cost superiority.

p-0076<figref idrefs="DRAWINGS">FIG. 18</figref> illustrates another embodiment of a sag-bending mold <b>600</b>. As can be seen, a single mold <b>600</b> can comprise several sections, such as first section <b>602</b> and second section <b>604</b>, for forming partial parabolic curves using a linear square root composite shaped upper surface. Although a shape similar to that of <figref idrefs="DRAWINGS">FIG. 7</figref> is shown repeated longitudinally, approximating a wave-like shape, other shapes can also be used or intermixed. For example, in another embodiment, the first section <b>602</b> can correspond to a shape similar to that of <figref idrefs="DRAWINGS">FIG. 7</figref>, while the second section <b>604</b> can correspond to a shape similar to <figref idrefs="DRAWINGS">FIG. 9</figref>. In certain embodiments, only two such sections are joined, while in other embodiments, three, four, five, or any practical number of such sections can be formed into a single mold.

p-0077In certain embodiments, a transition surface can be present between joined sections to provide a gradual curvature section, if desired. For example, in certain embodiments, there can be a flat portion <b>606</b> on the upper surface of mold <b>600</b> between sections <b>602</b> and <b>604</b>. As stated above, the flat portion <b>606</b> can be minimized to inhibit waste glass.

p-0078After forming a glass sheet using a mold similar to that shown in <figref idrefs="DRAWINGS">FIG. 18</figref>, or described in reference thereto, the glass sheet can be separated, divided, or cut into the correct number of partial parabolic glass shapes. Thus, while two such shapes may be formed by the mold illustrated in <figref idrefs="DRAWINGS">FIGS. 13-15</figref>, and <figref idrefs="DRAWINGS">FIG. 18</figref> illustrates a mold which may produce at least four such partial parabolic portions, other molds can produce six, seven, twelve, or any number of partial parabolic portions depending on how many linear square root composite shaped portions are joined.

p-0079<figref idrefs="DRAWINGS">FIG. 19</figref> illustrates another embodiment of a mold similar to that of <figref idrefs="DRAWINGS">FIG. 18</figref>, except that mold <b>620</b> is formed with four joined linear square root composite shaped portions. Portion <b>622</b> can correspond to a linear square root composite curve portion similar to portion <b>212</b> of <figref idrefs="DRAWINGS">FIG. 7</figref>, whereas portion <b>624</b> can correspond to portion <b>222</b> of <figref idrefs="DRAWINGS">FIG. 7</figref>. Portion <b>626</b>, however, can correspond to portion <b>212</b> of <figref idrefs="DRAWINGS">FIG. 9</figref>, except that it has been rotated to join with portion <b>622</b> with a slope matching the slope of portion <b>622</b> at the intersection therebetween. In other words, portions <b>622</b> and <b>626</b> are joined as a smooth curve where they have a common tangent direction at the joining. Similarly, portion <b>628</b> has been oriented to be smoothly joined with portion <b>624</b>. A single mold such as mold <b>620</b> can thus be used to create four partial parabolic glass shapes which can be separated by dividing them, such as by cutting, at the location of portion joins.

p-0080<figref idrefs="DRAWINGS">FIG. 20</figref> illustrates another embodiment of a mold <b>630</b> similar to that of mold <b>620</b> except that portion <b>632</b> corresponds to portion <b>212</b> of <figref idrefs="DRAWINGS">FIG. 8</figref>, and portion <b>634</b> corresponds to portion <b>222</b> of <figref idrefs="DRAWINGS">FIG. 8</figref>. Portions <b>636</b> and <b>638</b> can be joined to their respective portions <b>632</b>, <b>634</b> in a manner similar to that described above with respect to <figref idrefs="DRAWINGS">FIG. 19</figref>.

p-0081As can be seen, various other portions can be joined similarly to make other permutations. So long as the linear square root composite curved portions are joined smoothly as shown, or with a transition portion permitting a smooth joining, numerous possibilities for molds can be constructed. Additionally, more than four portions can be joined, and permutations of multiple-portion molds can be themselves smoothly joined to form molds which can produce any number of portions, even or odd, as desired for the embodiment. Thus, while two- and four-portion molds are shown, three-portion molds, eight-portion molds, or seventeen-portion molds, together with any other number, can all be formed using the techniques and advances described herein.

p-0082For illustrative purposes, the following description of method <b>700</b> may refer to elements mentioned above in connection with <figref idrefs="DRAWINGS">FIGS. 1-18</figref>. In practice, portions of method <b>700</b> may be performed by different elements of the described system, e.g., mold <b>300</b>, upper surface <b>302</b>, or any other component, whether or not illustrated. It should be appreciated that method can include any number of additional or alternative tasks, the tasks shown in <figref idrefs="DRAWINGS">FIG. 19</figref> need not be performed in the illustrated order, and method <b>700</b> may be incorporated into a more comprehensive procedure or process having additional functionality not described in detail herein.

p-0083Method <b>700</b> describes a method of forming a partial parabolic glass sheet, from a flat or planar glass sheet or curved into the desired shape. Initially, a planar glass sheet can be positioned on or above a sag-bending mold having a linear square root composite shape on an upper, glass-supporting surface <b>702</b>. Subsequently, the glass sheet can be heated to a first temperature <b>704</b>. The first temperature can be sufficient to cause the glass sheet to sag under its own weight or, in some embodiments, a downward force additionally can be applied. In either case, the planar glass sheet can be caused to deform to follow the shape of the glass-supporting surface of the sag-bending mold.

p-0084The curved glass sheet can then be cooled below the first temperature <b>706</b>. The cooling can be sufficient to harden the glass and ensure that it will retain its shape once removed from the mold. The curved glass sheet can then be removed from the mold <b>708</b>. In certain embodiments, the curved glass sheet can be removed once only partially cooled to its final rest temperature. In other embodiments, the curved glass sheet can be cooled entirely to its final rest temperature before separating the curved glass sheet from the mold. The curved glass sheet can now be in the shape of a partial parabola appropriate for use in a solar concentrator or other application.

p-0085The curved glass sheet, after separated from the sag-bending mold, can optionally be divided or separated into discrete partial parabolic portions, sections, or segments. As described above, a single mold having a linear square root composite curve shape can be used to form several partial parabolic portions from a single planar glass sheet, advantageously with less localized defects and greater throughput than a parabolic mold.

p-0086While at least one exemplary embodiment has been presented in the foregoing detailed description, it should be appreciated that a vast number of variations exist. It should also be appreciated that the exemplary embodiment or embodiments described herein are not intended to limit the scope, applicability, or configuration of the claimed subject matter in any way. Rather, the foregoing detailed description will provide those skilled in the art with a convenient road map for implementing the described embodiment or embodiments. It should be understood that various changes can be made in the function and arrangement of elements without departing from the scope defined by the claims, which includes known equivalents and foreseeable equivalents at the time of filing this patent application.