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TECHNICAL FIELD 
       [0001]    This disclosure relates generally to thermoplastic polyolefin (TPO) membrane roofing materials and methods and more particularly to TPO outside corner patches for sealing around vents and other structures that protrude from a roof structure. 
       BACKGROUND 
       [0002]    It is common for commercial and other roofs that are substantially flat to seal the roof with a waterproof membrane such as polymer coated membranes, more commonly referred to as thermoplastic polyolefin membranes or simple TPO membranes. Almost all such roofs include various protrusions that project upwardly from the roof deck such as, for instance, vents, ductwork, air conditioning units, and the like. Providing a water-tight seal around such protrusions, and particularly where the corners of a protrusion meet the flat roof deck, can be a challenge. More specifically, it is possible to wrap the protrusion at least partially with a skirt of TPO membrane with the bottom edge portion of the skirt flaring out to cover and be heat sealed to the roof membrane. However, this requires that the skirt be slit at the bottom of the corners of the protrusion, which leaves a region where the corners meet the flat roof unsealed and subject to leaks. 
         [0003]    Corner pieces made from TPO have been developed to address this problem. For example, the Firestone® ReflexEON® inside/outside corner patch is a molded piece of TPO plastic with the general shape of a right angle corner permanently molded in. The molded corner is placed around the bottom corner of a protrusion and the patch is heat sealed to the surrounding TPO membranes to seal the corner. In contrast, GenFlex® TPO reinforced outside corners are factory fabricated corners made from high performance TPO roofing membrane. These are generally made by slitting a square piece of TPO membrane from its center to a corner and then spreading the membrane out at the slit to cause the opposite corner to form a loose pleat. The gap between the spread edges of the slit is then filled in with another piece of TPO membrane, which is heat sealed in place to form a unitary corner patch. In use, the loose pleat is applied around the bottom corner of a protrusion and the patch is heat sealed to surrounding TPO membranes on the roof and the protrusion to form a water-tight seal. 
         [0004]    Other examples of attempted solutions can be found in U.S. Pat. Nos. 4,700,512; 4,799,986; 4,872,296; and 5,706,610. It also has been common in the past for installers of membrane roofs to custom make their own corner patches on-site by heating, stretching, cutting, and otherwise manipulating small pieces of TPO membrane. Corner patches and other solutions in the past have not been entirely satisfactory for a number of reasons including that they do not fit well around corners, they must be “bunched up” to fit a corner properly, thus jeopardizing the ability for form a reliable seal, and/or they contain heat sealed joints that can fail and result in a leak. There is a need for a corner patch that addresses satisfactorily the shortcomings and problems of the prior art. 
       SUMMARY 
       [0005]    Briefly described, a patch is disclosed for flat TPO sealed roofs that seals the outside bottom corners of roof protrusions such as vents, ductwork, air conditioning units, where the corners meet the flat roof. In one embodiment, the patch is made of a circular blank of TPO material that is vacuum formed to produce a plurality of radially extending flutes or peaks and valleys in the patch. This is referred to herein as a daisy wheel configuration. The number of flutes, the depth of each flute, and the radius of the blank are optimized according to methods of the invention so that the patch fits an outside bottom corner of a roof protrusion perfectly when the flutes are spread out. The patch can then be heat sealed to surrounding TPO membranes on the protrusion and the roof to provide a water-tight seal where corners of protrusions meet the flat roof. The TPO daisy wheel corner patch of this disclosure also can be optimized for corners that are not orthogonal; i.e. where the sides of the protrusion and the roof do not form right angles with respect to each other. This has not generally been possible with prior art prefabricated corners and has required tedious custom fabricating of corner patches on sight for acceptable results. The patch of this invention also is easily and efficiently packaged because the daisy wheel shape of the patches allows them to be nested together in a compact stack. 
         [0006]    Thus, an improved prefabricated TPO corner patch is now provided that fits a corner for which it is designed perfectly to provide a reliable water-tight seal, that is compact and efficient to stack, store, and transport, and that can be optimized for orthogonal and other outside corner shapes commonly encountered in flat or semi-flat commercial roofs. These and other aspects, features, and advantages will be better understood upon review of the detailed description set forth below when taken in conjunction with the accompanying drawing figures, which are briefly described as follows. 
     
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         [0007]      FIG. 1  is a perspective view of a section of a flat TPO sealed roof with a protrusion and illustrates one preferred application of the TPO outside corner patch. 
           [0008]      FIG. 2  is a perspective view of a TPO outside corner patch that embodies principles of the disclosure in a preferred form. 
           [0009]      FIG. 3  a perspective view of a circular TPO blank from which the corner patch of this disclosure is molded illustrating design variables for optimizing the number and depth of flutes for a particular corner. 
           [0010]      FIG. 4  shows a generic protrusion with a corner patch and illustrates how a design circumference is determined for a patch of a give radius. 
           [0011]      FIG. 5  is a graph illustrating the results of the optimization methodology of the present disclosure. 
       
    
    
     DETAILED DESCRIPTION 
       [0012]    Referring now in more detail to the drawing figures, wherein like reference numerals indicate like parts throughout the several views,  FIG. 1  illustrates a section  11  of a flat roof having a protrusion  13 . The protrusion is illustrated as a generic square upward projection from the roof deck. In reality, such projections take many forms and protrusion  13  may represent, for example, a chimney, a vent pipe, a duct, and air conditioning platform or unit, or otherwise. In any event, the protrusion  13  and the flat roof deck form outside corners  20  where the corners of the protrusion meet the roof deck. In the illustrated embodiment, the outside corners  20  are orthogonal; that is, the faces of the protrusion and the roof deck all meet at approximately right angles. However, the outside corner patch of this disclosure is not limited to use with orthogonal outside corners but may be optimized for non-orthogonal outside corners. 
         [0013]    The flat portion of the roof  11  is covered and sealed with a TPO membrane  14  as is known in the roofing art to prevent water from leaking into the building below. A cutout (not visible) is formed in the membrane at the location of the protrusion and the peripheral edges of the cutout extend up to the bottom of the protrusion. In order to seal along these bottom edges, a skirt or apron  16  of TPO membrane material is wrapped around and sealed to the protrusion  13  with the bottom of the skirt  16  flaring out to overly the membrane  14 . More particularly, the skirt  16 , when installed, includes an upper portion  17  that covers at least the lower section of the protrusion and flaps  18  that flare outwardly to overly and cover the membrane  14 , to which the flaps  18  are thermally welded to form a watertight seal. In order to allow the flaps  18  to extend outwardly, the TPO membrane forming the skirt  16  is slit during installation at the bottom corners of the protrusion, as indicated by reference numeral  19 . This leaves an outside corner  20  where the corners of the protrusion and the end of the slit meet the roof deck that is subject to leaks unless properly sealed. Outside corner patches  21  according to the present disclosure are applied to seal these outside corners  20 , as detailed below. 
         [0014]    An outside corner patch  21  according to the present disclosure is applied at each of the outside corners  20  of the protrusion to form a watertight seal at these corners. Referring to the foreground outside corner in  FIG. 1 , the outside corner patch  21  comprises a specially formed circular piece of TPO membrane material that has been fluted, as detailed below, to conform to the shape of the outside corner when the patch is spread out. In this illustration, the corner patch  21  is applied beneath the upper portion  17  of the skirt and beneath the two adjacent flaps  18 . It will be understood, however, that the patch also may be applied over the top of the upper portion  17  of the skirt and over the top of the two adjacent flaps  18  if desired. In either event, the corner patch  21  is thermally welded to the TPO material of the skirt  16  and the roof membrane  14 , as indicated at  22 , thus forming a watertight seal at the bottom outside corner of the protrusion. Thermal welding or heat sealing of TPO corners and other members to membranes is well known in the commercial roofing trade and thus the details of this process need not be discussed in detail here. 
         [0015]      FIG. 2  illustrates a preferred configuration of the outside corner patch of this disclosure before being applied to the outside corner of a protrusion, as illustrated in  FIG. 1 . The patch  21  is generally circular in shape with a central region  26  and a periphery  27  and is radially fluted to define an array of radially extending peaks  28  and corresponding radially extending valleys  29 . This forms the daisy wheel configuration of the patch. The peaks and valleys expand in amplitude from substantially zero amplitude at the central region  26  of the patch to a maximum amplitude at the periphery  27  of the patch. The patch  21  can be fabricated in a variety of ways. Preferably, however, a circular cutout of standard TPO membrane material is heated and vacuum formed to generate the daisy wheel configuration with a predetermined number of peaks and valleys. Other possible fabrication methods might include injection molding, thermoforming, pressure molding, or similar known techniques. The patch shown in  FIG. 2  is illustrated with  10  peaks and  10  valleys defining the daisy wheel configuration. However fewer or more peaks and valleys might be selected based upon the optimization techniques described in detail below. 
         [0016]    For installation of the outside corner patch of this disclosure, the patch is positioned with its central region  26  aligned with and covering the corner where the faces of the protrusion meet the flat roof. The flutes of the patch are then spread out substantially flat as the patch is conformed to the contour of the outside corner. More specifically, the flutes are spread out until the patch lies flat against both of the faces of the protrusion and also lies flat against the flat roofing membrane in the region of the corner. With the number of flutes and the sizes of the flutes optimized for the three dimensional shape of the outside corner, the patch conforms perfectly to the faces of the protrusion and the roof when fully spread out. The patch can then be thermally welded or heat sealed to the underlying or overlying, as the case may be, TPO material of the upper portion  17  of the skirt, the flaps  18 , and the roof membrane  14  thus forming a watertight seal at the outside corner of the protrusion. 
         [0017]    As mentioned above, in order for the outside corner patch of this disclosure to conform perfectly to an outside corner, its configuration, i.e. the number and sizes of the flutes should be optimized for the shape of the outside corner and the diameter of the patch. Most outside corners are orthogonal, but the patch may also be optimized for non-orthogonal outside corners if desired. The optimization methodology described below is for an orthogonal outside corner.  FIG. 3  illustrates the design variables that enter into the optimization process. The starting circular blank of TPO material  31  from which the patch is to be formed has a center O, a periphery  33  and can be divided into pie-shaped sections  34 , each of which will be deformed into a generally cone-shaped peak or a valley of the final fluted patch, as illustrated by phantom line  36 . An imaginary plunge circle  37  may be constructed as an aid in deriving the optimization algorithms. The variables shown in  FIG. 3  that are relevant to the optimization process of this invention are defined as follows.
       n: number of flutes (total of peaks plus valleys)   r b : radius of circular TPO blank   r p : radius of plunge circle   α: flute blank angle   h: depth of draw   β: flute depth angle   a, b, c, d, and e identify various useful points on the construction       
 
         [0025]    With these optimization variables identified, and with reference to  FIG. 3 , we see that for triangle oac: 
         [0000]      sin(α/2)= ab/ 2/ oa=ab/ 2 /r   b    
         [0000]      Thus:  ab= 2 r   b  sin(α/2)   (1) 
         [0000]      where: α=2π /n    (2) 
         [0026]    Assume that a plunge circle will generate arc aeb when the flat blank is deformed so that the edge of the flute conforms to the plunge circle. Then, for triangle acd, we can see from the Pythagorean theorem for right triangles that: 
         [0000]    
       
      
       ad 
       2 
       =ac 
       2 
       +cd 
       2  
      
     
         [0000]      or:  r   p   2 =( ab/ 2) 2   +cd   2  but  cd+h=r   p    
         [0000]      so:  r   p   2 =( ab/ 2) 2 +( rp−h ) 2    
         [0000]    solving this equation for r p  gives: 
         [0000]        r   p =(( ab)   2 /4+ h   2 )/2 h    (3) 
         [0000]      and: sin(β/2)= bc/db=ab/ 2 /r   p    
         [0000]      so that: β=2 sin −1 ( ab/ 2 r   p )   (4) 
         [0027]    Hence, for a given depth of draw “h,” the plunge circle radius r p  can be calculated from equation 3. Then, the plunge circle circumference is: 
         [0000]      2πr p    
         [0000]    and the length of the flute edge that will follow the contour of the plunge circle when the blank is deformed is: 
         [0000]      ↑/2π×2πr p  or just βr p    
         [0000]    Finally, the total length of the perimeter edge of a fluted patch with n flutes, which we shall designate the “fluted circumference” or c f , is given by the total of the lengths of each individual flute, or: 
         [0000]      c f =nβr p    (5) 
         [0000]    Now, referring to  FIG. 4 , which shows a fluted circular patch stretched flat and conformed to an outside orthogonal corner, and considering that the radius of the fluted patch is equal to the radius of the blank r b , we can determine, using the equation below, the total length of the perimeter of a fluted patch required for the patch to conform perfectly to the corner. We shall call this perimeter length the “design circumference” or simply the “target.” 
         [0000]      (2 πr   r )+¼(2 πr   b )=5/4 (2 πr   b )   (6) 
         [0000]    The design circumference also can be derived by considering that A in  FIG. 4  is ¾ of a circle while B and C are each ¼ of a circle. Adding the circumferences of each of these partial circles gives: 
         [0000]      ¾(2 πr   b )+¼(2 πr   b )+¼(2 πr   b )=5/4(2 πr   b ) 
         [0028]    Hence, optimization routines can be run for a blank of a given radius by selecting various values of flute draw h and, for each value of h, varying the number of flutes n until the combination of h and n generate a fluted circumference c f  that is equal or very close to the design circumference given by equation 6.  FIG. 5  illustrates, in the form of a graph, the results of such an iteration to determine the optimum combination of flutes n and flute draw h required for a corner patch having a 4 inch diameter radius to conform perfectly to an outside orthogonal corner. The design circumference or target calculated from equation 6 is represented by the dark horizontal line on the graph. Each curve of the graph represents the fluted circumference c f  for one of the flute draw values shown in the box at the upper right of the graph for various values of the number of flutes n. It will be noted that only the data points on each graph represent a realistic combination of h and n since n must be an even integer. 
         [0029]    It can be seen from  FIG. 5  that the following combinations of number of flutes n and flute draw h generate, for a four inch radius blank, a fluted circumference that is very close the design circumference: 
         [0000]      n=12 and h=0.69 inch 
         [0000]      n=16 and h=0.5 inch 
         [0000]      and n=20 and h=0.4 inch 
         [0000]    Either of these combinations would result in a fluted patch that would conform to an outside orthogonal corner when stretched out flat. However, due to manufacturing considerations, and to produce a relatively rigid and robust final product, the first combination of n=12 and h=0.69 is considered most optimal. 
         [0030]    A four inch radius TPO blank was formed according to the above optimization methodology with 12 flutes and a flute draw of 0.69 inches and was tested on an orthogonal outside corner of a protrusion. The test patch proved to conform perfectly to the corner when placed with its center directly at the corner and its flutes stretched out flat to cover the deck and contiguous sides of the protrusion. Of course, patches of radii other than 4 inches such as, for instance, 2, 6, or 8 inches, can be optimized according to the forgoing methodology so that the radius of the starting TPO blank is not a limitation of the methodology or the invention. 
         [0031]    The invention has been described herein in terms of preferred embodiments and methodologies considered by the inventors to represent the best mode of carrying out the invention. However, numerous additions, deletions, and modifications of the illustrated embodiments might be made by those of skill in the art without departing from the spirit and scope of the invention as set forth in the claims. For example, the patch has been described within the context of flat commercial roofing. However, the invention is not limited to flat roofs or commercial roofing but may be adapted for sealing corner protrusions in non-flat roofs. Indeed, the invention may be applied in non-roofing scenarios such as in sheet metal structures, tub and shower basins, and the like where it is desired to seal outside corners of protrusions.

Summary:
An outside corner patch for TPO roofing is formed from a circular piece of TPO membrane material being vacuum formed to define an array of flutes that extend from the center of the piece toward its edges. The flutes form ridges and valleys that generally are shaped as conical sections with the apex of the conical sections located at the center of the patch. The number and size of the flutes is optimized in such a way that, when the flutes are stretched flat, the patch conforms to and fits flat against the surfaces of an outside corner formed by the intersection of a roof deck with an upward protrusion from the roof. The TPO outside corner patch is applied over the corner and thermally welded to surrounding TPO membranes on the roof deck and the protrusion to form a watertight seal at the outside corner.