Patent Publication Number: US-6981935-B2

Title: Suction roll with sensors for detecting temperature and/or pressure

Description:
FIELD OF THE INVENTION 
     The present invention relates generally to industrial rolls, and more particularly to rolls for papermaking. 
     BACKGROUND OF THE INVENTION 
     Cylindrical rolls are utilized in a number of industrial applications, especially those relating to papermaking. Such rolls are typically employed in demanding environments in which they can be exposed to high dynamic loads and temperatures and aggressive or corrosive chemical agents. As an example, in a typical paper mill, rolls are used not only for transporting a fibrous web sheet between processing stations, but also, in the case of press section and calender rolls, for processing the web sheet itself into paper. 
     A papermaking machine may include one or more suction rolls placed at various positions within the machine to draw moisture from a belt (such as a press felt) and/or the fiber web. Each suction roll is typically constructed from a metallic shell covered by a polymeric cover with a plurality of holes extending radially therethrough. Vacuum pressure is applied with a suction box located in the interior of the suction roll shell. Water is drawn into the radially-extending holes and is either propelled centrifugally from the holes after they pass out of the suction zone or transported from the interior of the suction roll shell through appropriate fluid conduits or piping. The holes are typically formed in a grid-like pattern by a multi-bit drill that forms a line of multiple holes at once (for example, the drill may form fifty aligned holes at once). In many grid patterns, the holes are arranged such that rows and columns of holes are at an oblique angle to the longitudinal axis of the roll. 
     As the paper web is conveyed through a papermaking machine, it can be very important to understand the pressure profile experienced by the paper web. Variations in pressure can impact the amount of water drained from the web, which can affect the ultimate sheet moisture content, thickness, and other properties. The magnitude of pressure applied with a suction roll can, therefore, impact the quality of paper produced with the paper machine. 
     Other properties of a suction roll can also be important. For example, the stress and strain experienced by the roll cover in the cross machine direction can provide information about the durability and dimensional stability of the cover. In addition, the temperature profile of the roll can assist in identifying potential problem areas of the cover. 
     It is known to include pressure and/or temperature sensors in the cover of an industrial roll. For example, U.S. Pat. No. 5,699,729 to Moschel et al. describes a roll with a helically-disposed fiber that includes a plurality of pressure sensors embedded in the polymeric cover of the roll. However, a suction roll of the type described above presents technical challenges that a conventional roll does not. For example, suction roll hole patterns are ordinarily designed with sufficient density that some of the holes would overlie portions of the sensors. Conventionally, the sensors and accompanying fiber are applied to the metallic shell prior to the application of the polymeric cover, and the suction holes are drilled after the application and curing of the cover. Thus, drilling holes in the cover in a conventional manner would almost certainly damage the sensors, and may well damage the optical fiber. Also, during curing of the cover often the polymeric material shifts slightly on the core, and in turn may shift the positions of the fiber and sensors; thus, it is not always possible to determine precisely the position of the fiber and sensors beneath the cover, and the shifting core may move a sensor or cable to a position directly beneath a hole. Further, ordinarily optical cable has a relative high minimum bending radius for suitable performance; thus, trying to weave an optical fiber between prospective holes in the roll may result in unacceptable optical transmission within the fiber. 
     SUMMARY OF THE INVENTION 
     The present invention is directed to sensing systems for industrial rolls that can be employed with suction rolls. As a first aspect, the present invention is directed to an industrial roll comprising: a substantially cylindrical shell having an outer surface and an internal lumen; a polymeric cover circumferentially overlying the shell outer surface; and 
     a sensing system. The sensing system includes: a plurality of sensors embedded in the cover, the sensors configured to sense an operating parameter of the roll; and a signal-carrying member serially connected with and extending between the plurality of sensors. The signal-carrying member follows a helical path over the outer surface of the shell, wherein the signal-carrying member extends between adjacent sensors extends over more than one complete revolution of the shell outer surface (and, preferably, an intermediate segment of the signal-carrying member extends over more than a full revolution of the roll between adjacent sensors). 
     As a second aspect, the present invention is directed to an industrial roll comprising: a substantially cylindrical shell having an outer surface and an internal lumen; a polymeric cover circumferentially overlying the shell outer surface, the cover including an internal groove that defines a helical path; and a sensing system, wherein the sensing system includes a plurality of sensors embedded in the cover that are configured to sense an operating parameter of the roll and a signal-carrying member serially connected with and extending between the plurality of sensors. The signal-carrying member resides in the groove and follows the helical path in the shell outer surface. 
     As a third aspect, the present invention is directed to an industrial roll, comprising: a substantially cylindrical shell having an outer surface and an internal lumen; a polymeric cover circumferentially overlying the shell outer surface; and a sensing system including a plurality of sensors embedded in the cover, the sensors configured to sense an operating parameter of the roll; and a signal-carrying member serially connected with and extending between the plurality of sensors. At least one of the plurality of sensors is configured to slide along and relative to the signal-carrying member. 
     As a fourth aspect, the present invention is directed to an industrial roll, comprising: a substantially cylindrical shell having an outer surface and an internal lumen; a polymeric cover circumferentially overlying the shell outer surface, wherein the cover and shell include a plurality of through holes extending from an outer surface of the cover to the shell lumen, such that the lumen is in fluid communication with the environmental external to the cover outer surface; and a sensing system comprising: a plurality of sensors embedded in the cover, the sensors configured to sense an operating parameter of the roll; and a signal-carrying member serially connected with and extending between the plurality of sensors, the signal-carrying member following a helical path over the outer surface of the shell. The cover further comprises at least one blind drilled hole located over one of the plurality of sensors. 
     As a fifth aspect, the present invention is directed to a method of calculating the axial and circumferential positions of sensors on an industrial suction roll. The method comprises the steps of: providing as input variables (a) one of the diameter and circumference of the roll and (b) an angle defined by a hole pattern in the industrial roll and a plane perpendicular to the longitudinal axis of the roll; selecting a value for one of an axial or circumferential position of a sensor; and determining the other of the axial or circumferential position of the sensor based on the values of the diameter or circumference of the roll, hole pattern angle and axial or circumferential position. 
     Each of these aspects of the invention (as well as others) can facilitate the employment of a sensing system within a suction roll cover, thereby overcoming some of the difficulties presented by prior sensing systems. 
    
    
     
       BRIEF DESCRIPTION OF THE FIGURES 
         FIG. 1  is a gage view of a suction roll and detecting system of the present invention. 
         FIG. 2  is a gage perspective view of a shell and cover base layer formed in the manufacture of the suction roll of  FIG. 1 . 
         FIG. 3  is a gage perspective view of shell and cover base layer of  FIG. 2  being scored with a drill. 
         FIG. 4  is a gage perspective view of a groove being formed with a lathe in cover base layer of  FIG. 3 . 
         FIG. 5  is an enlarged partial gage perspective view of an optical fiber and sensor positioned in the groove formed in the cover base layer as shown in  FIG. 4 . 
         FIG. 6  is a greatly enlarged side section view of a sensor and optical fiber of  FIG. 5 . 
         FIG. 7  is a gage perspective view of the topstock layer being applied over the cover base layer, optical fiber and sensors of  FIGS. 3 and 5 . 
         FIG. 8  is a gage perspective view of the topstock layer of  FIG. 7  and shell and cover base layer of  FIG. 3  being drilled with a drill. 
         FIG. 9  is an enlarged top view of a typical hole pattern for a suction roll of  FIG. 1 . 
         FIG. 10  is a schematic diagram exhibiting the derivation of formulae employed in some embodiments of methods of determining axial and circumferential positions of sensors according to the present invention. 
         FIG. 11  is a flow chart illustrating steps in determining axial and circumferential positions of sensors according to methods of the present invention. 
     
    
    
     DETAILED DESCRIPTION OF THE INVENTION 
     The present invention will now be described more fully hereinafter, in which preferred embodiments of the invention are shown. This invention may, however, be embodied in different forms and should not be construed as limited to the embodiments set forth herein. Rather, these embodiments are provided so that this disclosure will be thorough and complete, and will fully convey the scope of the invention to those skilled in the art. In the drawings, like numbers refer to like elements throughout. Thicknesses and dimensions of some components may be exaggerated for clarity. 
     Referring now to the figures, a suction roll, designated broadly at  20 , is illustrated in  FIG. 1 . The suction roll  20  includes a hollow cylindrical shell or core  22  (see  FIG. 2 ) and a cover  24  (typically formed of one or more polymeric materials) that encircles the shell  22 . A sensing system  26  for sensing pressure, temperature, or some other operational parameter of interest includes a helical optical fiber  28  and a plurality of sensors  30 , each of which is embedded in the cover  24 . The sensing system  26  also includes a processor  32  that processes signals produced by the sensors  30 . 
     The shell  22  is typically formed of a corrosion-resistant metallic material, such as stainless steel or bronze. A suction box (not shown) is typically positioned within the lumen of the shell  22  to apply negative pressure (i.e., suction) through holes in the shell  22  and cover  24 . Typically, the shell  22  will already include through holes that will later align with through holes  82  and blind-drilled holes  84 . An exemplary shell and suction box combination is illustrated and described in U.S. Pat. No. 6,358,370 to Huttunen, the disclosure of which is hereby incorporated herein in its entirety. 
     The cover  24  can take any form and can be formed of any polymeric and/or elastomeric material recognized by those skilled in this art to be suitable for use with a suction roll. Exemplary materials include natural rubber, synthetic rubbers such as neoprene, styrene-butadiene (SBR), nitrile rubber, chlorosulfonated polyethylene (“CSPE”—also known under the trade name HYPALON), EDPM (the name given to an ethylene-propylene terpolymer formed of ethylene-propylene diene monomer), epoxy, and polyurethane. In many instances, the cover  24  will comprise multiple layers ( FIGS. 2 and 7  illustrate the application of separate base and topstock layers  42 ,  70 ; additional layers, such as a “tie-in” layer between the base and topstock layers  42 ,  70  and an adhesive layer between the shell  22  and the base layer  42 , may also be included). The cover  24  may also include reinforcing and filler materials, additives, and the like. Exemplary additional materials are discussed in U.S. Pat. No. 6,328,681 to Stephens and U.S. Pat. No. 6,375,602 to Jones, the disclosures of which are hereby incorporated herein in their entireties. 
     The cover  24  has a pattern of holes (which includes through holes  82  and blind drilled holes  84 ) that may be any of the hole patterns conventionally employed with suction rolls or recognized to be suitable for applying suction to an overlying papermaker&#39;s felt or fabric and/or a paper web as it travels over the roll  20 . A base repeat unit  86  of one exemplary hole pattern is illustrated in  FIG. 9 . The repeat unit  86  can be defined by a frame  88  that represents the height or circumferential expanse of the pattern (this dimension is typically about 0.5 to 1.5 inches) and a drill spacing  90  that represents the width or axial expanse of the pattern. As is typical, the columns of holes  82 ,  84  define an angle θ (typically between about 5 and 20 degrees) relative to a plane that is perpendicular to the longitudinal axis of the roll  20 . 
     Referring back to  FIG. 1 , the optical fiber  28  of the sensing system  26  can be any optical fiber recognized by those skilled in this art as being suitable for the passage of optical signals in a suction roll. Alternatively, another signal-carrying member, such as an electrical cable, may be employed. The sensors  30  can take any form recognized by those skilled in this art as being suitable for detecting the operational parameter of interest (e.g., stress, strain, pressure or temperature). It is preferred, as described below, that the sensors  30  be of a configuration that permits them to slide (at least for a short distance) along the optical fiber  28 . Exemplary fibers and sensors are discussed in U.S. Pat. No. 5,699,729 to Moschel et al. and U.S. patent application Ser. No. 09/489,768, the contents of each of which are hereby incorporated herein by reference in their entireties. 
     The processor  32  is typically a personal computer or similar data exchange device, such as the distributive control system of a paper mill, that can process signals from the sensors  30  into useful, easily understood information. It is preferred that a wireless communication mode, such as RF signaling, be used to transmit the data from the sensors  30  to the processing unit  32 . Other alternative configurations include slip ring connectors that enable the signals to be transmitted from the sensors  30  to the processor  32 . Suitable exemplary processing units are discussed in U.S. Pat. No. 5,562,027 to Moore and U.S. patent application Ser. No. 09/872,584, the disclosures of which are hereby incorporated herein in their entireties. 
     The suction roll  20  can be manufactured in the manner described below and illustrated in  FIGS. 2–9 . In this method, initially the shell  22  is covered with a portion of the cover  24  (such as the base layer  42 ). As can be seen in  FIG. 2 , the base layer  42  can be applied with an extrusion nozzle  40 , although the base layer  42  may be applied by other techniques known to those skilled in this art. It will also be understood by those skilled in this art that, although the steps described below and illustrated in  FIGS. 3–6  are shown to be performed on a base layer  42 , other internal layers of a cover  24  (such as a tie-in layer) may also serve as the underlying surface for the optical fiber  28  and sensors  30 . 
     Referring now to  FIG. 3 , the base layer  42  of the cover  24  is scored or otherwise marked, for example with a multi-bit drill  46 , with score marks  44  that correspond to a desired pattern of holes  82 ,  84  that will ultimately be formed in the roll  20 . The score marks  46  should be of sufficient depth to be visible in order to indicate the locations where holes will ultimately be formed, but need not be any deeper. 
     Turning now to  FIG. 4 , a continuous helical groove  50  is cut into the base layer  42  with a cutting device, such as the lathe  52  illustrated herein. The groove  50  is formed between the score marks  44  at a depth of about 0.010 inches (it should be deep enough to retain the optical fiber  28  therein), and should make more than one full revolution of the outer surface of the base layer  42 . In some embodiments, the groove  50  will be formed at the angle θ defined by the holes  82 ,  84  and will be positioned between the columns of holes. In most embodiments, the angle θ is such that the groove  50  encircles the base layer  42  multiple times; for example, for a roll that has a length of 240 inches, a diameter of 36 inches, and an angle θ of 10 degrees, the groove  50  encircles the roll twelve times from end to end. 
     Referring now to  FIG. 5 , after the groove  50  is formed in the base layer  42 , the optical fiber  28  and sensors  30  of the sensor system  26  are installed. The optical fiber  28  is helically wound within the groove  50 , with the sensors  30  being positioned closely adjacent to desired locations. The fiber  28  is retained within the groove  50  and is thereby prevented from side-to-side movement. 
     It may be desirable to shift the positions of the sensors  30  slightly to precise locations on the base layer  42 . Because the optical fiber  28  is retained within the groove  50  and its relative inflexibility (i.e., it may break at a relatively high bending radius) may prevent bending a portion of the fiber  28  out of the groove in order to position a sensor  30 , in some embodiments the sensor  30  may be free to slide short distances along the fiber  28 . One exemplary design is illustrated in  FIG. 6 . As can be see therein, the sensor  30  includes a plurality of bending elements  60  (typically formed of glass or nylon) that are positioned in a staggered relationship. The fiber  28  threads between the bending elements  60  to form a series of merging undulations  62 . In this regard the sensor  30  resembles sensors described in U.S. patent application Ser. No. 09/489,768 identified above. That sensor is typically constructed with an epoxy or other filling material  63  that fills the gaps between the bending elements  60  and the undulations  62  and maintains the positional relationship between them (i.e., it maintains the undulations  62  in alignment with the bending elements  60  and holds the bending elements  60  in line with one another). In the sensor  30  of the present invention, it is preferred that an epoxy or other material be used to fill the volume between the bending elements  60  and the undulations  62 , but that such filling material not bond to the undulations  62 , thereby enabling the bending elements  60  (which are typically attached to a common substrate  64 ) to slide along the fiber  62 . This may be carried out, for example, by selecting a filling material (such as an epoxy) that does not chemically bond to the fiber  28 , or by coating the fiber  28  with a coating (such as a mold release) that prevents the filling material  63  from bonding to the fiber  28 . Such a slidable configuration would enable the positioning of the sensor  30  to be adjusted slightly relative to the fiber  28  to a desired precise position while not overstressing the fiber  28  through undue bending. 
     Once the sensors  30  are in desired positions, they can be adhered in place. This may be carried out by any technique known to those skilled in this art; an exemplary technique is adhesive bonding. 
     Referring now to  FIG. 7 , once the sensors  30  and fiber  28  have been positioned and affixed to the base layer  42 , the remainder of the cover  24  is applied.  FIG. 7  illustrates the application of a top stock layer  70  with an extrusion nozzle  72 . Those skilled in this art will appreciate that the application of the top stock layer  72  can be carried out by any technique recognized as being suitable for such application. As noted above, the present invention is intended to include rolls having covers that include only a base layer and top stock layer as well as rolls having covers with additional intermediate layers. Application of the top stock layer  70  is followed by curing, techniques for which are well-known to those skilled in this art and need not be described in detail herein. 
     Referring now to  FIG. 8 , after the top stock layer  70  is cured, the through holes  82  and the blind drilled holes  84  are formed in the cover  24  and, in the event that through holes  82  have not already been formed in the shell  22 , are also formed therein. The through holes  82  can be formed by any technique known to those skilled in this art, but are preferably formed with a multi-bit drill  80  (an exemplary drill is the DRILLMATIC machine, available from Safop, Pordenone, Italy). Care should be taken not to drill through holes  82  over the locations of sensors  30 ; instead, blind-drilled holes  84  can be drilled in these locations. 
     Because the hole pattern may define the path that the optical fiber  28  (and, in turn, the groove  50 ) can follow, in some rolls conventional placement of the sensors  30  (i.e., evenly spaced axially and circumferentially, and in a single helix) may not be possible. As such, one must determine which axial and circumferential positions are available for a particular roll. Variables that can impact the positioning of sensors include the size of the roll (the length, diameter and/or circumference) and the angle θ defined by the hole pattern. Specifically, the relationships between these variables can be described in the manner discussed below. 
     The length of the fiber extending from an origin point on the roll to a particular axial and circumferential position can be modeled as the hypotenuse of a right triangle, in which the axial position serves as the height of the triangle and the total circumferential distance covered by the fiber serves as the base of the triangle (see  FIG. 10 ). This relationship can be described as:
 
sin θ= a/FL ; and  Equation 1
 
cos θ= Xdπ/FL   Equation 2
 
     wherein: 
     FL=fiber length from origin to sensor position; 
     a=axial distance from origin to sensor position; 
     d=diameter of the roll; 
     X=number of revolutions of fiber around the circumference of the roll; and 
     θ=angle defined by suction hole pattern relative to plane through axis of roll. 
     Solving equations 1 and 2 for FL, then substituting yields:
 
 Xdπ /cos θ= a /sin θ  Equation 3
 
     Because (sin θ/cos θ) can be simplified to tan θ, the expression can be reduced to
 
 a=Xdπ (tan θ)  Equation 4
 
     Thus, for any axial position a, the corresponding circumferential position (expressed in the number revolutions, which can be converted into degrees by multiplying by 360) can be calculated; the reverse can be performed to calculate the axial position from a given circumferential position. 
     An alternative method for calculating the axial and circumferential positions employing some practical measurements used in suction rolls can also be used. For a specific roll with a designated hole pattern, the following variables can be assigned: 
     α=angular position on the roll; 
     z=axial position on the roll; 
     d=drill spacing; 
     N=number of frames in the circumference of a roll (this is a whole number); and 
     B=number of frames required for a diagonal row of holes to move in the axial direction the distance of one drill spacing. 
     For an optical fiber  28  that follows the drill pattern on a drilled roll,
 
α=( B/N )( z/d )  Equation 5
 
     with α being given in revolutions (again, multiplying α by 360 degrees gives the angular position in degrees). Thus, for a given drilled roll defined by a diameter, a length and a hole pattern, B, N and d are known. The circumferential position can then be calculated for a given axial position; alternatively, the axial position can be calculated for a given circumferential position. 
     Those skilled in this art will recognize that the aforementioned methods of calculating axial position and circumferential position may be performed using different forms of variables as demonstrated, and that other forms may also be used that consider the diameter and/or circumference of the roll and the angle at which the fiber travels in its helix. 
     In some embodiments, the calculation can be performed with a computer program designed and configured to receive data input of the type described above and, using such data, calculate axial and circumferential positions for sensors. Such a program is exemplified in  FIG. 11 . As an initial step, input variables regarding the configuration of the roll (typically one of diameter or circumference of the roll) and the angle of the hole pattern (typically either the angle itself or a similar property, such as the drill spacing and the numbers of frames required to complete a circumference and to move the pattern one full drill spacing) are provided. Next, one of a circumferential position or an axial position is selected. The computer program can then determine the other of the circumferential or axial position of the sensor. This information can be used to determine whether the combination of axial and circumferential positions is suitable for use with the roll. 
     Inasmuch as the present invention may be embodied as methods, data processing systems, and/or computer program products, the present invention may take the form of an entirely hardware embodiment, an entirely software embodiment or an embodiment combining software and hardware aspects. Furthermore, the present invention may take the form of a computer program product on a computer-usable storage medium having computer-usable program code embodied in the medium. Any suitable computer readable medium may be utilized including, but not limited to, hard disks, CD-ROMs, optical storage devices, and magnetic storage devices. 
     Computer program code for carrying out operations of the present invention may be written in an object oriented programming language such as JAVA®, Smalltalk or C++. The computer program code for carrying out operations of the present invention may also be written in conventional procedural programming languages, such as “C”, or in various other programming languages. Software embodiments of the present invention do not depend on implementation with a particular programming language. In addition, portions of computer program code may execute entirely on one or more data processing systems. 
     The present invention is described above with reference to block diagram and/or flowchart illustrations of methods, apparatus (systems) and computer program products according to embodiments of the invention. It is understood that each block of the block diagram and/or flowchart illustrations, and combinations of blocks in the block diagram and/or flowchart illustrations, can be implemented by computer program instructions. These computer program instructions may be provided to a processor of a general purpose computer, special purpose computer, or other programmable data processing apparatus to produce a machine, such that the instructions, which execute via the processor of the computer or other programmable data processing apparatus, create means for implementing the functions specified in the block diagram and/or flowchart block or blocks. 
     These computer program instructions may also be stored in a computer-readable memory that can direct a computer or other programmable data processing apparatus to function in a particular manner, such that the instructions stored in the computer-readable memory produce an article of manufacture including instruction means which implement the function specified in the block diagram and/or flowchart block or blocks. 
     The computer program instructions may also be loaded onto a computer or other programmable data processing apparatus to cause a series of operational steps to be performed on the computer or other programmable apparatus to produce a computer implemented process such that the instructions which execute on the computer or other programmable apparatus provide steps for implementing the functions specified in the block diagram and/or flowchart block or blocks. 
     It should be noted that, in some alternative embodiments of the present invention, the functions noted in the blocks may occur out of the order noted in the figures. For example, two blocks shown in succession may in fact be executed substantially concurrently or the blocks may sometimes be executed in the reverse order, depending on the functionality involved. Furthermore, in certain embodiments of the present invention, such as object oriented programming embodiments, the sequential nature of the flowcharts may be replaced with an object model such that operations and/or functions may be performed in parallel or sequentially. 
     The use of the equations set forth above can be demonstrated with the following examples. 
     EXAMPLE 
     In this example, it is assumed that the roll has the dimensions set forth in Table 1, and that the hole pattern is that illustrated in  FIG. 9 . 
     
       
         
           
               
               
               
             
               
                   
                   
               
               
                   
                 Dimension 
                 Quantity 
               
               
                   
                   
               
             
            
               
                   
               
            
           
           
               
               
               
               
            
               
                   
                 Diameter 
                 36 
                 inches 
               
               
                   
                 Axial Length of Roll between Outermost 
                 238 
                 inches 
               
               
                   
                 Sensors 
               
               
                   
                 Frame 
                 0.725 
                 inches 
               
               
                   
                 Drill Spacing 
                 1.405 
                 inches 
               
               
                   
                   
               
            
           
         
       
     
     The diameter and frame measurements indicate that the variable N above is 156, and for the hole pattern of  FIG. 9 , the variable B is 9. Thus, for this roll, Equation 5 yields:
 
α=0.041z  Equation 6
 
This equation can then be used to calculate axial and circumferential coordinates for sensors.
 
     If the circumferential spacing is maintained to be the same as a typical roll (usually 21 sensors over a 360 degree circumference, or about 17.14 degrees between sensors), a set of circumferential and axial positions can be calculated (Table 2). 
     
       
         
           
               
               
               
               
             
               
                   
               
               
                   
                 Total Angle 
                 Simple Angle 
                 Axial Position 
               
               
                 Sensor No. 
                 (degrees) 
                 (degrees) 
                 (inches) 
               
               
                   
               
             
            
               
                   
               
            
           
           
               
               
               
               
            
               
                 1 
                 0.000 
                 0.000 
                 0.0 
               
               
                 2 
                 377.143 
                 17.143 
                 25.55 
               
               
                 3 
                 754.286 
                 34.286 
                 51.10 
               
               
                 4 
                 1131.429 
                 51.429 
                 76.65 
               
               
                 5 
                 1508.572 
                 68.572 
                 101.70 
               
               
                 6 
                 1885.714 
                 85.714 
                 127.25 
               
               
                 7 
                 2262.857 
                 102.857 
                 152.80 
               
               
                 8 
                 2640.000 
                 120.000 
                 178.35 
               
               
                 9 
                 3017.144 
                 137.144 
                 203.90 
               
               
                 10 
                 3394.286 
                 154.286 
                 229.45 
               
               
                   
               
            
           
         
       
     
     It can be seen from the “Total Angle” calculation that, for each subsequent axial position, the angle increases by a full revolution of the roll. This corresponds to a full loop of the optical fiber  28  around the roll between adjacent sensors  30 . It can also be seen that, for this embodiment, the sensors  30  would be positioned over less than a full circumference of the roll  20  (only about 154 degrees), so some portions of the circumferential surface of the roll  20  would not have sensors  30  below them. In addition, there are fewer sensors  30  (ten, as opposed to the more typical  21 ) spaced relatively evenly along the length of the roll  20 . 
     If, rather than the circumferential spacing of a conventional roll being maintained, the conventional axial spacing of 11.9 inches is maintained, Equation 2 gives the circumferential positions shown in Table 3. 
     
       
         
           
               
               
               
               
             
               
                   
               
               
                   
                 Total Angle 
                 Simple Angle 
                 Axial Position 
               
               
                 Sensor 
                 (degrees) 
                 (degrees) 
                 (inches) 
               
               
                   
               
             
            
               
                   
               
            
           
           
               
               
               
               
            
               
                 1 
                 0.0 
                 0.0 
                 0.0 
               
               
                 2 
                 175.785 
                 175.785 
                 11.9 
               
               
                 3 
                 351.570 
                 351.570 
                 23.8 
               
               
                 4 
                 527.335 
                 167.335 
                 35.7 
               
               
                 5 
                 703.140 
                 343.140 
                 47.6 
               
               
                 6 
                 878.925 
                 158.925 
                 59.5 
               
               
                 7 
                 1054.711 
                 334.711 
                 71.4 
               
               
                 8 
                 1230.496 
                 150.496 
                 83.3 
               
               
                 9 
                 1406.281 
                 326.281 
                 95.2 
               
               
                 10 
                 1582.066 
                 142.066 
                 107.1 
               
               
                 11 
                 1757.851 
                 317.851 
                 119.0 
               
               
                 12 
                 1933.636 
                 133.636 
                 130.9 
               
               
                 13 
                 2109.421 
                 309.421 
                 142.8 
               
               
                 14 
                 2285.206 
                 125.206 
                 154.7 
               
               
                 15 
                 2460.991 
                 300.991 
                 166.6 
               
               
                 16 
                 2636.776 
                 116.776 
                 178.5 
               
               
                 17 
                 2812.562 
                 292.562 
                 190.4 
               
               
                 18 
                 2988.347 
                 108.347 
                 202.3 
               
               
                 19 
                 3164.132 
                 284.132 
                 214.2 
               
               
                 20 
                 3339.917 
                 99.917 
                 226.1 
               
               
                 21 
                 3515.702 
                 275.702 
                 238.0 
               
               
                   
               
            
           
         
       
     
     In this embodiment, all axial positions are satisfied. All angular positions are not, and in addition, the angular positions are not in circumferential order, so detecting of sensors may be more difficult. 
     The foregoing is illustrative of the present invention and is not to be construed as limiting thereof. Although exemplary embodiments of this invention have been described, those skilled in the art will readily appreciate that many modifications are possible in the exemplary embodiments without materially departing from the novel teachings and advantages of this invention. Accordingly, all such modifications are intended to be included within the scope of this invention.