Abstract:
A reciprocating motion disk for mixing wastewater in a tank of a treatment plan is optimized for geometry, along with cycling speed and stroke length, to cause effective mixing velocity throughout the tank.

Description:
BACKGROUND OF THE INVENTION 
       [0001]    This application claims benefit of provisional application No. 62/304,042, filed Mar. 4, 2016. 
     
    
       [0002]    This invention is concerned with efficient mixing of liquids and slurries using a reciprocating motion to oscillate a mixing disk. In particular the invention is directed to increasing mixing performance and efficiency with optimized geometry in a mixing disk in a reciprocating mixer. An important application of the invention is in wastewater treatment, in the mixing of wastewater slurries, especially in digesters, such as anaerobic digesters. 
         [0003]    Linear motion mixers, i.e. reciprocating motion mixers, are described in U.S. Pat. Nos. 7,685,896, 7,399,112, 7,364,351, 7,278,781, 7,029,166, 6,830,369, 6,007,237, 5,100,242 and 5,052,813, as well as U.S. Pub. No. 2015/182927. These mixers, at least in the context of the current invention, typically are used in very large vessels, such as, for example, a mixing disk of six feet operating in a vessel about 50 feet in diameter. The vessel or tank can be much larger. 
         [0004]    As discussed in some of the above patents and publication, several recognize the mixing disk geometry as having an effect on the efficiency and performance of the reciprocating mixer disk. Some of the patents, such as U.S. Pat. Nos. 6,830,369 and 7,685,896, provide a formula for a target range of effectiveness, the formula including, effectively, the ratio of the overall circular area defined by the entire disk to the open area at the center of the disk, wherein stroke length and cycle duration are also considered in the formula. 
       SUMMARY OF THE INVENTION 
       [0005]    According to the current invention, it has been discovered that optimum ratios of the areas of the disk should be different from what is taught in the prior art, especially for use in municipal wastewater treatment applications including anaerobic digesters, aerobic digesters, holding tanks, and other treatment tanks and vessels. 
         [0006]    The mixing disk is essentially a flat annular plate, circular or polygonal in shape surrounding an open central area (“open area”), with the annulus of the ring having a specific “footprint” or closed area, designed to produce a high velocity flow through the open area during linear reciprocating travel of the mixing disk. If the open area is too large relative to the closed area (also considering stroke length and frequency), fluid velocity through open area can be too low for effective mixing. In an ideal design that center fluid velocity and flow rate are maximized. 
         [0007]    Whether the annular ring of closed area surrounding the open area is planar or formed with a cone angle, it is the projected area of the disk or closed area that is important in considering optimum mixing performance. The following equation is followed for optimizing disk parameters, where CPM is frequency in cycles per minute, A closed  is the closed area of the disk, and A open  is open area at the center of the disk: 
         [0000]      CPM×stroke length×( A   closed   +A   open )/ A   open ,
 
         [0008]    the resultant mathematical value being greater than 1527 and less than 8750, or more broadly, between about 1530 and about 8750. 
         [0009]    Typically in municipal sewage treatment operations, the stroke length, i.e. the distance traveled from the topmost position to the bottommost position of the mixing disk, will be in the range of 6 inches to 24 inches. Typically the frequency will be in the range of about 25 to 35 CPM. A typical overall diameter of the mixer disk can be about four or six feet to eight feet. 
         [0010]    Since the ratio in the above formula is a ratio of areas, the same ratio can be obtained by the quotient of the square of the outer diameter of the disk and the square of the inner diameter of the annular closed area. This assumes the area of the shaft hub and the radial spokes, which is minimal, is disregarded and simply counted as part of open area. For greater accuracy, the hub and spokes area can be factored into the calculation of the open area, i.e. subtracted out, which will shift the number range somewhat. The claims herein assume the hub and spokes area is not subtracted out. 
         [0011]    The same formula and optimal range is applicable for non-circular mixer disk shapes, such as polygonal, as well as circular. The term essentially circular may be used herein, with the intention of including polygonal shapes, such as pentagonal, hexagonal, octagonal, etc. The term also includes non-circular round shapes such as elliptical. 
         [0012]    The invention thus optimizes the performance of a linear motion reciprocating mixing disk, using the parameters developed according to the invention for application to any type of shaped, open-center reciprocal mixer disk. These and other objects, advantages and features of the invention will be apparent from the following description of a preferred embodiment, considered along with the accompanying drawings. 
     
    
     
       DESCRIPTION OF THE DRAWINGS 
         [0013]      FIG. 1A  is a perspective view showing one example of a reciprocating mixer disk geometry. 
           [0014]      FIG. 1B  is a plan view of the same disk. 
           [0015]      FIGS. 2A and 2B  are perspective and plan views showing another disk geometry. 
           [0016]      FIGS. 3A and 3B  are perspective and plan views showing another mixing disk, with a polygonal shape. 
           [0017]      FIG. 4  is a diagram with plan and section views showing disk geometry factors, i.e. dimensions. 
           [0018]      FIGS. 5-8  are schematic plan views showing a polygonal disk and identifying areas of the disk that can be involved in calculations regarding disk geometry. 
       
    
    
     DESCRIPTION OF PREFERRED EMBODIMENTS 
       [0019]    The drawings illustrate several different general configurations of reciprocating mixer disk to which the principles of the invention can be applied.  FIGS. 1A and 1B  show one pattern in the form of a mixing disk  10  comprised of an annular ring  12  of closed area that is shown flat, planar but which could be conical if desired, as in some mixer disks of the prior art. The flat ring  12  has outer and inner radii, R o  and R i  respectively, defining a width, W disk =R o R i ,closed area  14  (references to closed area herein include only the ring  12 ). In the center is an open area  16 . If the solid ring  12  is sloped, in a conical shape, it is the projected area that is important to the invention, that is, the area that would be seen as the closed area  12  in plan view, such as seen in  FIG. 1B . 
         [0020]    The mixing disk has a hub  18  to which a reciprocating mixer shaft is attached concentrically, secured into either the hub side seen in  FIG. 1A  or the opposite side. A series of spokes or struts  20 , eight of which are shown in the examples here, secure the closed-area ring  12  to the hub  18 . The mixing disk  10 , for most purposes according to the invention, is of a large diameter, e.g. four or six feet to seven or eight feet, and can be formed of stainless steel or other corrosion-resistant metal, such as painted steel. 
         [0021]      FIGS. 2A and 2B  show another design of a mixing disk  22 , similar to that of  FIG. 1A  but with struts  24  secured to the annular disk  12  in a different way, in which the end of each strut is forked with a slot, fitting closely over the thickness of the ring  12 . The struts can be welded or otherwise securely affixed to the disk. 
         [0022]      FIGS. 3A and 3B  show a different design wherein the closed area annular mixing disk  12   a  is polygonal rather than circular, in this case octagonal. The formula for optimum area applies no matter the shape selected, so long as a closed generally ring-shaped area surrounds an open central area. Here, as seen in  FIG. 3B , the closed area  12   a  is defined by eight trapezoid shapes. The area of each segment is defined by the width, times the average length of major sides, shown as L 1  for the outer length and L 2  for the inner length of the segment. This area can also be calculated using the radii R o  and R i  and the number of sides, which defines the segment angle involved. The open area  16   a  of the polygonal mixer disk  24  is simply calculated as the area of an octagon (in this example), made up of eight triangles. The spokes  20  may or may not be included, as explained further below. The hub  18  area may also be included or not as discussed below. 
         [0023]    In all these different geometries, the sum of the projected closed area and the open area is divided by the open area to form the ratio of interest, basically the ratio of the entire area within the disk&#39;s outer diameter, to the open area, or area of the circle or polygon inward of the closed annular ring. This ratio is multiplied by frequency in CPM and stroke length in inches. 
         [0024]    As noted above, the numerical value resulting from this calculation should be in the range of about 1530 to about 8750. This results in a mixer disk design with a greater ratio of projected closed area to total, overall area, as compared to mixer disks of the prior art. 
         [0025]    More specifically, the above design number is preferably between about 2000 to 5000, and more preferably, one mixer disk design has the above numerical value in the range of about 2400 to about 3750. As an example, this would correspond to a circular mixer disk reciprocating at 30 CPM with a stroke of 20 inches, and a ratio of outer disk radius, R o  to inner ring radius R i  of 2.0 to 2.5 (ratio of squares being 4.0 to 6.25). 
         [0026]    Testing was conducted in a liquid-filled tank, with mixing disks reciprocated in the tank as stated above, and with R o /R i  ratios in a range between 1.5 and 2.5. Average velocities of liquid movement were measured within the tank, and it was found that for velocities of one inch per second, two inches per second, and three inches per second, a strong correlation was found between the percentage of tank volume being at each of these average velocities and the ratio of outer radius R o  to inner radius R i . Ratios of 2.0 to 2.5 (squared range 4.0 to 6.25) were found very effective, with radius ratios of 2.2 to 2.3 being most effective. The ratio range of 2.0 to 2.25 corresponds to a range of 2904 to 3038 for the calculated design factor number described above. 
         [0027]      FIGS. 4 through 8  are diagrams showing disk geometry factors and relationships of various areas. The following definitions apply in those drawings: 
         [0028]    Velocity Definitions: 
         [0029]    VM=maximum disk velocity 
         [0030]    VU=disk velocity upper limit=e.g. 43.982 in/s (24″ stroke @35 CPM) 
         [0031]    VL=disk velocity lower limit=e.g. 7.854 in/s (6″ stroke @25 CPM) 
         [0032]    VM range (VL≦VM≦VU) 
         [0033]    Polygon Definitions: 
         [0034]    n=(number of equal length sides) 
         [0035]    R i =inner perpendicular distance from polygon center to side 
         [0036]    R o =outer perpendicular distance from polygon center to side 
         [0037]    R x =x perpendicular distance from polygon center to side 
         [0038]    X f =disk width displacement factor (0.20&#39;xf≦0.40) 
         [0039]    Area Definitions: 
         [0040]    A hs =area occupied by hub and spokes 
         [0041]    A open  (area open)=A i −A hs    
         [0042]    A closed  (area closed)=A o -A i    
         [0043]    A x =(area displaced thru A open ) 
         [0044]    A do  (area displaced away from A open )=A o -A x    
         [0045]    A f =A open /A x  (0.35≦af≦0.65), broadly 0.25 to 0.75 
         [0046]    In the example below the number of polygonal sides is assumed at one million, i.e. essentially infinite, in effect circular. Dimensions are assumed for the purpose of this example. In the example the area occupied by the disk hub and spokes A hs  is subtracted out of the full area within the inner polygon to calculate the true open area. However, all of this hub and spokes area can be ignored, as noted above, and as assumed relative to the claims. That is, the open area can be more approximately calculated by simply using the entire polygonal or circular area inward of the solid annulus. 
         [0000]    
       
         
               
             
               
               
             
               
               
               
             
           
               
                   
               
               
                 CALCULATING DISK GEOMETRY-OPEN AREA METHOD 
               
               
                   
               
             
             
               
                   
               
             
          
           
               
                 n := 1000000 
                 number of equal length polygon sides (3 ≦ n ≦ ∞) 
               
               
                   
                 when n = ∞ the disk is circular in shape 
               
               
                 R i  := 22 · in 
                 inner distance from polygon center to side 
               
               
                 R o  := 42 · in 
                 outer distance from polygon center to side 
               
               
                 A hs  := 176.715 · in 2   
                 area occupied by disk hub and spokes 
               
               
                 W disk  := R o  − R i  = 20 · in 
                 disk width between inner and outer polygons 
               
               
                   
               
               
                 
                   
                     
                       
                         
                           A 
                           i 
                         
                         := 
                         
                           
                             n 
                             · 
                             
                               R 
                               i 
                               2 
                             
                             · 
                             
                               tan 
                               ( 
                               
                                 π 
                                 n 
                               
                               ) 
                             
                           
                           = 
                           
                             1520.53 
                             · 
                             
                               in 
                               2 
                             
                           
                         
                       
                     
                   
                 
                 area inner polygon 
               
               
                   
               
               
                 
                   
                     
                       
                         
                           A 
                           o 
                         
                         := 
                         
                           
                             n 
                             · 
                             
                               R 
                               o 
                               2 
                             
                             · 
                             
                               tan 
                               ( 
                               
                                 π 
                                 n 
                               
                               ) 
                             
                           
                           = 
                           
                             5541.77 
                             · 
                             
                               in 
                               2 
                             
                           
                         
                       
                     
                   
                 
                 area outer polygon 
               
               
                   
               
               
                 A open  := A i  − A hs  = 1343.82 · in 2   
                 open area allowing fluid passage 
               
               
                 A closed  := A o  − A i  = 4021.24 · in 2   
                 closed area displacing fluid 
               
               
                 a f  := 0.559 0.35 ≦ a f  ≦ 0.65 
                 disk area displaced thru open area factor range 
               
               
                   
               
               
                 
                   
                     
                       
                         
                           A 
                           x 
                         
                         := 
                         
                           
                             
                               A 
                               open 
                             
                             
                               a 
                               f 
                             
                           
                           = 
                           
                             2.404 
                             × 
                             
                               
                                 10 
                                 3 
                               
                               · 
                               
                                 in 
                                 2 
                               
                             
                           
                         
                       
                     
                   
                 
                 area displaced thru open area 
               
               
                   
               
               
                 
                   
                     
                       
                         
                           R 
                           x 
                         
                         := 
                         
                           
                             
                               
                                 A 
                                 x 
                               
                               · 
                               
                                 
                                   ( 
                                   
                                     n 
                                     · 
                                     
                                       tan 
                                       ( 
                                       
                                         π 
                                         n 
                                       
                                       ) 
                                     
                                   
                                   ) 
                                 
                                 
                                   - 
                                   1 
                                 
                               
                             
                           
                           = 
                           
                             27.66 
                             · 
                             in 
                           
                         
                       
                     
                   
                 
                 displaced distance from polygon center to side 
               
               
                   
               
               
                 A do  := A o  − A x  = 3137.81 · in 2   
                 area displaced away from open area 
               
               
                 X := R x  − R i  = 5.662 · in 
                 offset distance from inner polygon 
               
               
                   
               
               
                 
                   
                     
                       
                         
                           x 
                           f 
                         
                         := 
                         
                           
                             X 
                             
                               W 
                               disk 
                             
                           
                           = 
                           
                             
                               0.283 
                                
                               
                                   
                               
                                
                               0.20 
                             
                             ≤ 
                             
                               x 
                               f 
                             
                             ≤ 
                             0.40 
                           
                         
                       
                     
                   
                 
                 approximate disk width displacement factor range 
               
               
                   
               
               
                 A i _infinite_sides := π · R i   2  = 1520.53 · in 2   
                 area outer polygon with infinite sides (circular) 
               
               
                 A o _infinite_sides := π · R o   2  = 5541.77 · in 2   
                 area inner polygon with infinite sides (circular) 
               
               
                 A x _infinite_sides := π · R x   2  = 2403.96 · in 2   
                 area displaced with infinite sides (circular) thru open area 
               
             
          
           
               
                 A open  = 1.344 × 10 3  · in 2   
                 A x  − A open  = 1.06 × 10 3  · in 2   
                 A closed  − A x  = 1.617 × 10 3  · in 2   
               
               
                   
               
             
          
         
       
     
         [0047]      FIG. 7 , as well as  FIG. 4 , demonstrates the effect of the disk moving through liquid. As the disk moves, the volume of liquid directly in the path of the disk annulus is divided between that which enters the opening (along with liquid in the path of the opening) and liquid that flows outwardly, to clear the outer edge of the disk. The sectional view in  FIG. 4  shows this, as does the plan view, and denotes the radius from center to the division point as R. This is the displaced distance from the polygon center. Similarly, A x  is the area displaced through the open area of the disk, as denoted in  FIG. 7 . The value of R x  is dependent on disk velocity. 
         [0048]    The formulas and sample calculations above reflect that the behavior of liquid flow and liquid velocity in the tank induced by the movement of the disk is relatively complex. However, the formula given above, which involves CPM, stroke length, area enclosed and area open, and the ranges given for the resulting number calculated, are effective if the area occupied by the hub and spokes is simply ignored. That is, open area is simply considered as the full area defined within the inner part R i  of the disk. 
         [0049]    As noted above, the velocity of movement of the disk has an effect on how much liquid in the disk&#39;s path is drawn into and through the opening, i.e. area A x . The factor A f  is defined as A open /A x , and preferably is in the range of  0 . 25  to  0 . 75 , more preferably 0.35 to 0.65. Typically A f  will be about 0.5, plus or minus 10%. 
         [0050]    Linear motion reciprocating mixers with mixing disk meeting the above criteria produce a high velocity through the central opening and mix liquids such as municipal wastewater treatment sludge (particularly in anaerobic digestion) more efficiently than with prior art mixing disks. It should be understood, however, that the invention applies to mixing disks for other liquids and slurries as well, in a wide range of viscosities and densities. 
         [0051]    The above described preferred embodiments are intended to illustrate the principles of the invention, but not to limit its scope. Other embodiments and variations to these preferred embodiments will be apparent to those skilled in the art and may be made without departing from the spirit and scope of the invention.