M E M O R A N D U M DATE: 1 September, 1999 TO: Jm Russell FROM: Peter Tkack RE: Aalyss of wde ply tube wdg as compared to Kova Kore CC: Larry McMlla The goal of ths report s to aalyze the spral tube wdg process ad partcular, how a 14 wde ply tube compares to a Kova Kore. I geeral, two thgs stad out: 1. The spral wdg speed of 173 bfm produces tubes at a rate of 13 tpm (a hgh Kova Kore speed).. The beam stregth s reduced to 88% of Kova Kore levels (14% extra s eeded to match KK levels). Fgure 1. Example wde ply spral wder from 1965 patet
Appedx Calculatos ad assumptos: To beg the aalyss, a descrpto of the Kova Kore ad wde ply tube beg maufactured s made. Example case: ID KK Ø4.00 (madrel sze) W wde 14 s the wdth of the bottom ply of paper the tube. L KK 150 s the tube legth. Speed KK the speed of good rug Kova Kore s 1 13 tpm (we wll use 13). Nomeclature: ID OD t w d umber of ples where the ply umber goes from =1,, 3,...,. the sde dameter of the tube or the madrel dameter. the outsde dameter of the tube. the thckess or calper of the th ply of paper the tube. the wdth of the th ply of paper the tube. the dameter of the th ply of paper the tube. ptch the lear dstace a tube travels wth oe revoluto. α I the agle of the th ply of paper to the tube wth 0 beg covolute. Speed the speed of the outsde surface of the belt ply. A vsual descrpto ca be foud fgure. as follows.
Ptch thckess thckess 1 Ply = 1 Ply = α 1 α Web wdth 1 Fgure 1. Tube makg omeclature Ply = Web wdth
Geometrc Aalyss (Geeral Case): Sce at the wder belt, all of the ples are movg uso, they all share the ptch of the outer ply. ptch = ptch1 = ptch = ptch3 =... ptch eq. 1 It should also be oted that the OD s the madrel dameter plus the dametrcal thckess of the ples. [ ] OD = ID + t 1 + t + t 3 +... + t. eq. whch may be rewrtte as... OD = ID + t = 1.. eq. 3 For ay termedate ply, the dameter s... dameter = ID + t k= 1.. k eq. 4 Now we should look at the belt ply or outer ply to calculate the ptch of the tube. The subscrpt for the outer ply s so the dameter wll be d ad the agle wll be α, etc. The ply agle for the outer ply (α ) s ted to the OD (d ) ad the belt ply wdth (wdth ) by the geometry of the ply. πd α wdth Assumg a butt jot o the outer ply, α s calculated from the wdth ad dameter by lookg at a rght tragle. πd wdth = πd s (α ) eq. 5 α wdth
or solvg for α. α wdth = s 1 eq. 6 πd The ptch ca be the calculated from α, the wdth ad the theory of smlar tragles. wdth = ptch cos (α ) The we ca solve for ptch. ptch = wdth / cos(α ) eq. 7 wdth Now that we have calculated the ptch ad the agle for the belt ply, we ca calculate the uder ply wdths ad agles,.e. wdth 1 to wdth -1 ad α 1 to α -1. Usg a smlar tragle from the same method as above we ca calculate α. α ptch ( ) ta α = ptch πd α 1 πd Ptch the we ca solve for α. 1 ptch α = ta. πd eq. 8 We ca rewrte equato 5 wth the varable stead of the belt ply umber to let us solve for the termedate ply wdth. wdth = πd s (α ) eq. 9 Now that we have a relatoshp for the ply wdths, agles, thckess, ad dameters, we may calculate the varous ply speeds by lookg at the legth of ply wrap for each ply,.e. the plylegth. Sce we kow the ptch ad the crcumferece for each ply, the plylegth ca be calculated usg Pythagorea Theorem.
That s... {(πd ) plylegth = (( πd ) + ptch ) + ptch } ptch. eq. 5 ptch Sce the wder speed s refereced to the belt ply or the wder belt... πd Speed = Speed eq. 6 The speed of the lower ples s ted to the belt ply by the dametrcal posto. speed = speed (( πd ) + ptch ) ( πd ) + ptch ( ) Tube Costructo Rules:. eq. 7 1. The geeral rule Rock Hll s that we buld up a core wall wth 5 pot paper for heavy walled cores of less tha e-ch dameter. Usg much thcker paper ca result formato ad crackg problems ad gog below 5 pot creases the umber of ples whch must be mataed ad creases the chace for breakout.. The belt ply wdth s the startg pot ad for ths study s 14 wde. 3. The ples below that drop as ecessary 1 / 8 cremets. 4. If the O.D. of the tube does ot come up exactly to the specfcato, the the thrd or fourth from the belt ply may be exchaged for 30 pot ples or a ply may be removed. These ples are usually mouted a posto, whch allows for quck replacemet f the O.D. vares durg a ru.
Example Calculato: (0.375 wall wde ply carpet core) Costruct a core whch has a I.D. of Ø4.000 wth a 0.375 wall, (resultg a theoretcal Ø4.750 O.D.) wth a 14 top ply. Top Ply To start the calculato, we wll be workg wth the belt ply. As per the costructo assumptos, the belt ply s 14 wde ad s at a wall of 0.375. Sce the top ply s at a dameter of 4.750, ad sce... α wdth = s 1. Eq. 8 πd The ply agle s the arcs of (14 / π 4.750 ) = 69.74. Body Ples The body ples are calculated the same way that the Belt Ply s calculated. The ply thckesses are frst developed (chagg 5 pot for 30 pot at approprate places order to reach the correct OD). Equato 8 s used to determe each ew agle ad equato 6 to determe the wdth for that agle. The theoretcal ply gap s the dfferece betwee the zero ply gap ply wdths ad the actual ply wdths (whch step cremets of 1 / 8 ). Ply No. Ply gap Theoretcal Ply Ply Actual Ply ply wdth dameter agle Ply wdth calper 15 0.000 14.000 4.75 69.75 14.00 0.05 14-0.130 13.870 4.70 69.94 14.00 0.05 13-0.60 13.740 4.65 70.14 14.00 0.05 1-0.391 13.609 4.60 70.34 14.00 0.05 11-0.53 13.477 4.55 70.53 14.00 0.05 10-0.530 13.345 4.50 70.73 13.88 0.05 9-0.66 13.13 4.45 70.93 13.88 0.05 8-0.795 13.080 4.40 71.13 13.88 0.05 7-0.98 1.947 4.35 71.33 13.88 0.05 6-1.06 1.813 4.30 71.53 13.88 0.05 5-1.196 1.679 4.5 71.73 13.88 0.05 4-1.331 1.544 4.0 71.93 13.88 0.05 3-1.341 1.409 4.15 7.13 13.75 0.05-1.477 1.73 4.10 7.33 13.75 0.05 1-1.613 1.137 4.05 7.54 13.75 0.05
Speed Aalyss: Sce at the wder belt, all of the ples are movg uso, they all share the ptch of the outer ply. ptch = ptch1 = ptch = ptch3 =... ptch eq. 1 ptch = wdth cos ( α ) = 14 cos(69.74 ) = 40.43" eq. 9 ( πd ) + ptch ). = ( 4.75) + 40.43 = 43. 10 plylegth = π eq. 5 Sce the wder speed s refereced to the belt ply or the wder belt... Speed = Speed eq. 10 The speed of the lower ples s ted to the belt ply by the dametrcal posto. For the Kova Kore, the speed terrupted ad averages to produce 13 tpm at 150 each. To match ths wth a spral mache, the tube would have to have a lear speed of 150 /tube 13tubes/m 1 /ft = 16.5 feet/m eq. 11 To match ths o a spral wder at a spral-wdg agle of 69.75, you would have to ru at a belt speed of Belt speed = Lear speed plylegth / ptch eq. 1 Belt speed = 16.5 43.10 / 40.43 = 173. bfm. A wde ply spral wder would eed to ru 173 bfm order to match a 13 tpm Kova Kore mache.
Beam Stregth Aalyss: The beam stregth of a spral tube s reduced due to the agle of the fbers away from the logtudal drecto. For cylder board, the MD to CD stregth rato s assumed to be Stregth Stregth MD CD cylder 65 = 35 eq. 13 If the ples are oreted a dagoal patter 69.74 from logtudal, the the effectve stregth ca be solved for usg the equato of a ellpse. x a + b y = 1 equato of a ellpse eq. 14 x y x y + = 1 = + eq. 15 a b 65 35 35 The rato of x to y s y? 69.7 65 x = ta( 69. 74 ) x eq. 16 y y = x = 0. 369x eq. 17 ta(69.74 ) Smultaeously solvg equatos 15 ad 17 for x results ( / ta(69.74 )) x + x = 1 eq. 18 65 35 x (0.369x) = + = x 65 35 1 65 1 0.136 + =.000348 35 eq. 19
1 x = = 874 0.000348 eq. 0 x = 53.61 eq. 1 Solvg for y gves x y = = 19.79 eq. ta ( 69.74 ) The stregth the 69.74 drecto s the square root of the sum of the squares x + y = 19.79 + 53.61 = 57.15 eq. 3 The reducto from the covolute wrapped core s the rato of the two WdePlyStregth CovoluteStregth = 57.15 65 = 0.879 = 87.9% 88% eq. 4 Therefore, to match the beam stregth of the Kova Kore tube, the wde ply tube would have to add materal. 1 = 1.138 eq. 5 87.9% A crease of 1.138 1 = 0.138 = 13.8% 14% eq. 6 14% more materal stregth s eeded a wde ply tube to match the beam stregth of a Kova Kore tube.