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Technical Consultancy in Textiles and NonwovensArticles of Dr. N.BalasubramanianYarn diameter Specific volume and Packing densityDevelopments in DraftingEnd breakage distributionTesting of man made fibres yarns and filamentsBackward and Forward Feed in combing and noil

Yarn diameter specific volume and packing density

N.Balasubramanian

Retd. Jt. Director (BTRA) and Consultant

Abstract
Relationship between yarn diameter and count by equations given by different authors is compared. A new equation is proposed taking into account the variation in specific volume with count. Methods for determining yarn diameter and packing density are critically reviewed and their relative merits discussed. Influence of fibre properties and process parameters on yarn diameter and specific volume are discussed. Packing density is found to be higher at core and reduces towards the surface with yarns produced by all spinning systems. While rotor yarns have been found to have higher diameter and lower packing density, contradictory results are found in regard to air-jet yarns vis a vis ring yarns. Packing density is found to be lower at core and increases, reaches a maximum and drops rapidly towards surface in rotor, compact, siro- spun and worsted ring yarns. Packing density in ring yarns is found to be lower than that given by �open� and �close� packing
Introduction
For successful quality monitoring, it is necessary to know the important Quality parameters of yarn and the factors affecting them and their interrelationships. These aspects are briefly covered in this article
Yarn diameter and Count
Yarn diameter is an important quality of yarn. Diameter and its variation affect appearance and sale value. Diameter also affects the cover of woven fabric and stitch length and tightness of knitted fabric. Weft insertion rate of air jet weaving machines are affected by diameter. Relationship between yarn count and diameter is dependent upon specific volume of yarn. Specific volume in turn determines the absorbency and dyeing behavior of yarn. Specific volume, �v� is the ratio of the volume of yarn to that of the same weight of water. Specific volume of yarn depends upon the raw material, type of spinning system, twist factor and spinning parameters. If
d is diameter (apparent diameter assuming yarn to have a circular cross-section) in inch of ring yarn
d1 diameter in cm
r1 is radius of yarn in cm
C English count (Ne)
T Tex count
v specific volume in cm3/gm
ρ density of yarn in gm/cm3
then v = 1/ ρ = (π r2 105) / T
= 314159.3 r12 / T
= 78539.8 d12 / T ......1
v = 858 C d2 ....2
(1/d) = 29.3√(C/v) .....3
Assuming specific volume as 1.1. Pierce1 gave the following equation for yarn diameter. (1/d) = 28√C .....4
(1/d) = 28 √C (1/d1) = 280.24 / √(T v) .....5
El. Mogahzy2,3 has given the following empirical equation relating count and yarn diameter for ring yarns.
d1 = -0.10284 + 1.592/√C .....6
Relationship between yarn diameter and count for ring yarns by Mogahzy equation and by Pierce�s equation are given in Fig 1. Not only diameter is found to be higher but also the rate of reduction in diameter with count is steeper with Mogahzy than Pierce. This is because specific volume increases in Mogahzy equation from 1.18 to 2.13 as count reduces from40s to 10s while Pierce has assumed specific volume to be the same irrespective of count. As finer fibres are used in finer count, specific volume (inverse of density) is likely to increase with reduction in count but the order of increase from 1.18 to 2.13 found with Mogahzy equation appears to be high. Specific volume seldom exceeds 2. From actual experimental studies, where �v� is found to vary from 1.5 to 1.75, following equation is given connecting v with count.
v = 1.75 - 0.0036 (C - 10) ......7
By inserting this value in equation 2, relation between diameter and count given by d = (√1.786-0.0036C)/(29.3√C ) .......8
Relation between diameter and count with this equation is also shown in Fig 1.
The modified equation helps to raise the diameter to a higher level with a higher rate of reduction with count than with Pierce equation.
yarn diameter and count
Fig 1 Yarn diameter vs Count
Equations 3 and 4 can also be used to determine specific volume �v� of yarn from measurements of diameter and count.
Specific volume from twist
If twist factor is k and t is twists per inch then,
k = t/√C.
If fibres follow helical path and θ is angle surface fibre makes with axis and r is radius of yarn in inch then as shown in Fig 2
tan θ =2Πrt= 2Πrk√C
=Π(√v)k/29.3
=0.107k(√v)
v = tan2θ /(.0114 k2) .......9
v = (tan2θ ×7957.7)/ r12 ×T ......10
Yarn specific volume can be obtained from helix angle of surface of fibre and twist factor in case of ideal yarn where fibres follow helical path from any of the above equations 9 or 10.
Helical path
Fig 2: Yarn with fibres following helical path
Diameter
As yarn is not perfectly circular, diameter measurements have to be made in two perpendicular planes or, even better, at a number of places around circumference. Narkhedkar and Kane4 have reviewed some methods for measuring yarn diameter. The simplest method is to allow a parallel beam of light to fall on yarn, and image is then projected on a screen or monitor and diameter measured by a graduated scale. Alternately, yarn image is viewed under a microscope fitted with CCD camera, transferred to monitor of PC and image analysis is used. From pixel concentration, yarn body outline is determined. Presence of wild fibres and hairiness makes the yarn boundary hazy and adds to errors in measurement. Van Issum and Chamberlin5 developed a method for determining free diameter of yarn by photographic integration. Electronic equipments are also available for measuring yarn diameter and its variability. OM module incorporating optical sensor of Uster Evenness tester 5 �S800 measures yarn diameter by projecting two parallel light beams perpendicular to each other on yarn. In addition diameter variation, roundness, density and surface structure of yarn are measured. QQM is a hand held portable device, by Investa UNI, with two optical sensors of 2mm width through which yarn is passed. Good correlation is found between yarn diameter measured by OM module of Uster, and QQM and laboratory measurement based on image analysis6. Thread flattening occurs in fabric due to interlacement of yarns in weaving and yarn diameter measurements by microscopic methods may not give accurate estimates of yarn diameter. Hamilton7 therefore developed an instrument (shown in Fig 3) where yarn is found under tension on to a spindle in the form of coils. Yarn gets flattened and assumes an elliptical shape. Major ellipse axis Y, is obtained from number of coils per unit length and minor axis X is found with the help of a feeler placed on wound coils. The flattening of yarn obtained by this instrument may not conform to what is obtained by interlacement of threads in weaving as the conditions are much different.
Yarn diameter measurement
Fig 3 : Diameter of yarn by coil winding (Hamilton)
Hamilton further found that with increase in twist, major diameter increases, minor diameter reduces and yarn density increases first and then falls. While increase of yarn density with increase of twist is expected, the fall of the same beyond a point is not on expected lines. This may be the outcome of the instrumental method for determining yarn diameter. Barella8 found considerable differences between diameter measurements made by his instrument and Van Issum with a more closer agreement with Hamilton�s method. Because of hairiness, photographic methods tend to give higher results for yarn diameter and specific volume. Tsai and Chu9 found that cross-section of both ring and rotor yarns to be elliptical with irregular outline with ellipticities of1.09 and 1.07 respectively. Onions10 et al; measured thickness and compressibility of worsted yarns by applying load ranging from 1 to 100 gm.
Influence of fibre properties
Packing density and fibre density are the main factors determining yarn diameter. Yarn diameter will be higher with fibres of lower density because of higher fibre diameter. The Table below gives density and diameter of multifilament yarns from Polypropylene, HDPE, and Polyester11.
Table 1 Influence of fibre density on yarn diameter
Fibre Type Density of fibre g/cm3 Diameter of 75 den multifilament yarn, mm Diameter of 150 den multifilament yarn, mm
Polypropylene0.910.1080.151
HDPE0.950.1060.149
Polyester1.380.0880.124
Finer fibres compact better than coarser fibres and result in lower yarn diameter. Further, coarser fibres offer greater resistance to bending during twisting leading to higher diameter. Longer fibres result in lower diameter because of reduced hairiness and better compaction. However, contradictory results are reported for jute/polyester blends by Debnath and Sengupta12 .Diameter of jute/polyester blend is higher than 100 % jute yarn though jute is coarser and has higher density than polyester. Hamilton13 found that yarn bulk density increases from 0.82 in Indian cotton to 0.98 gms/cm3 in Sea Island. Higher diameter will therefore be found with short staples. Fibres with lower density like propylene will result in bulkier yarn with higher diameter as shown in Table 1. Fibre crimp is a major factor affecting thickness and compressibility10. Fibres with higher crimp will result in larger yarn diameter. Marked increase in thickness and compressibility is found with acrylic yarns after bulking.
. Packing Density and Specific volume
Packing density and specific volume are important parameters as they determine the bulkiness, feel, warmth and dyeing characteristics of yarn. They also influence the amount of yarn that can be put on a package. A number of methods have been proposed for determining packing density. International Standard IN 22-103-01-01 (Technical University of Liberec, Czechoslovakia) describes two methods for determining yarn packing density. The two methods are also discussed by Kremenakova14.
Direct Method
Yarn cross-section is prepared by embedding yarn and taking thin cross-sections without disturbing the fibre position. Yarn is coated with a substance like polyvinyl acetate to prevent disturbance of fibres during cross-sectioning. After drying,yarn is embedded in a medium, dried and thin sections of 25 �m are prepared by a microtome. Some of the embedding substances used include Perspex in chloroform, nitrocellulose in ether/ethanol (later covered with wax), molten wax and paraffin. The cross-section is viewed under a microscope. Template is placed on cross-section with circular rings to indicate yarn periphery. Total fibre area is mapped by using PC software like LUCIA Metlab after separation of individual images. Yarn area is obtained from diameter of ring.
Packing Density = (Fibre area)/(Yarn area)
This method can also be used for determining radial packing density by placing a template with concentric rings separated by a known width as shown in Fig 4. In the figure cross-section has been divided into 5 zones For more detailed investigation of radial density, yarn cross-section can be divided into more zones up to 20. In this case, some fibres will be lying on the border contributing to both zones.
Yarn cross-section
Fig 4 Yarn cross-section divided in number of radial zones.
Fibre cross-section in each radial zone is mapped and divided by yarn area in the zone to get radial packing density of that zone. About 50 cross-sections are examined to get a reliable package density.
Secant Method
For fibres with irregular cross-sections like viscose, 3 lobe polyester,secant method is used. Centre of gravity of fibre is determined and fibre cross-section are reconstructed around it after determining equivalent fibre diameter .Equivalent Fibre diameter is estimated from the linear density and mass density of fibre assuming the cross-section to be circular and fibre cross-sectional area estimated from the following equations
df = 1/5 √(( t(f))/(πρ(f))) and Sf = (π d(f)2)/4
Where
d(f) = Fibre diameter in mm
T(f) = Fibre linear density in Decitex
ρ(f)= Fibre mass density in g/cm3
Sf = Fibre cross-sectional area
As the fibres follow helical path, fibre cross-sectional area is corrected as per yarn twist and distance of annular ring from centre of yarn by multiplying by secant of helix angle.
Packing density in the whole yarn, as well as radial packing density, is estimated from the ratio of fibre area to yarn area. Packing density of ring yarn is found to vary from .4 to .6.
Radial Packing density
Fig 5 Radial packing density of Ring and Rotor yarn
Fig 5 shows typical radial packing density of Ring yarns for 20 tex yarn. Packing density is high at the core and reduces rapidly after a radius of 0.08mm. This method is however not recommended for fibres like cotton and wool which have a high variation in linear density from fibre to fibre. Kremenakova 14 postulates that the border area between yarn core and surface lies at packing density of 0.15 and determines yarn diameter at this point from radial packing density curve.
Packing density from tracer fibre technique
.
Tracer fibre image

Morton15 determined zonal distribution of fibres in yarn by tracer fibre technique. A small quantity of (0.1%) of coloured fibre is introduced in the mixing. Yarn is optically dissolved in a liquid of same refractive index as fibre so that path of coloured fibre can be traced under microscope. Image of yarn in 2 perpendicular directions by use of a mirror placed at 450 will result in accurate mapping of tracer fibre profile. A typical trace of tracer fibre is given in Fig 6. Number of turns per unit length made by a fibre can be ascertained and the length contributed by fibre to different radial zones of yarn can be estimated. Image acquisition software can be used to get an accurate mapping of tracer fibre. To get % fibre contributing to each zone, length is corrected for helix angle in the zone by multiplying by secant tan-1(2πr/z), where r is zone radius(.1r, .2r �) and z is turns per unit length. By examining about 100 tracer fibres, mean zonal distribution can be found. If density of fibre packing is uniform throughout yarn section, % of fibre in each zone will vary as per volume of the zone. Actual zonal distribution was compared against theoretical (assuming uniform packing density) by Morton. A typical figure is shown in Fig 7.Zonal contribution is much lower than theoretical in the zone close to surface indicating that packing density reduces steeply near surface.
Zonal Distribution
Fig 7 : Comparison of actual and theoretical zonal distribution of fibre
Ishtiaque16 has developed the following formula for estimating packing density from tracer fibre examination.
Packing Density = (2 π n F Z2)/(((√(1+(π D Z)2) ))- 1)
Where
n = number of fibres in cross-section
F= Fibre cross-sectional area, mm2
Z = Average number turns of helix in tracer fibre (turns/mm)
D = yarn diameter in mm
Dogu17 gives the following equation for radial packing density of with filaments migration
vr =(n dx/dr)/(πr cosθ)
Where
vr = packing density at radius r
n = number of starting points at axis
Θ = twist angle of fibre at radius r.
dr/dx = Rate of migration
Hickie and Chaikin18 determined radial packing density by taking a large number of microscopic photographs of yarn cross-section. They found a low packing density at core and packing density increases and reaches a maximum at about one quarter of radius from core in worsted yarn and reduces thereafter. Neckar19 used Omest tester to determine radial packing density of cotton yarn by mathematical evaluation of tracer fibre photographs. The instrument has a measuring cell and a specialized camera. Two perpendicular images of tracer fibre are recorded in the instrument side by side by means of a prismatic system.. The radial density results for 25.2 tex and 56.6 tex obtained by this method are given in Fig 8.
Radial packing density from tracer fibre
Fig 8 : Radial packing density from trace fibre analysis
Fig 8 shows that packing density initially increases from core reaches a maximum and then comes down rapidly towards surface as found by Hickie and Chaikin. Maximum packing density lies at a higher radius in finer yarn than in coarser yarns. Theoretical model for radial packing density was proposed and results obtained from the same compared with actual measurements.
Packing density from diameter and fibre density
Packing density = Yarn volume/Fibre volume
If
T= Yarn count in tex
C= yarn count Ne
d= Diameter of yarn in cm then
yarn volume in 1 cm length = (Π d2)/4
Fibre volume in cm length =(T/(105ρ) Packing density = (1.27 *10-5*T)/(d2*ρ)
=(749 *10-5)/(d2*ρ*C) Only problem withis method is measurement of diameter of yarn accurately.
Out of the two techniques, packing density from cross-section is likely to give more accurate results as no assumptions are involved and 100% fibres have been covered. Further radial packing density is low at core and increases, reaches a maximum and drops towards surface with tracer fibre technique even in ring yarns. This trend is not found in ring yarns with cross-section method where packing density reduces continuously from core to surface . Only drawback of cross-section method is that it is more tedious and time consuming.
Displacement technique
Carnaby20 developed an instrument for measuring yarn specific volume by fluid displacement technique. A length of yarn is made to hang in a glass tube which is then filled with small glass beads. From the weight of beads for filling the tube with and without yarn, specific volume of yarn is estimated.
Effect of twist
Jaoudi21 et al; found that yarn density increases with twist and asymptotically approaches a value. Twist and count have the maximum influence on packing coefficient. Hearle and Merchant22 also found that with increase in twist, specific volume of yarn (inverse of density) reduces and asymptotically approaches a value (1.25 cm3/gm) which is still higher than fibre specific volume (0.88 cm3/gm), in nylon staple fibre yarns. This is because entrapped air is not fully expelled with increase in twist. Initial fall in specific volume with twist is greater for finer yarns than in coarser yarns. Kremenakova et al;23 found that with polypropylene, packing density increases slowly with twist and reaches a limiting value of 0.7 to 0.8. With cotton increase of packing density with twist is more rapid. Barella24 found that yarn density approaches fibre density at the time of break during loading in tensile load tester.
Effect of fibre properties
As discussed earlier yarns made from finer fibres and higher density have higher packing density. Krenakova et al;25 confirmed that packing density is higher and diameter lower with finer fibres. Kremenakova et al26; found higher packing density with yarns made from polyester fibre with scallop oval shape and high shape variability. Packing density is least with polyester fibre of hex channel shape.
Process factors
Yarn packing density increases with increase in spinning tension as the fibres get closely packed. Lower strand width at the delivery by the use of finer roving and lower ring frame draft will increase packing density because twist flows closer to nip. However Tyagi et al;27 found that packing density increases with spinning draft initially and afterwards drops.
Compact Yarns
Strand width at delivery nip of front rollers determines the extent to which twist flows to the nip and the size of spinning triangle. If the strand width is reduced spinning triangle diminishes, twist flows right up to nip and increases compactness of yarn.. Compact spinning is developed on this principle. Basal and Oxenham28 and Tyagi et al;27 found the diameter to be lower and density higher of compact yarn compared to ring yarn. Yilmaz29 et al; found that packing density is high near the core and reaches a maximum at one fifth radius and thereafter reduces parabolically towards surface in compact yarns. About 15 � 30 % higher packing density is found in compact yarns (.55 to .70) than ring yarns (.5 to .6). As in the case of ring yarns, packing density increases with twist in compact yarns. Yarns from 3 different pneumatic compact spinning had no significant difference in packing density but fibre number is found to vary between the systems. Diameter of both carded and combed mechanical compact yarns (from RoCos) is lower than ring spun conventional carded and combed yarn30. Difference in diameter of compact and ring yarn increases as count becomes coarser and decreases with increase in twist. Reduction in diameter is more with pneumatic compacting systems than mechanical compacting systems. The former brings down the strand width to a higher extent. Indigenous mechanical compact system, MCS �Positive is also effective in reducing diameter of yarns compared to ring spinning31.Semi positive nip is however not that effective in compacting and reduces diameter only slightly. Wu32 et al;, who reconstructed yarn structure from the path of tracer fibres, found that packing density is more uniform in compact yarn than in ring yarns up to 70% yarn radius. Further, packing density is lower in compact yarn than ring yarns in this core region. Beyond 70 % radius , compact yarn has a packing density higher than ring yarns. While packing density of ring yarn starts reducing from 35 % of radius, that of compact yarn starts reducing only after 70 % of radius. Further fibres in compact yarn have a larger twist angle than ring yarns. These results are at variance with that of Yilmaz et al;29. This suggests that reconstruction of yarn from tracer fibres may give erroneous results.
Rotor yarns
Rotor spinning is primarily intended for coarse and medium counts to achieve high production rates and improved yarn evenness. EL Magahzy2,3 has given the following empirical equation connecting yarn diameter and count in rotor yarns.
d1 = - 0.16155 + (1.951/√C) .....11
Relationship between count and diameter for rotor yarns by EL Magahzy are also shown plotted in Fig 1. Rotor yarn will be found to have higher diameter than ring yarns in the coarse count range and the difference decreases as count becomes finer. Basu33 et al (SITRA) found that rotor yarns are 10 % bulkier than ring yarns and found the following equation gives a good fit with actual measurements.
(1/d) = 23.5 √C ....12
(1/ d1) = 252/√T .....13
In Fig 9 variation of yarn diameter with count by equations given by Mogahzy and SITRA are compared. Mogahzy�s equation gives higher diameter than SITRA in coarser counts and the difference diminishes with increase in count. Higher diameter with rotor yarns is because of higher specific volume arising from the fact that twisting takes place at much lower tension than in ring spinning.
Count vs diameter in rotor
Fig 9 : Variation of diameter with yarn count in rotor yarn
Radial packing density of rotor and ring yarns is compared in Fig 6. Rotor yarns have a significantly lower density near the core region, increases to a maximum and then decreases rapidly towards surface. Beyond 0.1mm the two yarns have nearly same packing density. Jiang34 et al found packing density in rotor yarns to be low at core and increases continuously and reaches a maximum at one third or one fourth of radius from yarn axis. Similar results are also reported by Neckar345 et al;. Afterwards packing density reduces steeply towards the surface. With increase of twist, fibre scatter is reduced and packing density increases and maximum packing density shifts towards core. Packing density in rotor yarns also reduces as yarn becomes finer. Neckar35 et al; found that for the same count, rotor yarns have about 15 � 19 % lower number fibres than ring yarns primarily because of poor alignment of fibres caused by radial and twist migration. Radial packing density of both rotor and ring yarns are non uniform. Packing of yarns is more towards core in rotor yarn than ring yarns. Higher rotor speed and diameter result in more compact yarn with lower diameter because of higher tension36. Increase of card draft and corresponding reduction in rotor draft reduces yarn diameter37
Air-Jet Yarn
Air-jet technology is suitable for medium counts with polyester and polyester/cotton blends with potential for high production rates and unique yarn properties. EL Mogahzy has given the following empirical equation connecting yarn diameter and count in air-jet yarns.
d1 = - 0.09298 + 1.5872/√C .....14
Diameters of air-jet and ring yarns, as per Mogahzy equation are compared in Fig 10. Air-jet yarns will be found to have slightly higher diameter than ring yarn.
Air-jet and Ring yarns
Fig 10 Comparison of diameter of air-jet and ring yarns by Mogahzy equation

However, Tyagi et al;38 and Kumar37 et al; found air jet yarn to have a lower diameter and higher packing density than ring yarns while rotor yarn has the maximum diameter and lowest packing density. Helix angle is found to be lowest in air-jet yarns and highest in rotor yarn. Increase in card draft and corresponding reduction in air jet draft increases yarn diameter and lowers packing density. This is confirmed by Tyagi et al;38 This may be because of reduced strand width which brings down the number of wrapper fibres. Increase in second nozzle pressure improves packing density38.
Air vortex Spinning
Air vortex spinning is a modification of air jet spinning designed for 100 % cotton and cotton rich blends. Basal and Oxenham39 found that yarn diameter reduces with delivery speed in air vortex spinning because of longer time for which fibre bundle is exposed to whirling air force. Kilichand and Okur40 found diameter to be lower and density higher in compact yarn than in ring yarns while that of vortex yarn falls in between. While regenerated fibres have lower yarn diameter than cotton in ring and compact, no such difference is found between the two in vortex yarns. Roundness of compact yarn is better than ring yarns because of lower strand width at delivery. Vortex yarns have a much lower roundness than ring and compact yarns. Failure mechanism in a tensile test in different types of yarns was investigated by Cybulska et al;41. Yarn diameter is found to be lowest and diameter unevenness highest in yarns made by all spinning systems at the time of break
Friction spun yarn
Friction spinning is suitable for coarse counts with high production rates. Ishtiaque et al;42 found yarn diameter reduces and packing density increases with increase in suction pressure in core-sheath friction spun as well as rotor yarns. Core-sheath friction spun yarns have lower diameter and helix angle than rotor yarns. In another study Ishtiaque43found that packing density increases with friction drum speed, decrease in delivery rate and increase in throttle diameter up to 52mm. Packing density of core sheath spun friction yarn increases as yarn becomes coarser and with increase in friction ratio44. As in rotor yarn, packing density is higher in core and reduces towards surface. Fibre to fibre friction has a significant influence on packing density of sore-sheath friction spun yarn45. With increase in fibre friction, packing density increases.
Siro spun yarn
Siro spiniing is nothing but double rove spinning common in olden times, except that that the space between rovings is adjustable to get optimum yarn quality. Ishtiaque46 et al found that unlike ring yarns packing density reaches a maximum at one third of yarn radius from the core in siro spun yarns. Packing density of Siro spun yarn is higher than ring yarn. Johari47 found increase in packing density and reduction in diameter with increase in spacing between rovings.
Plied Yarn
Plied yarn is expected to have a higher packing density than single yarn especially if it is z over z. Ishtiaque etal;48 have confirmed this. Further they found that packing density of single yarn in doubled yarn increases significantly upon doubling possibly because untwisting and retwisting reduces air pockets. However measurement of packing density in plied yarn is difficult as yarn does not have a circular cross-section. Moreover shape of cross-section varies from place to place.
Splicing
In modern winding machines splicing is invariably done in place of knot for joining broken ends. Splicing technology has improved over the years to minimize diameter and strength difference of a splice and normal yarn. Das et al49 showed that yarn twist and splice air pressure have a significant influence on splice diameter, % increase in diameter and retained packing density. Fibre friction has no significant in influence on these properties.
Packing of fibres in yarn
Schwarz50 postulated two types of packing of fibres in yarn viz; circular or open packing and polygonal or close packing, shown in Fig 11 and 12.
open packing
Fig 11 : Open Packing

In open packing, fibres follow on concentric circles over a central fibre in a number of layers. The first layer consists of a single fibre and second layer has 6 fibres all of them touching each other as well as central fibre. The third layer is formed by fibres touching the circle that contains the second layer of fibres. Build up of layers one over other proceeds in this manner. Open packing gives a circular yarn with air spaces between layers from 2nd layer onwards. As shown by Hearle et al; 51 radius of yarn with n layers is given by (2n � 1)rf where rf is fibre radius. Packing density of open packing is around 0.76 as shown in Table 2.
Table 2 : Packing density of open packed yarns with different layers
Number of layersNumber of fibresPacking density
3190.76
4370.755
5620.765
6930.768
Petrulis52 considered two modifications of open packing with different amount of gaps between filaments in some ring layers. With wider gap between filament packing density is slightly lower. However actual packing density of yarn is much lower than that given by open packing. This means that actual yarns have more yarn pockets than open packing.
Close packing
Fig 12 : Close Packing

Close packing gives a hexagonal outline with all fibres touching each other. Yarn is not circular and closer to elliptical. For yarn with n layers, major radius of yarn is (2n � 1) rf and minor radius is (1.732n � 0.732 )rf. Close packing, as its name indicates, has a higher packing density. Table 3 gives packing density with close packing for different layers. With increase in number of layers above 5 packing density becomes stable and reaches a value of around 0.87
Table 3 Packing density with close packing
No of LayersTotal number of FibresPacking Density 319.836
437.852
561.84
691.87
7127.874
8169.873
9217.879
Potrulis and Petrulyte53 developed a hypothetical model of close packing for estimating packing density. This allows calculation of a vector for each filament and cross-sectional layer in which filament lies. Yarn is assumed to have a circular cross-section with fibres close packed Packing density varies from.75 for 2 layers to 0.898 for 12 layers. Another method, considering yarn with close packing with comb type cells and infinite number of layers was also examined. In this case packing density works out to 0.906. Balakrihna Iyer and Phatarford54 found yarn diameter to be nearly constant with small number of fibres in the twist factor range 20 � 40. Packing density was found to be 0.907. However, actual measurements show packing density varies from .45 to .6 in cotton yarns. This means that fibres are packed more loosely than either close or open packing.
References
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