deriving an empirical formula to determine the optimum level of false-twist in mechanically-crimped...

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published online 23 September 2011Textile Research JournalTasnim N. Shaikh and Someswar S. Bhattacharya

Textured Polyester YarnDeriving an Empirical Formula to Determine the Optimum Level of False-Twist in Mechanically-Crimped

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Original article

Deriving an empirical formula todetermine the optimum level offalse-twist in mechanically-crimpedtextured polyester yarn

Tasnim N Shaikh and Someswar S Bhattacharya

Abstract

Flat polyester yarn is bulked by the mechanical crimp texturizing process. The magnitude of the false-twist, introduced

together with the torsional rigidity of the supply yarn, has a significant effect on the transformation process. The

torsional rigidity of the supply yarn is in turn influenced by its fineness and bending characteristics. It is therefore

deemed necessary to establish a mathematical relationship between the optimum false-twist level and the yarn fineness

for materials with known bending characteristics. Such a relationship may help in producing a yarn with optimum bulk

properties, while minimizing resource utilization and waste generation. This paper reports research work done in this

direction.

Polyester yarn is chosen for this investigation as it has a wide range of applications. Fully drawn multifilament polyester

yarns, with different linear densities, are textured on a single-head mechanical crimp texturing machine. The results are

analysed with the help of polynomial curve fitting (Polyfit) using MATLAB. This mathematical tool facilitated the deri-

vation of the desired formula (polynomial equation) for the calculation of optimum false-twist level for fully drawn

multifilament polyester yarns undergoing mechanical crimp texturizing.

Keywords

Bulk, mechanical crimp texturizing, optimum false-twist, yarn fineness, Polyfit

Introduction

The mechanical crimp texturizing concept deals withthe twisting of the flat filament yarn after attaining acrimpy configuration by false-twisting. The productyarn obtained has crimpy- constituent filaments boundtogether by a real twist. This has brought newly engi-neered yarn toward a closer resemblance with the prefer-able spun yarn structure. The real twist (pre-twist) is usedfor preventing mobility of the crimpy structure againstthe stresses imposed in the forthcoming processes.1

Details of the process and the effect of process var-iables on newly designed textured yarn characteristicshave already been given in detail in earlier publica-tions.1,2 Therefore, only the production principle ofmechanical crimp texturizing has been reviewed(Figure 1).

It is quite apparent from the production principle(Figure 1) that twist levels (pre-twist and false-twist)

used during the course of texturizing are the determi-nant factors for the stability and bulk characteristics ofnewly engineered yarn apart from its mechanicalbehavior. Outcomes of earlier experiments1,2 also sub-stantiate this argument. Yarn produced with low pre-twist is soft and executed higher bulk and instability ascompared to that produced with higher pre-twist.1

Thus pre-twist has an inverse relationship to the bulkand instability of mechanical crimp textured yarn.

Textile Engineering Department, The Maharaja Sayajirao University of

Baroda, India.

Corresponding author:

Tasnim N Shaikh, Textile Engineering Department, The Maharaja Sayajirao

University of Baroda, Kalabhavan, Vadodara, Gujarat 390001, India

Email: [email protected]

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On the contrary the degree of crimpiness of the tex-tured yarn is directly influenced by false-twist. Highercrimpiness attained at higher false-twist level does notallow close locking of filaments and thereby resulted inthe notable open (voluminous) and looser (low crystal-line) structure.2 Thus false-twist has a major influenceon the bulk and crystallinity of mechanical crimp tex-tured yarn.

The mechanical crimp texturizing process deals withbending deformation of flat continuous filaments underthe action of torque caused by the high level of false-twisting. The extent of crimpiness gained by the feederyarn is thereby purely dependent on raw-materialtorsional rigidity (equation 1) and bending stiffness(equation 2) characteristics.

Torsional rigidity, R ðmNmÞ ¼ EnT2=� ð1Þ

Where E¼ shape factor (¼1 for circular c.s. filamentand higher for multilob fibers), n¼ shear modulus(¼0.29–0.42N/tex for nylon and ¼0.62N/tex for poly-ester), T¼ linear density in tex, r¼density, 1.14 g/cm3

for nylon and 1.39 g/cm3 for polyester.

Bending stiffness, B ðmNmÞ ¼ ð1=4�Þð�ET2=�Þ ð2Þ

Where �¼ shape factor (¼1 for circular cross-sectionand higher for multilob fibers), E¼ tensile modulus inN/tex (nylon¼ 1.7–3.3N/tex and polyester¼ 4.5N/tex), T¼ linear density in tex, r¼density, 1.14 g/cm3

for nylon and 1.39 g/cm3 for polyester.

It is apparent from these relationships (equation 1and 2) that for identical raw-material (polyester/nylon)characteristics, linear density of yarn (T) plays animportant role in defining its bending as well as tor-sional deformation. Thereby, bulkiness of the productyarns varies mainly with type and fineness of supplyyarn at an applied distortion force (false-twist) duringmechanically crimp texturizing.3 Thus only false-twist(distortion-force) is varied for constant supply yarnparameters to derive experimentally its optimumvalue. The optimum false-twist refers to, ‘the false-twist at which highest possible product yarn bulk isattained without causing any physical/mechanicaldamage to the filament/yarn’. This is the basic themeof the present research work described here.

However, the optimum twist level also depends onthe applied filament yarn tension.3 Parent yarn tensionand bulking-zone length are the major process param-eters inducing filament yarn tension during this newconcept of texturizing and thereby affect the resultantcrimped yarn bulk characteristics. Initially, the impactof these parameters were experimentally investigatedand assessed. Based on these test results, the optimalvalue for each tension controlling parameter was iden-tified. Then texturizing of fully drawn multifilamentpolyester yarn was conducted at the optimum filamentyarn tension. Parent yarn parameters, that is, parentyarn fineness, filament fineness and pre-twist only, arevaried. A selection of parent yarn variables are depen-dent on their rigidity against bending and torsionaldeformation, which can affect the value of optimum

Figure 1. Production principle of mechanical crimp texturizing.

2 Textile Research Journal 0(00)




false-twist. Finally, mathematical modulation of thedesired bulky end product for known type and finenessof supply yarn is done on the basis of outcomes of theseexperiments.

Experimental

Materials and Methodology

Bulking-zone length refers to the centre to centre dis-tance between the twist-trapper wheel and twist-trapperpin of the magnetic pin-twister (false-twister).1

McIntosh4 has found that changing the bulking-zonelength would allow a much wider choice of tensions.So, the product yarn bulk achieved is different eventhough textured under identical conditions. Thus opti-mization of the bulking-zone length is needed beforeidentifying the optimum false-twist for different fine-ness parent yarns for the newly designed mechanicalcrimping apparatus.

Experiments are conducted with 100 den/48 filssupply yarn by varying only the bulking-zone lengthfrom a minimum possible 1 inch, to a maximum possi-ble 4 inch. Bulking-zone length is varied with a regularincrement of 1 inch, during experimentation. The min-imum length chosen is 1 inch, as below this excess ten-sion results in frequent end-break. Whereas goingbeyond 4 inch, results in poor texturizing values forthe product yarn.

Higher input filament yarn tension diminishes bulkand also results in poor mechanical properties of theproduct yarn.3 Therefore, a preferable input tensionvalue needs to be identified in this research. This canhelp in achieving a product with good mechanical qual-ity along with the preservation of the bulk. Input parentyarn tension value is varied with additive tensioner andmeasured with mechanical tensiometer. For this

purpose parent yarns of 50 den/36 fils and 100 den/48fils are textured at the optimum bulking-zone length.The parent yarn input tension is varied between 0.05 gf/den (0.45 gf/tex) to 0.09 gf/den (0.81 gf/tex) (based onparent yarn linear density).

After optimizing the laboratory apparatus set-up forprevailing tension during mechanical crimping, trialsare conducted to identify the optimum false-twistlevel for yarns. Yarn parameters are varied in termsof total fineness, filament fineness and pre-twist. Fullydrawn circular cross-section polyester yarns of 100 den/48 fils, 150 den/72 fils, 200 den/96 fils, 250 den/120 filsand 300 den/144 fils are used to study the effect of yarnfineness on optimum false-twist. Trials are carried outwith 60 den/6 fils, 30 den/14 fils, 75 den/36 fils, and50 den/36 fils fully drawn circular cross-section polyes-ter multifilament yarns to diagnose the impact of fila-ment fineness on optimum false-twist. However,selection of yarn fineness for different filament finenessis restricted by the availability of raw material from themanufacturer. Experiments for optimization of false-twist with variable pre-twist are conducted only withconstant 100 den/48 fils supply yarn to avoid undueoverlapping of raw-material variables effect.

The value of underfeed used is mainly influenced bythe ductility and mechanical properties of the feederyarn. It is selected to give 25–28 per cent residualparent yarn extension.5 All the parent yarns are texturedin each division with conditions of 250m/min deliveryspeed and the bulked yarn winding tension of 0.075 gf/den (0.675 gf/tex) (based on parent yarn linear density).Sen et al.6 have suggested this winding tension for build-ing suitable package for mechanical bulked yarn.

Properties of all the parent yarns used throughoutthe study are given in Table 1. Fully drawn polyestermultifilament yarns with circular cross-section and0.9 per cent spin-finish were used throughout the

Table 1. Properties of the Parent Yarns used for the study

Description

of parent yarn dpf Color

Per cent

Extension

Tenacity

(cN/dtex)

Per cent Boiling

Water Shrinkage

30 den/14 fils 2.14 White 34 4.75 5.50

50 den/36 fils 1.39 Black 24 3.50 5.00

60 den/6 fils 10 White 28 4.40 7.00

75 den/36 fils 2.08 Peach 35 4.00 7.18

100 den/48 fils(1� 100/48) 2.08 White 35 4.56 2.00

150 den/72 fils 2.08 Green 24 3.50 3.00

200 den/96 fils (2� 100/48) 2.08 White 35 4.56 2.00

250 den/120 fils 2.08 White 31 3.12 2.40

300 den/144 fils (3� 100/48) 2.08 White 35 4.56 2.00

dpf¼ denier per filament.

Shaikh and Bhattacharya 3




study to avoid undue overlapping of minor contribut-ing shape factor and frictional properties of parentyarn.

Textured yarns were tested for their per cent bulk,per cent instability, per cent boiling water shrinkageand mechanical properties. They were conditioned for24 hours at standard atmosphere for tropical regions,7

that is, 65%� 2% relative humidity and 27�C� 2�Ctemperature before testing. An Erma scope projectionmicroscope was used to study the structural character-istics of textured yarn. Magnification of the order of100� was used for this purpose.

Mechanical properties were checked on an Instrontensile tester 1121 model, using a gauge length of500mm and cross-head speed of 300mm/min.8

Looking at the simulation with the air-jet texturedyarn in imparting stability to the textured structure,1,2

the DuPont method9 was used to measure the stabilityof curls. Burnip et al.10 have introduced the concept ofbulk factor (y) for measuring the bulk of false-twisttextured yarns. Hence bulk is the outcome of crimpcharacteristics attained by flat yarn on mechanicalcrimp texturizing, as well as false-twist texturizing.The same method was adopted for measuring thebulk of the newly engineered yarn. This method hasalready been mentioned briefly in earlier publication.1

Optimization of process tension

Optimization of bulking zone length

The graphical representation of the results (Figure 2)verifies the importance of the bulking-zone length. Thepresence of well mingled small size high frequency

curls, in the structure of textured yarn produced atthe one inch bulking-zone length, have caused morecontraction due to higher crimping of the flat feederyarn. This has attributed toward a higher rise in percent denier as well as bulk of product yarn (Figure2a). These results have shown good agreement to thefindings of Wilson et al.5 for false-twist texturizing.

These results may be associated with the processundergone by filaments between the false-twist spin-dle-pin and the twist-trapper wheel at the texturizingzone. When the pre-twisted filament bundle leaves thetwist-trapper wheel under tension, opening of the pre-twist starts right from the point of exit. As soon asuntwisting gets completed, retwisting begins in theopposite direction due to false-twisting. Enough yarntension is maintained during this course of transitionto develop the desired torque at a given level of false-twisting. This facilitates the filaments to follow a helicalpath at a certain angle (depends on magnitude of twistand denier per filament) to the filament yarn longitudi-nal axis. However at the point of completion of untwist-ing and beginning of retwisting, the yarn becomes slackin the bulking zone. This is due to the release of theextra length of twist contraction. The situation becomesmore crucial for the long length bulking-zone. As lengthreleased on untwisting is more, gives higher drop in yarntension. More drop in yarn tension before retwisting,delays the generation of the desired magnitude oftorque required for bending deformation. This delayresults in poor crimpiness, declination of texturizingvalues (Figure 2a) and per cent elongation (Figure 2b).

The texturizing quality purely depends on the bend-ing deformation of individual filaments at an estab-lished torque in the texturizing zone. For a smaller

Drop in Tenacity (%)

Increase in denier

Boiling water shrinkage (%)

Instability (%)

1 1.5 2 2.5 3 3.5 4

4

6

8

10

12

14

16

18

Tex

turi

sin

g p

rop

erti

es

Bulk factor

1 1.5 2 2.5 3 3.5 4

10

15

20

25

30

35

Mec

han

ical

pro

per

ties

Extension (%)

Drop in tenacity (%)

(%)

Bulking zone length (inches)Bulking zone length (inches)

(a) (b)

Figure 2. Effect of bulking zone length on mechanical crimp textured yarn properties.





bulking-zone, that is, 1 inch comparatively more torquedevelops in the texturizing zone. The more the torque,the more bending deformation there is, which increasescrimpiness. Higher differential tension develops in fila-ments present in the centre to the surface layers of yarnmatrix at higher torque. This causes frequent migra-tion, and increased reversals. The degree of intermin-gling of curls increases within the yarn matrix. Thehigher drop in tenacity with increased crimpinessis expected. However, a smaller drop in tenacity (%)is recorded at the smaller bulking-zone lengths. This ismainly attributed to the presence of higher crossingcurls in the product yarn structure (Figure 2).Altogether this results in a product with preferable tex-turizing properties as well as mechanical properties(Figure 2). Therefore preference is given to the shorterbulking-zone length for texturizing.

Optimization of Parent Yarn Input Tensionat Bulking-zone

The input parent yarn tension value is optimized exper-imentally for 50 den/36 fils and 100 den/48 fils parentyarns. The texturizing carried out at the input tensionof 0.075 gf/den (0.675 gf/tex) (based on parent yarnlinear density) has executed better results for both theyarns. The texturizing carried out below this valueresulted in balloon formation at the bulking-zone; thishas caused an increase in end-breaks. Working with ahigher input tension value than this results in excesstension, which leads to filament fraying followed byrupture.

Optimization of False-twist

The extent of deviation from the yarn axis on crimpingcan make the product yarn performance differently. Itbecomes interesting to know the influence of majorparent yarn parameters on textured yarn with respectto crimping media (false-twist). Yarn fineness, filamentfineness and pre-twist (binding force for constituent fil-aments) are the expected variables. They define theextent of bending deformation undergone by flatfeeder yarn during the course of texturizing.

In the absence of a past record for the novel concept,the minimum false-twist employed is calculated as perHeberlein’s advanced formula (equation 3). Heberleinput forward an advanced empirical formula (equation3), registered under US Patent 2,904,952, in September1959. It is used for determining the optimum twist K(twist per meter) in false-twist texturizing for yarns withdifferent linear density D (denier).11,12 This selection isbased on the simulation found in crimping

methodology adopted by both the texturizing technol-ogy, except the use of heat.

K ¼ 800þ275, 000

Dþ 60ð3Þ

The Heberlein’s formula used for the study is derivedfor the false-twist texturizing process. In that process,deformation of the flat filament is carried out in theheated status. In the present study, deformation is car-ried out without heating the filament. Bending rigidityand torsional rigidity offered by the same type of fila-ment yarn is higher in the cold state than the heated flatfilament yarn at their glass transition temperature (Tg),which is mainly attributed to the increased mobility ofthe molecules at their glass transition temperature(Tg).3,13 More torque is required due to the increasedresistance to deformation, although the mechanicalmedia used for the deformation is the same. Undersuch conditions, the optimum twist value for the newconcept is higher than the calculated value for the false-twist texturizing concept. So, the optimum false-twistvalue calculated by equation 3 is defined as the mini-mum false-twist. The minimum false-twist is increasedgradually with the uniform increment of 100 twist permeter until the breakage of constituent filaments. Thefalse-twist value at which filament rupture begins isrecorded as the highest false-twist (Tmax. (tpm)) for allthe yarns under consideration. Rupture of constituentfibers/filaments begins on twisting the yarn beyond itsoptimum twist level. This is due to the obliquityeffect.7,13 Therefore the allowable optimum false-twistlevel K (tpm) is defined as follows: ‘Optimum false-twist is the highest possible false-twist at which constit-uent filaments of parent yarn execute the maximumpossible crimp without getting damaged/ruptured’.

Microscopical examination of the product yarn wascarried out at 100� magnification for confirmation ateach stage. The significance of such practically derivedvalues was verified by carrying out texturizing at thehighest possible delivery rate (350m/min for laboratoryapparatus) for all yarns. The number of end-breaks perhour, occurrence of broken filaments and regularity ofoptimum crimpiness in textured yarn structure wererecorded. These parameters were used to evaluate theperformance. A total of five randomly (from differentlayers of the package) selected yarn segments, eachapproximately 10m long were observed under a projec-tion microscope (100�) for this purpose.

Effect of Yarn Fineness on OptimumFalse-twist Level

Single end, two ends and three ends of fully drawn100 den/48 fils polyester yarn (i.e. 100 den/48 fils,





200 den/96 fils, and 300 den/144 fils), as well as 75 den/36fils, 150 den/72 fils, and 250 den/120 fils fully drawn poly-ester yarnswere texturedon laboratory apparatus at con-stant pre-twist (twist factors 24 tex1/2.turns/cm).Constituent filament fineness is kept constant, that is,2.08 den to avoid undue overlapping of effects.

The maximum false-twist, Tmax (tpm), as well as theoptimum false-twist K (tpm) were derived for each yarnunder consideration as per the methodology describedearlier.

Confirmation of the optimum twist, K, and maxi-mum twist, Tmax, by microscopical examination of100 den/48 fils textured yarn has been illustrated inFigure 3. It is quite apparent from these photographsof microscopical views that the minimum degree ofcrimpiness is attained at the lowest selected false-twist(F1), as per Heberlein’s formula. The degree of crimpi-ness is increased with the constant increment of 100 tpmin false-twist level up to sample F15. However, a furtherrise in false-twist of 100 tpm caused the partial ruptureto the yarn sample F16. So, the next trial was conductedwith half of the increment, (i.e. 50 tpm). The best pos-sible texturizing effect is observed with this sample F17.In order to study the impact of false-twist changebetween sample F16 and F17 texturizing was also con-ducted with the increment of 5 tpm. Deterioration inthe texturizing quality instead of improvement in thecrimping was observed in the form of filament breakageand fusing (sample F18). Partial or complete rupturewas recorded between false-twist levels of 3990 tpm -3998 tpm for six trials conducted for the confirmationof Tmax. These values of false-twist are also confirmedat the highest possible delivery rate as mentioned

previously. The regularity of the texturizing effect atoptimum false-twist was executed by wrapping boards(5 inch� 5 inch) prepared from different layers ofbobbin under constant tex=2 tension (Figure 4).

The experimentally measured optimum false-twist(K) and maximum false-twist (Tmax) values for the dif-ferent yarn fineness (D) under consideration arereported in Table 2 and shown graphically in Figure 5.

It can be seen (Table 2 and Figure 5a) that the opti-mum twist values recorded have shown a decline withthe increase in yarn fineness. Even though yarn size hasbeen increased by a constant value (50 den), there is nouniform reduction in optimum twist value from 100 denyarn to 300 den yarn. A proportionately higher reduc-tion was observed for fine denier yarn as compared tothe coarser one in the group. Thus the reduction inoptimum twist value is not linear for all the yarnsunder consideration although composed of identical fil-ament fineness (Table 2).

Effect of Filament Fineness on OptimumFalse-twist Level

Fully drawn polyester yarns with different denier perfilament, that is, 10 (60 den/6 fils), 2.14 (30 den/14 fils),2.08 (75 den/36 fils), and 1.39 (50 den/36 fils), are tex-tured on lab apparatus with a constant pre-twist (twistfactors 24 tex1/2 turns/cm). In order to diagnose theimpact of filament rigidity on the yarn bending defor-mation, the false-twist level is increased gradually, asper the methodology adopted in the previous phase ofexperimentation. Empirically derived optimum false-twist (K) and maximum false-twist (Tmax) values for

Figure 3. Microscopical view (100�) for the mechanical crimp textured 100 den/48 fils. Yarn produced at different false-twist level

(Fi tpm) used during study.





all the yarns under consideration are reported in Table3 and also shown graphically in Figure 5a.

The optimum twist values recorded have shown adecline with the increase in yarn fineness irrespectiveof the constituent filament fineness. Finer 60 den yarnhas exhibited higher optimum false-twist although,composed of coarser filaments (10 den). Comparatively

low optimum false-twist is recorded for coarser 75 denyarn although, composed of finer constituents(2.08 den). A similar trend is also observed for the restof the yarns in the group, that is, between 60 den yarnand 50 den yarn, as well as between 50 den yarn and30 den yarn. Thus the optimum twist value for thegiven type of raw material remained exclusively depen-dent on the parent yarn linear density and is indepen-dent of constituent filament fineness for mechanicalcrimp texturizing.

Various relationships patented/established for theevaluation of optimum false-twist ‘K’ for the false-twist texturizing process, also focused only on yarnfineness ‘D’.11,12 Thus good correlation is foundbetween both the systems using a similar crimping tech-nique. However, for identical yarn fineness a significantdifference in optimum false-twist values is observedbetween two systems in the present study. This ismainly attributed to the freedom of molecular move-ment during deformation as mentioned earlier.

Effect of Pre-twist on Optimum False-twist Level

Pre-twist is a real twist present in the structure ofthe feeder yarn, subjected to false-twisting action

Figure 4. Photographic View of Wrapping boards prepared for Mechanical Crimp Textured 100d/ 48 fils. Yarn at Optimum false-

twist (3938 tpm).

Table 2. Experimentally derived values of maximum false twist

level (Tmax) and optimum false-twist (K) tpm for different parent

yarn fineness (D)

Yarn specificationsMaximum FT Optimum FT

Denier, den Fils/cs dpf Tmax (tpm) K (tpm)

75 36 2.08 4238 4132

100 48 2.08 3994 3937

150 72 2.08 3020 2925

200 96 2.08 2598 2533

250 120 2.08 2307 2240

300 144 2.08 2110 2097

Constant pre-twist factor¼ 24 tex1/2.turns/cm. cs¼ cross-section,

dpf¼ denier per filament, FT¼ false-twist, tpm¼ twist per meter.





at the bulking-zone. It is used to bind the constituentcrimpy filaments together at the delivery end. Eventhough it is low in magnitude and removed by false-twist at the bulking-zone, it is desirable to study itsinfluence on yarn bending deformation at the bulking-zone.

Fully drawn 100 den/48 fils polyester multifilamentyarn is textured at five different pre-twist levels, thatis, twist factor (tex1/2.turns/cm) of 2, 4, 6, 12 and 24.The same strategy is adopted for the measure ofoptimum false-twist for feeder yarn with different pre-twist as described previously. Results of the studyare given in Table 4 and presented graphically inFigure 5b.

No significant difference (% critical difference <1) isobserved in the optimum false-twist value reported for100 den/48 fils polyester yarn, textured at different pre-twist level (Table 4). Thus, pre-twist has not shown anypeculiar trend or impact on the optimum false-twistlevel attained for a selected supply yarn texturedunder the identical conditions (Figure 5b). An increase

in pre-twist has reduced the apparent product yarnbulk. This is mainly due to the increased compactnessof the product yarn structure at higher locking twist(Figure 6). However, the degree of crimpiness attainedby product yarn constituents remains the same, as theoptimum false-twist induced is identical in the bulking-zone (Table 4).

Development of an empirical formula

The optimum false-twist value for the given type of rawmaterial remained exclusively dependent on the parentyarn linear density for the novel concept of mechanicalcrimp texturizing, the conclusion derived based on ear-lier discussion. So, it is imperative to develop the math-ematical relationship between these two variables forthe mechanical crimp texturizing process. Accuratemathematical evaluation of optimum false-twist for agiven supply yarn, before starting with actual produc-tion process, would definitely add to the process econ-omy, as well as help in minimizing process waste atconsiderably reduced efforts.

The polynomial curve fitting (Polyfit) program ofMATLAB was used (Appendix 1) as a mathematicaltool. Since this program square roots the square ofdeviation of the theoretical value from the actual one,the error associated in the estimation is less. The desiredempirical formula was derived by making use of thistool and experimental results (Tables 2 and 3).

Polynomial Curve Fitting (Polyfit) Program

In the program, p¼ polyfit(x,y,n) finds the coefficientsof a polynomial p(x) of degree ‘n’ that fits the data,p(x(i)) to y(i)), in a least squares sense. The result ‘p’

Figure 5. Scatter diagram for optimum false-twist (K) and maximum false-twist (Tmax).


level (Tmax) and optimum false-twist (K) tpm for different parent

yarn filament fineness

Yarn specificationsMaximum FT Optimum FT

Denier, den Fils/cs dpf Tmax (tpm) K (tpm)

30 14 2.14 5940 5792

50 36 1.39 5012 4887

60 6 10 4660 4534

75 36 2.08 4238 4132

Constant pre-twist factor¼ 24 tex1/2.turns/cm. cs¼ cross-section, dpf ¼

denier per filament, FT¼ false-twist, tpm¼ twist per meter.





is a row vector of length ‘n+1’ containing the polyno-mial coefficients in descending powers

PðxÞ ¼ p1xn þ p2x

n�1 þ . . .þ pnxþ pnþ1 ð3Þ

where,

‘P(x)’ is the dependent variable. In the present case, it isthe optimum false-twist level ‘K’ in tpm (twist permeter) evaluated from the known value of yarn

denier (D). These notations for the variables areused to maintain consistency with earlier relation-ships established for false-twist texturizing.

‘X’ is known as an independent variable. In the presentstudy, yarn denier (D) is the independent variable.

‘n’ is the degree of polynomial. Its value can be chosensuch that the theoretically plotted curve fits veryclose to the curve formulated by using the practicalfeed data.

p1, p2,. . ., pn+1 are variable coefficients.

The ‘p¼ polyfit (D, K, n)’ program (Appendix 1)was used for the known values of ‘D’ and ‘K’ derivedfrom the practical results (Tables 3 and 4). Figure 7illustrates the desired close fit attained at degree ofpolynomial, n¼ 3. Where, small circles represent thepractical in-feed data.

Equation 4 represents polynomial formula derivedon the substitution of values of coefficients p1, p2, p3and p4, obtained after confirmation of close fit.

KðtpmÞ ¼ 7151:7� 53:9Dþ 0:2D2 � 0:000255D3 ð4Þ

The equation was also verified for its accuracy by thesame program (Appendix 1). The basic mathematical


level (Tmax) and optimum false-twist (K) at different pre-twist

level

Pre-twist factor

(tex 1/2.turns/cm)

Maximum FT

Tmax (tpm)

Optimum FT

K (tpm)

2 3989 3926

4 3996 3933

6 3981 3923

12 3992 3929

24 3998 3921

Parent yarn¼ 100 den (11.1 tex)/48 fils, underfeed¼ 25 %. tpm¼ twist

per meter, FT¼ false-twist.

Figure 6. Effect of pre-twist on textured yarn compactness. [Pre-twist factor (tex 1/2.turns/cm): i¼ 2, ii¼ 4, iii¼ 6, iv¼ 12, v¼ 24].





rule, ‘error should be as low as possible’ was opted forthis verification.

Conclusion

A novel concept of mechanical crimp texturizing wasdeveloped with the view of getting a favorable spunyarn like structure by a mechanical mode. The false-twist level used during mechanical crimp texturizingplayed a major role in deciding the bulkiness of theproduct yarn. Heberlein’s advance formula was usedfor the theoretical calculation of optimum twist ini-tially. Mechanical crimp texturizing was carried outwithout heating the filament to the glass transition tem-perature (Tg), unlike false-twist texturizing. Althoughthe mode of deformation force applied was the same,more torque was required to impart the desired crimpyconfiguration to the flat continuous filament feed yarn.This was visualized from the higher value of the opti-mum false-twist (K) recorded for different finenesssupply yarns (D), as compared to the calculated one

by Heberlein’s formula. This is mainly attributed tothe higher resistance offered by the molecular structureto defomation, when the flat filament yarn is fed undertension and textured at normal temperature.

It was also proven experimentally that for a givenraw-material (polyester in the present study), optimumfalse-twist (K) used during mechanical crimp texturiz-ing was mainly influenced by yarn fineness (D) andremained unaffected by filament fineness and pre-twist. So, the empirical formula was developed for theevaluation of optimum false-twist level (K) forknown polyester yarn fineness (D). The polynomialcurve fitting (Polyfit) program from MATLAB wasused as a mathematical tool for this purpose. Thiswas also verified successfully for the minimum possibleinduced error of theoretical estimation in comparisonto the practically derived values.

Funding

This research received no specific grant from any fundingagency in the public, commercial, or not-for-profit sectors.

Figure 7. Polyfit curve for optimum twist (K tpm) and yarn fineness (D).





References

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Appendix 1

Matlab program

. D¼ [30 50 60 70 100 150 200 250 300]

. K¼ [5792 4887 4544 4132 3940 2545 2533 2249 2057]

. p¼ polyfit (D, K, 3)

. f1¼p(1).�D.^3+p(2).�D.^2+p(3).�D+p(4)

. f1¼ 1.0e+003�

. 5.7144 4.9420 4.6065 4.3024 3.5620 2.7975 2.43492.2601 2.0592

. table¼ [D’ K’ f1’]

. table¼ 1.0e+003 *

. D.^3 K.^3 f.^3

. 0.0300 5.7920 5.7144

. 0.0500 4.8870 4.9420

. 0.0600 4.5440 4.6065

. 0.0700 4.1320 4.3024

. 0.1000 3.9400 3.5620

. 0.1500 2.5450 2.7975

. 0.2000 2.5330 2.4349

. 0.2500 2.2490 2.2601

. 0.3000 2.0570 2.0592

. Dnew¼ input(’Denier’);

. Knew¼(1).�D.^3+p(2).�D.^2+p(3).�D+p(4).




deriving an empirical formula to determine the optimum level of false-twist in mechanically-crimped...

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