The theoretical yarn unevenness of cotton considering the joint influence of fiber length distribution and fiber fineness

Abstract

The theoretical yarn unevenness has been discussed by considering the joint influence of fiber length distribution and fiber fineness. For fiber length distribution, the fiber length density function was derived from the specific Weibull parameters obtained from actual fiber length measurements using an USTER AFIS instrument. Meanwhile, fiber fineness was added to Suh’s model by expressing spun yarn irregularity in terms of fiber mass variation. Comparisons of the calculated value of theoretical yarn unevenness in this paper with the tested and previously researched ones for different cotton yarns have been made. It is shown that the theoretical yarn unevenness calculated in this paper could better reflect the effect of fiber length and fiber fineness on yarn unevenness. This might be the theoretical foundation for the prediction of theoretical cotton yarn unevenness on the basis of actual fiber length distribution and fiber fineness.

Keywords

theoretical yarn unevenness fiber length distribution fiber fineness density function mathematical expression

Research about yarn unevenness, an important evaluation indicator of yarn quality, has long been a classic topic in the field of textiles. The unevenness of yarn is a combination of theoretical yarn unevenness and additional yarn irregularity.^1–3 It is well known that theoretical yarn unevenness is caused by the random fiber alignment in the yarn, while the additional yarn irregularity is caused during processing. Undoubtedly, fiber properties, such as fiber length and fiber fineness acting as the two dominant characteristics of the raw material of the yarn, affect the yarn unevenness.^3–5 Due to the additional yarn irregularity depending on the processing, theoretical yarn unevenness is the best point for investigating the effect of actual fiber length and fiber fineness on yarn unevenness.

Many theoretical models have been put forward to express theoretical yarn unevenness. Basically, there are two main types of models. The first type is the models that provide mathematical expressions by considering the variation of the number of fibers in the yarn cross section and fiber fineness.^2–4,6 The most representative model among this type was put forward by Martindale,⁴ and its expression is
$CV = \sqrt{\frac{1}{\bar{n}} {1 + [CV (T)] 2}}$
(1)
where $\bar{n}$ is the expected number of fibers in the yarn cross section and CV(T) is the coefficient of variation of fiber fineness. This expression was based on an assumption that the number of fibers in the yarn cross section followed the Poisson distribution, as pointed out by Spencer-Smith and Todd.⁷ Though this model has been widely employed in numerous mills and researches, it does not take the fiber length effect into account. The second type is the model based on Suh’s theory,^5,8 in which the fiber length is considered as having an important impact on yarn unevenness. Theoretical yarn unevenness from Suh’s theory can be expressed by
${CV [S_{T} (Δ)]} 2 = \frac{E_{l \leq Δ} [\frac{l 2 (3 Δ - l)}{3 (l + Δ)}] + E_{l > Δ} [\frac{Δ 2 (3 l - Δ)}{3 (l + Δ)}]}{\bar{n} Δ \cdot E_{l} (\frac{Δ l}{l + Δ})} = \frac{\int_{0}^{Δ} \frac{l 2 (3 Δ - l)}{3 (l + Δ)} f (l) d l + \int_{Δ}^{\infty} \frac{Δ 2 (3 l - Δ)}{3 (l + Δ)} f (l) d l}{\bar{n} Δ \cdot \int_{0}^{\infty} \frac{Δ l}{l + Δ} f (l) d l}$
(2)
where l is fiber length, f(l) is the fiber length density function, Δ is length interval in the yarn, and S_T(Δ) is the total length of all the fibers found within length interval Δ. Suh’s earlier work worked very well with the actual length distribution data they have used. However, the variation of fiber fineness was ignored in this theory.

The purpose of this paper is to discuss the theoretical yarn unevenness by considering the joint influence of fiber length and fiber fineness. Fiber length measurements from AFIS have been applied and the fiber length density function was gained by the specific Weibull parameters obtained from the actual measurements. Meanwhile, the fiber fineness has been added to Suh’s model by expressing fiber mass variation. Then, theoretical values from Martindale’s work (without fiber length distribution effects), Suh’s model (without fiber fineness variation), and the calculated results with actual fiber length distributions in this paper were experimentally estimated and comparisons are made.

Model development

Assumptions

It is well known that yarn unevenness is defined as the unevenness of yarn linear density along its length.^3,9 In this paper, theoretical yarn unevenness refers to the coefficient of variation of the yarn weight among each length interval in the yarn, which is in accordance with the test theory of commonly used evenness tester. In order to calculate the theoretical yarn unevenness with the given characteristics of fiber length and fiber fineness, several assumptions related to the yarn are made and shown as follows:
All fibers in the yarn are fully extended in parallel along the yarn’s length axis.^2–5,8–10

Fiber fineness among different fibers can vary but be independent of fiber length.^11–13

Theoretical derivation of the model

If Δ(mm) is the length interval in the yarn and l(mm) the fiber length; the fiber length distribution function in the yarn is F(l) and density function is f(l); S(mm) is the length of the fiber with the length l to be found within the length interval Δ in the yarn; N is the total number of fibers found within length interval Δ; T(tex) is the fiber fineness; and G_i(g) the weight of the ith fiber within the length interval Δ in the yarn ( $i = 1, 2, \dots, n$ ), then $G_{i} = \frac{S_{i} T_{i}}{1000}$ ;¹⁴ the total weight of fibers within the length interval Δ is $G = \sum_{i = 1}^{N} G_{i}$ .

Due to fiber fineness being independent of fiber length, the expectation of the weight of the ith fiber within length interval Δ can be derived by
$E (G_{i}) = E (\frac{S_{i} T_{i}}{1000}) = \frac{1}{1000} E (S_{i}) E (T_{i})$
(3)

Similarly, the variance of the weight of the ith fiber within length interval Δ can be derived by
$D (G_{i}) = D (\frac{S_{i} T_{i}}{1000}) = (\frac{1}{1000}) 2 D (S_{i} T_{i}) = (\frac{1}{1000}) 2 [E (S_{i}^{2} T_{i}^{2}) - E 2 (S_{i} T_{i})] = (\frac{1}{1000}) 2 [E (S_{i}^{2}) E (T_{i}^{2}) - E 2 (S_{i}) E 2 (T_{i})] = (\frac{1}{1000}) 2 [E (S_{i}^{2}) E (T_{i}^{2}) - E 2 (S_{i}) E (T_{i}^{2}) + E 2 (S_{i}) E (T_{i}^{2}) - E 2 (S_{i}) E 2 (T_{i})] = (\frac{1}{1000}) 2 [D (S_{i}) E (T_{i}^{2}) + D (T_{i}) E 2 (S_{i})] = (\frac{1}{1000}) 2 [D (S_{i}) D (T_{i}) + D (S_{i}) E 2 (T_{i}) + D (T_{i}) E 2 (S_{i})]$
(4)

Then, the expectation of the total weight of fibers within the length interval Δ can be derived by
$E (G) = E (\sum_{i = 1}^{N} G_{i}) = \sum_{i = 1}^{N} E (G_{i}) = \frac{1}{1000} E (N) E (S) E (T)$
(5)

Similarly, the variance of the total weight of fibers within the length interval Δ can be derived by
$D (G) = E [D (\sum_{i = 1}^{N} G_{i} | N)] + D [E (\sum_{i = 1}^{N} G_{i} | N)] = E (N) D (G_{i}) + E 2 (G_{i}) D (N) = E (N) {(\frac{1}{1000}) 2 [D (S) D (T) + D (S) E 2 (T) + D (T) E 2 (S)]} + (\frac{1}{1000}) 2 D (N) E 2 (S) E 2 (T) = (\frac{1}{1000}) 2 {E (N) [D (S) D (T) + D (S) E 2 (T) + D (T) E 2 (S)] + D (N) E 2 (S) E 2 (T)}$
(6)

Finally, theoretical yarn unevenness, the coefficient of variation of the yarn weight among each length interval in the yarn, can be derived by
$[CV (G)] 2 = \frac{D (G)}{E 2 (G)} = \frac{(\frac{1}{1000}) 2 {E (N) [D (S) D (T) + D (S) E 2 (T) + D (T) E 2 (S)] + D (N) E 2 (S) E 2 (T)}}{(\frac{1}{1000} E (N) E (S) E (T)) 2} = \frac{E (N) [D (S) D (T) + D (S) E 2 (T) + D (T) E 2 (S)] + D (N) E 2 (S) E 2 (T)}{E 2 (N) E 2 (S) E 2 (T)} = \frac{1}{E (N)} {\frac{D (S)}{E 2 (S)} • \frac{D (T)}{E 2 (T)} + \frac{D (S)}{E 2 (S)} + \frac{D (T)}{E 2 (T)}} + \frac{D (N)}{E 2 (N)} = \frac{1}{E (N)} {[CV (S)] 2 • [CV (T)] 2 + [CV (S)] 2 + [CV (T)] 2} + [CV (N)] 2$
(7)

that is,
$CV (G) = \sqrt{\frac{D (G)}{E 2 (G)}} = \sqrt{{\frac{1}{E (N)} {[CV (S)] 2 • [CV (T)] 2 + [CV (S)] 2 + [CV (T)] 2} + [CV (N)] 2}}$
(8)
where CV(S) is the coefficient of variation of fiber length for the fibers found within length interval Δ; CV(T) is the coefficient of variation of fiber fineness; E(N) and CV(N) are the expectation and the coefficient of variation of the number of fibers found within length interval Δ in the yarn, respectively.

Determination of the unknown parameters

According to Suh’s theory,⁵ the expectation and the variance of S (the length of the fiber to be found within the length interval Δ in the yarn) are given by
$E (S) = E_{l} E_{S} [S | L = l] = E_{l \leq Δ} E_{S} (S | L = l \leq Δ) + E_{l > Δ} E_{S} (S | L = l > Δ) = \int_{0}^{Δ} \frac{Δ l}{l + Δ} f (l) d l + \int_{Δ}^{\infty} \frac{Δ l}{l + Δ} f (l) d l = E_{l} (\frac{Δ l}{l + Δ})$
(9)

$E (S 2) = E_{l} E_{S} [S 2 | L = l] = E_{l \leq Δ} E_{S} (S 2 | L = l \leq Δ) + E_{l > Δ} E_{S} (S 2 | L = l > Δ) = \int_{0}^{Δ} \frac{l 2 (3 Δ - l)}{3 (l + Δ)} f (l) d l + \int_{Δ}^{\infty} \frac{Δ 2 (3 l - Δ)}{3 (l + Δ)} f (l) d l$
(10)

$= E_{l \leq Δ} [\frac{l 2 (3 Δ - l)}{3 (l + Δ)}] + E_{l > Δ} [\frac{Δ 2 (3 l - Δ)}{3 (l + Δ)}]$

$D (S) = E_{l \leq Δ} [\frac{l 2 (3 Δ - l)}{3 (l + Δ)}] + E_{l > Δ} [\frac{Δ 2 (3 l - Δ)}{3 (l + Δ)}] - [E_{l} (\frac{Δ l}{l + Δ})] 2$
(11)

Then, the coefficient of variation of fiber length for the fibers found within length interval Δ in the yarn can be calculated by
$[CV (S)] 2 = \frac{D (S)}{E 2 (S)} = \frac{E_{l \leq Δ} [\frac{l 2 (3 Δ - l)}{3 (l + Δ)}] + E_{l > Δ} [\frac{Δ 2 (3 l - Δ)}{3 (l + Δ)}] - [E_{l} (\frac{Δ l}{l + Δ})] 2}{[E_{l} (\frac{Δ l}{l + Δ})] 2} = \frac{\int_{0}^{Δ} \frac{l 2 (3 Δ - l)}{3 (l + Δ)} f (l) d l + \int_{Δ}^{\infty} \frac{Δ 2 (3 l - Δ)}{3 (l + Δ)} f (l) d l}{[\int_{0}^{\infty} \frac{Δ l}{l + Δ} f (l) d l] 2} - 1$
(12)
where f(l) is the density function of the fiber length distribution by number. According to some previous research,^15–18 a mixture of two Weibull distributions can be adopted to fit the cotton fiber length distribution. So the density function of the mixture model, shown in equation (13), can be used to characterize the cotton fiber length distribution.
$f (l) = p \cdot \frac{β_{1}}{η_{1}} \cdot (\frac{l}{η_{1}}) β_{1} - 1 \cdot \exp [- (\frac{l}{η_{1}}) β_{1}] + (1 - p) \cdot \frac{β_{2}}{η_{2}} \cdot (\frac{l}{η_{2}}) β_{2} - 1 \cdot \exp [- (\frac{l}{η_{2}}) β_{2}]$
(13)
where p is the proportion of the first Weibull distribution, η₁ andη₂ are the scale parameters, and β₁ and β₂ are the shape parameters of the Weibull distribution, respectively.

Fiber fineness varies significantly along the fiber axis with natural cotton. In order to obtain the coefficient of variation of fiber fineness by number, YG002C fiber fineness meter was used. Experimental principle was shown as follows: after finishing the cotton sample, a single fiber was straight and placed on a glass slide, then this glass slide was placed in microscope stage. Through the objective lens (the magnification is 4×), the eyepiece and the camera, the longitudinal morphology of this fiber was shown on the screen of the computer. Thereafter, data collection by clicking on the opposite fiber edges to gain the diameter of the fiber was done with the longitudinal gauge of 0.2 mm. Each group of 30 fibers for one cotton sample was tested.

According to the relationship between fiber diameter and fiber fineness,¹² $d = 0.03568 \times \sqrt{\frac{T_{t}}{ρ}}$ (d is the fiber diameter, mm; T_t is fiber fineness, tex; ρ is the cotton fiber density, 1.54 g/cm³), fiber diameter could be converted into fiber fineness. Therefore, each group of fiber fineness could be gained. Then, the coefficient of variation of fiber fineness by number could be calculated. In this paper, the variance of fiber fineness is a gross oversimplification of the actual fiber fineness variance contributing to the yarn length segments.

According to Brown’s theory,¹⁹ the expected number of fibers to be found within length interval Δ in the yarn is given by
$E (N) = \frac{\bar{n}}{\bar{l}} (Δ + \bar{l})$
(14)
where $\bar{l}$ is the expected length of fibers in the yarn; $\bar{n}$ is the expected number of fibers in yarn cross section and it can be calculated by
$\bar{n} = \frac{T_{yarn}}{T_{fiber}}$
(15)
where T_yarn is yarn linear density (tex), T_fiber is fiber fineness (tex).

In order to obtain the value of CV(N), an assumption that the number of fibers in yarn cross section follows the Poisson distribution is still adopted.^4,5,7 Obviously, the number of fibers to be found within length interval Δ in the yarn also follows the Poisson distribution. Then,
$D (N) = E (N) = \frac{\bar{n}}{\bar{l}} (Δ + \bar{l})$
(16)

The coefficient of variation of the number of fibers found within length interval Δ in the yarn can be calculated by
$[CV (N)] 2 = \frac{D (N)}{E 2 (N)} = \frac{1}{E (N)} = \frac{1}{\frac{\bar{n}}{\bar{l}} (Δ + \bar{l})}$
(17)

Experimental verification

Four cotton slivers and their 10 kinds of resultant yarns from different cotton mills have been adopted. Before their relevant properties were tested, these samples were placed in a constant temperature and humidity laboratory for 24 hours. Their properties are shown in Table 1.
Table 1.
The properties of fibers and the resultant yarns

Cotton sample Average fiber fineness (tex) Yarn ID Yarn linear density (tex)

A 0.170 1# 9.72

2# 11.66

B 0.171 3# 14.58

4# 18.22

C 0.168 5# 12.96

6# 14.58

7# 18.22

D 0.165 8# 18.22

9# 19.44

10# 27.77

In this paper, the frequency histogram of the finished sliver was regarded as the same as that of the resultant yarns of each cotton sample. Fiber lengths of these four cotton samples have been tested by using an USTER AFIS instrument. The fiber length frequency histograms by number and their fitted curves of the mixture model are shown in Figure 1. Then, the parameters of this mixture model were obtained.
Figure 1.
Fiber length frequency histograms of four samples and their fitted curves of the mixture of two Weibull distributions.

Meanwhile, mean fiber length $\bar{l}$ and the coefficient of variation of fiber length for the fibers in the sliver, CV(L), also could be tested by AFIS. According to equation (12), the coefficient of variation of fiber length for the fibers found within the length interval Δ could be calculated. In this paper, Δ = 8 mm was assumed, in accordance with the actual test length commonly used in an Uster evenness tester. All the parameters about fiber length of the four cotton samples are listed in Table 2.
Table 2.
Results of several parameters of fiber length and CV(S) of four cotton samples

Cotton sample $\bar{l}$ (mm) CV(L) The parameters of the mixture model
[CV(S)]²

p η ₁ β ₁ η ₂ β ₂

A 25.10 0.0346 0.2719 27.8796 1.9923 27.0700 4.7577 0.1929

B 23.30 0.0391 0.9175 26.0911 2.9089 33.6874 8.6721 0.2042

C 22.40 0.0421 0.3857 23.4924 1.9328 27.8410 3.6555 0.2154

D 21.40 0.0452 0.3785 22.4907 1.7796 0.6215 26.6688 0.2274

Fiber fineness of the four cotton samples was tested by using a YG002C fiber fineness meter. The results of the coefficient of variation of fiber diameter, CV(d), and the coefficient of variation of fiber fineness, CV(T), are listed in Table 3.
Table 3.
Results of CV(d) and CV(T) of four cotton samples

Cotton sample [CV(d)]² [CV(T)]²

A 0.0090 0.0377

B 0.0126 0.0487

C 0.0105 0.0469

D 0.0134 0.0547

From Table 3, the relationship between the coefficient of variation of fiber diameter and fiber fineness could be given by [CV(T)]² ≈ 4[CV(d)]², which was in accordance with Martindale’s theory.⁴

According to equation (14), equation (15), and equation (17), the corresponding parameters about the number of fibers could be calculated and these are listed in Table 4.
Table 4.
Results of the parameters about the number of fibers of four cotton samples

Cotton sample Yarn ID Yarn linear density (tex) $\bar{n}$ E(N) [CV(N)]²

A 1# 9.72 57.18 75.40 0.0133

2# 11.66 68.59 90.45 0.0111

B 3# 14.58 85.26 114.53 0.0087

4# 18.22 106.55 143.13 0.0070

C 5# 12.96 77.14 104.69 0.0096

6# 14.58 86.79 117.79 0.0085

7# 18.22 108.45 147.18 0.0068

D 8# 18.22 110.42 151.70 0.0066

9# 19.44 117.82 161.86 0.0062

10# 27.77 168.30 231.22 0.0043

Comparison between the tested yarn unevenness and the calculated results

Yarn unevenness of 10 resultant yarns of four cotton samples were tested by and evenness tester and theoretical yarn unevenness was calculated from different models. The results are listed in Table 5 and comparisons have been made between the tested and the calculated results.
Table 5.
Comparison between the tested and calculated yarn unevenness

Cotton sample Yarn ID Yarn linear density (tex) Yarn unevenness (%)

Tested Calculated

Martindale’s Suh’s This paper

A 1# 9.72 13.84 13.47 12.46 12.83

2# 11.66 13.23 12.30 11.37 11.72

B 3# 14.58 12.24 11.09 10.13 10.50

4# 18.22 11.43 9.92 9.06 9.39

C 5# 12.96 13.01 11.65 10.62 11.05

6# 14.58 12.70 10.98 10.01 10.40

7# 18.22 11.75 9.83 8.96 9.30

D 8# 18.22 12.35 9.77 8.84 9.24

9# 19.44 12.02 9.46 8.55 8.96

10# 27.77 10.45 7.92 7.16 7.47

Even though the differences between Martindale’s, Suh’s, and this paper’s work are due to process variance (the additional yarn irregularity caused during the processing) and fiber fineness variance contributing to yarn irregularity, which cannot be easily separated, on the basis of the results in Table 5 it could also be concluded that:
All the calculated values of yarn unevenness were less than the tested ones because of the existence of the additional yarn irregularity caused during the processing.

For each cotton sample, variation tendency of different calculated values was in consistent with the one of the tested value. When fiber length distributions and fiber fineness of different yarns were the same, the larger the yarn linear density, the smaller the yarn unevenness, which is consistent with reality.

For different calculated values of yarn unevenness, the analyses were as follows: Martindale’s results were larger than the ones from Suh, because when yarn unevenness was calculated by Suh’s model, Δ = 8 mm was assumed, in accordance with the actual test length of commonly used in the Uster evenness tester, whereas in Martindale’s model, Δ → 0.

The results calculated in this paper were larger than Suh’s results because the effect of fiber fineness variance, CV(T), on yarn unevenness, ignored by Suh’s model, was taken into consideration in this paper.

In conclusion, the calculated values of the theoretical yarn unevenness obtained in this paper are consistent with reality, which shows that the method pointed to in this paper could be accepted to calculate the values of yarn unevenness in consideration of the actual test fiber length distribution and fiber fineness.

Conclusions

In this paper, the theoretical yarn unevenness was calculated by considering the joint influence of fiber length distribution and fiber fineness, in which a mixture of two Weibull distributions was adopted to gain the density function of fiber length distribution from different cotton samples. Comparisons were made between different calculated values of yarn unevenness and the tested ones. The results showed that for one cotton sample, variation tendency of the yarn unevenness calculated in this paper was in accordance with other calculated and tested ones. For different calculated values of yarn unevenness, the theoretical yarn unevenness calculated in this paper could reflect the joint influence effect of fiber length and fiber fineness on yarn unevenness better than other expressions, such as Martindale’s and Suh’s. This could be the theoretical foundation for the prediction of theoretical yarn unevenness from different cotton samples with different fiber length distribution and fiber fineness. To expand this research, how the different fiber characteristics such as fiber length and fiber fineness affect yarn unevenness can be further discussed.

Cotton sample	Average fiber fineness (tex)	Yarn ID	Yarn linear density (tex)
A	0.170	1#	9.72
2#	11.66
B	0.171	3#	14.58
4#	18.22
C	0.168	5#	12.96
6#	14.58
7#	18.22
D	0.165	8#	18.22
9#	19.44
10#	27.77

Cotton sample	$\bar{l}$ (mm)	CV(L)	The parameters of the mixture model	[CV(S)]²
A	25.10	0.0346	0.2719	27.8796	1.9923	27.0700	4.7577	0.1929
B	23.30	0.0391	0.9175	26.0911	2.9089	33.6874	8.6721	0.2042
C	22.40	0.0421	0.3857	23.4924	1.9328	27.8410	3.6555	0.2154
D	21.40	0.0452	0.3785	22.4907	1.7796	0.6215	26.6688	0.2274

Cotton sample	[CV(d)]²	[CV(T)]²
A	0.0090	0.0377
B	0.0126	0.0487
C	0.0105	0.0469
D	0.0134	0.0547

Cotton sample	Yarn ID	Yarn linear density (tex)	$\bar{n}$	E(N)	[CV(N)]²
A	1#	9.72	57.18	75.40	0.0133
2#	11.66	68.59	90.45	0.0111
B	3#	14.58	85.26	114.53	0.0087
4#	18.22	106.55	143.13	0.0070
C	5#	12.96	77.14	104.69	0.0096
6#	14.58	86.79	117.79	0.0085
7#	18.22	108.45	147.18	0.0068
D	8#	18.22	110.42	151.70	0.0066
9#	19.44	117.82	161.86	0.0062
10#	27.77	168.30	231.22	0.0043

Cotton sample	Yarn ID	Yarn linear density (tex)	Yarn unevenness (%)
A	1#	9.72	13.84	13.47	12.46	12.83
2#	11.66	13.23	12.30	11.37	11.72
B	3#	14.58	12.24	11.09	10.13	10.50
4#	18.22	11.43	9.92	9.06	9.39
C	5#	12.96	13.01	11.65	10.62	11.05
6#	14.58	12.70	10.98	10.01	10.40
7#	18.22	11.75	9.83	8.96	9.30
D	8#	18.22	12.35	9.77	8.84	9.24
9#	19.44	12.02	9.46	8.55	8.96
10#	27.77	10.45	7.92	7.16	7.47

Footnotes

Funding

This work was supported by the Natural Science Foundation of China (grant number 51173023) and the Chinese Universities Scientific Fund (grant number CUSF-DH-D-2014005).

References

. Spinning, China Textile Press, 2009.

Kuang

Yan

Jiang

. A new mathematical expression of yarn limit unevenness. Fiber Polym 2013; 14: 1943–1946.

Lin

Yan

. Effect of fiber length distribution and fineness on theoretical unevenness of yarn. J Text Inst 2011; 3: 214–219.

Martindale

. A new method of measuring the Irregularity of yarns with some observations on the origin of irregularities in worsted slivers and yarns. J Text Inst 1945; 2: 35–47.

Suh

. Probabilistic assessment of irregularity in random fiber arrays: effect of fiber length distribution on “variance-length curve”. Text Res J 1976; 45: 291–298.

Yan

Zhu

. A new approach to the theoretical yarn unevenness: a binomial distribution model. J Text Inst 2010; 8: 753–756.

Spencer-Smith

Todd

HAC

. A time series met within textile research. J R Stat Soc Suppl 1941; 7: 131–135.

Jiang

Kuang

. Simulation on fiber random arrangement in the yarn. J Text Inst 2014; 12: 1312–1318.

Zeidman

Suh

Subhash

. A new perspective on yarn unevenness: components and determinants of general unevenness. Text Res J 1990; 1: 1–6.

10.

Rao

. A mathematical model for the ideal sliver and its applications to the theory of roller drafting. J Text Inst 1961; 52: 570–600.

11.

Qiu

. Mathematical modeling of ideal sliver. J China Text Univ 1999; 3: 6–9.

12.

Zeidman

Batra

Sasser

. Determining short fiber content in cotton. Text Res J 1991; 61: 21–30.

13.

Cui

Calamari

Suh

. Theoretical and practical aspects of fiber length comparisons of various cottons. Text Res J 1998; 68: 467–472.

14.

Yao

. Textile materials, China Textile Press, 2009.

15.

Krifa

. Fiber length distribution in cotton processing: a finite mixture distribution model. Text Res J 2008; 8: 688–698.

16.

Cui

Rodgers

Cai

. Obtaining cotton fiber length distributions from the beard test method. Part 1. Theoretical distributions related to the beard method. Cotton Sci 2009; 4: 265–273.

17.

Kuang

Yang

. Study on the characterization of cotton fiber length distribution. Adv Mater Res 2013; 821: 398–401.

18.

Krifa

. A mixed Weibull model for size reduction of particulate and fibrous materials. Powder Technol 2009; 194: 233–238.

19.

Brown

Nhan

. Statistics for the number of fiber ends in a segment of a random assembly of aligned fibers. Text Res J 1985; 4: 206–210.

Cotton sample	Yarn ID	Yarn linear density (tex)	Yarn unevenness (%)
			Tested	Calculated
			Tested	Martindale’s	Suh’s	This paper
A	1#	9.72	13.84	13.47	12.46	12.83
A	2#	11.66	13.23	12.30	11.37	11.72
B	3#	14.58	12.24	11.09	10.13	10.50
B	4#	18.22	11.43	9.92	9.06	9.39
C	5#	12.96	13.01	11.65	10.62	11.05
	6#	14.58	12.70	10.98	10.01	10.40
	7#	18.22	11.75	9.83	8.96	9.30
D	8#	18.22	12.35	9.77	8.84	9.24
	9#	19.44	12.02	9.46	8.55	8.96
	10#	27.77	10.45	7.92	7.16	7.47