Modeling of yarn strength and its influencing factors in the pneumatic splicing process

Abstract

Yarn splicing strength is one of the most important indexes to evaluate yarn twist quality, and it determines final performance of the yarn. This paper establishes a prediction model of yarn splicing strength. The strength prediction model was used to predict the mechanical properties of yarn composed of different fibers during splicing. The effects of fiber properties, splicing and tensile strain on the splicing strength of yarn were described by using the characteristic parameters of the fiber and yarn. Finally, the interaction between parameters is analyzed by charts, and the optimal solution is obtained. The results show that the yarn splicing strength prediction model established by the mechanical analysis and geometric structure can reflect the mechanical state of the yarn in the pneumatic splicing chamber, and can be used to calculate the yarn splicing strength. Yarns with higher tensile modulus will obtain higher splicing strength under the action of lower twist and higher tensile stress. Therefore, the strength prediction model provides a theoretical basis for the study of the relationship among yarn strength and fiber properties, splicing, tensile strain and other parameters, and also provides guidance for the optimization of the modern spinning process. Furthermore, the influence of air pressure on the splicing strength is also discussed.

Keywords

Yarn splicing strength strength prediction model mechanical analysis pneumatic splicing

The splicing strength of yarn is one of the most important indexes of yarn quality, and it directly affects the spinning efficiency, the subsequent process of the yarn and even the final performance of the textiles. The splicing strength can be affected by many factors, such as fiber properties, yarn structure parameters and process parameters.¹ At present, the splicing strength is almost always tested after the pneumatic splicing process. For new fiber materials, the splicing strength of the yarn mainly relies on workers’ experience and the splice equipment. In addition, the test of the splicing strength is destructive, and it is necessary to stop the production process, which will affect the production efficiency of the enterprise and thus reduce the market competitiveness of enterprises. Therefore, it is significant to find a fast and accurate method to predict the yarn splicing strength based on the fiber properties, yarn structure and yarn process parameters.

The research method of the yarn splicing strength mainly include the finite element method (FEM), experimental method, theoretical method or combination of these methods. With the development of commercial finite element software, more and more researchers have investigated the yarn splice strength by the FEM.

Osman et al.² investigated the relationship among the splicing strength, the air inlet pressure and the cross-section area of the splicing chamber through ANSYS software. Results showed that the splicing strength increases with the increase of inlet pressure first and then decreases. In addition, increasing the cross-section area of the splicing chamber can increase the splicing strength. However, when the number of yarns is too large, the cross-section area does not affect the splicing strength any longer. Wang et al.³ developed a three-dimensional (3D) model of the splicing chamber with different cross-section shapes, and used FLUENT to simulate fluid flow in the splicing chamber. Results showed that the splicing quality of a circular cavity is better than that of a square or hexagon. Chang et al.⁴ established a splicing model using Pro/E, and simulated the splicing process using ANSYS. Results showed that the speed and the pressure in the splicing chamber increase with the increase of inlet pressure. A better splice joint can be obtained under the optimal parameters condition.

Although the FEM can provide a direct expression of the splicing process, its accuracy relies on the meshing of the model, while refining the meshing would be time-consuming. Some researchers investigated the splicing strength by the experimental method.

Das et al.⁵ conducted tensile tests for four different yarns (ring type, rotor type, friction type and air jet type). It is found that a higher inlet pressure increases the strength of the yarn splicing part, while the splicing time has little effect on the strength. Through experimental design and principal component analysis, Hassen et al.⁶ found that the effects of the yarn linear density and air flow duration on the twisting strength are more significant than that of the splice length. High-quality joints can be obtained under the optimal splicing time. In addition, different from traditional yarns, the splicing length of cotton/elastomer yarns has no significant effect on the splicing strength, due to their special structure. Moqeet et al.⁷ used MINITAB software to design splicing experiments. Results showed that a longer splicing length requires less splicing time, while a shorter splicing length requires more splicing time. In addition, the joint appearance will be better with a longer splicing length. Better yarn splicing strength, elongation and appearance can be achieved by optimizing the above due to their complex interactions. Berlin and Thangamani⁸ investigated yarn splicing strength by the experimental method. The MINITAB statistical results showed that the yarn splicing strength increased with the increase of yarn count. A higher compressed air pressure and shorter splicing time can greatly improve the yarn splice strength by increasing the overlap length, and this will not affect the appearance of the splicing.

From the above literature review, the experimental method is direct, but it is time and labor consuming, and it is not suitable for new fiber material exploration. Some scholars studied the yarn splicing strength by the theoretical method.

Gegauff⁹ developed the ideal cylinder model, assuming that yarn is composed of coaxial spirals, and he derived a simple mathematical relationship between the yarn strength and twist angle. Schwarz¹⁰ was one of the early scholars who investigated the yarn cross-section structure. He proposed two ideal stacking methods of circular fibers in yarns. One is the open-packing method, where the fibers are located between continuous concentric circles. The other is the close-packing method, where fibers are distributed in a hexagonal pattern. Platt¹¹ proposed an equation to describe the relationship between yarn strain and fiber strain. Also, the inhomogeneity and variability of fiber stress–strain properties are considered in the analysis of yarn strength.¹² Hearle¹³ modified the cylindrical geometric structure and proposed a yarn model with ideal coaxial helix geometry. The shape of the cross-section is circular. The number of fiber filaments contained in a unit area is constant, that is, the density of fiber stacking is constant. He also visualized the model. Results showed that the mechanical properties of yarn are affected by the twist, migration and other factors. Zhan et al.¹⁴ calculated the yarn strength by simulating the random arrangement of fibers in the yarn. According to the calculated strength value, the slip or break of the yarn is recorded. It is found that the yarn strength can be improved by a low twist. Under a high multiple twist, the increase of the initial fiber tension and fiber inclination angle lead to the decrease of yarn twisting strength.

With the rapid development of computer technology, more and more scholars have adopted advanced algorithms to predict the yarn twisting strength. Özkan et al.¹⁵ used the artificial neural network model and linear regression model to predict the splicing strength of yarns. Results showed that the artificial neural network model is more accurate than the linear regression model. The splicing strength is positively related to the splicing speed and the yarn number. Dong and Yu¹⁶ put forward a prediction model of worsted yarn quality based on a back propagation (BP) neural network. By establishing a BP neural network model, the yarn unevenness value (CV), breaking strength (BS) and ends down (ED) in splicing can be predicted. Compared with the multiple linear regression method, the prediction accuracy and convergence speed of the BP neural network are more reliable. Majumdar et al.^17,18 applied the adaptive network-based fuzzy inference system (ANFIS) to predict the elongation at break and the yarn unevenness value of spun yarn. Results showed that the prediction accuracy of the fuzzy neural model is better than that of the artificial neural network and linear regression model. Using fuzzy theory rules, they found that the longer the average fiber length, the less the staple fiber content, and the lower the yarn unevenness will be. In addition, it is also found that the effect of short fiber content on yarn unevenness is more significant than that of average fiber length. Nurwaha and Wang¹⁹ applied the ANFIS to the prediction of spinning strength, and compared it with the multiple linear regression model. Results showed that the prediction accuracy of the ANFIS is better than that of the multiple linear regression model. Zhang et al.²⁰ proposed a neural network combining an expert weighted neural network (EWNN) and particle swarm optimization (PSO) model to improve the application of soft computing technology in yarn strength prediction. Comparisons were made between the different models. It is concluded that both neural networks can be used to predict yarn strength.

From the above introduction, it is found that the research on yarn splicing strength has mainly focused on experimental investigation and FEM analysis. Meanwhile, the experimental method is limited by the laboratory experimental condition, and it is not suitable for the extreme conditions and new material exploration. In addition, it is also time-consuming and laborious. The FEM only simulates the flow field in the splicing chamber, but not yarn movement. While the FEM prediction accuracy relies upon the meshing of the model, finer meshing usually costs much more time.

This study established a mathematical model to predict the yarn splicing strength through the yarn material properties, fiber structure and splicing process parameters. The proposed prediction model is verified by the experimental data. Based on the validated model, the relationship between splicing strength and influencing factors is discussed in detail. Furthermore, the influence of different air pressures on the splicing strength is also discussed. This research will support the process planning of yarn splicing strength in the industry.

Model establishment

Principle of pneumatic splicing

As shown in Figure 1, the pneumatic splicer includes a splicing chamber, scissors, yarn guides and clamp levers. The yarn splicing process can be broken down into the following steps:

Yarn importing: the two raw yarns to be spliced are crossed into the splicing chamber, where the two ends of the yarn are fixed on the scissors and the clamp levers, respectively. One end of the yarn is imported by the top tube yarn, and the other end is imported by the bottom tube yarn.

Yarn clamping: the two yarn ends will be clamped and positioned by the gripper.

Yarn cutting: two yarns are pretreated according to the specified length. Since the end of the yarn held on the scissors will be cut off, the length of the yarn placed on the scissors usually is shorter, while the length of the yarn placed on the clamp is longer during the yarn importing process.

Yarn length adjustment: when the yarn guide rotates downward, the yarn in the splicing device is pulled out to the required length.

Yarn untwisting: when the gas enters the splicing room, the compressed air pushes the plunger down. The oscillator moves up and down under the action of the air flow. The fibers of the yarn are loosened by the beating of the oscillators.

Yarn twisting: the twisting process is followed by the untwisting process. The plunger goes down, and the untwisting air channel is closed, while the twisting air channel opens. The air flow is slow in the cylindrical inlet passage. However, when the air flow arrives in the accelerating passage, it becomes fast rapidly due to the abrupt change of the geometric dimension. The rotating air flow is generated by the collision of the gas with the wall. Two yarns are twisted under this condition.

Action reset: the splicing process is completed, and each mechanism returns to the initial state to prepare for the next splicing.

Figure 2 shows the 3D model of the splicing chamber. The structure of the splicing chamber is the same as that of Zhou et al.²¹ It is a symmetrical structure, which consists of an inlet channel, two acceleration channels, two outlet channels and a rotary channel. When the compressed air enters the splicing chamber from the air source, the tail yarn at both ends is spun quickly and twisted into a new twist joint. The diagram also includes the flow direction of compressed air.

Model assumption

In order to establish a theoretical connection, it is necessary to introduce basic assumptions. The yarn is considered as a homogeneous, isotropic geometric entity. Based on this principle, some assumptions are made as follows:

The yarn performance is evenly distributed in the length direction.

The fibers in the yarn have the same properties and the same size, and the cross-section is circular.

The fiber center line is a regular coaxial helix in the yarn.

Fibers are rotated and symmetrically distributed in the sectional view.

The yarn diameter is much larger than the fiber diameter.

Splicing, stretching and external force do not change the overall volume density of the yarn.

During the splicing process, the yarn is considered as a whole and the stress and the strain of the yarn follow Hooke’s law, while the interaction force of the fibers in the yarn is neglected in the model.

Modeling of yarn splicing strength

The key to the yarn splicing strength model is to determine the number, position and arrangement of fibers in the yarn and the change of the mechanical state of each fiber layer during the splicing process. The change of mechanical state of the fiber includes the effect of air splicing on the pre-strain of the fiber and the strain of each layer of the fiber induced by tensile action.

The research object is the air-twisted yarn, which can be regarded as a continuum, and will satisfy the theory of solid mechanics under the action of external forces. Therefore, the external force F can be predicted according to the cross-sectional area $A_{y}$ , tensile modulus $E_{y}$ and axial strain $ε_{y}$

F = A_{y} E_{y} ε_{y}

(1)

In this paper, it is assumed that the fiber is perfectly elastic, and is unable to withstand any stress. The fiber has an extremely low elastic modulus and extremely high fiber length-to-diameter ratio, which means that the shear force $v$ , bending moment $m$ and torque $t$ are basically zero. Under the tension of the yarn, the fiber only produces tensile force $p_{r}$ , which becomes the only stress acting on the fiber. Since the acting direction of $p_{r}$ is parallel to the fiber axis, the total load $P$ can be expressed as

P = \sum p_{r} \cos θ_{r}

(2)

Equation (2) represents the sum of cosine values of axial tension of each fiber, and the total resultant force can be expressed as the load generated by the axial stretching of the yarn.

If the tensile modulus of the fiber is $E_{r}$ and the total strain of the yarn is $ε_{r}$ , the yarn splicing strength $F$ model can be expressed as

F = \sum A E_{r} ε_{r} \cos θ_{r}

(3)

where A represents the area of the cross-section of the yarn and

θ_{r}

represents the angle between a single fiber and the yarn axis.

Hearle¹³ defined the ratio of the volume of a single fiber in the yarn to the volume of the whole yarn as the fiber-volume fraction $V_{f}$ . The relationship between the fiber-volume fraction $V_{f}$ and the twist T is as follows²²

V_{f} = 0.7 \times (1 - 0.78 \times ⅇ^{- 0.0495 T})

(4)

When the yarn is stretched, the fibers in the yarn are affected by the friction and adhesion of the surrounding fibers. The adhesion factor n refers to the amount of friction and adhesion applied to a single fiber.²² The greater the twist, the greater the adhesion factor

n = 0.0003355 \times s \times ⅇ^{- 91.94 ∕ T}

(5)

where

s

is a dimensionless parameter representing the ratio of fiber length to diameter.

When the yarn breaks, some of the fibers slip off. The ratio of the number of slippage fibers to the number of fibers out of the fracture is called the slippage rate $λ$ . The relationship among the slippage rate $λ$ , adhesion factor n and fiber length-to-diameter ratio $s$ is expressed by²²

λ = \frac{266.8}{s} \times ⅇ^{49.21 ∕ T}

(6)

By studying the stress of any single fiber, it is found that the effective fiber length coefficient $η$ in the yarn was related to friction factor $μ$ , adhesion factor n, fiber length-to-diameter ratio $s$ and slippage rate $λ$ , which can be expressed by Equation (7)²²

η = 1 - λ + \frac{\tanh [n s μ (1 - λ)]}{2 n s μ} \times \frac{[{(n s μ λ)}^{2} - 2]}{[1 + (1 - λ) {(n s μ λ)}^{2}]}

(7)

According to the continuum mechanics theory, the analysis is extended to the whole yarn structure mechanics analysis by force analysis of any fiber. When the tensile modulus of the fiber is $E_{r}$ , the tensile modulus of the yarn can be calculated as²³

E_{f} = \frac{3 v_{f} E_{r}}{4} \times η \times \frac{{(1 + \cos θ)}^{2}}{1 + \cos θ + \cos^{2} θ}

(8)

where

θ

is the twist angle of yarn. The yarn strength can be expressed as

F = \sum A E_{f} ε_{r} \cos θ_{r}

(9)

Yarn structures are arranged in the form of coaxial helices around each layer on the yarn shaft, where the spiral radius and spiral angle of the fibers in the same layer are the same. According to the fiber distribution of the cross-section, it can be divided into two cases: open-packing and close-packing.²⁴ For open-packing, the axis is surrounded by layers of concentric circles of the fiber layer. For close-packing, each fiber contacts closely, forming an arrangement with a hexagon as the outer contour line, as shown in Figure 3.

The close-packing method is adopted in this paper, and its configuration is shown in Table 1. The maximum number of fiber roots $n_{i}$ in each layer of the yarn body is determined by Equation (10). It is assumed that the path of the monofilament in the yarn is a coaxial spiral; its inclination angle $θ_{i}$ relative to the yarn shaft can be expressed by Equation (11)

n_{i} = 6 \cdot (ⅈ - 1) (ⅈ \geq 2)

(10)

\tan θ_{i} = \frac{6 \cdot [2 * (i - 1) + \frac{2 \sqrt{3}}{3}] \cdot r_{m}}{h}

(11)

h = \frac{100}{T_{tex}}

(12)

Table 1.

Configuration of fibers (close-packing)

Layer numbers ( $i$ )	Inclination angle of the layer ( $\tan θ_{i}$ )	Maximum number of fibers ( $n_{i}$ )
1	0	1
2	$\frac{6 \cdot [2 + \frac{2 \sqrt{3}}{3}] \cdot r_{m}}{h}$	6
3	$\frac{6 \cdot [4 + \frac{2 \sqrt{3}}{3}] \cdot r_{m}}{h}$	12
4	$\frac{6 \cdot [6 + \frac{2 \sqrt{3}}{3}] \cdot r_{m}}{h}$	18
:	:	:
$i$	$\frac{6 \cdot [2 * (i - 1) + \frac{2 \sqrt{3}}{3}] \cdot r_{m}}{h}$	$n_{i} = 6 \cdot (ⅈ - 1)$

In the table, $r_{m}$ is the radius of a single fiber and $h$ is the length of a spiral cycle of the fiber (pitch).

The number and position of fibers in the yarn are assumed to be packed in a close manner. A total of $ⅈ$ layers can be arranged on the cross-section, and the inclination angle of the fibers $θ_{i}$ can be determined by Table 1. The tensile modulus of the fiber is $E_{r}$ , and its force on the yarn axis is the cosine value of its own force under external load. Then the total yarn strength $F$ after splicing can be obtained by

F = 6 \sum_{1}^{ⅈ} (ⅈ - 1) \cdot A_{r} E_{f} ε_{r} \cos θ_{r} (ⅈ \geq 2)

(13)

where

ε_{r}

represents the fiber strain on any layer and

A_{r}

represents the area of the cross-section of the yarn.

The major effect of twist on the yarn structure and strength is mainly reflected in the yarn diameter D, twist angle $θ$ and yarn strain $ε_{r}$ . The mechanical state of the fibers in the yarn will change after the splicing process. The relationship between yarn strain and fiber strain under large strain can be expressed by²¹

ε_{r} = ε_{y} (\cos^{2} θ_{i} - v \sin^{2} θ_{i})

(14)

where the Poisson’s ratio

v

represents the ratio of radial strain to axial strain of the yarn. According to the principle of constant volume of yarn before and after stretching, the Poisson’s ratio

v

can be expressed by Equation (15)

v = \frac{1 - \frac{1}{\sqrt{1 + ε_{y}}}}{ε_{y}}

(15)

Combining Equations (13)–(15), the final yarn splicing strength can be calculated by

F = 6 \sum_{1}^{ⅈ} (ⅈ - 1) \cdot A_{r} E_{f} \cos θ_{i} {\cdot ε}_{y} (\cos^{2} θ_{i} - \frac{1 - \frac{1}{\sqrt{1 + ε_{y}}}}{ε_{y}} \sin^{2} θ_{i}) (ⅈ \geq 2)

(16)

Results and discussion

Comparison of the yarn splicing strength

From the discussion in the second section, it is found that the yarn splicing strength can be predicted by the splicing angle $θ_{i}$ and the yarn strain ( $ε_{r}$ ). The property of the yarn is the main factor affecting the strength of the twisted yarn. The fineness and modulus of the same material will result in better strength performance. It is assumed that the splicing process does not change the yarn density, and a higher splicing angle will lead to a larger radial size and a smaller tensile modulus. The experimental data in Zhang and Yao²⁵ are adopted for model validation. The measured yarn strength is from Jiangsu Xinguang Textile Co., Ltd in China. Twenty kinds of cotton ring spinning yarns with different counts and twists were measured by the electronic yarn strength meter.

It is assumed that the cross-sections of the yarn and the fiber are circular; the cross-section area of the yarn is denoted as $ρ_{y}$ and the cross-section area of the fiber is denoted as $s_{y}$ . The yarn special number is denoted as $N_{y}$ , the fiber specific number is denoted as $ρ_{f}$ , yarn density is denoted as $S_{f}$ and fiber density is denoted as $N_{ⅆ f}$ . They follow the following equations

N_{y} = ρ_{y} \times s_{y} \times 1000

(17)

N_{ⅆ f} = ρ_{f} \times S_{f} \times 10, 000

(18)

By studying the number of yarns and twists and combining the equations, Zhang and Yao²⁵ finally obtained the strength prediction model as follows

F = 16.19875 \times \frac{F_{f}}{N_{ⅆ f}} \times \frac{T^{- 0.0287} \times N_{y}^{0.8474}}{(1 - 0.78 \times ⅇ^{- 0.0495 T})}

(19)

where

F_{f}

is the strength of a single fiber, which is about 3–4 cN. The fiber fineness of the cotton was 1.67–2 dtex, and the average value of single fiber strength and the specific number of cotton fibers were taken respectively, that is,

F_{f}

was 3.75 cN and

N_{ⅆ f}

was 1.835 dtex.

The results predicted by this proposed model and Zhang and Yao’s model²⁵ are compared with the experimental data, as shown in Table 2. According to the analysis of the prediction error in Figure 4, the proposed prediction model is improved.

Table 2.

Predicted value and standard value of yarn splicing strength

Layer number	Twist ( $g / m)$	Actual count ( $tex)$	Predict count $(tex)$	Yarn splicing strength ( $cN$ )			Prediction error
Layer number	Twist ( $g / m)$	Actual count ( $tex)$	Predict count $(tex)$	Measured	Proposed model	Zhang and Yao’s model²⁵	Proposed model	Zhang and Yao’s model²⁵
3	132.4	41	37	123	60	129	0.51	0.04
4	121.8	58	61	162	90	219	0.44	0.35
5	102	91	91	238	178	286	0.25	0.2
6	82.1	125	127	291	299	350	0.03	0.2
7	76.8	159	169	400	377	449	0.06	0.12
8	67.6	210	217	497	602	610	0.21	0.23
9	63.8	258	271	578	702	647	0.21	0.12

The calculation and verification results show that the calculated value of yarn twisting strength increases with the increase of yarn number, which is consistent with the change trend of the predicted value of the model established by Zhang and Yao.²⁵ When the number of yarns is between 41 and 91, the error value of Equation (19) is less than that of Equation (16). However, when the number is 125–210, the splicing strength obtained by Equation (16) is more accurate than that obtained by Equation (19).

Therefore, the adjusted prediction model is expressed as follows

F = \{\begin{array}{l} 16.19875 \times \frac{F_{f}}{N_{ⅆ f}} \\ \times \frac{T^{- 0.0287} \times N_{y}^{0.8474}}{(1 - 0.78 \times ⅇ^{- 0.0495 T})} 3 \leq ⅈ \leq 5 \\ 6 \sum_{i = 1}^{j} (ⅈ - 1) \cdot A_{r} E_{f} \cos θ_{i} {\cdot ε}_{y} \\ (\cos^{2} θ_{i} - \frac{1 - \frac{1}{\sqrt{1 + ε_{y}}}}{ε_{y}} \sin^{2} θ_{i}) 6 \leq ⅈ \leq 9 \end{array}

(20)

Effect of yarn properties on splicing strength

In order to simplify the analysis, yarns composed of four different types of polyamide materials were selected as research objects, and their material properties are shown in Table 3.

Table 3.

Properties of materials²⁶

Materials	Tensile modulus $Y ∕ (GPa)$	Density $/ c m^{3}$
PA66	2.9	1.14
PA6	2.5	1.14
PA610	2.0	1.09
PA1010	1.6	1.05

PA: polyamide.

Under the action of tensile force and aerodynamic force, the stress and strain state of each yarn is the same, which is independent of its position. The splicing strength is equal to the product of the cross-sectional area, tensile modulus and tensile strain. Yarn with the same cross-sectional area is taken as an example for simulation analysis, assuming that its axial strain is 0.25 and the twist is 120 g/m. The predicted yarn splicing strengths of four different types of polyamide materials are shown in Figure 5.

According to Equation (8), when the fiber is spliced into the yarn structure, the overall tensile modulus will decrease in a certain proportion. As can be seen from Figure 5, with the same yarn cross-sectional area and material, the greater the overall yarn modulus, the higher the yarn strength will be.

Effect of twist angle on splicing strength

Splicing is the most important factor to enhance the yarn strength, and the twist degree is an objective index to describe the splicing degree. Substitute Equation (11) into Equation (16) to obtain the relationship between twist strength F and twist T. Taking polyamide PA66, density $ρ = 1.14 / c m^{3}$ , cross-sectional area A = 0.05 $m m^{2}$ and strain $ε_{r} = 0.25$ as an example, the variation of yarn splicing strength with the increase of splice was described. The results were calculated and plotted by MATLAB software, as shown in Figure 6.

According to the curve trend of the upper line chart, the splicing strength of yarns decreases with the increase of twist under the same tensile strain. The decrease in strength is due to the degree of inclination of the fiber to the yarn axis. The greater the twist, the greater the inclination angle and the less strength of fiber to the yarn axis, resulting in a decrease in overall yarn strength.

Effect of tensile strain on splicing strength

The tensile strain of yarn consists of two parts: one is pre-strain from the inner layer to the outer layer caused by air splicing; the other part is caused by axial tensile action on each layer of fiber strain. The influence of tensile strain $ε_{r}$ on the yarn strength is analyzed by simulation. The yarn object and calculation process adopted are consistent with the influence of twist on the yarn strength calculated in the Effect of twist angle on splicing strength section. The strain range is extended to 0–0.5, and four twists are taken from low to high as an example. The results are shown in Figure 7.

The simulation results show that the relationship between the tensile strain and the strength of the yarn increases linearly.

From the above discussion, it is found that the yarn splicing strength increases with the increase of the tensile modulus, while the yarn splicing strength decreases with the increases of the twist. The yarn splicing strength is linearly proportional to the tensile strain.

Flow field analysis in the splicing chamber based on FLUENT software

The prediction model of yarn splicing strength is shown in Equation (20), but there is a certain degree of prediction error. The prediction error may be caused by yarn unevenness, hairiness and so on caused by different inlet pressures during air splicing. However, these prediction errors cannot be quantitatively calculated by a mathematical model. The FEM will be used to analyze the influence of different inlet pressures on yarn stitching strength.

Fiber dynamics calculation

Aerodynamic force

The velocity at any point in the air splicing chamber can be decomposed into a tangential component rotating about the $x$ -axis parallel to the $oyz$ plane and an axial component moving along the $x$ direction.

Suppose that the velocity of any node on the fiber is $(v_{x}, v_{y}, v_{z})$ and the absolute velocity of the air flow at this node is $(v_{x}^{'}, v_{y}^{'}, v_{z}^{'})$ , then the relative velocity of the air flow to the node $(v_{r x}, v_{r y}, v_{r z})$ is

{\{v_{r x}, v_{r y}, v_{r z}\}}^{T} = {\{v_{x}^{'}, v_{y}^{'}, v_{z}^{'}\}}^{T} - {\{v_{x}, v_{y}, v_{z}\}}^{T}

(21)

The aerodynamic force $F_{D}$ acting on micro-segment fibers at the node is approximately calculated according to the following flow resistance formula around a cylinder in engineering fluid mechanics

F_{D} = \frac{1}{2} C_{D} ρ v_{r}^{2} A

(22)

where

C_{D}

is the drag coefficient around the flow and is related to the Reynolds number

R_{e}

ρ

is the air flow mass density and

A

is the area of the yarn cross-section.

The Reynolds number $R_{e}$ is defined as

R_{e} = \frac{v_{r} d}{λ}

(23)

where

λ

is the kinematic viscosity of air flow and

d

is the diameter of the fiber.

According to engineering fluid mechanics,²⁷ the relationship between the drag coefficient of flow around the cylinder and the Reynolds number is as shown in Figure 8.

Fluid–structure coupling solution

The free fluid–structure interaction calculation uses the basic theory of computational fluid dynamics and dynamics. First of all, the three conservation equations of fluid dynamics are as follows.

Mass conservation equation

\frac{\partial ρ}{\partial t} + \nabla \cdot (ρ \vec{v}) = 0

(24)

where

ρ

is fluid density and

\vec{v}

is the fluid velocity vector.

Momentum conservation equation

\frac{\partial (ρ \vec{v})}{\partial t} + \nabla \cdot (ρ \vec{v} \otimes \vec{v}) = - Δ p + \nabla \cdot τ + ρ \vec{v} + F

(25)

where

p

is static pressure, τ is the stress tensor and

ρ \vec{v}

and

F

are gravity and external excitation force, respectively. The stress tensor τ can be expressed as follows

τ = μ (\nabla \vec{v} + {(\nabla \vec{v})}^{T} - \frac{2}{3} \nabla \cdot \vec{v} I)

(26)

where

μ

is the viscosity of the fluid and

I

is the unit tensor.

Energy conservation equation:

\frac{\partial (ρ (h + \frac{1}{2} v^{2}))}{\partial t} - \frac{\partial ρ}{\partial t} + Δ \cdot (ρ \vec{v} (h + \frac{l}{2} {\vec{v}}^{2}))

= \nabla \cdot (λ \nabla T) + \nabla \cdot (\vec{v} \cdot τ) + \vec{v} \cdot (ρ \vec{v} + \vec{F}) + s_{E}

(27)

where

s_{E}

is the heat energy term.

The basic equation of dynamics is

[M] \{\ddot{u}\} + [C] \{\dot{u}\} + [k (u)] \{u\} = \{F (t)\}

(28)

where

[M]

is the structural mass matrix,

[C]

is the damping matrix,

[k]

is the stiffness matrix,

[F]

is the applied force vector,

\{\ddot{u}\}

is the node acceleration vector,

\{\dot{u}\}

is the node velocity vector and

\{u\}

is the node displacement vector.

By coupling the fluid dynamics equation with the nonlinear dynamics equation, the dynamics analysis of yarns in the splicing chamber channel can be carried out. Due to the length of the yarn and the narrow splicing chamber channel, the influence and interference of the yarn on the flow field cannot be ignored. Therefore, the method of direct coupling is adopted, and its continuity equation is as follows

\frac{\partial v}{\partial y} + \frac{\partial ω}{\partial z} = 0

(29)

The conservation equation of the yarn part can be derived from Newton's second law

ρ_{s} {\ddot{d}}_{s} = \nabla σ_{s} + f_{s}

(30)

where

ρ_{s}

is the yarn density,

σ_{s}

is the stress tensor,

f_{s}

is the volume force vector and

{\ddot{d}}_{s}

is the yarn acceleration vector.

The direct coupling method is solved by coupling the fluid–solid control equation to the same equation matrix, and the equation is as follows

[\begin{matrix} A_{f f} & A_{f_{s}} \\ A_{s f} & A_{s s} \end{matrix}] [\begin{matrix} Δ X_{f}^{k} \\ Δ X_{s}^{k} \end{matrix}] = [\begin{matrix} B_{f} \\ B_{s} \end{matrix}]

(31)

where

k

represents the iteration time step,

A_{f f}

Δ X_{f}^{k}

and

B_{f}

represent the system matrix to be solved and external force of the flow field respectively, similarly,

A_{s s}

Δ X_{s}^{k}

and

B_{s}

correspond to the terms of the yarn and

A_{f_{s}}

and

A_{s f}

represent the fluid–structure coupling matrices.

Establishment of the mathematical model

In the yarn splicing process, compressed air enters the splicing chamber through the air jet. Because the speed is very high and the direction of the air flow changes rapidly, strong air flow disorder in the high-speed small area is formed in the splicing chamber groove, so the fibers located in the air inlet are blown open and the fibers in other parts of the chamber begin to twine and twist under the action of eddy currents. Thus, the flow of compressed air in an air splicer can be assumed to be turbulent. In the cartesian coordinate system, the k-ε turbulence model is adopted. The k-ε turbulence model includes the standard k-ε turbulence model and the renormalization group (RNG) k-ε turbulence model. The standard k-ε model is not suitable for the calculation of rotational motion due to the swirling flow in the yarn splicing chamber, so the RNG k-ε turbulence model is selected

\frac{\partial}{\partial t} (ρ k) + \frac{\partial}{\partial X_{i}} (ρ k u_{i}) = \frac{\partial}{\partial X_{i}} (a_{k} μ_{eff} \frac{\partial k}{\partial X_{i}}) + G_{k} - ρ ε - Y_{m} + S_{k}

(32)

\frac{\partial}{\partial t} (ρ ε) + \frac{\partial}{\partial x_{i}} (ρ ε u_{i}) = \frac{\partial}{\partial x_{i}} (a_{ε} μ_{ⅇ_{f f}} \frac{\partial ε}{\partial x_{i}}) +

G_{1 ε} \frac{ε}{k} (G_{k} + G_{3 ε} G_{b}) - C_{2 e} ρ \frac{ε^{2}}{k} - R_{ε} + S_{ε}

(33)

where

G_{k}

is the turbulent kinetic energy generated by the average velocity gradient;

G_{b}

is the turbulent kinetic energy generated by buoyancy;

Y_{m} = 2 ρ ε M_{t}^{2}

is the effect of variable expansion on the total dissipation rate in compressible turbulence, where

M_{t} = \sqrt{\frac{k}{a^{2}}}

is the speed of sound;

a_{k}

and

a_{ε}

are the reciprocal of valid Prandtl numbers of

k

and

ε

, respectively;

S_{k}

and

S_{ε}

are custom design reference source terms.

Compared with the standard k-ε model, the main changes of the RNG k-ε model are as follows: (1) by modifying the turbulent viscosity, the rotation and swirl flow in the average flow are considered; (2) a term is added to the $ε$ equation to reflect the mainstream time-mean strain rate $E_{i j}$ , so that the generation term in the RNG k-ε model is not only related to the flow condition, but is also a function of the spatial coordinates in the same problem. Therefore, the RNG k-ε model can better deal with flows with high strain rates and a large degree of streamline bending.

Simulation calculation and data processing

Firstly, the solid model of the splicing chamber established with SolidWorks software was imported into ICEM CFD, the pre-processor of FLUENT, to divide grids and define boundary conditions, as shown in Figure 9. Then the grid files are imported into FLUENT for flow field analysis.

The velocity vector distributions of different slices in the rotating channel are presented in Figure 10. The other sections are symmetric about the central axis section (Z = 0), and the distance from the central axis section (Z = 0) is 4 mm (Z = 4), –4 mm (Z = −4), 6 mm (Z = 6) and –6 mm (Z = −6), respectively. Air forms two opposite flows at Z = 0 under different pressures. As the air flow moves axially, when the air flow reaches Z = 4 and Z = −4, the clockwise air flow gradually intensifies. When the air flow moves to Z = 6 and Z = −6, the strong clockwise air flow will prevail over the weak air flow and play a decisive role.

Three straight lines are taken in the splicing chamber, Line1, Line2, Line3, and the distribution is shown in Figure 11. Line1 is the central axis of the splicing chamber and Line2 and Line3 are symmetrical about Line1. The coordinates of the three lines are shown in Table 4. Then the tangential and axial velocity maps on Line1, Line2, Line3 are displayed, as shown in Figures 12 and 13.

Table 4.

Linear coordinates

Position	Line1	Line2	Line3
Starting coordinates
x-axis	–0.0946675	–0.09375779	–0.09403303
y-axis	–0.003356532	0.6506163	–0.6457875
z-axis	7.6973	7.684017	7.687644
Ending coordinates
x-axis	0.03685751	0.03095486	0.0428323
y-axis	–0.004051402	0.667047	–0.6513934
z-axis	–7.589767	–7.588865	–7.588865

As can be seen from Figure 12, tangential velocity is the maximum at the position corresponding to the inlet hole of the splicing chamber, and the direction is opposite. Tangential velocity at the outlet of the splicing chamber is basically zero. In the center plane of the splicing chamber (y = 0), the tangential velocity basically overlapped, and the maximum tangential velocity was in the center of the z direction of the splicing chamber (z = −0.054). According to the analysis of Figure 13, the axial velocity on the twist chamber axis is basically the same; the axial velocity of the splicing chamber corresponding to the splicing hole is maximum. The tangential velocity and the axial velocity at the symmetrical position in the same splicing chamber are symmetrically distributed along the central axis. Therefore, according to the different twist direction of the yarn, it should be reasonable to arrange the position of the twist hole.

By setting different inlet pressures, the maximum static pressure, maximum tangential velocity and maximum axial velocity of the splicing chamber were compared under different parameters. The simulation results are shown in Table 5.

Table 5.

Maximum static pressure, maximum tangential velocity and maximum axial velocity of the splicing chamber under different inlet pressures

Inlet pressure (MPa)	Maximum static pressure (MPa)	Maximum axial velocity (m/s)	Maximum tangential velocity (m/s)
0.1	8e + 4	6e + 2	1.15e + 3
0.2	2.4e + 5	7.2e + 2	1.5e + 3
0.3	1.3e + 5	1.25e + 3	1.7e + 3
0.4	4e + 5	1.3e + 3	2.25e + 3
0.5	4e + 5	1.6e + 3	2.3e + 3
0.6	6.5e + 5	1.6e + 3	2.6e + 3
0.7	1e + 6	1.7e + 3	2.8e + 3

According to the analysis in Table 5, the pressure, tangential velocity and axial velocity in the splicing zone all increase with the increase of inlet pressure, which indicates that the pressure of the splicing zone has a significant impact on the pressure and speed of the splicing zone. When the inlet pressure is low, it is not easy for the yarn to be inhaled into the splicing chamber and the splicing effect is not good. When the inlet pressure is high, the pressure in the splicing chamber also increases, which will affect the yarn untwisting. These fibers will not loosen easily because of the high pressure. The pressure should be controlled within a reasonable range. Tangential speed reflects the splicing strength: the greater the tangential speed, the greater the splicing strength of the yarn.

Conclusion

The splicing strength of a yarn is determined by the arrangement of the fibers, the material properties of the fibers, the splicing parameters and the force of the compressed air used for the splicing. In this paper, the strength of yarn subjected to air force is analyzed by numerical simulation. Through research and deduction, the following relationships and conclusions are found.

Under the axial tensile load of the yarn, the complex nonlinear relationship among yarn splicing strength F and cross-sectional area $A_{r}$ , tensile modulus $E_{r}$ , axial strain $ε_{r}$ and twist $T$ is obtained, and the prediction model of yarn splicing strength is established. By analyzing the fiber properties, production process parameters and yarn structure parameters, the final model can be used to predict the yarn splicing strength. The yarn splicing strength increases with the decrease of twist and with the increase of tensile stress. That is, yarns with a higher tensile modulus will obtain a higher splicing strength under the action of lower twist, lower twist angle and higher tensile stress.

Through FEM analysis, with the increase of inlet pressure, the tangential velocity and axial velocity of the splicing chamber increase. However, the inlet pressure should be controlled within a reasonable range. Tangential speed reflects the splicing strength of the yarn. The greater the tangential speed, the greater the splicing strength of the yarn.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the Major Technological and Equipment Projects (Grant No. 2102-320905-89-05-514710).

ORCID iD

Xia Ji

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