Fiber motion model based on yarn-forming mechanism in airflow

Abstract

With the development of the textile industry, requirements relating to the quality of spinning of fibers to form yarn are becoming increasingly stringent. This paper analyzes the force of fibers in airflow to develop a drafting model and a twisted structural model of airflow, and numerically simulates the internal flow field of the drafting and twisting structures to obtain the optimal parameters for the model and nozzle. The structure is designed to improve the spinnability of different fibers. A fluid–solid coupling simulation is used to analyze the movement of fibers in the drafting channel, and parameters of the model are analyzed and optimized. The proposed model provides theoretical support for improving the speed and quality of spinning.

Keywords

new spinning air spinning fiber model drafting channel fluid–solid coupling numerical simulation

Traditional textile manufacture involves the use of rollers for drafting. To increase the number of drafting stages, the speed of rotation of the roller should be increased. However, this increase in speed causes the roller to slip more frequently and disperses the transmission point of the fiber, which affects the evenness of the resulting yarn. For the new air jet vortex spinning, at least one of the two ends of the fiber is in the grip state during the twisting process. However, due to the pull-out structure between the holding points at the head and the end, the distance between them is limited, so the adaptation of the two gripping points to the fiber is worse. Tumbler spinning relies on a high-speed rotating tumbler to implement twisting. It can currently reach a speed of 70,000 rpm. To further improve speed, more stringent requirements are imposed on the material of the tumbler, its bearing capacity, and the performance of the entire machine.

Scholars worldwide have studied the above problems in detail. Tyagi et al. studied the effects of structural parameters of the nozzle on the final structure of the yarn in jet swirl spinning.¹ Xing et al. studied the effects of the structural parameters of the second nozzle on yarn quality in jet spinning,² and Zeng et al. simulated the internal flow field in jet spinning by changing the structural parameters of the nozzle to analyze the influence of parameters on the flow field.³ Zou et al. studied the influence of structural parameters of the nozzle on the internal flow field in jet swirl spinning.⁴ Zhang et al. examined the effects of the structural parameters of a cotton feeding pipe on the internal flow field in rotor spinning.⁵ These studies have focused on optimizing only the prevalent structure and have not attended to practical problems in it. Chiba et al. simplified the fiber into an ellipsoid model and studied its motion in jet swirl spinning.⁶ Chattopadhyay et al. studied the tension caused by a rotating spinning cup on the fiber in rotor spinning.^7,8 Eldeeb et al. simulated and analyzed the three-dimensional (3D) flow field of air jet spinning, which is closer to the actual situation than the two-dimensional (2D) case.⁹

The above studies assume a simplified fiber, which is a significant departure from practice. Moreover, research on parallel straightness of the fiber in the spinning process is lacking, where this directly affects the strength and uniformity of the fiber strip.

In view of the above problems, this paper suggests that the entire process of spinning be carried out using airflow, wherein the fiber is drawn and twisted by high-speed jet airflow to avoid the limitations incurred by the roller and high-speed rotating parts of the machine. At the same time, the draft model can improve spinnability by changing the structure of the nozzle. By setting different values of the parameters, the processes of drafting and twisting are simulated and analyzed to obtain the best model to improve the speed of spinning while reducing the time needed for drafting. A numerical simulation of fluid–solid coupling was carried out to obtain the optimal spinning structure to straighten the fiber in airflow and improve the quality of spinning.

Movement of fibers in airflow

Force of fiber in air stream when stretched in parallel

During the movement of the parallel extensor fiber in air stream, relative motion between the fiber and the air stream occurs. The surface of contact during the relative motion produces a force parallel to it, called the viscous force. The movement of fibers in air depends partly on the viscous force on the surface of the fiber F_S and the dynamic pressure F_d at the head of the fiber, as shown in Figure 1.

Figure 1.

The force of the straight, parallel fiber in airflow.

The size of viscous force is related to such properties of the fluid as density, temperature, and velocity, and the viscous force Fs can be expressed as

F_{S} = - 6 π η r ν

(1)

where η is the coefficient of viscosity of air, 1.983 × 10^–5Pa/s, r is the radius of the object (m), and ν is the velocity of the fluid (m/s).

When studying dynamic pressure at the end of the fiber, for convenience of calculation, we assume that the head and end of the fiber are planar, that is, the direction of action of pressure due to airflow is perpendicular to the plane of the head and end of the fiber. Then, the dynamic pressure F_d of the fiber is

F_{d} = σ_{c} \cdot S_{d}

(2)

where σ_c is the impact stress of airflow at the tip of the fiber (Pa) and S_d is the plane area of its head (m²). According to the theory of fluid mechanics

σ_{c} = \frac{1}{2} ρ v^{2}

(3)

where ρ is the density of air (kg/m³) and ν is the rate of airflow (m/s).

Because the viscous force here provides the power for the fiber, we assume it has a positive value. Then, the total force F on the fiber in airflow is

F = 6 π η r ν + \frac{1}{2} ρ ν^{2} S

(4)

Force of non-parallel unstretched fibers in airflow

In the analysis of non-parallel unstretched fibers, each fiber is divided into n segments. Assuming that the mass of each segment is concentrated at the head, the motion of the fiber in air can be represented by the position of the end of the head. Ganser¹⁰ provides the formula of resistance F as

F = \frac{π}{8} ρ d^{2} C_{D} V^{2}

(5)

When the fiber moves in air, neither of its ends is held and the fiber moves in the flow field in a free state. The main force on the fiber is due to the air, and is represented as resistance. The magnitude of the resistance is related to parameters of airflow, the cross-sectional area of the force on the fiber, and the orientation of the fiber.¹⁰ According to the theory of fluid mechanics, the stress on the fiber in air can be expressed as

F_{i} = \frac{π}{8} ρ d^{2} C_{D} (V_{ai} - V_{fi}) | V_{ai} - V_{fi} |

(6)

where ρ is the density of air, V_ai and V_fi are components of the velocity of airflow at the tip of the fiber in section i, d is the equal spherical diameter of the fibers with value

\sqrt[3]{\frac{3}{2} d_{f}^{2} l}

, d_f is the diameter of a section of the fiber, l is the length of the fiber segment, and C_D is the non-dimensional coefficient of resistance of the flow field.

The value of C_D is determined according to the expression for C_D and the Reynolds number R_e as proposed by Hölzer et al.¹¹

\begin{matrix} C_{D} = \frac{8}{R_{e}} \frac{1}{\sqrt{Φ_{⊥}}} + \frac{16}{R_{e}} \frac{1}{\sqrt{Φ}} + \frac{3}{R_{e}} \frac{1}{\sqrt{Φ^{\frac{3}{4}}}} + 0 . 421^{0.4 (- log Φ) 0.2} \frac{1}{Φ_{⊥}} \end{matrix}

(7)

where Φ and

Φ_{⊥}

are sphericity and lateral sphericity, respectively, and can be solved as follows

Φ = \frac{s}{S}

(8)

Φ_{⊥} = \frac{s_{max}}{s_{⊥}}

(9)

where s is the surface area of a sphere of equal volume of each fiber segment, and its value is πd², S is the surface area of each fiber segment, and its value is πd_f l, s_max is the maximum sectional area of the sphere of equal volume of each fiber segment, and its value is, ¼_Πd² and

s_{⊥}

is the effective area of the fiber under the force of airflow.

Figure 2 shows the projection diagram of fiber segments in different morphology in the airflow field.

Figure 2.

Effective area of different fiber segments subjected to airflow (a) the first morphology; (b) the second morphology.

The projected area of the first morphology (Figure 2(a)) is $S_{⊥} = d_{f} l$ and that of the second morphology (Figure 2(b)) is $S_{⊥} = d_{f} l sin α$ .

The formula to calculate the Reynolds number R_e is as follows

R_{e} = \frac{ρ (V_{ai} - V_{fi}) d}{μ}

(10)

where μ is the coefficient of dynamic viscosity of gas.

According to Newton’s second law of motion, the dynamic equation of the ith root of the fiber tip in the flow field can be expressed as

F_{i} = m_{i} \frac{d^{2} r_{i}}{d t^{2}}

(11)

where m_i is the mass of the ith fiber, t is the time of the fiber in the airflow field and r_i is the position of the ith root of the fiber tip in the airflow field.

The solution of the dynamic equation reflects the position of the fiber in the flow field.

Draft model of airflow

Airflow draft model

A satisfactory airflow drafting model needs to satisfy the following conditions:

There is a difference in speed before and after the direction of motion of the whisker, and the corresponding number of drafting stages can be achieved by stretching and drawing the whisker based on this difference.

The point of transmission of the fiber should be concentrated as much as possible to ensure the evenness of the yarn.

Fibers in the whiskers should be as straight as possible before entering the twisting stage, which is conducive to improving the strength of the yarn and reducing hairiness.

To satisfy the above conditions, we establish a draft model as shown in Figure 3. The model is a cylinder as a whole, and the diameter of the internal channel decreases gradually under a certain curve relationship, so as to ensure that the airflow speed increases gradually along the fiber movement direction, so as to achieve the purpose of drafting, that is, V₁<V₂<V₃.

Figure 3.

Model of drafting channel.

The velocity difference between the fiber head and the fiber tail is ensured. Moreover, the stable gradient of the velocity of airflow causes the speed of the fiber to change and stabilize. Because the fiber is always in tension, it is useful for the hook fiber to reach the parallel straightening state during motion.

Because the fiber is light and easy to move through airflow, the velocity of airflow in the drafting process can be considered to approximate to that of the fiber. The theoretical number of drafting stages E can be calculated according to the velocity of airflow V_in at the inlet of the drafting channel and the velocity of airflow V_out at the outlet as

E = \frac{V_{out}}{V_{in}}

(12)

To ensure high spinning speed, the speed of airflow at the outlet of the draft passage should be equal to the speed of drawing of the yarn. V_out should be set to 450 m/s. To satisfy the requirement of a large number of drafting stages, E should be set to 150. V_in should be 3 m/s.

The draft channel uses a circular section. Because the size of the outlet of the pipeline should match that of the inlet of the twisting part of the hollow spindle, the diameter of the outlet of the draft channel in this paper was set to 1 mm. According to the formula for flow calculation in equation (13), the diameter of the inlet of the pipeline was set to 12 mm. As gas flow in the pipeline was caused by negative pressure at its outlet, the negative pressure was set to –0.15 MPa and the inlet pressure was set to atmospheric pressure, 0.1 MPa. According to pressures at the inlet and outlet, the length of the pipeline was calculated to be 50 mm using equation (14).

Q_{in} = Q_{out} = v \times s = c \times Δ P

(13)

c = \frac{3 π d_{1}^{3} d_{2}^{3}}{128 η L (d_{1}^{2} + d_{1} d_{2} + d_{2}^{2})} P

(14)

where Q_in is the inflow of gas (m³/s), Q_out is the amount of gas flowing out (m³/s), v is airflow velocity (m/s), s is the cross-sectional area of the pipeline (m²), c is its conduction (m³/s),

Δ P

is the pressure difference between the ends of the pipe (Pa), d₁ is the diameter of the inlet (m), d₂ is the diameter of the exit (m), η is the coefficient of viscosity of air (1.983 × 10^–5Pa/s), L is the length of the pipe (m), and P is the average pressure of gas in the tube (Pa).

Numerical simulation of flow field

Governing equations

In practical problems of fluid flow, when the flow velocity reaches the sonic level, compressibility in the flow process needs to be considered. In the design and calculation of the draft channel in the previous section, the speed of airflow in the channel can reach that of sound and even exceed it. Therefore, airflow in this paper was assumed to mirror the flow of a compressible fluid. Its basic governing equation is as follows

\frac{\partial (ρ u_{x})}{\partial x} + \frac{\partial (ρ u_{y})}{\partial y} + \frac{\partial (ρ u_{z})}{\partial z} = 0

(15)

\begin{matrix} \nabla (\frac{ρ u_{x}}{u}) # x 0003 D; - \frac{\partial p}{\partial x} # x 0002 B; \frac{\partial τ_{xx}}{\partial x} # x 0002 B; \frac{\partial τ_{yx}}{\partial y} # x 0002 B; \frac{\partial τ_{xz}}{\partial z} # x 0002 B; ρ f_{x} \nabla (\frac{ρ u_{y}}{u}) # x 0003 D; - \frac{\partial p}{\partial y} # x 0002 B; \frac{\partial τ_{xy}}{\partial x} # x 0002 B; \frac{\partial τ_{yy}}{\partial y} # x 0002 B; \frac{\partial τ_{yz}}{\partial z} # x 0002 B; ρ f_{y} \nabla (\frac{ρ u_{z}}{u}) # x 0003 D; - \frac{\partial p}{\partial z} # x 0002 B; \frac{\partial τ_{xz}}{\partial x} # x 0002 B; \frac{\partial τ_{yx}}{\partial y} # x 0002 B; \frac{\partial τ_{zz}}{\partial z} # x 0002 B; ρ f_{z} \end{matrix}

(16)

\nabla (\frac{ρ}{u} T) = \nabla (\frac{k}{c_{p}} grand (T)) + S_{T}

(17)

p = ρ RT

(18)

Equation (15) is the continuity equation, equation (16) is the conservation of momentum equation, equation (17) is the conservation of energy equation, and equation (18) is the ideal gas state equation. ρ is fluid density, u is its velocity, τ is the viscous stress tensor, f is the force due to volume, T is the temperature of the gas, k is its heat transfer coefficient, c_p is specific heat capacity, S_T is the viscous dissipation term, and R is the molar constant of the gas.

Establishing turbulence models

Reynolds proposed that the state of fluid flow can be divided into laminar flow and turbulence, where the state in between is a transition state. The Reynolds number can be used to determine the state of the fluid. The formula to calculate it was obtained through experiments

R_{e} = \frac{ρ vd}{μ}

(19)

where ρ is fluid density, v is its velocity, μ is the viscous stress tensor, d is the characteristic length, and, for cylindrical structures, d is diameter.

Reynolds’ experiment showed that when R_e < 2000, the flow state is laminar in the tubular channel. When R_e is between 2000 and 4000, the flow is in the transition state. At R_e > 4000, the flow state is turbulent. In the model used in this paper, owing to the high velocity of airflow, R_e >4000 was calculated and the airflow was turbulent.

Of the turbulence models, the k-ɛ model is the most commonly used. It contains three further models: the standard k-ɛ model, the renormalization group (RNG) k-ɛ model, and the realizable k-ɛ model. For different problems involving fluids, different turbulence models should be selected for analysis. The RNG k-ɛ model and realizable k-ɛ model are improvements over the standard k-ɛ model, and are often applied to solve more complex problems. As the draft is implemented according to the gradient of velocity of airflow, the twisting process depends on the rotation of airflow. Of the three turbulent transport equations of the model, the one which can realize the k-ɛ model contains parameters related to the gradient of velocity and the rotation rate. Thus, the theoretically realizable k-ɛ model is most suitable for solving the model developed in this paper. Through the analysis of the calculation results, three turbulence models are compared, and the results show that they are consistent with the theoretical predictions. The transport equation of the realizable k-ɛ model is

\begin{matrix} \frac{\partial}{\partial t} (ρ k) + \frac{\partial}{\partial x_{j}} (ρ {ku}_{j}) = \frac{\partial}{\partial x_{j}} [(μ + \frac{μ_{t}}{σ_{k}}) \frac{\partial k}{\partial x_{j}}] + G_{k} \\ + G_{b} - ρ ɛ - Y_{M} \end{matrix}

(20)

\begin{matrix} \frac{\partial}{\partial t} (ρ ɛ) + \frac{\partial}{\partial x_{j}} (ρ ɛ u_{j}) = \frac{\partial}{\partial x_{j}} [(μ + \frac{μ_{t}}{σ_{ɛ}}) \frac{\partial ɛ}{\partial x_{j}}] \\ + ρ C_{1} S ɛ - ρ C_{2} \frac{ɛ^{2}}{k + \sqrt{v ɛ}} \\ + C_{1 ɛ} \frac{ɛ}{k} C_{3 ɛ} G_{b} \end{matrix}

(21)

C_{1} = max [0.43, \frac{η}{η + 5}]

(22)

η = S \frac{k}{ɛ}

(23)

S = \sqrt{2 S_{ij} S_{ij}}

(24)

where G_k is the turbulent kinetic energy due to the gradient of velocity, G_b is the turbulent kinetic energy caused by buoyancy, and Y_M is the contribution of the fluctuating expansion of the compressible fluid to its total rate of dissipation. C₁ _ɛ is 1.44, C₂ is 1.9, σ_k is 1, and σ_ɛ is 1.2.

Turbulent viscosity μ_t is given by

μ_{t} = ρ C_{μ} = ρ C_{μ} \frac{k^{2}}{ɛ}

(25)

where C_μ is related to the rate of rotation and the rotational angular velocity of the system.

Numerical simulation

The numerical simulation of the model of flow field in this paper was carried out using ANSYS Fluent software. The general steps for flow field analysis using Fluent are as follows:

Use Fluent’s built-in geometry to create modules or import geometric models, and obtain the model of the area of calculation of the flow field through the corresponding commands.

Numerically discretize the region in the model for flow field calculation, including grid division, initial conditions, and boundary conditions.

Set the analysis step and start the calculation, including defining the material, selecting the turbulence model, and setting the number of iterations.

Complete the calculation, and display the calculated results in a cloud diagram, flow diagram, and vector diagram of each indicator. Analyze the results.

This section conducts a numerical simulation-based analysis of the flow field, and selects parameters for the optimal model and working conditions. As shown in Figure 4, three main parameters affecting the flow field were extracted and set in this section: the vertical distance d between the exit of the air jet hole and the inlet of the hollow spindle, angle α between the axis of the air jet hole and the horizontal axis, and gas pressure P injected by the air jet hole.

Figure 4.

Parameters affecting the flow field.

d is the vertical distance between the exit of the air jet hole and the inlet of the hollow spindle, α is the angle between the axis of the air jet hole and the horizontal axis, and P is the gas pressure injected by the air jet hole. As shown in Table 1, different parameter values were set and 27 models were generated. The above steps were executed for all models and a numerical simulation was carried out.

Table 1.

Settings of the parameters

Angle α (°)	Distance d (mm)	Gas pressure P (MPa)
60	0	0.5
70	1	0.55
80	2	0.6

The results of calculation of all 27 models were visualized through a post-processing module and reflected in various images for comparative observation and analysis. Figure 5 is a post-processed image of the results of simulations of the models at angle α = 60°, 0 mm from d, and at an injection pressure of 0.6 MPa. Figure 5 shows the distribution cloud diagram of gas pressure inside the model. In the spinning process, only when the twisting chamber is under negative pressure, the airflow at the entrance of the draft passage will flow to the twisting chamber through the draft passage. The airflow is the basis for the fiber to enter the draft passage when it leaves the carding roller, and the fiber will enter the twisting chamber with the airflow after entering the passage. Thus, the analysis of pressure distribution in the cloud should focus on the pressure in the twisting chamber and the range of pressure distribution. Figure 5 shows that pressure in the twisting room of the model was negative but its distribution was not uniform.

Figure 5.

Cloud diagram of pressure distribution.

Figure 6 shows the distribution cloud diagram of gas velocity inside the model. For the drafting channel presented in this paper, we hope that the fiber can reach the full parallel stretch state during the transportation of the draft passage, which depends on the constant acceleration of airflow and the uniform distribution of airflow. The constant acceleration of airflow can ensure that there is a speed difference between the head end and the tail end of the fiber, and the uniform distribution of airflow can make the fiber speed change point centralized. Therefore, when analyzing the velocity distribution cloud diagram, we should focus on the distribution of the velocity gradient in the draft channel. It can be seen from Figure 6 that there is a velocity gradient in the draft passage of the model, and the distribution is relatively uniform, but the gradient distribution range is small; too small a distribution range may lead to the fiber in the draft passage not having enough area for parallel straightening.

Figure 6.

Cloud diagram of velocity distribution.

Figure 7 shows the diagram of airflow, that is, its trajectory. Because the fiber was very light and the velocity of airflow in the model was high, the fiber easily moved in the flow field of the model. Therefore, the trajectory of airflow is important for the movement of the fiber. Backflow or turbulence in the flow process can hinder yarn formation and transport. The turbulence of airflow causes the tension in the yarn to fluctuate such that it can easily break. Figure 7 shows that airflow in the draft passage of the model was smooth, and a strong vortex was formed in the vortex room to twist the yarn. However, a small amount of airflow into the hollow spindle had backflow, which is not conducive to the movement of the yarn in the direction of spinning.

Figure 7.

Cloud diagram of airflow.

The results of the simulation of each model after post-processing were used to obtain a pressure diagram and a velocity diagram. The airflow diagram of each model’s simulation diagram was divided into a total of 27 groups. These groups were analyzed to find the one that best satisfied the requirements of spinning. Owing to the large space occupied by the images, the authors do not enumerate them, but only analyze typical problems reflected in the images.

Some of the responses in a set of simulation diagrams are analyzed. The cloud diagram of pressure distribution in Figure 8 shows that the flow field in some models was uneven. As air flowed from a position with high pressure to one with low pressure, the uneven distribution of pressure led to unsteady airflow and affected the motion of the fiber. As shown in Figure 9, the cloud diagrams of some models featured turbulence in airflow and too weak an eddy current in the vortex chamber. Turbulence in airflow is not conducive to yarn movement and can even break it.

Figure 9.

Cloud diagram of low vortex strength airflow.

Figure 8.

Cloud diagram of non-uniform pressure distribution.

An analysis and comparison of the groups of simulation diagrams is provided in Table 2.

Table 2.

Summary of analysis and comparison

Analysis item	There is a problem	Perfect state	Purpose
Pressure	Distribution is heterogeneous	Vortex chamber has negative pressure and uniform distribution	Smooth separation of the fiber from the carding roller and its entry into the swirl chamber through airflow
Velocity	Small velocity gradient	There is a velocity gradient and a gradient distribution norm	Drafting is carried out and the fibers are sufficiently parallel and straight in the drafting channel
Filament line	Disorder, backflow	The overall streamline is smooth and stable, and the swirl in the swirl chamber is prominent	Ensure that the spinning process is smooth so that the fiber bundles can be fully twisted in the swirl chamber

The models corresponding to the simulation diagrams that did not meet the given requirements were eliminated. The model that best met the design requirements stated in this paper had α = 70°, distance = 0 mm, and air injection pressure = 0.6 MPa. The simulation results are shown in Figures 10 to 12. The structural parameters and those of the working conditions of this group of models were regarded as optimal.

Figure 10.

Cloud diagram of optimal pressure distribution.

Figure 11.

Cloud diagram of optimal velocity distribution.

Figure 12.

Cloud diagram of optimal airflow.

The results of the numerical simulation of the internal flow field of the optimal structural model show that the outlet flow velocity of the draft channel was 430 m/s and the inlet flow velocity of the draft channel was 3 m/s. According to equation (12), the theoretical number of drafting stages ranged from 150 to 200. This was in line with expectations of high-speed spinning and improved number of draft stages.

Numerical simulation of fluid–solid coupling

Theoretical model of fluid–solid coupling

Because the fiber was light and soft, and the speed of airflow was high, the movement of the fiber responded immediately to changes in airflow. Therefore, when studying its movement in airflow, it is necessary to reasonably set the characteristics of the fiber to accurately reflect the movement of the fiber in the calculation process. In research on the motion of the fiber, some scholars have simplified it into easy-to-study models, such as ellipsoid, cylindrical, and chain models. Some researchers have developed 2D models to reflect features of 3D models. However, these models are different from the empirical situation and the results of their simulation are thus limited. In this paper, the 3D model closest to the empirical situation of the fiber was chosen. It could better reflect the flexibility and deformation of the fiber in the simulation process. As shown in Figure 13, the diameter of the fiber was 20 µm and its length was 25 mm. The properties of the fibers were set in the corresponding module for subsequent analysis.

Figure 13.

Fiber model.

The small elongation of the fiber during its movement in airflow can be considered linear elastic within the range of deformation, and its constitutive equation is

\begin{matrix} {\begin{matrix} \begin{matrix} σ_{x} \\ σ_{y} \\ σ_{z} \end{matrix} \\ \begin{matrix} σ_{xy} \\ σ_{yz} \\ σ_{zx} \end{matrix} \end{matrix}} = \frac{E}{(1 + μ) (1 - 2 μ)} [\begin{matrix} 1 - μ & μ & μ & 0 & 0 & 0 \\ μ & 1 - μ & μ & 0 & 0 & 0 \\ μ & μ & 1 - μ & 0 & 0 & 0 \\ 0 & 0 & 0 & \frac{1}{2} (1 - 2 μ) & 0 & 0 \\ 0 & 0 & 0 & 0 & \frac{1}{2} (1 - 2 μ) & 0 \\ 0 & 0 & 0 & 0 & 0 & \frac{1}{2} (1 - 2 μ) \end{matrix}] {\begin{matrix} \begin{matrix} ɛ_{x} \\ ɛ_{y} \\ ɛ_{z} \end{matrix} \\ \begin{matrix} γ_{xy} \\ γ_{yz} \\ γ_{zx} \end{matrix} \end{matrix}} \end{matrix}

(26)

where E is Young’s modulus, μ is Poisson’s ratio, ${\begin{matrix} \begin{matrix} σ_{x} \\ σ_{y} \\ σ_{z} \end{matrix} \\ \begin{matrix} σ_{xy} \\ σ_{yz} \\ σ_{zx} \end{matrix} \end{matrix}}$ is the stress vector, and ${\begin{matrix} \begin{matrix} ɛ_{x} \\ ɛ_{y} \\ ɛ_{z} \end{matrix} \\ \begin{matrix} γ_{xy} \\ γ_{yz} \\ γ_{zx} \end{matrix} \end{matrix}}$ is the strain vector.

The equation of motion of the fiber is

ρ_{s} \frac{\partial^{2} u_{s}}{\partial t^{2}} = \nabla \cdot σ_{ij} + f_{s}^{B} + \sum f_{s}^{i}

(27)

where ρ_s is the fiber density, u_s is its displacement vector; σ_ij is the Cauchy stress tensor, and f_s^B is the volume force acting on the fiber. Here, it is equal to the gravitational force on the fiber. As the fiber is light, it can be assumed to be zero.

\sum f_{s}^{i}

is the force between fibers. As the fiber enters the draft channel as a single fiber, this can be simplified to zero. From this, we can get

ρ_{s} \frac{\partial^{2} u_{s}}{\partial t^{2}} = \nabla \cdot σ_{ij}

(28)

Numerical simulation of fluid–solid coupling

To implement the calculation, the solution platform shown in Figure 14 was built on the ANSYS Workbench. The three modules A, B, and C in the solution platform are, respectively, the geometric modeling module, the model mesh division module, and the flow field analysis module for the draft channel. E is the geometric module of the fiber, and D is the module that shares the draft channel model and flow field analysis data with the fiber model to conduct two-way fluid structure coupling analysis and to view the results.

Figure 14.

Solution platform.

In this paper, 3D fluid–solid coupling analysis of the motion deformation of the fiber in airflow was carried out. The sequence of modeling and boundary conditions are as follows:

The flow field model in the draft channel was established, the fiber model was suppressed in the model, and the flow field model was meshed. The mesh near the fiber model was encrypted and the meshing was started at the same time. In other words, the flow field and the fiber were meshed in real time during the calculation

Boundary conditions were set for the coupling surface of the convection field and the fiber, including the transfer of force and displacement. The boundary conditions for properties of the gas, inlet and outlet conditions, and conditions for the wall were then set, as were such parameters as time step and solution time.

The flow field model was suppressed, the fiber model was meshed, and the boundary conditions of the coupling surface with the flow field in the region of the fiber were set. Following this, boundary conditions for the fiber were set, including its properties, conditions on the wall, time steps, solution time, and the number of iterations.

The purpose of the numerical simulation of fluid–solid coupling while considering the fiber was to observe whether a single fiber could be straightened to be sufficiently parallel during its motion by simulating it in the draft channel. The fiber was found to be fully straightened in parallel during the drafting process.

The fiber model established above was one of straightening form, but considering only the motion of the fiber in the flow field cannot help explain many problems. Many kinds of fiber forms enter the flow field. In this paper, according to the hook in the form of the fiber’s morphology, they can be divided into four kinds: the fiber in straight state, the fiber in small extent front hook state, the fiber in large extent front hook state, the fiber in large extent rear hook state. The fiber in straight state has been modeled in the previous description. As shown in Figure 15, other forms of fibers are modeled. According to theoretical analysis, it is more difficult for the fiber of the front hook to reach a parallel unbent state than that of the back hook. Therefore, if the fiber in large extent rear hook state can reach the state of parallel extension in the process of movement, then the fiber in small extent front hook state must also reach the state of parallel extension. If the analysis results show that the fiber in large extent rear hook state fails to reach the parallel stretching state in the process of movement, this paper aims to supplement the analysis of the fiber in small extent front hook state.

Figure 15.

3D model of different forms of fiber.

When the fiber leaves the carding roller, it may enter the flow field not at the central axis but along the edge of the channel. The different positions of the fiber are also thought to affect its movement and deformation in the flow field. Therefore, in this paper, the numerical simulation of fluid–solid coupling was carried out for flow fields entering from the central axis and the position deviating from it, respectively, to analyze the movement and deformation of fibers entering from different positions with different initial forms.

The results of fibers entering from different positions were taken as a set of diagrams to observe the time taken for their movement in the draft channel and changes in their shapes during it. Figure 16 shows the results of the fiber in the unbent state, Figure 17 shows the results of the fiber with a small degree of the pre-hooked state, Figure 18 shows the results of the fiber with a large degree of the pre-hooked state, and Figure 19 shows the results of the fiber with a large degree of the post-hooked state.

Figure 16.

Fiber in straight state. (a) Enter the flow field from the middmle position and (b) Enter the flow field from the position deviating from the middle.

Figure 17.

Fiber in small extent front hook state. (a) Enter the flow field from the middmle position and (b) Enter the flow field from the position deviating from the middle.

Figure 18.

Fiber in large extent front hook state. (a) Enter the flow field from the middmle position and (b) Enter the flow field from the position deviating from the middle.

Figure 19.

Fiber in large extent rear hook state. (a) Enter the flow field from the middmle position and (b) Enter the flow field from the position deviating from the middle.

The gray spots in Figures 16 to 19 represent the velocity vector of airflow. With a gradual decrease in diameter of the draft passage, the velocity of airflow in the duct increased and the velocity vector became denser. When the fiber entered the denser area, its motion was dominated by the dense velocity vector. Therefore, the velocity vector in the results of the simulation of the fiber’s motion in the later stage was concealed to facilitate the observation of its motion. The calculation was stopped when the fiber touched the outlet of the drafting channel. By analyzing and comparing the above results, the following conclusions can be drawn:

The velocity of airflow in the draft channel increased, and the fiber velocity increased with an increase in airflow velocity.

The time taken for the fibers to move in different states in the draft channel was more or less the same, and did not increase or decrease due to different fiber morphologies.

Fibers entering the flow field in a straight state remained in it during movement because airflow was in an accelerated state, the speed of the fiber head end is always greater than the speed of the fiber tail end, and the fiber was always in a stretched state during movement. Thus, the stretched state remained unchanged.

Regardless of the state of the fiber and the position from which the fiber entered the flow field, the fiber moved toward the axis at the center of the draft channel. This is because the velocity of airflow at the center axis of the channel was greater than that at the edge of the same section. Pressure at the higher velocity of airflow was low, and higher pressure at the edge pushed the fiber toward the center axis.

The fiber in the initial state was in large extent rear hook state and could be straightened to be parallel during movement. This is because the end of the fiber was in the area of higher speed of airflow, whereas the end of the hook was in the area of lower speed of airflow. This means that the fiber was pulled forward by the front of the high-speed airflow, and the tail hook was straightened naturally. Therefore, the small extent rear hook state fiber during motion was able to obtain parallel straightening to a greater extent. This is also consistent with past theoretical reasoning.

Fibers in the form of a small forward hook were straightened in parallel during movement. Theoretical analysis of this paper suggests that the fiber of the front hook is not easy to straighten in parallel because of the action of airflow with higher velocity in the central axis and slight radial fluctuation of airflow at the edge.

The fibers in the form of a large forward hook could not obtain parallel straightening when they moved in the draft channel. Although the front hook was slightly straightened in the area with lower speed of airflow, with the increase in this speed, the hook was no longer straightened and reached the exit quickly. It was assumed that if the draft channel had been long enough, the fiber hook would also have been straightened. Supplementary airflow should also be able to straighten the fiber of the hook over a short distance.

Conclusions

The force of the fiber in the airflow was analyzed, and an airflow drafting model was established. Airflow numerical simulation was used to determine more suitable nozzle structure parameters. By changing the structure of the nozzle, the spinnability of the nozzle to different length fibers was improved. A drafting model was used to help improve the number of drafting stages and spinning speed was improved by increasing the gas flow rate. The spinning process was controlled by airflow, the fiber was straightened to be parallel in air, and the quality of yarn was thus improved.

Footnotes

Declaration of conflicting interests

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

Funding

The authors received no financial support for the research, authorship, and/or publication of this article.

ORCID iD

Xin rong Li

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