The model of rotating spinning is shown in Figure 3. Cartesian coordinate system oxyz is stationary relative to rotating container. This non-inertial coordinated system rotates around the axis oy at angular velocity. The origin o is at the center of rotation. The axis oz coincides with the nozzle axis.

In the process of rotating spinning, the flow solution is subject to pressure, centrifugal force, Coriolis force, viscous force, and gravity. The fluid motion can be regarded as steady motion. The continuity equation and momentum equation in the rotating frame are given as ∇ · U = 0 U · ∇ U = 1 ρ ∇ · P - w × ( w × r ) - 2 w × U (1) where U is relative velocity vector, p is stress tensor, p = −p + T , T is partial stress tensor, p is pressure, w is angular velocity, ρ is density of the solution, r is position vector, w × ( w × r ) is centrifugal force, and 2 w × U is Coriolis force.

Because the solution used in rotating spinning experiment is power-law fluid, the constitutive equation can be written as T = 2 k | I 2 | n − 1 2 D I 2 = 2 t r ( D 2 ) (2) where k is consistency index, n is the rheological index, D is the strain rate tensor, and I ₂ is the invariant of the strain rate tensor.

To obtain analytical solution, we simplify the fluid motion in the nozzle to one-dimensional laminar flow. The fluid motion on plane orz is shown as Figure 4A.

The Coriolis force can be ignored because it is counteracted by the pressure gradient along r axis of spinning solution in the container and nozzle tube. Figure 4B shows the forces acting on the micro-unit obtained from the flow field. Because of the existence of free flow surface in the container, the pressure gradient along z axis can be ignored. Momentum Eq. 1 can be given as follows: ρ w 2 z = ∂ T r z ∂ r (3) where z is the axial position, w is angular velocity, and ρ is density of the solution.

The power-law fluid flows along the axis z direction. The constitutive Eq. 2 is simplified as follows: T r z = k ( d u d r ) n (4) where u is the flow velocity of spinning solution, r is the radial position, and k and n are the rheological indexes.

In addition, the following boundary conditions at the container wall should to be satisfied as z = 0 , u = 0 ; r = D 2 , u = 0 (5) where D is the container diameter.

Substituting Eq. 4 into Eq. 3 and combined with Eq. 5, flow field distribution in the container is deduced as follows: u 1 = n n + 1 ( ρ ω 2 z k ) 1 n ( ( D 2 ) n + 1 n − r n + 1 n ) (6)

The average flow velocity of spinning solution in the container is expressed as u 1 a = n 1 + 3 n [ ρ ω 2 z k ] 1 n ( D 2 ) n + 1 n (7)

As shown in Figure 5A, the container outlet, nozzle inlet, and pipe wall are taken as the control bodies. According to Eq. 7, the average velocity of container outlet V ₁ is calculated as V 1 = n 1 + 3 n [ ρ ω 2 L k ] 1 n ( D 2 ) n + 1 n (8) where L is the distance from the container outlet to the rotation center.

Based on the mass conservation equation of steady flow V ₁ A ₁ = V ₂ A ₂, the average velocity in straight pipe inlet V ₂ can be written as V 2 = n d 2 ( 1 + 3 n ) [ ρ ω 2 L k ] 1 n ( D 2 ) 3 n + 1 n (9)

In the nozzle tube, there is a pressure gradient along the axial direction. Momentum equation can be simplified as ρ w 2 z + ∂ p ∂ z = ∂ T r z ∂ r (10)

The boundary conditions at the nozzle inlet and the wall can hold as z = L + D − d 2 cot α 2 , u 2 a = V 2 ; r = d 2 , u = 0 (11) where α is taper and d is the nozzle diameter.

Substituting Eq. 4 into Eq. 10 and combined with boundary conditions (11), the distribution of flow field in the straight tube of nozzle is obtained as follows: u 2 = n 1 + n [ ρ ω 2 z + c k ] 1 n ( ( d 2 ) n + 1 n − r n + 1 n ) (12)

The average velocity is given as u 2 a = n 1 + 3 n [ ρ ω 2 z + c k ] 1 n ( d 2 ) n + 1 n (13) where c is pressure drop. According to the boundary condition (11), it can be given as follows: c = ∂ p ∂ z = ρ ω 2 [ ( ( D d ) 3 n + 1 − 1 ) L − D − d 2 cot α 2 ] (14)

It is found that the pressure drop is related to the rheological parameters of solution, rotation angular velocity, and nozzle structure.

The nozzle straight tube outlet and nozzle elbow outlet are taken as the control bodies, as shown in Figure 5B, and the average velocity at the nozzle straight tube outlet V ₃ is expressed as follows: V 3 = n 1 + 3 n [ ρ ω 2 k ( ( D d ) 3 n + 1 L + S ) ] 1 n ( d 2 ) n + 1 n (15)

According to the mass conservation of the steady flow, the average velocity at the nozzle outlet V ₄, the Coriolis force F _k, and centrifugal force F _C on the jet are expressed as follows: { V 4 = V 3 ⁡ cos ⁡ θ i + V 3 ⁡ sin ⁡ θ j F K = 2 ω V 3 ( − sin ⁡ θ i + cos ⁡ θ j ) F C = ω 2 ( L + D − d 2 cot α 2 + S + R ⁡ sin ⁡ θ ) i + ω 2 R ( 1 − cos ⁡ θ ) j (16)

Therefore, the power at the nozzle outlet can be written as P = ω 2 n 1 + 3 n [ ρ ω 2 k ( ( D d ) 3 n + 1 L + S ) ] 1 n ( d 2 ) n + 1 n · [ ( L + D − d 2 cot α 2 + S ) cos ⁡ θ + R ⁡ sin ⁡ θ ] (17)

The main parameters of the curved-tube nozzle are θ, S, R, d, and α. Bending angle θ, curvature radius R, and nozzle diameter d are selected as the optimized design variables considering the influence of various parameters on the fluid motion during the spinning process. According to actual spinning conditions, other parameters are set as constants, and the optimization objective function is established as follows: max f ( x ) = max P ( x ) (18)

The design variables of the model can be written as x = { bending angle θ curvature r a d i u s R   nozzle d i a m e t e r d } T (19)

The system parameters of the optimization model are shown in Table 3.

The simplified fitness function can be written as f ( x ) = d − 2 [ ( 40 − d 2 ) cos ⁡ θ + R ⁡ sin ⁡ θ ] (20)

The process of genetic algorithms is shown in Figure 6. The crucial sections of genetic algorithms are fitness function, encoding, and initial population. Design parameters (θ, R, d) are encoded in a particular bit string, namely, “chromosomes.” Each chromosome corresponds to an individual and individuals form populations.

The main operations of genetic algorithms are selection, crossover, and mutation. The value of output power is regarded as the individual adaptability. Selection operation can determine whether chromosomes generate crossover and mutation according to fitness. There are many kinds of selection methods such as roulette, rank, and tournament. In this paper, we choose roulette to process selection operation. Roulette selection operator is expressed as P i = f i ∑ i = 1 n f i (21) where p is the probability that can be selected and f is the value of outlet power.

Crossover operation exchanges the fragments of nozzle structure parameter coding to form the next generation. Mutation changes one or more gene values in the coding of nozzle structural parameters. The strongest individuals are finally retained after several generations of elimination.

It can be found from Figure 7A that structure parameters are non-linear and non-monotonic to the outlet power. The fitness function is theoretical model searching for the best combination of design parameters. It should be verified by numerical simulation and corresponding experiments of rotating spinning to avoid unreliable conclusion.

To explore the best structure of the curved-tube nozzle, the three structure parameters (θ, R, d) are optimized to maximize the objective function Eq. 21 based on the genetic algorithm. After multiple parameter adjustment and iterative operation, the basic setup properties of genetic algorithm are population size of 100 individuals, crossover probability 0.8, and 0.1 mutation rate. In Figure 7B, the calculation process of genetic algorithm can be seen. The maximum value of fitness function has been obtained after about 50 generations of evolution. The best design parameters corresponding to the maximum value of objective function 161.2 are shown in Table 4.

According to the dynamic model of the rotating spinning system, it can be seen that the flow field distribution is related to the bending angle, curvature radius, and nozzle diameter. Therefore, simulation experiments under different combinations of design parameters have been carried out by utilizing the finite-element CFD method.

The three-dimensional motion model of the spinning solution is established as shown in Figure 8. The solid structure such as container wall and nozzle wall can be simplified by the fluid simulation software ICEM.

Nozzle outlet, solution inlet, nozzle wall, and tank wall are the four parts in the simulation model. The container diameter is 10 mm, the overall length is 60 mm, and the nozzle straight tube is 12 mm long. The unstructured grid division method is adopted, and the maximum grid size is 0.6 mm. The boundary layer is divided into four layers meshing as hexahedral with 0.01 mm initial height and 1.1 increase rate.

The boundary conditions of rotating spinning motion model mainly include inlet boundary, outlet boundary, wall, dynamic mesh, and solution rheological parameters. The inlet boundary is velocity inlet, the hydraulic diameter is 6 mm, the outlet boundary is pressure outlet, and the hydraulic diameter is 2 mm.

The dynamic mesh is set as the rotating reference system, the rotating axis is z axis, and the rotating angular velocity is 4,000 rpm. The wall is set to move the wall relative to the grid area rotation speed of 0, and the rotation axis is z axis.

Different simulations of solution motion are established with bending angle within 0–90°, curvature radius within 3∼8 mm, and nozzle diameter within 0.5∼1 mm. Figure 9–Figure 11 show simulation results of rotating spinning under different combinations of design structure parameters.

Figure 9 and Figure 10 show the velocity contours of spinning solution in the nozzle tube and nozzle outlet, respectively. With the decrease of the nozzle diameter, the flow rate of the solution in the nozzle tube increases. This phenomenon shows that the relationship between nozzle diameter and compression effect is inversely proportional; the smaller the nozzle diameter, the better compression effect can be produced, resulting in more rapid flow velocity.

The maximum value of outlet velocity is concentrated at the tube axis when the bending angle of the nozzle is 10.8°. With the increase of bending angle, the flow field distribution gradually deviates from the tube axis and the outlet velocity gradually decreases. When the bending angle is 90°, the flow field in nozzle becomes chaotic and an obvious low-velocity region is produced, which reflects the negative influence of excessive bending angle on the flow field distribution.

Furthermore, the flow field distribution is more uniform when curvature radius is 8 mm. However, larger curvature radius consumes more solution kinetic energy, which leads to the decrease of the outlet velocity.

Figure 11 is a statistical analysis for the outlet velocity distribution along the radius direction. Figure 11A shows a significant linear relationship between the nozzle diameter and the outlet velocity. The larger the velocity, the greater the outlet velocity. Figure 11B and Figure 11C reflect the significant influence of curvature and bending angle on the deviation of flow field distribution in the rotating spinning process. Compared with nozzle diameter, these two parameters cannot increase the outlet velocity. However, the nozzle with the curvature of 8 mm and bending angle of 10.8° can reduce the deviation of velocity distribution at the outlet, make the flow field distribution more uniform, and benefit the stability of jet.

In conclusion, the best combination of structure parameters for curved-tube nozzle is bending angle 10.8°, curvature radius 8 mm, and nozzle diameter 0.5 mm, which is consistent with theoretical optimization results. It can effectively counteract flow field inhomogeneity and greatly improve the outlet velocity.

In the process of rotating spinning, angular velocity, solution rheological characteristics, structural parameters, and other factors will affect the final experimental results. To verify the theoretical optimization results, comparative experiments have been carried out with the same concentration PEO spinning solution and the same rotational speed. The electron microscopy has been applied to study the fiber diameter and morphology of nanofibers prepared by the ordinary straight nozzle and the curved-tube nozzle.

The rotating spinning equipment and the nozzles used in the experiment are shown in Figure 12. The equipment can rotate at high speed by frequency conversion speed regulation, up to 6,000 rpm. We use two kinds of nozzle to prepare nanofibers: one is straight tube with nozzle diameter 0.5 mm; another is curved tube with bending angle 10.8°, curvature radius 8 mm, and nozzle diameter 0.5 mm. The rotating spinning experiment was carried out with 6% polyethylene oxide aqueous solution at the motor speed of 4,000 rpm.

SEM images of PEO nanofibers with different nozzles are shown as Figure 13 and Figure 14. It can be found that the diameter distribution of nanofibers prepared by straight nozzle is relatively dispersive in the range of 1,000∼1,200 nm. Also, the surface quality of nanofibers is poor. In comparison, the diameter of nanofibers prepared by curved-tube nozzle is mostly in the range of 800∼1,000 nm, the diameter distribution of nanofibers is more concentrated, and the morphology of nanofibers is more uniform. In conclusion, the overall quality of nanofibers prepared by curved nozzles has been greatly improved.