# Numerical Simulation on the Acoustic Streaming Driven Mixing in Ultrasonic Plasticizing of Thermoplastic Polymers

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Simulation

#### 2.1. Mathematical Modeling

#### 2.1.1. Thermodynamic Equations

**T**and pressure

**p**. In the first law of thermodynamics, entropy per unit mass

**s**and density per unit mass

**ρ**are independent variables. The internal energy per unit mass

**ε**is calculated using Equation (1).

**dε**and

**dT**and

**dp**can be established by the standard Legendre transformation, as shown in Equation (2).

**c**is the isobaric heat capacity per unit mass;

_{p}**α**is the isobaric thermal expansion coefficient; and

_{p}**k**is the isothermal compression coefficient.

_{T}#### 2.1.2. Constitutive Equation

**μ**is shear rate dependent, exhibiting Newtonian fluid behavior at low shear rates and power-law fluid behavior at high shear rates. The viscosity model is shown in Equation (3).

**μ**,

_{0}**μ**,

_{inf}**λ**, and

**n**are material coefficients. $\dot{\gamma}$ is the shear rate, 1/s;

**μ**is the viscosity at zero shear rate, Pa s;

_{0}**μ**is the viscosity at infinite shear rate, Pa s;

_{inf}**λ**is the relaxation time, s; and

**n**is the power law index.

#### 2.1.3. First Order Thermo-Viscosonic Equation

**g**can be expressed as

**g = g**, where

_{0}+ g_{1}**g**is the value of the zero-order state and

_{0}**g**is the acoustic perturbation [25,26]. If the acoustic perturbation

_{1}**g**oscillates at the angular frequency

_{1}**ω**of the acoustic excitation, the field

**g**can be represented as Equations (4) and (5).

**1**is the unit tensor.

#### 2.1.4. Second Order Thermo-Viscosonic Equation

**g**can be expressed as

**g = g**, with

_{0}+ g_{1}+ g_{2}**g**containing the oscillation term and the time constant term. Assuming second-order field time averaging, the second-order thermal viscous acoustic equations for mass, momentum, and energy are represented as Equations (9)–(11).

_{2}**Re**, which can be calculated using Equation (12).

**ρ**,

**μ**are the flow velocity (m/s), density (kg/m

^{3}), and viscosity (Pa s) of the melt, respectively. The characteristic length

**d**(m) of a circular pipe is the diameter. According to the calculation,

**Re**is substantially lower than the transition value of 2300, indicating that the melt flow state in UPMIM process is laminar, which fits the model setting conditions.

#### 2.1.5. Total Force of Fluorescent Particles

**F**generated by the acoustic wave on the particle and the Stokes drag force

^{rad}**F**produced by the acoustic streaming flow. When migrating at velocity

^{drag}**v**

**in a fluid with flow velocity**

_{p}**v**

**, a spherical particle with radius**

_{m}**a**, density

**ρ**, and compression property

_{p}**k**is subjected to the acoustic radiation force

_{p}**F**and Stokes drag force

^{rad}**F**, which may be estimated using Equations (13) and (14).

^{drag}**k**is the melt compressibility and the pre-factors ${f}_{1}$ and ${f}_{2}$ are given by

_{0}#### 2.2. Numerical Modeling

#### 2.3. Calculation Scheme

## 3. Experimentation

#### 3.1. Material Properties

#### 3.2. Methodology

## 4. Results and Discussions

#### 4.1. Acoustic Streaming Characteristics

^{−}

^{4}N near the vortex, as indicated in Figure 4b. The acoustic radiation force is positively associated with the distance from the top end, with a maximum value of 1.52 × 10

^{−}

^{6}N near the bottom end, as demonstrated in Figure 4c. The Stokes drag force is two orders of magnitude of the acoustic radiation force. Hence, the total force is approximately equal to the Stokes drag force.

#### 4.2. Acoustic Streaming Driven Mixing

#### 4.3. Analysis of the Influence Mechanism

^{−5}N to 6.86 × 10

^{−5}N while maintaining the distribution pattern when the ultrasonic amplitude increased from 20 μm to 120 μm, as illustrated in Figure 10.

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 2.**Specimen preparation and characterization. (

**a**) Experimental procedures. (

**b**) Segmentation and characterization of the fluorescence intensity.

**Figure 3.**Melt stream velocity distribution in (

**a**) the plasticizing chamber, (

**b**) radial and (

**c**) axial directions along the dash lines (ultrasonic frequency, 20 kHz; ultrasonic amplitude, 80 μm; melt height, 6 mm; fluorescent particles distribution, uniform).

**Figure 4.**Forces on fluorescent particles (ultrasonic frequency, 20 kHz; ultrasonic amplitude, 80 μm). (

**a**) Seeding the fluorescent particles. (

**b**) Stokes drag force distribution. (

**c**) Acoustic radiation force distribution.

**Figure 5.**Trajectory of fluorescent particles (ultrasonic frequency, 20 kHz; ultrasonic amplitude, 80 μm): (

**a**) 0 s; (

**b**) 1 s; (

**c**) 2 s; (

**d**) 3 s; (

**e**) 4 s; (

**f**) 5 s.

**Figure 6.**Influence of the ultrasonic amplitude on the trajectory of the fluorescent particles in numerical simulation (ultrasonic frequency, 20 kHz; the ultrasonic amplitudes are all peak-to-peak values of the sinusoidal wave. These are defined according to the allowable range of the ultrasonic vibration system used in the experiment for comparison): (

**a**) particle seeding; (

**b**) particle tracing.

**Figure 7.**Influence of the ultrasonic amplitude on the mean gray value of the ultrasonic plasticized fluorescent specimens in the experimentation (ultrasonic frequency, 20 kHz; plasticizing pressure, 20 MPa; ultrasonic action time, 6 s; holding pressure, 20 MPa; holding time, 6 s).

**Figure 8.**Melt stream velocity distribution under various ultrasonic amplitudes (ultrasonic frequency, 20 kHz): (

**a**) 20 μm; (

**b**) 40 μm; (

**c**) 60 μm; (

**d**) 80 μm; (

**e**) 100 μm; (

**f**) 120 μm.

**Figure 9.**Influence of ultrasonic amplitude on the melt stream velocity in (

**a**) radial and (

**b**) axial directions.

**Figure 10.**Total force on fluorescent particles at various amplitudes (ultrasonic frequency, 20 kHz): (

**a**) 20 μm; (

**b**) 40 μm; (

**c**) 60 μm; (

**d**) 80 μm; (

**e**) 100 μm; (

**f**) 120 μm.

**Figure 11.**Trajectory of fluorescent particles at various ultrasonic amplitudes (ultrasonic frequency, 20 kHz): (

**a**) 20 μm; (

**b**) 40 μm; (

**c**) 60 μm; (

**d**) 80 μm; (

**e**) 100 μm; (

**f**) 120 μm.

Density | Acoustic Velocity | Melt Point | * μ_{0} | ** μ_{inf} | Power Index | Particle Size |
---|---|---|---|---|---|---|

0.9 g/cm^{3} | 1623 m/s | 170 °C | 2000 pa s [27] | 10 pa s [28] | 0.38 | 200 mesh |

_{0}is the viscosity at zero shear rate. ** μ

_{inf}is the viscosity at infinite shear rate.

Formula | Density | Particle Size | Excitation Peaks | Emission Peak |
---|---|---|---|---|

BaMg_{2}Al_{16}O27:Eu^{2+} | 5.1 g/cm^{3} | 200 mesh | 395 nm | 450 nm |

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**MDPI and ACS Style**

Wu, W.; Zou, Y.; Wei, G.; Jiang, B.
Numerical Simulation on the Acoustic Streaming Driven Mixing in Ultrasonic Plasticizing of Thermoplastic Polymers. *Polymers* **2022**, *14*, 1073.
https://doi.org/10.3390/polym14061073

**AMA Style**

Wu W, Zou Y, Wei G, Jiang B.
Numerical Simulation on the Acoustic Streaming Driven Mixing in Ultrasonic Plasticizing of Thermoplastic Polymers. *Polymers*. 2022; 14(6):1073.
https://doi.org/10.3390/polym14061073

**Chicago/Turabian Style**

Wu, Wangqing, Yang Zou, Guomeng Wei, and Bingyan Jiang.
2022. "Numerical Simulation on the Acoustic Streaming Driven Mixing in Ultrasonic Plasticizing of Thermoplastic Polymers" *Polymers* 14, no. 6: 1073.
https://doi.org/10.3390/polym14061073