**1. Introduction**

Currently, ultrafine powders are widely used in various fields, and the powder separation technique has gradually occupied an important position in industry. The main production equipment for ultrafine powders is the turbo air classifier. The classification performance of the classifier directly affects the e fficiency of powder production. Therefore, many researchers [1–7] have conducted extensive studies on the theoretical analysis, flow field simulation, structural optimization, and other aspects of pneumatic grading equipment, and have made progress by obtaining many valuable results and providing the basis for the optimization of classifiers, performance enhancements, and fine separations. The main factors a ffecting the classification sharpness index and performance during the classification process are the rotor cage speed and the air inlet velocity inside the classifier [6–8]. According to the principle of classification, a material is subjected to inertial centrifugal force and air drag force at the same time during the classification process. Some researchers analyzed the e ffect of the rotor cage rotary speed on the classification sharpness index, using the Fluent software, and obtained a reasonable parameter combination for classification. Gao, Yu, and Liu [9,10] found that increasing the rotor cage rotary speed resulted in a finer product, but the higher speed caused the flow field to become uneven, and increased the classification sharpness index. Through the study of the classifier airflow velocity, Diao et al. [11] found that increasing the air inlet velocity could improve the

classification efficiency. However, the inertial centrifugal forces of small- and large-sized materials are different. The distribution of small-diameter materials in the flow field is relatively uniform, but large-particle-sized materials are easily moved to the outside of the flow field. This leads to a high concentration outside the flow field and reduces the classification performance.

Based on above researches, in order to further study the classification performance, many scholars have found that it is meaningful to conduct in-depth research on the evaluation index of classification performance. Some scholars [12–14] pointed out that the Whiten's efficiency curve equation needs to be revised. Hence, the parameters in the Whiten's equation were correlated with the operating conditions of the air classifier as well as the material characteristics. The fish-hook phenomenon was demonstrated in a circulating-air classifier. Based on the experimental data, a process model was developed to predict the bypass fraction within the classifier [15]. Xing et al. [16] measured and analyzed the vortex swirling between rotor blades, using the particle image velocimetry (PIV) technique. They found changes in the regulation of the classification efficiency and cut-size, and optimized the operating parameters to achieve the minimum cut-size. However, in the actual classification process, the agglomeration and inclusion of fine particles in the coarse particles would cause a decrease of the classification sharpness index. Some researchers [17] demonstrated secondary airflow and found that when the ratio of the secondary airflow to the main airflow was maintained at 0.168, the classification was optimum. Based on the narrow particle size distribution experimental system, the best rotor cage speed difference between two turbo air classifiers was found, and the results showed that with a decreasing rotation speed difference, the productivity of the narrow-level product decreased and the uniformity increased. Nevertheless, many evaluation indexes of classification performance can accurately judge the grading performance, but it is very troublesome in actual production. Therefore, it is especially important to propose an efficient and simple evaluation index [18–22].

Due to limitations, such as the processing cost and other factors, it is difficult for enterprises and research institutes to produce a variety of grading wheels for structural parameters. Therefore, in the production process, one or more optimal processes are obtained through manual adjustment and matching of various process parameters. In the above studies, scholars mainly studied the effect of a single factor (rotation speed, airflow velocity, etc.) on the classification performance; however, research on the influence of various factors on the classification performance is rare. Consequently, this study applied the combination of process parameters as variables in numerical simulations and material experiments to obtain the optimum process parameters of the KFF ('KFF' is a code for a vertical turbo air classifier type) series turbine air classifier. Thus, in the production process, using the different manually controlled process parameters, one or more optimal process parameters can be obtained. In addition, a new evaluation index, the relative classification sharpness index, is proposed. The test results showed that it is the same as other classification performance evaluation indicators. It can be used to determine whether the classification status is good, simple, and easy, and has a certain guiding effect on industrial production.

#### **2. Details of the Calculation Methodology**

## *2.1. Description of the Equipment*

The equipment for the experiment comprised of a KFF series turbo air classifier, high-pressure induced-draft fan, cyclone collector, pulse bag-filter, and electrical control system. The sketch of the KFF series turbo air classifier is shown in Figure 1a. The schematic diagrams of the vertical turbo air classifier with corresponding geometric parameters are shown in Figure 1b,c. The air supply system consisted of induced-draft fans. The induced-draft fan is "pumping" at the end of the turbo air classifier, providing the transport power for the particles. Firstly, the material is sent to the main classifier by the feeding system, and effective classification of the material is achieved by adjusting the rotor cage speed and matching it with a reasonable secondary air inlet velocity. Under the action of centrifugal force, the coarse powder is collected along the wall of the cylinder and the fine powder continues to be classified by the airflow into the next grading machine so that a reasonably super-fine material is collected when it enters the cyclone separator or dust removal system.

**Figure 1.** Diagram of the experiment equipment (**a**) and the 3D view of the geometry (**b**) and dimensions (**c**) of the turbo air classifier.

## *2.2. Particle Cutting Size Calculations*

The force diagram at the edge of the classifier rotor cage after the material particles enter the classifier is shown in Figure 2. According to the theoretical method in the literature [15,16], we know that the particle is mainly influenced by the centrifugal and drag forces when it is delivered at the inlet edge of the rotor cage. In this experiment, the radial velocity of the airflow moving around the rotor cage is denoted as *Vr*, and the rotational speed of the rotor cage generating the centrifugal force is *n*. It was assumed that the tangential and radial velocities in the circumferential and vertical directions are uniform and that the particles are spherical. The mathematical definitions of the forces are given in Equations (1)–(3):

$$m\frac{D\mathbf{V}\_{pr}}{Dt} = F\_c - F\_d\tag{1}$$

$$\mathbf{F}\_d = \frac{1}{2} \mathbf{C}\_D (\mathbf{Re}) \ast \rho\_{air} \ast \pi \ast \left(\frac{d\_p}{2}\right)^2 \ast (\mathbf{V}\_{pr} - \mathbf{V}\_r) \ast \left|\mathbf{V}\_{pr} - \mathbf{V}\_r\right| \tag{2}$$

$$F\_{\mathbb{C}} = m \frac{\mathbf{V}\_{T}^{2}}{r} = \frac{4}{3} \ast \pi \ast \left(\frac{d\_{p}}{2}\right)^{3} \ast \rho\_{p} \ast \frac{\mathbf{V}\_{T}^{2}}{r} \tag{3}$$

**Figure 2.** Particle-influencing forces at the inlet of the rotor cage.

It was assumed that there is no slip between the particle and the air tangential velocity. When the centrifugal and fluid drag forces reach equilibrium on the particle at the outer periphery of the rotor cage, and if the radial velocity of the particle is zero, the size of the particle is called the cut size (*d*50). The *d*50 can be expressed as follows from Equations (1)–(3):

$$d\mathfrak{s}\_0 = \frac{3C\_\mathrm{D}\rho\_{air}\mathbf{V}\_r^2r}{4\mathbf{V}\_\mathrm{T}^2\rho\_p} \tag{4}$$

By testing the classifier inlet air volume, the corresponding airflow radial velocity can be calculated:

$$\mathbf{V}\_{\mathbf{r}} = \frac{\mathbf{Q}}{120\pi rh} \tag{5}$$

Using the known rotor cage speed, the tangential velocity of the particles at the outer edge of the grading wheel can be calculated:

$$\mathbf{V}\_{\rm T} = \frac{2\pi rn}{60} \tag{6}$$

The following equation can be derived from Equations (4)–(6):

$$d\_{50} = \frac{3}{64} \frac{\mathcal{C}\_{\rm D} \rho\_{air} Q^2}{\pi^4 r^3 n^2 h^2 \rho\_p} \tag{7}$$

where:

> *dp*: Particle diameter (μm);

**V***r*: Radial velocity of airflow at the outer cylindrical periphery (*m*/*s*);

**V**T: Tangential velocity of airflow at the outer cylindrical periphery of the rotor cage (*m*/*s*);

**<sup>V</sup>***pr*: Particle radial velocity (m/s);

ρ*air*: Density of airflow (kg/m3);

ρ*p*: Particle density (kg/m3);

*r*: Radius of rotor cage (mm);

*h*: Blade height (mm);

*m*: Mass of particle (kg/m3);

*n*: Rotor cage speed (rpm);

*Q*: Total volumetric flow rate of air (m3/s);

*C*D: Drag coefficient;

Re: Reynolds number; and

*d*50: Cut size of classification (μm).
