*2.2. Description of a Multifactorial Experimental Design*

To achieve the objectives of the study, the method of a multifactorial experimental design was applied. The analysis of the design of the grinder and a review of scientific and technical literature on grain grinding made it possible to identify a number of factors (Table 1) which allow us to most fully describe the process of grinding in the proposed centrifugal–rotary grinder. Below is a brief description of the selected factors (*x*1*–x*7) and how they vary:



**Table 1.** Grinder experimental factors and the levels of their variations.

**Figure 4.** Geometry of the inserts with appropriate dimensions labeled.

**Figure 5.** Geometry of the opening of the separating surface. h1—size between the parallel planes of two adjacent knives.

Barley with 14% moisture content was used as the ground material. To reduce the amount of research, a type 27-2 matrix of the fractional factor experiment was used.

The optimization criteria included the power consumption *y*<sup>1</sup> (kW) and the grinder performance (capacity) *y*<sup>2</sup> (kg/s). The measurement of power consumption was carried out using a Mercury 221 electrical energy meter connected to a PC via USB in CAN/RS-232/RS485 (Figure 6). The indicators were monitored instantly using a K-505 measurement kit.

**Figure 6.** A set of measuring equipment used for the research. 1—personal computer; 2—K-505; 3—Mercury 221 electric energy meter with USB-CAN/RS-232/RS485 adapter; 4—frequency converter Hyundai N700-220HF; 5—circuit breaker

The performance (capacity) of the experimental installation *y*<sup>2</sup> was determined by monitoring the mass of the ground material per unit of time under a steady operating mode.

Zoo-technical requirements for mixed fodder concentrate according to GOST 9268-2015, GOST R 51550-2000, and other guidelines highlight a number of criteria for assessing the quality of the resulting product. However, the criteria directly dependent on the design and operating mode of the grain grinder. Some criteria are the percentage of particles greater than 3 mm *y*<sup>3</sup> after grinding, the grinding coarseness *y*4, and the presence of whole grains in the grinding results *y*5.

#### **3. Results and Discussion**

The results of the sieve analysis reveal that the most critical parameter for the compliance of the finished product with the zoo-technical requirements is the percentage of particles greater than 3 mm, *y*3, which in our results is up to 60%. Screen sizing also shows the complete absence of whole grains *y*<sup>5</sup> in all samples. Calculations of the grain size (grinding coarseness) *y*<sup>4</sup> shows that grains are within the range of 1.5–3.2 mm. However, the use of this indicator as an optimization criterion at this stage is not informative, since the grain size depends largely on the percentage of particles greater than 3 mm. Thus, in this study, the most significant criterion for assessing the quality of the product is the percentage of particles greater than 3 mm, *y*3. The critical particle size in different experiments depends on many factors (type of material, intended use, and subsequent processing) [19,20].

The processing of the results of the multifactorial experiment was carried out using the StatGraphics software package. A multivariate analysis of variance ANOVA (Table 2) and the regression results in Equations (1)–(3) show that the models obtained are statistically significant and describe the current processes with a reliability of at least 95%.

$$\begin{aligned} y\_1 &= 3.89 + 0.12\mathbf{x}\_1 + 0.2\mathbf{x}\_2 - 0.15\mathbf{x}\_3 + 0.3\mathbf{x}\_5 - 1.3\mathbf{x}\_6 + 0.28\mathbf{x}\_1\mathbf{x}\_3 + 0.12\mathbf{x}\_1\mathbf{x}\_5 - 0.28\mathbf{x}\_2\mathbf{x}\_4 + 0.1\\ &+ 0.14\mathbf{x}\_3\mathbf{x}\_6 - 0.18\mathbf{x}\_3\mathbf{x}\_7 + 0.12\mathbf{x}\_4\mathbf{x}\_7 \end{aligned} \tag{1}$$

$$\begin{aligned} \mathbf{y}\_2 &= 0.02\mathbf{1} + 0.004\mathbf{x}\_1 + 0.003\mathbf{x}\_2 - 0.001\mathbf{x}\_3 + 0.002\mathbf{x}\_6 + 0.002\mathbf{x}\_1\mathbf{x}\_2 - 0.001\mathbf{x}\_1\mathbf{x}\_3 + 0.001\mathbf{x}\_2\mathbf{x}\_3 \\ &+ 0.001\mathbf{x}\_2\mathbf{x}\_3 + 0.001\mathbf{x}\_2\mathbf{x}\_4 \end{aligned} \tag{2}$$

$$\begin{cases} y\_3 = 13.1 + 1.71\mathbf{x}\_1 - 3.38\mathbf{x}\_2 - 5.6\mathbf{x}\_3 + 2.4\mathbf{x}\_4 + 5.75\mathbf{x}\_5 + 8.88\mathbf{x}\_6 - 2.6\mathbf{x}\_7 + 3.01\mathbf{x}\_1\mathbf{x}\_4 - 1.6\mathbf{x}\_2\mathbf{x}\_6 \\ -1.68\mathbf{x}\_2\mathbf{x}\_6 - 8.16\mathbf{x}\_3\mathbf{x}\_6 + 3.2\mathbf{x}\_3\mathbf{x}\_7 \end{cases} \tag{3}$$


**Table 2.** Results of the model's analysis of variance (ANOVA).

The search for a compromise solution aimed at reducing the amount of energy consumed *y*<sup>1</sup> and the percentage of particles exceeding 3 mm *y*3, while increasing the performance (capacity) *y*<sup>2</sup> of the centrifugal–rotary grain grinder, was carried out using the StatGraphics software package (Table 3). The optimal values of the factors, obtained for the compromise solution in the selected range of factor variation, are presented in Table 4.

**Table 3.** Optimization criteria for the three factors, calculated using the StatGraphics software package.


**Table 4.** Optimal values of the factors obtained using the StatGraphics software package.


The conducted analysis of the influence of factors on electricity consumption (1), and the performance of the centrifugal–rotary grinder (2), as well as the content of particles larger than 3 mm after grinding (3) reveals the following:

One of the most significant factors influencing the performance of the grinder *y*<sup>2</sup> is the feed of grain *x*<sup>1</sup> (Figure 7d). At the same time, an increase in the grain feed *x*<sup>1</sup> leads to an increase in all indicators *y*1, *y*2, and *y*3. This is a consequence of an increased volume of material being transported by the lower disk of the grinder and an increased speed of transportation through the centrifugal–rotary grinder.

**Figure 7.** Two-dimensional sections of the response surface.

The increased rotation frequency of the lower disk *x*<sup>2</sup> leads to a directly proportional increase in the linear velocity of the work units, *v* = 2π*nR* (n—rotation frequency, R—disc radius), at all stages, and an increase in the centrifugal inertial force. *F* = 8*m*π2*n*2*R* (m—mass, n—rotation frequency, R—disc radius). As a result, the number of contacts of the grain material with the work units of the grinder is increased, while the time of its exposure in the work area is decreased. Thus, with increased values of the factor *x*2, a decrease in the percentage of particles larger than 3 mm is observed with a simultaneous increase in the performance of the centrifugal–rotary grinder *y*2. However, this leads to increased work for grinding and transporting the material, which is reflected in the total power consumption *y*1. Bitra [21] notes that increasing speed affects the effective specific energy of hammers in different ways, depending on the type of raw material. In our experiments, the effective specific energy increases at certain rate and then decreases. Similar dependencies were shown by Moiceanu et al. [6].

Reducing the gap (clearance) of the separating surface *x*<sup>3</sup> naturally reduces the grinder performance *y*<sup>2</sup> and increases the power consumption *y*1. As a result, there is an increase in the time required for the removal of the ground material from the work unit, as well as an increase in the mass of the transported ground material along the separating surface of the grinder. At the same time, reducing the gap of the separating surface *x*<sup>3</sup> does not lead to a logical decrease in the percentage of particles larger than 3 mm *y*3. This phenomenon is explained by the fact that when there is a small gap of the separating surface, the particles of the material are mostly reflected from this "smooth" surface and then move along it until they pass through it. Likewise, a larger gap of the separating surface *h* (Figure 5) has an effect on the third grinding stage, since the knives of the separating surface have a larger approach angle and, as a result, have a lower reflectivity.

Changing the number of knives at the first grinding stage *x*<sup>4</sup> does not have a significant effect on the power consumption *y*<sup>1</sup> and grinder capacity *y*2. An analysis of Equation (3), characterizing the quality of the product obtained, shows that reducing the number of knives from 9 to 3 at the first stage *x*<sup>4</sup> allows for a reduction of the percentage of particles over 3 mm in the finished product *y*<sup>3</sup> to 4.8%.

Reducing the number of knives from 18 to 9 at the second stage *x*<sup>5</sup> contributes to a reduction of particles over 3 mm *y*<sup>3</sup> in grinding by more than 11% and a reduction of the power consumption *y*<sup>1</sup> by 0.6 kW. This is understood as a result of an increase in the speed of transportation of the grain material through the grinder and a decrease in the mass accelerated by the lower disk.

Replacing the "old" knives *x*<sup>6</sup> with "new" ones is the main factor affecting energy consumption, and saves at least 2.6 kW of electricity. This can be explained by the fact that with the use of "old" knives, grinding takes place mainly by impact rather than cutting, resulting in a decrease in the number of particles larger than 3 mm *y*3. At the same time, the performance (capacity) *y*<sup>2</sup> is reduced and the power consumption of the bulk grinder *y*<sup>1</sup> is increased. Kovác et al. [ ˇ 22] noticed that the basic factors affecting the properties of the ground material and the unit energy consumption in milling process are, among others, the knife angle and number of knives.

The presence of a special insert *x*<sup>7</sup> installed in the distribution bowls of the lower disk of the centrifugal–rotary grinder does not have any significant effect on the power consumption *y*<sup>1</sup> and the performance *y*2. However, its installation increases the content of particles larger than 3 mm *y*<sup>3</sup> by 5.2%.

An analysis of the interactions of factors in the regressions, Equations (1)–(3), was carried out using (among others) two-dimensional sections (Figure 7), and shows the following results:

In general, installing a special insert *x*<sup>7</sup> into the distribution bowl of the grinder, in conjunction with the opening of the separating surface *x*3, has negative effects on the percentage of particles larger than 3 mm *y*3, and on the performance of the centrifugal–rotary grinder *y*2. Thus, at a minimum gap (clearance) of the separating surface *x*<sup>3</sup> = −1, when the insert is installed, the percentage of particles larger than 3 mm is more than 25% and without the insert this value is 15%. However, when the gap is *x*<sup>3</sup> = 0.6 (*h* = 30.5 mm), particles larger than 3 mm are not observed in any case. Also, an increase in the gap (clearance) of the separating surface *x*<sup>3</sup> with the insert *x*<sup>7</sup> installed in the distribution bowl leads to a drop in the performance of the centrifugal–rotary grinder by 0.025 kg·s<sup>−</sup>1, whereas without this insert the capacity increases by 0.008 kg−<sup>1</sup> (Figure 7a);

The interaction of the factors *x*<sup>1</sup> and *x*<sup>2</sup> is most significant for regulating the performance of the grinder (*y*2); reducing the feed and rotation frequency of the lower disk of the grinder leads to increased performance (Figure 7b);

The interaction of the factors of the condition of knives *x*<sup>6</sup> and the gap (clearance) of the separating surface *x*<sup>3</sup> is most significant for the percentage of particles larger than 3 mm (*y*3). At the same time, regardless of the angle of sharpening of the knives, the number of particles larger than 3 mm *y*<sup>3</sup> decreases with an increasing gap of the separating surface. The deterioration of the sharpening of the knives leads to a twofold increase in power consumption *y*1, which is more than 5 kW (Figure 7c). Branco et al. [23] suggested that knives should be replaced or sharpened periodically to ensure high efficiency in grinding.

Some of the determining parameters that affects the power consumption *y*<sup>1</sup> during grinding are the interactions of factors *x*<sup>1</sup> and *x*3, as well as *x*<sup>2</sup> and *x*4. When solving a compromise problem of increasing the grinder performance *y*<sup>2</sup> and reducing its power consumption *y*1, it is necessary to increase the values of the factors *x*3, *x*2, and *x*4, while decreasing the value of *x*<sup>1</sup> (Figure 7d,e).

With an increase in the feed *x*<sup>1</sup> and an increase in the number of knives at the first grinding stage *x*4, the percentage of particles larger than 3 mm *y*<sup>3</sup> increases. As a result, the number of knives should be minimized (Figure 7f).

The search for a compromise solution was made using the StatGraphics software package, with equal significance of optimization criteria with the desired result shown in Table 4. The results of the optimization are shown in Table 4. These results suggest the compromised optimum is achieved by selecting the maximum values of the grain feed *x*<sup>1</sup> (in the selected range of factor variation, Table 1), the rotation frequency *x*2, and opening of the separating surface *x*3. Furthermore, "new" knives *x*<sup>6</sup> should be used without inserts in the distribution bowl *x*7, where nine knives at the second stage *x*5, and six at the first stage *x*<sup>4</sup> (Table 4) should be employed. Under these optimum conditions, it is possible to achieve a power consumption *y*<sup>1</sup> of 2.59 kW, a grinder capacity *y*<sup>2</sup> of 0.032 kg−1, and the complete absence of particles larger than 3 mm *y*<sup>3</sup> after grinding. At the same time, the obtained values (Table 4) correspond in general to only 87.8% of the desired results, which indicates the insufficiency of the selected ranges for the factors to achieve the most optimal values of the power consumption *y*1, performance (capacity) *y*2, and quality of the resulting product, in terms of the content of particles exceeding 3 mm in size *y*3.

#### **4. Conclusions**

The analysis of the results of this study (using the method of a multifactorial experimental design) into the operation of a centrifugal–rotary grinder suggests that the grinder is underutilized in the selected range of factor variation. The installation of special inserts in the distribution bowl of the lower disk (*x*7) generally has a negative impact on the quality of the resulting product in terms of the content of particles larger than 3 mm. The number of knives installed at the second stage of the grinder (*x*5), the gap (clearance) of the separating surface (*x*3), and the technical condition of the knives (*x*6) are among the most important factors influencing the power consumption and the quality of the resulting product. A reduction in the number of knives at the first stage (*x*4) has a positive effect on all the selected optimization criteria. Finally, by varying the factors in the selected range, it is possible to obtain a product corresponding to medium and coarse grinding.

Summarizing the results of the study, we can conclude that, in further research the material feed (*x*1) and rotor speed (*x*2) should be increased, and the range of variation of the opening of the separating surface (*x*3) should be extended. Further, the installation of a special insert (*x*7) in the distribution bowl of the lower disk should be abandoned, or additional research related to changes in its shape and size should be carried out. Finally, the number of knives at the first stage (*x*4) and at the second stage (*x*5) should be reduced, and "new" knives (*x*6) should be used in all cases.

**Author Contributions:** Conceptualization, A.M.; Formal analysis, A.M. and A.B.-K.; Funding acquisition, A.M.; Investigation, P.A.S., A.Y.I., A.V.P. and I.I.I.; Methodology, P.A.S., A.Y.I., A.V.P. and I.I.I.; Writing—original draft, P.A.S., A.Y.I., A.V.P. and I.I.I.; Writing—review & editing, A.M. and A.B.-K.

**Funding:** This research received no external funding.

**Conflicts of Interest:** The authors declare no conflict of interest.
