**3. Results**

#### *3.1. Fan Characteristics*

The numerical calculations were completed after 7200 time steps, which corresponds to 20 rotations of the rotor. The flow is established after approximately 1000 steps. The total pressure increase Δ*p* was used as a criterion for flow stabilization. The velocity contours of the resolved flow are shown in Figure 5.

**Figure 5.** Velocity contours of resolved flow.

On the basis of the obtained results, the basic parameters of the fan were calculated to determine its characteristics. The formula was used to calculate the mechanical power

$$N\_m = M\omega \tag{11}$$

where *M* is the torque on the rotor expressed in [Nm] and *ω* is the angular velocity expressed in [rad·s<sup>−</sup>1]. The effective power was calculated as

$$N\_u = \Delta p \dot{V} \tag{12}$$

where Δ*p* is the pressure increase and *V*˙ is the volume flow rate behind the rotor. Due to the low compression, the compressibility of the medium is not taken into account. The efficiency was calculated according to the equation

$$\eta = \frac{N\_u}{N\_m} \tag{13}$$

In Figure 6, it can be seen that points 1 ÷ 4 present a pressure increase significantly different from the rest of the measurement points and reach Δ*p* = 185.2 ÷ 215.7 Pa and their amplitude is approximately approximately 20 Pa, while the amplitude at the points 5 ÷ 20 is only 2 Pa. The results of the calculations of pressure increase and efficiency are shown in Figure 7.

**Figure 6.** Pressure increase.

**Figure 7.** Pressure increase and efficiency characteristics.

#### *3.2. Aerodynamic Noise Characteristics*

The acoustic power of the fan during its operational conditions was determined. Therefore, distributions of the fluid pressure and velocity around the fan in the duct in successive moments was calculated. Next, the FW-H analogy was used to determine the sound pressure values in points on the sphere around the fan. In this case, reflections from walls of the duct were not taken into account because the radiated power is a parameter of the acoustic source. At low sound pressures and the assumption of unidirectional coupling between the flow and the acoustic field, the reflections do not have much of an impact on the power of the source itself. The considerable size of the sphere was necessary to be able to treat the acoustic wave as locally plane in the receivers. Using the FW-H analogy, a time domain acoustic signal was obtained on the surface of a sphere of radius *R* = 3 m. On this area, 510 receivers were placed in which the acoustic signal was obtained. Sound pressures were determined at the measurement points, from which the sound intensity was calculated assuming that the wave was locally flat. The integral of the intensity along the surface of the sphere gives the sound power value. The acoustic pressure was determined in each of the receivers, and assuming a locally plane wave, on this basis the sound intensity was calculated. The receivers on the sphere were evenly distributed and each was assigned to a sphere surface element. Integration was performed using the rectangle rule (the value of the intensity in the receiver multiplied by the surface element assigned to a given node) [38]

$$SWL = 10\log\_{10}\left(\frac{P}{P\_0}\right) \tag{14}$$

where *P*<sup>0</sup> is the reference power equal to 10−12*W* and *P* is the power of sound expressed by the formula

$$P = \oint\_{A} I dA = \oint\_{A} \frac{p^2}{\rho\_0 c} dA \approx \frac{\sum\_{i} A\_i p\_i^2}{\rho\_0 c} \tag{15}$$

where *A* is the surface area, *prms* is the root mean square of the sound pressure, *ρ* = 1.1225 kg/m3 is the density of air, *c* = 340 m/s is the speed of sound. On the basis of the calculated values of the sound pressure, calculations were carried out to obtain the sound pressure level.

$$SPL = 20\log\_{10}\left(\frac{p\_{rms}}{p\_{ref}}\right) \tag{16}$$

In addition, a Fourier analysis was carried out to verify the blade pass frequency calculated from the relationship

$$BPF = \frac{RPM \times z}{60} \tag{17}$$

where *pref* is the reference pressure equal to 2 · <sup>10</sup>−<sup>5</sup> Pa. The results are shown in Figure 8, showing the blade pass frequency of 300 Hz and its harmonics.

The results obtained are compared with the fan characteristics in Figure 9. From the results obtained, it can be concluded there is a significant increase in the sound power level in the stall area compared to the working area, which is up to 10 dB. In the operating area, the sound power level is in the range 79.3 ÷ 90.9 dB (see Table 4) and can be approximated by a parabola with a local minimum. In experimental work [39], a similar character of sound power level was obtained.

**Figure 8.** FFT of acoustic signal.


**Table 4.** SWL results.

From the resulting sound pressure level distribution shown in Figure 10, it can be seen that there are negligible differences in cases 5–20. A common feature of all cases is a higher sound pressure level on the upstream side and around the X axis at the height of the blades. In cases 1–4, the sound pressure level is much higher than in the other cases, and it is related to greater pressure fluctuation in the stall area.

**Figure 9.** Pressure increase, efficiency, and sound power level characteristics.

**Figure 10.** *Cont*.

**Figure 10.** Sound power level distribution: (**a**–**r**) are the measurement points 1–18.

#### *3.3. Sensitivity of the Fan Parameters to the Change of the Blade Angle*

The study investigated the sensitivity of the fan characteristics to a change in the blade angle. For this purpose, we carried out additional numerical calculations for the blade angle *θ* = 21◦ (Figure 11).

**Figure 11.** Geometry of the model with marked angles *θ* = 20◦ and *θ* = 21◦.

The sensitivity of a characteristic function to changes in a design parameter can be defined as the partial derivative of the function describing that characteristic, with respect to that parameter. The sensitivity coefficients, in normalized form, can be found as

$$S\_{\mathbf{x}}^{y} = \frac{\partial y\_{i}/y\_{i}}{\partial \mathbf{x}\_{j}/\mathbf{x}\_{j}}\tag{18}$$

where *S<sup>y</sup> <sup>x</sup>* is the normalized sensitivity coefficient, *yi* is model dependent variable and *xj* is input parameter.

$$S\_{\theta}^{\Delta p} = \frac{\partial p/p}{\partial \theta/\theta} \approx \frac{\Delta p/p}{\Delta \theta/\theta} = 0.2416\tag{19}$$

$$S\_{\theta}^{\eta} = \frac{\partial \eta / \eta}{\partial \theta / \theta} \approx \frac{\Delta \eta / \eta}{\Delta \theta / \theta} = 0.1042\tag{20}$$

$$S\_{\theta}^{\rm SWL} = \frac{\partial SWL/SWL}{\partial \theta/\theta} \approx \frac{\Delta SWL/SWL}{\Delta \theta/\theta} = -5.9558 \cdot 10^{-5} \tag{21}$$

The values from points 8 ÷ 16, which constitute the work area, were used for the calculations. Two single (∗) symbols define the beginning of a new stall zone (see Figure 12).

**Figure 12.** Pressure increase, efficiency, and sound power level characteristics for blade angles 20◦ and 21◦.

#### **4. Conclusions**

Computer simulations using CFD techniques were carried out to investigate the noise generated by the axial fan. The URANS time-averaging method and the FW-H analogy implemented in Ansys Fluent were used in the calculations. Using these methods, the sound pressure was calculated on a sphere with a radius of 3 m from the sound source, from which the acoustic characteristics were determined and an FFT analysis was performed. The analysis was carried out for twenty characteristic points for two different blade angles. The main conclusions of the numerical analysis are presented below.

On the basis of the torque and total pressure increase analysis, the stall area can be verified for the operating points 1 ÷ 4 and normal work in points 5 ÷ 20. In the stall region, the total pressure increase is Δ*p* = 185.2 ÷ 215.7 Pa, while in the normal operating area, it is 79.8 ÷ 99.8 Pa.

Fourier analysis, calculated using the FW-H acoustic analogy of the acoustic pressure, made it possible to determine the blade pass frequency equal to 300 Hz and its harmonics, which allows to confirm the accuracy of the numerical simulations.

As expected, the determined characteristics show that the fan efficiency increases with increasing total pressure increase.

The difference in the indicated sound power level in the test area is just 1 dB. However, attention should be paid to the obtained characteristics of the sound power level, which in the studied area has a local minimum, which encourages additional considerations on determining the optimal operating point for which the emitted noise is the lowest.

The determined sound power level on a sphere with a radius of 3 m from the noise source indicates higher emission from the upstream, which may be caused by turbulent flow caused by the curved pipeline. The common feature of the presented results is a greater level of generated noise around the X axis at the height of the blades.

From the simulations carried out for a variable blade angle, it can be seen that, for angle *θ* = 21◦ at certain points of the characteristic curve, the fan efficiency increases from 2% to 8%. A sound power level at an angle of *θ* = 21◦ emits 1 dB less at certain points than for angle *θ* = 20◦.

For blade angle *θ* = 21◦, the characteristic point that defines the stall zone has moved, and for this, the angle is located between the points 1 ÷ 6.

The generated noise is influenced by many more design and operational parameters of the axial fan. The aim of the article was to show the change in the angle of the rotor blades in the entire exploitation area, i.e., with a variable flow rate. The calculations consist of examining more than 20 measurement points for one parameter change, which require a lot of computational time, but they are planned for the second part of the research. Further experiments on the real object will be aimed at verifying the numerical model and extending the research with parameters influencing the generated noise.

**Author Contributions:** Conceptualization, D.R. and I.C.; methodology, I.C.; software, D.R.; validation, I.C., D.R.; formal analysis, D.R.; investigation, D.R.; resources, D.R.; data curation, D.R.; writing—original draft preparation, D.R. and I.C.; writing—review and editing, D.R. and I.C.; visualization, D.R.; supervision, I.C.; project administration, I.C. All authors have read and agreed to the published version of the manuscript.

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

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Data sharing not applicable.

**Acknowledgments:** This research was supported in part by PLGrid Infrastructure. This research was supported by national subvention no. 16.16.130.942.

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

### **Abbreviations**

The following abbreviations are used in this manuscript:

