Research on Multi-Optimal Project of Outlet Guide Vanes of Nuclear Grade Axial Flow Fan Based on Sensitivity Analysis

Abstract

Nuclear grade axial flow fans are widely used in nuclear power plants for ventilation and heat dissipation and have the advantages of high efficiency and high flow rates. A nuclear grade axial flow fan with OGVs (outlet guide vanes) can recover the kinetic energy of the dynamic impeller outlet winding to increase the ventilator pressure, thus improving the ventilator efficiency; therefore, the OGVs play an essential role in the performance of the axial flow fan. Based on accurate numerical simulations, an MRGP approximation model was developed to analyse the factors affecting the OGVs duct and optimise the guide vane structure, combined with the Sobol method for sensitivity analysis. The experiments and numerical simulations show that the total pressure of the optimised model increases by 154 Pa, and the noise decreases by 4.1 dB. The multi-objective optimisation method using the parametric approach and combining it with the MRGP model is highly reliable. It provides a key design direction for optimising nuclear grade axial flow fans.

1. Introduction

Large size and significant flow rate are the characteristics of nuclear grade axial flow fans; their defect is in their flow field because the dynamic-static coupling duct, rotation duct, and guide vane structure are more complex and cause backflow, vortexes, and other flow phenomena. The blade root of the flow appears separated. In the design phase of the OGVs, less attention was given to the degree of OGVs [1], and the design of the OGVs because the axial distance of the moving blades is unreasonable, which is due to the pattern of flow at the axial distance [2], resulting in a decline in fan performance. This has a negative impact on noise. Therefore, it is essential to optimize the design of the OGVs.

The performance of axial fans is influenced by many factors [3,4,5]. Cai et al. [6] presented their experimental investigations and numerical simulations of the performance of a deflected-blade axial flow fan. They showed that compared to radial blades, forward-swept blades could increase pressure by 13.1%, boost flow by about 5%, reduce noise by 2–4 dB, and increase efficiency by 3%. Corsini et al. [7,8,9] studied the flow fields of two subsonic axial fans with 35° forward-swept blades and unswept blades. The results demonstrated that the forward-swept blades were more stable, especially at low flow speeds, with a stall delay effect. Bellows’ results show that forward-swept blades on low-speed axial flow fans can improve aerodynamic performance and the potential for a wide range of applications.

The noise emitted by the axial flow fan is a key area of research [10]. Therefore, in addition to obtaining the required aerodynamic performance, operating with low noise is one of the design considerations. Sharland [11] investigated the relationship between turbulent boundary layer noise and trailing edge noise. Amiet [12] predicted the combined model of the turbulent boundary layer noise and trailing edge noise. Zenger et al. [13] explored the interrelationships between noises and found that turbulent boundary layer noise was negligible compared to other noise sources.

For the optimisation of axial flow fans, researchers have used various optimisation techniques to reshape and optimise the impeller blades. Liu Fei et al. [14] investigated the internal flow problem in an axial flow fan with front and rear guide vanes using numerical calculation methods and obtained a relatively reasonable position for the front guide vanes. Zhang Rui et al. [15] used the spoke-flow control technique of evenly spreading strips on the inner sidewall of the inlet tapered tube of an axial flow pump, carried out simulations and analyses based on the FBM-CC model, and found that the efficiency of the axial pump could be improved to a certain extent and the formation of the “saddle area” could be effectively suppressed. Seo et al. [16] used the response surface method (RSM) to obtain the optimum blade bending angle for a low-speed axial fan to optimize the fan’s efficiency. Huang Yougen et al. [17] used a multi-objective genetic algorithm combined with an approximate model to optimise the blade shape of an axial fan, resulting in a 10% increase in fan efficiency. Jinxin CHENG [18] used a Bezier surface as a parametric method to control the suction surface of the blade, significantly reducing the design variables, ensuring a smooth blade surface, and successfully improving the aerodynamic performance of an axial compressor. Ma Pengfei [19] proposed a combined optimisation method for axial flow blades that included experimental design and velocity gradient algorithm, which improved the optimisation efficiency by 3.07% and 0.87%, respectively, compared to the conventional direct experimental design or sequential quadratic programming algorithm. The analysis was completed and the results concluded that the optimised vanes increased the pump’s efficiency by 2.91% at the optimum operating point. Adjei Richard Amankwa [20] used parameterised free-form deformation (FFD) for the aerodynamic design optimisation of the static vanes, using a B-sample based FFD control body to achieve a 6.1% reduction in total pressure loss. Li et al. [21] developed a 1.5 D and non-constant computational fluid dynamics (CFD) model to rapidly and accurately predict the aerodynamic performance of large dynamic-vane adjustable axial fans. Yao [22] explained the basic variable operating characteristics of a dynamic adjustable vane large axial flow fan in combination with field operations.

Regarding the degree of influence of each parameter on the optimisation problem, Hongyi Wang [23] used the MRGP model to estimate the 4000 h failure probability of an industrial robot drive, yielding a total system failure probability of 3.56%, with the most influential being an IGBT bond wire break failure probability of 1.84%. Using the Sobol metric and its calculation method to estimate the primary and total metrics for the global sensitivity of each failure mode, we conclude that operating relative humidity, PCB resin board thickness, ripple current, and cyclic temperature loading range have the most significant impact on the uncertainty of the system output. The MRGP method compensates for the inability of the AK-MCS and AK-SYS methods to address failure-related reliability issues. The Sobol metric and its calculation method do not require the tremendous computational effort of the MCS method, for example.

The above methods are effective in optimising the performance of large axial flow fans, but researchers have not fully considered the discreteness of the optimisation parameters and the two-by-two coupling between the optimisation parameters, so there is still some room for improvement in these methods. In this study, by parametrically designing the rear guide vane of an axial fan and selecting the total pressure and noise value as the optimisation object, the MRGP approximation model is established to optimise the parameters of the OGVs, and the influence of each parameter on the optimisation object is elucidated through the Sobol method. Finally, the changes in the flow field characteristics before and after the improvement are analysed, and the acoustic characteristics are investigated through numerical simulations and prototype verification. In addition, the MRGP approximation model combined with the Sobol method is used for multi-objective design optimisation, which reduces the computational effort and solves the reliability problem, while ensuring the feasibility, accuracy, and universality of the method and adapting it to the design optimisation of similar machinery. In addition, most large axial flow fans are calculated through simulations or engineering experience and are not experimentally verified. This article not only proves the effectiveness of optimisation through simulation calculations, but also tests the vibration, noise, and aerodynamic performance of a nuclear grade axial flow fan, further demonstrating the feasibility of the optimisation method.

2. Numerical Simulation Calculation Methods and Model Validation

In this research, the optimisation object was a nuclear grade axial flow fan, as shown in Figure 1. The structure is mainly composed of a cowl, moving blades, outlet guide vanes, motor, and diffuser. The structural parameters are as follows: impeller diameter, D = 1050 mm; number of moving blades, Z₁ = 22; number of OGVs, Z₂ = 23; rotating speed, n = 1485 r/min; flow rate, Q = 70,200 m³/h; total pressure, p_tf = 1700 Pa; and sound pressure level, SPL =112 dB.

2.1. Numerical Simulation Calculation Methods

2.1.1. Meshing

To improve calculation efficiency, the geometric model of the fan is simplified, and the entire axial fan duct is divided into four ducts: the inlet duct, the rotation duct, the OGVs duct, and the outlet duct. A simplified diagram of the turbine duct is shown in Figure 2a.

The core components of the axial flow fan, the moving blades, and the OGVs are divided into structural grids by TurboGrid to improve calculation accuracy. The rotation duct grid is shown in Figure 2b, and the OGVs duct grids are shown in Figure 2c, while the more regular shape of the inlet and outlet ducts are divided into O-block grids.

For other ducts, due to the complex and irregular shape of the structure, the grid is divided by an unstructured grid, the grid density is increased, and the first node should be placed on the bottom of the viscous area, where y⁺ needs to be ≤5. The empirical formula of y⁺ is given by Equation (1) [24]:

$$Y_{\mathrm{wall}}=6{\left(\frac{V_{\mathrm{ref}}}{v}\right)}^{-\frac{7}{8}}{\left(\frac{L_{\mathrm{ref}}}{2}\right)}^{\frac{1}{8}}{y}^{+}$$

(1)

where Y_wall is the height of the first layer of the boundary layer grid in mm; V_ref is the reference speed in m/s; L_ref is the reference length in m; v is the fluid kinematic viscosity in m²/s, and y⁺ is a dimensionless parameter, indicating the boundary point between the viscous bottom and logarithmic layers. According to the empirical formula, the height of the first layer of the boundary layer grid should be less than 0.72 mm.

After grid quality has reached the required level, the independence of the calculation grid also needs to be verified. As shown in Figure 3, when the total grid number of the axial flow fan reaches about 5.5 × 10⁶, the value of the total pressure of the axial flow fan obtained from the numerical simulation is unchanged. Therefore, taking into account calculation accuracy and time, the number of grids for the entire basin of the fan is 5,570,210.

2.1.2. Aerodynamic Performance Calculation Method

For this study, the maximum blade tip speed u of the axial flow fan is less than 0.3 Mach; the simulation can therefore be carried out using a mathematical model of the three-dimensional incompressible flow field. The main boundary conditions for axial fans are set as pressure inlet and pressure outlet. The steady calculation adopted a multiple reference system model (MRF), the working fluid is air, and the impeller duct setting is to rotation duct. The solution adopted the pressure-based implicit solver. And the k-ω turbulence model was used to solve the three-dimensional Reynolds-averaged Navier–Stokes (N-S) equation. The SST k-ω turbulence model [25,26] is a two-equation model derived from a combination of k-ε and k-ω. The combined model uses the k-ω model in the wall region where the viscous influence is more pronounced. In contrast, the k-ε model is used in the region where the turbulent flow is fully developed, thus avoiding the impact of minor disturbances in the incoming flow on the simulation calculations. The SST k-ω turbulence model gradually changes from the standard k-ω model inside the boundary layer to the high Reynolds number k-ε model outside. The SST k-ω turbulence model achieves a gradual change from the standard k-ω model inside the boundary layer to a high Reynolds number k-ε model outside the boundary layer. The SIMPLE (Semi-Implicit Method for Pressure-Linked Equations) algorithm was applied to couple the pressure and velocity. The index parameters of each aerodynamic performance were set to the second-order upwind. When the value of each calculation residual is less than the convergence residual of 1 × 10⁻⁵, the convergence is considered.

The equation of the SST k-ω turbulence model is as follows:

$$\frac{\partial k}{\partial t}+\nabla \cdot (uk)=P_k-{\beta }^{\ast }k\omega +\nabla \cdot \left[\left(\nu +{\sigma }_k{\nu }_T\right)\nabla k\right]$$

(2)

$$\frac{\partial \omega }{\partial t}+\nabla \cdot (u\omega )=\delta S^2-\beta {\omega }^2+\nabla \cdot \left[\left(\nu +{\sigma }_{\omega }{\nu }_T\right)\nabla \omega \right]+2\left(1-F_1\right)\frac{{\sigma }_{\omega 2}}{\omega }\nabla k\cdot {(\nabla \omega )}^T$$

(3)

$${\nu }_T=\frac{a_{1}k}{\max \left(a_1\omega ,S{F}_2\right)}$$

(4)

where $P_k=\min \left[{\nu }_T(\nabla \times u)\cdot {(\nabla \times u)}^T,10{\beta }^{\ast }k\omega \right]$, S is the absolute value of vorticity, and the constants of the SST k-ω turbulence model in this study are σ_k1 = 0.85, σ_k2 = 1.0, σ_ω1 = 0.5, σ_ω2 = 0.856, β₁ = 0.075, β₂ = 0.0828, a₁ = 0.31, δ₁ = 0.5532, and δ₂ = 0.4403, β^* = 0.09.

2.1.3. Calculation Method of Sound Field Prediction

The FW-H method [27] of large eddy simulation (LES) combined with acoustic analogy theory is used for noise simulation. The steady calculation results are used as the initial flow field of unsteady calculation, and the time required for impeller rotation 1° is used as the time step Δt, namely Δt = 1/(6n). Therefore, the time step in this study is set to 0.0001 s to obtain the flow field information in the near field. FLUENT solved the FW-H equation by calculating the time-domain integral and area integral to get the far-field noise of the fan. The FW-H equation is as follows:

$$\frac{1}{C_0^2}\frac{{\partial }^2{p}^{\prime }}{\partial t^2}-\frac{{\partial }^2{p}^{\prime }}{\partial x_i^2}=\frac{{\partial }^2}{\partial x_i\partial x_j}\left[T_{ij}H(f)\right]-\frac{\partial }{\partial x_i}\left[n_{i}p\delta (f)\mathsf{\Delta }f\right]+\frac{\partial }{\partial t}\left[p{v}_n\delta (f)\mathsf{\Delta }f\right]$$

(5)

where C₀ is the sound velocity of the undisturbed fluid, p′ is the sound pressure, x_i is the space fixed coordinate system, v_n is the normal velocity, and T_ij is the Lighthill stress tensor. The three terms on the right side of the equation represent the quadrupole, dipole, and monopole.

2.2. Model Accuracy Verification

2.2.1. Verification of Aerodynamic Performance

According to the “GB/T 1236-2017 Standard Airway Performance Test for Industrial Fans” [28], the aerodynamic performance of a large axial fan in this study was tested using a C-type air chamber. Figure 4 shows the air performance test device diagram for the test and the photos of the field test device. A comprehensive tester of atmospheric pressure, temperature, and air humidity was arranged at the front end of the inlet collector (air velocity is zero). The corresponding air parameters of the axial fan under different operating conditions were recorded. The performance test was carried out at room temperature and atmospheric pressure. The flow rate is changed during the test from 40,000 m³/h to 85,000 m³/h by adjusting the inlet throttling device. The pressure difference and the specific value of the inlet and outlet pressure of the Pitot hydrostatic pipe under nine different flow conditions are read. The data for the different operating conditions are shown in

Table 1. Data for different working conditions.

Group	1	2	3	4	5	6	7	8	9
Q (m3/h)	40,516.5	57,319.8	65,316.5	67,684.8	70,263.1	73,718.2	77,661.4	81,414.0	84,769.2
ptf (Pa)	1432.7	1747.5	1978.4	1895.4	1781.7	1582.1	1143.2	442.7	207.5

.

To improve the reliability of the experimental data, this paper uses standard errors to analyse the experimental data; the standard measuring error calculation formula is:

$$\sigma =\sqrt{\frac{\sum {(p_i-\overline{p})}^2}{n-1}}$$

(6)

where σ is the standard error values, n is the number of measurements, p_i is the deviation of the measured values, and $\bar{p}$ is average values.

The flow rates at 10 sets of design operating points were tested and the results are shown in

Table 2. Total pressure data at the design flow.

Group	1	2	3	4	5	6	7	8	9	10
pi	1780.7	1779.2	1783.9	1785.7	1781.3	1781.1	1778.9	1784.7	1781.0	1780.4
$(p_{\mathrm{i}}-\bar{p})$2	0.98	6.35	4.90	16.37	0.15	0.37	7.98	9.32	0.47	1.68

.

Therefore, the standard error values can be calculated by the formula:

$$\sigma =\sqrt{\frac{\sum {(\overline{p}-p_i)}^2}{n-1}}=\sqrt{\frac{48.57}{9}}=5.40$$

The error analysis for the other operating points is similar to it. According to this method, it is known that the measurement error values are within reasonable limits.

The total pressure-flow performance curve and the efficiency-flow performance curve shown in Figure 5 were obtained by comparing the performance results of the prototype predicted by the numerical calculation model with the experimental values.

2.2.2. Sound Field Verification

Referring to the noise test standard, “GB/T 2888-2008 Methods of noise measurement for fans blowers compressors and roots blowers” [28], noise receiving points were set up at 45° from the centre of the outlet surface, at a horizontal position of one times the diameter (1050 mm). Once the noise value at the adjusted operating point has stabilised, take a reading of the sound level meter and switch to the next operating point. Figure 6 shows a schematic diagram of the noise monitoring point setup, and Figure 7 shows a test setup diagram. The test device is a RINO NL-42 high-precision sound level meter. The device has a sampling frequency of 20–8000 Hz, a range of 25–130 Hz, and a sensitivity of 0.01 dB.

The differences of the test and simulation results are shown in

Table 3. Comparison of test and simulation under different working conditions.

Q (m3/h)	40,516.5	57,319.8	65,316.5	67,684.8	70,263.1	73,718.2	77,661.4	81,414.0	84,769.2
Simulated value (dB)	107.7	109.4	109.9	110.2	109.5	109.4	109.8	109.5	109.4
Experimental value (dB)	108.5	110.3	110.8	110.9	110.4	110.3	110.5	110.6	111.2
Difference (%)	0.74	0.82	0.81	0.63	0.82	0.81	0.63	0.99	0.72

.

2.2.3. Summary

As can be seen from the data, the performance curve of the fan predicted by the SST k-ω model has a specific error compared with the test value, as can be seen from the external characteristics curve (Figure 5), when the flow rate of 75,000–85,000 m³/h near the design flow rate numerical simulation results and the test values are almost the same. As the pressure decreases, the simulated flow rate at each operating point is lower than the test value. Still, the maximum error does not exceed 5%, which aligns with the engineering calculation error requirements. As shown in Table 1, the noise simulated by the FW-H method using LES combined with acoustic analogy theory is very close to the experimental noise, with a maximum error of about 1%. Therefore, the method is highly reliable and can reference similar fan calculations.

3. Prototype Performance Analysis

The vibration characteristics of a typical fan used in nuclear power generation cannot be ignored during its own operation. Strong vibrations can subject rotating machinery to additional unfavourable loads, generate huge noise, and, in severe cases, even cause extreme consequences such as equipment failure, endangering the operational safety of nuclear power plants. At the same time, as the axial flow fan works, the irregular flow and vibration phenomenon caused by the structural design destroys the flow energy and leads to pressure pulsation between the blades and the flow channel, especially in the inflow area of the guide vane, resulting in energy loss and waste under working conditions, thus affecting the stability and safety of the equipment work [1]. Therefore, the simulation and analysis of the vibration characteristics and aerodynamic properties of the critical parts of the fan are of direct importance for the optimisation of pressure pulsation in practical engineering. To obtain primary data on the vibration characteristics of the fan’s essential parts and identify the vibration- and noise-inducing mechanisms, corresponding experimental studies are required. The frequency domain data are acquired by the B&K Time Data Recorder software and the B&K 4535-B triaxial acceleration sensors are used for the vibration signal acquisition during the operation of the nuclear grade axial flow fan. As shown in Figure 8a, the measurement points were installed with triaxial acceleration sensors in the OGVs duct, dynamic-static coupling duct, and rotation duct. The B&K 4535-B triaxial accelerometer is shown in Figure 8b.

As shown in Figure 9, according to the analysis of the test data, the triaxial acceleration of the fan showed significant fluctuations from 0 to 250 Hz and 450 to 600 Hz in the 0 to 600 Hz band in the OGVs duct, dynamic-static coupling duct, and rotation duct near the motor. The radial acceleration fluctuates the most, while the remaining two fluctuate less. In addition, the fluctuations in radial acceleration in the 0–600 Hz band in the static lobe region are generally more significant than those in the remaining two areas, especially in the 450–600 Hz range, which is presumed to be caused by the vibration of the fluid passing through the OGVs duct.

To get a clearer view of the internal flow field characteristics of the axial flow fan, the blade flow surfaces of the fan impeller with radial cross-sections of 0.1, 0.5, and 0.9 span were chosen for the study. Figure 10 shows a schematic diagram of the radial section of the fan impeller.

Figure 11 shows the flow diagram of the prototype fan with the different span, on which it can be seen that before the optimisation of the leaf shape, there is an apparent vortex coming off the trailing part of the moving blades in the A area on the 0.1 span, and the airflow is turbulent in the d dynamic-static coupling duct; as shown in the B area in the figure, there are vortices and backflow phenomena in the front abdomen of the OGVs on the 0.5 span, the airflow speed is deficient, there is flow separation at the trailing edge of the blade root, and the flow field condition is not good; as shown in the C area, the OGVs at 0.9 span are at the end of the same giant vortex. In summary, the fan has a poor flow field in the OGVs duct, shedding vortices and swirls, and the OGVs structure has a significant impact on the fan’s performance and noise.

Comprehensive analysis of the acceleration and vibration characteristics of the fan and internal flow analysis shows that the OGVs duct is the worst of the three ducts. To further analyse the impact of the OGVs on the fan’s performance, the following analysis of the static pressure distribution and pressure pulsation at the different span of the OGVs was conducted.

From Figure 12, it can be seen that the static pressure distribution of the different OGVs spans are the same; the separation points of the three spans of the OGVs are all at 0.25, very close to the front, while the gradient of the pressure surface static pressure curve is enormous, in 0.0–0.4 streamwise, and C_p decreases from 0.4 to −0.8, indicating that there is a large number of shedding vortices and apparent pressure pulsations in the flow field in this region. C_p is the pressure coefficient. The equation of the C_p is as follows:

$$C_p=\frac{p}{0.5\rho u^2}$$

(7)

where p is the static pressure in Pa; ρ is the density in m³, and u is the leaf tip linear velocity in m/s.

For the axial flow fan that increases the OGVs, the vortex, backflow, and other complex flow phenomena in the OGVs duct may lead to rapid changes in fluid pressure within the fan continuously over time, i.e., pressure pulsation. Vortex shedding noise and the pressure pulsation value related to the blade surface per unit area of the pulse of the lift can reflect the sound radiation pressure. Therefore, reducing the value of the pressure pulsation can effectively mitigate the acoustic radiation pressure. The relationship is as follows:

$$P=\frac{x_i}{4\pi R^{2}C}{\iint }_S\left\{-\frac{\partial F_T}{\partial t}\left(y_i,t\right)\right\}dS\left(y_t\right)$$

(8)

where F_T is the pressure pulsation value (or pulsation lift per unit area on the blade surface), x_i is the coordinates of the observation point, y_i is the coordinate value of the source point (i = 1, 2, 3), and R is the distance between the source and the observation point. Usually, R is much larger than the blade size, so $R=\left|x_i-y_i\right|\approx \left|x_i\right|$; C is the acoustic velocity, and S is the blade surface area.

The effect of the OGVs on the pressure pulsation over the OGVs duct is analysed below. Pressure pulsation monitoring points are set at the 0.1, 0.5, and 0.9 spans at the OGVs inlet, pressure surface, suction surface and outlet, respectively, and the monitoring points are set up as shown in Figure 13.

From the time domain graph of pressure pulsation, information, such as the amplitude of pressure changes, can be analysed and the key locations of pressure changes can be determined. The following is an analysis of the pressure pulsation time domain diagram.

Figure 14 shows the time-domain analysis of the pressure pulsation at each monitoring point over one cycle. x-axis is the number of cycles, and z-axis is the pressure coefficient C_p. From the waveform, the pressure pulsation at P3 is very consistent, with peaks around −0.12, while the remaining points have very irregular pressure pulsation waveforms with no consistent peaks; numerically, the amplitudes at P1−P3 and P10−P11 are small, both around 0.3, while the amplitudes at P4−P9 are high, with P9 having the largest amplitude at 0.728. The irregular waveforms and large amplitudes indicate the existence of abnormal pressure pulsations in the duct.

Through the aerodynamic performance, the acceleration vibration characteristics, internal flow field, static pressure distribution, and pressure pulsation static pressure distribution can be analysed if the fan’s performance meets the standard. Still, the uneven airflow at the OGVs led to significant noise. The aerodynamic performance also has some room for improvement involving the above problems, the OGVs for parameterisation, and performance optimisation.

4. OGVs Parameterisation and Its Optimisation Method

4.1. OGVs Parameterisation

The analysis of the flow field above shows that the overall performance of the fan is up to standard, but that there is a significant separation vortex in the area of the rear guide vane, as well as further reflux in the abdomen of the OGVs This model has a large number of motorised parts and OGVs and a high grid consistency. Changing the chord length of the OGVs will alter the original grid consistency, so the chord length is set to a constant value during the parameterisation process.

To harmonise with the engineering guide vane type line design, the following method was chosen to describe the OGVs type line. In Figure 15, O(x,y) is the centre of the OGVs type line circle, A(x₁,y₁) is the start of the type line, and B(x₂,y₂) is the end of the type line. R is the radius of the OGVs type line, σ_a is the axial distance between the moving blades and the OGVs, and θ is the OGVs incidence. The starting point A of the type line is determined by changing the axial distance σ_a. In the case of keeping the chord length unchanged, by changing the centre position O and radius R counterclockwise to draw a starting point for A, chord length is a fixed value of the arc, namely the type line, the endpoint is point B. The angle between the chord and the y-axis formed by connections A and B is the incidence θ.

Some range control of the variables is required to reduce the calculation time. The variable range of some parameters was adjusted, i.e., y₁ for a constant value, R = 164~264 mm, σ_a = 40~160 mm, −20 ≤ x ≤ 20 mm, and −20 ≤ y ≤ 20 mm. Some of the values are shown in

Table 4. OGVs structure parameters summary table.

Point	σa (mm)	R (mm)	x (mm)	y (mm)
1	100.00	214.10	0.00	0.00
2	114.1	205.08	−7.06	−5.67
3	85.13	212.64	14.76	−2.92
4	131.26	187.35	−6.99	−11.41
·····	·····	·····	·····	·····

.

4.2. Optimal Design of MRGP Combined with Sobol Method

The axial distance σ_a from the OGVs to the moving blade, the radius R, the circular horizontal coordinate x, and the circular vertical coordinate y are used as input variables, the total pressure and the noise value of the fan are used as output values, the corresponding proxy model is established using a multidimensional response Gaussian model, and the sensitivity analysis of the variables is carried out using the Sobol method to establish a combined MRGP. This is used to select a combined MRGP and Sobol method of active learning sensitivity for nuclear class axial fans.

The MRGP model extends the Kriging model, which can only deal with a single response output for multiple variables. In contrast, for a multi-dimensional response output, multiple Krining models need to be constructed, which does not consider the coupling of response values. Therefore, MRGP models for multivariate multi-response problems are gaining attention.

The MRGP model was proposed by Zhen [29]. M functions are united into a specific implementation of a multi-response Gaussian process:

$$G(x)=f{(x)}^{\mathrm{T}}B+Z(x)$$

(9)

The polynomial regression function f(x) is the same as the Kriging model, but the regression coefficient B is an n × m matrix, and Z(x) is expanded to a 1 × m row vector. Z(x) covariance is represented by an m × m covariance matrix, i.e., Σ^m:

$${\displaystyle \sum {}^m}=\left[\begin{array}{ccc}{\sigma }^2(g_1,g_2)& \cdots & {\sigma }^2(g_1,g_m)\\ ⋮& \ddots & ⋮\\ {\sigma }^2(g_m,g_1)& \cdots & {\sigma }^2(g_m,g_m)\end{array}\right]$$

(10)

In this study, the MRGP model is selected as the approximation function, using 65 sets of variables obtained through Opt LHD combined with numerical simulation calculations with the sample data of the corresponding responses to build the approximation model, of which the first 60 sets are used as the sample matrix data and the last five sets as the test data.

To verify the model’s accuracy, it is necessary to check the model’s accuracy before solving it by means of the relevant accuracy-test formula. This study uses a global sensitivity analysis based on the Sobol method, a Monte Carlo method based on analysis of variance (ANOVA), which expresses the degree of influence of all parameters on the model results in terms of the total variance D, which can be described as:

$$D=\int f^2(x)\mathrm{d}x-f_0^2$$

(11)

Partial variance D_i is used to indicate the influence degree of a single parameter or the influence of parameters on the model results, which can be expressed as:

$$D_{i_1,i_2,\cdots ,i_t}=\int f_{i_1,i_2,\cdots ,i_3}^2\mathrm{d}{x}_{i_1}\mathrm{d}{x}_{i_2}\cdots \mathrm{d}{x}_{i_s}$$

(12)

By solving the first-order and global sensitivity parameters of each influencing factor, the Sobol method is used to evaluate the sensitivity of the approximate function obtained by MRGP. The first-order sensitivity indicates the influence of a single factor on the total pressure and noise of the fan. The overall sensitivity suggests the result of a single factor on the total pressure and noise of the fan and reflects the interaction with other influencing factors. Therefore, when the first-order sensitivity of a variable is very different from the overall sensitivity, it can be concluded that the factor has a significant interaction. The sensitivity meter are shown in

Table 5. Axial flow fan total pressure sensitivity meter.

Variable Name	First-Order Sensitivity	Overall Sensitivity	Absolute Difference
σa	0.456899	0.498651	0.041752
R	0.169560	0.185055	0.015495
x	0.195959	0.213866	0.017907
y	0.086202	0.094079	0.007877

and

Table 6. Axial flow fan noise sensitivity meter.

Variable Name	First-Order Sensitivity	Overall Sensitivity	Absolute Difference
σa	0.235408	0.242367	0.006959
R	0.087362	0.089945	0.002583
x	0.100964	0.103948	0.002984
y	0.044414	0.045727	0.001313

.

Based on the MRGP model, the whole optimisation process was established through Matlab. The results are shown in Figure 16 below.

4.3. Optimal Results

Table 7. Structural parameters corresponding to the optimal solution.

Parameter	σa (mm)	R (mm)	x (mm)	y (mm)	Total Pressure (Pa)	Noise (dB)
Original	100	214.1	0	0	1488	110.4
Optimised	145.37	197.4	−19.31	8.49	1642	106.3
Δ	45.37	−16.7	−19.31	8.49	154	−4.1

shows the structural parameters corresponding to the optimal scheme and the total pressure and noise at the operating point for design. It can be seen that at the design operation point, the total pressure of the fan increases by 154 Pa and the noise decreases by 4.1 dB. Figure 17 shows a comparison of the leaf type.

5. Result Analysis

5.1. Numerical Analysis before and after Optimisation

As shown in Figure 18a, in the OGVs inlet direction, the airflow is separated at the ventral surface of the guide vane because of the influence of the air admittance attack angle, which leads to obvious separation vortices and backflow, occupying a larger flow space in the vane channel. Figure 18b shows the optimised flow diagram. After optimising the OGVs, it can be seen from the chart that the backflow phenomenon is reduced, and there is no apparent vortex in the ventral part of the optimised OGVs; the flow turbulence is improved, the airflow in the runner is more uniform, and no evident separation flow is produced.

In order to have a better macroscopic probe of the flow separation phenomenon on the blade surface, this section uses the Q-criterion surface to illustrate the flow vortex state of different blades. The Q criterion can be expressed as follows:

$$Q=\frac{1}{2}\left({\omega }_{ij}^2-{\sigma }_{ij}^2\right)$$

(13)

$${\omega }_{ij}=\frac{1}{2}\left(\frac{\partial u_i}{\partial x_j}-\frac{\partial u_j}{\partial x_i}\right)$$

(14)

$${\sigma }_{ij}=\frac{1}{2}\left(\frac{\partial u_i}{\partial x_j}+\frac{\partial u_j}{\partial x_i}\right)$$

(15)

where σ_ij is the rate of change of strain, ω_ij is the vortex amplitude, and u_i and u_j are the components of the flow field velocity in the i and j directions, respectively.

As shown in Figure 19a, due to the turbulent airflow before the optimisation of the OGVs shape, there is a large amount of backflow phenomena in the OGVs duct, and large vortices exist on the ventral surface of the OGVs. In contrast, the vortices shed at the top of the OGVs mix with the vortices leaking from the moving blades, intensifying the development of vortices in the dynamic-static coupling duct, and the vortex is difficult to dissipate as the flow develops. As shown from Figure 19b, after optimisation of the OGVs, the number of vortices in the dynamic-static coupling duct, and the OGVs duct is significantly reduced.

According to vortex theory, turbulent vortices are the main contributors to fan noise. For isentropic flow at low Mach numbers, the vortex sound equation is:

$$\left\{\frac{1}{c_0^2}\frac{{\partial }^2}{\partial t^2}-{\nabla }^2\right\}B=div(\omega \times v)$$

(16)

where B represents the total enthalpy of the fluid, w represents the vortex vector of flow in the flow field, and v represents the yes vector. As the stretching and breaking of vortices is a fundamental factor in the generation of the flow field, the study of vortices is a priority in fan design as unreasonable turbulent vortices can produce unstable pressure pulsations and consequently losses in the flow field.

Figure 20 compares the vorticity at each section of the spans before and after optimisation. As shown in Figure 20a,b, at 0.9 span, after the shedding vortex at the trailing edge of the moving blades before optimisation enters the OGVs duct, some air at the top of the OGVs flows from the pressure surface to the suction surface and then breaks away from the vane. As shown in Figure 20c,d, at 0.5 span, the vortex at the top of the vane further develops along the OGVs driven by the main stream, forming a stable vortex in the abdomen of the OGVs, resulting in a deterioration of the flow field. As shown in Figure 20e,f, at 0.1 span, the combined effect of the vortex coming off the trailing edge of the leading vane and the incidence of the trailing vane results in significant vortex generation in the dynamic-static coupling duct. The optimised OGVs have a more reasonable air admittance attack angle, and the separation vortices appearing at the OGVs tip are significantly reduced. At the same time, the size of the vortices in the dynamic-static coupling duct at 0.1 span is also considerably reduced because of the increased spacing between the moving blades and OGVs.

As shown in Figure 12, the pressure gradient on the pressure surface before the optimisation is more significant, the pressure pulsation on the surface of the OGVs caused by the turbulent boundary layer on the surface of the OGVs itself will produce aerodynamic noise, and the trailing edge of the blade will create shedding vortexes, which will also cause random pressure pulsations.

As shown in Figure 21, after optimisation, the pressure gradient on the pressure surface of the OGVs is more stable, the separation point is shifted back to the rear of the OGVs, and the impact of the impeller wake is on the leading edge of the OGVs position. The pressure drops smoothly, and the pressure pulsation on the surface of the OGVs is reduced. The smooth pressure gradient enables the boundary layer of the OGVs to be damaged less significantly, the separation vortex is diminished, and the aerodynamic noise generated by the pressure pulsation on the surface of the OGVs is reduced. At the same time, the optimised OGVs increase the airflow’s kinetic energy into the fan’s pressure energy, reducing the energy loss of the airflow’s frictional collision with the OGVs, thus increasing the total pressure and efficiency of the fan.

The frequency domain pulsation of the pressure before and after optimisation is shown in Figure 22. Comparing the frequency domain plots before and after optimisation, it can be seen that there is no significant change in the pressure pulsation in the inlet duct of the OGVs, with the blade frequency still being the dominant frequency. The amplitude at points P5–P9 is significantly reduced after optimisation, with the amplitude at points P7–P9 being particularly pronounced, decreasing from a maximum of 0.11 to 0.03. At the same time, the pulsation at 200–600 Hz is also significantly reduced. There is no significant change in the fluctuation of the pressure pulsation at P1–P4, and the pulsation and amplitude at P10–P12 at 0–600 Hz are reduced, from 0.02 to 0.01. In summary, the analysis indicates that the matching of the fan moving blades and OGVs before the optimisation is worse than the optimised fan, resulting in the rear guide vane being affected by the airflow from the automatic vane, thus increasing the noise of the fan and reducing the efficiency of the fan. After optimisation of the OGVs, the impact of the airflow on the OGVs is effectively reduced, and the flow of the nuclear grade axial flow fan is improved.

According to GB/T 3767-1996, “Acoustics-Determination of sound power levels of noise sources using sound pressure-Engineering method in an essentially free field over a reflecting plane” [30]. For noise analysis modelled by LMS Vritual.Lab software, a hemispherical domain field unit containing 19 domain points was established, as shown in Figure 23; for the rated operating conditions 19 domain points in the centre frequency point near the sound pressure distribution cloud, the location of the ring line A in the figure is the fan outlet 45° doubled diameter in the region. From Figure 23b, it can be found that the sound pressure value in the area where the ring line A is located after optimisation is reduced more obviously by about 3–4 dB, indicating that the optimised fan structure has a better noise reduction effect. From the previous vortex theory, it is clear that large-scale vortices will eventually break up into small-scale vortices and thus form unstable pressure pulsations affecting the acoustic performance. Therefore, combined with the Q-criteria and analysis in the previous section, it is clear that the optimised fan noise will be reduced.

5.2. Experimental Result Analysis

As shown in Figure 24, a diagram of OGVs before and after optimisation, the optimised fan was proofed to facilitate the analysis of the results before and after optimisation. The modified axial flow fan was subjected to aerodynamic performance tests, noise tests, and triaxial acceleration tests. Figure 25 shows the fan performance curves before and after optimisation. It can be seen that the air performance curve of the optimised fan follows the same trend as the original fan performance curve. The complete and static pressures of the optimised fan have been increased at all operating points. At the operating design point, the total pressure of the fan is increased by approximately 154 Pa and the efficiency rises by 3.4%. There is a slight increase in fan efficiency at low flow rates, and only a 1.14% increase in efficiency. However, at high flow rates near the operating point, efficiency increases by 2–3.2%.

Table 8. Test of the noise value under each working condition.

Working Condition	Volume Flow (m3/h)	Original SPL (dB)	Optimised SPL (dB)	Optimisation Percentage (%)
1	41,341.5	108.5	104.8	3.4
2	57,821.7	110.3	106.4	3.5
3	65,617.4	110.8	106.7	3.7
4	67,842.4	110.9	106.7	3.8
5	70,661.5	110.4	106.3	3.7
6	74,354.8	110.3	106.0	3.9
7	78,423.4	110.5	106.3	3.8
8	81,935.4	110.6	106.5	3.7
9	85,241.2	111.2	107.0	3.8

shows the noise values of the axial flow fan structure before and after optimisation under different working conditions. From the table, it can be seen that the noise of the axial flow fan has been significantly reduced after optimisation, with optimisation percentages in the range of 65,000 m³/h to 85,000 m³/h, between 3.7% and 3.9%; under minor flow conditions, the noise value of the fan is only reduced by 3.7 dB, and with the increase in flow, the noise of the fan can be reduced by 4.3 dB after optimisation. When the air volume exceeds 57,000 m³/h, the change in noise with the air volume is more minor. At the design working point, the fan noise was reduced from 110.4 dB to 106.3 dB, a reduction of 4.1 dB, with a noticeable noise reduction effect.

For a more accurate analysis of the noise, B&K microphones were used to obtain the noise spectrum; Figure 26 shows the B&K experimental equipment. The test results show that the original and optimised SPL spectral distribution peaks at the fundamental frequency and its multiples. This is because the discrete component of the fan noise contributes less to the far-field noise than the turbulent broadband noise in the frequency range tested, and therefore the fan noise changes from discrete to turbulent. The reduction in noise is more pronounced in the low frequency range of 20–160 H. Waterfall plots of the noise characteristics are shown in Figure 27a,b. The motor rotates at 25 Hz, the moving blades BPF at 545 Hz, and twice that at 1090 Hz. Figure 27a,b compare the noise waterfall plots for the original fan and the optimised one. After optimisation, the noise of the fan is reduced in the low frequency region around the motor frequency and the noise level at the blade passing frequency of 545 Hz is also significantly reduced, indicating that the fan is well optimised for noise.

As shown in Figure 28, optimised vibration acceleration is still dominated by radial acceleration. In the range of 450–600 Hz, the optimisation of the OGVs resulted in a significant reduction in the magnitude of pulsations in the OGVs duct, with a decrease in amplitude from 0.47 to 0.28 mm/s² at the vane-passing frequency and an unsatisfactory maximum magnitude of 0.52 mm/s² at the rotational frequency multiplier. The axial and circumferential vibration acceleration magnitudes in the dynamic-static coupling duct are significantly reduced. The vibration acceleration at the rotational frequency is reduced from 0.31 to 0.17 mm/s², while the pulsations are decreased considerably. The rest of the monitoring points showed no significant change in acceleration vibration.

6. Conclusions

This study carried out a parametric approach based on multi-objective optimised nuclear grade axial flow fans. Its feasibility and effectiveness were tested through experiments and numerical simulations.

(1)

Through the experimental and numerical simulation analysis of the nuclear grade axial flow fan before and after optimisation, the fan’s total pressure and noise performances were effectively improved under different operating conditions. At the design working point, the total pressure was increased by 154 Pa, the noise was reduced by 4.1 dB, and the noise was significantly reduced.

(2)

The parametric method combined with the MRGP model approach was applied to the optimised design of the OGVs of a nuclear grade axial flow fan to verify the feasibility of coupling the multiple Gaussian model with a multi-objective algorithm for the design of a nuclear grade axial flow fan. Furthermore, the Sobol method is used to determine the weight of each influencing factor on the noise and total pressure, clarify the coupling between parameters and response with fewer design variables, and provide a focused design direction for the optimised design of nuclear class axial fans. The optimisation process is simple, has fewer variables, and has applications that extend to similar fluid machinery optimisation.