Multiparameter and Multiobjective Optimization Design Based on Orthogonal Method for Mixed Flow Fan

: Optimization design of an impeller is critical for the energy performance of a fan. This paper takes the promotion of fan e ﬃ ciency and pressure rise as the optimization objectives to carry out multiparameter and multiobjective optimization research. Firstly, an experimental test bench is built to measure the energy performance of the original fan and verify the accuracy of the numerical method. Then, the hub outlet angle of impeller β 1 , the impeller outlet angle increment ∆ β 1 , the wrap angle ϕ , the hub outlet angle of di ﬀ user β 2 , and the di ﬀ user outlet angle increment ∆ β 2 are set as the optimal parameters to conduct orthogonal optimal design. The results show that the e ﬃ ciency of the optimal fan increases by 11.71%, and the pressure rise increases by 50.15%. The pressure and velocity distributions in an optimal fan are uniform, the internal ﬂow separation is weakened, and the inﬂuence of tip leakage ﬂow is reduced, which makes for the improvement of energy performance for the fan.


Introduction
As the fan is widely applied in national defense construction and industrial production, it plays a significant role in the national economy and daily life [1]. However, the efficiency of the fan is relatively low, resulting in a huge waste of energy. Therefore, the investigation into fan optimization is of great significance for energy conservation, emission reduction and the improvement of energy allocation.
Due to the complicated geometry of the fan, the gas flow characteristics and internal mechanism is complex. There are multiplex phenomena such as flow separation and vortex loss [2,3], which relates to the performance and efficiency of the fluid machinery. In order to promote the efficiency, it is necessary to study the flow characteristics and structure optimization. As multiple structural parameters need to be considered for the fan design and each parameter has a certain value range, which has different influence degrees on the aerodynamic performance of the fan.
In terms of experimental research, advance technology is used to observe the internal flow pattern of the fan. Comparing the flow field distribution under different design parameters, optimization for the aerodynamic performance is realized. The jet phenomenon in the impeller flow channel is experimentally observed [4], and the results reveal that the Reynolds number and the rotational speed have an important effect on the instability and uniformity of flow field. Later, hot-wire anemometer is adopted to measure the internal flow field of the fan [5], and the smoke visualization technology the optimal combination of impellers was obtained, and the aerodynamic performance of the fan was significantly improved.
The aerodynamic performance of the fan with different structural parameters has been studied through experimental research and numerical simulation in the above literature review. On this basis, different optimization algorithms are applied to the fan design, but with various optimization goals. Furthermore, now researches mainly focus on single-target optimization studies, while multiparameter and multiobjective optimization studies on the fan are relatively rare.
In this paper, investigations on the multiparameter and multiobjective optimization design of the fan are conducted. The hub outlet angle of impeller β 1 , the impeller outlet angle increment ∆β 1 , the impeller wrap angle ϕ, the hub outlet angle of diffuser β 2 , and the diffuser outlet angle increment ∆β 2 are set as optimal parameters, and the pressure rise and efficiency are chosen as optimal objectives. With the conduction of orthogonal optimization design, the optimal fan is obtained. The influence degree of geometric parameters on the aerodynamic performance are studied, the optimal design scheme is obtained, and the internal flow mechanism is explored.

Physical Model of the Fan
In this paper, the fan is constructed with modeling software BladeGen (ANSYS Inc., Pittsburgh, PA, USA), and Table 1 presents the structural parameters of the fan.  Figure 1 shows the geometric structure of the fan, which mainly includes the inlet pipe, outlet pipe, impeller, and diffuser. In order to ensure the stability of the numerical calculation, equal-diameter inlet pipe and outlet pipe are added. The fan in the present work is a mix-flow fan with rated flow rate of 18 L/s and rated power of 500 W, and it is manufactured by Tsinghua University.

Computational Mesh of the Fan
In this paper, ICEM 17.0 (ANSYS Inc., Pittsburgh, PA, USA) and Turbogrid 17.0 (ANSYS Inc., Pittsburgh, PA, USA) are employed to generate the structured mesh of each component [27], and grid encryption is performed at the junction of the impeller blade and hub, and at the junction of the diffuser blade and hub. The mesh generation and mesh details of the impeller and diffuser are shown in Figure 2a

Computational Mesh of The Fan
In this paper, ICEM 17.0 (ANSYS Inc., Pittsburgh, PA, USA) and Turbogrid 17.0 (ANSYS Inc., Pittsburgh, PA, USA) are employed to generate the structured mesh of each component [27], and grid encryption is performed at the junction of the impeller blade and hub, and at the junction of the diffuser blade and hub. The mesh generation and mesh details of the impeller and diffuser are shown in Figure 2a

Numerical Method
In this paper, the computational fluid dynamics software CFX 17.0 (ANSYS Inc., Pittsburgh, PA, USA) is adopted for numerical simulation of the fan [28]. The SST k-ω turbulence model is chosen to

Numerical Method
In this paper, the computational fluid dynamics software CFX 17.0 (ANSYS Inc., Pittsburgh, PA, USA) is adopted for numerical simulation of the fan [28]. The SST k-ω turbulence model is chosen to solve the three dimensional time-averaged Navier-Stokes equation, and the model constants are given as β' = 0.09, α = 5/9, β = 0.075, σ k = 2 and σ ω = 2. The impeller is set as the rotating domain; the inlet pipe, diffuser and outlet pipe are set as static domains. The connection method between the impeller and the diffuser is connected with frozen rotor technology.

Boundary Conditions
According to the numerical method in Section 3.1, the boundary conditions for calculations [29] are shown in Table 2.

Mesh Independence Verification
It is important to choose a reasonable mesh for numerical simulation, which can save time and computing resources with the premise of ensuring calculation accuracy. Therefore, this paper uses five groups of mesh to conduct the grid independence test [30]. The mesh number distribution of each component is presented as follows, and the mesh independence verification obtained is shown in Table 3. The final number of selected mesh is 99,900 for inlet pipe; 1,625,886 for impellers; 940,401 for diffuser; 104,400 for outlet pipe; and the total number is 2,770,587. In addition, the average value of y+ on the impeller blade is less than 5.

Simulation Validation
In order to verify the accuracy of the numerical method, a test bench and fan are set up for experimental research, as shown in Figure 3. The experimental research is carried out under a temperature of 25 • , pressure of 1 atm, and relative humidity of 50%. The fluid medium is air with a density of 1.185 kg/m 3 and viscosity of 18.1 Pa·s.

Simulation Validation
In order to verify the accuracy of the numerical method, a test bench and fan are set up for experimental research, as shown in Figure 3. The experimental research is carried out under a temperature of 25°, pressure of 1 atm, and relative humidity of 50%. The fluid medium is air with a density of 1.185 kg/m 3 Figure 4 presents a comparison between the numerical calculations and the experimental results. As can be seen in Figure 4, the red curve is the numerical simulations, and the black curve stands for the experimental measurement results. The average error between the numerical and experimental results is 2.84%, which verifies the accuracy of the numerical method.  As can be seen in Figure 4, the red curve is the numerical simulations, and the black curve stands for the experimental measurement results. The average error between the numerical and experimental results is 2.84%, which verifies the accuracy of the numerical method.

Structural Optimization Based on Orthogonal Method
In this paper, the orthogonal method is applied in fan optimization. Selecting representative points and the results of comprehensive response can be obtained through a small group of experiments. The characteristics of orthogonal method are that the number of experiments is small, while the conclusions are reliable [31]. Based on the experimental study and numerical research, this paper selects the following five parameters and four levels for orthogonal method optimization. The five parameters are the hub outlet angle of impeller β1, the impeller outlet angle increment Δβ1, the impeller wrap angle φ, the hub outlet angle of diffuser β2, and the diffuser outlet angle increment Δβ2. The optimization goals are the efficiency and the pressure rise of the fan.

Structural Optimization Based on Orthogonal Method
In this paper, the orthogonal method is applied in fan optimization. Selecting representative points and the results of comprehensive response can be obtained through a small group of experiments. The characteristics of orthogonal method are that the number of experiments is small, while the conclusions are reliable [31]. Based on the experimental study and numerical research, this paper Energies 2020, 13, 2819 7 of 15 selects the following five parameters and four levels for orthogonal method optimization. The five parameters are the hub outlet angle of impeller β 1 , the impeller outlet angle increment ∆β 1 , the impeller wrap angle ϕ, the hub outlet angle of diffuser β 2 , and the diffuser outlet angle increment ∆β 2 . The optimization goals are the efficiency and the pressure rise of the fan.
The expression of pressure rise is as follows [28]: where PR is pressure rise, P inlet is inlet pressure and P outlet is outlet pressure.

Analysis of Orthogonal Experiments
According to the orthogonal experiments in Section 4.1, the range analysis of the fan efficiency and pressure rise is performed with the following expression [31]: where η i is the efficiency of each orthogonal experiment, K i is the sum of η i , and R stands for the range.
The results are shown in Tables 6 and 7. In order to simultaneously consider the promotion of fan efficiency and pressure rise, this paper conducts a multiobjective optimization study and defines the variable F for range analysis. F can be calculated by formula (5) [31]: where ∆PR and η are calculated by formula (6) and formula (7), respectively.
As for multiobjective optimization research, different weight coefficients need to be given to the optimization goals. In this paper, the weight of the efficiency λ 2 is taken as 0.8 since the fan efficiency is the principal object to be considered, and the weight of pressure rise λ 1 is taken as 0.2. Range analysis is performed according to the value of F, and the new range analysis is shown in Table 8. The ranking of the impact degree for the five factors is the impeller wrap angle ϕ, the hub outlet angle of diffuser β 2 , the hub outlet angle of impeller β 1 , the impeller outlet angle increment ∆β 1 , and the diffuser outlet angle increment ∆β 2 .
Therefore, considering comprehensively increasing the efficiency and pressure rise of the fan, the impact degree of the impeller wrap angle ϕ is the largest, followed by the hub outlet angle of diffuser β 2 , the hub outlet angle of impeller β 1 , the impeller outlet angle increment ∆β 1 , and the diffuser outlet angle increment ∆β 2 . Finally, the optimization parameters are selected as follows: The hub outlet angle of impeller β 1 is 50 degrees, the impeller outlet angle increment ∆β 1 is 8 degrees, the impeller Energies 2020, 13, 2819 9 of 15 wrap angle ϕ is 60 degrees, the hub outlet angle of diffuser β 2 is 80 degrees, and the diffuser outlet angle increment ∆β 2 is 8 degrees.

Performance Comparison
According to the orthogonal experiments in Section 4.2, the optimal structure of the fan is obtained. The efficiency of original fan is 41.21%, while the efficiency of optimal fan is 52.92%, which is improved by 11.71%. Furthermore, the pressure rise of original fan is 7152 Pa, while the pressure rise of the optimal fan is 10,739 Pa, which is promoted by 50.15%. Figure 5 demonstrates the pressure distribution of the pressure side in the original and optimal fan. It can be seen from the figure that the pressure gradually decreases from the leading edge to the trailing edge of the blade. This is because the air flows from the inlet to the impeller and impacts the leading edge, causing the generation of local high-pressure region. It is also displayed that the pressure gradually increases from the bottom to the top of the blade, because the interaction force between the blade and the airflow increases with the peripheral speed. The pressure change on pressure side is more obvious from 0.8Q d to 1.2Q d . Compared with the original fan, the pressure distribution in the optimal fan is uniform, the local high-pressure region is reduced, and the pressure gradient is smaller, which indicates that the airflow is smoother in the passage.

Simulation Result of Pressure
Energies 2020, 13, x FOR PEER REVIEW 10 of 17 distribution in the optimal fan is uniform, the local high-pressure region is reduced, and the pressure gradient is smaller, which indicates that the airflow is smoother in the passage.   Figure 6 illustrates the pressure distribution of the suction side in the impeller. It can be seen from the figure that local high-pressure regions exists at the leading edge in the original fan, and the area becomes more obvious as the flow rate increases. While the pressure distribution is more uniform within the optimal fan, and the flow separation does not easily occur as the pressure gradient is smaller, which is in line with the design expectations.  Figure 6 illustrates the pressure distribution of the suction side in the impeller. It can be seen from the figure that local high-pressure regions exists at the leading edge in the original fan, and the area becomes more obvious as the flow rate increases. While the pressure distribution is more uniform within the optimal fan, and the flow separation does not easily occur as the pressure gradient is smaller, which is in line with the design expectations.

Simulation Result of Velocity
It can be inferred from Figure 5 and Figure 6 that the pressure on the pressure side is higher than that on the suction side, and the local high-pressure region exists at the leading edge of the blade. Due to the large pressure difference at the leading edge and the existence of tip clearance, the gas velocity is relatively large and the flow pattern is more complicated. Therefore, the velocity distribution of different spans under the design flow rate are compared.
As for the optimal fan in Figure 7, the flow patterns at different spans are smoother than that in the original fan. It can be seen at span 0.1 that flow pattern at the inlet of optimal fan is more even, indicating that the flow condition has been improved. As shown at span 0.5, the pressure gradient inside the optimal fan is smaller. Furthermore, there are no turbulent vortices that occurred in the flow channel, and no obvious flow separation phenomenon exists in the impeller, indicating that the flow state is ideal. As described at span 0.9, the large vortex area appeared in the original fan with irregular distribution. However, the vortex area is significantly reduced and the flow velocity is relatively smooth within the optimal fan. Figure 7 confirms that the optimal fan exhibits better aerodynamic performance.

Simulation Result of Velocity
It can be inferred from Figures 5 and 6 that the pressure on the pressure side is higher than that on the suction side, and the local high-pressure region exists at the leading edge of the blade. Due to the large pressure difference at the leading edge and the existence of tip clearance, the gas velocity is relatively large and the flow pattern is more complicated. Therefore, the velocity distribution of different spans under the design flow rate are compared.
As for the optimal fan in Figure 7, the flow patterns at different spans are smoother than that in the original fan. It can be seen at span 0.1 that flow pattern at the inlet of optimal fan is more even, indicating that the flow condition has been improved. As shown at span 0.5, the pressure gradient inside the optimal fan is smaller. Furthermore, there are no turbulent vortices that occurred in the flow channel, and no obvious flow separation phenomenon exists in the impeller, indicating that the flow state is ideal. As described at span 0.9, the large vortex area appeared in the original fan with irregular distribution. However, the vortex area is significantly reduced and the flow velocity is relatively smooth within the optimal fan. Figure 7 confirms that the optimal fan exhibits better aerodynamic performance.

Simulation Result of Tip Leakage Flow
Since the pressure on the pressure side of the blade is higher than that on the suction side, the airflow on the pressure side will flow through the tip clearance between the blade and shroud under the effect of pressure difference, resulting in the occurrence of tip leakage flow [32][33][34]. In addition, the existence of the tip leakage flow will affect the flow stability of the mainstream in the flow channel. Therefore, the impact of the tip leakage flow is expected to be reduced in the optimal fan. Figure 8 shows the velocity distribution of the tip clearance area near the leading edge of the blade. As for the original fan under different flow conditions, the tip leakage vortex is located at the

Simulation Result of Tip Leakage Flow
Since the pressure on the pressure side of the blade is higher than that on the suction side, the airflow on the pressure side will flow through the tip clearance between the blade and shroud under the effect of pressure difference, resulting in the occurrence of tip leakage flow [32][33][34]. In addition, Energies 2020, 13, 2819 12 of 15 the existence of the tip leakage flow will affect the flow stability of the mainstream in the flow channel. Therefore, the impact of the tip leakage flow is expected to be reduced in the optimal fan. Figure 8 shows the velocity distribution of the tip clearance area near the leading edge of the blade. As for the original fan under different flow conditions, the tip leakage vortex is located at the top of the suction side. With the decrease of flow rate, the range of the leakage vortex gradually spreads to the hub along the radial direction, and the countercurrent area expands, which in turn affects the mainstream. While it can be seen that the leakage vortex dose not diffuse to the flow channel in the radial direction for the optimal fan, therefore, less influence is added on the mainstream.
Energies 2020, 13, x FOR PEER REVIEW 13 of 17 top of the suction side. With the decrease of flow rate, the range of the leakage vortex gradually spreads to the hub along the radial direction, and the countercurrent area expands, which in turn affects the mainstream. While it can be seen that the leakage vortex dose not diffuse to the flow channel in the radial direction for the optimal fan, therefore, less influence is added on the mainstream. As shown in Figure 9, the streamline near the impeller inlet expands to the flow channel and blends with the mainstream for the original fan under a flow rate of 0.8Qd. While the streamline near the impeller inlet is uniform for the optimal fan and the streamline expands along the tip clearance with regular distribution. When the flow rate increases to 1.0Qd, the streamline near the impeller inlet has a significant mixing effect with the mainstream for the original fan, while the distribution in the optimal fan is smooth. As the flow rate changes to 1.2Qd, the streamline expands into the next stage along the tip clearance for the original fan, and the flow velocity is faster. While the streamline distribution in optimal fan is still even in the current and next passage. It can be inferred that the optimal fan had a certain ability in the suppression of the tip leakage flow, which creates small impact on the flow stability of the impeller. As shown in Figure 9, the streamline near the impeller inlet expands to the flow channel and blends with the mainstream for the original fan under a flow rate of 0.8Q d . While the streamline near the impeller inlet is uniform for the optimal fan and the streamline expands along the tip clearance with regular distribution. When the flow rate increases to 1.0Q d , the streamline near the impeller inlet has a significant mixing effect with the mainstream for the original fan, while the distribution in the optimal fan is smooth. As the flow rate changes to 1.2Q d , the streamline expands into the next stage along the tip clearance for the original fan, and the flow velocity is faster. While the streamline distribution in optimal fan is still even in the current and next passage. It can be inferred that the optimal fan had a certain ability in the suppression of the tip leakage flow, which creates small impact on the flow stability of the impeller.

Conclusions
In this paper, the numerical solving method of the Reynolds time-averaged Navier-Stokes equation is adopted to investigate the flow characteristics and internal flow field of the fan. Comparing with the experimental results, the accuracy of the numerical method is verified. On this basis, the orthogonal method is employed to conduct the multiparameter and multiobjective optimization of the fan to improve the energy performance of the fan. The main work can be concluded as follows.
(1) An experimental test bench is set up for the measurement of the energy performance. Then the numerical calculations are compared with experimental results to verify the accuracy of the numerical method.
(2) Five important parameters and two optimization targets are set, and the orthogonal method is adopted for the optimization design of a fan. The influence degree of geometric parameters on the energy performance of the fan is obtained, and then an optimization scheme is proposed.
(3) The results show that the efficiency of the optimal fan increases by 11.71%, and the pressure rise increases by 50.15%. The internal flow separation phenomenon in optimal fan is weakened, and the impact of the tip leakage flow is reduced.