Multi-Point, Multi-Objective Optimisation of Centrifugal Fans by 3D Inverse Design Method

: In this paper, we present the design and optimization of a centrifugal fan with requirements of maximizing the total-to-static pressure rise and total-to-static efﬁciency at two operating points and the maximum torque provided by the motor power using a 3D inverse design method, a DOE (design of experiment) study, an RSM (response surface model) and a MOGA (multi-objective genetic algorithm). The fan geometry is parametrized using 13 design parameters, and 120 different designs are generated. The fan performances of all the designs at two operating conditions are evaluated through steady-state CFD simulations. The resulting design matrix is used to create an RSM based on the Kriging method and MOGA is used to search the design space using the RSM and ﬁnd the optimal design.


INTRODUCTION
Centrifugal fans are used in many applications where relatively high pressure rise is required in the compact size.The application can vary from household appliances, industrial to air-conditioning and data center cooling applications.In many of these applications the fans are required to meet multi-point requirements in terms of both pressure rise and efficiency.In the literature, several work have been published to show the aerodynamic optimisation of centrifugal fans using both numerical and experimental methods.[1] [2] [3] In this paper we started with the design of a baseline design which meets the pressure rise requirements at two operating points using 3D inverse design method.The fan aerodynamic performance (pressure-rise and efficiency) is evaluated through steady-state CFD simulations.The meridional geometry and blade loading parameters are then parameterized using 13 parameters and around 120 different blade geometries are generated with a wide range of variation of these 13 design parameters.Pressure rise and efficiency values of these 120 designs will be evaluated using CFD simulations.A design matrix consisting of all the design and performance parameters of these 120 designs is obtained.Kriging will be used to create a RSM using the design matrix data and MOGA will be used to search the optimal designs in the design space based on RSM.The final optimal design is selected from the Pareto Front and its performance is validated using CFD simulations at two operating points.The flow chart of the design and optimization process is shown in Figure 1.

3D inverse design method
The 3D inverse design method, also known as Circulation Method in the literature, is described in details in [4].It has been used in the design of all types of fans including axial fans [5] and centrifugal blowers [6].The blade geometry is calculated iteratively based on the prescribed blade loading distribution ( (  )  , the derivative of   along the meridional direction) on the meridional geometry.  is the circumferentially averaged bound circulation and is defined by Eq. ( 1) below.For incompressible potential flow, the blade loading (the pressure difference between the pressure surface and suction surface of the blade) is defined by Eq. (2).
Using the inverse design method, the blade meridional geometry and the blade thickness distribution need to be defined first.LE and TE   are used as inputs which will fix the work coefficient (specific torque/power) based on Euler's Turbomachinery Equation.The streamwise blade loading distribution (

𝜕𝑚
) is defined using a three-segment method shown in the Figure 2. The loading distribution is defined by two curves at hub and shroud.For each curve, two points on the meridional location (NC and ND) divide the curve into three parts.The first and last curves are parabolic and the middle curve is a straight line.The slope of the middle straight line is defined by a parameter called SLOPE.The loading value at LE (m = 0) is defined by DRVT which controls the incidence at the LE.Senlis (France), 27 -29 June 2022 The last input parameter required is the stacking condition which is a spanwise wrap angle distribution at a fixed streamwise location.For fans, it is common to stack at the TE.Most of the centrifugal fans are made from sheet metal and the blade has to be 2D and axial filament.However, development in additive manufacturing can make it easier to design centrifugal fan blades with 3D geometry.The 3D inverse design method used in this study makes it quite easy to design either in 3D or in 2D.In this paper, a 2D geometry is used for the blade optimization and a zero stacking at TE has been used for all the designs.Once all the inputs are specified, the 3D inverse design method computes the blade shape for a given specified blade loading.

Design specifications and baseline blade generation
The design specifications are shown in Table 1.The fan is required to meet the minimum pressure rise requirements at two operating points (OP 1 and OP 2 ).The maximum torque is set by the motor provided and the main target is to maximize the efficiency at two operating points.The geometrical constraints are that the fan diameter cannot exceed the maximum value provided and the blade has to be 2D and axial-filament.
The meridional geometry of the baseline design is shown in Figure 3.The blade thickness distribution is defined as 5 mm and the No. of blades is 10.The LE/TE   and streamwise blade loading are shown in Figure 4.It is noted that LE/TE   is normalized by the blade tip radius and tip speed.Once the solver is converged, the 3D blade from the inverse design method will be converted to a 2D/axial-filament blade using axial filament modification which axially maps the wrap angle distribution of the hub to all the other spanwise layers.Senlis (France), 27 -29 June 2022

Mesh generation
The structured mesh is generated using ANSYS TurboGrid for the impeller blade (around 500,000 hexahedra elements) and unstructured mesh is generated for the inlet cone and inblock/outblock geometries (around 500,000 tetrahedra elements).The total number of elements is around 1,000,000.The near wall element size is 0.03 mm for the impeller and 0.05 mm for the inblock/outblock which result in the maximum y+ value on the walls of less than 1.0.The No. of prism layers for the inflation is 10.As it can be seen in the Figure 5, the gap between the stationary inlet cone and rotating impeller shroud casing is well resolved.

CFD setup
Steady-state CFD simulations (RANS) are performed by using commercial software ANSYS CFX.
The inlet boundary condition is mass flow rate and the outlet boundary condition is opening with 1 atm static pressure.The impeller domain is rotational with the fixed RPM.Periodic boundary conditions are used for all domains to save the computational cost.The interface between the stationary domain and rotational domain is set as Frozen Rotor.The turbulence model used is SST k-.High resolution (2 nd -order) advection scheme is used.The convergence criteria for the continuity and momentum equations is set as RMS < 1.0e-04.The fan pressure rise and torque values are monitored and will reach to a constant value once the solution is converged.The complete CFD model is shown in Figure 6.

CFD results
Once the CFD simulations are converged, the pressure rise, torque and efficiency values can be obtained for both operating points (OP 1 and OP 2 ) and are summarized in Table 2.It can be seen that the baseline design almost meets the pressure rise requirements at both operating points and does not exceed the maximum torque.The efficiencies for OP 1 and OP 2 are 57.7 % and 44.2 % respectively.After examining the detailed flow field, it is found that the flow behaves well at hub and mis-span locations but separates near the shroud part of the blade at OP 1 and OP 2 which can be seen in Figure 7. Senlis (France), 27 -29 June 2022

DOE
For each of the 13 design parameters, a large variation of range around the baseline value is specified which allows a large design space to be explored.120 designs are generated using OLHS (optimal Latin hypercube sampling) method.For each design, the blade generation, meshing and CFD (two operating points) are performed automatically in ANSYS Workbench as shown in Figure 9.

Optimisation
The terms response surface model (RSM), surrogate model, approximation model and meta model are used as synonyms in the literature.The RSM is a mathematical model and constructed based on data from known designs (usually from DOE) and provides fast approximation and evaluation of objectives for different design parameters at new design points.
Kriging is a method of interpolation which was first proposed by a South African statistician and mining engineer Danie G. Krige and used to predict the location of unknown mineral resources.The basic idea of Kriging is that the value at an unknown point should be the average of the known values at its neighbors, weighted by the neighbor's distance to the unknown point.
MOGA (NSGA-II, Non-dominated Sorting Genetic Algorithm-II) is a multi-objective genetic algorithm which was first proposed by Deb et al. [7] It is well-suited for highly non-linear design spaces, each objective is treated separately and a Pareto front is constructed by selecting feasible non-dominated designs.
All the DOE results can be used to create a RSM using Kriging approximation method.The calculated CoP (Coefficient of Prognosis) of the output parameters for the Kriging RSM is around 0.85.The optimisation will be performed by searching the design space with the given constraints and objectives and the performance of different designs can be quickly evaluated through Kriging RSM model.The constraints and objectives used in the optimisation are listed in 0 0.5  3).This parameter is used to control the fan P-Q curve slope (shape) and should be kept as close as possible to the target value which is around -5 °.
10,000 designs are generated using MOGA (NSGA-II) and the resulting Pareto Front and selected optimal design is shown in Figure 11.An obvious trade-off between the efficiencies at two operating points can be observed.
The CFD performance of the optimal design is compared with the baseline and shown in Figure 12.The pressure rise of the optimal design is improved significantly at both operating points compared to the target and baseline values while maintaining the desirable P-Q slope (shape).The efficiencies of the optimal design are improved by around 20 % at both operating points compared to the baseline design.The torque values are reduced and do not exceed the maximum constraint.The comparison of the meridional geometry between the optimal design and the baseline is shown in Figure 13.The improvement in efficiencies can also be explained by comparing the flow field of two designs (Figure 7 and Figure 14).Especially the flow near the shroud of the optimal design is improved significantly and now fully attaches on the blade surface compared that of the baseline design.

CONCLUSIONS
In this paper, a systematic optimisation methodology using inverse design method, DOE and MOGA has been presented and is used to optimize the aerodynamic performance of a centrifugal fan at two operating points.The pressure rise and efficiencies at two operating points are improved significantly and the improvement is validated using steady-state CFD simulations.

Figure 1 :
Figure 1: Flow chart of the design and optimization process

Figure 2 :
Figure 2: Streamwise blade loading defined using three-segment method

Figure 7 :
Figure 7: Baseline relative velocity vector at OP 1 and OP 2

Figure 8 :
Figure 8: Meridional geometry parametrisation Figure 9: Automated DOE workflow in ANSYS WorkbenchOnce the CFD simulations are finished, a design matrix of 120 designs with all the design parameters and performance parameters are obtained.All the CFD results are shown in Figure10.It can be seen that a number of designs in the DOE meet the minimum pressure rise requirements at both operating points with good efficiencies compared to the baseline design.The torque values of almost all the DOE designs do not exceed the maximum constraint.

Figure 10 :
Figure 10: CFD results (pressure rise, efficiency and torque) of DOE designs

Figure 11 :
Figure 11: Pareto Front and selected optimal design

Figure 13 :
Figure 13: Comparison of meridional geometry for baseline and optimal design

Figure 14 :
Figure 14: Optimal design relative velocity vector at OP 1 and OP 2

Table 1 :
Design specifications and constraints

Table 2 :
Baseline CFD results

Table 3 .
It is noted that 'Angle' is defined by Eq. (

Table 3 :
Optimisation constraints and objectives