Structural optimization of S-CO2 Brayton cycle compressor impeller based on evolutionary algorithm

As the critical components for marine low-speed diesel engine flue gas waste heat recovery (WHR) supercritical carbon dioxide (S-CO2) Brayton cycle system, the structure of the compressor impeller is optimized by the evolutionary algorithm (EA) based on the co-simulation of the CAESES, ANSYS CFX and Opti Slang. The law of impeller pressure ratio, efficiency and power consumption in S-CO2 Brayton cycle (SCBC) as a function of rotational speed, inlet temperature, pressure and impeller structural parameters are revealed, and the method of improving SCBC efficiency for marine low-speed diesel engine flue gas waste heat recovery is studied. The optimized impeller structure is greatly enhanced in aerodynamic performance and safety, and the isentropic efficiency is increased by 2.54%, the pressure ratio is increased by 35.64%, and the temperature rise is increased by only 4.6%. A 100kW S-CO2 compression cycle test bench was set up to verify the simulation-optimized impeller results. The final results show that the optimized impeller structure, aerodynamic performance and safety are greatly improved. It provides theoretical support for selecting and optimising compressor impellers for marine low-speed diesel engine flue gas waste heat recovery S-CO2 Brayton cycle.


Introduction
With the progress of new materials and manufacturing processes, the SCBC power generation system has gradually become a hot spot for research on waste heat recovery of ship main engines again [1].Echogen Power Systems (EPS) in the United States designed EPS100 heat recovery system for flue gas waste heat recovery of ship main engine, the temperature of waste heat supply was 532°C, the input power of waste heat was 33.3 MW, the total output value of its power output was 8.6 MW, the net output value was 8MW, and the efficiency of the power output can reach 24% [2].
The SCBC operates near the critical point (P=7.38MPa,T=304.13K) and the high density, low viscosity, and low compressibility of the S-CO2 work mass make the S-CO2 compressor outstandingly efficient and low power consumption [3].In contrast to other Brayton cycles, the S-CO2 power generation cycle accounts for about 30% of the work of compression of the turbine output, while the helium cycle accounts for about 45%, and the gas turbine cycle is even higher at 50% to 60% [4].However, the high pressure and density of the S-CO2 working mass and the drastic change in physical properties near the critical point make the existing compressor inlet state parameters difficult to control, and there are also technical challenges such as efficient sealing and gap leakage reduction.Therefore, it is crucial to carry out research on S-CO2 compressors to improve compressor efficiency and stability and to verify technical reliability for S-CO2 closed power generation cycle.The impeller, as the only element in the centrifugal compressor that does work on the gas, is critical, and its performance will determine the system waste heat recovery effect by affecting the SCBC.As a critical component in the SCBC for flue gas heat recovery in marine low-speed engines.The compressor is involved in the SCBC through an isentropic compression process, and the compressor is mainly related to the cycle efficiency and net recovery work by affecting the pressure ratio.Compared to axial flow compressors, centrifugal compressors are simple in structure and high in efficiency, and are more suitable for S-CO2 power cycles.It will be critical to carry out component-level experimental studies to test the operating characteristics of the compressor and its ancillary equipment.
SCHEMMANN [5] simulated the three-dimensional flow field and hydrostatic analysis of centrifugal compressor blades under ANSYS Workbench platform and combined it with a genetic algorithm for optimal design analysis, which showed that the maximum stress of the blades was reduced by 40% while ensuring the same efficiency.TONG [6] combined the advantages of an artificial neural network and genetic algorithm to optimize the design of centrifugal pump impeller and simulate it with the redesigned worm casing as a whole, using CFD 3D fluid simulation results as the optimized initial population, resulting in a 1.5% increase in efficiency at the design operating point and 6% increase at all non-operating points.HILDEBRANDT [7] carried out and experimentally verified the optimization of centrifugal compressors through 3D flow field simulations only and simulations combining 3D hydrodynamics and hydrostatics, respectively, showing that the difference between the optimal values is large and that the performance improvement of the compressor obtained by the limitation of aerodynamic optimization alone is significantly greater than that obtained by the two limitations of fluid optimization and safety, and the working range is also greatly reduced.SALVIANO [8] used the Bezier curve fitting method to parametrically model the impeller blade of a centrifugal compressor based on the idea of "uniform design" and combined CFD and genetic algorithm to optimize the parametric model, which improved the impeller blade shape and improved the isentropic efficiency of the compressor; For the structural optimization of centrifugal impeller.TANG [9] constructed a fluid-solid coupling simulation analysis model of centrifugal compressor impeller based on ANSYS-CFX.TANG [10] developed a design and performance prediction model for the S-CO2 compressor, analysed the effects of the number of blades and the inlet and outlet angles of the blades on the performance of the compressor, and optimized the performance of the current compressor under normal conditions using a simulated annealing algorithm.  , and the compressor inlet works near the critical point, S-CO2 shows strong characteristics of a non-ideal gas, small changes in pressure and temperature will cause large changes in density, specific heat capacity and other properties, and often accompanied by condensation at the leading edge of the blade, and the existing compressor design and performance verification methods may no longer be applicable [11].Therefore, it is more urgent to establish an experimental database of S-CO2 centrifugal compressors of different power classes by experimental methods and then to verify the existing pneumatic design theories and numerical simulation methods by calibration.Figure 1 shows the recommended technical route to be taken for S-CO2 turbomachinery of different power classes proposed by Fleming [12] et al.
In this paper, the impeller structure of SCBC compressor is optimized based on EA algorithm through the joint simulation of CAESES, ANSYS CFX and Opti Slang; the changes of impeller pressure ratio, efficiency and power consumption with speed, inlet temperature, pressure and impeller structure parameters in SCBC are revealed, which completes the research on the method of improving the efficiency of SCBC for flue gas waste heat recovery of the marine low-speed engine.The optimized impeller structure, aerodynamic performance and safety are greatly improved, which provides theoretical support for the design and optimization of compressor impeller selection for SCBC, a key equipment for flue gas waste heat recovery of marine low-speed engines.

Impact of compressor pressure ratio on the system
The marine low-speed diesel engine flue gas waste heat recovery system is shown in Figure 1.The work mass absorbs heat from flue gas in the flue gas heat exchanger (heat exchanger) of the main engine of the ship, and becomes high temperature and high-pressure supercritical work mass through isobaric heat absorption (1-2) process; the work mass enters the turbine for isentropic expansion (2-3) work process, and then enters the high temperature recuperator (HTR) and low temperature recuperator (LTR) for reheating, and heats the compressed work mass to increase the cycle (3-4, 4-5); after the LTR, the mass will be compressed in a split stream, respectively, in the main compressor and re-compressor to complete the isentropic compression (6-7, 5b-8b) process, where the main compressor with a cooler before the completion of the inter-cooling process (5a-6) to reduce compressor power consumption; the compressed mass in the HTR, LTR to complete the reheat and convergence process (7-8a, 8-1), and then the next working cycle starts.In SCBC, the cycle pressure ratio is the compressor pressure ratio.As shown in Figure 3, the efficiency and net recovered work of SCBC are improved by increasing the pressure ratio, and the improvement slows down with the increase of the pressure ratio.From the cycle parameter optimization [13], it is known that the compressor inlet pressure and inlet temperature should be as close as possible to the supercritical point and higher than the critical point, so in this model, the selected compressor inlet pressure is 7.709 MPa, and when the cycle pressure ratio is 1.7, the maximum pressure is 13.098 MPa, which can meet the design pressure of 15 MPa and the power range of the maglev motor, and the net output work can is improved.The increase in the circulating pressure ratio can improve the circulating efficiency, but for safety reasons, the maximum circulating pressure should meet the setting pressure of the circulating equipment, and the increase of circulating equipment pressure resistance will bring the cost increase.

Parameters of the compressor impeller
The SCBC compression test bench is a simplification of the S-CO2 Brayton thermal cycle, replacing the isobaric heat absorption of waste heat absorption and the adiabatic expansion of the turbine with the adiabatic expansion of the expansion valve, focusing on the performance of the core equipment in the Brayton cycle for flue gas waste heat recovery of marine low-speed engines and its influence on the Brayton thermal cycle, the performance state change of the circulating mass near the critical point and the control of the cycle system through the S-CO2 Brayton cycle compression test bench.The S-CO2 Brayton cycle compression test bench focuses on the core equipment performance and its influence on the Brayton thermal cycle, the performance state change of the circulating mass near the critical point and the control of the cycle system, and also provides the basis for the overall design of the Brayton cycle for the flue gas waste heat recovery of the marine low-speed engine.
The compression cycle test bench and schematic diagram are shown in Figure 4, which consists of the compressor, pressure-reducing valve and cooler, etc.The main compressor in this circuit is designed to work at 30,000 r/min, with a pressure ratio of 1.2 and a mass flow rate of 3.45 kg/s.The compressor in this test is a single-stage centrifugal compressor, the bearing form is electromagnetic bearing, the sealing method is a two-stage grate ring labyrinth seal, the driving method is high-speed permanent magnet motor drive, and the shaft system is a single shaft.The specific parameters of the magnetic levitation motor are shown in Table 1.The design efficiency of the Magnetic levitation motor is 76%.The impeller after the cold set grate ring is shown in Figure 5, and the optimization target is a centrifugal compressor semi-open type with a split blade ternary flow impeller.Its main structural parameters are shown in Table 2, the compressor impeller material is titanium alloy TC11, through fiveaxis continuous machining, after the completion of machining through dynamic balance test, the dynamic balance amount of 0.3g• mm, plane 1 dynamic balance amount of 0.123g• mm, plane 2 dynamic balance amount of 0.174g• mm.The compressor structure is shown in Figure 6, the impeller and magnetic levitation motor connection is a threaded connection, the dynamic sealing form of the impeller is a two-stage grate tooth seal, magnetic levitation motor through the front radial, rear radial magnetic bearing suspension motor rotation shaft-driven impeller high-speed rotation, the magnetic levitation motor by the front radial, rear radial and thrust bearing support.

Simulation settings
As shown in Figure 8, the structural parameters of the full parametric modeling of the ternary flow impeller are mainly divided into the wheel disc, wheel cover, main blade and slave blade, through which the impeller structure is parameterized, and then the multi-physics field fluid-structure coupling calculation of the impeller is carried out by ANSYS CFX; the impeller structural parameters are subjected to sensitivity analysis, and then the next set of structural parameters is given to CAESES by the algorithm for seeking the best, and a new impeller structure is generated, which is automatically searched for the structural parameters with the optimal target amount by the evolutionary algorithm to complete the automatic seeking of the impeller structural parameters.The optimization steps include firstly, parametric modeling of the impeller leaf shape and hub curve by module in Opti Slang: constructing a simulation sample set by using the Latin super-liberation method in Opti Slang, which has the advantages of small sample size and high representativeness; obtaining the mapping relationship between each optimization parameter and the optimization target by using the 3D and mechanical simulation results to establish an approximate model, and performing sensitivity analysis and verification of the metamodel of optimal prognosis (MOP) model.In this paper, the compressible ideal carbon dioxide is used as the working mass, the turbulence model is Shear Stress Transport (SST) k-ω model the wall and impeller surfaces are adiabatic without slip conditions, and the internal heat transfer is set to Total Energy.to obtain the compressor characteristic curve, the calculation is gradually increased from low speed to rated speed.The specific inlet and outlet boundaries are: compressor inlet temperature 305K, total inlet pressure 7.9MPa, the outlet is the average static pressure and mass flow rate is 3.45kg/s.
Computational iteration step is 200, and the convergence residual is 10 -4 .The higher the accuracy of the determination of convergence, the longer the required computation, the higher the accuracy of the solution.The convergence conditions are detection volume remains smooth and the global flux is conserved.
The full parametric model of the impeller in CAESES is shown in Figure 9. Since the impeller is a sixmaster and six-slave semi-open centrifugal impeller, the impeller and its airway can be simplified to a 1/6 impeller single runner model as shown in Figure 10; the grid number of the single runner model: 299918, grid size: 10mm, which has been tested for network irrelevance.The sizes of the grid are test in three ways, 6mm, 10mm, 14mm, the grid number of the single runner model are 935308, 299918, 89576.Considering the computing resources, speed and model accuracy, the 10mm grid is selected.The error between 6mm and 10mm is 4.82%, and the error between 10mm and 14mm is 12.23%.Figure 10.CFX-Pre of the impeller.

Flow-solid coupling analysis of the original impeller
The real 3D flow is decomposed into two relative quasi-positive exchange surfaces (usually called S1 and S2 flow surfaces) in the impeller mechanical quasi-ternary flow theory, which simplifies the numerical calculation by decomposing the complex ternary flow into two types of binary problems [14].
To analyze the velocity variation in the impeller channel, the relative Mach number was chosen [15].The physical meaning of the Mach number is the compressibility of the fluid, and in the flow process, the larger the Mach number indicates that the compressibility of the fluid is also greater, subsonic incompressible flow: M< 0.3; subsonic compressible flow: 0.3 ≤ M ≤ 0.8 transonic flow: 0.8 ≤ M ≤ 1.2, so in the field of aerodynamics, the Mach number can better characterize the flow than the simple fluid velocity.
For rotating machinery, its internal flow is very complex, especially at high speeds.The distribution of each flow field of the impeller at the rated speed of 30,000 rpm and rated flow rate of 3.45 kg/s is given in Figure 11-13.Among them, Figure 11 shows the relative Mach number, velocity vector and entropy distribution clouds of the flow surface at 20%, 50% and 80% leaf height S1.From the crosssectional relative Mach number distribution clouds, it can be seen that there is a small-scale surge near the leading edge of the blade inlet, and the velocity drops sharply after the surge.Downstream of the impeller, there is a low-speed fluid convergence into the trailing area, which can be seen in the secondary shunt blade on the low-speed fluid inhibition, the work mass in the leading edge of the blade there is an acceleration process, from the region of high relative Mach number to the rear of the upper impeller appeared in the low energy region of low Mach number.From the relative Mach number cloud diagram of the S1 flow surface at different blade heights, it can be seen that S-CO2 is always a subsonic incompressible flow state, which is consistent with its incompressibility to reduce compressive power dissipation.
Contour of M rel at 20% Span Contour of M rel at 50% Span Contour of M rel at 80% Span Velocity Vectors at 20% Span Velocity Vectors at 50% Span Velocity Vectors at 80% Span Contour of s at 20% Span Contour of s at 50% Span Contour of s at 80% Span As can be seen from the cloud plot of the relative Mach number distribution in the meridional plane in Figure 12, the flow begins to gradually decelerate after the inlet surge passes, and then gradually separates from the boundary layer.After the surge, the low-velocity fluid appears at the top of the impeller and then expands to near the middle of the impeller, where a low-velocity zone is formed, and this part the fluid flows out at the exit with the convergence of low-velocity fluid in the wake region.

Multi-objective optimization model
The optimization schemes for solving multi-objective optimal value problems are mainly of three main types: gradient methods, natural inspiration methods (including genetic algorithms, particle swarm algorithms, etc.), and adaptive surface methods [16].Genetic algorithms are suitable for solving optimization problems for impeller geometry parameters due to their high efficiency, global nature, and multi-parameter optimization [17].
In the genetic algorithm, it is necessary to analyze the fitness of new individuals after recombination and mutation in each generation, such as the three-dimensional flow field and mechanical analysis of each individual in each generation, which is bound to increase the computational difficulty and increase the simulation time.By using the MOP module in Opti Slang, it not only accurately establishes the mapping relationship between the optimization parameters and the objective function, but also performs the nonlinear sensitivity analysis for each optimization parameter, eliminates the interference of insensitive parameters to the establishment of the approximate model, and improves the prediction quality of the model.3D flow field calculations are required for the initial population of individuals one by one, and physical results such as isentropic efficiency, pressure ratio, and power are provided to MOP as raw data [18].For nonlinear outputs, non-convergent resultant inputs will reduce the CoP values, which may lead to larger deviations from the actual values obtained by approximating the model and affect the optimization results.The optiPlug is connected to ANSYS and Opti Slang respectively to facilitate the exchange and editing of input and output data from these two environments [19].

Sensitivity analysis
The structural parameters of the impeller as well as the impeller speed, inlet temperature and pressure are selected as optimization variables, and the compressor power consumption, efficiency and pressure ratio are the optimization objectives, which are first scanned by DOE for global sensitivity analysis.The global parameter sensitivity analysis is performed by removing the unimportant variables from the model through MOP to improve the prediction quality of the model.The results are shown in Figure 14.The parameter with the greatest influence on the compressor pressure ratio is the impeller speed, which reaches 92.4%, followed by the compressor inlet pressure with a weight of 17.5%; in addition, the amount of optimization.The Kriging method is a regression algorithm for spatial modeling and prediction (interpolation) of stochastic processes based on the covariance function.The residuals of the calculated variable speed on the optimized quantity are shown in Figure 15 and reflect the error between the predicted and actual values of the model.The CoP of the rotational speed on temperature rise, pressure ratio, and power consumption are 99.87%,93% and 90%, respectively.It indicates that the predicted values have small errors with the actual values and the model accuracy is high.The Pareto solution set is obtained by the algorithm for multi-objective model seeking, and several sets of preferred solution elements near the initial design point are selected according to the principle of ensuring that the energy head size is not lower than the initial design.Different impeller threedimensional models were established with corrected parameters, and the inlet and outlet compression ratios and isentropic efficiency of the impellers of different three-dimensional models were compared by numerical simulation to obtain the final optimization results.The Pareto front surface obtained by the evolutionary algorithm and the particle swarm algorithm for the optimization is shown in Figure 16 and 17, respectively.

Optimization results
The results of the multi-objective optimization are shown in Table 4, and the temperature rise does not change much; the main differences are in the pressure ratio and power consumption; the pressure can be increased by changing the structural parameters and other conditions (rotational speed, etc.), but it needs to bear the increase in power consumption required for the pressure increase.At the design operating point, the isentropic efficiency increases by 2.54%, the pressure ratio increases by 35.64%, and the temperature rise increases by only 4.6%.

Conclusion
Successful multi-objective geometry optimization of compressor impeller, a key component of SCBC, by coupling CAESES, ANSYS and Opti Slang.The model is optimized and validated using the evolutionary algorithm in Opti Slang, and finally, the optimization is compared with the original model and analyzed for the conclusion.The complete optimization system based on CAESES, ANSYS coupled with Opti Slang is easy to operate and efficient.Optimized model, aerodynamic performance, as well as safety, greatly improved.At the design speed, the optimization degree of both isentropic efficiency and pressure ratio is affected by the flow rate; at the design operating point, the isentropic efficiency is increased by 2.54%, the pressure ratio is increased by 35.64%, and the temperature rise is increased by only 4.6%.

Figure 2 .
Figure 2. The process of the S-CO2 recompression Brayton cycle.

Figure 3 .
Figure 3. Influence of compressor pressure ratio on SCBC.

Figure 5 .
Figure 5.The impeller of the compressor.Figure 6.The structure of the compressor.

Figure 6 .
Figure 5.The impeller of the compressor.Figure 6.The structure of the compressor.

Figure 8 .
Figure 8.The parameters of the impeller.

Figure 9 .
Figure 9. Parametric model of the impeller.Figure 10.CFX-Pre of the impeller.

Figure 13 Figure 13 .
Figure13shows a cloud of the import and export Mach number, pressure and entropy distribution.S-CO2 is accelerated by the impeller rotation, and the velocity, pressure and entropy values are enhanced to some extent.

Speed/Size Compressor Turbine Seal Bearing Generator Shafting Power/MW 75000/5cm 30000/14cm 10000/40cm 3600/1.2m Single Centrifugal Multiple Single Axial Multiple Single Runoff Multiple Single Axial Multiple Gas bearing Oil-lubricated bearing Electromagnet bearing Hydrostatic bearing Labyrinth seal Dry gas seal High speed pm motor Differentially-wound motor Gearbox -Synchronous motor Multiple shaft Single shaft 0.3 1 3.0 10 30 100 300 Figure 1. The
technical routes of S-CO2 turbomachinery with different power levels.S-CO2 centrifugal compressors have extremely high Reynolds number Re, Re can reach more than 1×10

Table 4 .
The optimization results.