Developing and correction methods for a one-dimensional axial flow compressor model

This paper establishes a one-dimensional model of an axial flow compressor using the Average Mid-Diameter Method and employs polynomials to adjust the off-design stagger angle and off-design loss models. Furthermore, a model stepwise correction strategy based on the Particle Swarm Optimization (PSO) algorithm is proposed. The accuracy of the corrected one-dimensional axial flow compressor model was verified and integrated into the overall engine model for validation. Simulation results indicate that the maximum error in calculating the characteristics of the established one-dimensional axial flow compressor model does not exceed 6%, and the maximum error in simulating the overall engine model remains below 4%. These findings validate the effectiveness of modeling and correction methods for a one-dimensional axial flow compressor model targeting aero-engine component-level model.


Introduction
Mathematical models of aircraft engines play a crucial role in the analysis of engine performance, performance prediction, and control plan design.With the advancement of aerospace engineering technology, the demands on mathematical models for aircraft engines have been steadily increasing.Among the vital components of an aircraft engine, the axial flow compressor is of particular significance.Traditional modeling methods heavily rely on component test characteristics, which have gradually become insufficient to meet the computational requirements associated with variable guide vanes, variable inlet areas, and similar complexities.Additionally, full three-dimensional computational fluid dynamics (CFD) simulations, while accurate, are computationally intensive and not suitable for engine modeling.
Hence, it is necessary to further establish and correct the one-dimensional mechanism model of the axial flow compressor based on the component map characteristics to achieve refined modeling of the axial flow compressor while ensuring the calculation accuracy of the model.Presently, there is limited research in China concerning the one-dimensional modeling of axial flow compressors.Among the existing studies, Wu Hu and colleagues have explored one-dimensional methods for the geometric stability of axial flow compressors [1] .Wang Hejian and colleagues have developed a multi-stage transonic compressor performance prediction program [2] .Zhong Yongjian and others have integrated one-dimensional performance calculations with multi-objective optimization algorithms to enhance the design-point performance of two compressors [3] .Chen Jiegui has employed a one-dimensional method IOP Publishing doi:10.1088/1742-6596/2764/1/012006 2 based on mean mid-diameter to assess the off-design performance of multi-stage axial flow compressors with adjustable inlet guide vanes [4] .However, these one-dimensional modeling methods are primarily employed for preliminary compressor design and performance estimation.They emphasize the consistency of computational results with experimental trends, and even with model refinements, the overall accuracy remains limited, rendering them unsuitable for engine modeling.
In this paper, a one-dimensional performance model of an axial flow compressor using the Average Mid-Diameter method [5] was established.Applying polynomial corrections to the off-design stagger angles and off-design loss models based on design point data.Furthermore, a stepwise correction approach was proposed for axial flow compressor modeling, employing the PSO algorithm.Finally, the effectiveness of proposed modeling and correction methods was validated through comparisons with test results.

Model calculation process
The compressor model established in this paper utilizes a row-by-row calculation approach.The calculation process for a single row is illustrated in figure 1. Step 1: Mass flow rate as an input.Calculate the angle of attack angle i and stagger angle  based on the inlet velocity triangle, blade structure, and characteristics.
Step 2: Initial guess value for the axial velocity of the exit airflow 2a

Off-design stagger angle model
The stagger angle is predominantly influenced by the angle of attack and the relative Mach number of the incoming airflow.In conjunction with blade characteristic curves [6] , and by referencing classical off-design stagger angle model [7][8] this paper divides the calculation of off-design stagger angle into two segments based on the magnitude of the angle of attack.
First, calculate the critical angle of attack, In the equation, In the equation, 0  represents the reference stagger angle, 3 k , 4  k and 5 k are correction factors.

Reference loss model
The reference loss model [9][10] utilizes the Denton/Traupel loss model, which accounts for various loss mechanisms and avoids extensive reliance on empirical data.While this model has broad applicability, it may require calibration and adjustments for accuracy.
Calculate the loss coefficients, In the equation, profile Calculate the stator loss work,

2.2.3
Off-design loss model When the blade row deviates from the design point, it can cause flow separation at the trailing edge, resulting in increased flow losses [11] .Therefore, the off-design loss model in this paper, building upon IOP Publishing doi:10.1088/1742-6596/2764/1/0120064 the reference loss model, primarily accounts for the impact of attack angle and inlet Mach number on the loss coefficients.The formula for calculating the off-design loss coefficients is given in Equation 6.
In the equation, 6 k , 7 k and 8 k are correction factors.

Correction method
In this paper, the PSO algorithm [12] is employed to correct the model.PSO is a simulation of cooperative behavior in birds' foraging and movement, serving as an iterative optimization tool and a populationbased multi-objective stochastic optimization technique.The system starts with a set of random solutions and iteratively searches for the optimal values.Since the performance of different compressor models can vary significantly, there can be considerable differences in the values of reference model loss coefficients and correction coefficients.Using the PSO algorithm directly for the entire model correction may yield sub-optimal results.Therefore, this paper proposes a method for stepwise correction of the one-dimensional compressor model.Initially, the PSO algorithm is used to correct the reference loss model, aligning the design point simulation results of this model with the results from the three-dimensional model on a cross-sectional basis.Once the design point is aligned, the PSO algorithm is employed again to optimize the values of correction coefficients, ensuring that the off-design performance of the compressor matches.During this phase, adjustments to the reference loss model may be considered within a limited range.This approach reduces the particle dimension involved in the PSO optimization process, significantly improving optimization speed and effectiveness.

Calculation results of one-dimensional model of multi-stage axial flow compressor with adjustable guide vanes
To validate the applicability of the one-dimensional modeling and correction approach, a five-stage axial flow compressor with adjustable inlet guide vanes and two adjustable stator rows was selected as the test case.Using the methods outlined above, a one-dimensional model of this axial flow compressor was established and corrected.Initially, the Denton/Traupel loss model empirical coefficients were adjusted using the PSO optimization algorithm to match the design point.The comparison of the one-dimensional model's computed results for rotor inlet and exit conditions before and after correction with 3D simulation data is shown in figure 2. Although there were some discrepancies in the results before correction, they were not substantial, indicating that the Denton/Traupel loss model was reasonably suitable for this axial flow compressor.After correction, the discrepancies in rotor inlet and exit conditions were within 2%, and the errors in pressure ratio, isentropic efficiency, and design point alignment were 0.17% and 1.76%, respectively, demonstrating improved design point alignment.When comparing the calculated results of the established and corrected one-dimensional compressor model in this study with the test data, the maximum relative error in pressure ratio occurs at 95% speed, which is 5.7%.The maximum relative error in isentropic efficiency occurs at 100% speed, amounting to 2.6%.Overall, the computed results exhibit not only a consistent trend but also relatively small errors when compared to the test data, demonstrating higher precision than many one-dimensional computational programs.

Calculation results of the engine model
The mathematical model for the entire turbo-shaft engine of the axial flow compressor described above is established.The engine primarily comprises seven components: the inlet duct, axial flow compressor, centrifugal compressor, combustion chamber, gas turbine, power turbine, and nozzle.Among these components, the centrifugal compressor, gas turbine, and power turbine are modeled using test characteristics, while the axial flow compressor is modeled using the one-dimensional model developed in this paper.The engine's test conditions are at standard atmospheric conditions at sea level, with the fuel quantity set to its design value.The power turbine's speed is varied at 85% and 100% of the design value, while simultaneously adjusting the axial flow compressor's guide vane angles during the bench test.The errors between the calculations from the engine model and the experimental engine's output power, gas generator rotor speed, and total temperature at the inlet of the power turbine are shown in figure 5. From figure 5, it can be observed that the relative errors of the engine model simulation do not exceed 4%, meeting the accuracy requirements for the relative error of the engine model calculation.

Conclusion
This paper investigates the modeling and correction methods of a one-dimensional axial flow compressor model for the purpose of engine mathematical modeling.Simulation results were compared with test data, leading to the following conclusions: (1) The use of polynomial-corrected off-design models and a stepwise correction approach based on the PSO algorithm significantly improve the accuracy of the one-dimensional axial flow compressor model compared to models found in most previous literature.
(2) The established axial flow compressor model can be integrated into the engine model, and when compared to test data, the calculation error of the engine model does not exceed 4%.

Figure 1 .C , absolute angle 1  , relative velocity 1 W , relative angle 1  1 P , static pressure 1 P , total temperature * 1 T , static temperature 1 T
Figure 1.Calculation process for a single blade row.The input parameters for each blade row are obtained from the exit flow of the preceding row and include axial velocity 1a C , absolute velocity 1 C , absolute angle 1  , relative velocity 1 W , relative


represents the profile loss coefficient, trailing  represents the trailing edge loss coefficient, shock  represents the shock loss coefficient, tip  represents the tip clearance loss coefficient, and axial  represents the axial gap loss coefficient.Calculate the rotor loss work, '

Figure 2 .
Figure 2. Comparison of errors in the inlet and outlet parameters of the rotor before and after correction at the design point.The adjustment patterns for the zero-level inlet guide vanes and the first two rows of stators under different speeds for this axial flow compressor are shown in figure 3.These adjustment profiles were fed into the corrected one-dimensional model.Building upon the modified reference loss model, the PSO optimization algorithm was further employed to optimize the correction coefficients 1 8( ... ) k k in order to adjust the off-design stagger angle and off-design loss models.This process completes the comprehensive correction of the one-dimensional model.

Figure 3 .
Figure 3. Compressor guide vane adjustment pattern.The comparative test results of the corrected model and experimental data are depicted in figure 4.When comparing the calculated results of the established and corrected one-dimensional compressor model in this study with the test data, the maximum relative error in pressure ratio occurs at 95% speed, which is 5.7%.The maximum relative error in isentropic efficiency occurs at 100% speed, amounting to 2.6%.Overall, the computed results exhibit not only a consistent trend but also relatively small errors when compared to the test data, demonstrating higher precision than many one-dimensional computational programs.

Figure 4 .
Figure 4. Comparison of corrected model calculated characteristics and test characteristics.

Figure 5 .
Figure 5. Error between corrected engine model calculations and experimental data.
Calculate the blade work u L and loss work f L based on the inlet and outlet velocity triangles and loss model.Step 5: Calculate the isentropic work i L based on thermodynamic state parameters.

1
Ma represents the relative Mach number of the incoming airflow, 1