Mathematical Problems in Engineering

Volume 2017 (2017), Article ID 6263274, 18 pages

https://doi.org/10.1155/2017/6263274

## Multipoint and Multiobjective Optimization of a Centrifugal Compressor Impeller Based on Genetic Algorithm

^{1}Department of Mechanics, Tianjin University, Tianjin 300072, China^{2}School of Mechanical and Aerospace Engineering, Kingston University London, London SW15 3DW, UK

Correspondence should be addressed to Zhengxian Liu

Received 25 July 2017; Accepted 6 September 2017; Published 15 October 2017

Academic Editor: Filippo Ubertini

Copyright © 2017 Xiaojian Li et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The design of high efficiency, high pressure ratio, and wide flow range centrifugal impellers is a challenging task. The paper describes the application of a multiobjective, multipoint optimization methodology to the redesign of a transonic compressor impeller for this purpose. The aerodynamic optimization method integrates an improved nondominated sorting genetic algorithm II (NSGA-II), blade geometry parameterization based on NURBS, a 3D RANS solver, a self-organization map (SOM) based data mining technique, and a time series based surge detection method. The optimization results indicate a considerable improvement to the total pressure ratio and isentropic efficiency of the compressor over the whole design speed line and by 5.3% and 1.9% at design point, respectively. Meanwhile, surge margin and choke mass flow increase by 6.8% and 1.4%, respectively. The mechanism behind the performance improvement is further extracted by combining the geometry changes with detailed flow analysis.

#### 1. Introduction

The reliance on numerical methods in the aerodynamic design process of turbomachinery components has considerably increased in the last decades. Nowadays, Computational Fluid Dynamics (CFD) codes have matured to a level where they are capable of not only providing a substantial insight into the three-dimensional flow field in turbomachines, but also calculating aerodynamic performances of the machines [1–3]. Meanwhile, higher pressure ratio demand, impeller efficiency, and compressor map width are in severe trade-off relations for transonic impellers [4]. Essentially, the aerodynamic design of transonic impellers is a multiobjective problem. It is a challenging task to design a high pressure ratio centrifugal impeller for high efficiency and wide operating range at the same time [5].

In order to obtain better designs and reduce design cost, automated design optimization of centrifugal impeller has received a widespread attention in recent years. Guo et al. [6] conducted an automated design optimization of a high pressure ratio centrifugal impeller by integrating an evolution algorithm, 3D blade parameterization method, CFD solver technique, and data mining technique. Verstraete et al. [7] combined a genetic algorithm with an artificial neural network (ANN) to optimize a centrifugal compressor. Hyun-Su et al. [8] carried out the optimal design of impeller for a centrifugal compressor under the influence of flow-induced vibration using fluid-structure interaction and response surface method (RSM). Although much progress has been made in this area, the research on the multiobjective design optimization of a blade is still insufficient especially for very high pressure centrifugal impellers, which has to take into account the trade-off among objective functions. In addition, most published optimization design techniques were performed at one single operating point, usually design point, with the danger of serious deterioration of the performance at off-design conditions such as poorer surge margin and smaller swallowing capacity.

The optimization for the whole speed line is even more challenging, especially near surge condition (certain impellers require good efficiency near the surge and most compressors need good surge margin), as CFD convergence is often not guaranteed under or near such a condition. Demeulenaere et al. [9] performed a multipoint optimization of a turbocharger compressor wheel. They simulated entire compressor stage including diffuser and housing with significantly increased computational time and effort. Their criterion of surge improvement is higher pressure ratios and the method is open to questions. Pini et al. [10] did a shape optimization of a supersonic turbine cascade at off-design conditions. These two studies propose a pseudoobjective function by summing up all the penalty terms and the original performance objectives with weighting factors, which is not strictly multiobjectives at multipoints. The design variables are so strong and complex to affect the performances that the more variables must be considered in the optimization design. However, there are less analytical expressions available to directly correlate the design variables with the performance at present. Data mining techniques are considered to be able to provide a possible way to extract some useful information from the design space and make the optimization problems in an accessible way. By detecting the features of the data set such as parameter correlations, data mining method can help to gain the mechanism over the performance improvement by optimized designs. Guo et al. [6, 11] applied SOM-based data mining to optimization results of turbomachinery. Jeong et al. [12] conducted a data mining for aerodynamic design space. However, limited research on data mining in turbomachinery design means that the process of data mining and data mining results are still not very clear.

The purpose of this paper is to develop a multipoint and multiobjective design optimization method for high pressure ratio impellers to achieve better aerodynamic performances at both design and off-design conditions and with a wider operating range. The remaining of this paper is organized as follows: first, a time series autoregressive (AR) model is developed to predict surge point from the CFD simulation of a single impeller flow passage and validated by experimental results. A self-organization map (SOM) is then carried out on samples of CFD results to explore the relation between objective functions and design parameters. A total of 27 key design variables selected by the SOM are then employed in the follow-up multiobjective optimization. Finally, the results from the optimization are shown and discussed, and some conclusions and remarks are drawn.

#### 2. Time Series Based Surge Detection

Surge is instability of the centrifugal compressors and is associated with strong unsteadiness of the inlet and outlet pressures and temperatures of the compressor. Therefore, monitoring the time based signal by adopting a Fast Fourier Transform analysis makes it possible to state the instant when the compressor starts to surge and to highlight the typical frequency peak related to surge occurrence [13–15]. However, there is no universal standard for the upper limit of pressure pulsation amplitude in surge detection, making it inconvenient to use. In CFD simulation, though a very accurate CFD model will be the perfect candidate near surge for the rise of numerical instabilities due to the large temporal and spatial gradients related to the actual flow physics. In real design environment, often only a single impeller flow passage is employed in numerical simulation in order to reduce computational time and effort. As a result, the numerical calculation may become unstable and not convergent before or after real surge because of the simplification made to the real compression system in CFD model. This makes surge detection of a new impeller design or judging the surge margin relative to the baseline impeller difficult.

##### 2.1. Autoregressive Model

Here we propose a new method for surge detection in CFD; it is based on autoregressive (AR) statistical pattern recognition algorithms [16] by monitoring model residual variances. As this paper was first written, we found a similar approach which was employed for early surge warning in axial compressors [17].

The slope of pressure rise is a reliable indicator of compressor stability. If the slope is positive the compressor will be unstable. The maximum or peak compressor pressure ratio thus defines the stability limit. Numerical studies of axial compressors [18, 19] showed that this criterion gives good results in predicting compressor stall. Centrifugal compressors may be able to operate into the left of the peak pressure ratio, but compressors will be in minor surge in this condition [20], and compressor outlet and inlet flow conditions will be pulsating and unsteady. Just like in experimental detection of surge, the unsteadiness of numerical simulation results could be utilized to find the peak compressor pressure ratio and thus help to find stability limit of a compressor.

In the surge detection, one may use the frequencies or amplitude of the unsteady signal. These two parameters are however dimensional so their values depend directly on compressor size and speed, making them unsuitable as universal thresholds in surge detection. By contrast, the time series model methods, such as AR and ARMA, are based on monitoring model residual variances, which is independent of specified compressor impellers and speed.

Autoregression is a data processing technique that is commonly used in constructing a model from time sequence data for extracting underline trends. For a stationary time series (), autoregressive process produces the following results:where is model coefficients, the random component of the model, and the variance. If is not zero-mean, then the process can be similarly carried out by introducing a new time series where is the mean of . The model coefficients, , and the variance, , can be calculated by the following method: Autocovariance function,

Coefficients are calculated by where The variance is given by By monitoring the variance of compressor unsteady signal, one may judge whether the compressor surges or becomes unstable.

##### 2.2. Surge Detection Method

For the outlet boundary condition, mass flow rate condition is not appropriate as the convergence of CFD cannot be guaranteed near surge condition, resulting in premature sinusoidal waves of flow parameters. Thus, a static pressure condition is adopted and the signal of mass flow rate at LE is monitored. It tends to be a periodic wave but not sinusoidal.

The rise of static pressure at outlet boundary is flattened out near surge condition; a dichotomy or sectional method is applied to the determination of the maximum outlet static pressure near surge condition. This method will be demonstrated in the next section with an example. The flow chart of surge detection is presented in Figure 1. An initial static pressure is specified at impeller outlet before a steady calculation. If this calculation is not convergent, the CFD simulation is switched to unsteady calculation, and the temporal mass flow rate signal at LE is monitored. The AR model is then applied to this time series signal to determine whether the compressor surges at the set pressure level and this is followed by a dichotomy method to find a new static pressure between previous converged steady and present unsteady pressures at the outlet. The detection process continues until a preset minimum pressure rise is reached, and the last steady point is regarded as the last stable point before surge.