# Optimization of a Centrifugal Compressor Using the Design of Experiment Technique
## 1 Introduction  
Design and optimization of rotating machines by means of the numerical optimization methods has become an established technique in all industries for many decades. In centrifugal compressors, the impeller plays a most important role in machine performance and this has attracted significant interest of researchers' and developers over several decades. Not only will a good impeller design bring significant performance benefits itself, but a good impeller will deliver a more uniform flow to the diffuser hence improving the performance of the whole compressor. In the preliminary design stage, inlet and outlet angles and the main dimensions of the impeller are obtained. This design will be followed by detail design where the complete gas path and the geometry are specified which includes hub and shroud contours and the blade curvature for the vaned components [1]. During detail design some adjustments are repeatedly necessary as the acceptable performance in a relatively wide design range is constrained by some limitations [2]. Thermodynamics and fluid mechanic principles, mechanical, vibration, and the manufacturing constraints and the cost dictate these limitations. Design and optimization of rotating parts by means of the numericaloptimization methods has become an established technique for recent decades. Numerical optimization techniques help compressor designers obtain optimum efficiency or maximum pressure ratio [3] or widener stable operating range [4]. Evaluating the effect of different aspects in the optimization process simultaneously imposes heavy computational cost, hence some techniques are developed to reduce the problem exploring space. The Design of Experiment (DoE) technique is one of this methods. The DoE technique helps the designer limit the calculations needed to find the effects of input variables on the output variables. In this technique the shortcoming of inaccuracy of using an approximation function is compensated by reducing the number of calculations. A combination of mathematical and statistical methods are used for constructing the design space [5]. Many researchers have targeted the optimization of centrifugal compressors. Many parameters are considered in this regard. For example optimization techniques were implemented to optimize the blade profile [6], meridional profiles [7; 8], blade sweep angle [9], the splitter [10], the vaneless diffuser [11] and there are many others. In the metamodeling technique to build a surrogate model, DoE can be employed to generate appropriate training samples [12]. [PERSON] et al. optimized the geometry of a return channel for a centrifugal compressor through the artificial neural network (ANN) method which is trained using a database of 3D CFD solutions created with a DoE technique [13]. [PERSON] et al. also optimized the meridional geometry of an impeller. They used Latin-hypercube sampling of DoE to generate design points for implementing the Neural Network (NN) method [14]. [PERSON] and [PERSON] designed a compressor diffuser using DoE to see the effects of four parameters and their interactions on the blade efficiency [15]. [PERSON] et al. presented a parametric study for an optimization problem using DoE. They identified the most influential geometrical factors affecting impeller performance and optimized three different impellers [16]. [PERSON] et al. optimized a centrifugal compressor impeller using DoE and response surface models to increase pressure ratio, isentropic efficiency and to reduce work input of a compressor [17]. [PERSON] et al. introduced a response surface method for constructing a surrogate model to parametrize the design space for aerodynamic optimization of a transonic fan [18]. [PERSON] et al. considered the \(\beta\) angle distribution to optimize a centrifugal compressor impeller. They implemented the Kriging interpolation to construct the approximation function [19]. In this article three main characteristics of an impeller are considered for investigation of their effects on performance.