## 4 Optimization  
### Optimization Methodology  
Optimization is an effective tool for selecting the best design among a variety of available options through identifying design variables and design objectives. Figure 8 shows a schematic of the design optimization process in this study. The initial design geometry was obtained via meanline design, and the performance of the design geometry was evaluated via CFD. The building of a database using design of experiments (DOE) and obtaining and optimizing a metamodel were performed using Minitab software. Metamodel optimization allows special insight into system performance and allows the system to be examined across a wide range of design conditions at a relatively modest computational cost [13]. Optimization starts with building a database using DOE. The number of regression model coefficients increases rapidly as the number of design parameters increases. Various DOEs have been proposed to obtain a highly accurate response surface function with a minimum number of experiments under given conditions. In this study, the Box-Behnken design method was adopted because it can derive quadratic regression equations, easily create orthogonal blocks, and use fewer design points than 3k-factorial design or central composite design [14]. The boundaries of the design space were normalized to the range selected for each design parameter so that the existing geometry value was set to 0 and the maximum range of change for the design variables was set to \(\pm 1\). For 4 design variables, 25 test cases were designed to obtain the equation relating the design parameters and the objective function. In each test case, two design variables were fixed at 0, the existing geometry value, and the other two variables were set to \(\pm 1\). Figure 7: Performance prediction curves from numerical analysis: (**a**) expansion ratio vs. mass; (**b**) expansion ratio vs. efficiency; (**c**) expansion ratio vs. power; (**d**) expansion ratio vs. expander outlet temperature. A meta-model was built using the DOE points to construct the function \(F\) correlating the objective functions \(U_{j}\) with the design parameters \(X_{i}\) as follows:  
\[U_{j}=F_{j}(X_{i})\qquad\quad i=1,\ \ N\quad\quad j=1,\qquad M \tag{3}\]  
In the present study, the response surface method (RSM) was used to obtain an optimal design. It is widely used among approximate model-based design optimization techniques. The RSM is a technique that approximates the response surface function from the value of the objective function obtained after performing 3D CFD on a set of design variables. The response surface function \(U_{j}\) can be expressed as a quadratic polynomial using the design parameters \(X_{i}\) and \(X_{j}\) as follows:  
\[U_{j}=\beta_{O}^{j}+\sum_{i=1}^{n}\beta_{i}^{j}X_{i}+\sum_{i=1}^{n}\beta_{ii}^{ j}X_{i}^{2}+\sum_{i<k}^{n}\beta_{ik}^{j}X_{i}X_{k} \tag{4}\]  
where \(\beta_{O}\), \(\beta_{i}\), \(\beta_{ii}\), and \(\beta_{ik}\) are the polynomial coefficients, and these undefined coefficients are determined via least square regression using the 3D CFD results. A detailed analysis of the effect of parameters and their interactions on the response was also performed through the surface plots using Minitab software. In the response surface method, the reliability of the response surface function can be checked, and this reliability can be increased by removing insignificant undefined coefficients, which is called response surface model pooling. Response surface model reduction is achieved through analysis of variance (ANOVA) and regression analysis. The analysis of variance is used to determine whether each term of the response surface function is significant, and the significance is confirmed under the condition that the significance probability \(p\) is <0.05. Insignificant terms are removed in the order of block, alternating term, and square term, and analysis of variance is performed. After the analysis of variance is finished, regression analysis is performed.