2.2. Response Surface Method Creating the design space points by Box-Behnken technique for a problem with three parameters varying at three levels which are named, -1, 0 and 1 is shown in Fig. 1. Each point of designed experiments suggests a combination of three parameters. For this problem, an imaginary cube around the central point (means all parameter at their 0 level) suggests the black points on the cube edges as the design space. The design points, in fact are lying on a sphere of radius V2 (Montgomery, 2017). This design does not include the cube vertices (unlike the CCD design technique), which can be advantageous when these points are impossible to examine based on the problem constraints. Evaluating the effect of design space points on the objective functions could be handled using Response Surface Method (RSM). This method includes mathematical or statistical techniques to analyze the problems in which the objective, called response, is influenced by many parameters (Montgomery, 2017). It approximates the relation between the design parameters and response by surface fitting to optimize the objective or response. Fig. 2. Optimization process in this research. Fig. 3. Meridional view of cases I, II and III. Fig. 4. Impeller cast model, test rig and the schematic. Fig. 6. Mesh independency study. Fig. 5. Impeller and volute grids for Case I. Overall efficiency is then calculated using Eq. (5) in which c and ware the impeller calculated torque and impeller rotational speed, respectively. ets Fig. 7. Experimental and numerical results. Fig. 8. Experimental results comparison for all three cases. Table 3 Cases from DoE design Space Table 2 Optimization parameters variation boundaries Box-Behnken technique is used to construct DoE design space. This method with 5 parameters, each at three levels, results in the total number of 46 design points. As it was mentioned, this technique reduces the number of required experiments noticeably to identify the best case while the algorithm like full factorial, required 3° design points in this problem (Montgomery et al., 2009). Table 5 Impellers with maximum head, efficiency and multi-objective optimized result: Fig. 13. Streamline near hub and near shroud spans for the optimal cases and Case 0.
