1. Introduction
Pumps play an important role in industrial production, urban drainage, sewage treatment, and other fields, and they consume about 15% of the total energy consumption [
1]. Mixed flow pumps have a smaller volume with a lighter weight compared to centrifugal pumps. They also have lower shaft power compared to axial-flow pumps [
2]. Due to these characteristics, mixed flow pumps have increasingly gained recognition and have been widely used in recent years. Like most pumps, widening the high-efficiency region has been challenging, and therefore some in-depth research is required to attempt to meet the challenge.
In recent years, with the improvement of science and technology, computational fluid dynamics (CFD) has been widely used in the field of mixed flow pump optimization [
3]. Compared with experiment-based pump optimization, CFD-based optimization has the advantages of less cost involvement and more convenience. The shape of the impeller and vane diffuser can be easily modified by varying the design parameters in CFD-based optimization [
4]. However, CFD analysis cannot directly give the ideal results. To obtain the desired performance, a trial and error process is needed, which requires a lot of time and extensive experience. Over the past years, researchers have introduced various algorithms in CFD-based optimization systems to solve this problem, including the use of the design of experiment (DOE) to discretize the design space [
5,
6], surrogate models to construct the approximate model between the design parameters and objective function [
7,
8], and optimization algorithms to search the global optimization solution in the design space [
9,
10].
One of the challenges faced by the above optimization system is how to accurately describe the three-dimensional shape of the blade with as few design parameters as possible. With the improvement of computer technology and design methods, the use of inverse design method (IDM) to parameterize the blade has become an ideal method [
11], which uses circulation, stacking, and blade loading to parameterize the blade [
12]. Compared with the use of geometric parameters for impeller parameterization, the number of design parameters used in IDM is fewer, and the design parameters are more closely related to hydraulic performance. More importantly, compared with the traditional design, which makes use of the conformal transformation method to obtain the blade angle, in IDM, the design parameters directly control the blade angle and are therefore much likely to achieve better results [
13]. The effectiveness of IDM has been verified by Zangeneh and Goto [
14,
15] and has been extensively applied in mixed flow pump [
16,
17], centrifugal pump [
18,
19], and axial-flow pump [
20,
21] optimization designs.
Although the abovementioned optimizations have achieved satisfactory results, there are still some limitations. For example, most studies only use the spanwise distribution of impeller exit circulation (SDIEC) to control the impeller Euler’s head, while the influence of its distribution form on pump efficiency, cavitation, and other performances of the impeller have been ignored. As a result, during the optimization process, constant distribution of SDIEC is adopted and only the stacking condition and blade loading are used as design variables. However, the theoretical derivation of Chen [
22] and Lang et al. [
23] shows that the distribution form of SDIEC has an important influence on impeller performance and should be considered in the impeller design. The correctness of the above theory has been confirmed through a series of experimental studies [
24,
25]. The superiority of nonlinear SDIEC in mixed flow pump optimization has also been verified in the author’s previous work [
26].
In this paper, to quantitatively study the influence of SDIEC on the optimization results, the mixed flow pump impeller was optimized in two different cases by using a comprehensive optimization system. In the first case, the influence of SDIEC was ignored, while in the second case, the influence of SDIEC was considered. First, the mixed flow pump design specification and optimization strategy were introduced and the accuracy of the CFD calculation was verified. Thereafter, the strategy was applied to optimize the mixed flow pump impeller in two different cases. Finally, the optimization results were compared, and the influence of SDIEC on impeller internal flow field was analyzed.
6. Conclusions
In this study, the influence of SDIEC on the optimization results of mixed flow pump impeller was quantitatively studied by using the combined optimization system. During the optimization process, the impeller was parameterized by the inverse design method, the optimization objectives were the pump efficiencies at 1.2Qdes and 0.8Qdes, and the constraints were the pump head and efficiency at 1.0Qdes. The conclusions of this research are as follows:
- (1)
In the first case, the influence of SDIEC was ignored in the optimization process and only the stacking condition and blade loading were used as design variables, but satisfactory results were still obtained. Taking optimized impeller F1 as an example, the pump efficiencies at 1.2Qdes, 1.0Qdes, and 0.8Qdes are 80.32%, 87.62%, and 80.54%, respectively. These values correspond to 6.48%, 2.41%, and 0.06% improvement over the baseline impeller.
- (2)
In the second case, the influence of SDIEC was considered in the optimization process, and the stacking, blade loading, and circulation were used as the design variables and the upper limit of optimization was further improved. Taking optimized impeller S2 as an example, the pump efficiencies at 1.2Qdes, 1.0Qdes, and 0.8Qdes are 81.08%, 88.87%, and 81.75%, respectively. These values correspond to 0.76%, 1.24%, and 1.21% improvement over the impeller F1.
- (3)
SDIEC also has a significant influence on the blade loading distribution of the optimized impeller. In impeller F1, the blade loading at the shroud and hub are aft-loaded and fore-loaded, respectively. While in impeller S2, the blade loading at the shroud and hub are both fore-loaded.
Based on the above findings, to further improve the optimization upper limit of the mixed flow pump, it is necessary to consider the influence of SDIEC in the optimization process. Additionally, the research results can provide guidance for the optimization of other rotating machinery.