1. Introduction
In the process of liquefaction, storage, and transportation of liquefied natural gas (LNG), it is necessary to reduce the pressure of high-pressure LNG. In the traditional process, a Joule–Thomson (J–T) valve is usually used to complete this. However, when reducing the pressure through the throttle valve, it is very easy to vaporize and flash, reducing the liquefaction efficiency and increasing the total energy consumption. In recent years, the cryogenic liquid expander has gradually replaced the J–T valve. The use of a cryogenic liquid expander can reduce the production of flash steam, and can recover additional high pressure to generate electricity and improve energy efficiency. The research in references [
1,
2] shows that the use of a cryogenic expander can increase the liquefaction rate by 5%.
In hydraulic machinery, cavitation occurs easily, which affects the operational stability of the whole unit. A cryogenic liquid expander is similar to conventional hydraulic machinery, cavitation may still occur [
3]. The cavitation experiment of the cryogenic medium is difficult to operate, and the related research is less. Hord [
4,
5] carried out a series of detailed experiments with airfoils, blunt bodies and other experimental bodies of different sizes, and obtained the cavitation flow images of liquid nitrogen and liquid hydrogen, which has become an important index to verify the accuracy of numerical simulations.
Scholars have carried out a large number of numerical simulation studies on the cavitation flow simulation. Hsiao [
6] proposed a multiscale two-phase flow model based on a coupled Eulerian/Lagrangian to capture the sheet cavitation formation and bubble cloud shedding on a hydrofoil. Du et al. [
7] proposed a new cavitation model considering the evolution of bubble number density as an important factor. Örley [
8] considered the compressibility of all phases in order to accurately capture the pressure wave dynamics of collapse events. Kähler [
9] used the Boltzmann simulation method to calculate the cavitation flow of a liquid moving past a constraint. For the simulation of cryogenic fluid cavitation, Utturkar [
10] used a modified cavitation model to study the steady-state cavitation characteristics of a cryogenic hydrofoil by calibrating the correlation model coefficients. Tailan et al. [
11,
12,
13] introduced thermodynamic terms into the normal temperature cavitation model, which can better predict the temperature and pressure drop in the low-temperature cavitation zone. In the field of rotating machinery, research on cryogenic medium cavitation is mainly focused on the inducer in a cryogenic pump [
14,
15,
16].
The flow in the draft tube of rotating machinery is also the focus of research. Liu [
17] used the dynamic grid method to catch the pressure fluctuations in a pump–turbine draft tube. In the study by Zhang [
18], the vortex identification methods were reviewed to reveal the complex vortex structures in hydroturbines. Arispe [
19] used the computational fluid dynamics (CFD) method to obtain a draft tube geometry that improved the hydrodynamic performance. The pressure fluctuations generated by vortex ropes in the draft tube were studied when hydraulic turbines operated at off-design conditions [
20].
Multiobjective evolutionary algorithms are widely used in runner optimization design systems [
21,
22,
23]. The optimization strategy usually consists of a runner design method, design of experiment (DOE), CFD analysis, response surface methodology, and multiobjective genetic algorithm method. This strategy was successfully used in the design of a pump–turbine runner [
24,
25], after the optimization, the performance of the runner was obviously improved. Because of its simplicity, this optimization strategy could be used in the development of fluid machines.
Because the outflow from the runner contains a strong swirl component of velocity, the cavitation in the draft tube is serious. The flow in the draft tube is closely related to the flow condition of the runner, hence, optimizing the shape of the runner blades can achieve the purpose of minimizing the cavitation in the draft tube. This paper carries out the optimal design of the runner to restrain the cavitation in the draft tube and reduce the production of gas using the energy loss coefficient to describe the character of the draft tube.
5. Optimization Result
Figure 12 presents the black plots of the multiobjective optimization results, and the blue plots are the results on the Pareto front. Models A, B, and C (the red plots) were selected on the Pareto curve for further analysis. A comparison between the predicted values of the genetic algorithm and the numerical simulation results of CFD is shown in
Table 4. There are some errors in the head and energy loss coefficients of the three models, but the results of different runners have the same trend, which verifies that the genetic algorithm can provide a reference for the design of runners.
5.1. Effect of Blade Loading
Table 5 shows the runner blade loading of the optimized models chosen on the Pareto curve. As shown in
Figure 13, the blade loading distributions, which show great similarity, are aft-loading and fore-loading on the hub and shroud for runners A, B, and C.
Figure 14 shows the runners’ shapes.
The pressure and gas volume fraction distribution in the draft tube of the three runner models and the original model are shown in
Figure 15 and
Figure 16. When the low-pressure area in the draft tube is large, the cavitation is serious, and the gas distribution area is large. Through the calculation of the numerical simulation results, when the cavitation is serious, the energy loss coefficient in the draft tube is larger, when the energy loss coefficient is small, the distribution of the low-pressure area in the draft tube is smaller, and the cavitation can be better suppressed. It is proven that the energy loss coefficient can reflect the severity of cavitation in the draft tube.
According to the calculation results, the cavitation of the original model is the most serious, the energy loss coefficient of the draft tube is the largest, the inhibition effect of model A is the best, and the energy loss coefficient is the smallest.
According to the optimization results and CFD analysis, it can be found that runners with aft-loading and fore-loading on the hub and shroud have better performance considering the effects on the runner head and draft tube cavitation.
5.2. Effect of The Blade Lean
Based on the optimization results, more runners with the same blade loading distribution and different blade lean were designed to be investigated.
Table 6 shows the blade loading distribution, with the blade lean ranging from −10° to 10°, as shown in
Figure 17.
Figure 18 shows the gas volume fraction in the draft tube, and the energy loss coefficients are also listed in
Table 6.
The results show that the runners with large positive or negative blade lean angles have large energy loss coefficients and the gas volume fraction distribution in the draft tube is worsening. When the blade lean angles are between 0° and 5°, the cavitation in the draft tube is improved, which can also be inferred from the optimization results on the Pareto curve. Therefore, large blade lean angles are not recommended to be used for the expander runner considering the cavitation in the draft tube.
6. Conclusions
A strategy for cavitation suppression in an LNG cryogenic expander draft tube by a multiobjective optimization method was employed in the present study. The process combines the runner design method, DOE, RSM, CFD analysis and genetic algorithm.
During the optimization, runners with different design parameters were obtained using the Latin hypercube test. The second-order RSM was used to describe the relationship between runner design parameters and objective functions, and then the final optimization models were obtained using the NSGA-II genetic algorithm.
The accuracy of the simulation of cryogenic liquid cavitation is verified by comparing with the experimental results of Hord, and the cavitation model is applied to the simulation of the cryogenic expander to obtain the cavitation flow in the draft tube.
The energy loss coefficient of the draft tube was used to evaluate the flow in the draft tube. Through the numerical simulation calculation, it was proven that the energy loss coefficient of the draft tube can be used to evaluate the cavitation severity of the draft tube. The more serious the cavitation is, the greater the energy loss coefficient is, as shown in
Figure 16.
During the optimization process, the single runner head and the energy loss coefficient of the draft tube were taken as the optimization objectives. After optimization, the head is obviously increased, and the energy loss coefficient is reduced. The low-pressure area in the draft tube is obviously reduced and the cavitation is restrained to a certain extent. The runners on the Pareto curve have similar blade loadings. Considering the expander head and cavitation in the draft tube, it is recommended to design the expander runner with aft-loading and fore-loading on the hub and shroud.
The effects of blade lean angles for the cavitation in the draft tube were studied. It was found that large positive and negative angles are not recommended, and the runners with blade lean angles ranging from 0° to 5° have better performance for cavitation in the draft tube.