Performance optimization of flywheel using experimental design approach

Introduction

Flywheels have been used for a long time as mechanical energy storage devices; one of the earliest forms is a potter’s wheel. Today, the flywheels are a practical and attractive solution for energy storage technologies. The recent improvements in materials, magnetic bearings, power electronics, and the introduction of high speed electric machines, has re-emerged, the flywheel, as a promising for energy storage applications (Bakay et al. 2010; Bolund, Bernhoff, and Leijon 2007; Conteh and Emmanuel 2016; Erhan and Ozdemir 2021; Haruna et al. 2011; Liu and Jiang 2007; Murakami et al. 2007; Šonský and Tesař 2019). Compared to a chemical battery, the FESS has great advantages in durability and lifetime, especially in hot or cold environments (Takahashi, Amei, and Itoh 1990; Takahashi, Itoh, and Andoh 1989), that makes it an appealing candidate for a wider use in renewable energy systems and become an attractive battery alternative. The storage capacity of flywheel depends on the rotational speed and the moment of inertia; which they are closely related to the form of rotor.

Many research works have been done on flywheel geometry. A study of flywheel geometry by Shinde et al. (2017) showed that flywheel with triangular cross-sectional geometry is more efficient than compared to the rectangular and circular cross-section of geometry. A similar study was done by Pawar et al. (2018); after performing a series of finite element analysis, it was observed that with the change in flywheel geometry and its material, there is a significant effect on the performance with a reduction in weight of the flywheel. More recently, a study conducted by Vardaan and Kumar (2022) where they carried out the analysis of the thresher machine flywheel. The authors concluded that the amount of kinetic energy stored by wheel-shaped structure flywheel is greater than any other shape of the flywheel. Therefore, it can be noted that the effect of geometry plays a significant role in flywheel performance.

The purpose of this study is to demonstrate how to find the most efficient geometry in order to increase the performance of the flywheel. A virtual 3D model of the flywheel is designed and analyzed, which provides a database of the design for mathematical modeling and optimization of the flywheel.

Flywheel modelling & simulation

In the present study, among parameters defining the flywheel geometry, three of them were taken under considerations which seem to have significant influence on the flywheel performance. As demonstrated elsewhere (Chiriță et al. 2017), the outer diameter, the rim width and the rim height have a remarkable effect on the specific energy accumulated by the flywheel. However, an increasing of the thickness of the disc leads to a decrease in the specific energy. The geometry parameters are given in Table 1 and illustrated in Figure 1.

Figure 1:

Section view of 3D model of the flywheel.

In order, to optimize the performance of the flywheel, different configurations have been created and analyzed by using SolidWorks software. The flywheel was considered of Titanium alloy material (Ti-3Al-8V-6Cr-4Mo-4Zr) with the mechanical properties as noted in Table 2, rotating at 70,000 rpm, which correspond to an angular velocity of 7330.38 rad/s.

A total of 27 configurations of the flywheel geometry has been created, simulated and optimized. Therefore, using the capabilities of Evaluate tool in SolidWorks, the flywheel mass (m) and the moment of inertia (I) have been defined. Then, the kinetic energy (E_c) stored in the flywheel (Eq. (1)) and the specific energy (E_spec) in Eq. (2), were calculated with the formulas below:

The present study was conducted such that highlights the varying effect of the geometry parameters, on the mechanical stress of the flywheel. SolidWorks Simulation is powerful Finite Element Analysis (FEA) software, that allowed to determine the maximum stresses in the flywheel for each analyzed 3D model configurations.

The geometry parameter values, the mass, the moment of inertia, the kinetic and specific energy and the maximum of Von-Mises stress values are given in Table 3.

Mathematical modelling

In order to investigate the effect of geometry parameters on the flywheel performance under high speed rotation, the modelling by the mathematical method of design of experiments seems adequate. By using the functionalities of MODDE, a software package that is used by scientists, engineers and statisticians alike to help understand complex processes and products (Umetrics™ 2017), the study was conducted using the No-conventional design of experiments, which allow to use the obtained data to establish a mathematical model that makes possible to evaluate the degree of the effect of each parameter separately and their simultaneous interaction, on the accumulated energy and the mechanical stress of the flywheel. Then, according to the desired limit of response, the geometry parameters will be optimized to reach the best possible solution.

The effect of geometry parameters

Figure 2 is a graphical presentation of the estimated coefficients of the mathematical models. From the vertical bars of the diagram results that, in general, the effects of parameters separately appear to have a positive act on the increase of the specific energy. The rim diameter (Dr) and rim height (Hr) presents a relatively large effect compared to that of width rim (Wr). Although, the effects of parameters separately might increase the maximum value of Von-Mises stress. The rim diameter shows the highest effect, followed by rim height and relatively less important effect of width rim.

Figure 2:

Diagram representation of the parameters effects.

On the other hand, the interaction effect of Dr*Wr and Dr*Hr. leads to an increasing of the specific energy. However, the simultaneous increase of Wr*Hr, may negatively affect on this last and even on Von-Mises stress by increasing its maximum value. The highest and lowest interaction effect on Von-Mises stress are shown successively by Dr*Hr and Dr*Wr interaction of parameters.

Optimization of parameters

The aim of this part is to design a flywheel with a high energy storage density and supporting high speed. By observing the effect of Wr and Hr separately, which is positive on the specific energy and their interaction (Wr * Hr) which seems has a negative effect; for this reason, care has to be taken in manipulating the values of design parameters. Therefore, it is appropriate to apply the optimization of geometry parameters in the design of flywheel; by using the capabilities of Optimizer tool on MODDE, which helps on how to find the optimal conditions or best compromise as a set point and alternatively the most robust set point.

In general, flywheel designs are subject to size constraints in addition to stress constraints. Actually, the load on the flywheel is mainly centrifugal force. Therefore, the objective of optimization is based on a combination of reducing the stress constrains applied to the flywheel and reaching the highest level of energy storage density (Table 5).

Table 5, shows the optimized values of geometry parameters, with the goal of maximizing the specific energy while minimizing Von-Mises stress, such a way that the high value of the stress constraints, shall be less than the yield strength of the flywheel material. The table exposed the design parameter values which allow to obtain the highest possible value of specific energy under minimum values of applied stress constraints and without exceeding the Yield strength. It is noted that the optimized values of rim diameter and rim height, reached their maximum levels, however, it reached its lowest level for the rim width. As demonstrated previously, the simultaneous increase of geometry parameters can have a negative effect on responses.

The optimized values of design parameters of the flywheel, at constant rim diameter values of 105 and 115 mm, are shown in Table 6. The specific energy exhibited a high value for the configuration M₁₁₅, about 16% more than for M₁₀₅. This result was achieved with a relatively little decrease of the rest of the parameters. A remarkable increase of von-mises stress value is noted for M₁₁₅, without certainly exceeding the yield strength.

In Table 7, the optimized values of design parameters of the flywheel are obtained at constant rim width values of 5 and 7 mm. From the table, it is noted that the high value of the specific energy was recorded by the configuration M5. However, the rise in the constant value of rim width, leads to a relative decline in response of specific energy, in M7. For the rest of parameters, no change was noted of the rim diameter, and a considerable decrease in rim height. On the other hand, the maximum stress value of both configurations, do not show a significant difference.

In Table 8, the optimizing was carried out at constant rim height values of 15 and 25 mm. The optimized values of design parameters of the flywheel show a high value of specific energy in the configuration M₂₅. Compared to M₁₅, only the rim width was readjusted to 7.33 mm, with the change of the constant value of rim height from 15 to 25 mm, however, the rim diameter remains similar. As for Von-Mises stress, it was relatively highest in M₂₅.

Conclusions

Flywheel mechanism can effectively improve the mechanical performance of the system, and the flywheel’s energy storage density can be significantly improved by highlights the degree of the effect of the design geometry parameters. This paper presents a mathematical method for optimization of geometry parameters of the flywheel which offer many possibilities to optimize according to the desired limit of response.

The results obtained in the present study can be summarized by the following conclusions:

1. The 3D model configurations were created and simulated using the modeling capabilities and powerful Finite Element Analysis (FEA) of SolidWorks software, which reveal a great efficiency in the study of different design cases.

2. The outstanding modeling capabilities of MODDE based on the Design of Experiments (DoE) approach, allow to establish a mathematical model of the response, which easily explains the degree of the effect of the design parameters separately and their simultaneous effect.

3. The analysis of mathematical models of the responses of specific energy and Von-Mises stress, has demonstrated that the rim diameter presents a high influence on the accumulated energy by the flywheel; followed by the rim height and the rim width. However, their interaction may adversely affect the responses.

4. The use of the capabilities of Optimizer tool on MODDE, can be very helpful on how to find the optimal values of design parameters based on a combination of primarily reaching the highest level of energy storage density and reducing the stress constrains exercised centrifugal force and by applied to the flywheel.