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| Fig. 1: Schematic of a rotating flywheel. (Source: A. Kashefi) |
Flywheel energy storage has a wide range of applications in various industries such as wind generators, marine technologies, aeronautical vehicles, etc. [1-3] In simple words, kinetic energy is stored in flywheels with a determined angular velocity. The geometric structures and material features of flywheels used in today's industry are complicated and required advanced numerical solvers to analyze.
In this article, we study flywheels using a simplified, but yet useful, model. By this model, we are able to obtain useful information as well as a valuable understanding of constrains imposed to designing flywheels. Particularly, we address how material properties used in production of flywheels affect the storable amount of energy. To answer this question, we compute the stress distribution in a flywheel as a function of its angular velocity. Furthermore, the angular velocity identifies the kinetic energy in the flywheel. In this way, a mathematical relationship between the kinetic energy stored in the flywheel and the yield stress of the flywheel material is determined.
Let us consider a flywheel with the inner radius of Ri and outer radius of Ro, as shown in Fig. 1. The flywheel rotates with the angular velocity of ω. Moreover, the flywheel material has a density of ρ with the Poisson ratio of υ. An analytical solution to the tangential and radial distribution of the flywheel can be obtained using the concepts of linear elasticity. [4] We further analyze the problem under the following assumptions. First, the outer radius should be large compared to the flywheel thickness. Second, the flywheel thickness should be constant. Third, the flywheel experiences constant stresses over its thickness. Here, we list the final equations. The radial stress (Sr) of the flywheel at the radius of r is computed as:
| Sr = | 1 8 |
ρ ω 2 (3 + υ) (Ri2 + Ro2 – | Ri2
Ro2 r2 |
– r2 | ) |
Similarly, the tangential stress (St) of the flywheel at the radius of r is calculated as
| St = | 1 8 |
ρ ω 2(3 + υ) (Ri2 + Ro2 + | Ri2
Ro2 r2 |
– r2 | 1 + 3 υ 3 + υ |
) |
The maximum radial and tangential stresses are, respectively, determined as
| max (Sr) = | 1 8 |
ρ ω 2 (3 + υ) (Ri2 + Ro2 – | 2Ri Ro | ) |
| max (St) = | 1 4 |
ρ ω 2 [ (3 + υ)Ro 2 + (1 - υ)Ri2 | ] |
The stored kinetic energy (E) in the flywheel is computed as
| E = | 1 2 |
J ω 2 |
where J is the moment of inertia of the flywheel. In the next section we use these equations to analyze a specified problem.
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| Fig. 2: Radial stress distribution of the flywheel with the the density of 7750 kg/m3, the Poisson ratio of 0.27, the angular velocity of 3000 rpm, and the inner and outer radius of, respectively, 0.1 m and 0.4 m. (Source: A. Kashefi) |
The equations provided in the previous section is used for analysis of a desired design as long as the underlying assumptions are satisfied. In this section, we focus on a specific case for a better understanding of the problem and the associated limitations in the energy that can be stored in a flywheel under the constraints of material properties such as yield stress. Let us start with a simple example. We consider a steal flywheel with the density of 7750 kg/m3, the yield stress of 240 MPa, and the Poisson ratio of 0.27. The inner and outer radius of the flywheel are, respectively, 0.1 m and 0.4 m.
Figs. 2 and 3, depict the radial and tangential stress distribution in the flywheel for the angular velocity of 3000 rpm. As can be observed in Fig. 2, the radial stress at r = 0.1 m and r = 0.4 m is zero, while the maximum radial stress happens at approximately r = 0.2 m with the value of 28.225 MPa. According to Fig. 3, the tangential stress experiences its maximum and minimum values, respectively, at r = 0.1 m and r = 0.4 m with the values of 101.732 MPa and 28.302 MPa. Comparing Fig. 2 and Fig. 3, the tangential stress is always greater than the radial stress.
Next, we discuss the role of angular velocity in the magnitude of the maximum tangential stress. As can be realized from the equations given in the previous section, the maximum stress is proportional to the squared angular velocity. Interestingly, the stored rotational energy in a flywheel is also proportional to the squared angular velocity. Hence, to increase the stored energy, one needs to enhance the angular velocity of the flywheel. However, the enhanced angular velocity leads to an increase in the stress caused in the flywheel material; and this introduces the probability of yielding.
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| Fig. 3: Tangential stress distribution of the flywheel with the the density of 7750 kg/m3, the Poisson ratio of 0.27, the angular velocity of 3000 rpm, and the inner and outer radius of, respectively, 0.1 m and 0.4 m. (Source: A. Kashefi) |
For the given design and using the equations given for computing the maximum tangential stress, the maximum possible angular velocity is 4606 rpm. In a real engineering world; however, the design comes with a safety factor. Note that the above computed angular velocity only cares the yielding stress criterion. Other design constrains should be considered for a full analysis.
As another calculation, we solve the same problem but now with another material such as aluminum with the yield stress of 276 MPa and the density of 2700 kg/m3. Given these material properties, the maximum possible angular velocity is 8374 rpm. Comparing this value with that one computed using steel, the importance of the choice of the flywheel material is reflected. Note that the kinetic energy of the flywheel is also a function of the moment of inertia, and thus a function of the material density. Hence, by keeping the geometric features of the flywheel constant, changing the flywheel material leads to a different yield stress and density, affecting the energy storage. Mathematically, this fact can be represented by the following equation
| E M |
≤ | 2 σy ρ |
Ro2 +
Ri2 (3 + ν) Ro2 + (1 - ν) Ri2 |
where σy and M are the yield stress and mass of the flywheel, respectively. From this equation, we realize how the yield stress controls the energy storage per unit mass of the flywheel.
© Ali Kashefi The author warrants that the work is the author's own and that Stanford University provided no input other than typesetting and referencing guidelines. The author grants permission to copy, distribute and display this work in unaltered form, with attribution to the author, for noncommercial purposes only. All other rights, including commercial rights, are reserved to the author.
[1] R. Sebastian, and R. Peña Alzola, "Flywheel Energy Storage Systems: Review and Simulation for an Isolated Wind Power System," Renew. Sustain. Energy Rev. 16, 6803 (2012).
[2] S. M. Mousavi G. et al., "A Comprehensive Review of Flywheel Energy Storage System Technology," Renew. Sust. Energy Rev. 67, 477 (2017).
[3] X. Li, and A. Palazzolo, "A Review of Flywheel Energy Storage Systems: State of the Art and Opportunities," J. Energy Storage, 46, 103576 (2022).
[4] R. G. Budynas and J. K. Nisbett, Shigley's Mechanical Engineering Design, 11th Ed. (McGraw-Hill, 2019).