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| Fig. 1: A schematic depicting a series hybrid drivetrain for vehicles. (Image source: B. Wu.) |
Hybrid-electric vehicles are a compelling alternative to conventional combustion- engine vehicles because by incorporating an energy storage system within the drivetrain, they are able to augment the energy delivered by the engine using an electric motor powered by the energy storage system. Two examples of such hybrid systems are series hybrids (in which the engine serves only as an electricity generator for the electric motors) and parallel hybrids (in which the engine and the electric motors work together to propel the vehicle), depicted in Fig. 1 and Fig. 2, respectively. [1] The energy storage system can be used to power auxiliary loads, prevent the engine from idling, and also contribute to vehicular acceleration thus reducing the required output of the engine and allowing for higher fuel efficiency. [1]
The possible energy storage systems within such drivetrains can take many forms, but this report will focus on two types: chemical energy storage systems (i.e. batteries) and mechanical energy storage systems (i.e. flywheels, such as the example in Fig. 3). When a chemical energy storage system is connected via an external circuit, electrons will flow from one electrode to another; it is the flow of the electrons that produces a current (which can be used to do work). [2] By contrast, mechanical energy storage systems such as flywheels rely on the conservation of angular momentum to store rotational kinetic energy. The speed at which a flywheel spins is directly proportional to how much energy it is storing; speeding up or slowing down the flywheel will increase and decrease the amount of energy within the system, respectively. [3]
This report aims to explore the viability of both types of energy storage systems within hybrid vehicle drivetrains by calculating the energy density (J/kg) of both a metal-based flywheel and a Lithium-Ion battery. By comparing the energy densities of both such systems, this report will draw conclusions on whether or not one particular system can provide significant practical advantages over the other. While energy densities for Lithium-Ion batteries are well-known and can be extracted from literature, a theoretical upper bound for the amount of energy that can be stored in flywheels per unit mass will be derived and compared against energy densities for Lithium- Ion batteries.
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| Fig. 2: A schematic depicting a parallel hybrid drivetrain for vehicles. (Image source: B. Wu.) |
The goal of this section is to derive an upper bound expression that represents the energy density that can be stored within a flywheel. The desired expression should result in a number that can be expressed in units of Joules/kilogram, which is consistent with the SI units for describing energy density.
We begin by identifying an expression that represents the maximum amount of energy that a flywheel can store. [4]
| E | = | 1 2 |
I ω2 | = | 1 2 |
M r2 ω2 |
where I represents the moment of inertia of the flywheel, M represents the mass of the flywheel, r represents the radius of the flywheel, and ω represents the angular velocity of the flywheel. Note that this formulation is only correct for a flywheel with all of its mass at the rim; for flywheels that are uniform disks, then the right side of the equation needs to be multiplied by a constant of 3/5:
| E | = | 3 10 |
M r2 ω2 |
For calibration, let us calculate the maximum amount of energy that can be stored in a flywheel of the same specification as the one in Fig. 3. According to the Vauxhall Astra (Model Year 2014) user manual, the A16XHT engine has a maximum rotational speed of 6000 rpm and a diameter of 0.38 m (therefore giving it a radius of 0.19 m). [7] Additionally, among automotive applications, the vehicle with the heaviest flywheel uses one with a mass of 57 kg, so we set this as the upper bound of our calculations. [8]
First, we calculate the rotation frequency ω
| ν | = | 6000 min-1 60 sec min-1 |
= | 100 sec-1 |
| ω | = | 2 π ν = 628 sec-1 | ||
Next, we calculate the energy density:
| E M |
= | 3 10 |
(ω r)2 | = | 3 10 |
(628 sec-1 × 0.19 m)2 | = | 4271 J kg-1 |
Finally, we multiply both sides of this equation by the mass of the flywheel to get an upper bound on the energy that it can store:
| E | = | M × (E/M) | = | 57 kg × 4271 J kg-1 | = | 2.43 × 105 J |
By contrast, Gasoline has a density of 2.57 kg gal-1 and an energy density of 4.2 × 107 J kg-1. [9] The amount of energy present in a gallon of gasoline is
| 4.2 × 107 J kg-1 × 2.57 kg gal-1 | = | 1.08 × 108 J gal-1 |
However, the limitation of the above expression that calculates the maximum amount of energy that a flywheel can store is that it assumes that flywheels can spin at speeds approaching infinity. Therefore, to apply a reasonable physical constraint on flywheel systems, the following expression is required:
| E M |
< | f σ ρ |
where E represents the maximum energy that can be stored in a flywheel, M is again the mass of the flywheel, f is an order-1 factor (that, in nominal situations, can be approximated as roughly 1) that depends on the shape of the flywheel, σ is a the yield stress of the flywheel material, and ρ is its mass density. [3] Of particular interest is the σ/ρ term, which can also be referred to as the specific tensile strength or energy density of the flywheel. [3] Intuitively, this expression indicates that a flywheel cannot spin too fast because otherwise the centrifugal forces acting upon it will tear the flywheel completely apart.
This last inequality results in an expression for energy density, as desired: Both sides of the inequality are in units of energy/mass (Joules/kilogram). Furthermore, the inequality indicates that the upper bound of energy density for any flywheel will be the quantity f × σ/ρ.
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| Fig. 3: A cutaway depiction of a dual mass metal-based flywheel from an Opal A16XHT model engine. (Source: Wikimedia Commons) |
Let us now evaluate this theoretical upper bound and compare it to a conventional chemical energy storage system.
It is reasonable to assume that any flywheel capable of storing enough energy to power a vehicle propulsion system would need to be made out of a material with a high tensile strength and yield stress, such as a high-strength steel (commonly referred to as HSLA; High-Strength Low-Alloy).
A popular HSLA formulation for structural applications has a minimum nominal yield stress of 460 Megapascals (4.6 × 108 Pa). [5] Furthermore, steels exhibit small variations in their mass densities due to differences in the alloys used; for the sake of simplicity, we assume a typical mass density at approximately 7,800 kg m-3 for the following calculation. [6] We further assume that f = 1, which is standard for a flywheel that is perfectly circular. The specific tensile strength of this flywheel will be the theoretical upper bound for energy density, and it is calculated as
| E M |
= | 4.6 × 108 Pa 7800 kg m-3 |
= | 5.90 × 104 J kg-1 |
By contrast, a Lithium-Ion battery typically has a known energy density of 300 Wh/kg, which convert at 1 Wh = 3.6 kJ to
| E M |
= | 300 Wh kg-1 × 3600 J Wh-1 | = | 1.08 × 106 J kg-1 |
This indicates that Lithium-Ion batteries are, for equivalent mass, able to store more than 18 times the energy compared to a flywheel constructed out of HSLA. Therefore, flywheels are not a practical alternative alternative to Lithium-Ion battery storage in hybrid drivetrains because the additional weight incurred by a flywheel.
While flywheels may seem like a compelling mechanical alternative to battery systems that currently exist on vehicles with hybrid drivetrains, a comparison of the respective energy densities of both types of storage systems reveals that the energy density of these flywheels falls short. The fundamental physical principle of flywheel energy storage rotational kinetic energy imposes limitations on the achievable maximum energy density due to practical constraints on the rotation rate of a flywheel.
The specific tensile strength of high-strength materials such as HSLA steel will set an upper bound on the rotational speed of flywheels; if this bound is exceeded, then structure failure of the flywheel can occur. Despite the use of these high-strength materials, which in theory should allow the flywheel to spin at a faster rate (and thus store more energy), the energy density of flywheels remains substantially lower than that of gasoline and Lithium-Ion batteries. For instance, Lithium-Ion batteries widely employed in current hybrid systems exhibit an energy density of over 18 times that of a flywheel of equivalent mass. As such, this represents a core challenge facing the adoption of flywheel technology in hybrid vehicles: the substantial weight required for flywheels to match the energy storage of existing battery systems would likely compromise the overall efficiency and practicality of hybrid drivetrains. While flywheels will still find niche applications in many mechanical systems, these limitations make them impractical as a direct replacement for Lithium-Ion batteries in hybrid automotive propulsion.
© Brian Wu. 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] Y. Chen et al., "A Review of Lithium-Ion Battery Safety Concerns: The Issues, Strategies, and Testing Standards," J. Energy Chem. 59, 83 (2020).
[2] H. Li, "Practical Evaluation of Li-Ion Batteries," Joule 3, 911 (2019).
[3] G. Canbolat and H. Yaşar, "Performance Comparison For Series and Parallel Modes of a Hybrid Electric Vehicle," Sakarya J. Sci. 23, 43 (2019).
[4] X. Li and A. Palazzolo, "A Review of Flywheel Energy Storage Systems: State of the Art and Opportunities," J. Energy Storage 46, 103576 (2021).
[5] B. E. Layton, "A Comparison of Energy Densities of Prevalent Energy Sources in Units of Joules Per Cubic Meter," Int. J. Green Energy 5, 438 (2008).
[6] O. Bamisile et al., "Development and Prospect of Flywheel Energy Storage Technology: A Citespace-Based Visual Analysis," Energy Rep. 9, Suppl. 10, 494 (2023).
[7] "Vauxhall Astra Owners Manual, Model Year 2014," Opel Automobile GmbH, KTA-2685/80VX, July 2013.
[8] M. Hedlund et al., "Flywheel Energy Storage for Automotive Applications," Energies 8, 10636 (2015).
[9] B.E. Layton, "A Comparison of Energy Densities of Prevalent Energy Sources in Units of Joules Per Cubic Meter," Int. J. Green Energy 5, 438 (2008).
