There are two types of FESS rotors that can be taken into account for a high specific energy density: a flat disk rotor out of isotropic material (shape 1) and a cylindrical design with fiber reinforced plastics with fibers oriented in circumferential direction (orthotropic) (shape 2). Contact online >>
There are two types of FESS rotors that can be taken into account for a high specific energy density: a flat disk rotor out of isotropic material (shape 1) and a cylindrical design with fiber reinforced plastics with fibers oriented in circumferential direction (orthotropic) (shape 2).
The equation to determine the energy content Ekin of a flywheel for a solid disk is
where m represents the mass of the flywheel, r the radius and ω the angular velocity.
For isotropic materials, the energy density is limited by the maximum radial strength in the rotor and directly influences the energy density (energy-to-mass-ratio Em that can be written as:
Thin rings, constructed out of orthotropic materials like fiber composites, show two specific properties in comparison to disk-shaped isotropic materials:
The maximum stress occurs in circumferential direction, which is in the fiber direction of the composite materials. Since carbon fibers show an enormous strength in the fiber direction, they are an optimal material to take the circumferential stress. Due to the slender ring thickness, radial stress that normally limits the flywheel performance and that would be taken mostly by the resin is negligible.
The momentum of inertia grows differently (see eq. [4]) in comparison to the mass (see eq. [5]) concerning the radii ratio ri/ra
Energy density depending on radii ratio ri/ra
If CNTs can be produced in length of cm instead of mm they would be suitable for FESS and could increase energy density of a FESS up to 2,900 W h/kg since they reach a yield strength of around 30 GPa (Yu 2000). For further calculations of shape 2, a radii ratio ri/ra of 0.9 was set. That allows us to use a simple equation for calculating the limiting circumferential stress for thin rims (Feldhusen 2001):
For thin rims, with a radii ratio ri/ra of 0.9, the shape factor decreases to K=0.45 and thus the energy density is calculated according to
To see how downscaling affects the energy content, the rotational velocity and the losses due to gas and motor friction, two concepts with the following dimensions relations are considered:
The CFRP-ring and the Laval-disk-with-rim shapes can be seen in Figure 3 where the Laval-disk-with-rim is assumed to be a disk with the same momentum of inertia and volume to allow the use of a less complex air loss estimation. Whenever disk shape is mentioned in the following, they imply the properties of the Laval-disk-with-rim.
Schematic shape 1: ring (left), shape 2: Laval-disk-with-rim (top right) and a solid disk (bottom right) used for the simplified air loss estimation
The mass specific energy densities are shown in Table 2. They are calculated after eqs [3] and [7] and take into account the maximum feasible rotational velocity that is limited by the yield strength. Furthermore, the mass specific energy densities are independent of the absolute dimensions. As can be seen, the energy density of the isotropic material is highest at the disk shape due to the increase of the shape factor K. The orthotropic materials show higher absolute energy densities since the strain-to-density ratio is higher. Energy densities for composite materials in disk shapes are not examined. In this case, the radial stress would be the limiting factor, which would only be determined by the comparatively low stength of the resin.
Energy densities of different flywheel materials at diskand ring shape (Torayca, T1000 Data Sheet; Torayca, T300 DataSheet)
Maximum rotational speed vs. diameter for disk (left) and ring (right)
With eq. [1], the absolute energy content of the flywheel geometries is calculated for a maximum achievable rotational velocity (see Figure 5). The highest energy content can be achieved with CNTs in a ring shape. Since the volume between ring and disk shapes is equal, the CNTs also show the best energy content-to-volume ratio. Wolfram shows the second highest energy content but only in a disk shape. Steel, in a disk shape and the T1000 composite ring, shows equivalent energy contents. The least favorable materials are titanium and aluminum when it comes to energy density regarding the volume since these materials are too light to reach adequate values.
Energy content vs. diameter for disk shape (left) and ring shape (right)
Smaller flywheels have a smaller ratio between mass (correlated to stored energy) and surface (correlated to gas friction), and hence, the gas friction which is proportional to the surface increases. The gas friction (Pgas) can be calculated (Kolk 1997) assuming molecule to housing impacts only and no molecule to molecule impacts that would cause a viscous fluid. Since a micro-FESS would run in a highly evacuated vacuum with a rest pressure of less than 1∗10−4mbar, the assumption of molecule to housing impacts would be valid up to a flywheel to housing distance of 1 m.
When a molecule hits the spinning rotor it transfers its momentum:
molecule per time unit hits a specific rotor surface, where
is equal to the density of molecules at pressure pG. cˉ is the average molecule velocity is governed by the Maxwell equation:
Thus, the transferred angular momentum of inertia due to molecule impacts is
The gas friction losses PGas for a solid disk is thus
RG: molar gas constant
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