On Specific Energy Capacity of Flywheel Energy Storage

Applied Mathematical Sciences, Vol. 8, 2014, no. 124, 6181 - 6190

HIKARI Ltd, www.m-hikari.com

http://dx.doi.org/10.12988/ams.2014.47594

On Specific Energy Capacity of

Flywheel Energy Storage

D. V. Berezhnoi

Kazan Federal University

420008 Kazan, Russia

D. E. Chickrin

Kazan Federal University

420008 Kazan, Russia

A. F. Galimov

Kazan Federal University

420008 Kazan, Russia

Copyright © 2014 D. V. Berezhnoi, D. E. Chickrin and A. F. Galimov. This is an open access

article distributed under the Creative Commons Attribution License, which permits unrestricted

use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

The paper introduces basic methods of computational investigation for specific

energy capacity of flywheel energy storage. In addition to the traditional

estimation of energy capacity on the kinetic energy specific potential energy

estimation of elastic strain is added. The possibilities of the use of various

structural materials in the manufacture of flywheels is analyzed, some

recommendations on the form of flywheel are given. "Extended" estimation of

energy capacity which is used in this paper gives greater variability in the design

of flywheel energy storage. In some cases it allows the reduction of the rotational

speed of the rotor part of the structure.

Keywords: flywheel, specific energy capacity, kinetic and potential energy.

6182 D. V. Berezhnoi, D. E. Chickrin and A. F. Galimov

1. Introduction

With the development of modern technologies in industry and transport many

mobile devices appear, that is why the problem of energy storage becomes crucial.

Creation of new materials allows not only improving the traditional

electrochemical batteries that already exist, but look for new methods of

accumulation and storage of energy, including mechanical ones [2,3,7]. These

include the so-called static and dynamic mechanical energy storage.

In static mechanical energy accumulators elastic element works on stretching

(compressive), which is preferable, on twisting (shifting), or on bend, but such

batteries’ accumulated specific energy are relatively small. However, they have a

firm place in many machines and mechanisms, due to its valuable properties:

energy storage stability, high efficiency and long life. One of the simple and

promising technologies is flywheel energy storage. Flywheel accumulates energy

in the form of rotational kinetic energy. In other words, it is a massive rotating

body that is used as kinetic energy (inertial energy storage device) storage [8].

However, even modern technologies do not use all the possibilities of a flywheel.

If rotates rapidly, it can accumulate kinetic energy that is easy to increase, and,

furthermore, to use, which turns a flywheel into electromechanical battery.

In this paper we present some results of the estimation of accumulated flywheel

energy, including kinetic and potential energy of strain for different materials.

Also we give some advices on the choice of materials for a flywheel.

2. Evaluation of the potential and kinetic energy accumulated

during disk rotation around its axis

2.1. Common relations

Let us consider the problem of finding the stresses in a homogeneous disk with

density [5], which rotates with constant angular velocity . In this case, the

solution does not depend on the angle of rotation , but depends only on the

current disk radius r . We know that centrifugal force applies at each point of the

rotating body, proportional to the distance to rotation axis. Then the radial

component of the potential mass forces is 2r and the equilibrium equation can

be written as follows

2rr

rrr rr

,

where rr is radial stress, is tangential stress. Ratios between the

components of the strain tensor and the displacement vector components in polar

coordinates can be written as follows:

1 1 1, , ,

2

r r r

rr r

u u uu u u

r r r r r r

from which

On specific energy capacity of flywheel energy storage 6183

, , 0,r r

rr r

u u

r r

due to

, 0.r ru u r u

We represent the ratios between stress and strain in accordance with the

generalized Hooke's law. With respect to the plane stress state, these ratios can be

written as follows

2 2 2 2, .

1 1 1 1

r r r r

rr rr rr

u u u uE E E E

r r r r

where is Poisson ratio, E is Young’s modulus of disk material.

Substituting these relations into the equilibrium equation, we get the equation in

terms of displacements 2 2

2

2 2

1 1.r r ru u u

rr r Er r

(1)

It is known, that general solution of equation (1) can be represented as the sum

of homogeneous equation solution 0

ru and heterogeneous equation particular

solution 1

ru .

We search for the general solution of equation in the form 0 n

ru r . Substitute it

into homogeneous equilibrium equation we obtain

2 21 0nn r .

Hence 1n and 0 1

1 2ru C r C r ,

where constants 1C and

2C can be defined from boundary conditions. General

solution of heterogeneous equation 1 n

ru Ar .

Substitute it into (1) we have

2 2 2

21 0

1

nEn Ar r

,

from which 2

213,

8n A

E

.

Thus, solution of the equation (1) have the form 2

1 2 3

1 2

1

8ru C r C r r

E

,

and stress ratios

2 2 2 2 2 2

1 2 1 2

3 1 3, .

1 1 8 1 1 8rr

E E E EC C r r C C r r

E E

Strain potential energy

2 21 12

2 2rr rr rr rr

V V

dV dVE E

.

6184 D. V. Berezhnoi, D. E. Chickrin and A. F. Galimov

2.2. Aperture disk

The most simple static conditions for central aperture disc on inner and outer

surfaces are

0 10, 0rr rrr r .

Stress expressions

2 2 2 2 2 2 2 2 2 2

0 1 1 0 1 11 1 , 1 1 ,rr r k r r r r r k r r r r

where

2 2 0

0 1

1

3 1 3, , .

8 3

rr k

r

Strain potential energy

1

1

2

2 2

0 0

12

2

rb

rr rr

kr

rdrdzdE

2 2 2 4 2 2 2

0 1

2

2 1 1 7 6 2 17 6.

3 1

b r k k k

E

We define disk rotation kinetic energy

4 4 2 4 2 2 2 4 2

1 0 1 1 1 01 2 1

4 4 3

b r r b k r r b k rK

.

The ratio of strain potential energy to kinetic energy has the form

4 2 2 2

0

2 2

1 7 6 2 17 6 3

3 1 1

k kK

E k

.

Since the mass of the disk is

2 2

1 1m br k ,

the ratio of potential energy to mass

2 4 2 2 2

0

2

2 1 7 6 2 17 6

3 1

k km

E

,

and the ration of kinetic energy to mass

2

02 1

3

kK m

.

Stress rr is positive and reach its maximum when 0 1r r r :

2max

0 1rr k .

Stress also positive for all values of r and reach its maximum when 0r r :

2

max 2

0 0

3 12 (1 ) 2

3

kk

.

It is always max max

rr . Therefore, the strength condition can be written, for

example, according to the first strength theory: max

y ,

or limiting

On specific energy capacity of flywheel energy storage 6185

0 2

3

2 3 1

y

k

.

Then expressions for strain potential energy, rotation kinetic energy, its ratios

and specific potential and kinetic energy are as follows:

22 2 2 4 2 2 2

1

22 2

1 3 1 7 6 2 17 6,

6 1 3 1

yb r k k k

E k

4 2

1

2

1,

3 1

yb k rK

k

24 2 2 2

2 2 2

1 7 6 2 17 6 3

3 1 1 3 1

y k kK

E k k

,

22 4 2 2 2

22 2

3 1 7 6 2 17 6

6 1 3 1

y k km

E k

,

2

2

1

3 1

y kK m

k

. (2)

Thus, specific energy capacity of rotating axis aperture disk is

2

2

1

3 1

yk

e m K m mk

2 4 2 2 22

22 2

3 1 7 6 2 17 6.

6 1 3 1

yk k

E k

,

2.3. Solid disk

Equations for finding stresses in solid disk 0 0r have the form

2 2 2 2

0 1 0 11 , 1 .rr r r r r r r

In this case we search for radial displacement

22 3

1

1

8ru C r r

E

,

and expressions for stresses are

2 2 2 2

1 1

3 1 3, .

1 8 1 8rr

E EC r C r

E E

The most simple static conditions on outer surface are as follows

1 0rr r ,

and stresses defined according to formulas

2 2 2 2

0 1 0 11 , 1 .rr r r r r r r

6186 D. V. Berezhnoi, D. E. Chickrin and A. F. Galimov

Both stresses are positive and it increase while reaching disk axis. On the disk

axis (when 0r ) both stresses turn to maximum and become equal to each other max max

0rr .

Hence tangential stresses equal zero on any plain, including symmetry axis of

the disk 0r , then rr and are primary stresses. If we have plain stressed

condition, then 0zz . We can use IV theory strength condition to estimate value

of stress, which destroy material. In this case we have

2 2 2

max max max max max

01 2 .rr zz zz rr y

Then expressions for strain potential energy, rotation kinetic energy, its ratios

and specific potential and kinetic energy are as follows:

2 2 2

1

2

2 7 6,

3 1

yb r

E

2

12

3

ybrK

,

2

2

7 6 3

3 1

yK

E

.

2 2

2

2 7 6

3 1

ym

E

,

2

3

yK m

. (3)

Specific energy capacity of rotating solid disk is

22

2

2 7 62.

3 3 1

y ye m K m m

E

2.4. Constant-strength disk

If the disk does not have a constant cross-sectional height, we can choose such a

form that circumferential and radial stresses rr in all points of the disk will

be constant and equal. In this case, disk section form can be defined according to

the formula [6] 2 2

20

r

h h e

.

In order to make the stress of the disk (which form we define through the

formula above) constant, it is necessary to apply load on outer surface, which

cause radial load .

We can use IV theory strength condition to estimate value of stress, which

destroy material. In this case we have

On specific energy capacity of flywheel energy storage 6187

2 2 2 2 2 2

1 2 1 2 1 2 2 .zz zz y

Then strain potential energy is

2 2

2

0 12

2 2

0 0 0

12

2

r

yh e r

rr rr rdrdzdE

2 2

2 2 11

22 2

20 0

2

0

12 1 2 1,

y

y

rrr

yy y

eh he rdr

E E

and kinetic energy is

2 2

2 22 22 1

0 1 1

22 2 222 1

22 2 3

0 0 2

0 0 0 0

2 1 1 2,

2

r

yy

y

rrh e r r

y ye rK r rdrdzd h e r dr h

where the mass

2 2

2 22 22 1

0 1 1

22

2

0 0 2

0 0 0 0

12 2 .

r

yy

y

rrh e r r e

m rdrdzd h e rdr h

Then expressions for strain potential energy, rotation kinetic energy and its

ratios are as follows:

2 21

2 21

2

2 2 2

1

1 1

1 1 2

y

y

r

y

r

y

eK

E e r

,

2 1ym

E

,

2 21

2 21

2 2 2

1

2

1 1 2

1

y

y

r

y y

r

e rK m

e

.

If angular velocity approaches infinity , then first and third expressions

become more simple

1yK

E

,

yK m

. (4)

We can wright specific energy capacity of rotating solid cylinder

2 11 1 .

y y y ye m m K m

E E

6188 D. V. Berezhnoi, D. E. Chickrin and A. F. Galimov

3. Numerical results

Evaluations for three disks (aperture disk, solid disk and constant-strength disk)

were conducted to estimate specific energy capacity of some construction

materials, which mechanical characteristics can be seen in table 1. In table 2 we

have specific energy capacities of disks of different shapes (for potential energy

m , kinetic energy K m and general energy e m ). The disks made of materials,

which mechanical characteristics can be seen in table 1.

Table 1. Mechanical characteristics of some construction materials.

NN Material Young’s

modulus E

MPa

Poisson’s

ratio

Yield stress

y MPa

Density 3kg m

1 High carbon steels 215000 0.3 1155 7900

2 Titanium alloys 120000 0.3 1245 4800

3 Boron carbide 472000 0.3 5687 2550

4 Composites polymer

CFRP

150000 0.25 1050 1600

5 Polyurethane

elastomers (eiPU)

3 0.48 51 1250

Table 2. Specific energy capacities of disks of different shapes MJ kg made of

materials, which can be seen in table 1.

NN Thin ring Solid disk Constant-strength

disk m K m e m m K m e m m K m e m

1 0,003 0,073 0,0756 0,002 0,088 0,090 0,0006 0,146 0,147

2 0,009 0,129 0,138 0,005 0,157 0,162 0,0019 0,259 0,261

3 0,087 1,115 1,202 0,054 1,35 1,41 0,019 2,23 2,25

4 0,015 0,328 0,344 0,011 0,404 0,414 0,0034 0,656 0,659

5 1,92 0,02 1,94 0,821 0,023 0,844 0,361 0,041 0,401

4. Analyzing results

The ratio between specific energy capacity e m and specific strength âð of

flywheel material have the form

yek

m

,

On specific energy capacity of flywheel energy storage 6189

where k is flywheel shape coefficient, which represents its effectiveness. We can

find shape coefficient k using expressions (2-4), which coincides with

expressions [1,4].

Structural formulas for disk shapes are

2

2

1 2

1

,, , ,

,

y y y ke m K m Ï m k k Ï K

E E k

,

where functions 1 ,k and 2 ,k are different for each type of shape.

Analysis of the ratio K for each flywheel shapes shows, that the ratio of

potential energy to kinetic energy is greater for aperture disk and smaller for

constant-strength. This is due to the fact that the main body of the aperture disk

locates away from the axis of rotation. Under equal conditions that increases

kinetic energy. For constant-strength disk the situation is vice versa.

The same conclusion can be made analyzing the expression for specific potential

energy m , i.e. accumulated potential energy reaches its maximum for aperture

disk and minimum for constant-strength disk. It is because the more elastic

material works away from rotation axis, the more specific potential elastic energy

accumulated.

However, we may made the opposite conclusion while estimating specific

potential energy K m . Here the disk strength is the main characteristic, which the

highest for constant-strength disk. Full accumulated specific energy K m m is

greater for disks with mass concentration near rotation axis for all materials,

except rubber. The reason for this is that ratio âð E becomes too huge and

percentage of specific potential energy in the full specific energy dominates. But

still general accumulated specific energy too small, because expression âð is

small. However, calculations show that for rubber-like materials (for instance, for

polyurethane elastomers (eiPU) [9]), energy accumulation derives from potential

strain energy.

5. Conclusion

We should mention in conclusion that we analyze energy capacity of flywheels

of different shapes in this paper. Similar studies have been conducted previously,

but they gave evaluation accumulated rotation kinetic energy only. We calculate

and accumulated elastic strain potential energy. Calculations were made for some

of the classical types of flywheels for which we can obtain the exact value of

energy capacity. The same evaluation of complex forms of the flywheels

(including composite and combined ones) can be carried out in the known

numerical packages of strength analysis, in the ANSYS, in particular.

We need specific strength of the flywheel material and the form coefficient in

order to calculate flywheels’ specific kinetic energy. We use the ratio of ultimate

material tensile strength to its Young's modulus to estimate elastic strain potential

energy. It gives us wide opportunities of flywheel energy storage devices design,

6190 D. V. Berezhnoi, D. E. Chickrin and A. F. Galimov

i.e. in some situation accumulated strain potential may exceeds greatly kinetic

energy of rotation. At the same time the shape of the flywheel influence on its

energy capacity (energy capacity of strain potential energy) in opposite manner. In

addition, if the flywheel accumulates more strain potential energy (than kinetic

energy), it will make possible to reduce the rotational speed and acceleration. In

its turn that will positively affect safe operation and service life and allow to

abandon the sealed housing, which creates vacuum in the rotation area of the

flywheel (or at least use lower order vacuum).

Acknowledgements. The work was performed according to the Russian

Government Program of Competitive Growth of Kazan Federal University as part

of OpenLab “Makhovik”. The reported study was also supported by RFBR,

research project No. 12-01-00955, 12-01-97026, 13-01-97059, 13-01-97058.

References

[1] K.V. Belykh, N.M. Filkin, To a question of calculation flywheel drives of

kinetic energy, materials of the international scientific and practical

conference «Upgrade and scientific researches in a transport complex»,

Perm, 26–28 april 2012. V.1. Perm: PNIPU, (2012), 281-289.

[2] T.O. Drum, Energy storage-flywheel, 2011. http://www.resilience.org/

stories/2011-10-05/energy-storage-flywheel.

[3] N.V. Gulia, Flywhee engines, М.: Mashinostroenie, 1976.

[4] N.V. Gulia, Energy accumulation, М.: Nauka, 1980.

[5] V.V. Matveev, G.V. Pisarenko, A.P. Yakovlev, The reference manual on

resistance of materials, Kiev: Naukova Dumka, 1988.

[6] Y.N. Rabotnov, Resistance of materials, М.: Fizmatgiz, 1962.

[7] Alternative energy storage methode. http://www.mpoweruk.com/alternatives

.htm

[8] Flywheel. http://ru.wikipedia.org/wiki/Маховик#.D0.A1.D1.83.D0.BF.D0

.B5.D1.80.D0.BC.D0.B0.D1.85.D0.BE.D0.B2.D0.B8.D0.BA.

[9] Materials Data Book, Cambridge University Engineering Department, 2003.

Received: July 11, 2014