Energy storage via high-energy
density composite flywheels
Brian C. Fabien
[email protected]
Department of Mechanical Engineering
University of Washington
Seattle, WA 98195
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Presentation overview
Introduction - Motivation - Basic Principle
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Presentation overview
Introduction - Motivation - Basic Principle
Stacked-ply disk design problem
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Presentation overview
Introduction - Motivation - Basic Principle
Stacked-ply disk design problem
Disk construction
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Presentation overview
Introduction - Motivation - Basic Principle
Stacked-ply disk design problem
Disk construction
Testing and evaluation
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Presentation overview
Introduction - Motivation - Basic Principle
Stacked-ply disk design problem
Disk construction
Testing and evaluation
Conclusion
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Some energy storage technologies
Lead acid battery: 18 Wh/kg
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Some energy storage technologies
Lead acid battery: 18 Wh/kg
Nickel-cadmium battery: 31 Wh/kg
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Some energy storage technologies
Lead acid battery: 18 Wh/kg
Nickel-cadmium battery: 31 Wh/kg
Hydrostorage: 300 Wh/m3
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Some energy storage technologies
Lead acid battery: 18 Wh/kg
Nickel-cadmium battery: 31 Wh/kg
Hydrostorage: 300 Wh/m3
Composite flywheels: 100 to 1000 Wh/kg
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Some energy storage technologies
Lead acid battery: 18 Wh/kg
Nickel-cadmium battery: 31 Wh/kg
Hydrostorage: 300 Wh/m3
Composite flywheels: 100 to 1000 Wh/kg
Compressed air: 2000 Wh/m3
Energy storage via high-energy density composite flywheels – p.3/34
Some energy storage technologies
Lead acid battery: 18 Wh/kg
Nickel-cadmium battery: 31 Wh/kg
Hydrostorage: 300 Wh/m3
Composite flywheels: 100 to 1000 Wh/kg
Compressed air: 2000 Wh/m3
Gasoline: 14000 Wh/kg
Energy storage via high-energy density composite flywheels – p.3/34
Some energy storage technologies
Lead acid battery: 18 Wh/kg
Nickel-cadmium battery: 31 Wh/kg
Hydrostorage: 300 Wh/m3
Composite flywheels: 100 to 1000 Wh/kg
Compressed air: 2000 Wh/m3
Gasoline: 14000 Wh/kg
Hydrogen: 38000 Wh/kg
Energy storage via high-energy density composite flywheels – p.3/34
Flywheel Energy Storage (FES)
FES usage
- Electrical load leveling
- Batteries for electrical vehicles
- Pulsed power supplies
FES design challenges
- Bearings
- Drive/generator
- Containment
- Flywheel
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Flywheel Energy Storage - Basic ideas
A kinetic energy storage device
Maximum energy density:
1
1
2
α=
Iωf
Wh/kg
2
3600ρV
I - moment of inertia of the disk (kg-m2 ),
ωf - failure speed of the disk (rad/s),
ρ - density (kg/m3 ),
V - volume (m3 ).
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Flywheel Energy Storage - Schematics
Bearings
Renewable
Power
Source
Motor/Generator
Flywheel
T
00000
11111
11
00
T’
Motor coils
Motor magnets
Flywheel
B’
Housing
11
00
00000
11111
B
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Stacked-ply Flywheel
tangential
ply
radial
ply
Alternate layers of radial and tangential plies
Can fiber angle variation improve performance?
How can such disks be constructed?
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Constitutive relationship
1
fiber
direction
ro
r
σθθ
ri
σrr
2

σ11

 σ22
τ12
φ
Ply stiffness relationship

 


Q11 Q12 0
ǫ11

 


=
 Q21 Q22 0   ǫ22 

γ12 k
0
0 Q66 k
k
|
{z
}
[Q]k
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The k-th ply stiffness


 


ǫrr
Q̄rr Q̄rθ Q̄r6
σrr

 



 σθθ  =  Q̄rθ Q̄θθ Q̄θ6   ǫθθ 
γrθ k
Q̄r6 Q̄θ6 Q̄66 k
τrθ k
|
{z
}
[Q̄(φ)]k
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Stiffness components
Q̄rr
Q̄rθ
Q̄22
Q̄66
Q̄r6
Q̄θ6
Q11 cos4 φ + Q22 sin4 φ + (2Q12 + 4Q66 ) cos2 φ sin2 φ
Q12 (cos4 φ + sin4 φ) + (Q11 + Q22 − 4Q66 ) cos2 φ sin2 φ
Q11 sin4 φ + Q22 cos4 φ + (2Q12 + 4Q66 ) cos2 φ sin2 φ
(Q11 + Q22 − 2Q12 − 2Q66 ) cos2 φ sin2 φ
+ Q66 (cos4 φ + sin4 φ)
= (Q11 − Q12 − 2Q66 ) cos3 φ sin φ
− (Q22 − Q12 − 2Q66 ) cos φ sin3 φ
= (Q11 − Q12 − 2Q66 ) cos φ sin3 φ
− (Q22 − Q12 − 2Q66 ) cos3 φ sin φ.
=
=
=
=
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Ply properties
Q11
Q22
Q12 = Q21
E11
=
1 − ν12 ν21
E22
=
1 − ν12 ν21
ν21 E11
=
1 − ν12 ν21
Q66 = G12
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Laminate constitutive relations




ǫrr
σ̃rr




 σ̃θθ  = [A(φ)]  ǫθθ 
γrθ
τ̃rθ
1
1
◦
[A(φ)] = λ
[Q̄(90 )]
[Q̄(φ)] + [Q̄(−φ)]
+(1 − λ)
| {z }
2
2
|
{z
}
tang. stiffness
radial stiffness
total thickness of tangential reinforcement plies
.
λ=
total thickness of laminate
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Laminate constitutive relations (cont.)

ǫrr

 ǫθθ
γrθ
Effective strain-stress



Srr Srθ 0
σ̃rr
 


=
  Srθ Sθθ 0   σ̃θθ 
0
0 S66
0
|
{z
}

[S(φ)]
where [S(φ)] = [A(φ)]−1 is the effective compliance matrix




σ̃rr
σ11




 σ22  = [T(φ)]k [Q̄(φ)]k [S(φ)]  σ̃θθ  .
0
τ12 k
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Coordinate trasformation

cos2 φ

T(φ) = 
sin2 φ
− cos φ sin φ
2
sin φ
cos2 φ
cos φ sin φ

2 cos φ sin φ

− 2 cos φ sin φ  .
cos2 φ sin2 φ
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Stress state
Equilibrium:
d
dr (rσ̃rr ) − σ̃θθ
+ ρω 2 r2 = 0
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Stress state
Equilibrium:
d
dr (rσ̃rr ) − σ̃θθ
+ ρω 2 r2 = 0
Compatibility: r dǫdrθθ + ǫθθ − ǫrr = 0
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Stress state
Equilibrium:
d
dr (rσ̃rr ) − σ̃θθ
+ ρω 2 r2 = 0
Compatibility: r dǫdrθθ + ǫθθ − ǫrr = 0
States:
σ̃rr
x1 =
,
2
2
ρω ro
σ̃θθ
x2 =
,
2
2
ρω ro
x3 = φ,
x4 = r/ro = τ
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Stress state
Equilibrium:
d
dr (rσ̃rr ) − σ̃θθ
+ ρω 2 r2 = 0
Compatibility: r dǫdrθθ + ǫθθ − ǫrr = 0
States:
σ̃rr
x1 =
,
2
2
ρω ro
σ̃θθ
x2 =
,
2
2
ρω ro
x3 = φ,
x4 = r/ro = τ
Control variable: =⇒ derivative of fiber angle: u = dφ/dτ
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Stress state
Equilibrium:
d
dr (rσ̃rr ) − σ̃θθ
+ ρω 2 r2 = 0
Compatibility: r dǫdrθθ + ǫθθ − ǫrr = 0
States:
σ̃rr
x1 =
,
2
2
ρω ro
σ̃θθ
x2 =
,
2
2
ρω ro
x3 = φ,
x4 = r/ro = τ
Control variable: =⇒ derivative of fiber angle: u = dφ/dτ
Nondimensional compliance: Sθθ = E11 Sθθ and
Srθ = E11 Srθ .
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State equations
ẋ1 =
ẋ2 =
x2 − x1 − x24 /x4 ,
′
− x1 x4 Srθ u − Srr − Srθ x24
′
+x2 x4 Sθθ u + Sθθ / [x4 Sθθ ] ,
ẋ3 = u,
ẋ4 = 1.
′ = dS /dφ, and
where ẋi = dxi /dτ, i = 1, 2, 3, 4, Sθθ
θθ
′ = dS /dφ.
Srθ
rθ
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Flywheel Performance
Energy density:
α=
ωf2
14400
ro4
ro2
4
− ri
− ri2
[Wh/kg]
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Flywheel Performance
Energy density:
α=
ωf2
14400
ro4
ro2
4
− ri
− ri2
[Wh/kg]
Design problem: Find the radial-ply fiber orientation that
will maximize ωf .
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Failure theories: When does the disk fail?
Maximum stress failure criterion
Maximum strain failure criterion
Tsai-Wu failure criterion
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Optimization results
Compare 4 designs:
Benchmark design: φ(τ ) = 0.
Jσ , maximum stress failure criterion.
Jǫ , maximum strain failure criterion.
JTW , Tsai-Wu failure criterion.
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Fiber orientation
1.6
1.4
Benchmark
Jσ
1.2
Jε
JTW
Fiber angle, φ (radians)
1
0.8
0.6
0.4
0.2
0
−0.2
−0.4
0
0.1
0.2
0.3
0.4
0.5
r/ro
0.6
0.7
0.8
0.9
1
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Benchmark design
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Maximum stress design
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Maximum strain design
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Tsai−Wu design
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Effective radial stress
0.2
0.18
0.16
Effective radial stress
0.14
0.12
0.1
0.08
0.06
Benchmark
J
σ
0.04
J
ε
J
0.02
0
TW
0
0.1
0.2
0.3
0.4
0.5
r/ro
0.6
0.7
0.8
0.9
1
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Effective tangential stress
0.55
Benchmark
Jσ
Jε
0.5
Effective tangential stress
JTW
0.45
0.4
0.35
0.3
0
0.1
0.2
0.3
0.4
0.5
r/ro
0.6
0.7
0.8
0.9
1
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Radial ply stress σ11
0.6
Radial ply stress, σ11
0.5
Benchmark
J
0.4
σ
J
ε
J
0.3
TW
0.2
0.1
0
0
0.1
0.2
0.3
0.4
0.5
r/ro
0.6
0.7
0.8
0.9
1
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Tangential ply stress σ11
0.75
Benchmark
Jσ
0.7
Jε
Tangential ply stress, σ11
JTW
0.65
0.6
0.55
0.5
0.45
0
0.1
0.2
0.3
0.4
0.5
r/ro
0.6
0.7
0.8
0.9
1
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Energy density
Disk
Maximum Maximum Tsai-Wu
design
stress
strain
Benchmark
150
150
142
Jσ
168
168
106
161
161
157
Jǫ
JTW
166
166
156
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Flywheel construction
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Fiber layout
Benchmark design
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Fiber layout
Optimized design
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Tangential strain
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Conclusions
Fiber angle optimization can improve energy density
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Conclusions
Fiber angle optimization can improve energy density
Failure criteria is significant
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Conclusions
Fiber angle optimization can improve energy density
Failure criteria is significant
Optimized radial plies can be constructed
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Conclusions
Fiber angle optimization can improve energy density
Failure criteria is significant
Optimized radial plies can be constructed
Behavior as predicted by classical lamination theory
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