flywheel-energy-storage

name	flywheel-energy-storage
description	Flywheel energy storage systems — moment of inertia, energy density, burst speed, material selection (steel/composite), bearing losses, vacuum housing, charge/discharge cycles, UPS applications, grid storage, safety factors.
metadata	{"priority":7,"promptSignals":{"phrases":["flywheel energy storage","flywheel design","flywheel moment of inertia","FESS","flywheel UPS","flywheel burst speed"],"minScore":3}}

Flywheel Energy Storage Systems — Complete Skill

Energy Storage Fundamentals

Stored kinetic energy: E = (1/2) × I × ω² [J; I = mass moment of inertia [kg·m²]; ω = rotational speed [rad/s]] E = (1/2) × I × (2π × N/60)² [N in RPM]

Usable energy (between ω_max and ω_min): E_usable = (1/2) × I × (ω_max² - ω_min²) [J] Depth of discharge: η_DoD = 1 - (ω_min/ω_max)² = 1 - (N_min/N_max)² Typical DoD: 75% (ω_min = 0.5 ω_max) → E_usable = 75% × E_max

Specific energy (energy density) by shape: Disk: E_m = (1/2) × σ_max / ρ [J/kg; uniform stress approximation] Annular ring: E_m ≈ (σ_max/ρ) × (R_o² + R_i²) / (R_o² + R_i²)... → lower than solid disk Optimal hollow disk: E_m_max = K × σ_max / ρ [K = shape factor; K ≈ 0.5 for solid disk; K ≈ 0.6 for optimized rim]

Shape factor K for common geometries:

• Solid disk (uniform): K = 0.606
• Thin ring: K = 0.500
• Uniform stress disk (Stodola): K = 1.0 (theoretical maximum; variable thickness)
• Constant thickness disk in practice: K ≈ 0.606

Material Selection

Figure of merit for flywheel materials: FM = σ_allow / ρ [specific strength; higher is better for energy density]

Material	σ_u [MPa]	ρ [kg/m³]	σ_allow/ρ [kJ/kg]	E_specific [Wh/kg]
4340 Steel	1,000	7,800	64	27
Maraging 300	1,800	8,000	112	39
Titanium Ti-6Al-4V	900	4,430	101	35
Carbon fiber/epoxy (UD)	1,500	1,600	469	130
Kevlar 49/epoxy	1,250	1,400	446	124
E-glass/epoxy	600	1,900	158	44

Composite flywheels: significantly higher energy density; brittle fracture → debris containment critical Steel flywheels: lower energy density; more predictable failure (ductile); common for cost-sensitive applications

Stress Analysis

Rotating disk (Lamé equations — uniform disk): σ_r = (3+ν)/8 × ρ ω² (R_o² + R_i² - R_i²R_o²/r² - r²) [radial stress] σ_θ = (3+ν)/8 × ρ ω² (R_o² + R_i² + R_i²R_o²/r² - (1+3ν)/(3+ν) × r²) [hoop stress]

Peak stress (solid disk, r = 0): σ_max = (3+ν)/8 × ρ ω² × R_o² [at center for solid disk] For ν = 0.3: σ_max = 0.413 × ρ ω² R_o²

Burst speed: ω_burst² = σ_u / (0.413 × ρ × R_o²) [for solid disk, uniform thickness] Safety factor on burst: SF_burst = ω_burst / ω_max ≥ 1.5 (minimum); 2.0 (recommended)

Composite flywheel hoop stress: σ_θ = E_θ × ε_θ = E_θ × (u/r) [u = radial displacement; E_θ = hoop modulus] For filament-wound composite ring: σ_θ ≈ ρ ω² r² (approximate for thin ring)

Bearing and Loss Considerations

Bearing types for flywheel:

• Rolling element (ball/angular contact): σ friction loss ≈ 0.001–0.01 of stored energy per revolution; contact fatigue limits life
• Active magnetic bearings (AMB): near-zero friction; requires power (≈ 100–500 W for large flywheels); preferred for high-cycle FESS
• Superconducting bearings (HTS): zero-loss; requires cryogenic cooling; highest efficiency

Windage loss (in air): P_windage = C_M × ρ_air × ω³ × R_o⁵ [W; C_M = torque coefficient ≈ 0.01–0.05 depending on geometry/Re] Vacuum operation: p < 0.1 Pa → windage ≈ 0; required for high-efficiency FESS

Total idle losses: P_idle = P_bearing + P_windage + P_eddy_current [typically 0.5–2% of rated power] Self-discharge time: τ = E_stored / P_idle [hours to fully discharge from idle losses]

Charge/Discharge Dynamics

Motor-generator (M/G): Charging: motor drives flywheel to ω_max; P_charge = T_motor × ω Discharging: flywheel drives generator; P_discharge = T_gen × ω Efficiency: η_roundtrip = η_charge × η_discharge ≈ 0.85–0.95 (depending on M/G efficiency and bearing losses)

Response time: Instantaneous response (sub-millisecond); excellent for power quality applications Ramp rate: limited by M/G current rating and thermal limits

Application duty cycles: UPS: discharge 100% in 10–30 seconds; recharge in 60–120 seconds Frequency regulation: 1–5% SOC swings; high cycle count (>100,000 cycles per year) Load leveling: 50% DoD; 4–8 cycles per day

Containment and Safety

Burst containment design: Steel flywheel failure: radial cracks → fragments eject radially → require heavy steel containment (adds mass) Composite flywheel failure: delamination → smaller debris → lighter containment possible but still critical

Containment design pressure: P_containment = KE_burst / V_containment [approximate; V = containment volume] Actual design: fragments analysis (fragment mass × V_fragment²) per military fragment containment standards

Containment weight (typical): Steel containment: 1.5–2.5× flywheel weight Composite gravel/sand containment: composite flywheels often use sand bed or Kevlar wrap

Safety standards: Maximum rotational speed limited to 80% of analytically predicted burst speed by most safety authorities Ground isolation: flywheel room must be evacuated during operation (personnel safety from potential burst)

System Design Example

UPS flywheel, 100 kW × 15 s = 1500 kJ: Required energy: E = 100,000 × 15 = 1,500,000 J with η = 0.90: E_stored = 1,667 kJ Steel flywheel at N_max = 20,000 RPM, ω_max = 2094 rad/s σ_max = SF_burst × σ_allow: select R_o from σ_max = 0.413 × ρ ω² R_o² I_required = 2 × E / ω_max² = 2 × 1,667,000 / 2094² = 0.760 kg·m² Solid disk: I = 1/2 × m × R_o² → m = 2I/R_o² (solve with R_o from stress constraint)

Standards

Standard	Scope
ANSI/AIAA S-090	Flywheel energy storage module design
IEC 62477	Safety requirements for power electronic converters
IEEE 1679	Recommended practice for characterizing FESS
ASME Section VIII	Pressure vessel for vacuum housing
MIL-STD-810	Environmental test (for military FESS)

Output

Provide: flywheel geometry (R_o [m], R_i [m], thickness profile), material (type, σ_allow [MPa], ρ [kg/m³], specific strength [kJ/kg]), mass [kg] and moment of inertia I [kg·m²], operating speed range N_min–N_max [RPM], stored energy E_max [kJ] and usable energy E_usable [kJ], energy density [Wh/kg], peak hoop/radial stress [MPa] vs. σ_allow, burst speed ω_burst [RPM] and safety factor SF_burst, bearing type and idle losses P_idle [W], vacuum level required [Pa], roundtrip efficiency η [%], self-discharge time [hours], containment design basis, and applicable standard (IEEE 1679, ANSI/AIAA S-090).