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贝兹极限 · Cp_max = 16/27 ≈ 0.593
Betz limit · Cp_max = 16/27 ≈ 0.593
Cp=4a(1−a)² · a=1/3→Cp_max=16/27=0.593 · 不可超59.3% · 流管
the maximum fraction of wind kinetic energy a turbine can extract
风功率 · P = ½ρAv³
Wind power · P = ½ρAv³
P=½ρAv³ · 立方律 · ½ρv³=612.5W/m²@10m/s · 可提取P·Cp
wind power grows with the cube of wind speed — speed dominates everything
叶尖速比 · λ = ωR/v
Tip-speed ratio · λ = ωR/v
λ=ωR/v · Cp-λ峰@λ_opt≈7 · Cp_max≈0.48<贝兹 · 变速跟踪
why there is an optimal tip-speed ratio that maximizes Cp
风力发电机功率曲线
Wind Turbine Power Curve
切入3/额定12/切出25 · v³升→平段→归零 · 桨距削峰
Wind Energy · four operating regions
风电容量因子
Wind Capacity Factor
CF=实际/(额定×8760)=0.35 · 陆0.25–0.45海更高 · 间歇
Wind Energy · actual energy ÷ rated potential
风切变廓线
Wind Shear Profile
v(h)=v_ref(h/h_ref)^α · α=0.14 · 100m→8.28m/s · 塔高风强
Wind Energy · wind speed rises with height
风速威布尔分布
Weibull Wind-Speed Distribution
f(v)=(k/c)(v/c)^(k−1)e^(−(v/c)^k) · k=2/c=8→均值7.09 · 右偏
Wind-speed frequency follows a Weibull distribution — shape k and scale c set the mean, mode and power-relevant speed
转子扫掠面积与功率
Rotor Swept Area & Power
A=πR² · P∝D² · D翻倍→功率×4 · D80→1.39MW/D160→5.54MW
Power scales with the square of rotor diameter — double the diameter, quadruple the power
风电场尾流损失
Wind-Farm Wake Losses
Jensen δv/v=(1−√(1−Ct))(D/(D+2kx))² · 5D≈18%亏损 · 间距越大越小
An upstream turbine leaves a slower wake — downstream machines see less wind and produce less (Jensen/Park model)
风轮推力系数 · Ct(a)=4a(1−a)
Rotor thrust coefficient · Ct(a)=4a(1−a)
Ct=4a(1−a) · Ct_max=1@a=0.5 · a=1/3→Ct=8/9 · 推力/功率峰异位
Thrust and power peak at different operating points — drag the axial induction factor a
年发电量 AEP · 功率曲线 × 风速分布
Annual Energy Production · power curve × wind distribution
AEP=8760·ΣP(v)f(v)≈6100MWh · 中风速贡献大 · CF=AEP/(额定8760)
2 MW turbine · Weibull site (k=2) · drag scale c to see AEP and capacity factor
偏航误差功率损失 · P = P_max·cos³(γ)
Yaw misalignment loss · P = P_max·cos³(γ)
功率≈P_max·cos³γ · 15°损10%/30°损35% · 偏航对风
Off the wind by γ, only the axial component drives the rotor — drag the yaw angle γ