18.11.13_wind turbine optimization
TRANSCRIPT
DESIGN OPTIMIZATION DESIGN OPTIMIZATION FORMULATION FOR FORMULATION FOR
VERTICAL AXIS WIND TURBINEVERTICAL AXIS WIND TURBINE
AMANDEEP
SAVI TAKKAR LAXMAN RAVESH 1
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Schematic diagram
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Material Properties
• 𝞼 yield strength of mild steel = 250 MPa• 𝞼 yield strength of stainless steel = 210 MPa• 𝞼 yield strength of HDPE = 15 MPa• E stainless steel (young’s modulus) = 200GPa• E Mild steel (young’s modulus) = 210GPa• Density of mild steel = 7.85*10-6 kg/mm3
• Density of stainless steel = 8*10-6 kg/mm3
• Density of HDPE =0.93*10-6 kg/mm3
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Design Variables
=
Db
HLc
btc
Ds
ts
Ls
tb
X1X2X3X4X5X6X7X8X9
=
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OBJECTIVE FUNCTIONOBJECTIVE FUNCTION
Our aim is to minimize the Rotating mass of wind turbine for a required minimum power.
F(x)= Rotating mass of wind turbine
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• Rotating mass = mass of connectors + mass of blades+ mass of shaft
= {8*Lc*(2btc-tc2) * (7.85*10-6 )} + 4*(π*Db/2)*H*tb*{0.93*10-6}
+ (π*Ds *ts*Ls*(8*10-6 )
F(x)=
Then the objective function {8*X3* (2*X4*X5-X52) * (7.85*10-6 )}
+ 4*(π*X1/2)* X2*X9*{0.93*10-6} + (π*X6 *X7*X8*(8*10-6 )
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Design assumptions/considerations
• Minimum air velocity considered for the calculation of power generated by the wind turbine is taken as 1 m/s, so that in practical, minimum power requirement will be met at any wind speed higher than 1 m/s.
• Force exerted on blades by wind is considered to be UDL over the blade acting horizontally.
• Force acting at the connector end is taken as perpendicular to the end of connector.
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Design wind speed calculation• Standard followed – IS-875 (Part III).• V = Vb*k1*k2*k3*k4 = 26.7 m/s.
• Vb = 47 m/s for Delhi being 3rd basic wind zone region.
• k1 = 0.71 (considering mean probable design life of structure for 5 years).
• k2= 0.80 (for height less than 10 m under terrain category 4).
• k3 = 1 for flat contoured area.
• k4 = 1 for non-cyclonic Delhi region.9
CONSTRAINTSCONSTRAINTS
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Design of hollow shaft
R2=Lc2 – H2/4
Torque T=F*(R+0.5Ds)where F=ρairAV2
T = 0.5*ρAV2*[sqrt(4Lc2-H2)+0.5Ds]
J =(π/4)Ds3ts (Assuming Ds>>ts)
τ =0.39 X 10-9 Db*H*V2[sqrt(4Lc2-H2)+0.5Ds]/(Ds
2*ts)
τ = T*Ds /(J*2)
(τallowable stress / τ) - 1 ≥ 0
τallowable stress
[0.39 X 10-9 *X1*X2*7.13*108 *[sqrt{4(X3)2 –(X2)2+0.5*X6}]/(X6)2*X7
Where
τallowable stress =62.87N/mm2 ; FOS = 1.67
R
H
Lc
- 1 ≥ 0
2Ɵ
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(Due to wind)
Buckling load on shaft• Dead load on shaft = weight of connectors + weight of blades• P = 8*Ac*Lc *ρ(mild steel) *g + 4*π*0.5Db*tb*H*ρHDPE*g• P= 8*(2*X4*X5-X52)*X3*7.85*10-6*g + 4*π*0.5*X1*X9*X2*0.93*10-6*g
• End conditions are “Effectively held in position at both ends but not restrained against rotation”
Effective length = 1*Ls ; Pcritical load= π2EI/(Le2)
I = π/8*Ds3*ts = π/8*X63*X7 (Assuming Ds>>ts)
Pcritical load= π2*E* π/8*X63*X7 X82
Allowable load = Pcritical load / FOS (Where FOS = 2)
(Pallowable load / P) -1 ≥0
π2*200*103* π/8*X63*X7X82*2*(8*(2*X4*X5-X52)*X3*7.85*10-6*g + 4*π*0.5*X1*X9*X2*0.93*10-6*g)
- 1 ≥ 0
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Design of connectors• Total bending moment= sqrt(M1
2 + M22 )
• M1 =0.5*ρAV2 *Lc = 0.5*1.225*10-9*X1*X2*X3*7.13*108
• M2 = weight of blade * radius of arm=(0.5)*(3.14/2)*Db*H*tb*ρHDPE*g*Lc*cosθ where cosθ=R/Lc
(R=sqrt{Lc2 – H2/4})
=(3.14/4)*X1*X2*X9*0.93*10-6*g*sqrt{X32-X22/4}
• M = sqrt(M12 + M2
2)
Max. M1 due to wind force & M2 due to weight of blades
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b
b
tc
σ1 = M*y/I where y=b+tc/4 =X4+X5/4 andI=(5/24)b3tc =(5/24)*X43*X5 (b>>tc)
σ1 = sqrt(M12+M2
2)*(X4+X5)(5/24)*X43*X5*4
Direct compressive stress on the connector = W*sinθAc
W = 0.5*0.5*π*Db*tb *H* ρHDPE *g Where sinθ = H/(2Lc) Ac = 2*b*tc – tc
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σ2 = 0.5*0.5*π*X1*X9*X2*0.93*10-6*g*X2(2*X4*X5-X52 )*2*X3
σ = σ1 + σ2
hence,(σallowable stress/ σ) -1 ≥ 0 σallowable stress = 150 N/mm2 ; FOS = 1.67 14
Design of blades
UDL on blade with F = ρair AV2
Maximum bending moment in case of U.D.L = F*H/8 (at centre)M = ρair AV2 * H/8 = 1.225* 10 -9 *X1*X2*7.13*108*X2
8σ=M*y/I y=(Db/2)-2Db/3π=0.29Db = 0.29*X1I=(0.11*128)Db3tb=(0.11*128) *X13 *X9 (Db>>tb )σ= 1.225* 10 -9 *X1*X2*7.13*108*X2* 0.29*X18*(0.11*128) *X13 *X9Hence (σallowablestress/σ) -1 ≥ 0 where σallowablestress =8.982 MPa FOS = 1.67
Centroid
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Power constraint
• POWER = Cp *½(ρair*Α*V3) P = 0.16*0.5*1.225*10-9*Db*H*1012
= 0.16*0.5*1.225*10-
9*X1*X2*1012
P ≥ 3*106
0.16*0.5*1.225*10-9*X1*X2*1012
3*106
- 1 ≥ 0
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Other constraints
• Ds/ts ≥ 15Ds / ts – 15 ≥ 0
X6 / X7 - 15 ≥ 0
• b/tc ≥ 15b/tc – 15 ≥ 0
X1/X5 – 15 ≥ 0
• Db/tb ≥ 15
X1/X9 – 15 ≥ 017
OPTIMIZATION PROBLEMOPTIMIZATION PROBLEMFind X =
Which minimizes the objective function
{8*X3* (2*X4*X5-X52) * (7.85*10-6 )} + 4*(π*X1/2)* X2*X9*{0.93*10-6} + (π*X6 *X7*X8*(8*10-6 )
min. F(X) =
Subjected to,
g1(X) :
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g2(X) :
g3(X) :
g4(X) :
g5(X) :
g6(X) : X6 / X7 - 15 ≥ 0 g7(X) : X1/X5 – 15 ≥ 0
Variable bounds : Xi ≥ 0 for all i = 1 to 9 19
g8(X) : X1/X9 – 15 ≥ 0
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Objective Function
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Constraints
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Function Tool
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Output
THANK YOU
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