Propeller or Rotor in Axial Flight

NACA 0012

Standard C81 Table

CL M

alpha 0 0.2 0.3 0.4 0.5 0.6

-180 0 0 0 0 0 0

-172.5 0.78 0.78 0.78 0.78 0.78 0.78

-161 0.62 0.62 0.62 0.62 0.62 0.62

-147 1 1 1 1 1 1

-129 1 1 1 1 1 1

-49 -1.18 -1.18 -1.18 -1.18 -1.18 -1.18

-39 -1.18 -1.18 -1.18 -1.18 -1.18 -1.18

-21 -0.8 -0.8 -0.81 -0.83 -0.85 -0.85

-16.5 -1.007 -1.007 -0.944 -0.96 -0.965 -0.965

-15 -1.19 -1.19 -1.09 -1.055 -0.99 -0.98

-14 -1.333 -1.333 -1.22 -1.096 -1 -0.97

-13 -1.334 -1.334 -1.28 -1.12 -1 -0.96

-12 -1.255 -1.255 -1.26 -1.13 -1 -0.947

-11 -1.161 -1.161 -1.19 -1.12 -0.994 -0.93

-10 -1.055 -1.055 -1.01 -1.082 -0.985 -0.91

-8 -0.844 -0.844 -0.88 -0.907 -0.922 -0.87

-6 -0.633 -0.633 -0.66 -0.684 -0.741 -0.77

-4 -0.422 -0.422 -0.44 -0.456 -0.494 -0.544

-2 -0.211 -0.211 -0.22 -0.228 -0.247 -0.272

0 0 0 0 0 0 0

2 0.211 0.211 0.22 0.228 0.247 0.272

4 0.422 0.422 0.44 0.456 0.494 0.544

6 0.633 0.633 0.66 0.684 0.741 0.77

8 0.844 0.844 0.88 0.907 0.922 0.87

10 1.055 1.055 1.1 1.082 0.985 0.91

11 1.161 1.161 1.19 1.12 0.994 0.93

12 1.255 1.255 1.26 1.13 1 0.947

13 1.334 1.334 1.28 1.12 1 0.96

14 1.333 1.333 1.22 1.096 1 0.97

15 1.19 1.19 1.09 1.055 0.99 0.98

16.5 1.007 1.007 0.944 0.96 0.965 0.965

21 0.8 0.8 0.81 0.83 0.85 0.85

39 1.18 1.18 1.18 1.18 1.18 1.18

49 1.18 1.18 1.18 1.18 1.18 1.18

129 -1 -1 -1 -1 -1 -1

147 -1 -1 -1 -1 -1 -1

161 -0.62 -0.62 -0.62 -0.62 -0.62 -0.62

172.5 -0.78 -0.78 -0.78 -0.78 -0.78 -0.78

180 0 0 0 0 0 0

CD M

alpha 0 0.18 0.28 0.38 0.48 0.62

-180 0.022 0.022 0.022 0.022 0.022 0.022

-175 0.062 0.062 0.062 0.062 0.062 0.062

-170 0.132 0.132 0.132 0.132 0.132 0.132

-165 0.242 0.242 0.242 0.242 0.242 0.242

-160 0.302 0.302 0.302 0.302 0.302 0.302

-140 1.042 1.042 1.042 1.042 1.042 1.042

-120 1.652 1.652 1.652 1.652 1.652 1.652

-110 1.852 1.852 1.852 1.852 1.852 1.852

-100 2.022 2.022 2.022 2.022 2.022 2.022

-90 2.022 2.022 2.022 2.022 2.022 2.022

-80 1.962 1.962 1.962 1.962 1.962 1.962

-70 1.842 1.842 1.842 1.842 1.842 1.842

-60 1.662 1.662 1.662 1.662 1.662 1.662

-50 1.392 1.392 1.392 1.392 1.392 1.399

-30 0.562 0.562 0.562 0.562 0.562 0.562

-21 0.332 0.332 0.332 0.332 0.332 0.332

-16 0.155 0.155 0.181 0.207 0.235 0.257

-15 0.102 0.102 0.148 0.181 0.209 0.233

-14 0.038 0.038 0.099 0.146 0.18 0.212

-13 0.0264 0.0264 0.0455 0.094 0.148 0.191

-12 0.022 0.022 0.03 0.06 0.111 0.164

-11 0.0196 0.0196 0.0232 0.038 0.078 0.135

-10 0.0174 0.0174 0.0189 0.0259 0.053 0.105

-9 0.0154 0.0154 0.0159 0.0187 0.0351 0.077

-8 0.0138 0.0138 0.0138 0.0147 0.022 0.053

-7 0.0122 0.0122 0.0122 0.0123 0.0141 0.035

-6 0.011 0.011 0.011 0.011 0.011 0.0212

-5 0.01 0.01 0.01 0.01 0.01 0.0132

-4 0.0093 0.0093 0.0093 0.0093 0.0093 0.01

-3 0.0088 0.0088 0.0088 0.0088 0.0088 0.009

-2 0.0085 0.0085 0.0085 0.0085 0.0085 0.0085

-1 0.0083 0.0083 0.0083 0.0083 0.0083 0.0083

0 0.008 0.008 0.008 0.008 0.008 0.008

1 0.0083 0.0083 0.0083 0.0083 0.0083 0.0083

2 0.0085 0.0085 0.0085 0.0085 0.0085 0.0085

3 0.0088 0.0088 0.0088 0.0088 0.0088 0.009

4 0.0093 0.0093 0.0093 0.0093 0.0093 0.01

5 0.01 0.01 0.01 0.01 0.01 0.0132

6 0.011 0.011 0.011 0.011 0.011 0.0212

7 0.0122 0.0122 0.0122 0.0123 0.0141 0.035

8 0.0138 0.0138 0.0138 0.0147 0.022 0.053

9 0.0154 0.0154 0.0159 0.0187 0.0351 0.077

10 0.0174 0.0174 0.0189 0.0259 0.053 0.105

11 0.0196 0.0196 0.0232 0.038 0.078 0.135

12 0.022 0.022 0.03 0.06 0.111 0.164

13 0.0264 0.0264 0.0455 0.094 0.148 0.191

14 0.038 0.038 0.099 0.146 0.18 0.212

15 0.102 0.102 0.148 0.181 0.209 0.233

16 0.155 0.155 0.181 0.207 0.235 0.257

21 0.332 0.332 0.332 0.332 0.332 0.332

30 0.562 0.562 0.562 0.562 0.562 0.562

50 1.392 1.392 1.392 1.392 1.392 1.392

60 1.662 1.662 1.662 1.662 1.662 1.662

70 1.842 1.842 1.842 1.842 1.842 1.842

80 1.962 1.962 1.962 1.962 1.962 1.962

90 2.022 2.022 2.022 2.022 2.022 2.022

100 2.022 2.022 2.022 2.022 2.022 2.022

110 1.852 1.852 1.852 1.852 1.852 1.852

120 1.652 1.652 1.652 1.652 1.652 1.652

140 1.042 1.042 1.042 1.042 1.042 1.042

160 0.302 0.302 0.302 0.302 0.302 0.302

165 0.242 0.242 0.242 0.242 0.242 0.242

170 0.132 0.132 0.132 0.132 0.132 0.132

175 0.062 0.062 0.062 0.062 0.062 0.062

180 0.022 0.022 0.022 0.022 0.022 0.022

0.7 0.75 0.8 0.9 1

0 0 0 0 0

0.78 0.78 0.78 0.78 0.78

0.62 0.62 0.62 0.62 0.62

1 1 1 1 1

1 1 1 1 1

-1.18 -1.18 -1.18 -1.18 -1.18

-1.18 -1.18 -1.18 -1.18 -1.18

-0.85 -0.71 -0.68 -0.64 -0.64

-0.965 -0.795 -0.76 -0.7 -0.7

-0.98 -0.83 -0.79 -0.72 -0.72

-0.97 -0.84 -0.805 -0.73 -0.73

-0.96 -0.85 -0.815 -0.735 -0.735

-0.94 -0.85 -0.82 -0.74 -0.74

-0.923 -0.85 -0.81 -0.74 -0.74

-0.9 -0.845 -0.805 -0.73 -0.73

-0.84 -0.82 -0.77 -0.695 -0.695

-0.75 -0.77 -0.72 -0.593 -0.593

-0.578 -0.627 -0.603 -0.396 -0.396

-0.313 -0.35 -0.395 -0.2 -0.2

0 0 0 0 0

0.313 0.35 0.395 0.2 0.2

0.578 0.627 0.603 0.396 0.396

0.75 0.77 0.72 0.593 0.593

0.84 0.82 0.77 0.695 0.695

0.9 0.845 0.805 0.73 0.73

0.923 0.85 0.81 0.74 0.74

0.94 0.85 0.82 0.74 0.74

0.96 0.85 0.815 0.735 0.735

0.97 0.84 0.805 0.73 0.73

0.98 0.83 0.79 0.73 0.73

0.965 0.795 0.76 0.7 0.7

0.85 0.71 0.68 0.64 0.64

1.18 1.18 1.18 1.18 1.18

1.18 1.18 1.18 1.18 1.18

-1 -1 -1 -1 -1

-1 -1 -1 -1 -1

-0.62 -0.62 -0.62 -0.62 -0.62

-0.78 -0.78 -0.78 -0.78 -0.78

0 0 0 0 0

-50 -40 -30

0.72 0.77 0.82 0.92 1

0.022 0.022 0.022 0.022 0.022

0.062 0.062 0.062 0.062 0.062

0.132 0.132 0.132 0.132 0.132

0.242 0.242 0.242 0.242 0.242

0.302 0.302 0.302 0.302 0.302

1.042 1.042 1.042 1.042 1.042

1.652 1.652 1.652 1.652 1.652

1.852 1.852 1.852 1.852 1.852

2.022 2.022 2.022 2.022 2.022

2.022 2.022 2.022 2.022 2.022

1.962 1.962 1.962 1.962 1.962

1.842 1.842 1.842 1.842 1.842

1.662 1.662 1.662 1.662 1.662

1.392 1.392 1.392 1.392 1.392

0.562 0.562 0.562 0.562 0.562

0.332 0.332 0.332 0.342 0.342

0.274 0.292 0.305 0.342 0.342

0.252 0.271 0.282 0.298 0.298

0.233 0.249 0.26 0.293 0.293

0.216 0.231 0.239 0.272 0.292

0.198 0.211 0.22 0.252 0.291

0.17 0.192 0.202 0.232 0.275

0.145 0.176 0.186 0.213 0.254

0.122 0.159 0.172 0.199 0.232

0.101 0.14 0.155 0.183 0.214

0.082 0.111 0.139 0.169 0.192

0.0615 0.082 0.12 0.14 0.17

0.038 0.054 0.088 0.111 0.14

0.0167 0.03 0.0575 0.095 0.112

0.0102 0.0175 0.0355 0.086 0.102

0.0086 0.0117 0.024 0.081 0.098

0.0083 0.0091 0.0175 0.078 0.096

0.008 0.008 0.0137 0.078 0.095

0.0083 0.0091 0.0175 0.078 0.096

0.0086 0.0117 0.024 0.081 0.098

0.0102 0.0175 0.0355 0.086 0.102

0.0167 0.03 0.0575 0.095 0.112

0.038 0.054 0.088 0.111 0.14

0.0615 0.082 0.12 0.14 0.17

0.082 0.111 0.139 0.169 0.192

0.101 0.14 0.155 0.183 0.214

0.122 0.159 0.172 0.199 0.232

0.145 0.176 0.186 0.213 0.254

0.17 0.192 0.202 0.232 0.275

0.198 0.211 0.22 0.252 0.291

0.216 0.231 0.239 0.272 0.292

0.233 0.249 0.26 0.293 0.293

0.252 0.271 0.282 0.298 0.298

0.274 0.292 0.305 0.342 0.342

0.332 0.332 0.332 0.342 0.342

0.562 0.562 0.562 0.562 0.562

-50 -40 -30

1.392 1.392 1.392 1.392 1.392

1.662 1.662 1.662 1.662 1.662

1.842 1.842 1.842 1.842 1.842

1.962 1.962 1.962 1.962 1.962

2.022 2.022 2.022 2.022 2.022

2.022 2.022 2.022 2.022 2.022

1.852 1.852 1.852 1.852 1.852

1.652 1.652 1.652 1.652 1.652

1.042 1.042 1.042 1.042 1.042

0.302 0.302 0.302 0.302 0.302

0.242 0.242 0.242 0.242 0.242

0.132 0.132 0.132 0.132 0.132

0.062 0.062 0.062 0.062 0.062

0.022 0.022 0.022 0.022 0.022

-1.5

-1

-0.5

0

0.5

1

1.5

-20 -10 0 10 20 30 40 50

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

-20 -10 0 10 20 30 40 50

Atmospheric Inputs Geometric Inputs

Density 0.00238 slugs/ft^3 Nb, number of blades 3

V_∞ 0.00000 mph Disk Diameter 8.00

V_∞ 0.00000 ft/s Disk Radius 4.00

V_∞ 0.00000 knots Disk Area 50.27

V_∞ 0.00000 m/s c, Chord 0.42

RPM 2000 rev/min σ, Solidity 0.09931268

RPS 33.333333 rev/s

rad/s 209.43951 rad/s

V_tip 837.75804 ft/s

λ, Inflow Ratiov_i v_i Thrust Ct Ct, hover onlyPower (ideal)

(mph) (ft/s) lbs check (ft*lb/s)

0 0 0 0 0 0 0 0

1 0.003501 2 2.93 2.06 0.000025 0.000025 6.03

2 0.007003 4 5.87 8.23 0.000098 0.000098 48.27

3 0.010504 6 8.80 18.51 0.000221 0.000221 162.91

4 0.014006 8 11.73 32.91 0.000392 0.000392 386.17

5 0.017507 10 14.67 51.43 0.000613 0.000613 754.23

6 0.021008 12 17.60 74.05 0.000883 0.000883 1,303.32

7 0.02451 14 20.53 100.79 0.001201 0.001201 2,069.62

8 0.028011 16 23.47 131.65 0.001569 0.001569 3,089.34

9 0.031513 18 26.40 166.62 0.001986 0.001986 4,398.69

10 0.035014 20 29.33 205.70 0.002452 0.002452 6,033.87

11 0.038515 22 32.27 248.90 0.002967 0.002967 8,031.08

12 0.042017 24 35.20 296.21 0.003531 0.003531 10,426.53

13 0.045518 26 38.13 347.63 0.004144 0.004144 13,256.41

14 0.04902 28 41.07 403.17 0.004806 0.004806 16,556.94

15 0.052521 30 44.00 462.83 0.005517 0.005517 20,364.31

16 0.056023 32 46.93 526.59 0.006277 0.006277 24,714.73

17 0.059524 34 49.87 594.47 0.007086 0.007086 29,644.40

18 0.063025 36 52.80 666.47 0.007944 0.007944 35,189.53

19 0.066527 38 55.73 742.58 0.008852 0.008852 41,386.32

20 0.070028 40 58.67 822.80 0.009808 0.009808 48,270.96

0.0040

0.0050

0.0060

0.0070

0.0080

0.0090

0.0100

Po

wer

Co

eff

icie

nt,

Cp

0.0000

0.0010

0.0020

0.0030

0.0040

0.000 0.010 0.020 0.030 0.040 0.050 0.060 0.070 0.080P

ow

er

Co

eff

icie

nt,

Cp

Thrust Coefficient, Ct

Power Corrections

Κ, Induced Power correction factor1.100

ft Cd_0 0.010

ft

ft^2

ft

Power (ideal) Power (ideal)Cp,simple Cp_i Cp_0 Cp_correctedFM Cp/σ Ct/σ

(HP) (kW) induced profile

0 0 0 0.0000000 0.000124 0.0001241 0 0.00125 0

0.01097 0.00818 0.0000001 0.0000001 0.000124 0.0001242 0.0006911 0.001251 0.00

0.08777 0.06545 0.0000007 0.0000008 0.000124 0.0001249 0.0054992 0.001258 0.000988

0.29621 0.22088 0.0000023 0.0000025 0.000124 0.0001267 0.0182969 0.001276 0.002222

0.70212 0.52357 0.0000055 0.0000060 0.000124 0.0001302 0.0422063 0.001311 0.00395

1.37133 1.02260 0.0000107 0.0000118 0.000124 0.0001359 0.078941 0.001369 0.006172

2.36967 1.76706 0.0000185 0.0000204 0.000124 0.0001445 0.1282996 0.001455 0.008888

3.76294 2.80602 0.0000294 0.0000324 0.000124 0.0001565 0.1881246 0.001576 0.012098

5.61698 4.18858 0.0000440 0.0000484 0.000124 0.0001725 0.2548329 0.001737 0.015801

7.99762 5.96382 0.0000626 0.0000688 0.000124 0.0001930 0.3243084 0.001943 0.019998

10.97067 8.18083 0.0000859 0.0000944 0.000124 0.0002186 0.3927791 0.002201 0.024689

14.60197 10.88868 0.0001143 0.0001257 0.000124 0.0002498 0.4573789 0.002516 0.029874

18.95732 14.13647 0.0001484 0.0001632 0.000124 0.0002873 0.5163203 0.002893 0.035553

24.10257 17.97328 0.0001886 0.0002075 0.000124 0.0003316 0.568779 0.003339 0.041725

30.10353 22.44820 0.0002356 0.0002591 0.000124 0.0003833 0.6146458 0.003859 0.048391

37.02602 27.61030 0.0002898 0.0003187 0.000124 0.0004429 0.654265 0.004459 0.055551

44.93588 33.50868 0.0003517 0.0003868 0.000124 0.0005110 0.6882229 0.005145 0.063205

53.89892 40.19241 0.0004218 0.0004640 0.000124 0.0005881 0.717199 0.005922 0.071352

63.98096 47.71060 0.0005007 0.0005508 0.000124 0.0006749 0.7418752 0.006796 0.079994

75.24785 56.11231 0.0005889 0.0006478 0.000124 0.0007719 0.7628859 0.007772 0.09

87.76538 65.44664 0.0006868 0.0007555 0.000124 0.0008797 0.7807955 0.008857 0.10

0.080 0.090 0.100

Michael Duffy AE 6070, HW 2 Sep. 10th, 2006

Ct/Cp_ Ideal

1000

285.5993

142.7997

95.19978

71.39983

57.11987

47.59989

40.7999

35.69992

31.73326

28.55993

25.96358

23.79994

21.96918

20.39995

19.03996

17.84996

16.79996

15.86663

15.03154

14.27997

Airspeed, V 250 ktas Temperature, T 44.756o F at the given alt for the given day type (NOT USED, BUT FOR YOU INFO)

Pressure Altitude, h 4000 ft Delta 0.8637

Type of Day std std/trop/hot Theta 1.0694

Temperature, T 95o F Sigma 0.8076

Density 1.920E-03 slugs/ft3

m 3.937E-07

n 2.051E-04 ft2/sec

M 0.365

a, Speed of Sound 1154.546 ft/sec

Airspeed, V 224.671639 kcas

USER INPUTS QUIK CALCS

INPUT HERE:

OUTPUT HERE:

at the given alt for the given day type (NOT USED, BUT FOR YOU INFO)

INSTRUCTIONS

Propeller calculator instructions:

1) Open MS Excel before loading the spreadsheet, should have a blank workbook.

2) Click Tools -> Add-ins -> Check the 'SOLVER' Check box

3) Click Tools -> Options -> 'Calculation' Tab -> Check the 'Iteration' Check Box -> then hit ok

4) Click Tools -> Macro -> Visual Basic Editor (or hit Alt + F11), this will open Visual Basic Editor

5) In Visual Basic Editor Click Tools -> References.. -> Check the 'SOLVER' check box

6) Now you can open 'PROPELLER_OR_ROTOR_IN_AXIAL_FLIGHT.xls' in MS Excel

7) When prompted -> Click on 'Enable Macros'

8) Change the Blue values to you design, and click on the 'Calculate Button'

Note: If you get 'NaN' in any cell you can either restart the program by closing and reopening, or put zero's in cells 'N7, N8, T3, and T4' this will fix almost any problem, then you can start turning things back on until you find the problem.

Air Properties: Rotor Properties:

Density 0.00191967 slugs/ft^3 Blade Pitch 0 deg.

Temp 95 °F Blade Pitch 0 rad.

ν, Kinematic Viscosity0.00020507 ft^2/s θ_tip 6.208 deg. For Ideal Twist

μ, Dynamic Viscosity3.9367E-07 lb.s/ft^2 θ_tip 0.10835 rad.

a, Speed of Sound1,154.55 ft/s θ_.75 14.452 deg. For Linear Twist

θ_twist -20 deg.

Blades, Nb 4 Nb

Radius, R 23.5 ft 7.1628

r_root cut out 0 ft 0

Diam. 47 ft 14.3256

Root Chord, c_root31.12122 in 0.7904789

Tip Chord, c_tip10.37374 in 0.263493

n_elements 20

Ω,RPM 264.15 rev/min

Ω,RPS 4.4025 rev/s

Ω,rad/s 27.66172 rad/s

V_tip 650.0505 ft/s

M_tip 0.56 compressible

X_tsr #DIV/0! tip speed ratio

V_∞ 0.000 mph 0

V_∞ 0.000 ft/s

λc 0.0000 (V∞)/ΩR

Twist Dist 1 1 = linear, 0 = Ideal

Chord Dist 1 2 = constant, 1 = linear taper, 0 = ideal taper

element numbery_location of elementdy Chord Chord Twist dist. Twist dist. θ θ

ft ft in ft deg. deg. deg. rad.

Ideal Input Local Blade Pitch

1 1.175 1.175 31.12122 2.593435 124.16 28.45 28.45 0.49658

2 2.35 1.175 30.02924 2.502437 62.08 27.45 27.45 0.47913

3 3.525 1.175 28.93727 2.411439 41.39 26.45 26.45 0.46167

4 4.7 1.175 27.8453 2.320442 31.04 25.45 25.45 0.44422

5 5.875 1.175 26.75333 2.229444 24.83 24.45 24.45 0.42677

6 7.05 1.175 25.66135 2.138446 20.69 23.45 23.45 0.40931

7 8.225 1.175 24.56938 2.047448 17.74 22.45 22.45 0.39186

8 9.4 1.175 23.47741 1.956451 15.52 21.45 21.45 0.37441

9 10.575 1.175 22.38544 1.865453 13.80 20.45 20.45 0.35695

10 11.75 1.175 21.29346 1.774455 12.42 19.45 19.45 0.33950

11 12.925 1.175 20.20149 1.683458 11.29 18.45 18.45 0.32205

12 14.1 1.175 19.10952 1.59246 10.35 17.45 17.45 0.30459

13 15.275 1.175 18.01755 1.501462 9.55 16.45 16.45 0.28714

14 16.45 1.175 16.92557 1.410465 8.87 15.45 15.45 0.26969

15 17.625 1.175 15.8336 1.319467 8.28 14.45 14.45 0.25223

16 18.8 1.175 14.74163 1.228469 7.76 13.45 13.45 0.23478

17 19.975 1.175 13.64966 1.137471 7.30 12.45 12.45 0.21733

18 21.15 1.175 12.55768 1.046474 6.90 11.45 11.45 0.19988

19 22.325 1.175 11.46571 0.955476 6.53 10.45 10.45 0.18242

20 23.5 1.175 10.37374 0.864478 6.21 9.45 9.45 0.16497

Important note: Must have: -Tools-Options-Calculation Tab – Iteration check box checked on for this program to work: -Tools-Addi-ins-Solver =>Add-in must be checked -In Visual Basic Tools -References-Solver Must be checked Hit F9 to Recalculate or Calculate Button Items in BLUE are INPUTS Items in BLACK are Calculated via magic

Total:

0.00

5.00

10.00

15.00

20.00

25.00

30.00

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

An

gle

(d

eg

)

x/R

Angle (deg)

EFF. ALPHA INFLOW ANG LOCAL PITCH

Airfoil Propeties: Correction Factors:

Cl Cl_α Cd Prandtl Tip Loss 1

Section Lift CoeffecientSection Drag Coeffecient Prandtl Root Loss 1

For Ideal Twist 0.0000 #DIV/0! 0.0080 Table Look Up

5.7 0.0087 Polynomial or Constant

For Linear Twist Compressiblity 1 1 = on, 0 = off

Table Look up 0 1 = table look up, 0 = linear approx.

m

m

m

m

m Propeller calculator instructions:


2) Click Tools -> Add-ins -> Check the 'SOLVER' Check box





7) When prompted -> Click on 'Enable Macros'

m/s 8) Change the Blue values to you design, and click on the 'Calculate Button'


2 = constant, 1 = linear taper, 0 = ideal taper

φ φ α α r dr σ, Solidity σ, Solidity V_tan X_loc

deg. rad. deg. rad. ft/s

Inflow Angle Effective Angle of Attack local solidityThrust Weighted local speed ratio

24.35 0.42503 4.10 0.07155 0.05 0.05 0.140513 1.76E-05 32.503 #DIV/0!

20.96 0.36584 6.49 0.11329 0.1 0.05 0.135583 6.78E-05 65.005 #DIV/0!

18.09 0.31574 8.36 0.14593 0.15 0.05 0.130653 0.000147 97.508 #DIV/0!

15.86 0.27675 9.60 0.16747 0.2 0.05 0.125722 0.000251 130.010 #DIV/0!

14.14 0.24684 10.31 0.17993 0.25 0.05 0.120792 0.000377 162.513 #DIV/0!

12.80 0.22332 10.66 0.18600 0.3 0.05 0.115862 0.000521 195.015 #DIV/0!

11.68 0.20392 10.77 0.18794 0.35 0.05 0.110932 0.000679 227.518 #DIV/0!

10.72 0.18718 10.73 0.18723 0.4 0.05 0.106001 0.000848 260.020 #DIV/0!

9.87 0.17226 10.58 0.18470 0.45 0.05 0.101071 0.001023 292.523 #DIV/0!

9.09 0.15872 10.36 0.18079 0.5 0.05 0.096141 0.001202 325.025 #DIV/0!

8.38 0.14629 10.07 0.17576 0.55 0.05 0.09121 0.00138 357.528 #DIV/0!

7.72 0.13480 9.73 0.16979 0.6 0.05 0.08628 0.001553 390.030 #DIV/0!

7.11 0.12411 9.34 0.16303 0.65 0.05 0.08135 0.001719 422.533 #DIV/0!

6.54 0.11409 8.91 0.15559 0.7 0.05 0.07642 0.001872 455.035 #DIV/0!

6.00 0.10468 8.45 0.14755 0.75 0.05 0.071489 0.002011 487.538 #DIV/0!

5.49 0.09582 7.96 0.13896 0.8 0.05 0.066559 0.00213 520.040 #DIV/0!

5.02 0.08757 7.43 0.12976 0.85 0.05 0.061629 0.002226 552.543 #DIV/0!

4.60 0.08030 6.85 0.11957 0.9 0.05 0.056698 0.002296 585.045 #DIV/0!

4.36 0.07612 6.09 0.10630 0.95 0.05 0.051768 0.002336 617.548 #DIV/0!

9.37 0.16349 0.08 0.00148 1 0.05 0.046838 0.002342 650.050 #DIV/0!

0.0937 0.0750

0.9 1 0.000000

0.020000

0.040000

0.060000

0.080000

0.100000

0.120000

0.140000

0.160000

0.180000

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

INF

LO

W R

AT

IO

x/R

INFLOW RATIO

INFLOW GUESS INFLOW FROM MOMENTUM

Output:

1 = on, 0 = off Ct Cp/σ Ct/σ Thrust

1 = on, 0 = off 0.01066 0.0125 0.1421 15,000.32

Check Convergence:







Don't Use Yet

f_tip f_root F v_inflow v_inflow v_inflow a_induction factorλ, Inflow

Prandtl mph mph ft/s (V+vi)/ΩR

local speed ratio using lift curve using tables

89.40 0.25 0.43 10.03 0.0000 14.711 #DIV/0! 0.022631

49.20 0.61 0.63 16.98 3.7759 24.902 #DIV/0! 0.038308

35.89 1.12 0.79 21.72 7.6822 31.853 #DIV/0! 0.049001

28.91 1.81 0.89 25.18 7.6069 36.927 #DIV/0! 0.056807

24.31 2.70 0.96 27.92 8.7068 40.950 #DIV/0! 0.062995

20.90 3.84 0.99 30.20 9.7362 44.289 #DIV/0! 0.068132

18.21 5.28 1.00 32.08 10.6893 47.051 #DIV/0! 0.072380

16.03 7.12 1.00 33.58 11.5763 49.248 #DIV/0! 0.075760

14.19 9.50 1.00 34.70 12.4099 50.894 #DIV/0! 0.078292

12.60 12.60 1.00 35.47 13.1983 52.024 #DIV/0! 0.080031

11.19 16.71 1.00 35.92 13.9483 52.680 #DIV/0! 0.081039

9.89 22.25 1.00 36.07 14.6644 52.898 #DIV/0! 0.081375

8.68 29.93 1.00 35.94 15.3526 52.710 #DIV/0! 0.081086

7.51 40.90 1.00 35.55 16.0138 52.143 #DIV/0! 0.080214

6.37 57.32 1.00 34.92 16.6524 51.223 #DIV/0! 0.078798

5.22 83.49 1.00 34.08 17.2726 49.985 #DIV/0! 0.076894

4.03 129.42 0.99 33.08 16.0627 48.511 #DIV/0! 0.074627

2.77 224.16 0.96 32.10 14.5092 47.081 #DIV/0! 0.072426

1.38 499.20 0.84 32.11 12.6474 47.100 #DIV/0! 0.072456

0.00 4992.00 0.00 73.11 25.9158 107.233 #DIV/0! 0.164962

Converged

Prandtl Tip/Root Loss

Calculate

1.0000

0.9 1

INFLOW FROM MOMENTUM

0.0000

0.2000

0.4000

0.6000

0.8000

1.0000

1.2000

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Cl

x/R

Cl AND Cd

Cl Cd


U_vel_vector Re Re Mach Cl_α Cl Cd dL=

ft/s check Local Section Lift CoeffecientSection Drag Coeffecientlbf

1 blade

35.68 451,193 451,193 0.03 5.70 0.4080 0.00920 1.519

69.61 849,464 849,464 0.06 5.71 0.6468 0.01138 8.845

102.58 1,206,237 1,206,237 0.08 5.72 0.8348 0.01406 23.889

135.15 1,529,310 1,529,310 0.11 5.74 0.9607 0.01629 45.925

167.59 1,822,012 1,822,012 0.14 5.76 1.0359 0.01775 73.157

199.98 2,085,390 2,085,390 0.17 5.78 1.0756 0.01851 103.748

232.33 2,319,646 2,319,646 0.20 5.81 1.0927 0.01876 136.192

264.64 2,524,813 2,524,813 0.23 5.85 1.0953 0.01866 169.266

296.92 2,700,969 2,700,969 0.25 5.89 1.0883 0.01834 201.851

329.16 2,848,233 2,848,233 0.28 5.94 1.0739 0.01786 232.858

361.39 2,966,717 2,966,717 0.31 5.99 1.0536 0.01725 261.254

393.60 3,056,504 3,056,504 0.34 6.06 1.0283 0.01655 286.103

425.81 3,117,657 3,117,657 0.37 6.12 0.9986 0.01580 306.589

458.01 3,150,216 3,150,216 0.39 6.20 0.9650 0.01501 322.018

490.22 3,154,212 3,154,212 0.42 6.29 0.9278 0.01421 331.814

522.44 3,129,669 3,129,669 0.45 6.38 0.8872 0.01341 335.480

554.67 3,076,622 3,076,622 0.48 6.49 0.8423 0.01262 332.454

586.94 2,995,159 2,995,159 0.51 6.61 0.7906 0.01183 321.441

619.34 2,885,693 2,885,693 0.53 6.75 0.7171 0.01092 296.418

658.84 2,777,356 2,777,356 0.56 6.90 0.0102 0.00867 4.317

BET

3,795.14

0.0000E+00

1.0000E-04

2.0000E-04

3.0000E-04

4.0000E-04

5.0000E-04

6.0000E-04

7.0000E-04

8.0000E-04

9.0000E-04

1.0000E-03

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

Ct

- T

hru

st

Co

eff

x/R

Ct - Thrust Coeff

MOMENTUM THEORY SMALL ANGLE

0.00000

0.00200

0.00400

0.00600

0.00800

0.01000

0.01200

0.01400

0.01600

0.01800

0.02000

0.9 1

Cd

dD= dT= dQ= dP_i= dP_p= dP=

lbf lbf ft*lb ft*lb/s ft*lb/s ft*lb/s

1 blade n blades n blades induced profile n blades

0.034 5.479 3.090 81.428 4.056 85.483

0.156 32.818 31.110 822.781 37.786 860.567

0.402 90.333 109.987 2893.303 149.131 3042.434

0.779 175.860 249.985 6525.477 389.551 6915.028

1.254 282.534 448.638 11619.906 790.188 12410.094

1.785 403.104 697.028 17923.103 1357.891 19280.995

2.338 531.587 982.728 25100.542 2083.411 27183.953

2.884 663.090 1290.908 32761.279 2947.471 35708.750

3.402 793.121 1605.309 40483.704 3921.913 44405.617

3.872 917.278 1909.428 47847.743 4970.332 52818.076

4.277 1031.359 2187.635 54462.887 6050.869 60513.756

4.605 1131.554 2426.015 59987.815 7119.946 67107.761

4.851 1214.522 2612.985 64144.302 8135.378 72279.681

5.010 1277.417 2739.749 66727.434 9058.751 75786.184

5.082 1317.867 2800.637 67613.527 9856.928 77470.455

5.072 1333.824 2793.407 66768.282 10502.171 77270.453

4.982 1322.976 2719.733 64263.673 10968.829 75232.502

4.809 1280.079 2586.868 60339.421 11217.809 71557.231

4.512 1180.868 2414.788 55683.350 11113.846 66797.196

3.669 14.650 406.306 1827.138 9411.977 11239.115

BET

63.77 15,000.32 31,016.34 747,877.10 110,088.23 857,965.3

66,724.74 N

0.7 0.8 0.9 1

SMALL ANGLE EXACT

0.000

10000.000

20000.000

30000.000

40000.000

50000.000

60000.000

70000.000

80000.000

90000.000

0 0.1 0.2 0.3 0.4 0.5

Po

wer

- ft

*lb

/s

x/R

Power - ft*lb/s PROFILE POWER INDUCED POWER

dP= dP= dCt dCp dCt dCp λ, Inflow

HP kW check: check: (V+vi)/ΩR

0.155423864 0.11589956 3.8927E-06 9.3439E-08 3.6173E-06 8.0182E-08 0.022631

1.564667895 1.16677265 2.3319E-05 9.4066E-07 2.2064E-05 8.4060E-07 0.038308

5.531697319 4.12498598 6.4186E-05 3.3256E-06 6.1677E-05 3.0606E-06 0.049001

12.57277808 9.37551901 1.2496E-04 7.5586E-06 1.2135E-04 7.0949E-06 0.056807

22.56380648 16.8258276 2.0075E-04 1.3565E-05 1.9634E-04 1.2903E-05 0.062995

35.05635376 26.1415185 2.8642E-04 2.1075E-05 2.8149E-04 2.0233E-05 0.068132

49.42536942 36.8564917 3.7772E-04 2.9714E-05 3.7251E-04 2.8725E-05 0.072380

64.92500012 48.4145643 4.7116E-04 3.9032E-05 4.6591E-04 3.7938E-05 0.075760

80.73748627 60.2059332 5.6355E-04 4.8538E-05 5.5846E-04 4.7388E-05 0.078292

96.03286469 71.6116949 6.5177E-04 5.7734E-05 6.4700E-04 5.6574E-05 0.080031

110.0250104 82.0456362 7.3283E-04 6.6145E-05 7.2849E-04 6.5018E-05 0.081039

122.0141105 90.9859066 8.0402E-04 7.3353E-05 8.0020E-04 7.2294E-05 0.081375

131.417601 97.9980883 8.6297E-04 7.9006E-05 8.5972E-04 7.8041E-05 0.081086

137.7930623 102.752269 9.0766E-04 8.2839E-05 9.0498E-04 8.1986E-05 0.080214

140.8553731 105.035834 9.3641E-04 8.4680E-05 9.3428E-04 8.3948E-05 0.078798

140.4917328 104.764667 9.4774E-04 8.4462E-05 9.4614E-04 8.3852E-05 0.076894

136.7863674 102.001577 9.4004E-04 8.2234E-05 9.3890E-04 8.1740E-05 0.074627

130.1040556 97.0185776 9.0956E-04 7.8217E-05 9.0881E-04 7.7823E-05 0.072426

121.4494471 90.5648372 8.3906E-04 7.3014E-05 8.3858E-04 7.2685E-05 0.072456

20.43475471 15.238194 1.0410E-05 1.2285E-05 1.3605E-05 1.2104E-05 0.164962

MOMENTUMSMALL ANGLE APPROXEXACTBET

1,559.94 1,163.24 0.01066 0.00094 0.01060 0.00092

0.5 0.6 0.7 0.8 0.9 1

x/R

TOTAL POWER

Solver v_i_diff dT dCt

mph check: check:

0.000000 0.000000 7.2081 2.2013E-06

0.000000 0.000000 41.3067 1.8586E-05

0.000000 0.000000 101.3757 5.6756E-05

0.000000 0.000000 181.6647 1.1553E-04

0.000000 0.000000 279.2467 1.8993E-04

0.000000 0.000000 391.9727 2.7470E-04

0.000000 0.000000 516.1073 3.6553E-04

0.000000 0.000000 646.2080 4.5893E-04

0.000000 0.000000 776.4017 5.5164E-04

0.000000 0.000000 901.4045 6.4049E-04

0.000000 0.000000 1016.6928 7.2240E-04

0.000000 0.000000 1118.3340 7.9460E-04

0.000000 0.000000 1202.9454 8.5466E-04

0.000000 0.000000 1267.7528 9.0048E-04

0.000000 0.000000 1310.7793 9.3035E-04

0.000000 0.000000 1331.4136 9.4277E-04

0.000000 0.000000 1332.4326 9.3605E-04

0.000000 0.000000 1328.8260 9.0640E-04

0.000000 0.000000 1403.8016 8.3644E-04

0.000000 0.000000 7659.5607 5.4425E-08

MOMENTUM

0.000000 22,815.43 0.01050

I USED THIS PROGRAM TO DESIGN A ROTOR WHICH FOLLOWED THE CONSTRAINTS IN THIS PROBLEM.

min power

-60 -50 -40 -30

Cp

/σ

Twist from Root to Tip

Cp/σ Held Constant: Thrust Weighted Solidity = 0.1 Thrust = 15,000lb Tip Speed is 650 ft/s Rotor Dia = 47 ft Root Cut-Out = 0 ft Hover Tip and Root Losses Included Cl corrected for Compressibility

Cp

/σ

Cp/σ

You are asked to design a rotor that will carry 15000 lbf at 4000 feet pressure altitude at 95 deg F. Due to

geometric limitations, the rotor diameter is 43 feet. The designer decides to use a thrust weighted solidity of 0.1, and has chosen the number of blades to be 4. The designer also decides to use a 3 bladed rotor for this

design. Because the company has prior experience with NACA 0012 airfoils, a NACA0012 airfoil is also

chosen. The tip speed is selected to be 650 feet/sec.

Determine the best combinations of linear taper ratio and linear twist that will minimize power consumption.

Use combined Blade Element Momentum theory, and Prandtl’s tip loss model F(r). The NACA 0012 airfoil

properties are supplied below as a table of alpha and Mach number. Please use these.

1. AE 6070 students only:

The designer seeks your advice on changing solidity (between 0.05, 0.075, or 0.10), and the number of blades

(between 2, 3, 4, or 5). Repeat the BEMT simulations for subsets of these combinations and give your

recommendation. Note that a very small value of sigma may cause the rotor to stall.

min power

3-Blades

c_r/c_t

1

2

3

4

-60 -50 -40 -30


-60 -50 -40 -30

Cp

/σ


Cp/σ

-60 -50 -40 -30

Cp

/σ


Cp/σ

Data:

2

2

2

3

3

3

4

4

4

5

5

5

Data:

20.6

22

26

30

min power

0.0094

0.00945

0.0095

0.00955

0.0096

0.00965

Cp

/σ

Cp/σ

0.00102

0.00103

0.00104

0.00105

0.00106

0.00107

0.00108

0.00109

0.0011

0.00111

0.00112

0.04 0.05 0.06 0.07 0.08

Cp

σ, Solidity, Thrust Weighted

Cp

0.009350 2 4 6

Taper Ratio Root/Tip


Data: Sea Level Taper (r:t, 1:1)

θ_twist θ_.75 Ct Cp/σ

0 11.137 0.008608 0.00778

-5 11.124 0.008608 0.007551

-10 11.091 0.008608 0.007381

-15 11.038 0.008608 0.00727

min power -20 10.966 0.008608 0.007219

-25 10.875 0.008608 0.007227

-30 10.764 0.008608 0.007296

-35 10.629 0.008608 0.007421

-40 10.466 0.008608 0.007602

Data: 4k, 95 Taper (r:t, 1:1); c = 20.57"


0 13.134 0.010658 0.010445

-5 13.118 0.010658 0.01016

-10 13.085 0.010658 0.009937

-15 13.035 0.010658 0.009774

-20 12.969 0.010658 0.009671

min power -25 12.887 0.010658 0.009629

-30 12.789 0.010658 0.009646

-35 12.674 0.010658 0.009723

-40 12.541 0.010658 0.009858

-50 12.204 0.010657 0.010293

Data: 4k, 95 Taper (r:t, 1:1); c = 27.43"


0 13.242 0.010658 0.010603

-5 13.218 0.010658 0.0103

-10 13.177 0.010658 0.01006

-15 13.119 0.010658 0.009881

-20 13.045 0.010658 0.009764

4-Blades

3-Blades

0.007

0.0075

0.008

0.0085

0.009

0.0095

0.01

0.0105

0.011

-20 -10 0

0.0098

0.01

0.0102

0.0104

0.0106

You are asked to design a rotor that will carry 15000 lbf at 4000 feet pressure altitude at 95 deg F. Due to

geometric limitations, the rotor diameter is 43 feet. The designer decides to use a thrust weighted solidity of 0.1, and has chosen the number of blades to be 4. The designer also decides to use a 3 bladed rotor for this

design. Because the company has prior experience with NACA 0012 airfoils, a NACA0012 airfoil is also

chosen. The tip speed is selected to be 650 feet/sec.

Determine the best combinations of linear taper ratio and linear twist that will minimize power consumption.

Use combined Blade Element Momentum theory, and Prandtl’s tip loss model F(r). The NACA 0012 airfoil

properties are supplied below as a table of alpha and Mach number. Please use these.

1. AE 6070 students only:

The designer seeks your advice on changing solidity (between 0.05, 0.075, or 0.10), and the number of blades

(between 2, 3, 4, or 5). Repeat the BEMT simulations for subsets of these combinations and give your

recommendation. Note that a very small value of sigma may cause the rotor to stall.

min power -25 12.955 0.010658 0.009709

-30 12.849 0.010658 0.009715

-35 12.726 0.010658 0.009781

-40 12.586 0.010658 0.009908

-45 12.425 0.010658 0.010093

3-Blades 4-Blades

c_r/c_t Cp/σ min c_r/c_t Cp/σ min

1 0.009709 1 0.009629

2 0.009604 2 0.009519

3 0.009545 3 0.009468

4 0.009511 4 0.00945

Comparison

0.0094

0.0096

0.0098

-20 -10 0

0.0094

0.0096

0.0098

0.01

0.0102

0.0104

0.0106

-20 -10 0

0.0094

0.0096

0.0098

0.01

0.0102

0.0104

0.0106

-20 -10 0

Data: 4k, 95 Taper (r:t, 1:1); θ_twist = 0 deg.

Blades, N chord, c σ, Solidity, Thrust Weightedθ_.75 Ct

2 20.56 0.05 20.105 0.010658

2 30.87 0.075 15.686 0.010658

2 41.17 0.1 13.464 0.010658

3 13.71938 0.05 19.868 0.010659

3 20.57907 0.075 15.46 0.010658

3 27.43 0.1 13.242 0.010658

4 10.28 0.05 19.772 0.010658

4 15.43 0.075 15.353 0.010658

4 20.57 0.1 13.134 0.010658

5 8.23163 0.05 19.699 0.010658

5 12.34744 0.075 15.29 0.010658

5 16.46342 0.1 13.069 0.010658

Data: 4k, 95 θ_twist = -25 deg.

Root Chord, c_rootTip Chord, c_tipTaper r/t

20.6 20.6 1

22 20.2 1.089109

26 18.9 1.375661

30 17.61111 1.70347

36 15.68189 2.295641

42 13.76462 3.0513

min power 48 11.84382 4.052748

54 9.92046 5.443296

60 7.95 7.54717

Comparison

Min power consumption is found at a linear twist of ratio. I only proceed with this iterative process 3 times, I could go back and continue to optimize the twist with this new taper ratio, but this would take a long time. I noticed that as my linear twistrate was decreased the taper ratio wanted to go up. I noticed that the taper was getting a bitun-reasonable and stopped at 6.4:1. I plot this blade in CATIA to see how it looks in terms of feasibility.

0.08 0.09 0.1 0.11

Solidity, Thrust Weighted

8 10 12

feasibility.


Ct/σ Thrust

0.085997 15000.72

0.085993 15000.04

0.085996 15000.59

0.085994 15000.21 25.0%

0.085992 14999.86

0.085992 14999.97

0.085997 15000.77

0.085991 14999.75

0.08599 14999.57

Taper (r:t, 1:1); c = 20.57" Data: 4k, 95 Taper (r:t, 2:1); c_r = 41.5", c_t = 13.833"

Ct/σ Thrust θ_twist θ_.75 Ct Cp/σ Ct/σ Thrust

0.106631 15000.23 0 13.108 0.010659 0.010137 0.106548 15000.69

0.106627 14999.73 -5 12.996 0.010658 0.009885 0.106546 15000.33

0.10663 15000.17 -10 12.863 0.010659 0.009698 0.106547 15000.58

0.10663 15000.07 -15 12.71 0.010658 0.009575 0.106541 14999.74

0.106629 15000.06 -20 12.539 0.010658 0.009519 0.106543 14999.98

0.106628 14999.85 -25 12.349 0.010658 0.009527 0.10654 14999.59

0.106627 14999.76 -30 12.14 0.010658 0.009601 0.106538 14999.29

0.106625 14999.37 -35 11.911 0.010658 0.00974 0.10654 14999.48

0.106625 14999.5 -40 11.659 0.010658 0.009943 0.106541 14999.69

0.10662 14998.77 -45 11.378 0.010658 0.010205 0.10654 14999.58

-50 11.053 0.010658 0.010514 0.106538 14999.32

Taper (r:t, 1:1); c = 27.43" Data: 4k, 95 Taper (r:t, 2:1); c_r = 44.1", c_t = 22.05"

Ct/σ Thrust θ_twist θ_.75 Ct Cp/σ Ct/σ Thrust

0.106615 14999.88 0 13.205 0.010658 0.010283 0.106624 15000.04

0.106613 14999.62 -5 13.085 0.010658 0.010013 0.10662 14999.47

0.106617 15000.17 -10 12.946 0.010659 0.009812 0.106628 15000.62

0.106617 15000.12 -15 12.787 0.010658 0.009675 0.106627 15000.4

0.106617 15000.15 -20 12.609 0.010658 0.009605 0.10662 14999.48

4-Blades

3-Blades

0.106616 14999.95 -25 12.414 0.010659 0.009604 0.106629 15000.71

0.106614 14999.77 -30 12.199 0.010658 0.009668 0.106625 15000.16

0.106609 14999.06 -35 11.964 0.010658 0.009798 0.10662 14999.52

0.106615 14999.79 -40 11.708 0.010659 0.009996 0.106629 15000.77

0.106616 14999.94

0.82%

0.89%

0.81%

0.64%

0.0094

0.0095

0.0096

0.0097

0.0098

0.0099

0.01

0 0.5 1 1.5 2 2.5

Cp

/σ w

/Op

tim

ized

Tw

ist

for

Min

Po

wer

Taper Ratio (c_r/c_t)

Cp/σ w/Optimized Twister for Min Power

Cp/σ Ct/σ Thrust Cp

0.022223 0.213355 14999.54 0.001111

0.014498 0.142101 14999.74 0.001087

0.010921 0.106551 14999.95 0.001092

0.021489 0.213171 15000.52 0.001074

0.014049 0.14211 15000.09 0.001054

0.010603 0.106615 14999.88 0.00106

0.021161 0.213367 15000.35 0.001058

0.013831 0.142143 14999.41 0.001037

0.010445 0.106631 15000.23 0.001045

0.020941 0.213164 15000 0.001047

0.013701 0.142112 15000.24 0.001028

0.010349 0.106578 14999.53 0.001035

θ_twist = -25 deg. Data: 4k, 95 θ_twist = -20 deg.

θ_.75 Ct Cp/σ Ct/σ Thrust Root Chord, c_rootTip Chord, c_tipTaper r/t

12.877 0.010658 0.009616 0.106474 15000.07 36 15.68189 2.295641

12.8 0.010658 0.009583 0.10627 14999.61 42 13.76462 3.0513

12.623 0.010658 0.009549 0.106303 14999.05 48 11.84382 4.052748

12.458 0.010659 0.009523 0.106309 15000.49 54 9.92046 5.443296

12.23 0.010659 0.009496 0.106286 15000.86 57 8.959844 6.361718

12.022 0.010658 0.009479 0.106211 15000.21 60 7.95 7.54717

11.835 0.010658 0.009473 0.10615 14999.72 63 6.972974 9.034882

11.667 0.010658 0.009479 0.106105 14999.94 66 6.010943 10.97997

11.524 0.010659 0.009508 0.106247 15000.78

Min power consumption is found at a linear twist of -20 deg. and a taper ratio of 6.36 root to chord ratio. I only proceed with this iterative process 3 times, I could go back and continue to optimize the twist with this new taper ratio, but this would take a long time. I noticed that as my linear twist rate was decreased the taper ratio wanted to go up. I noticed that the taper was getting a bit

reasonable and stopped at 6.4:1. I plot this blade in CATIA to see how it looks in terms of

0.2%

1.7%

Data: 4k, 95 Taper (r:t, 3:1); c_r = 41.5", c_t = 13.833" Data:

θ_twist θ_.75 Ct Cp/σ Ct/σ Thrust θ_twist

0 13.137 0.010659 0.009973 0.106574 15000.49 0

-5 12.962 0.010658 0.009748 0.10657 14999.88 -5

-10 12.766 0.010658 0.009588 0.106573 15000.32 -10

-15 12.549 0.010658 0.009495 0.106571 15000.09 -15

-20 12.312 0.010658 0.009468 0.106569 14999.83 -20

-25 12.055 0.010658 0.009508 0.106567 14999.5 -25

-30 11.778 0.010658 0.009617 0.106571 15000.08 -30

-35 11.478 0.010658 0.009791 0.106567 14999.48 -35

-40 11.153 0.010658 0.010031 0.106571 15000.09 -40

-45 10.794 0.010659 0.010331 0.106577 15000.93

Data: 4k, 95 Taper (r:t, 3:1); c_r = 55.3", c_t = 18.433" Data:

θ_twist θ_.75 Ct Cp/σ Ct/σ Thrust θ_twist

0 13.221 0.010658 0.010104 0.106633 14999.74 0

-5 13.041 0.010658 0.009864 0.106637 15000.37 -5

-10 12.839 0.010658 0.00969 0.106635 14999.97 -10

-15 12.617 0.010658 0.009583 0.106635 14999.99 -15

-20 12.375 0.010658 0.009545 0.106632 14999.6 -20

-25 12.114 0.010658 0.009577 0.106637 15000.28 -25

-30 11.832 0.010658 0.009676 0.106633 14999.82 -30

-35 11.529 0.010659 0.009845 0.106638 15000.44 -35

-40 11.226 0.010663 0.010086 0.106683 15006.72 -40

-45 10.84 0.010658 0.010377 0.106634 14999.83 -45

3 3.5 4 4.5

w/Optimized Twister for Min Power

θ_twist = -20 deg.

θ_.75 Ct Cp/σ Ct/σ Thrust

12.442 0.010658 0.009475 0.106279 14999.9

12.282 0.010659 0.009437 0.106215 15000.78

12.1405 0.010658 0.009411 0.106154 15000.21

12.017 0.010659 0.009397 0.106108 15000.42

11.961 0.010659 0.009394 0.106081 15000.51

11.918 0.010658 0.009409 0.106243 15000.32

11.872 0.010658 0.009416 0.106273 14999.4

11.828 0.010658 0.009421 0.106257 15000.35

4k, 95 Taper (r:t, 4:1); c_r = 47.52", c_t = 11.88"


13.177 0.010659 0.009873 0.106618 15000.44

12.958 0.010658 0.009667 0.106609 14999.19

12.718 0.010658 0.009529 0.106615 15000.01

12.456 0.010658 0.009455 0.106612 14999.62

12.174 0.010658 0.00945 0.106617 15000.26

11.871 0.010659 0.009512 0.106618 15000.42

11.546 0.010658 0.009641 0.106613 14999.73

11.198 0.010658 0.009838 0.106612 14999.56

10.823 0.010658 0.010101 0.106617 15000.24

4k, 95 Taper (r:t, 4:1); c_r = 63.4", c_t = 15.85"


13.246 0.010659 0.009983 0.106551 15000.46

13.022 0.010658 0.009763 0.106542 14999.11

12.778 0.010659 0.009613 0.106553 15000.7

12.512 0.010659 0.009528 0.106552 15000.59

12.225 0.010658 0.009511 0.106545 14999.54

11.918 0.010658 0.009564 0.106542 14999.1

11.591 0.010658 0.009687 0.10655 15000.31

11.24 0.010658 0.009879 0.106545 14999.63

10.863 0.010658 0.010138 0.106546 14999.69

10.45 0.010658 0.010457 0.106545 14999.54

Propeller or Rotor in Axial Flight

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