Propeller design calculation

Propeller design calculation Here we describe the steps in a simple propeller design calculation, using lifting line theory. A table with an example is given afterwards. This corresponds to chapter 5.1 in Minsaas “Propeller Theory”, but be aware that the example in chapter 5.1 have several numerical errors, and the different lines in the example table don’t necessarily come in order of calculation. The propeller design calculation consists of three steps: 1. Calculation of thrust and torque 2. Check of risk of cavitation 3. Determination of camber and pitch distribution Calculation of thrust and torque 1. Select propeller diameter and RPM. A Bp-, diagram or experience might be used. 2. Select blade section thickness and camber distribution, using for instance tabulated data in books such Abbott & von Doenhoff “Theory of wing sections” 3. Design radial chord length distribution c(r) 4. Design radial thickness distribution t(r) 5. Find radial wake distribution w(r) from model tests or from empirical data 6. Design circulation distribution. The following generic type of distribution is frequently used: m, In this case, the design is about selecting values of k, a, and m. ,,,kxxaxxsinsin2,,,, ,7. Calculate mean induced tangential velocity U at all radii: U,TTmean2,r VwU(1)2,，UTA8. Solve to find mean axial induced velocity U at all radii = A,2rn -U2UAT UTmean9. Find mean hydrodynamic angle of attack , at all radii: tan,, iiUAmean 10. Find correction factors for finite number of blade to determine values of U and U at the TA blades. Two alternative methods: a. Goldstein factors (used in the example below) b. Induction factors UUUTTmeanAmeantan,,11. Calculate , , and : = = UUTAi,,,,(zx,),(zx,),UAii UU22AT12. Calculate the resulting total velocity at each radius: ,) = ( + + (2rn - )VV,22 13. Calculate the lift of each section: dLV,,,, dL14. Calculate the corresponding lift coefficient: ,CL21,Vc,2 t,,15. Calculate the drag coefficient of each section: 212CC,，DF,,c,, 2116. Calculate the drag of each section: dDVCcdr,,,D2 U,,T 17. Calculate total thrust of each section: d2dsin,,,,,,,TrnrdDi,,2,, U,,A18. Calculate torque of each section: ddcos,,QVrrdD,,，，Ai,,2,, 19. Integrate (sum up) to find total thrust and torque 20. Are the thrust according to required thrust? a. No: Go to step 6 and adjust the circulation distribution. Change of RPM or diameter, is also possible. b. Yes: Proceed to cavitation check Example calculation A spreadsheet containing the formulas and numbers behind this calculation is available on the subject web pages Main input data: Vs 18 knots T 1080 kN RPM 150 n= 2.5 Hz Circulation distribution D 6 m parameters: x0.2 boss k= 20 a= 0.1 r0.6 m boss m= 0.4 z 4 blades 3 Water Density 1025 kg/m 2Kin.viscosity 1.19E-06 m/s Calculation of thrust and torque: x=r/R 0.204 0.284 0.364 0.444 0.524 0.604 0.684 0.764 0.844 0.924 r 0.612 0.852 1.092 1.332 1.572 1.812 2.052 2.292 2.532 2.772 xx 0.005 0.105 0.205 0.305 0.405 0.505 0.605 0.705 0.805 0.905 w 0.149 0.137 0.129 0.123 0.118 0.113 0.110 0.107 0.104 0.102 t 0.219 0.198 0.178 0.158 0.138 0.118 0.097 0.077 0.057 0.037 c 1.287 1.488 1.636 1.738 1.795 1.804 1.757 1.642 1.435 1.07 3.47 11.72 15.21 17.58 19.17 20.02 20.06 19.14 17.03 13.15 , V7.882 7.992 8.068 8.126 8.172 8.210 8.242 8.270 8.294 8.316 A U0.903 2.188 2.217 2.100 1.941 1.759 1.556 1.329 1.070 0.755 Tmean U0.988 2.855 3.604 4.101 4.432 4.613 4.635 4.471 4.067 3.274 Amean 42.435 37.469 31.597 27.116 23.648 20.869 18.556 16.555 14.746 12.983 ,i 1 1 1 0.997 0.993 0.984 0.964 0.924 0.841 0.66 ,(x,z,,i) U0.903 2.188 2.217 2.106 1.955 1.787 1.614 1.438 1.273 1.144 T U0.988 2.855 3.604 4.114 4.464 4.688 4.808 4.838 4.836 4.960 A 42.44 37.47 31.60 27.13 23.69 20.95 18.71 16.85 15.31 14.10 ,i V? [m/s] 12.414 15.483 18.837 22.327 25.897 29.520 33.180 36.867 40.576 44.306 Lift [N/m] 44191 185934 293683 402246 508889 605893 682192 723201 708294 596990 C0.109 0.254 0.247 0.226 0.206 0.188 0.172 0.158 0.146 0.139 L Rn 1.35E+07 1.94E+07 2.60E+07 3.27E+07 3.92E+07 4.49E+07 4.91E+07 5.10E+07 4.90E+07 3.99E+07 C2.85E-03 2.68E-03 2.56E-03 2.47E-03 2.40E-03 2.35E-03 2.32E-03 2.30E-03 2.32E-03 2.39E-03 F C7.64E-03 6.79E-03 6.23E-03 5.83E-03 5.53E-03 5.31E-03 5.14E-03 5.04E-03 5.00E-03 5.11E-03 D dr 0.252 0.24 0.24 0.24 0.24 0.24 0.24 0.24 0.24 0.228 dD [N] 783 1192 1779 2485 3277 4107 4894 5530 5812 5018 dT [N] 528 725 932 1133 1316 1468 1570 1603 1534 1223 D dQ[Nm] 354 806 1655 2946 4718 6950 9512 12131 14194 13491 D dT [N] 7691 34692 59102 84781 110528 134335 153500 164510 162426 130789 dQ [Nm] 4952 23935 41983 61591 81846 101151 117304 127468 127823 105425 Resulting performance: Thrust 1042.4 kN Torque 793.5 kNm Power 12464 kW Vs 18 knots RPM 150 K0.126 T K0.0159 Q J0.544 A 0.682 , Cavitation check To check for cavitation, we need to estimate the local velocity at each blade section (radius). To do so, we need to select thickness and velocity distributions. Mostly, standard profiles for which properties are known are selected. For the example, we have chosen NACA a08 camber distribution and NACA 16 thickness distribution. Other data we need are: Draugt to propeller shaft: 7 m Water vapour pressure: p=1500 Pa v Atmospheric pressure p=101325 Pa a Calculation procedure: ppghr,，,,()av1. Calculate cavitation number: ,,21V,,2 ,,vvtc,,,,2. Calculate velocity due to thickness: , where value for 111，,,，,,,,,,VV0.12,,,,0.12,, v,, is taken from the table in Appendix I in Abbott & Doenhoff. ,,V,,0.12 ,,vv,v,,,,3. Calculate velocity due to camber: where the value for is taken ,,CL,,,,VVV,,,,,1C,1CLL from the table in Appendix II in Abbott & Doenhoff. ,,,vv,,4. Calculate total max velocity at the suction side of each section: 1VV,，，x,,,,,VV,,,, 2,,Vx5. Check for cavitation. If then we will get cavitation. Often, a margin is 1,,,,,V,,, 2,,Vxintroduced, for instance: 10.8,,,,,,V,,, Example of cavitation check: 2.077 1.316 0.876 0.614 0.449 0.340 0.265 0.211 0.172 0.141 , v/V 1.194 1.152 1.124 1.104 1.088 1.075 1.063 1.054 1.045 1.039 0.030 0.071 0.069 0.063 0.057 0.052 0.048 0.044 0.041 0.039 ,v/V V15.200 18.930 22.469 26.050 29.655 33.267 36.858 40.461 44.065 47.763 x 2(V/V)-1 0.499 0.495 0.423 0.361 0.311 0.270 0.234 0.204 0.179 0.162 x? Cavitation? No No No No No No No No Yes Yes Determination of camber and pitch distribution The point here is correction of the geometry for the fact that the propeller blade sections aren’t foil sections alone in linear motion, but operate in vicinity of other blades in a helical motion. This can be taken into account directly by lifting surface calculations, but can also be taken into account in an approximate manner, as described in Minsaas “Propeller Theory” chapter 6.1. For the purpose of this example, we use the following correction formulas: 23Camber correction factor: k,1.6946，0.5048x,4.0012x，4.3283xc 3Correction factor for angle of attack: k,1，1.46xa zckCorrection factor for effect of thickness: ,2.5cos,tixD The basis for the calculation is that the propeller blade sections are designed to lift purely by camber. It is fairly straight forward to modify the procedure to allow for a combination of angle of attack and camber. Calculation procedure: fff1. Calculate max camber of each radius: where is the ,,,kCcL，，，，cccNACAa08NACAa08 max camber value found in the tabulated data for NACA a08 camber profile. 2. Calculate correction of ideal angle of attack due to 3-D effects: ,,1.54Cki3L, ktt3. Calculate correction of ideal angle of attack due to thickness effects: ,,tc Pr4. Calculate resulting geometric pitch distribution: ,,，，,,,,tan，，iit3DR0 Example calculation of camber and pitch distribution: kc 1.668 1.614 1.557 1.509 1.483 1.494 1.553 1.675 1.873 2.159 ka 1.0124 1.0334 1.0704 1.1278 1.2101 1.3217 1.4672 1.6511 1.8778 2.1518 kt 7.760 6.931 6.380 5.806 5.228 4.649 4.055 3.428 2.733 1.872 f/c 0.0118 0.0267 0.0250 0.0222 0.0199 0.0183 0.0174 0.0172 0.0178 0.0195 0.1695 0.4046 0.4068 0.3934 0.3843 0.3827 0.3887 0.4019 0.4229 0.4594 ,i3 1.3205 0.9222 0.6942 0.5278 0.4020 0.3041 0.2239 0.1608 0.1086 0.0647 ,t P/D 0.6173 0.7173 0.7341 0.7434 0.7492 0.7526 0.7536 0.7529 0.7522 0.7576