Tuesday, December 28, 2010

Model Propeller Constants - Weick, Durand and Lesley

"Aircraft Propeller Design", written by Fred E. Weick and published by McGraw Hill Book Company in 1930 makes reference to wind tunnel test results for a series of propellers with pitch to diameter ratios between 0.5 and 1.1, see pages 105 to 108 . The tests of these 13 propellers performed by W. F. Durrand and Everett Lesley are reported in "TESTS OF THIRTEEN NAVY TYPE MODEL PROPELLERS", NACA TR 237, 1926 .   You can find out more about the tests at Durand Lesley Propeller Collection and download the test report at NACA TR 237 .

In 1994 I, Chris Stoddart, developed and posted to the internet a mathematical model based on these tests and Fred Weick's Representative Section / Blade Element theory, weick.tk.  The model made use of MiniTK a free version of the TK Solver, a mathematical modeling and problem solving software system based on a declarative, rule-based language, commercialized by Universal Technical Systems, Inc. , see TK_Solver at Wikipedia  and UTS .  The free version can still be used as a DOS application in Windows and with Linux.  You can download MiniTK from  MINITK and the Weick.tk mathematical model from Weick.TK .
Weick.TK allows the examination of many interactions on propeller performance on parameters such as efficiency, power and thrust as effected by pitch, diameter, velocity, and rotational speed and other factors.  The program can select an optimal propeller to match a flight condition. It gives reasonable answers to queries ranging from rubber power to control line speed.

The Rule, Variable, and Unit sheets of Weick TK appear below.

Rule Sheet
Status Rule
Comment ;Program - Weick.tk, Written by Chris Stoddart, July 23, 1994


Satisfied eta=TAN(PHI)/TAN(PHI+GAMMA)
Satisfied C_p=QGF*B*WR*J^2*((TAN(PHI)+TAN(GAMMA))/(TAN(PHI)*SIN(PHI)))*C_L
Satisfied C_s=J/C_p^0.2
Satisfied C_s = V*(rho/(P*(N/(2*pi()))^2))^.2
Satisfied eta=C_t*J/C_p
Satisfied BETA=ATAN((4*PoverD)/(3*PI()))
Satisfied TAN(PHI)=V/((3*D/8)*N) ;-- use without inflow
Satisfied ALPHA=BETA-PHI
Satisfied C_L=0.355+4.3*ALPHA-3*(ALPHA+0.075)^2 ;-- use without inflow
Satisfied GAMMA=0.47*ALPHA+0.0073/(ALPHA+0.075) ;-- use without inflow
Satisfied GAMMA=ATAN(1/LovrD)
Satisfied T=(rho*C_t*N^2*D^4)/(2*pi())^2
Satisfied P=(rho*C_p*N^3*D^5)/(2*pi())^3
Satisfied Q=P/N
Satisfied M_0=T/P
Satisfied J=V/(D*N/(2*pi()))
Satisfied PoverD = Pitch/D
Satisfied WR = Chord/D
* Undetermined condition IF OPT = 'yes THEN ALPHA = .8/57.3

Variable Sheet
Status Input Name Output Unit Comment





File Name: weick.tk July 23, 1994

.579 D
m Diameter

.722 Pitch
m Geometric pitch


Chord .06948 m Chord at 3/4 distance to tip

2 B

Number of Blades

.12 WR

Width ratio at 3/4 distance to tip


PoverD 1.24697754749568
Aerodynamic Pitch over Diameter

8 N
rev/s Angular velocity

5 V
m/s Forward velocity


T 1.06051082397811 N Thrust


P
w Absorbed Power


Q .129313391262165 N*m Absorbed Torque


M_0 .163155511381247
Figure of Merit [T/P in N/w]








J 1.07944732297064
Advance ratio V/(N*D)


eta 81.5777556906235 % Efficiency


C_t .120262549489696
Thrust Coefficient


C_p .159132947457634
Torque Coefficient


C_s 1.55901190738972
Speed/Power Coefficient


BETA .486760145073151 rad Geometric Angle of Attack


PHI .429595614242357 rad Relative Wind Angle


ALPHA .0571645308307935 rad Aerodynamic Angle of Attack


C_L .548405092943241
"Lift Coefficient" Correlation


LovrD 12.1526639689179
"Lift/Drag ratio" Correlation


GAMMA .0821015133831694 rad Angle who's tangent is Drag over Lift

1.226 rho
kg/m^3 Air density

.366372781256872 QGF

Overall torque grading factor (0.366)


OPT

To find optimal Pitch given D,HP,N,V





enter 'yes, otherwise leave blank

Units Sheet
From To Multiply By Add Offset Comment
m in 39.37

ft in 12

m mm 1000

kg g 1000

rad/s RPM 9.549296585513721

m/s ft/s 3.280833333333333

w hp .001342281879194631

rad deg 57.29577951308232

N*m lbf*in 8.850748065226473

lbf*in ozf*in 16

lbf ozf 16

w J/s


N lbf .2248090247334889

N/w lbf/w .224809024733

lbf/w ozf/w 16

rev/s rad/s 6.283185307179586

ft/s MPH .6818181818181818

w HP .001342281879194631

lbf*in gf*cm 1153.16

- % 100

lbf gmf 454

ozf*in gmf*cm 72.07250000000001