1. RENEWED LIFTING LINE
THEORY
DESIGN OF SHIP PROPELLERS
AUTHOR: DOCTOR G. PEREZ GOMEZ
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2. PURE LIFTING LINE THEORY
-PROPELLER BLADES REPLACED BY
LIFTING LINES AND ASSOCIATES
FREE VORTICES.
-THE PROPELLER BLADES ANNULAR SECTIONS
ARE INDEPENDENTS
-THE Z LIFTING LINES SHALL BE REPLACED
BY n.
Lim. nC(n)=ZC(z)
C(n)– 0
n --- infinite
C Circulation at radial station r
2

3. CALCULATION OF VORTICES INDUCED
VELOCITIES
-RADIAL VORTICES
DO NOT INDUCE
VELOCITIES AT THE
PROPELLER DISCK.
-HELICOIDAL VOR-
TICES ARE PLACED
ON REVOLUTION
SURFACES PASING
BY THE ENDS OF
ELEMENTAL RADIAL
VORTICES.
- THE REVOLUTION SURFACE CONTRACTION
HAS BEEN IGNORED IN THE FIGURE.
INDUCED VELOCITIES PRODUCED BY HELICOIDAL VORTI-
CES PLACED ON A CYLINDER OF RADIO r
W=WxIo+WrRo+WfiFio cylindrical coordinates
W induced velocity vector
Io, Ro, Fio, unitary vectors in the directions x, r,
and perpendicular to r
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4. INDUCED VELOCITIES PRODUCED BY
HELICOIDAL VORTICES
-THE FLUID IS INCOMPRESIBLE
div W =dWx/dx +dWr/dr +1/rdWfi/dfi =0
-OUT OF HELICOIDAL VORTICES THE FLUID IS IROTATIONAL
(rotW=0)
dWfi/dr-1/r dWr/dfi =1/r dWx/dfi - dWfi /dx =
dWr/dx -dWx/dr =0
-THE VECTOR FLUID VELOCITY, V MUST BE TANGENT TO
HELICOIDAL VORTICES, DUE THAT ROTATIONAL AND
IROTATIONAL REGIONS ARE INMISCIBLES.
-DUE TO THE EXISTING AXIAL SIMILITUDE
dWx/dfi = dWr/dfi =dWfi /dfi =0
-THE INDUCED VELOCITIES ARE CALCULATED USING BIOT –
SAVART FORMULA
4

5. INDUCED VELOCITIES PRODUCED BY
HELICOIDAL VORTICES
-THE INDUCED VELOCITY IN A POINT Q IT IS CALCULATED USING BIOT-
SAVART FORMULA.
G ( r) is C(r) , dl is a differential element of length on the vortex, s is
position vector of dl respect to Q.
-FROM A VALUE OF x, W DOES NOT CHANGE WITH x.
d Wx/d x =d Wr/d x =d Wfi/d x=0
-FROM ALL THE ABOVE IS CONCLUDED
Wx= cte
Wr=cte
Wfi=cte
Wr=0
-IF Q IS VERY FAR FROM THE VORTEX, r IS INFINITE AND W IS ZERO AND
SO Wx, AND Wfi.
THE HELICOIDAL VORTICES ONLY INDUCE VELOCITIES TO POINTS
PLACED IN THE INNER REGION OF THE SURFACE WHERE THEY ARE
PLACED
-IF THE REVOLUTION SURFACE WERE A PURE CYLINDER, THE INDUCED
VELOCITIES AT THE PROPELLER DISCK WOULD BE JUST THE HALF THAT
AT THE INFINITE DOWN STREAM.
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6. CALCULATION OF COMPONENTS
TANGENTIAL AND AXIAL OF INDUCED
VELOCITIES
-THE ELEMENTAL LENGTH OF VORTEX MN, CAN BE REPLACED
BY MP+PN WHEN BIOT-SAVART FORMULA IS APPLIED.
DOING THE SAME WITH ALL THE POSSIBLES SEGMENTS MN
IT IS POSSIBLE TO REPLACE THE HELICOIDAL FREE VORTICES
BY A SET DE STRAIGHT VORTICES AND A SET OF CIRCULAR
VORTICES BOTH PERPENDICULARS
-βio IS THE HYDRODYNAMIC PITCH ANGLE.
tanβio = ( MP/PN
MP=PNtan βi0== (2πr)/n tanβio
MP IS THE DISTANCE BETWEEN TWO CONSECUTIVES
CIRCULAR VORTICES, AND THE NUMBER OF THIS PER UNIT
OF LINEAL LENGTH IS 1/MP
-IN THE FIGURE ARE REPRESENTED THE COMPONENTS OF
INDUCED VELOVITIES, Wx, AND Wt
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7. CALCULATION OF THE COMPONENTS
OF INDUCED VELOCITIES AT THE
INFINITE DOWN STREAM.
-IN THE FIGURE ARE REPRESENTED THE
STRAIGTH VORTICES PLACED ON THE
CYLINDER OF RADIOUS r.
-THERE IS THE INTEGRATION BOUNDARY TO
APPLY STOKES THEOREM AT THE INFINITE
DOWN STREAM
2π(r-dr/2)Wfi=nГn=ZГz
Wfi=ZГz/(2πr).
I IS THE INTENSITY OF CIRCULAR VORTICES
PER UNIT OF LENGTH.
APPYLING STOKES : Wxdx=Idx
I=ZГz/(2πr tanβio);Wx=I
REPLACING THE VALUE OF tan bio IT IS OBTAINED THE
FOLLOWING EQUATION:
AFTER KNOW Bio ,Wx SALL BE ALSO KNOWN
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8. CALCULATION OF INDUCED VELOCITY
ON A CONTROL POINT Q, BY ALL FREE
VORTICES
-THE FREE VORTICES INNERS TO Q DO NOT INDUCE VELOCITIES.
-THE INDUCED VELOCITIES CORRESPONDING TO FREE VORTICES
PLACED ON REVOLUTIONS SURFACES BELONGUING TO THE
SAME BLADE ANNULAR ELEMENT ARE OPPOSITE.
-ON THE CONTROL POINTS Q ONLY INDUCED VELOCITIES ARE
PRODUCED BY THE REVOLUTION SURFACES ORRESPONDING
TO THE ANNULAR ELEMENT WHERE POINT Q IS PLACED..
-KN0WING THE INDUCED VELOCITY AT THE INFINITE DOWN
STREAM, IT IS NECESSARY TO CALCULATE THE VALUES OF
INDUCED VELOCITIES AT THE PROPELLER DISCK.
THE CONTRACTION OF REVOLUTION SURFACES MUST BE
CALCULATED.
THIS IS A ESENTIAL CHARACTERISTIC OF RENEWED LIFTING
LINE THEORY.
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9. CALCULATION OF INDUCED VELOCITIES AT
THE PROPELLER DISCK
CALCULATION OF FLUID VEIN CONTRACTION
-AS FIRST APROXIMATION THE INDUCED VELOCITIES AT THE
PROPELLER DISCK SHALL BE HALF OF THE VALUES AT THE
INFINITE DOWN STREAM.
-NEXT THE CONTINUITY EQUATION SHALL BE APPLIED TO
OBTAIN THE RADII (X0c) OF THE REVOLUTION SURFACES AT THE
INFINITE DOWN STREAM
THE FIRST MEMBER OF THE EQUATION CORRESPOND TO THE
PROPELLER DISCK
-Wa ARE THE AXIAL COMPONENTS OF INDUCED VELOCITIES
-THE ABOVE EQUATIONS MUST BE CALCULATED DEPARTING
FROM THE CONSECUTIVE RADIO TO THE PROPELLER HUB.
-AT THE INFINITE DOWN STREAM THE FIRST RADIOUS IS NULL.
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10. CALCULATIONS OF INDUCED VELOCITIES AT
THE PROPELLER DISCK
-AFTER HAVING SOLVE THE ABOVE EQUATIONS, THE VALUES OF
RADII ( X0c), SHALL BE KNOWN ,AND THEN THE NEW INDUCED
VEOCITIES AT THE INFINITE DOWN STREAM SHALL BE
CALCULATED.
-THE AXIAL COMPONENTS OF INDUCED VELOCITIES AT THE PRO-
PELLER DISCK WILL BE CALCULATED APPLYING AGAIN CONTINUITY
EQUATION.
-THE TANGENTIAL COMPONENTS OF INDUCED VELOCITIES AT THE
PROPELLER DISCK SHALL BE CALCULATED APPLYING CONSERVATION
OF KINETIC MOMENT BETWEEN THE PROPELLER DISCK AND THE
INFINITE DOWN STREAM
=
-FINALLY THE VELOCITIES POLIGONOM AT THE PROPELLER
DISCK (OF PAG. 6), SHALL BE KNOWN .
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11. GENERALIZATIONS FOR THE CASE OF TIP
LOADED PROPELLERS
MODIFICATIONS ON THE RADIAL
LOADINGDISTRIBUTION
-THIS TYPE OF PROPELLER IS CHARACTERISED TO HAVE A NON
NULL LOAD AT THE BLADES TIP
-TO DO THIS POSSIBLE THEY HAVE TIP PLATES (BARRIER
ELEMENTS AT THE BLADES TIPS)
-DUE TO THE ESPECIAL LOADING DISTRIBUTION, THE INDUCED
VELOCITIES ARE SMALLER THAN IN THE CASE OF A ALTERNATIVE
CONVENTIONAL PROPELLER.
-IN THIS FIGURE ARE SHOWN THE EFECTS OF TIP PLATES
ON A TWODIMENSIONAL PROFILE. THE CIRCULATION
ALONG THE SPAN IS COMBINATION OF A LINEAL
DISTRIBUTION PLUS A PARABOLIC ONE.
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12. LOADING RADIAL DISTRIBUTION OF A
TIP LOADED PROPELLER
IN THE FIGURE IS REPRESENTED A TYPICAL RADIAL LOADING DISTRIBUTION
OF A TIP LOADED PROPELLER.
- THE CIRCULATIONS
C Q VALUES AT THE HUB
(C(M)), AND AT THE TIP
(C(N)) ARE EQUALS.
T N - IN A CONTROL POINT X
M THE VALUE OF THE
A CIRCULATION AQ IS AT+TQ
Xh X 1 X
AT=C(T)=C(M)=C(N)
.
- THE INDUCED VELOCITIES DUE TO MN ARE NULL
- OF COURSE THE DIMENSIONS OF TIP PLATES MUS BE ADEQUATES TO
SUPORT THE CIRCULATION C(N)
-THE EXISTENCE OF THE TIP PLATES MAKE POSSIBLE TO REDUCE THE
MAGNITUDES OF INDUCED VELOCITIES AND SO TO INCREASE THE
PROPELLER OPEN WATER EFFICIENCY
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13. CALCULATIONS TO BE DONE DURING
THE DESIGN PROCESS
-DURING THE DESIGN PROCESS SOMME ITERATION MUST BE DONE TO
OBTAIN THE CONVERGENCE OF DESIGN PROPELLER THRUST (TTA) AND
THE PROPELLER PROPULSIVE EFFICIENCY (EEP).
-THE HULL V- EHP CORRESPONDENCE SHALL BE KNOWN.
-AT THE BEGINNING, EEP CAN BE ASSUMED 0.65 .
IN EACH ITERATION THE INITIAL VALUE OF EEP SHALL BE THE ONE
CORRESPONDING TO THE PREVIUS ITERATION.
-THE SHIP SPEED (V) - PROPULSION POWER (BHP) CURVE TO BE
USED IN ANY ITERATION SHALL BE :
BHP= EHP/EEP
-BE MCR THE MAX. CONTINUOS RATING OF ENGINES POWER.
-BE PPA . MCR/100 THE DESIGN POWER FOR THE PROPELLER/S.
-TO THIS POWER THE SHIP ESPEED SHOUL BE VVA. AND THE SHIP
ADVANCE RESISTANCE R.
- THE DESIGN PROPELLER THRUST (TTA) SHOULD BE :
TTA=R/((1-t).NL)
- t IS THE SUCTION COEFF.
-NL IS THE NUMBER OF SHAFT LINES.
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14. CALCULATIONS TO BE DONE DURING
THE DESIGN PROCESS
-THE CIRCULATION RADIAL DISTRIBUTION MUST BE ADAPTED TO THE
ADEQUATE PROPELLER THRUST.
-AFTER THE CALCULATIONS OF INDUCED VELOCITIES, IT WILL BE
POSSIBLE TO CALCULATE THE RADIAL THRUST DISTRIBUTION.
THE POLYGONON IS PLACED AT THE
PROPELLER DISCK
TCI(X0) =ρ ZV*(X0)Ci(X0)cosβio(X0)
TCI IS IDEAL THRUST OF ANNULAR
SECTION.
TCI útil(X0) = TCI (X0)-ZRv(X0) Sin(βio(X0))
TCI util(X0) IS THE REAL PROPELLER THRUST
Rv(X0) IS THE VISCOUS RESISTANCE OF ANNULAR SECTION
-Rvnsinβio(Xh)
Tcal IS THE REAL TOTAL PROPELLER THRUST.
Rvn IS THE COMPONENT DUE TO HUB VISCOUS
REST.
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15. CALCULATIONS TO BE DONE DURING
THE DESIGN PROCESS
-TCal MUST BE EQUAL TO TTA.
- IF { Tcal-TTA} <= 0.001 TTA THEN THE CIRCULATION RADIAL
IS CORRECT.
- IF NO, IT IS NECCESARY TRANSFORM Ci(X0).
Cin(X0) = Ci(X0)[1+(TTA-Tca)/TTA].
- A NEW ITERATION SHOULD BE DONE TO CORRECT THE CALC.
PROPELLER THRUST.
NEXT IT IS NECCESARY TO CORRECT THE ASSUMED
VALUE OF EPP (BHP)
M(X0) IS THE MOMENT REQUESTED BY A GENER.
ANNULAR SECT.
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16. CALCULATIONS TO BE DONE DURING
THE DESIGN PROCESS
THE Mcal REQUESTED BY THE PROPELLER IS:
+ Rvn.cosβi0(XH)D/2(XH)
EEQ=TTA.VVA(1-w)/(2π(RPM.RPMA/6000).Mcal)
EEPcal=EEQ(1-t)/(1-w).ETAM
EEOcal =EEQ/EER
IF [EEP-EEPcal] <=0.0001 THE HYDRODYNAMIC CALC.
HAVE FINISHED.
IF NO EEP=(EEPcal+EEP)/2
NEW V-BHP CURVE SHOULD BE OBTAINED AND THEN NEW
ITERATION PROCESS MUST BE PERFORMED.
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17. RADIAL DISTRIBUTIONS OF
GEOMETRICAL PITCHES AND CAMBERS
- A MEAN LINE MUST BE CHOOSED TO DEFINE THE GEOMETRY OF
PROPELLER BLADES ANNULAR SECTIONS.
THEY SHALL BE KNOWN:
(f/Cr)o, Clio, αo, αlo, αto
- FROM FORMER CALCULATIONS SHALL BE KNOWN THE RADIAL
DISTRIBUTION OF CL= L/(0.5ρV*^2 Cr)
- A RADIAL DISTRIBUTION OF a COEF. SHALL BE CHOOSED AND Cli
COEF. SHALL BE DEFINED.
Cli=CL/a
- TWODIMENSIONAL APPROACH TO GEOMETRICAL PITCHES AND
CAMBERS
γ=GEOMETRICAL PITCH ANGLE
ϒ=βio +αi +(a-1)Cli/(2π)
αi=Cli/Clio αio
(f/Cr)=(f/Cr)o CLi/Clio
- TO OBTAIN TRIDIMENSIONAL PITCHES AND CAMBERS IT IS
NEEDED TO INTRODUCE CORRECTIONS IN PITCHES (Δ1a) AND
CAMBERS (Kc). THIS SHALL BE DONE USING NEW CASCADES
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THEORY

18. RADIAL DISTRIBUTIONS OF
GEOMETRICAL PITCHES AND CAMBERS
- ACCORDING NEW CASCADES THEORY
Δ1a = (Cli/Clio αto αio (αlo/αio –A(αlo + αto)))/αlo +
+2Aαto (αto + αlo))
IN THE CASE OF CONVENTIONAL PROPELLERS Δ1a MUST BE
MULTIPLIED BY 0.575
A=4πr/(Z Cr) sin(βio)/Clio
B=1/(αto +αlo) –A
Kc = (1+B( αio+Δ1a(Clio/Cli) ) )/(1-Bαto)
- γtrid= = βio + αi + a Δ 1α + (a-1) Cli/(2π)
-- (f/Cr)trid=(f/Cr) Kc
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19. REFERENCES
1 Pérez Gómez, G., Souto Iglesias, A., López
Pavón, C., González Pastor, D., ¨Corrección y
recuperación de la teoría de Goldstein para el
proyecto de hélices ¨ . Ingeniería Naval. Nov.
2004.
3 Pérez Gómez, G., ¨Utilidad de la teoría renovada
de las líneas sustentadoras para realizar el
diseño de hélices con carga en los extremos de
las palas, y para estimar el rendimiento de
cualquier hélice al efectuar su anteproyecto ¨.
Ingeniería Naval. Marzo 2007.
5 Pérez Gómez, G., “ De las hélices TVF, a la última
generación de hélice CLT”. Ingeniería Naval.
Noviembre 2009
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