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RENEWED LIFTING LINE THEORY

DESIGN OF SHIP PROPELLERS

AUTHOR: DOCTOR G. PEREZ GOMEZ

PURE LIFTING LINE THEORY

-PROPELLER BLADES REPLACED BYLIFTING LINES AND ASSOCIATES FREE VORTICES.

-THE PROPELLER BLADES ANNULAR SECTIONSARE INDEPENDENTS

-THE Z LIFTING LINES SHALL BE REPLACED BY n. Lim. nC(n)=ZC(z)C(n)– 0n --- infiniteC Circulation at radial station r

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CALCULATION OF VORTICES INDUCED VELOCITIES

-RADIAL VORTICESDO NOT INDUCEVELOCITIES AT THE PROPELLER DISCK.

-HELICOIDAL VOR-TICES ARE PLACED ON REVOLUTIONSURFACES PASINGBY THE ENDS OF ELEMENTAL RADIALVORTICES.

INDUCED VELOCITIES PRODUCED BY HELICOIDAL VORTI-CES PLACED ON A CYLINDER OF RADIO r

W=WxIo+WrRo+WfiFio cylindrical coordinatesW induced velocity vectorIo, Ro, Fio, unitary vectors in the directions x, r,and perpendicular to r

THE REVOLUTION SURFACE CONTRACTIONHAS BEEN IGNORED IN THE FIGURE.

-

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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

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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. dWx/dx =dWr/dx =dWfi/dx=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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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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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 STREAM2π(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 THEFOLLOWING EQUATION:

AFTER KNOW Bio ,Wx SALL BE ALSO KNOWN

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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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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 THEINFINITE DOWN STREAMTHE 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.9

CALCULATIONS OF INDUCED VELOCITIES AT THE PROPELLER DISCK

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-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 THEPROPELLER DISCK SHALL BE CALCULATED APPLYING CONSERVATION OF KINETIC MOMENT BETWEEN THE PROPELLER DISCK AND THEINFINITE DOWN STREAM

-FINALLY THE VELOCITIES POLIGONOM AT THE PROPELLERDISCK (OF PAG. 6), SHALL BE KNOWN .

=

GENERALIZATIONS FOR THE CASE OF TIPLOADED 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 ALTERNATIVECONVENTIONAL PROPELLER.

-IN THIS FIGURE ARE SHOWN THE EFECTS OF TIP PLATESON A TWODIMENSIONAL PROFILE. THE CIRCULATION ALONG THE SPAN IS COMBINATION OF A LINEAL DISTRIBUTION PLUS A PARABOLIC ONE.

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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 Q VALUES AT THE HUB (C(M)), AND AT THE TIP (C(N)) ARE EQUALS.

N - IN A CONTROL POINT X THE VALUE OF THE A CIRCULATION AQ IS AT+TQ 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

C

Xh 1

M

.

T

X

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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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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.

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TCI(X0) =ρ ZV*(X0)Ci(X0)cosβio(X0)

THE POLYGONON IS PLACED AT THEPROPELLER DISCK

TCI IS IDEAL THRUST OF ANNULARSECTION.

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.

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)

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M(X0) IS THE MOMENT REQUESTED BY A GENER. ANNULAR SECT.

CALCULATIONS TO BE DONE DURING THE DESIGN PROCESS

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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 NEWITERATION PROCESS MUST BE PERFORMED.

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 THEORY 17

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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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.

2 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.

3 Pérez Gómez, G., “ De las hélices TVF, a la última generación de hélice CLT”. Ingeniería Naval. Noviembre 2009

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