HELSINKI UNIVERSITY OF TECHNOLOGY SHIP HYDRODYNAMICS LABORATORY
OTANIEMI FINLAND REPORT NO 12
ON THE DESIGN OF NON-OPTIMAL PROPELLERS BY
MAX HONKANEN
TKK OFFSET 1977
ISBN 951-751-035-7
ISSN 0356-1313ABSTRACT
The need of designing propellers, whose circulation distri-bution departs from the optimum, is usually determined by the cavitation characteristics. The question of the shape of the circulation distribution has been usually solved by adopting an experimentally verified shape to be "scaled" to give the correct thrust and torque.
In this paper, a new method of modifying the circulation distribution is presented. It is based on a scheme of reducing the propeller loading near the hub and the tip by certain percentages. These define an ideal efficiency distribution along the propeller radius departing from Betz's optimum condition. A usual lifting line calculation is then performed resulting in a circulation distribution that gives improved cavitation characteristics to the propeller.
Finally, a design example has been worked out using both the Eckhardt-Morgan design method and the induction factor method by Lerbs together with the non-optimal circulation distribution. The results show clearly the favorable effect on the cavitation characteristics with a negligible loss in the propeller efficiency.
CONTENTS Page ABSTRACT 1 CONTENTS 2 INTRODUCTION 3
IDEAL EFFICIENCY DISTRIBUTION 4
2.1 The Form of the Distribution 4 2.2 Determination of the Constants 5
EXAMPLE CALCULATIONS 7
3.1 Data of the Ship 7
3.2 Propeller Design Calculations 8 i. CONCLUSIONS 9 ACKNOWLEDGEMENTS 9 NOMENCLATURE 12 REFERENCES 13 2
1. INTRODUCTION
The power requirements of high-speed ships have been the cause of an increased cavitation of ship propellers and of propeller excited vibrations. The concept of good propeller performance therefore also includes the cavitation characte-ristics of the propeller in addition to the efficiency. The more heavily the propeller is loaded, the more important the
cavitation characteristics become. The application of the Betz minimum energy loss condition, Reference [li, results in an optimum circulation distribution that tends to con-centrate the thrust near the blade tips. Such a thrust gra-ding may not have the best qualities from the cavitation's point of view.
Several methods of obtaining a non-optimal circulation distri-bution exist. The method of Hill, [21, is based on modifying the thrust grading curve. The difference between the modified and the optimal thrust grading curves then defines a new ra-dial distribution of the hydrodynamic pitch angle. Lerbs, when introducing his induction factor method in [31, used a similar technique on the circulation distribution itself fixing the shape of it and determining a "scale factor" with the aid of the induction factor method. The design method of Eckhardt and Morgan, [14], is also capable of dealing with non-optimal propellers by using a pre-determined shape of the hydrodynamjc pitch angle distribution.
All these methods have one disadvantage in common: the impro-vement of the cavitation characteristics is related to a "non-physical" quantity like the circulation distribution or to an empirical well behaving distribution shape. A Naval Architect who might not be quite familiar with the hydro-dynamic theories, needs a lot of trial and error working before he finds himself a suitable circulation distribution. The intention of this paper is to present a method that is related to the thrust grading curve by percentage reductions
¡4
at the tip and the root. The reduction of the sectional thrust will then obviously increase the local cavitation number thus improving the cavitation characteristics at the critical parts of the blade.
2. IDEAL EFFICIENCY DISTRIBUTION
2.1 The Forts of the Distribution
The radial distribution of the ideal efficiency of a pro-peller has to be constant for the minimum energv loss condi-tion as stipulated by Betz. The resulting circulacondi-tion distri-bution then gives the best possible efficiency to the screw. Now it is obvious that a non-optinal circulation distribution will, on the contrary, be a result of a non-linear distri-bution of the ideal efficiency. Let us assume that the pro-peller blade extends front O.2R to R. The first possible non-linear radial ideal efficiency distribution is obviously a parabolic one:
n(x) = nEa +
b(x - 0.2)2 + c(x - 1)2] (1)where
constant ideal efficiency
x = rIE = non-dimensional radius of the blade a,b,c = constants to be determined
The Bets nimimum energy loss condition is, of course, found by letting a 1 and b o = O. The constant ideal efficiency stay be determined as a first approximation from Kramer's
charts.
The constants a, b, o are determined in such a way that the sectional ideal efficiency is greater than Tì by amounts q1
and q2 at the tip and the root, respectively, and that the mean efficiency over the blade is
ni,
thusn.(l) (1 + q1»). 1 n1(0.2) (1 + q2»). 1 f n.(x)dx (1 - O.2)n 0.2 1
The ideal efficiency distribution is now alternatively expressed in terms of the coefficients q1 and q2 as
n(x)
r.[]. - q1 - q25
+ -(2q1 + - 0.2)2
FIG.1: VELOCITY DIAGRAM
We want to determine the reduction coefficients q, and q2 (2)
(3)
+ + 2q2)(x - 1)2]
2.2 Determination of the Constants With the aid of the simplified velocity diagram of 'ig. 1, the hydrodynamic pitch angle, the induced velocity factor as de-fined in [5] and the sectional thrust coefficient in a non-viscous fluid are
t an B tanß = ni(x)/1 - w(x)
1-w
tanB(tanB1 - tanß) a' -1 4: tan2B1 = 7T3x3Ka'(l - a') wherew(x) wake fraction at radius x w overall wake fraction K Goldstein's K-function
dkTt
dn(x)
k.
-i n1(x)
According to [41, the coefficients k1 are usually of the order 5. They may be found by differentiating (4):
dkT' - (1 - 2a')da'
i'
a'(l - a')2tanß - tanB + tan2ß1tanß da'
(1 + tanß)
d(tanß1)
d(tan8)
dn(x)
tanß1 (6)
In the first formula of (6) it has been assumed that the Goldaten's K-function remains constant.
A substitution of the results into the definition of k in (5) yields
k
1 - tan2ß1+ tanB1[(l - tan2ß1)(l - tan2ß) + 4 tan8tanß] (1 + tan2B1)(l + tan8tanß)(tanßi - tanß)
(7) The unknown constants q1 and q2 may now be computed from
(5) by first calculating k1and k2 from (7) at the tip and the root correspondingly. In practical calculations, the tip may be approximated by O.975R and the root by O.2R because these sections usually define the limits of the blade in the design calculations.
6
in such a way that the corresponding sectional thrust coeffi-cients are reduced by given amounts pl and p2 at the tip and the root. Let us first write
3. EXAMPLE CALCULATIONS
3.1 Data of the Ship
To illustrate the method, a design example has been worked out. The data of the ship have been reproduced from [5] and are as follows: Length L 207.0 in Beam B = 27.14m Draught T 8.5 in Block coefficient CB = 0.64 Shaft immersion = 4.88 m Number of propellers N = 2 Screw diameter D = 6.1 in Boss diameter Db 1.1 15 Number of blades Z 5
Expanded area ratio aE 0.67
Speed of revolution n 1.933 rIs Delivered horsepower per screw P0 14600 kW Effective horsepower per screw 9570 kW
Speed oI the ship V 11.83 m/s
Wake fraction w = 0.17
Thrust deduction fraction t 0.18
The ship is a twin-screw passenger vessel, and the geometry of the propellers is according to Troost-BB series as
de-fined in [6h The resistance curve of the ship is given by 7
The wake distribution is assumed to be
Speed of the ship, knots 22.50 23.00 23.50 24.00 Effective horsepower, lip 21400 23300 25600 28600
x 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.85.
8
3.2 Propeller Design Calculations
Two lifting-line calculations have been performed in order to illustrate the method: the first one is in essentials according to Eckhardt-Morgan's design method using Goldstein's K-function in determining the circulation distribution, and the second one is in accordance with Lerb's induction factor method. The lifting-surface corrections of both calculations have been made according to Morgan and Silovic, [7], as presented in polynomial form in [61. The calculations are performed with a series of computer programs reported in [8].
The basic calculations were performed with an optimum circu-lation distribution using the Bets minimum energy loss condition. In order to improve the cavitation characte-ristics, the loading was then reduced by 25 % at the tip
and by lO at the root. The blade sections were all designed for the shockless entry condition.
The main results of the propeller design together with the obtained ship speeds are summarized in Table 1 below.
TABLE i: MAIN RESULTS OF THE PROPELLER DESIGN CALCULATIONS
As it could be expected, the induction factor method of Lerbs resulted in a slightly better design. From the propulsion's point of view, the departure from the optimum condition resul-ted in a reduction of l.1 % in the efficiency and 0.3 % in the speed.
Item
Optimum circulation Non-optimum circulation Goldstein Lerba Goldstein Lerbs "ean pitch ratio 1.120 1.112 1.116 1.113
Pitch ratio p07 1.121 1.103 1.129 1.116
Dpen efficiency
0.708
0.718
0.6990.707
Speed of ship, rn/s 12.02 12.0611.98
12.029
The geometric data of the designs are summarized in Fig. 2 and the hydrodynamic data in Fig. 3. The calculated values at each section have been smoothed to continuous curves by using a weighted least squares fit. The cavitation margin
is defined as the percentage difference of the true cavita-tion number and the critical cavitacavita-tion number.
4. CONCLUSIONS
The first conclusion that may be drawn from the example cal-culation is, that the induction factor method seems to produce better propeller designs both from the propulsion's and from the cavitation's point of view.
The second conclusion is, that even à considerable departure from the Bets minimum energy loss condition does not lead to unacceptable decrease of the propeller efficiency. Thus, a slight sacrifice of the efficiency might be justified in many cases to improve the cavitation characteristics of the screw.
The third Conclusion to be drawn is, that the method of de-termining the non-optimal circulation distribution by first defining percentage reductions of the thrust grading curve at the blade tip and the root and then calculating a non-linear ideal efficiency distribution works as desired. The favorable effect on the cavitation characteristics is particularly pronounced in connection with the K-method.
ACKNOWLEDGEMENTS
The author wishes to thank Mr. K. Sundelin of Messrs. Prorissi-valu Ltd for permitting the use of Reference [81. Thanks are also due to Prof. V. Kostilainen of Ship Hydrodynamics Laboratory for checking the paper and for publishing it.
P LOO
oo
Kappa method, opttnum circutaiiøn
u Kappa method, non.opflmum cfrculaf ion
Lerbs's me4ho4, opt!muwi círcuaBon
e
Lerbs's m4 hod, non-optt'nurn circulaiion
LoadIng reduced 25Xaf tip and tO%al root
0.3 -01
-Geametric date
G g 6 Trcosf 88 I I t I I h i i I 0.1 O. 0.3 0.4 0.5 0.6 0.? 0.8 0.9 0.08 0.0?046
405
0.04 0.03 a.oa 0.01lo
.03
Hqdradqnarnjc data
Kappa method Lerbi's method
L.
Opt
G- difi ibtI0
Reduced loading a tip an4 ,ot
Ciru!ai'jo,, ds#rjbf
ion G Cavja- tion L
/
i
loo .05 50 2 0.3 04' ¿ 0.8(9
1.o xNOMENCLATURE
.aE Expanded area ratio
a' Induced velocity factor
a,b,c Constants of the ideal efficiency distribution B Beam of the ship
CB Block coefficient c Section chord D Propeller diameter Db Boss diameter G Non-dimensional circulation F/(rDVA) Shaft immersion
Ratio of thrust reduction to ideal efficiency reduction
Sectional thrust coefficient T'/(pn2D)
kQ' Sectional torque coefficient Q'/(Pn2D5) L Length of the ship
Section camber N Number of propellers n Speed of revolution Delivered horsepower Effective horsepower p Pitch ratio
Reductions of the sectional thrust at R and O.2R Coefficients of the ideal efficiency distribution R Radius of the propeller
T Draft of the ship
t Thrust deduction fraction, section thickness y Speed of the ship
VA Speed of advance w Wake fraction
w(x) Wake fraction at radius x
x Non-dimensional radius of the blade Z Number of blades
13
a Angle of attack B Geometric pitch angle 3. Hydrodynamic pitch angle
Ideal efficiency
n1(x) Ideal efficiency at radius x K Goldstein's K-function
REFERENCES
ri] Bets, A.: Screw Propellers with Minimum Loss of Kinetic Energy. Reprinted in Vier Abhandlungen zur Hydro- und Aerodynamik by L. Prandtl and A. Betz,
1929.
Hill, J.G.: The Design of Propellers. Trans. SNAME,
Vol. 57, l99.
Lerbs, H.W.: Moderately Loaded Propellers with a Finite Number of Blades and an Arbitrary Distribution of Cir-culation. Trans. SNAME, Vol.
60, 1952.
Eckhardt, M.K., Morgan, W.B.: A Propeller Design Method. Trans. SNAME, Vol. 63,
1955.
O'Brien, T.P.: The Design of Marine Screw Propellers. Hutchinson Scientific and Technical, Third impression,
1969.
[61 van Oossanen, P.: Calculation of Performance and Cavi-tation Characteristics of Propellers Including Effects of Non-Uniform Flow and Viscosity. Publication no
157,
Netherlands Ship Model Basin, Wageningen,l974.
[71 Morgan, W.B., Silovic, V.: Propeller Lifting-Surface Corrections. Trans. SNANE, Vol.
