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PROPELLER DESIGN AND CAVITATION
Prof. Dr. S. Beji
2
Dimensional Analysis The thrust ๐ of a propeller could depend upon:a) Mass density of water, ๐.
b) Size of the propeller, represented by diameter ๐ท.
c) Speed of advance, ๐๐ด.
d) Acceleration due to gravity, ๐.
e) Speed of rotation, ๐.
f) Pressure in the fluid, ๐.
g) Viscosity of the water, ๐.
๐ = ๐(๐๐, ๐ท๐ , ๐๐ด๐ , ๐๐ , ๐๐ , ๐๐, ๐๐)
๐๐ฟ
๐2
1
=๐
๐ฟ3
๐
๐ฟ ๐๐ฟ
๐
๐๐ฟ
๐2
๐1
๐
๐๐
๐ฟ๐2
๐๐
๐ฟ๐
๐
Dimensional Analysis Multiplying and equating the powers
๐ = 1 โ ๐ โ ๐๐ = 1 + 3๐ โ ๐ โ ๐ + ๐ + ๐๐ = 2 โ 2๐ โ ๐ โ 2๐ โ ๐
Substituting ๐ and ๐ in the expression for ๐ gives ๐ = 2 + ๐ + ๐ โ ๐ then
๐ = ๐๐ท2๐๐ด2 ๐๐ท
๐๐ด2
๐๐๐ท
๐๐ด
๐๐
๐๐๐ด2
๐ ๐
๐๐ด๐ท
๐
where ๐ = ๐/๐ is the kinematic viscosity. In general we can write
๐12๐๐ท
2๐๐ด2 = ๐
๐๐ท
๐๐ด2 ,
๐๐ท
๐๐ด,
๐
๐๐๐ด2 ,
๐
๐๐ด๐ท
Note that since the disc area of the propeller , ๐ด0 = ๐ 4 ๐ท2, is proportionalto ๐ท2, the thrust coefficient can also be written in the form
๐12๐๐ด0๐๐ด
2
Dimensional Analysis If the model and ship quantities are denoted by the suffixes M and S,
respectively, and if ๐ is the linear scale ratio, then ๐ท๐ ๐ท๐ = ๐
If the propeller is run at the correct Froude speed of advance, ๐น๐๐ = ๐น๐๐,๐๐ด๐๐๐ด๐
= ๐1/2
The slip ratio has been defined as 1 โ ๐๐ด ๐๐ . For geometrically similarpropellers, therefore, the nondimensional quantity ๐๐ท ๐๐ด must be the same
for model and ship. Thus, as long as ๐๐ท ๐๐ด2 and ๐๐ท ๐๐ด are the same in ship
and model
๐ โ ๐ท2๐๐ด2
๐ท๐๐ท๐
= ๐,๐๐ด๐๐๐ด๐
= ๐1/2,๐๐๐๐
=๐ท๐
2
๐ท๐2 โ
๐๐ด๐2
๐๐ด๐2 = ๐3
๐๐๐ท๐๐๐ด๐
=๐๐๐ท๐๐๐ด๐
,๐๐๐๐
=๐ท๐๐ท๐
โ๐๐ด๐๐๐ด๐
=1
๐โ ๐1/2 =
1
๐1/2, ๐๐ = ๐๐ โ ๐
1/2
Dimensional Analysis The thrust power is given by ๐๐ = ๐ โ ๐๐ด, so that
๐๐๐๐๐๐
=๐๐๐๐
โ๐๐ด๐๐๐ด๐
= ๐3.5,๐๐
๐๐=๐๐ท๐๐๐
โ๐๐๐๐ท๐
= ๐3.5 โ ๐0.5 = ๐4
If the model results were plotted as values of
๐ถ๐ =๐
12๐๐ท
2๐๐ด2 , ๐ถ๐ =
๐12๐๐ท
3๐๐ด2
to a base of ๐๐ด ๐๐ท or ๐ฝ, therefore, the values would be directly applicable to theship. But the above coefficients have the disadvantage that they becomeinfinite for zero speed of advance, a condition sometimes occurring in practice.Since ๐ฝ or ๐๐ด ๐๐ท is the same for model and ship, ๐๐ด may be replaced by ๐๐ท.
๐พ๐ =๐
๐๐2๐ท4, ๐พ๐ =
๐
๐๐2๐ท5
๐ฝ =๐๐ด๐๐ท
, ๐0 =๐๐๐๐ท0
=๐๐๐ด
2๐๐๐0=
๐ฝ
2๐โ๐พ๐
๐พ๐0, ๐ =
๐0 โ ๐๐ฃ๐๐2๐ท5
where ๐๐ = ๐ โ ๐๐ด is the thrust power and ๐๐ท0 = 2๐๐ โ ๐0 is the delivered power
of the open water propeller.
Dimensional Analysis
Propeller-Hull Interaction Wake: Previously we have considered a propeller working
in open water. When a propeller operates behind themodel or ship hull the conditions are considerablymodified. The propeller works in water which is disturbedby the passage of the hull, and in general the water aroundthe stern acquires a forward motion in the same directionas the ship. This forward-moving water is called the wake,and one of the results is that the propeller is no longeradvancing relatively to the water at the same speed as theship, ๐, but at some lower speed ๐๐ด, called the speed ofadvance.
Propeller-Hull Interaction Froude wake fraction
๐ค๐น =๐ โ ๐๐ด๐๐ด
, ๐๐ด =๐
1 + ๐ค๐น
Taylor wake fraction
๐ค =๐ โ ๐๐ด๐
, ๐๐ด = ๐(1 โ ๐ค)
Resistance augment fraction and thrust deduction fraction
๐ =๐ โ ๐ ๐
๐ ๐=
๐
๐ ๐โ 1, ๐ = (1 + ๐)๐ ๐
๐ก =๐ โ ๐ ๐
๐= 1 โ
๐ ๐
๐, ๐ ๐ = 1 โ ๐ก ๐
Power Definitions
Brake power is usually measured directly at the crankshaft coupling bymeans of a torsion meter or dynamometer. It is determined by a shoptest and is calculated by the formula ๐๐ต = (2๐๐)๐๐ต, where ๐ is therotation rate, revolutions per second and ๐๐ต is the brake torque, ๐ โ ๐.
Shaft power is the power transmitted through the shaft to the propeller. For diesel-driven ships, the shaft power will be equal to the brake power for direct-connect engines (generally the low-speed diesel engines). For geared diesel engines (medium- or high-speed engines), the shaft horsepower will be lower than the brake power because of reduction gear โlosses.โ Shaft power is usually measured aboard ship as close to the propeller as possible by means of a torsion meter. The shaft power is given by ๐๐ = (2๐๐)๐๐.
Power Definitions
Delivered power is the power actually delivered to the propeller.There is some power lost in the stern tube bearing and in any shafttunnel bearings between the stern tube and the site of the torsionmeter. The power actually delivered to the propeller is thereforesomewhat less than that measured by the torsion meter. This deliveredpower is given the symbol ๐๐ท.
Thrust power is the power delivered by propeller as it advances through the water at a speed of advance ๐๐ด, delivering the thrust ๐. Thethrust power is ๐๐ = ๐๐๐ด.
Effective power is defined as the resistance of the hull, ๐ , times the ship speed ๐, ๐๐ธ = ๐ ๐.
Propulsive efficiency is defined as
๐๐ท =๐๐ธ๐๐ท
=๐๐ธ๐๐
๐๐๐๐ท0
๐๐ท0๐๐ท
=๐ ๐
๐๐๐ด๐0๐๐ =
๐ 1 โ ๐ก ๐
๐๐(1 โ ๐ค)๐0๐๐ =
(1 โ ๐ก)
(1 โ ๐ค)๐0๐๐
๐๐ท = ๐๐ป๐0๐๐ , ๐๐ป =(1 โ ๐ก)
(1 โ ๐ค), ๐๐ =
๐๐ท0๐๐ท
Different Design Approaches Case 1: ๐๐ท, ๐ ๐๐, ๐๐ด are known and ๐ท๐๐๐ก. is required.
In this case the unknown parameter ๐ท๐๐๐ก may be eliminated from the diagrams by plotting ๐พ๐/๐ฝ
5 versus ๐ฝ instead of ๐พ๐ versus ๐ฝ as follows.
๐พ๐๐ฝ5
=๐
๐๐2๐ท5
๐๐ท
๐๐ด
5
=๐๐3
๐๐๐ด5 =
2๐๐๐ ๐2
2๐๐๐๐ด5 =
๐๐ท๐2
2๐๐๐๐ด5
๐พ๐ 1 4๐ฝโ 5 4 =
๐๐3
๐๐๐ด5
1 4
= 0.1739 ๐ต๐1
which are the variables used in charts between pages 192-196. On these charts the optimum ๐0 and 1/๐ฝ are read off at the
intersection of known ๐พ๐ 1 4๐ฝโ 5 4 value. Afterwards ๐ท๐๐๐ก is
computed using the optimum 1/๐ฝ value as ๐ท๐๐๐ก = ๐๐ด/๐๐ฝ.
Different Design Approaches Case 2: ๐๐ท, ๐ท, ๐๐ด are known and ๐ ๐๐ is required.
In this case the unknown parameter ๐ may be eliminated from the diagrams by plotting ๐พ๐/๐ฝ
3 versus ๐ฝ instead of ๐พ๐ versus ๐ฝ as follows.
๐พ๐๐ฝ3
=๐
๐๐2๐ท5
๐๐ท
๐๐ด
3
=๐๐
๐๐ท2๐๐ด3 =
2๐๐๐
2๐๐๐ท2๐๐ด3 =
๐๐ท
2๐๐๐ท2๐๐ด3
๐พ๐ 1 4๐ฝโ 3 4 =
๐๐
๐๐ท2๐๐ด3
1 4
= 1.75 ๐ต๐2
which are the variables used in charts between pages 197-201. On these charts the optimum ๐0 and 1/๐ฝ are read off at the
intersection of known ๐พ๐ 1 4๐ฝโ 3 4 value. Afterwards ๐ is
computed using the optimum 1/๐ฝ value as ๐ = ๐๐ด/๐ฝ๐ท.
Typical Propeller Design Given DataService speed: ๐ = 21 ๐๐๐๐ก๐ = 10.8 ๐/๐
Effective power with model-ship correlation allowance: ๐๐ธ = 9592 ๐๐
Estimated propulsive efficiency: ๐๐ท = ๐๐ธ/๐๐ท = 0.75
Immersion depth of propulsion shaft: โ = 7.5 ๐
Estimated delivered power at 21 ๐๐๐๐ก๐ : ๐๐ท = ๐๐ธ/๐๐ท = 9592/0.75 = 12789 ๐๐
Empirically determined wake fraction: ๐ค = 0.20
Empirically determined thrust deduction fraction: ๐ก = 0.15
Estimated relative rotative efficiency: ๐๐ = 1.05
Design CalculationSelected propeller diameter (with adequate clearance): ๐ท = 6.4 ๐
Selected number of blades (from consideration of vibration forces): ๐ = 4
Calculation of velocity of advance (wake speed): ๐๐ด = ๐ 1 โ ๐ค = 10.8(1 โ
Typical Propeller Design
Wageningen B-Series charts: ๐พ๐ 1 4 โ ๐ฝโ 3 4 =
๐๐ท
2๐๐๐ท2๐๐ด3
1 4
=
12477
2๐โ1.025โ 6.4 2โ 8.64 3
1 4= 0.52039
Using the charts given on pages 197 (B4-40), 198 (B4-55), and 199 (B4-70) for
๐พ๐ 1 4 โ ๐ฝโ 3 4 = 0.52039 from the optimum propeller efficiency curve we get
Expanded blade area ratio: 0.40 0.55 0.70
1/๐ฝ at optimum efficiency: 1.260 1.275 1.290
๐ ๐๐ = 60 โ ๐๐ด/๐ท โ 1/๐ฝ : 102.1 103.3 104.5
Corresponding ๐/๐ท: 1.110 1.090 1.070
Open water efficiency ๐0: 0.673 0.670 0.663
We proceed to choose the blade area ratio by applying a cavitation criterion given by Keller.
Typical Propeller DesignThe required thrust is
๐ =๐
1 โ ๐ก=
๐ ๐
1 โ ๐ก ๐=
๐๐ธ1 โ ๐ก ๐
=9592
(1 โ 0.15) โ 10.8= 1044.9 ๐๐
The Keller area criterion for a single-screw vessel gives๐ด๐ธ๐ด0
=1.3 + 0.3 โ ๐ โ ๐
๐0 โ ๐๐ฃ โ ๐ท2+ ๐
where ๐0 โ ๐๐ฃ = ๐๐๐ก๐. โ ๐๐ฃ๐๐๐๐ข๐ + ๐๐โ = 98100 โ 1750 + 1025 โ 9.81 โ 7.5 =171764 ๐/๐2 = 171.8 ๐๐/๐2, and ๐ is a constant varying from 0 to 0.2. Taking ๐ = 0.2,
๐ด๐ธ๐ด0
=1.3 + 0.3 โ 4 โ 1044.9
171.8 โ 6.4 2+ 0.2 = 0.571
Interpolating between ๐ด๐ธ/๐ด0 = 0.55 and ๐ด๐ธ/๐ด0 = 0.70 for ๐ด๐ธ/๐ด0 = 0.571 we get ๐ = 103.5 ๐๐๐, ๐/๐ท = 1.087, and ๐0 = 0.669. Accordingly the ๐๐ท value becomes:
๐๐ท =1 โ ๐ก
1 โ ๐คโ ๐0 โ ๐๐ =
1 โ 0.15
1 โ 0.20โ 0.669 โ 1.05 = 0.746
This compares well with the value of 0.750 assumed. If a larger difference had been found, a new estimate of the power would have to be made, using ๐๐ท = 0.746.
Typical Propeller Design Another Example of Calculating Optimum ๐ ๐๐A somewhat different example of calculating optimum ๐ ๐๐ is given below. A common problem for the propeller designer is the design of a propeller when the required propeller thrust ๐ (e.g. from a model test equal to the resistance corrected for the thrust deduction) and the propeller diameter ๐ท (e.g. from the available clearance) are known. The advance velocity of the propeller may be estimated from the ship speed and wake fraction. The unknowns are the required power and the engine ๐ ๐๐. Suppose that the following data is given for a container vessel.Thrust: ๐ = 142 ๐ก๐๐๐ = 142000 โ 9.81 ๐ = 1393000 ๐ = 1393 ๐๐Ship speed: ๐ = 21 ๐๐๐๐ก๐ = 10.8 ๐/๐ Advance velocity (Wake speed): ๐๐ด = 21 1 โ 0.2 = 16.8 ๐๐๐๐ก๐ = 8.64 ๐/๐ Selected propeller diameter (with adequate clearance): ๐ท = 7.0 ๐Selected number of blades (from consideration of vibration forces): ๐ = 4Immersion depth of propulsion shaft: โ = 5.0 ๐
Design CalculationWe begin by considering Kellerโs formula for ๐ด๐ธ๐ with ๐ = 0 for the given ship
๐ธ๐ด๐ =๐ด๐ธ๐ด0
=1.3 + 0.3 โ 4 โ 1393
(98.1 โ 1.75 + 1.025 โ 9.81 โ 5.0) โ 7.0 2= 0.485
For single skrew wake field ๐ = 0.2, it may be less if the wake field is good.
Typical Propeller DesignObviously ๐ is an important parameter affecting the value of ๐ด๐ธ๐ . If it were selected as 0.2, ๐ธ๐ด๐ would be 0.685 instead of 0.485. Now that we have taken ๐ = 0 and consequently the minimum ๐ธ๐ด๐ must be 0.485. To be on the safe side let us select ๐ธ๐ด๐ = 0.55. Already we have selected ๐ = 4 therefore our propeller now is B4-55. Consequently we can use the ๐พ๐ โ ๐พ๐ โ ๐ฝ diagram of B4-55 series. We know the thrust ๐ and the diameter ๐ท, hence seek for the ๐ ๐๐. To use the diagram we need to know both ๐พ๐ and ๐ฝ; however at present we can only calculate ๐พ๐/๐ฝ
2 as follows๐พ๐
๐ฝ2=
๐
๐๐2๐ท2=
1393000
1025 โ 8.652 โ 72= 0.3707
Now when we select a ๐ฝ โ ๐/๐ท pair in the diagram it must be such that ๐พ๐/๐ฝ2 =
0.3707. This may be done either by trial-and-error followed by linear iteration or by the graphical method. The graphical method gives the result directly but in order to use it we must draw the curve corresponding to ๐พ๐ = 0.3707๐ฝ2 for a range of ๐ฝ values. Wherever this curve crosses a ๐พ๐ value in the diagram of B4-55, that particular ๐พ๐ โ ๐ฝ pair corresponding to a definite ๐/๐ท ratio in the diagram is a correct ๐พ๐ โ ๐ฝ pair. Supposing that we have established the following table for various ๐/๐ท ratios using the ๐พ๐ โ ๐พ๐ โ ๐ฝ diagram (B4-55)
Typical Propeller Design
Our aim now is to select the propeller with maximum open-water efficiency. The first examination indicates it may be ๐/๐ท = 1 but to make sure we try the close neigborhood of it; ๐/๐ท = 0.95 and ๐/๐ท = 1.05. From these results we see that ๐/๐ท = 1 really gives the maximum efficiency hence we select it. Thus our final decision for the propeller is B4-55, ๐/๐ท = 1, ๐0 = 0.651, ๐ฝ = 0.699 so
that ๐ =๐๐ด
๐ฝ๐ท=
8.65
0.699โ7= 1.768 ๐๐๐ , ๐ = 60 โ ๐ = 106 ๐๐๐. Finally, since the
torque coefficient ๐พ๐ = 0.0310, ๐ = ๐๐2๐ท5๐พ๐ = 1025 โ 1.7682 โ 75 โ 0.0310,
๐ = 1669322 ๐ โ ๐. Power to be delivered ๐๐ท = 2๐๐๐/1000 = 18544 ๐๐
Typical Propeller Design Example of Calculating Optimum DiameterIn this example we suppose that from the beginning we have selected the engine and its RPM hence we are to select a diameter for maximum efficiency.
For a fast patrol boat the following data is given.
Advance velocity of propeller: ๐๐ด = 28 ๐๐๐๐ก๐ = 14.4 ๐/๐
Delivered power : ๐๐ท = 440 ๐๐
๐ ๐๐ = 720, ๐ = 720/60 = 12 ๐๐๐
๐ = 5
๐ธ๐ด๐ = ๐ด๐ธ/๐ด0 = 0.75
Typical Propeller DesignTo eliminate the unknown diameter we must use ๐พ๐/๐ฝ
5 as the parameter.
๐พ๐
๐ฝ5=
๐
๐๐2๐ท5
๐๐ท
๐๐ด
5=
๐๐3
๐๐๐ด5 =
2๐๐๐โ๐2
2๐๐๐๐ด5 =
๐๐ท๐2
2๐๐๐๐ด5 =
440000โ122
2๐โ1025โ14.45=0.016
Applying any one of the approaches (graphical or interpolative) described in the previous problem we can establish the following table.
In this case the most efficient choice is ๐/๐ท = 1.4 and ๐ฝ = 1.19 so that ๐ท = ๐๐ด/๐๐ฝ, ๐ท = 14.4/(12 โ 1.19) = 1.01 ๐, ๐ท = 1๐. Since ๐พ๐ = 0.153, ๐ = ๐๐2๐ท4๐พ๐, ๐ =23500 ๐. If this thrust is not in accord with the resistance calculations of the boat calculations must be repeated for a different speed till the convergence is gained. Finally the cavitation possibility must be checked at least by using the Keller formula. In this case Keller formula gives minimum ๐ธ๐ด๐ = 0.59 hence OK.
๐ท/๐ซ ๐ฑ ๐ฒ๐ป ๐ฒ๐ธ ๐ผ๐ ๐ฒ๐ธ/๐ฑ๐
1.200 1.085 0.098 0.0237 0.713 0.016
1.300 1.140 0.124 0.0306 0.735 0.016
1.400 1.190 0.153 0.0387 0.747 0.016
1.350 1.165 0.138 0.0346 0.742 0.016