propeller design and cavitation prof. dr. s. · pdf filedesign calculation we begin by...
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PROPELLER DESIGN AND CAVITATION
Prof. Dr. S. Beji
2
Dimensional Analysis The thrust π of a propeller could depend upon:a) Mass density of water, π.
b) Size of the propeller, represented by diameter π·.
c) Speed of advance, ππ΄.
d) Acceleration due to gravity, π.
e) Speed of rotation, π.
f) Pressure in the fluid, π.
g) Viscosity of the water, π.
π = π(ππ, π·π , ππ΄π , ππ , ππ , ππ, ππ)
ππΏ
π2
1
=π
πΏ3
π
πΏ ππΏ
π
ππΏ
π2
π1
π
ππ
πΏπ2
ππ
πΏπ
π
Dimensional Analysis Multiplying and equating the powers
π = 1 β π β ππ = 1 + 3π β π β π + π + ππ = 2 β 2π β π β 2π β π
Substituting π and π in the expression for π gives π = 2 + π + π β π then
π = ππ·2ππ΄2 ππ·
ππ΄2
πππ·
ππ΄
ππ
πππ΄2
π π
ππ΄π·
π
where π = π/π is the kinematic viscosity. In general we can write
π12ππ·
2ππ΄2 = π
ππ·
ππ΄2 ,
ππ·
ππ΄,
π
πππ΄2 ,
π
ππ΄π·
Note that since the disc area of the propeller , π΄0 = π 4 π·2, is proportionalto π·2, the thrust coefficient can also be written in the form
π12ππ΄0ππ΄
2
Dimensional Analysis If the model and ship quantities are denoted by the suffixes M and S,
respectively, and if π is the linear scale ratio, then π·π π·π = π
If the propeller is run at the correct Froude speed of advance, πΉππ = πΉππ,ππ΄πππ΄π
= π1/2
The slip ratio has been defined as 1 β ππ΄ ππ . For geometrically similarpropellers, therefore, the nondimensional quantity ππ· ππ΄ must be the same
for model and ship. Thus, as long as ππ· ππ΄2 and ππ· ππ΄ are the same in ship
and model
π β π·2ππ΄2
π·ππ·π
= π,ππ΄πππ΄π
= π1/2,ππππ
=π·π
2
π·π2 β
ππ΄π2
ππ΄π2 = π3
πππ·πππ΄π
=πππ·πππ΄π
,ππππ
=π·ππ·π
βππ΄πππ΄π
=1
πβ π1/2 =
1
π1/2, ππ = ππ β π
1/2
Dimensional Analysis The thrust power is given by ππ = π β ππ΄, so that
ππππππ
=ππππ
βππ΄πππ΄π
= π3.5,ππ
ππ=ππ·πππ
βππππ·π
= π3.5 β π0.5 = π4
If the model results were plotted as values of
πΆπ =π
12ππ·
2ππ΄2 , πΆπ =
π12ππ·
3ππ΄2
to a base of ππ΄ ππ· or π½, 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 π½ or ππ΄ ππ· is the same for model and ship, ππ΄ may be replaced by ππ·.
πΎπ =π
ππ2π·4, πΎπ =
π
ππ2π·5
π½ =ππ΄ππ·
, π0 =ππππ·0
=πππ΄
2πππ0=
π½
2πβπΎπ
πΎπ0, π =
π0 β ππ£ππ2π·5
where ππ = π β ππ΄ is the thrust power and ππ·0 = 2ππ β π0 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, π, but at some lower speed ππ΄, called the speed ofadvance.
Propeller-Hull Interaction Froude wake fraction
π€πΉ =π β ππ΄ππ΄
, ππ΄ =π
1 + π€πΉ
Taylor wake fraction
π€ =π β ππ΄π
, ππ΄ = π(1 β π€)
Resistance augment fraction and thrust deduction fraction
π =π β π π
π π=
π
π πβ 1, π = (1 + π)π π
π‘ =π β π π
π= 1 β
π π
π, π π = 1 β π‘ π
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 ππ΅ = (2ππ)ππ΅, where π is therotation rate, revolutions per second and ππ΅ is the brake torque, π β π.
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 βlosses.β 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 ππ = (2ππ)ππ.
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 ππ·.
Thrust power is the power delivered by propeller as it advances through the water at a speed of advance ππ΄, delivering the thrust π. Thethrust power is ππ = πππ΄.
Effective power is defined as the resistance of the hull, π , times the ship speed π, ππΈ = π π.
Propulsive efficiency is defined as
ππ· =ππΈππ·
=ππΈππ
ππππ·0
ππ·0ππ·
=π π
πππ΄π0ππ =
π 1 β π‘ π
ππ(1 β π€)π0ππ =
(1 β π‘)
(1 β π€)π0ππ
ππ· = ππ»π0ππ , ππ» =(1 β π‘)
(1 β π€), ππ =
ππ·0ππ·
Different Design Approaches Case 1: ππ·, π ππ, ππ΄ are known and π·πππ‘. is required.
In this case the unknown parameter π·πππ‘ may be eliminated from the diagrams by plotting πΎπ/π½
5 versus π½ instead of πΎπ versus π½ as follows.
πΎππ½5
=π
ππ2π·5
ππ·
ππ΄
5
=ππ3
πππ΄5 =
2πππ π2
2ππππ΄5 =
ππ·π2
2ππππ΄5
πΎπ 1 4π½β 5 4 =
ππ3
πππ΄5
1 4
= 0.1739 π΅π1
which are the variables used in charts between pages 192-196. On these charts the optimum π0 and 1/π½ are read off at the
intersection of known πΎπ 1 4π½β 5 4 value. Afterwards π·πππ‘ is
computed using the optimum 1/π½ value as π·πππ‘ = ππ΄/ππ½.
Different Design Approaches Case 2: ππ·, π·, ππ΄ are known and π ππ is required.
In this case the unknown parameter π may be eliminated from the diagrams by plotting πΎπ/π½
3 versus π½ instead of πΎπ versus π½ as follows.
πΎππ½3
=π
ππ2π·5
ππ·
ππ΄
3
=ππ
ππ·2ππ΄3 =
2πππ
2πππ·2ππ΄3 =
ππ·
2πππ·2ππ΄3
πΎπ 1 4π½β 3 4 =
ππ
ππ·2ππ΄3
1 4
= 1.75 π΅π2
which are the variables used in charts between pages 197-201. On these charts the optimum π0 and 1/π½ are read off at the
intersection of known πΎπ 1 4π½β 3 4 value. Afterwards π is
computed using the optimum 1/π½ value as π = ππ΄/π½π·.
Typical Propeller Design Given DataService speed: π = 21 ππππ‘π = 10.8 π/π
Effective power with model-ship correlation allowance: ππΈ = 9592 ππ
Estimated propulsive efficiency: ππ· = ππΈ/ππ· = 0.75
Immersion depth of propulsion shaft: β = 7.5 π
Estimated delivered power at 21 ππππ‘π : ππ· = ππΈ/ππ· = 9592/0.75 = 12789 ππ
Empirically determined wake fraction: π€ = 0.20
Empirically determined thrust deduction fraction: π‘ = 0.15
Estimated relative rotative efficiency: ππ = 1.05
Design CalculationSelected propeller diameter (with adequate clearance): π· = 6.4 π
Selected number of blades (from consideration of vibration forces): π = 4
Calculation of velocity of advance (wake speed): ππ΄ = π 1 β π€ = 10.8(1 β
Typical Propeller Design
Wageningen B-Series charts: πΎπ 1 4 β π½β 3 4 =
ππ·
2πππ·2ππ΄3
1 4
=
12477
2πβ1.025β 6.4 2β 8.64 3
1 4= 0.52039
Using the charts given on pages 197 (B4-40), 198 (B4-55), and 199 (B4-70) for
πΎπ 1 4 β π½β 3 4 = 0.52039 from the optimum propeller efficiency curve we get
Expanded blade area ratio: 0.40 0.55 0.70
1/π½ at optimum efficiency: 1.260 1.275 1.290
π ππ = 60 β ππ΄/π· β 1/π½ : 102.1 103.3 104.5
Corresponding π/π·: 1.110 1.090 1.070
Open water efficiency π0: 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
π =π
1 β π‘=
π π
1 β π‘ π=
ππΈ1 β π‘ π
=9592
(1 β 0.15) β 10.8= 1044.9 ππ
The Keller area criterion for a single-screw vessel givesπ΄πΈπ΄0
=1.3 + 0.3 β π β π
π0 β ππ£ β π·2+ π
where π0 β ππ£ = πππ‘π. β ππ£ππππ’π + ππβ = 98100 β 1750 + 1025 β 9.81 β 7.5 =171764 π/π2 = 171.8 ππ/π2, and π is a constant varying from 0 to 0.2. Taking π = 0.2,
π΄πΈπ΄0
=1.3 + 0.3 β 4 β 1044.9
171.8 β 6.4 2+ 0.2 = 0.571
Interpolating between π΄πΈ/π΄0 = 0.55 and π΄πΈ/π΄0 = 0.70 for π΄πΈ/π΄0 = 0.571 we get π = 103.5 πππ, π/π· = 1.087, and π0 = 0.669. Accordingly the ππ· value becomes:
ππ· =1 β π‘
1 β π€β π0 β ππ =
1 β 0.15
1 β 0.20β 0.669 β 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 ππ· = 0.746.
Typical Propeller Design Another Example of Calculating Optimum π ππA somewhat different example of calculating optimum π ππ is given below. A common problem for the propeller designer is the design of a propeller when the required propeller thrust π (e.g. from a model test equal to the resistance corrected for the thrust deduction) and the propeller diameter π· (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 π ππ. Suppose that the following data is given for a container vessel.Thrust: π = 142 π‘πππ = 142000 β 9.81 π = 1393000 π = 1393 ππShip speed: π = 21 ππππ‘π = 10.8 π/π Advance velocity (Wake speed): ππ΄ = 21 1 β 0.2 = 16.8 ππππ‘π = 8.64 π/π Selected propeller diameter (with adequate clearance): π· = 7.0 πSelected number of blades (from consideration of vibration forces): π = 4Immersion depth of propulsion shaft: β = 5.0 π
Design CalculationWe begin by considering Kellerβs formula for π΄πΈπ with π = 0 for the given ship
πΈπ΄π =π΄πΈπ΄0
=1.3 + 0.3 β 4 β 1393
(98.1 β 1.75 + 1.025 β 9.81 β 5.0) β 7.0 2= 0.485
For single skrew wake field π = 0.2, it may be less if the wake field is good.
Typical Propeller DesignObviously π is an important parameter affecting the value of π΄πΈπ . If it were selected as 0.2, πΈπ΄π would be 0.685 instead of 0.485. Now that we have taken π = 0 and consequently the minimum πΈπ΄π must be 0.485. To be on the safe side let us select πΈπ΄π = 0.55. Already we have selected π = 4 therefore our propeller now is B4-55. Consequently we can use the πΎπ β πΎπ β π½ diagram of B4-55 series. We know the thrust π and the diameter π·, hence seek for the π ππ. To use the diagram we need to know both πΎπ and π½; however at present we can only calculate πΎπ/π½
2 as followsπΎπ
π½2=
π
ππ2π·2=
1393000
1025 β 8.652 β 72= 0.3707
Now when we select a π½ β π/π· pair in the diagram it must be such that πΎπ/π½2 =
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 πΎπ = 0.3707π½2 for a range of π½ values. Wherever this curve crosses a πΎπ value in the diagram of B4-55, that particular πΎπ β π½ pair corresponding to a definite π/π· ratio in the diagram is a correct πΎπ β π½ pair. Supposing that we have established the following table for various π/π· ratios using the πΎπ β πΎπ β π½ 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 π/π· = 1 but to make sure we try the close neigborhood of it; π/π· = 0.95 and π/π· = 1.05. From these results we see that π/π· = 1 really gives the maximum efficiency hence we select it. Thus our final decision for the propeller is B4-55, π/π· = 1, π0 = 0.651, π½ = 0.699 so
that π =ππ΄
π½π·=
8.65
0.699β7= 1.768 πππ , π = 60 β π = 106 πππ. Finally, since the
torque coefficient πΎπ = 0.0310, π = ππ2π·5πΎπ = 1025 β 1.7682 β 75 β 0.0310,
π = 1669322 π β π. Power to be delivered ππ· = 2πππ/1000 = 18544 ππ
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: ππ΄ = 28 ππππ‘π = 14.4 π/π
Delivered power : ππ· = 440 ππ
π ππ = 720, π = 720/60 = 12 πππ
π = 5
πΈπ΄π = π΄πΈ/π΄0 = 0.75
Typical Propeller DesignTo eliminate the unknown diameter we must use πΎπ/π½
5 as the parameter.
πΎπ
π½5=
π
ππ2π·5
ππ·
ππ΄
5=
ππ3
πππ΄5 =
2πππβπ2
2ππππ΄5 =
ππ·π2
2ππππ΄5 =
440000β122
2πβ1025β14.45=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 π/π· = 1.4 and π½ = 1.19 so that π· = ππ΄/ππ½, π· = 14.4/(12 β 1.19) = 1.01 π, π· = 1π. Since πΎπ = 0.153, π = ππ2π·4πΎπ, π =23500 π. 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 πΈπ΄π = 0.59 hence OK.
π·/π« π± π²π» π²πΈ πΌπ π²πΈ/π±π
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