VILUCE MARINE DESIGN PTY LTD REVISION 1.0 |DATE OF LATEST AMENDMENT: 30-MAR-19
Machinery and Propeller Design “MAFALDA” FERRY
Machinery and Propeller Design
i
ABSTRACT
Following an estimate of effective power required in the previous design phase, this report details the accurate
assessment of power requirements, the selection of suitable drive train machinery and the design of an appropriate
Marin BB Series propeller. The total powering requirement was found to be 2120 kW which is, as expected,
approximately double the effective power calculated by MAXSURF Resistance. The selected machinery matches that
of similar vessels from the published data. Design of the propeller to Marin BB definitions was hindered by the
relatively small space for the propeller between the shaft and the hull, however simple modifications and allowances
were made to be able to accommodate a propeller of sufficient diameter to provide the required thrust. The
resultant propulsion system consists of MTU 12V 2000 M72 main engine, ZF5050 V-drive gearbox and 1.29 m
diameter 5-blade propeller per demihull.
Machinery and Propeller Design
ii
Table of Contents
Abstract .............................................................................................................................................................................. i
1 Introduction .............................................................................................................................................................. 1
2 Powering Design Point .............................................................................................................................................. 1
3 Brake Power .............................................................................................................................................................. 2
4 Machinery ................................................................................................................................................................. 3
4.1 Main Engine ...................................................................................................................................................... 3
4.2 Gearbox ............................................................................................................................................................. 4
5 Propeller Design ........................................................................................................................................................ 5
6 Propeller Design Details ............................................................................................................................................ 6
6.1 Blade Lengths .................................................................................................................................................... 8
6.2 Thicknesses ....................................................................................................................................................... 8
6.2.1 Trailing and Leading edge thickness ....................................................................................................... 10
6.3 Offsets ............................................................................................................................................................. 10
7 Conclusion ............................................................................................................................................................... 12
Appendix A ...................................................................................................................................................................... 13
Appendix A.1 – MTU 12V 2000 M72 Datasheet ......................................................................................................... 13
Appendix B ...................................................................................................................................................................... 15
Appendix B.1 – Z5050V Gearbox ................................................................................................................................ 15
Appendix C ...................................................................................................................................................................... 17
7.1 Appendix C.1 – Values of V1 and V2 ............................................................................................................... 17
Machinery and Propeller Design
iii
List of Figures
Figure 1 Brake Power Requirement .................................................................................................................................. 2
Figure 2 V-Drive gearbox diagram .................................................................................................................................... 4
Figure 3 Propeller Principle Design ................................................................................................................................... 5
Figure 4 Thickness and Offset Dimensions ....................................................................................................................... 6
Figure 5 Definitions for Blade Lengths .............................................................................................................................. 6
List of Tables
Table 1 Propeller Details Input ......................................................................................................................................... 6
Table 2 Marin BB Series Dimensions ................................................................................................................................. 7
Table 3 Blade Lengths ....................................................................................................................................................... 8
Table 4 Blade Thicknesses for r/R = 0.25 and 0.6 ............................................................................................................. 8
Table 5 Blade Thicknesses ................................................................................................................................................. 9
Table 6 Leading Edge Thicknesses .................................................................................................................................. 10
Table 7 yFACE offsets for P > 0. All values in mm. ............................................................................................................. 10
Table 8 yBack offsets for P>0. All values in mm. ............................................................................................................... 11
Table 9 yFACE offsets for P<0. All values in mm. ............................................................................................................... 11
Table 10 yBACK offsets for P<0. All values in mm. ............................................................................................................ 11
Table 11 – V1 Values ....................................................................................................................................................... 17
Table 12 – V2 Values ....................................................................................................................................................... 18
List of Graphs
Graph 1 Trials Resistance Curve ........................................................................................................................................ 1
Graph 2: Blade thickness vs radius ................................................................................................................................... 9
Machinery and Propeller Design
1
1 INTRODUCTION
The determination of the propulsion system is a cyclic process requiring an estimation of propeller diameter and
RPM to obtain the required thrust of the propeller and match it with both off-the-shelf engine and gearbox options
and to enable the propeller to be designed to Marin BB definitions. Two pre-prepared Excel sheets were utilised
alongside with extensive market research of machinery available to a ferry operating out of Townsville, QLD. Upon
selection of the machinery, the principal dimensions of the propeller are found. These allow for a complete and
detailed design of the propeller as undertaken to provide the end point to an optimal drive train for this vessel.
2 POWERING DESIGN POINT
From the resistance vs speed curve from the previous design phase, a ‘Trials resistance curve’ was created by
effectively adding an additional 0.5 kn to each speed at which a resistance was calculated. This provides a margin for
error by assuming the resistance of the vessel is slightly higher than calculated. The design point for powering this
vessel for a operational speed of 25 kn would then be powering for a resistance calculated for the vessel at 25.5 kn.
With this increase in resistance, the Trials resistance curve was created, and the design point found as per Graph 1.
Graph 1 Trials Resistance Curve
The total resistance at the design point was found to be 82.58 Kn, or 41.29 kN per demihull.
0.00
20.00
40.00
60.00
80.00
100.00
120.00
10 15 20 25 30
Res
ista
nce
(kN
)
Speed (kn)
Speed vs Total Resistance
Machinery and Propeller Design
2
3 BRAKE POWER
Calculation of the brake power required the determination of a few key particulars, the first of which is the proposed
propeller diameter. Initially a diameter of 1.26 m was selected to ensure plenty of clearance between the hull and
propeller blade tips, however a propeller of this diameter was found to either not provide the required thrust or to
break the Marin BB definitions. The best results were found by maximising the propeller diameter to 1.29 m, leaving
plenty blade tip clearance before the rake of the blades is considered. Some assumptions were made in establishing
other important inputs, including the Taylor wake fraction, the thrust deduction fraction and the efficiencies of the
shaft transmission and gearbox. The Taylor wake fraction can typically range from 0.05-1.5 yet the professional
opinion is that the value is at the lower end of this spectrum for high-speed catamarans and so was set at 0.07. The
thrust deduction value was suggested to be a similar value of 0.07. The gearbox and shaft efficiency were chosen to
be 97% and 98% respectively as average values in the industry. Furthermore, a 10% de-rate was applied due to the
effects of the tropical climate of Townsville.
By iteratively testing rotation speeds for the propeller, it was also found that the most efficient rotation speed
occurs at the lowest RPM that allows the pitch to diameter ratio of the propeller to stay within range of the Marin BB
series. Overall, the smallest break power requirement for the vessel was obtained to be 889 kW per hull using a 1.29
m propeller rotating at 545 RPM. Any RPM lower than this requires an increase in blade pitch beyond the range of
the Marin BB Series, as demonstrated in Figure 1.
Figure 1 Brake Power Requirement
Machinery and Propeller Design
3
4 MACHINERY
4.1 MAIN ENGINE The propulsion machinery must supply 889 kN of brake power at 90% of its Maximum Continuous Rating (MCR). As
such, the MCR of the selected engine must be at least 1000 kW, as shown below.
𝑀𝑅𝐶𝑅𝑒𝑞𝑢𝑖𝑟𝑒𝑑 =889
0.9
= 988.12
≈ 1000 kW
Besides the base power requirement, the following factors were considered in the main engine selection:
• Size: The engine block must fit into the demihulls, which are 2.5 m wide each. The smaller the engine block,
the more accessible the engine room is for maintenance and the lighter the engine.
• Availability of parts: As Townsville is the operating area of the vessel, major shipping ports attract the
presence of most large engine manufacturers. This includes: MAN, Caterpillar, Wartsilla, MTU and Cummins.
• Efficiency and Ecology: The client expressed a strong preference for a vessel which is both environmentally
friendly and fuel efficient. A qualitative judgement was made on the relative consumption and
environmental impact of each engine considered over their lifetime
• Machinery Common to Industry: The published data reflects the accumulated knowledge and experience of
different naval architects and marine engineers from similar vessels to Mafalda. This limits the risk of
choosing an engine not appropriate for fast-ferry propulsion.
After extensive research, MTU 12V 2000 M72 was selected. This is a high-speed, 4-stroke diesel engine which
outputs 1060 kW at 1950 RPM in Category 1B of MTU. The reasoning for this selection is that this engine is widely
used in the high-speed aluminium ferry industry, popular for vessels in the 25 – 35 m range. Moreover, the engine is
only 1.385 m wide, allowing maximum access space in the engine room. Category 1B for MTU is defined to be ideal
for passenger ferry operations where the vessel operates at full load 60 – 80% of the time, with no large load
fluctuations. Selection the appropriate engine is necessary to optimize the lifespan of the engine under expected
operations. The data sheet for the engine can be found in Appendix A.1.
The total installed brake power of the vessel is 2120 kW, which exceeds published data vessels. This is due to
previous vessels underwent far more cycles in the design spiral, fairing the hull and reducing the total resistance of
the vessel. As explained in the resistance calculation report, the appendage resistance is considered to be a large
portion of the total resistance of the vessel and reducing it could reduce the total resistance considerably.
Machinery and Propeller Design
4
4.2 GEARBOX The transmission selection was simplified to be a selection of ZF Marine Group gearbox, as they have a big global
presence and provide a large range of compatibility with all major engines. At 90% MCR, the gearbox must be able to
handle 1755 ERPM and step it down to 545 SRPM, requiring a reduction ratio of 3.222. It must also withstand 954
kW (90% of 1060 kW) at 1755 ERPM and up to 1060 kW at 1950 ERPM (100% MCR).
The ZF 5050 V satisfies the requirements by a healthy margin up to 1114 kW at 2000 RPM with a reduction ratio of
3.222. This allows the machinery and the propeller to run at their most efficient speeds while avoiding over stressing
the gearbox. The gearbox weights 768 kg each, which is quite heavy, and size could be stepped down to save over
250 kg per gearbox, but it is not worth the risk to slightly exceed its suggested operational limits.
The reasoning for selecting a V drive gearbox is to allow the shaft to exit the hull far further forward in to the vessel,
increasing the propeller clearance and decreasing the shaft exit angle. Therefore, this gearbox selection addresses
the lack of propeller clearance and, more significantly, addresses the detrimental effect of the shaft exit angle as a
significant source of appendage resistance.
Figure 2 V-Drive gearbox diagram
Machinery and Propeller Design
5
5 PROPELLER DESIGN
The optimal Marin BB Series propeller is designed based on two key particulars, the diameter of the propeller and
the pitch to diameter ration (P/D) of the blades. The propeller needs a diameter of at least 1.26m to meet the P/D <
1.4 criteria of the series for the thrust required. As seen in Figure 3, the power delivered to the propeller is 816 kW
at the operational speed of 90% MCR of the engine which can be satisfied by a 1.29 m, 5 bladed propeller with a P/D
of 1.3455 and an area ratio (Ae/Ao) of 0.915, both of which are at the design boundaries of the series. However, as
they are within bounds. Thus, the designed propeller produce 44.53 kN of pull, safely more than the 44.40 kN
required pull, which takes into consideration the thrust deduction factor, determined at the design point.
Figure 3 Propeller Principle Design
Machinery and Propeller Design
6
6 PROPELLER DESIGN DETAILS
This section details the calculation of the blade lengths, thicknesses and offsets for the accurate construction of this
propeller to the Marin BB series definition. All constants and formulae are derived from the Ship Propulsion Notes by
Phil Helmore. These details describe the cross-section of the propeller at each discrete radius using the dimensions
shown in Figure 4 and 5 below.
Figure 4 Thickness and Offset Dimensions
Figure 5 Definitions for Blade Lengths
The input values for the calculations are provided in Table 1,
Table 1 Propeller Details Input
Input Value Unit
Z 5 -
D 1290 mm
P/D 1.3455 -
AE/AO 0.915 -
A 345 mm
P 1735.695 mm
Revs 1950 RPM
P 1060 kW
RR 3.222 -
G 7.5 g/cm^3
U 46
B 0.915
N 5
E0.25 1
E0.6 1.25
F0.25 2.146
C 1
R 605.21
M 7.55
F0.6 5.85
M0.6 6.88
C0.6 1.60
Machinery and Propeller Design
7
where:
Z = Number of blades
D = Diameter of propeller, in mm P/D = Pitch ratio
A = Rake of the propeller, in mm. Blade propellers lose negligible effect until tilted past 15o backwards, thus the rake of blade was calculated to be 345 mm
A = 𝑅 tan (𝜃)
= 1290 tan (15) = 345 mm
Revs = RPM of engine at 90% MCR P = Brake power of the engine, in kW
RR = Reduction ratio
G = Density of manganese aluminium bronze, which is the material selected for the propeller, anticorrosive marine metal, in g/cm3
U = Allowable stress of manganese aluminium bronze, in N/mm2 B = Ae/Ao N = Z
E0.25 = Through to C0.6 are constants as derived from the series definition and are found using P/D
M = 3.75𝐷
𝑃+ 2.8
𝑃
𝐷
F = P/D + 0.8
For the Marin BB Series, the following dimensions are provided,
Table 2 Marin BB Series Dimensions
r/R
CrZ/DAEAO ar/cr br/cr
- - -
0.2 1.6 0.581 0.35
0.3 1.832 0.584 0.35
0.4 2.023 0.58 0.351
0.5 2.163 0.57 0.355
0.6 2.243 0.552 0.389
0.7 2.247 0.524 0.443
0.8 2.132 0.48 0.486
0.9 1.798 0.402 0.5
0.95 1.434 0.318 0.5
Machinery and Propeller Design
8
6.1 BLADE LENGTHS To find the blade lengths, it is a simple matter of multiplying or dividing these dimensions by the appropriate input
values to isolate ar, br and cr. The results are provided in Table 3.
Table 3 Blade Lengths
Cr ar br
(mm) (mm) (mm)
377.7 219.5 132.2
432.5 252.6 151.4
477.6 277.0 167.6
510.6 291.1 181.3
529.5 292.3 206.0
530.4 278.0 235.0
503.3 241.6 244.6
424.5 170.6 212.2
338.5 107.7 169.3
6.2 THICKNESSES The blade thicknesses were calculated at 0.25R and 0.6R as per Lloyd’s Rules for the Classification and Construction
of Ships, as shown in Table 4. T0.25 and T0.6 were found to be 59.46 mm and 25.62 mm respectively.
Table 4 Blade Thicknesses for r/R = 0.25 and 0.6
Variable Value Unit
K 7994.03 -
L0.2 377.71 mm
L0.3 432.48 mm
L0.25 405.10 mm
T0.25 59.46 mm
T0.6 25.62 mm
Where:
K = 𝐺𝐵𝐷3𝑅2
675
L = Length of blade section at 25 (L0.25) and 60 (L0.6) percent radius, as appropriate
T = 𝐾𝐶𝐴
𝐸𝐹𝑈𝐿𝑁+ 100 √
3150𝑀𝑃
𝐸𝐹𝑅𝑈𝐿𝑁 in mm, for thicknesses of 25 and 60 percent radius
B = Developed area ratio E = 1 at 0.25R and 1.25 at 0.6R
Machinery and Propeller Design
9
The maximum thickness of each station is assumed to be linear between 0.2R to 0.6R, followed by a tangential curve
to a tip of 4 mm, as shown in Graph 2. The linear gradient was found by interpolation between T0.25 and T0.6.
Table 5 Blade Thicknesses
r/R T (mm)
0.2 64.3
0.25 59.5
0.3 54.6
0.4 45.0
0.5 35.3
0.6 25.6
0.7 19.6
0.8 12.7
0.9 7.6
0.95 5.6
1 4.0
Graph 2: Blade thickness vs radius
0.0
10.0
20.0
30.0
40.0
50.0
60.0
70.0
0.2 0.4 0.6 0.8 1
Tick
nes
s T
(mm
)
Percentage Radius
Blade Thickness from Base to Tip
Machinery and Propeller Design
10
6.2.1 Trailing and Leading edge thickness
The thickness of the trailing and leading edge has been set to a constant thickness of 4 mm beside the region of the
leading edge between 0.2R to 0.55R. Within this range, the leading edge thickness decreases from 0.55R linearly to
4mm.
0.2TLE = 0.2𝑇
6
= 10.7 mm The resulting leading edge thicknesses are summarised in table 6 as follows,
Table 6 Leading Edge Thicknesses
r/R TLE
(mm)
0.2 10.7
0.25 9.8
0.3 8.8
0.4 6.9
0.5 5.0
0.55 4.0
0.6 4.0
6.3 OFFSETS The offsets are calculated from the tabulated functions V1 and V2, which are simple non-dimensional functions of
the blade thickness, R and P, the proportion of the distance from the position of the maximum thickness. The offsets
have been calculated for 0.2R to 0.95R for both the face and back of the blades on both the leading side forward of
the point of maximum thickness where P is positive (Table 7 and Table 8), and the trailing side where P is negative
(Table 9 and Table 10).
Table 7 yFACE offsets for P > 0. All values in mm.
yFACE
r/R P1 P0.95 P0.9 P0.8 P0.6 P0.4 P0.2 P0
0.95 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.9 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.8 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.7 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.6 0.8 0.4 0.1 0.0 0.0 0.0 0.0 0.0
0.5 4.0 2.4 1.6 0.7 0.1 0.0 0.0 0.0
0.4 8.9 6.0 4.5 2.6 0.8 0.1 0.0 0.0
0.3 14.8 11.1 8.9 6.0 2.5 0.7 0.1 0.0
0.25 18.1 13.9 11.5 8.1 3.7 1.2 0.2 0.0
0.2 21.7 17.0 14.2 10.2 4.8 1.8 0.3 0.0
Machinery and Propeller Design
11
Table 8 yBack offsets for P>0. All values in mm.
yBACK
r/R P1 P0.95 P0.9 P0.8 P0.6 P0.4 P0.2 P0
0.95 4.0 4.1 4.2 4.3 4.6 4.8 4.9 4.9
0.9 4.0 4.2 4.5 4.9 5.6 6.1 6.4 6.5
0.8 4.0 4.7 5.4 6.6 8.6 10.0 10.7 11.0
0.7 4.0 5.6 7.0 9.4 12.9 15.3 16.6 17.0
0.6 4.8 7.6 10.0 14.0 19.6 23.0 24.9 25.6
0.5 8.0 11.9 15.1 20.4 27.5 31.8 34.4 35.3
0.4 12.9 17.9 21.7 28.0 35.9 40.7 43.8 45.0
0.3 18.8 24.6 29.1 36.0 44.6 49.9 53.5 54.6
0.25 22.1 27.7 32.3 39.8 48.8 54.6 58.2 59.5
0.2 25.7 30.4 35.3 43.0 52.7 59.3 63.1 64.3
Table 9 yFACE offsets for P<0. All values in mm.
yFACE
r/R -1P -0.95P -0.9P -0.8P -0.6P -0.4P -0.2P 0P
0.95 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.9 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.8 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.7 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.6 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.5 1.6 1.3 1.0 0.6 0.1 0.0 0.0 0.0
0.4 5.6 4.6 3.7 2.4 0.8 0.2 0.0 0.0
0.3 10.6 11.0 8.2 6.1 2.9 0.9 0.2 0.0
0.25 12.9 11.8 10.5 8.2 4.5 1.7 0.4 0.0
0.2 15.1 14.1 12.9 10.5 6.5 4.9 0.9 0.0
Table 10 yBACK offsets for P<0. All values in mm.
yBACK
r/R -1P -0.95P -0.9P -0.8P -0.6P -0.4P -0.2P 0P
0.95 4.0 4.1 4.2 4.3 4.6 4.8 4.9 4.9
0.9 4.0 4.2 4.5 4.9 5.6 6.1 6.4 6.5
0.8 4.0 4.7 5.3 6.5 8.5 9.9 10.7 11.0
0.7 4.0 5.3 6.5 8.7 12.3 14.9 16.5 17.0
0.6 4.0 6.1 8.1 11.7 17.9 22.2 24.8 25.6
0.5 6.5 9.1 11.6 16.4 24.6 30.6 34.2 35.3
0.4 12.5 14.9 17.5 22.6 31.9 39.1 43.6 45.0
0.3 19.4 23.5 24.7 30.3 40.0 47.6 52.9 54.6
0.25 22.7 25.1 28.1 34.0 44.3 51.9 57.5 59.5
0.2 25.9 28.2 31.4 37.6 48.5 58.4 62.2 64.3
Machinery and Propeller Design
12
The coordinates were calculated by means of the following formulas,
yFACE = V1(Tmax -TLE) for P > 0
yBACK = (V1 + V2)(Tmax – TLE) + TLE
yFACE = V1(Tmax -TTE) for P < 0
yBACK = (V1 + V2)(Tmax – TTE) + TTE
The values of V1 AND V2 can be found in appendix C
7 CONCLUSION
With specified propulsion machinery and a purpose-built Marin BB series propeller, this vessel is prepared to make
speed during sea trails. The main engine is a size larger than initially expected, leading also to a large gear box.
However, selected set up for the V-drive gearbox couple with a dimensionally small engine will allow for more
streamlined shafting and this is expected to considerable reduce drag and effectively increase the engine
performance and efficiency. Furthermore, the V-drive set up allows for sufficient propeller clearance as without it
the propeller would have to have been designed smaller and thus less efficient. The propeller details have been
calculated to provide a blueprint for construction of the optimal propeller for this vessel. The results of this design
phase match expectations for the most part and so support the resistance calculations of the previous phase and
assure the client that this vessel is on track to deliver to the design brief with satisfaction.
Machinery and Propeller Design
13
APPENDIX A
APPENDIX A.1 – MTU 12V 2000 M72 DATASHEET
Machinery and Propeller Design
14
Machinery and Propeller Design
15
APPENDIX B
APPENDIX B.1 – Z5050V GEARBOX
Machinery and Propeller Design
16
Machinery and Propeller Design
17
APPENDIX C
7.1 APPENDIX C.1 – VALUES OF V1 AND V2 Table 11 – V1 Values
r/R P1 P0.95 P0.9 P0.8 P0.6 P0.4 P0.2 P0
0.95 0 0 0 0 0 0 0 0
0.9 0 0 0 0 0 0 0 0
0.8 0 0 0 0 0 0 0 0
0.7 0 0 0 0 0 0 0 0
0.6 0.0382 0.0169 0.0067 0.0006 0 0 0 0
0.5 0.1278 0.0778 0.05 0.021 0.0034 0 0 0
0.4 0.2181 0.1467 0.1088 0.0637 0.0189 0.0033 0 0
0.3 0.2923 0.2186 0.176 0.1191 0.0503 0.0148 0.0027 0
0.25 0.3256 0.2513 0.2068 0.1465 0.0669 0.0224 0.0031 0
0.2 0.36 0.2821 0.2353 0.1685 0.0804 0.0304 0.0049 0
r/R -P1 -P0.95 -P0.9 -P0.8 -P0.6 -P0.4 -P0.2 -P0
0.95 0 0 0 0 0 0 0 0
0.9 0 0 0 0 0 0 0 0
0.8 0 0 0 0 0 0 0 0
0.7 0 0 0 0 0 0 0 0
0.6 0 0 0 0 0 0 0 0
0.5 0.0522 0.042 0.033 0.019 0.004 0 0 0
0.4 0.1467 0.12 0.0972 0.063 0.0214 0.0044 0 0
0.3 0.2306 0.24 0.179 0.1333 0.0623 0.0202 0.0033 0
0.25 0.2598 0.2372 0.2115 0.1651 0.0899 0.035 0.0084 0
0.2 0.2826 0.263 0.24 0.1967 0.1207 0.092 0.0172 0
Machinery and Propeller Design
18
Table 12 – V2 Values
r/R P1 P0.95 P0.9 P0.8 P0.6 P0.4 P0.2 P0
0.95 0 0.097 0.19 0.36 0.64 0.84 0.96 1
0.9 0 0.097 0.19 0.36 0.64 0.84 0.96 1
0.8 0 0.105 0.2028 0.3765 0.6545 0.852 0.9635 1
0.7 0 0.124 0.2337 0.414 0.684 0.866 0.9675 1
0.6 0 0.1485 0.272 0.462 0.72 0.879 0.969 1
0.5 0 0.175 0.3056 0.5039 0.7478 0.888 0.971 1
0.4 0 0.1935 0.3235 0.522 0.7593 0.8933 0.9725 1
0.3 0 0.189 0.3197 0.513 0.752 0.892 0.975 1
0.25 0 0.1758 0.3042 0.4982 0.7415 0.8899 0.9751 1
0.2 0 0.156 0.284 0.4777 0.7277 0.8875 0.975 1
r/R -P1 -P0.95 -P0.9 -P0.8 -P0.6 -P0.4 -P0.2 -P0
0.95 0 0.0975 0.19 0.36 0.64 0.84 0.96 1
0.9 0 0.0975 0.19 0.36 0.64 0.84 0.96 1
0.8 0 0.0975 0.19 0.36 0.64 0.84 0.96 1
0.7 0 0.0975 0.19 0.36 0.64 0.84 0.96 1
0.6 0 0.0965 0.1885 0.3585 0.6415 0.8426 0.9613 1
0.5 0 0.095 0.1865 0.3569 0.6439 0.8456 0.9639 1
0.4 0 0.0905 0.181 0.35 0.6353 0.8415 0.9645 1
0.3 0 0.08 0.167 0.336 0.6195 0.8265 0.9583 1
0.25 0 0.0725 0.1567 0.3228 0.605 0.8139 0.9519 1
0.2 0 0.064 0.1455 0.306 0.5842 0.7984 0.9446 1