The appendage resistance was calculated separately for each appendage rather than adding the wetted surface area to the bare hull and performing the analysis at once.
The expressions for resistance are given below:
Exposed Shafting, Stern Tubes and Bossings [2]
(
F)
and ε is the inclination of shaft relative to keel.Struts and Rudders [2]
, t/c is the thickness to chord ratio.
Bilge Keels [4]
Where CF is calculated with SB as the bilge keel wetted surface area and RnL
V2 2
1 T
D
AIR C A
R = ρAIR
References
1. Nordstrom, H.F., "Some Tests with Models of Small Vessels", Teknisk Tidskrift, Skeppsbyggnadskonst, 1936. Published in English as publication No.19 of the Swedish State Shipbuilding Experiment Tank, Göteborg, 1951.
2. Groot, D. de., "Weerstand en voortstuwing van motorboten", Schip en Werf, 1951.
Published in English in International Shipbuilding Progress, Vol.2, No.6, 1955.
3. Marwood, W.J., and Silverleaf, A., "Design Data for High Speed Displacement - Type Hulls and a Comparison with Hydrofoil Craft", Third Symposium on Naval Hydrodynamics, ONR ACR-65, 1960.
4. Beys, P.M., "Series 63 Round Bottom Boats", Davidson Laboratory, Stevens Institute of Technology, Report No. 949, 1963.
5. Yeh, H.Y.H., "Series 64 Resistance Experiments on High-Speed Displacement Forms", Marine Technology, July 1965.
6. Lindgren, H. and Williams, A., "Systematic Tests with Small, Fast Displacement Vessels, Including a Study of the Influence of Spray Strips", Proceedings of Diamond Jubilee International Meeting, Society of Naval Architects and Marine Engineers, 1968.
7. Marwood, W.J., and Bailey, D., "Design Data for High-Speed Displacement Hulls of Round-Bilge Form", British National Physical Laboratory, Ship Division, Report No.99, 1969.
8. Bailey, D., "The NPL High-Speed Round-Bilge Displacement Hull Series", Royal Institution of Naval Architects, Maritime Technology Monograph No.4, 1976.
9. Kafali, K., "The Powering of Round Bottom Motorboats", International Shipbuilding Progress, Vol.6, No.54, 1959.
10. Clement, E.P., "Graphs for Predicting the Resistance of Round Bottom Boats", International Shipbuilding Progress, Vol.11, No.114, 1964.
11. Oortmerssen, G. van, "A Power Prediction Method and Its Application to Small Ships", International Shipbuilding Progress, Vol.18, No.207, 1971.
12. Mercier, J.A., and Savitsky, D., "Resistance of Transom-Stern Craft in the Pre-Planning Regime", Davidson Laboratory, Stevens Institute of Technology, Report No.1667, 1973.
13. Clement, E.P., and Blount, D., "Resistance Tests of a Systematic Series of Planing Hull Forms", Transactions, Society of Naval Architects and Marine Engineers, Vol.71, 1963.
14. Holtrop, J. and Mennen, G.G.J., "A Statistical Power Prediction Method", International Shipbuilding Progress, Vol.25, No.290, 1978.
Propulsion
In carrying out the investigation, the following constants for operating conditions were carried throughout:
• Operating Medium: Sea Water (Salinity level 3.5%)
• Water Temperature: 180C
• Water Density: 1.0252 t/m3 (ITTC, 1963)
• Kinematic Viscosity: 1.10438 x 10-6 m2/s
• Sea State 4 or 5
For the vessel and propeller the following parameters were used:
• Wake Deduction Fraction (w): 0.03 Condition: 1.0 < FN∇ < 2.0
• Thrust Deduction Fraction (t): 0.015 Condition: 1.0 < F N∇ < 2.0
• Number of Propeller blades (Z): 4
• Rotation Rate (n): 8.1 rps (s-1)
With a set rotation rate and blade number, it is possible to determine the best propeller by varying P/D, BAR and D. Software was then developed within Microsoft Excel using the VBA programming language to handle multiple calculations and vary all the three propeller parameters as mentioned previously. The program takes advantage of the polynomial expression published in [5] for the Wageningen B-screw Series for Rn=2 x 106 ;
The various coefficients for C, s, t, u and v are shown in Table 6 and 7 below for thrust coefficient KT andKQ respectively.
Table 6: Coefficients for the KT polynomial representing the Wageningen B-Screw Series Propellers for a Reynolds number of 2 x 106.
Thrust Coefficient (KT)
n C s t u v
Table 7: Coefficients for the KQ polynomial representing the Wageningen B-Screw Series Propellers for a Reynolds number of 2 x 106.
Torque Coefficient (KQ)
n C s t u v
To make these results applicable for Reynolds Numbers greater than 2 x 106 a correction is applied, again as published in [5] and shown in table 8 and 9.
x
Table 8: Coefficients for the ∆KT polynomial for Reynolds number >2 x 106.
∆KT
Finally, the KT and KQ values for the propeller are the sum of the two components. It is specified the following conditions need to be adhered to before the above methods can be employed:
No. of blades 2 ≤ Z ≤ 7
Blade Area Ratio (BAR) 0.3 ≤ AE/AO ≤ 1.05 Pitch ratio (normally taken at 0.7R) 0.5 ≤ P/D ≤ 1.40
Table 9: Coefficients for the ∆KQ polynomial for Reynolds number >2 x 106.
∆KQ
6 -0.00059593 0 2 0 0 1
Given that the number of blades has been fixed at 4, the BAR and P/D ratios can be varied between these values. The diameter of the propeller is limited to between 1.5 metres and 1.95 metres; the specified propeller being 1.801 metres [6]. By
where T is the thrust produced (kN)
RT is the total resistance (bare hull + air+ appendages + added resistance in waves in sea state 4 where H1/3 = 2.5 m + additional frictional resistance due to fouling taken at 18%)
ρ is the water density (t/m3)
n is the rotation rate of the propeller (RPM in s-1) D is the propeller diameter (m)
KT is the thrust coefficient of the propeller
2 KQ
where J is the advance coefficient for the propeller
nD
J =VA and KQ is the torque
coefficient of the propeller. The advance velocity is VA=V(1-w) – with w being the wake
deduction fraction of the vessel. To widen the extent of analysis the data was filtered and reduced to only those developed thrust curves, which intersected the required thrust curves. The required thrust curves are directly calculated from the resistance of the vessel and its speed. The data was further filtered to those curves, which intersect within 1 knot of the specified service speed.
Having found the propeller characteristics we must firstly verify the results produced by the software, determine weather or not the thrust produced fits within the operational envelope of the specified engine and then apply cavitation theory to ensure there will be no ill effects due to this phenomenon. To verify the results the propeller characteristics found were used in the KT-KQ-J charts for the Wageningen B-screw series to find the thrust developed and open water efficiency. We can then proceed to find whether or not the developed thrust fits within the engine envelope at the required rotation rate. In doing so the following calculations, taken from Lewis (1988), were used, assuming a mechanical drive train for the following calculations:
Required brake power of the engine (kW)
M B
B η
P P
Required =
Brake power of engine accounting for gear box efficiency (kW)
GB
Power delivered to propeller (kW)
B A
D η
V P =T⋅
Efficiency of propeller in behind hull condition ηB=ηR⋅ηO
• ηM is the mechanical efficiency (taken as 0.97 [2])
• PB is the brake power
• η GB is the gear box efficiency taken as 0.98
• ηB is the efficiency behind the hull = ηO ηR
• ηO Open water efficiency of the propeller, obtained by selection of propeller
• PS is the shaft power
• ηS is the shaft efficiency (taken as 0.98 [2])
• PD is the delivered power to propeller
• ηR is the relative rotation efficiency (taken as 0.99 [4], Condition: 1.0 < FN∇ < 2.0 and the vessel’s FN∇ is approximately 1.55)
Finally to check for cavitation the following expressions are used in association with the Burrill’s Chart (1943),
where σ is the cavitation number
P0 – PV is the pressure at the shaft centre line kN/m2
P0 = PATM + ρgh, where g is acceleration due to gravity (m2/s) and h is the propeller immersion to shaft centreline.
PATM is taken as 98.1 kN/m2
where τ is the thrust-loading coefficient.
Using the values for τ and σ so calculated we can then observe any cavitation prediction from the chart as derived by Burrill (1943). The detailed results are shown in Appendix F. The Burril’s chart is shown below.
Figure 4: Burrill’s chart to determine cavitation.