Trans RINA, Vol 156, Part B2, Intl J Small Craft Tech, Jul-Dec 2014
MATHEMATICAL MODELING OF PROPELLER SERIES (DOI No: 10.3940/rina.ijsct.2014.b2.148)
M Bukarica, University of Belgrade, Serbia SUMMARY
Propeller series are used in the process of designing the ship propulsion system, propeller and engine. The emergence of computers has made it possible to create a program, an expert system, which greatly accelerates and improves this process. In order to create such a program, it is necessary to provide a mathematical model of the test results, i.e. describe them by some mathematical expressions. In addition to this mathematical model, such a program must also have a mathematical model of the ship resistance. Precision and reliability of results produced by the expert system for the selection of the ship propulsion system depends on precision of the mathematical models it uses.
Usual modeling procedures use standard mathematical functions and statistical methods. The aim of this paper is to present a "natural method" of mathematical modeling using spline functions, to compare them with each other, and to
point out their advantages and deficiencies. NOMENCLATURE J
Advance coefficient
Cavitation number Density of water (kg m-3) D
A0 Ae
Ae /A0 Expanded area ratio Z P
P/D n
va p0 T
KT Propeller efficiency
Expanded area of propeller blades (m2) Number of propeller blades
Diameter of propeller (m) Disk area of propeller (m2)
Pitch of propeller (m) Pitch ratio
Speed of advance of propeller (m s-1) Propeller rotational speed (s-1)
Static water pressure at propeller axis (N m-2) Thrust of propeller (N) Thrust coefficient
Q Torque of shaft propeller (Nm) KQ pv v
Vapor pressure of water (N m-2) Velocity of ship (m s-1)
Torque coefficient 1. INTRODUCTION
Fundamental empirical (experimental) tests in shipbuilding practice are tests of model ships and model propellers. Model ships are tested in the towing tank while model propellers are tested in the towing tank and the cavitation tunnel. In tests of ship models, the resistance of models is measured when they are towed at different speeds. In the tests of propeller models, torque and thrust are measured
at different water inflow
velocities and different rates of rotation of the model shaft. Water velocity and the rate of the shaft rotation are united in a new quantity named the advance coefficient. The ship model resistance depends on the towing speed. The torque and thrust of the model propeller depend on the advance coefficient.
These tests are repetitive, i.e. when they are executed again under the same conditions, the same results are
©2014: The Royal Institution of Naval Architects
obtained. This is because the nature of both dependences is deterministic, meaning that one quantity depends on another quantity according to certain law. The goal of testing is to determine this law. In reality, repeated testing in the tank and the cavitation tunnel will not give exactly the same results, but will range within small differences. This is because the measuring instruments have their own degree of accuracy, and because the flow around the model occurs in the turbulent boundary layer that cannot be repeated.
The term accuracy is generally used to indicate the
closeness of the agreement between the experimentally determined value of a quantity and its true value. The error is the difference between the experimentally determined value and the true value. The accuracy is said to increase as the error approaches zero. Only in rare instances is the true value of a quantity known. Unfortunately, in our case, the true values of resistance, thrust and torque are unknown. That means we cannot calculate the error precisely and we are
forced to estimate it. This estimate is
uncertainty, U. The ITTC has recommended a procedure of doing the tests and how to determine the uncertainty from the given results of measurement. In this methodology, all estimates are
confidence level, meaning that the true value of the quantity is expected to be within the U interval, about the experimentally determined value, 95 times out of 100.
We can generalize this case and say that the purpose of the tests is to experimentally determine how a quantity Y depends on another quantity X. Since we have found that this dependence is deterministic, quantity Y is named the dependent variable quantity and quantity X is named the independent variable quantity. The resistance of the ship model, the torque and the thrust of the propeller model are dependent variable quantities, and the speed of the ship model and the advance coefficient are independent variable quantities.
During an empirical test of a model, values of both dependent and independent quantities are measured at the
B-43 assumed to be made at a 95-percent
called an
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