3 Propeller geometry
3.5 Propeller outlines and area
The calculation of the blade width distribution is always made with reference to the cavitation criteria to which the propeller blade will be subjected. However, hav- ing once calculated the blade section widths based on these criteria it is necessary to fair them into a blade out- line. This can either be done by conventional draughting techniques or by the fitting of a suitable mathemati- cal expression. One such expression which gives good results is
c
D=K0(1−x)1/2+K1+K2(1−x)+K3(1−x)2 +K4(1−x)3+K5(1−x)4
where x is the non-dimensional radius and Kn, (n=0, 1,. . ., 5) are coefficients. There are four basic outlines in general use currently which describe the propeller blade shape:
1. the projected outline, 2. the developed outline, 3. the expanded outline, 4. the swept outline.
The projected outline is the view of the propeller blade that is actually seen when the propeller is viewed along the shaft centre line, that is normal to they–z- plane. Convention dictates that this is the view seen when looking forward. In this view the helical sections are defined in their appropriate pitch angles and the sec- tions are seen to lie along circular arcs whose centre is the shaft axis; Figure 3.11 shows this view together with the developed and expanded views. The projected area of the propeller is the area seen when looking forward along the shaft axis. From Figure 3.11 it is clear that the projected areaApis given by
Ap=Z R
rh
(θTE−θLE)rdr (3.10)
where the same sign convention applies forθas in the case of the skew angle andZis the number of blades.
Projected area is of little interest today. However, in the early years of propeller technology the projected area was used extensively on a thrust loading per unit projected area basis for determining the required blade
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40 Marine propellers and propulsion
Figure 3.11 Outline definition
area to avoid the harmful effects of cavitation. It will be noted that the projected area is the area in the plane normal to the thrust vector.
The developed outline is related to the projected out- line in so far as it is a helically based view, but the pitch of each section has been reduced to zero; that is the sec- tions all lie in the thwart-ship plane. This view is used to give an appreciation of the true form of the blade and the distribution of chord lengths. The developed and projected views are the most commonly seen represen- tations on propeller drawings; Figure 3.11 shows this view in relation to the projected outline.
To calculate the developed area it is necessary to integrate the area under the developed profile curve numerically if a precise value is required. For most pur- poses, however, it is sufficient to use the approximation for the developed areaADas being
ADAE
whereAEis the expanded area of the blade.
In the past several researchers have developed empir- ical relationships for the estimation of the developed area; one such relationship, proposed by Burrill for non-skewed forms, is
AD Ap
(1.067−0.229P/D) (3.11) In general, however, the developed area is greater than the projected area and slightly less than the expanded area.
The expanded outline is not really an outline in any true geometric sense at all. It could more correctly be termed a plotting of the chord lengths at their correct
radial stations about the directrix; no attempt in this outline is made to represent the helical nature of the blades and the pitch angle of each section is reduced to zero. This view is, however, useful in that it is sometimes used to give an idea of the blade section forms used, as these are frequently plotted on the chord lengths, as seen in Figure 3.11.
The expanded area is the most simple of the areas that can be calculated, and for this reason is the area most normally quoted, and is given by the relationship:
AE=Z R
rn
cdr (3.12)
In order to calculate this area it is sufficient for most pur- poses to use a Simpson’s procedure with 11 ordinates, as shown in Figure 3.12.
Blade area ratio is simply the blade area, either the projected, developed or expanded depending on the context, divided by the propeller disc areaAo:
Ap
Ao = 4Ap
πD2 AD
Ao = 4AD
πD2 AE
Ao = 4AD
πD2
⎫⎪
⎪⎪
⎪⎪
⎪⎬
⎪⎪
⎪⎪
⎪⎪
⎭
(3.13)
By way of example, the difference in the value of the projected, developed and expanded area ratio for the propeller shown in Figure 3.11 can be seen from Table 3.1. The propeller was assumed to have four blades and a constant pitch ratio for this example.
Propeller geometry 41
Figure 3.12 Evaluation of expanded area
Table 3.1 Example of comparative blade area ratios Projected Developed Expanded
area area area
Area ratio (A/Ao) 0.480 0.574 0.582
In each of the areas discussed so far the blade has been represented by a lamina of zero thickness. The true surface area of the blade will need to take account of the blade thickness and the surface profile on the suc- tion and pressure faces; which will be different in all cases except for the so-called ‘flat plate’ blades found in applications like controllable pitch transverse propul- sion units. To calculate the true surface area of one of the blade surfaces the algorithm of Figure 3.13 needs to be adopted.
This algorithm is based on a linear distance – that is between the successive points on the surface. This is suf- ficient for most calculation purposes, but higher-order methods can be used at the expense of a considerable increase in computational complexity.
The swept outline of a propeller is precisely what is conventionally meant by a swept outline in normal engineering terms. It is normally used only to repre- sent stern frame clearances. For the case of the highly skewed propeller a representation of the swept out-
line is important since the skew induced rake term, Figure 3.13 Algorithm for calculating surface area
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42 Marine propellers and propulsion
if not carefully controlled in design, can lead to con- siderable ‘overhang’ of the blade which, in turn, can lead to mechanical interference with the stern frame.
The swept outline is derived by plotting the rotation of each of the leading and trailing edge about the shaft axis.