Page:Aerial Flight - Volume 1 - Aerodynamics - Frederick Lanchester - 1906.djvu/330

a helical surface at right angles, equal to or greater than the peripteral area.

Now this is not an altogether clear proposition, for we are lacking definite information as to the distribution of the peripteral area, and it is evident that we might conceivably have overlapping in one place and clearance at other places. Moreover, the peripteral zone is not in reality a clearly defined region such as, as a matter of convenience, we have supposed. On the other hand, we only require an approximate solution; for even if we could gauge to a nicety the spacing of the blades required, we could not take advantage of our knowledge, for we are confined to whole numbers: we cannot employ fractional blades.

We will assume that if a propeller is designed so that no interference is to be anticipated at about the region of greatest efficiency, say 45 degrees, then no interference will take place at all. Furthermore, we will assume that the maximum thickness of the peripteral zone can be expressed in terms of the length of the blade according to the expression—

${\frac {{\frac {1+\epsilon }{1-\epsilon }}\ \kappa }{n}}.$

Referring to Table XIII., in which values are given as calculated from the plausible values of $\kappa$ and $\epsilon ,$ for ${\tfrac {1+\epsilon }{1-\epsilon }}\ \kappa$ and ${\frac {{\frac {1+\epsilon }{1-\epsilon }}\ \kappa }{n}}$ we may note that the latter varies from about unity for an aspect ratio $n=$ 3 to about ${\tfrac {3}{4}}$ for $n=$ 8. Taking the assumed angle of 45 degrees, and converting these into their circumferential equivalents, that is, multiplying by ${\sqrt {2}},$ we have 1.4 and 1.05. If we presume that the propeller is of the customary proportions, based on a 95 per cent, discard as to diameter, pitch, etc., the length of the blade (in the sense here employed) is approximately twice the radius at the point chosen, so that, expressing the circumferential spacing of the blades

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