Aerodynamics (Lanchester)/Chapter 9

by Frederick William Lanchester

Chapter IX.

 

On Propulsion, the Screw Propeller, and the Power Expended in Flight.

 

§ 197. Introductory.—The employment of the Newtonian method (§ 2) in the theory of propulsion has been already mentioned (§ 8). The application of this method, which constitutes the foundation of modern theory, owes its development principally to the work of Rankine and W. Froude.

In the general theory of propulsion we are not concerned with the machinery of propulsion, i.e., the form of propeller—paddle, screw, jet, or other known or unknown mechanism; we merely take account of the fact that forces are exerted between the propelled body and certain parts of the fluid, and investigate the conditions that obtain and the proportion of power that may be utilised and lost. The theory of propulsion on this broad basis is the common foundation of propeller theory generally, and the conditions deduced from the Newtonian principle are essential to every form of propeller. It is convenient, in the initial consideration of the problem, to introduce the notion of action at a distance, and to suppose the propulsive forces to consist of repulsions (or attractions), acting in the direction of motion between the propelled body and the particles of the fluid.

 

§ 198. The Newtonian Method as applied by Rankine and Froude.—It is supposed in the first instance that the fluid on which the propeller operates is at rest at the time the propulsive forces commence to act; this condition is intended to exclude any possible disturbances that may be set up in the fluid by the body in motion.

Let	F	be the sum of the propulsive forces.
„	m	„  mass of fluid handled per second.
„	$V$	„  velocity of vessel, the term vessel being used to denote the body propelled.
„	v	„  a uniform sternward velocity imparted to the fluid operated upon.

All in Absolute Units.

Then we have (§ 3) $\mathrm {F} =\mathrm {m} \ \mathrm {v} ,$ and the work done usefully per second is

and the energy left in the fluid per second, that is, lost power, is

or total energy per second

$=\mathrm {m} \ \mathrm {v} \ V+{\frac {\mathrm {m} \ \mathrm {v} ^{2}}{2}}$

or efficiency

This, according to Rankine,^[1] is the theoretical limit to the efficiency of a propeller. It will be shown subsequently that this assertion requires qualification.

If we depart from the simplicity of the assumption and suppose that the different portions of the fluid acted upon receive different velocities, the foregoing demonstration requires appropriate modification; the v of expression (1) and the v of expression (2) are not the same quantity, the v² in the latter expression becoming the mean square of the velocity v instead of the square of the mean. For a given value of m the efficiency must thus be less than if the velocity v were uniform over the mass, for the mean square is always greater than the square of the mean.

The full expression where the velocity is variable throughout the mass ${\mbox{m}}$ is,

${\frac {V}{V+{\frac {\mbox{v}}{2}}+{\frac {\Sigma \ {\mbox{m}}\ (x^{2})}{2\ {\mbox{m}}\ {\mbox{v}}}}}},$

(4)

where $x$ represents the velocity communicated to the particles of fluid in excess or in deficit of the mean, $x$ being accordingly plus or minus. It will be noted that since $x^{2}$ must be always positive the quantity ${\tfrac {\Sigma \ {\mbox{m}}\ (x^{2})}{2\ {\mbox{m}}\ {\mbox{v}}}}$ will be always positive, so that the efficiency will be less than if the mass were handled uniformly. The above expression is of but slight utility from a quantitative standpoint; it is given here as being conducive to exact thought and as being the more complete form of expression (3).

§ 199. Propulsion in its Relation to the Body Propelled.—In the preceding section the subject of propulsion has been treated in the abstract; it has been assumed that the body propelled is far away so that the fluid is unaffected by its presence, and that the fluid as a whole receives momentum.

Now we know from the Principle of No Momentum (§ 6) that, as a whole, the fluid does not receive momentum, and that if it receive momentum in one direction in one part it simultaneously receives equal and opposite momentum in some other part. The result of this is two-fold: (α) we know that the whole of the energy expended in the fluid does not appear as sternward motion, as assumed by Rankine; and (b), the problem becomes complicated by the reaction and motions produced in the fluid by the vessel itself as affecting the conditions under which the propeller is working.

For reasons stated in § 8 it is doubtful whether, under the conditions that ordinarily obtain, the error that arises from (α), neglecting the counter-current, is sensible; if, however, the proportion ${\mbox{v}}/V$ were to become considerable, the departure from theory would become serious. The validity of the present application of the Newtonian method depends definitely upon the fact that ${\mbox{v}}/V$ is small.

Considering next (b) the problem of propulsion in its full relation to the body propelled: if a vessel be towed through a fluid, the pull on the tow line being applied from without, the whole energy is, in the sense of § 198, usefully employed, and the condition of affairs is that tacitly taken in § 198 to represent unit efficiency.

The energy expended passes into the fluid and is swallowed up partly in overcoming viscous stress and partly in setting the fluid in motion; the first part vanishes at once into the thermodynamic system and is lost; the second part remains in the fluid as kinetic energy until in turn it is spent in overcoming viscosity, when by degrees it also vanishes.

Now the energy that is left in the kinetic form takes some time to disappear, and in the meantime it constitutes a wake current with a corresponding counterwake whose momenta are equal and opposite, the wake current being situated in the immediate track of the vessel, and the counterwake further afield and extending theoretically to the confines of the fluid region, but only sensible for a limited distance. This kinetic energy is not irretrievably lost; after the fluid has passed in effect out of dynamic connection with the vessel, there is ample time for the recovery of the energy if a suitable method could be devised, and then, by returning it to the source of power, it is evident that the vessel will be propelled with a less expenditure of power than previously, and the “efficiency” will become greater than that somewhat arbitrarily chosen as unity.

§ 200. A Hypothetical Study in Propulsion.—Let us consider the case in which the propeller is constrained to act on the wake current (or, as it is sometimes termed, “the frictional wake”), and in the first instance let us suppose that this current consists of a quantity of fluid moving en masse with a velocity $={\mbox{v}}_{1}.$ Now the force of propulsion is essentially equal, and of opposite sign, to the resistance experienced by the vessel, action and reaction being equal and opposite; consequently, on the Newtonian basis, the rearward momentum communicated by the propeller will be equal to the forward momentum communicated by the vessel, so that the conditions of propulsion will be satisfied if the propeller impart to the wake current a rearward velocity ${\mbox{v}}$ equal to ${\mbox{v}}_{1};$ that is to say, the fluid will be brought to rest.

Let us now re-calculate the efficiency as in § 198: we have work done usefully per second $={\mbox{F}}\ V={\mbox{m}}\ {\mbox{v}}\ V;$ and energy taken out of the fluid, that is, energy received per second, is $={\frac {{\mbox{m}}\ {\mbox{v}}_{1}^{2}}{2}};$ or, total energy expended per second $={\mbox{m}}\ {\mbox{v}}_{1}\ V-{\frac {{\mbox{m}}\ {\mbox{v}}_{1}^{2}}{2}}$, or,

This is greater than unity; the result being, as anticipated, that it is theoretically possible that a vessel should be propelled for a less expenditure of power than that by which it can be towed.

This important result, although not generally known,^[2] is not new; it was previously pointed out by Mr. W. Froude in the discussion on a paper by Sir F. C. Knowles (Proc. Inst. C. E. 1871). Froude evidently had also treated the matter quantitatively, since he mentions the theoretical possibility of a negative slip of a screw propeller, from the cause stated, equal to half the positive slip as ordinarily computed.

In the present hypothetical case the influence of the counterwake has not been taken into account, it forming no part of the Newtonian scheme; the conditions are too artificial for the omission to be a matter of any importance, apart from the fact already pointed out that the consequences of such an omission will not be serious.

It has been assumed that the whole resistance of the vessel is due to its skin friction. In marine propulsion “wave making” plays a prominent part; the resistance from this cause may be regarded as a force applied from without, since the waves travel away, taking their momentum with them; the consideration of wave making resistance would destroy the precise balance between ${\mbox{v}}$ and ${\mbox{v}}_{1},$ on which expression (5) is based; the matter of wave making has, however, no interest to us from an aerodynamic standpoint.

§ 201. Propulsion under Actual Conditions.—Under actual conditions neither of the hypotheses discussed in §§ 198 and 200 applies in its entirety. The requirements of the former hypothesis are most nearly met in the case of a paddle boat (with the paddles at the sides); the latter case is best exemplified in the stern-wheeler, a flat-bottomed type of craft whose propeller is particularly well placed for capturing the frictional wake. In the forms that succeed in practice the propeller is usually behind in the frictional wake, never in front; and the successful forms of propulsion are those in which a sufficient mass of water per second can be conveniently handled; thus, jet propulsion has become practically extinct. We are, therefore, led to appreciate the soundness of the Newtonian method.

There are many methods of mechanical propulsion, that is to say, there are several known mechanical devices for producing the reaction on the fluid which we have so far regarded as being accomplished by action at a distance. Firstly, we have the numerous devices employed by nature in the locomotive mechanism of birds, fishes, etc.; secondly, we have the primitive devices employed by man—the paddle, the oar, etc.; and finally, we have the two great inventions of marine engineering, the paddle wheel and the screw propeller. Of these various types of propeller, only two will be discussed as of interest in connection with the subject of the present work, the screw propeller and wing propulsion; the former alone being deemed suitable for treatment in the present volume, the latter being reserved for the section on “Avian Flight” which will form part of Vol. II.

§ 202. The Screw Propeller.—We will presume a general knowledge of the screw propeller, and proceed at once to the attack.

The theory of the screw propeller will be discussed on the basis of the peripteral theory of the foregoing chapters;
Fig. 124. this constitutes a new method which sheds considerable light on a hitherto somewhat obscure subject.

We shall in the following demonstration take the helical surface of uniform pitch as strictly the analogue of a plane in the foregoing theory, and we shall presume that the various propositions already proved in the case of the aerofoil apply mutatis mutandis to the helical equivalent. Thus the blade of a propeller becomes an aerofoil of a form suitable to glide in a helical path, the reaction on the blade (whose resolution is the torque and thrust) is the analogue of the weight, the helical surface at right angles to the blade reaction is the analogue of the horizontal plane, and concentric cylindrical surfaces represent vertical planes in the axis of flight.

We will begin by an examination of an element of a propeller blade represented by its section on one of the aforesaid cylindrical surfaces, of which we will suppose the development is given in Fig. 124. Now, on this development a helical surface will appear as an inclined straight line; let $O\ a$ represent the helical surface which we regard as the analogue of the horizontal, and ${\mbox{W}}$ (the analogue of $W$) the force at right angles thereto. Let $O\ b$ be the helical flight path, and $\gamma$ the gliding angle; then $\theta$, the angle cut off between $O\ b$ and the axis of $x$, will be the effective pitch angle; that is to say, the line $O\ b$ represents the helix along which the blade of the propeller will actually travel, and its pitch will be the effective pitch of the propeller.

Draw the line ${\mbox{F}}$ parallel to the axis of $y$ to represent the component of ${\mbox{W}}$ in the direction of motion of the vessel, cut off $O\ a={\mbox{W}},$ and draw $a\ c$ perpendicular to $y,$ draw $a\ b$ perpendicular to $O\ a,$ and $b\ d$ perpendicular to $O\ x.$

Then $O\ a$ being equal to ${\mbox{W}}$, we have $a\ c$ equal to ${\mbox{F}},$ the two triangles being equal in every respect. Let us denote $a\ b=f,$ and $b\ d=h.$

Now while the blade moves from to h the energy lost will be $={\mbox{W}}\ f;$ that is to say, we regard the matter as a case of gliding, to which it is strictly analogous. The energy utilised in propulsion during the same period will be $={\mbox{F}}\ h.$ (These quantities are indicated by the shaded areas in the figure.)

Now it follows from the construction that—

${\mbox{F}}={\mbox{W}}\cos(\theta +\gamma ),$

$f={\mbox{W}}\tan \gamma ,$

$h={\mbox{W}}\ {\frac {\sin \theta }{\cos \gamma }}$

 

§ 203. Conditions of Maximum Efficiency.—The conditions of maximum efficiency are attained for the element of the propeller under consideration, when (1) is solved for minimum value.

Firstly, we may note from Equation (1) (and it is otherwise self-evident) that $\gamma$ should be made as small as possible; that is to say, the propeller blade should be designed as an aerofoil for minimum gliding; we shall therefore take $\gamma$ from this point to denote the minimum gliding angle as independently ascertained. If the conditions of least gliding angle are infringed, the present theory continues to apply, but the result will be the best efficiency possible under the adverse conditions imposed, and not the real maximum.

Now $\gamma ,$ and therefore $\cot \gamma ,$ is constant in our expression; consequently we have to solve—

$\cot \gamma \ \sin \theta \ \cos \theta -\sin ^{2}\theta$

for maximum value where $\theta$ is the only variable. Differentiating in respect of $\theta$ we get—

$\cot \gamma \ \cos 2\ \theta -2\ \sin \theta \ \cos \theta =0$

transforming we get—

$\cot 2\ \theta =\tan \gamma ,$

which we may express in another way and say: The angle of greatest efficiency is 45 degrees, minus half the least gliding angle.^[3]

Thus, if we take the gliding angle in the case of air to be 10 degrees for any particular aspect proportions, the angle of greatest efficiency will be 40 degrees; or, taking the probable equivalent for water as 6 degrees, the appropriate angle becomes 42 degrees. The figures cited are probable figures for blades of about 4 : 1 ratio, as founded on experiment; it is known that the tabulated theoretical figures of § 181 are too low, from causes already discussed.

§ 204. Efficiency of the Screw Propeller, General Solution.—From Equation (1), § 202, we obtain

Efficiency $=1-{\frac {1}{\cot \gamma \ \cos \theta \ \sin \theta -\sin ^{2}\theta +1}}$

From which by transformation,

Efficiency $={\frac {\tan \theta }{\tan(\theta +\gamma )}}.$

This expression may be deduced directly from the conditions. Let Fig. 125 represent, by the lines $O\ a$ and $O\ b,$ the helices of horizontal and gliding path respectively; then, since the reaction ${\mbox{W}}$ is normal to the path $O\ a,$ the work done when the propeller is rotated through an angle represented by the line $O\ d$ will be ${\mbox{F}}\times b\ d;$
Fig. 125. but the work utilised is represented by ${\mbox{F}}\times b\ d;$ the efficiency is therefore ${\frac {b\ d}{a\ d}},$ that is ${\frac {\tan \theta }{\tan(\theta +\gamma )}}.$

The result is thus obtained in a more direct manner, all trigonometrical transformations being dispensed with; the original demonstration is, however, of a more exact nature and is based on a clearer conception of the conditions involved. The identity of the two methods may be demonstrated geometrically by showing that the shaded areas of Fig. 124 are proportional to the ordinates $b\ d$ and $b\ a$ of Fig. 125, a matter of perfect simplicity.

The present theory enables us to define the slip of the propeller as the difference between the ordinates $b\ d$ and $a\ d,$ the slip ratio being represented by ${\frac {a\ b}{a\ d}}.$ The term slip as here defined is not identical with the slip of the naval architect, which is derived from the mean pitch angle of the blades, a basis that can have no justification in theory. The conception of slip originated with the propeller of true helical form and then denoted the difference between the geometrical and effective pitch; when blades were given an increasing pitch the mean pitch angle was taken as the basis of calculation of the geometrical pitch; hence the present usage. The term slip, in its application to a screw propeller, is one that leads to confusion of thought; it is unscientific in its present usage, and would be better abolished.

Let us estimate the possible efficiency in the case of the maximum conditions of the preceding section. Employing equation, $E\ ({\mbox{efficiency}})={\frac {\tan \theta }{\tan(\theta +\gamma )}}$^[4] we have—

Air	$(\theta =40^{\circ })$		$E=70.4$ per cent.
Water	$(\theta =42^{\circ })$		$E=81.1$ per cent.

Now these efficiencies are only obtainable at the particular section of the blade where the angle $\theta$ is correct for maximum efficiency, and at all other points the efficiency must be less. A section of the blade here discussed is the section on a cylindrical surface which may be fully defined by its radius $r.$

In Figs. 126 and 127 we have plotted the calculated efficiency for different values of $r$ on the basis of the $\gamma$ values assumed for water and air. Abscissae represent radius in terms of pitch, the ordinates give corresponding efficiency values. The abscissae are also figured for values of $\theta .$ The efficiency falls to zero when $\theta =0,$ and again when $\theta +\gamma =90$ degrees, for in the first place $\tan \theta =0,$ and in the second $\tan(\theta +\gamma )$ becomes infinite.

§ 205. The Propeller Blade Considered as the Sum of its Elements.—Much of the faulty work of early writers on hydrodynamic problems has been due to the treatment of a body or surface as the equivalent of the sum of elements into which it may be arbitrarily divided, and this form of error is one against which it is important to be on guard.

We have already adopted in substance the principle of regarding an aerofoil as the sum of its sectional elements in the sense now contemplated in respect to the propeller blade (§ 192), but we do not suppose, in assessing the individual elements, that they are removed from their environment.^[5]

Let us examine our former procedure. We have an aerofoil whose aspect ratio is of considerable magnitude, and whose grading is specified, and we prove that the reaction due to each increment of length is proportional to the grading ordinate proper to that increment, irrespectively of the fore-and-aft dimension. The proof involves the tacit assumption that for the smooth curve form of grading specified, a geometrical similarity of section at all points involves a uniform pressure distribution.

Now, so long as the aspect ratio is great and the grading a smooth curve, there can be little question as to the propriety of this assumption; if, however, the aspect ratio be small, or the changes in the grading curve sudden, then the grading curve and the relative reaction curve ma}^ cease to coincide. Our assumption may therefore be considered as an approximation—a close approximation when the value of $n$ is great (say upwards of 4 or 5), and a rough approximation when the aspect ratio is small.

§ 206. Efficiency Computed over the Whole Blade.—On the basis of the efficiency curve (Figs. 126 and 127) and a knowledge of the radial distribution of the thrust reaction (the F of § 202), the computation of the efficiency for the whole blade is merely a matter of integration.

We have first to settle how much of the efficiency curve we propose to employ i.e., the radial limits of the blade length. If we make the blades too long, the efficiency at the extremities would be so low as to involve an extravagant expenditure of power; if, on the other hand, we confine the length of the blade to the region where the efficiency is about its maximum, in order to reap the benefit of the full value (as given in § 204), we encounter practical disadvantages in the increased propeller diameter required to deal with a given quantity of fluid (the proportion of the “disc” area utilised becoming small), and in the length (and consequent resistance) of the arm necessary to attach the blade to its boss.

In Figs. 126 and 127 we have taken the blade length,

indicated by the horizontal line, as confined to a zone in which the inferior limit of the efficiency is 90 per cent, of its maximum. It may be noted that this procedure gives for water a pitch almost exactly equal to the diameter; this is somewhat less than is customary, the generally accepted proportion being, pitch = 1${\tfrac {1}{4}}$ diameter (approximately). It would appear that designers unconsciously reject the whole of the curve where of less value than about 95 per cent, of the maximum possible. In spite of this fact, the efficiencies so far recorded leave much to be desired, and, apart from practical limitations, would appear to show that there is still considerable room for improvement. The defects in existing practice would seem, according to the present theory, to be found in the want of attention to the requirements of pterygoid section, in the low value of $n$ (aspect ratio) commonly adopted (possibly from practical requirements not included in the present hypothesis), and to undue fulness of plan-form towards the blade tips,^[6] and consequent excessive frictional loss.

The value of the total efficiency, having selected the blade limits, depends not only upon the efficiency curve, but also on the distribution of the thrust over the length of the blade, or the thrust grading, as it may be conveniently termed. If the range of efficiency were very great, we should have to specify the thrust grading before the total efficiency could be computed; but as the variation does not usually exceed 10 per cent, or so, and as the general character of the grading curve cannot be in doubt, we can arrive immediately at a close approximation.

On the 90 per cent, basis, if the thrust grading were uniform over the length of the blade, the mean efficiency would, for the character of the curve given (Figs. 126, 127), be 96 or 97 per cent, of its maximum. If, as must be the case, the thrust grading fall to zero at the extremities, the efficiency will be increased; hence we may take it as probably about 97${\tfrac {1}{2}}$ per cent, of its maximum. On the basis of § 204 this gives

For air	.  .  .  .  .	68.6 per cent.
For water	.  .  .  .  .	79.0 per cent.

We have no data of comparison in the case of air; in the case of water, so far as the author is informed, the highest actual efficiency recorded is somewhat over 70 per cent.

§ 207. Pressure Distribution.—It is evident that, according to the present theory, the propeller blade is amenable to precisely the same laws so far as its pressure-velocity relation is concerned as the ordinary aerofoil, and we presumably also have the two alternative types of fluid motion, the continuous and the discontinuous, according as the blade is given a pterygoid form (based on a helix) or whether a simple helical surface or sheet (the analogue of an aeroplane) is employed. We may read the appropriate pressure for air from either Table X. or XII., as the case may be.

A complication is introduced in the propeller blade by the fact that its different portions are moving at different velocities through the fluid, so that the pressures proper to least $\gamma$ vary at the different points along the length of the blade. This velocity, the $V$ of the propeller blade, will be given by the expression $V={\frac {\mbox{V}}{\sin \theta }},$ where ${\mbox{V}}$ is the velocity of the vessel, or, in terms of $r,$ we have

$V={\frac {\sqrt {(2\ \pi \ r)^{2}+p^{2}}}{p}}\ {\mbox{V}},$

where $p$ is the pitch, or

$V^{2}=(4\ \pi ^{2}\ r^{2}+p^{2})\ {\frac {{\mbox{V}}^{2}}{p^{2}}},$

and since $p$ and ${\mbox{V}}$ are constants the curve is of the form $y=x^{2}+$ const., when plotted (Fig. 128), where abscissae give radius in terms of pitch and ordinates give $V^{2}$ values. Now by § 185, for any aerofoil $P_{2}=V^{2}\times$ const., values of the constant being given in Table IX. Hence the curve (Fig. 128) may be taken to give the correct pressure value for all points along the blade, the pressure scale being determined by assigning a value to some convenient point from the Table.
Fig. 128.

§ 208. “Load Grading.”—Reference has already been made to the term thrust grading as representing the distribution of the sternward reaction along the length of the blade.^[7] We have now to discuss the considerations governing the form of the thrust grading curve, and also the curve of distribution of the normal reaction from point to point, which we may term the load grading.

Dividing the propeller disc into a number of concentric areas, we have, on the principle discussed in § 198, to distribute the momentum as nearly as may be possible in proportion to the mass of fluid passing through each annular element: that is, the force exerted by each small linear element of the blade should be proportional to the area it sweeps; that is to say, it will be proportional to $r.$

Our thrust grading curve on this basis would be that shown in Fig. 129 (α)—simply an inclined line. But we must complete this curve at the blade extremities. This has been done arbitrarily by a pair of ordinates, the thrust grading curve thus completed being the contour of the shaded area, the area itself representing the total thrust. But this “curve” infringes a condition already laid down, that the grading must constitute a smooth curve with no sudden changes of ordinate. Hence we must compromise, and we find that the combined conditions indicate a form such as that illustrated by Fig. 129 (b).
Fig. 129.

Now the reaction normal to the blade will at every point be equal to the sternward component multiplied by sec $(\theta +\gamma ),$ that is to say, if we multiply the ordinates of the thrust grading curve at every point by sec $(\theta +\gamma )$ we shall have the load grading curve Fig. 130 (compare Fig. 136), which represents the distribution of the pressure normal reaction along the length of the blade.

§ 209. Linear Grading and Blade Plan Form.—The linear grading for any radius is the quotient when the load value is divided by the pressure value for that radius ; thus the linear grading curve may be plotted from the other two, the ordinates being calculated by simple division (Fig. 130).

This linear grading is analogous to the aerofoil grading of § 192, and likewise represents the ordinates of the blade plan, i.e., the width of the blade from point to point for constant form of section; that is to say, all sections become geometrically similar.

Whether or not the similarity of sectional form is essential, as it would theoretically appear to be for best economy, must be regarded to a certain extent as an open question. The same query has arisen in the case of the aerofoil, but the objections in that case are partly concerned with aerodromic and other considerations.
Fig. 130. Should future experience show that flattening of the section (compare § 191) and diminution of pressure towards the extremities is advantageous from the aerodynamic point of view, the whole matter will require to be thoroughly reinvestigated before we can regard the theory of peripteral motion as complete.

§ 210. The Peripteral Zone.—Before we are in a position to discuss the conditions that regulate the number of blades permissible in a propeller we require to somewhat extend our knowledge of the dynamics of the periptery.

At first sight it might be supposed that the blades of a propeller in their helical paths are related in a similar manner to a number of superposed aeroplanes, and the law of maximum proximity will be the same in both cases. Such is not the case. Where we have to deal with a battery of superposed planes or aerofoils, whether vertically over one another, as in Fig. 131 (α), or (in order to better simulate the conditions) like a flight of steps (Fig. 131 (b)),
Fig. 131. the supporting reaction is continually derived from the virgin fluid, and the line of pressure reaction of any plane, or any component of it, only cuts the path of that plane once. In the case of the propeller, on the contrary, the component of the pressure reaction of any blade in the line of motion cuts the paths of that blade an in- definite number of times. We have here to deal with a fact new to our theory.

Let us suppose that we substitute for our propeller blade some device that acts directly on the fluid without involving the complexity of the cyclic or peripteral motion, and let us stipulate that this hypothetical device produce the same total reaction with the same expenditure of energy as the original aerofoil or propeller blade.

On the Newtonian basis we know (§ 3) that if $W$ be the total reaction, $m$ the mass per second of the fluid dealt with, and $v$ be the velocity imparted in the direction of the reaction—

$W=m\ v$  or  $v={\frac {W}{m}}.$

But the energy per second $={\frac {m\ v^{2}}{2}}$

Now, energy per second for pterygoid aerofoil handling mass per second $=m_{1}$ is

${\frac {m_{1}\ V^{2}\ (\beta ^{2}-\alpha ^{2})}{2}},$

and energy per second on the simple Newtonian basis for the same mass handled per second would be

${\frac {m_{1}\ V^{2}\ (\beta ^{2}+\alpha ^{2})}{2}},$

or, by (1), if $m_{2}$ is the mass per second dealt with by our hypothetical device, we shall have

${\frac {m_{2}}{m_{1}}}={\frac {m_{1}\ V^{2}\ (\beta +\alpha )^{2}}{m_{1}\ V^{2}\ (\beta ^{2}-\alpha ^{2})}}$ $={\frac {(\beta +\alpha )\ (\beta +\alpha )}{(\beta +\alpha )\ (\beta -\alpha )}}$

That is to say, the sectional area of the fluid stratum which would be acted upon will be—

${\frac {1+\epsilon }{1-\epsilon }}\ \kappa \ A$

Now, we may evidently regard the aerofoil, with its accompanying peripteral system, as the equivalent of the hypothetical device which we have temporarily assumed. The peripteral system actually constitutes a kind of tool or appliance by which the aerofoil is able to deal in effect with more air than actually comes within its sweep. This extended “sphere of influence” of the aerofoil will be termed the peripteral zone, and its cross-section, ${\frac {1+\epsilon }{1-\epsilon }}\ \kappa \ A$ is the peripteral area.

§ 211. The Screw Propeller: Number of Blades.—The number of blades in a propeller must be determined by the conditions of their non-interference. It is evident that if the peripteral areas of adjacent blades overlap, the total amount of fluid operated upon will be insufficient and the efficiency must diminish. We must therefore secure that the hehces on which the different blades are based are separated in effect by an area, measured on 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 in terms of the radius, we shall have for $n=$ 3, radius $\times$ 2.8, which, multiplied by ${\tfrac {180}{\pi }},$ gives 160 degrees apart, or two blades nearest whole number, and for $n=$ 8 we get 2.1, which gives 120 degrees, or three blades almost exactly.

In view of experience, it is evident that these results are lower than necessary; it is found advantageous to employ more blades than here stated. This discrepancy is perfectly explicable, for the assumptions we have made all tend towards a minimum value. The desirability of keeping the propeller circle as small as possible is probably responsible for the employment of a fourth blade in the marine propeller. Four blades doubtless give rise to some slight interference and loss of power, but not sufficient to be seriously detrimental.

If we take three blades as a standard for the marine propeller where $n=$ 3, the corresponding value when $n=$ 8 (a probably impracticable proportion) would be four blades almost exactly. Carrying the matter further on the same basis, if we design an aerial propeller, discarding below 95 per cent, of maximum, the blade length will be approximately 1.2 times the radius (at 45 degrees), so that the proportional number will be 5 blades for $n=$ 3, and 6${\tfrac {1}{2}}$ blades for $n=$ 8; that is to say, six blades can be carried. If we lower the discard point to 90 per cent, the conditions as to number of blades will become approximately the same as for the marine propeller discarding from the higher percentage; extending the comparison, it would be very difficult to distinguish the one propeller from the other, both being fashioned in accordance with the present theory for the same $n$ value.

It is probable that, owing to the much lighter pressures required, it will be practicable to employ greater values of $n$ in the aerial propeller than in the marine propeller: an aspect ratio of 6 or 8 does not appear to present any difficulty. This being the case, it is highly probable that the aerial propeller in practice may become almost as economical as, if not more economical than, its marine prototype. The whole question depends upon whether the gliding angle for the proportions of blade employed in the two types is in favour of the one or the other.

If the aspect ratio employed is as high as that here suggested, and if it is found advantageous to discard from as high a point as 95 per cent., it may with some confidence be predicted that it will be found advantageous to employ as many as six blades.

Table XIII.

$n.$	$\kappa \;{\tfrac {1+\epsilon }{1-\epsilon }}$	${\frac {\kappa \;{\tfrac {1+\epsilon }{1-\epsilon }}}{n}}$
3 4 5 6 7 8	2.85 3.45 4.13 4.69 5.29 5.98	.95 .86 .82 .78 .75 .748

§ 212. Blade Length. Conjugate Limits.—In §§ 206 and 211 the limits of the blade length have been assumed as determined by the rejection of those portions of the efficiency curve falling below some stated percentage of the maximum, but it has not been demonstrated that this course results in the highest efficiency over the whole blade.

It might, with some show of plausibility, be argued that since the outer portion of the blade has more work to perform, the region of maximum efficiency should be nearer the point of the blade than given by the method suggested, and that therefore the diameter in terms of the pitch should be less than that deduced. Let us investigate the point in question.

We will suppose that, with a certain stated pitch and therefore an efficiency curve of defined scale, we have (in accordance with the Newtonian principle) to operate on a given cross-sectional area of fluid. Then this area is represented by an annulus whose inner and outer boundaries are of the radii of the blade limits.

Let us, firstly, assume the straight line thrust grading curve of Fig. 129, so that the whole of the fluid within the annular area will be uniformly accelerated;
Fig. 132. and let us regard the thrust grading curve as representing the useful work of propulsion over one unit distance, and let the curve $a\ a\ a\ a$ (Fig. 132) represent the work expended in the same time. Let $E$ represent the efficiency as a variable, i.e., the ordinate of the efficiency curve, and let $w$ represent the useful work per unit length of the blade, i.e., the thrust grading ordinate (Fig. 132); then the ordinate $y$ of the curve $a\ a\ a\ a$ will be $y={\frac {w}{E}}.$

Now, if we suppose the limits of the blade length be moved from place to place, so that, however, the annular disc area of the propeller remains contant (which is our fundamental condition), it is a matter of simple geometry to show that the area of the thrust grading curve (of which $w$ is the ordinate) must remain constant; consequently, if we suppose any small variation to take place, and represent the various quantities as given on Fig. 132, we have, for the condition of minimum expenditure of energy—

$\Delta \ r_{1}\times y_{1}=\Delta \ r_{2}\times y_{2},$

or

${\frac {\Delta \ r_{1}}{\Delta \ r_{2}}}={\frac {y_{1}}{y_{2}}}={\frac {w_{2}}{E_{2}}}{\mbox{÷}}{\frac {w_{1}}{E_{1}}}={\frac {w_{2}\ E_{1}}{w_{1}\ E_{2}}}.$

But, since the area of the load grading curve is constant, we have

$\Delta \ r_{1}\ w_{1}=\Delta \ r_{2}\ w_{2}.$

that is

${\frac {\Delta \ r_{1}}{\Delta \ r_{2}}}={\frac {w_{2}}{w_{1}}},$

so that we have—

${\frac {w_{2}\ E_{1}}{w_{1}\ E_{2}}}={\frac {w_{2}}{w_{1}}},$  or  ${\frac {E_{1}}{E_{2}}}=1,$

or

$E_{1}=E_{2},$

proving that for the conditions of greatest economy, the blade limits are points of equal efficiency.

Hence, although different proportions may be chosen for the ratio $r_{1}/r_{2}$ (the inner and outer radial limits of the blades), there are appropriate conjugate values which are conducive to the maximum efficiency, and these values are determined by the points of equal efficiency on a curve plotted from the equation of efficiency (§ 204, Figs. 126, 127).

§ 213. The Thrust Grading Curve.—We know that the square-ended form of grading curve assumed in the preceding section is inadmissible, and in order that the principle of conjugate blade limits should apply strictly to a real propeller blade we must extend the demonstration to include other forms.

Let us suppose that we regard the thrust grading as made up of a number of small component distributions of load, each of which is strictly in accordance with the hypothesis (Fig. 133). Then we can approximate as closely as we please to a smooth curve, such as we know to be essential, by employing a great number of distributions of individually small magnitude.

But it is evident that each component distribution should in itself obey the conjugate law, hence we may formulate a rule for laying out the load grading curve as follows:—

On the efficiency curve (Figs. 126, 127), cut off by a horizontal line, the portion of the curve defined by the maximum blade limits selected.

Divide the maximum ordinate of the part so cut off (Fig. 133) into some convenient number of equal parts,
Fig. 133. and draw horizontal lines cutting the efficiency curve at $s\ s\ s.$ Draw a number of lines through the origin 0, making small angles with the axis of $x$ and with one another, to represent the component increments of the thrust grading distribution, and drop perpendiculars from the points $s\ s\ s$ to indicate the limits of each increment.

Through the intersections of the perpendiculars let fall, and the inclined lines drawn through the origin, draw the thrust grading curve.

It will be noted that the angular increments of the thrust distribution (the angles between the lines passing through the origin) need not bo equal to one another; it may be considered within the province of the designer to vary these as he may think fit. In general, owing to the desirability of utilising as uniformly as possible the whole of the fluid, the angle allotted to successive increments will be diminished as shown in the figure; the loss of efficiency resulting from modifying the curve in this manner, or even of making serious departures from the theoretical curve, is probably microscopic. The theory as here presented will prove of more value to the designer by letting him realise exactly what he is doing, than from its too rigid and literal application.

§ 214. On the Marine Propeller.—The marine propeller, which has already been made the subject of comparison, is of especial interest as representing in a concentrated form the experience of over half a century. Beyond the advantages to be gained from an examination of established practice as a quantitative check on our investigation, there is some reasonable probability that we may arrive at facts having a bearing on the future evolution of screw propeller design.

The conclusions to which theory has led have in many cases been already tested by appeal to experience, with, on the whole, satisfactory results, but we have so far made no comparison on the very vital subject of pressure-velocity relationship.

The pressures proper to the conditions of least resistance (minimum gliding angle), given in Tables X. and XII., will be very much greater where water is concerned, owing to the greater density. On the other hand, the coefficient $\xi$ is less, which will have a slight effect in the opposite direction.

The form of blade employed in the marine propeller is of necessity confined to low values of $n;$ long slender forms such as may be well suited to an air propeller will be too weak (unless made of disproportionate thickness) to stand the pressure required. In practice the value of $n$ employed is less than 3, usually very much less; we will therefore confine our attention to blades of this proportion with the knowledge that the results as to pressures and efficiency ought in general to be higher than those that obtain in practice.

Referring to § 181, we have the angle $\beta$ given by the expression ${\sqrt {\frac {2\ \xi \ C}{\kappa \ (1-\epsilon ^{2})}}}.$ Taking $\xi$ for water as = .01 and density = 64 lbs. per cubic foot we obtain, $\beta$ = .132 (radians) or, = 7.6°, and the theoretical minimum gliding angle is 3.95°, which in practice becomes 6° about. The $P_{2}/V^{2}$ relation (given for air in Table IX.) will become $\rho \ \kappa \ (\epsilon +1)\ \beta =$ 12.5.

The above are the data on the pterygoid basis; similarly on the plane basis, that is, for blades of perfect helical form, we have, taking $\xi =$ .005, on the principle explained in § 182, $\beta =$ .048, or, $\beta$° 2.75°, that is $\gamma$ (least value) = .096, or $\gamma$° = 5.5°. The $P_{3}/V^{2}$ relation is given by the expression—
${\frac {P_{3}}{V^{2}}}=c\ \beta \ C\ \rho$ (§ 186) = 4.55.

The above results may be tabulated as follows:—

Pterygoid basis.	${\begin{cases}\\\\\\\\\\\\\end{cases}}$	$\beta ^{\circ }$	.  .  .  .  .  .	7.6°
		$\gamma ^{\circ }\ {\bigg \{}$	(calculated)	3.95°
			(probable)	6°
		${\frac {P_{2}}{V^{2}}}$	.  .  .  .  .  .	12.5°
Plane basis.	${\begin{cases}\\\\\\\\\\\end{cases}}$	$\beta ^{\circ }$	.  .  .  .  .  .	2.75°
		$\gamma ^{\circ }$	.  .  .  .  .  .	5.52°
		${\frac {P_{3}}{V^{2}}}$	.  .  .  .  .  .	4.55°

The $P/V^{2}$ values given above are in absolute units. The pressure value is extended in Table XIV. as pounds per square foot for values of $V$ in feet per second.

§ 215. The Marine Propeller (continued).—Cavitation.—It is very questionable to what extent the plane basis of operation is applicable in the case of a screw propeller. There seems to be a grave theoretical objection that does not exist when the motion is rectilinear, as in the analogous case of a weight supported by an aeroplane.

When we have to deal with the propeller blade on the assumption of a discontinuous system of flow, it is evident that the fluid in the dead-water region must be following the blade in its spiral path and must consequently be subject to centrifugal force. So much is this the case that it is hardly possible to conceive of the same fluid remaining in the dead-water region for any considerable length of time, so that the resistance on this basis will be very greatly augmented, and it will not be fairly represented by the figures deduced from rectilinear theory.

Table XIV.

Pressures Proper to Greatest Efficiency
for Blade Velocity $=V.$

$V.$	Pterygoid Basis.	Plane Basis.
20 30 40 50 60 80 100 120 140	156 351 624 975 1404 2496 3900 5616 7650	56 126 224 350 504 896 1400 2020 2740

This objection does not apply to the calculations made on the pterygoid basis, or at the most only to a very small extent; the author is therefore disposed to think that the pterygoid basis is that indicated by the conditions as giving the correct data for propeller design, the alternative not having the importance that attaches to it in the main problem of flight.

Now the pressure values given in Table XIV. represent not only the pressures appropriate to different blades moving with the mean velocities given, but the pressures appropriate to different portions of the same blade according to its velocity at different points along its length. Moreover, the pressures are not definite plus values but represent the pressure difference between the face and back of the blade. But we know that under real conditions a fluid is only competent to bear a certain maximum negative pressure (that is, a certain absolute minimum) without giving rise to physical discontinuity; that is to say, the formation of a void. Consequently there will be some critical pressure which cannot be exceeded without destroying the peripteral system of flow. The velocity at which this critical pressure is reached marks the limit of speed at which full propeller efficiency can be obtained; for speeds involving any higher velocity the design of a screw propeller becomes a compromise.

The production of a physical discontinuity by the screw propeller was discovered by Messrs. Thornycroft,^[8] and is termed “cavitation”; the phenomenon is one that has given considerable trouble to naval architects where high speeds have to be developed. We will endeavour to estimate the critical value of $V$ and show what modifications of design are indicated when the said critical value is exceeded.

We do not know definitely how much of the total reaction is carried as pressure on the face, and how much as vacuum on the back of the blade, but assuming pterygoid form and neglecting the effect of thickness, it is probable that the reaction is equally divided. The influence of thickness will be to superpose a streamline system of flow on the peripteral system which will result in a general diminution of pressure on both faces, so that the reaction will be more than half borne by the vacuum on the back of the blade. If the peripteral system comprises any discontinuity it is probable that this will tend in the opposite direction.

On the whole, it is perhaps best to assume the equal division of the reaction; and in making this assumption, to bear in mind that if the blade is of heavy section, cavitation will probably commence at a lower velocity than that which theory leads us to expect.

Let us assume that the propeller is working under a head of 2 feet of water in addition to the atmospheric pressure, that is, let us take the total pressure to be 16 pounds per square inch. Converting this into square foot units, we have permissible vacuum 2,300 pounds per square foot, or maximum total reaction = 4,600, which corresponds to a velocity of approximately 108 feet per second. If we take the pitch as equal to 1${\tfrac {1}{4}}$ times the diameter, this velocity, at the extremity of the blade, corresponds to 39.7 feet per second for speed of vessel, that is, approximately, 27 miles per hour or 23${\tfrac {1}{2}}$ knots. In practice, the blade of a propeller is never brought to a point as we are now supposing; the end of the blade is always rounded and the pressure consequently less than contemplated by our theory. It is probable that from this cause cavitation does not commence to give trouble till a somewhat higher speed is reached.

It is evident that at speeds above the cavitation limit the design will need modification. It will be necessary to give an area to the blades in excess of that proper to greatest economy, the additional area being required first at the tip of the blade, and as the speed becomes higher the blade will become affected over a greater portion of its length. The secondary consequences of this will be that it will no longer pay to employ the outer blade extremities, and the diameter of the propeller in terms of its pitch will have to be diminished; in ordinary parlance, the screw will become of quicker pitch. Beyond this the aspect ratio of the blades will cease to be of the same importance, since we are unable to employ the higher pressures to which the greater values of n give rise. We may therefore expect to find the blade form becoming of more compact outline as higher speeds come into vogue.

It is manifest that for marine work at high speeds it is impossible to construct a fine pitch propeller to give any reasonable economy, for at a comparatively low vessel speed the velocity of the blades will begin to exceed the limiting value, and it will be necessary to add so much extra surface to reduce the pressure that the efficiency will be poor.

In aerial propellers we are fortunately not concerned with the phenomenon of cavitation.

§ 216. The Influence of the Frictional Wake.— It has already been shown that the efficiency of a propeller of any kind is increased by the fact of its operating on the frictional wake.

The efficiency that we have been discussing, the $E$ of the screw propeller, represents the efficiency on the basis of § 198; that is to say, the propeller is supposed to act on virgin water, and the towing efficiency is that taken as unity.

When we consider the wake as influencing the efficiency we have to adopt a convention. It is known that the wake is in reality a very disturbed region whose velocity varies greatly from point to point. Mr. R. E. Froude has shown that the mean wake velocity over the area swept by the propeller may be taken as its effective velocity without serious error, and he has also introduced the useful conception of a phantom ship having a speed equal to the actual velocity minus the mean wake velocity, that is, ${\mbox{V – v}}_{1}.$

The resistance of the phantom ship is supposed, at its velocity ${\mbox{V – v}}_{1},$ to be equal to that of the real ship at its velocity $={\mbox{V}},$ so that the propeller designed for the phantom ship on the basis of simple theory will be correct for the real ship to work in its frictional wake. The important fact is thus rendered apparent, that the form of the propeller proper to highest efficiency is independent of the existence or otherwise of a frictional wake.

Now the useful work is proportional to the actual velocity of the vessel, since the resistance (and therefore the thrust) is the same in both the phantom and the reality; consequently the useful work is in the relation—

${\frac {\mbox{Real}}{\mbox{Phantom}}}={\frac {\mbox{V}}{{\mbox{V – v}}_{1}}}.$

But the total work done in propulsion is the same in both cases, therefore if $E_{1}$ is the efficiency under real conditions, we have—

$E_{1}={\frac {\mbox{V}}{{\mbox{V – v}}_{1}}}\ E.$

The value of ${\mbox{v}}_{1}$ depends upon the lines of the vessel and position of the propeller, the essential point being the extent to which the frictional wake is led into the periptery of the propeller blades. It is manifest that ${\mbox{v}}_{1}$ is limited to a value less than the sternward component of the impressed velocity (compare § 200).

In the hypothetical case chosen in § 200, where it is assumed that the whole of the wake current is utilised by the propeller, we have—

Phantom ship velocity $={\mbox{V – v}}_{1},$ or by § 198—

$E={\frac {{\mbox{V – v}}_{1}}{{\mbox{V – v}}_{1}+{\frac {\mbox{v}}{2}}}}=$ ${\frac {{\mbox{V – v}}_{1}}{{\mbox{V –}}\ {\frac {{\mbox{v}}_{1}}{2}}}},\quad ({\mbox{for}}\ {\mbox{v}}_{1}={\mbox{v}}),$

but for real ship

$E_{1}={\frac {\mbox{V}}{{\mbox{V – v}}_{1}}}\ E,$

or

$E_{1}={\frac {\mbox{V}}{{\mbox{V – v}}_{1}}}\times {\frac {{\mbox{V – v}}_{1}}{{\mbox{V –}}\ {\frac {{\mbox{v}}_{1}}{2}}}}={\frac {\mbox{V}}{{\mbox{V –}}\ {\frac {{\mbox{v}}_{1}}{2}}}},$

which is the result already deduced in § 200 by the direct application of the Newtonian principle.

The device of the phantom ship is in reality merely a method of expressing a simple problem in relative motion in a palatable form; it is obvious that the argument treats the wake current as a favourable tidal current, or as the flow of a river, the ship's motion being credited in respect of its change of position relatively to some fixed mark; the method of the “phantom ship” presents the problem in a clear and precise form.

The question of wake influence is probably of less importance in connection with aerial flight than it is in the problem of marine propulsion.

§ 217. The Hydrodynamic Standpoint. Superposed Cyclic Systems.—It is of interest to form a mental picture of the hydrodynamic system of flow that accompanies a screw propeller.

It is evident that according to peripteral theory each blade of the propeller forms the axial core of a cyclic system and that the necessary condition of multiple connectivity is carried out by vortex filaments containing rotation trailing from the ends of each blade.

There will need to be as many cyclic systems as there are blades, so that if there are n blades the region will require to be n-ply connected. In the case, for example, of a two-bladed propeller the two cyclic paths are represented in Fig. 134 by

Fig. 134.	Fig. 135.

the two circuits drawn round the blades; the trailing vortices are shown diagrammatically.

It is possible that the inner end vortices are unnecessary, for the boss and shaft may be found to determine the connectivity of the region at the axis end of the blades (Fig. 135).

The vortex filaments are presumed to persist in the region of the wake till they have, metaphorically speaking, "taken root" in the fluid, so that the conditions of multiple connectivity are simulated. It would appear to be only necessary to suppose rotation to become generally distributed (through the agency of viscous stress) in the wake of the propeller to bring about the necessary condition. (Compare Chaps. III. and IV.)

In propellers giving rise to cavitation, or when air is sucked down owing to insufficient immersion, the dependent vortices become visible by their empty cores, and may be seen as interlacing helices following in the track of the extremity of each blade, like adherent strings of sea-weed.^[9]

The vortices from the external extremities of the blades are all of the same "hand" and consequently tend to wind round one another; they may be conceived to break up into spiral groups and perhaps sub-groups, as they are left behind in the propeller race, after the manner indicated already in the case of the aerofoil (Fig. 86).

 

§ 218. On the Design of an Aerial Propeller.—A few simple rules may be formulated for the design of an aerial propeller; these rules will be applicable mutatis mutandis to the marine propeller.

(1) From the conditions, assess the probable value of $\gamma$ (usually about 10 degrees), and (Fig. 136) plot the efficiency curve from the equation (§ 204). Any arbitrary scale may be employed.

(2) Decide on "discard point"; that is, the minimum percentage of maximum available efficiency, and so determine blade length.

(3) Draw the thrust grading curve, $b\ b\ b$ (Fig. 136), as in § 213 (Fig. 133). At this point the designer has to exercise his judgment; it is perhaps best to draw a trial curve freehand, the object being a smooth curve beginning and ending at zero, but in general character to simulate the truncated wedge form based on the Newtonian theory; then let fall perpendiculars from the conjugate points of equal efficiency, and draw radial lines through the origin to suit the freehand curve as nearly as possible; then correct the freehand curve to pass through the intersections.

(4) From the thrust grading curve $b\ b\ b$ (Fig. 136) derive the load grading curve $c\ c\ c;$ the ordinates being calculated by multiplying the thrust ordinates by the corresponding values of sec $(\theta +\gamma )$ (Fig. 136).

Fig. 136.

(5) Calculate the values of $V^{2}$ for different points along the blade (§ 207), and divide the values of the ordinates of the load grading curve $c\ c\ c$ by the corresponding values of $V^{2},$ and draw a curve representing the quotient. This is the linear grading curve $d\ d\ d$ and represents the relative height of the arched section at every point along the blade. (Compare § 192.)

(6) The "plan form" or "development" of the blade may now be laid out. If we proceed on the lines indicated by present theory, the plan form will be everywhere proportional to the linear grading; thus we have to settle the aspect ratio of the blade, lay off the maximum width, and draw a curve whose ordinates from point to point are proportional to the linear
Fig. 138. grading ordinates (Fig. 137 (a)). If we adopt this design of blade the sectional form, will be constant throughout the length, varying only in its scale; that is to say, the materialised $\alpha$ and $\beta$ angles will be everywhere the same (Fig. 137).

The theory may possibly be incomplete; as discussed in §§ 190, 191, 192, etc., there may be some unformulated objection to the pointed extremities to which present theory gives rise. If this is the case the section will become flatter towards the extremities, the linear grading remaining the same and the width of the blade becoming greater. If we take the elliptical aerofoil as our model we may derive the corresponding blade form by the construction given in Fig. 138, elliptical ordinates being substituted at every point for the corresponding parabolic or segmental ordinate.

(7) If such a modified plan form is adopted the sectional form

Fig. 137.

of the blade should be designed at every point to suit the width and grading, the angles $\alpha$ and $\beta$ having appropriate values assigned from point to point, according to the value of the constant $\epsilon$. It is possible that the constant $\epsilon$ ought to vary from point to point along the blade; if this is so it is a matter on which we have so far no information; for the present it should be taken as the aerofoil value of $\epsilon$ proper to the value of $n$ employed.

In Fig. 137 the blade is supposed continued to connect to the boss. Such continuation is always necessary, unless a boss of very large diameter is employed, the continuation being of stream-line section symmetrically disposed about the pitch helix. It will be observed that the linear grading falls with extreme rapidity as the inner "extremity" of the blade is approached, and thus the change of form from the pterygoid to the symmetrical stream-line section is very abrupt. It is probably advantageous to carry the pterygoid section beyond the theoretical blade limit and so merge it more gradually into the simple form. No cognisance of this structural feature has been taken in the hypothesis.

It should be remembered that the $\theta +\gamma$ spiral is at every point the analogue of the horizontal line, and the one from which the $\alpha$ $\beta$ angles are laid off; on this basis the setting out of the section is the same as for an ordinary aerofoil, but the full "dip" forward can be given to the section, since the possibility of the loss of equilibrium is no longer a factor (§ 138).

Parenthetically it may be remarked that the $\theta +\gamma$ series of spirals does not form a helix, for the angle $\gamma$ is constant at all points. This would usually be expressed by saying that the pitch of the blade increases towards the tip, but we know that the ordinary manner of defining the pitch by the mean angle of the blade is unscientific.

(7) Number of blades. The determination of the maximum number of blades permissible has been discussed in § 211, and it has been shown that this depends but little upon the

Fig. 139.

value of “n,” being chiefly dependent upon the relative length of the blade as compared to the diameter; the number of blades thus depends upon the discard percentage. For a 90 per cent, discard four blades are the appropriate number; if the discard be 95 per cent, six blades may be employed. In general the following rough-and-ready rule may be employed: Let $r_{1}$ and $r_{2}$ be the radii of the inner and outer extremities of the blades, then the number of blades permissible will be $=2.5\ {\frac {r_{2}+r_{1}}{r_{2}-r_{1}}},$ fractions being neglected.

(8) All that remains to be done is now to give a scale to the design that will render the propeller suitable for the intended load. To this end any convenient scale should first be assumed and the load calculated, for the actual value of $V$ (the velocity of flight), which it is intended to employ. The linear dimension will then be in the ratio to the dimension required as the square root of the calculated thrust is to the square root of the thrust required. That is to say, the scale unit will be in the inverse ratio.

An example of the design of an aerial propeller on the foregoing principles is given in Figs. 136, 137, 138, 139; the supposed data being as follows:—Velocity 70 feet per second; thrust = 100 lbs.; discard 90 per cent.; n = 6 ; $\gamma$ taken = 10 degrees.

§ 219. Power Expended in Flight.—The principles governing the expenditure of power in flight have, in this and the preceding chapters, been fully expounded, and it now only remains to draw certain elementary deductions.

The power essential to flight or thrust horse-power may be defined as that represented by the thrust multiplied by the velocity of flight, that is to say, the equivalent of the tow-rope expenditure. The actual power required will then be a thrust or essential power divided by the efficiency of the propulsion.

We have seen (§ 200) that the efficiency of propulsion may theoretically be greater than unity, so that the term essential must not be construed as meaning the minimum theoretically necessary on the Newtonian basis.

Now, the essential power will be given by the expression $W\ \gamma \ V,$ where $\gamma$ (assumed to come within the definition of a small angle) is expressed in circular measure. Now, if $\gamma$ is constant in respect of $V,$ as has been proved to be the case so long as the body resistance is regarded as negligible, or separately computed, the following deductions may immediately be made:—

(1) The energy required to travel from point to point is independent of the velocity and is constant.^[10]

(2) The power (horse-power) required is directly as the velocity.

(3) From (1) it follows that the maximum range of flight of a flying machine must depend upon the fuel carrying capacity, the energy value of the fuel, and the total efficiency of the prime mover and propelling mechanism, and is independent of the speed of flight.

(4) From (2) it follows that the velocity of flight is limited by the relation of horse-power to weight, and, other things being equal, is proportional to the horse-power per unit weight of the prime mover.

These conclusions are of considerable importance, and are illustrated in the Tables as follows:—

Table XV., column (1), gives, for values of $\gamma ^{\circ }=$ 6°, 7°, 8°, 9°, and 10°, the distance that could be run if an aerodrome had at its disposal the total energy of its own weight of hydrogen taken as giving 48,000,000 foot lbs. per lb. Column (2) gives the same information for petroleum spirit, taken as equal 16,000,000 foot lbs. per lb.; column (3) is based on the assumption that 25 per cent, only of the total heat is available, as representing the thermal efficiency of the petrol engine. Column (4) the distance after an allowance of 75 per cent, mechanical efficiency of engine and transmission, and a 66.6 per cent, efficiency of propulsion. Lastly, column (5) gives the actual range, or maximum possible distance, on the basis of columns (2) to (4) on the assumption that the aerodrome or flying machine carries 10 per cent, of its mean weight as fuel, i.e., petroleum spirit. The mean weight is, under these conditions, 5 per cent, less than the initial and 5 per cent, more than the final weight, the loss of weight being due to the fuel consumption.

As a maximum estimate of the range of flight conceivably possible without some fundamental discovery in fuel and prime movers we may take the following supposititious case. Using liquid hydrogen as fuel, and carrying 25 per cent, of the total mean weight, and assuming a yet-unheard-of thermal efficiency of 50 per cent., a total mechanical efficiency of 90 per cent., and a propeller efficiency of 70 per cent., with the minimum angle of $\gamma$ given in the Table (= 6°), the exhaustion of fuel will be complete after a flight of 6,800 miles distance.

The above estimate is based on an assumed development of the heat engine and other mechanical refinements not yet within sight, and indeed such as may never be realised. If we confine ourselves to existing appliances and existing methods it is doubtful whether the maximum range of flight can (without devoting the whole resources of the machine to the carrying of fuel), ever exceed 1,000 miles, and for the present this may be regarded as the probable extreme outside limit.

Table XV.

Possible Range of Flight on Basis of Computation given in Text.

(Column 5 gives computed range in miles, assuming propulsion by petrol motor and screw propeller, for fuel capacity equal one-tenth of total weight.)

$\gamma ^{\circ }.$	$\gamma .$	(1.)	(2.)	(3.)	(4.)	(5.)
6° 7° 8° 9° 10° 11° 12°	.105 .122 .140 .157 .175 .192 .210	86,600 74,500 65,000 57,900 52,000 47,400 43,300	28,866 24,833 21,666 19,300 17,333 15,800 14,433	7,216 6,208 5,416 4,825 4,333 3,950 3,608	3,608 3,104 2,708 2,412 2,166 1,975 1,804	360 310 270 251 216 197 180

§ 220. Power Expended in Flight (continued).—It has been shown that, neglecting body resistance, the power per unit weight requires to increase directly as the velocity. Table XVI. gives the calculated indicated horsepower per 100 lbs. weight for velocities ranging from 15 to 100 feet per second, the Table also being figured for the thrust horse-power for velocities from 30 to 200 feet per second. The value taken for the total efficiency is the same as employed in calculating column (4) of the preceding Table, i.e., 75 $\times$ 66.6 per cent. = 50 per cent., so that the velocities for a given horse-power value are in the ratio of 2 : 1.

The calculation has been made for values of $\gamma$ extending from 6 degrees to 12 degrees, as in the preceding Table; experiment would appear to show that 10 degrees is as low a value of $\gamma$ as can be obtained under practical conditions; it is, however, possible that with increased experience lower values may be obtained. An aerodrome whose $\gamma$ is greater than 12 degrees is certainly of inefficient design.

Table XVI.

Indicated Horse-power and Thrust Horse-power per 100 lbs. weight at different Velocities and for different Values of $\gamma$.

$\gamma ^{\circ }$.	Indicated horse-power per 100 pounds weight at velocities (ft./sec.)—
	15.	20.	25.	30.	35.	40.	45.	50.	60.	70.	80.	90.	100.
6° 7° 8° 9° 10° 11° 12°	0.57 0.66 0.76 0.85 0.95 1.05 1.14	0.76 0.87 1.02 1.14 1.27 1.40 l.53	0.95 1.11 1.27 1.42 1.59 1.75 1.91	1.14 1.33 1.53 1.71 1.91 2.10 2.29	1.33 1.55 1.78 2.00 2.22 2.44 2.67	1.53 1.77 2.04 2.28 2.54 2.80 3.05	1.72 2.00 2.29 2.56 2.86 3.14 3.44	1.91 2.22 2.51 2.86 3.18 3.50 3.82	2.28 2.66 3.06 3.42 3.82 4.20 4.58	2.66 3.10 3.56 4.00 4.44 4.88 5.34	3.06 3.54 4.08 4.56 5.08 5.60 6.10	3.4 4.0 4.6 5.1 5.7 6.3 6.9	3.8 4.4 5.1 5.7 6.3 7.0 7.6
	30	40	50	60	70	80	90	100	120	140	160	180	200
	Velocities (ft./sec.) at which thrust horse-power is required as above.

If we take account of body resistance, we know that the total resistance is greater the higher the velocity, for the body resistance increases as $V^{2},$ so that the angle $\gamma$ is no longer constant in respect of $V.$ We also know (§ 171) that the influence of the weight of the aerofoil, as additional to the load carried, is to place a lower limit on the velocity that may be usefully employed.

The Equation (5) of § 171 gives the condition of least resistance. The value of $\gamma$ can thus be calculated for any set of conditions, and the power data obtained from the Table.^[11] By plotting from the equations the conditions other than for least resistance may be examined with equal facility and the $\gamma$ values determined.

Before leaving the subject of power expenditure it is desirable to point out the extent to which the future of flight and the uses of a flying machine are circumscribed by economic considerations.

Leaving all attendant difficulties on one side, it is evident that the conveyance of goods by flying machine would be comparable, so far as power expenditure is concerned, with drawing them on a sleigh over a common road, so that where any other method of transport is possible, flight may be regarded as out of the question. In addition to this, the range of a flying machine must, unless after the manner of a soaring bird it derives its energy from wind pulsation, be strictly limited to a few hundred miles between each replenishment of fuel; and consequently we cannot at present regard aerial flight as a means of ocean transport, or even as a means of exploring inaccessible regions where the distance to be accomplished exceeds that stated.

Beyond this the velocity of flight is limited by the horse-power weight factor. If, as an example, we suppose that 25 per cent, of the weight of the machine is taken up by the motor itself, and if the motor weigh only 2${\tfrac {1}{2}}$ lbs. per horse-power, it is improbable, taking everything into account, that seventy miles per hour can be exceeded. Flying is comparable to hill climbing on a road automobile where $\gamma$ represents the gradient, this being about 1 in 5 or 1 in 6, and where the transmission efficiency is limited to about 66 per cent. The velocity limit rests entirely on the weight per horse-power, the aerodrome being presumed designed for least resistance; any continued improvement in prime movers, tending to a reduction of weight, will react in the direction of rendering higher aerial speeds practicable.

If it should be found possible to “soar” on a large scale after the manner of an albatros or gull, the limitation of range may, in certain exceptional cases, be partially or wholly removed.

It may be noted that on the liquid hydrogen estimate of maximum possible range, no allowance has been made for the possible power to be derived directly from the expansion prior to combustion. We have also omitted to discuss the possible increase of thermal efficiency theoretically available by the employment of the low temperature of the boiling point of hydrogen as a refrigerator, that is, as the temperature at which the heat engine discards. The use of liquid hydrogen is at present too daring and distant a suggestion to be taken quite seriously; it has here been put forward merely as representing the maximum known fuel value in a possibly available form.

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1. “Miscellaneous Scientific Papers,” Rankine, XXXIII.
2. The author has known this result received with open incredulity by persons considered to be authorities on propulsion.
3. It is worthy of remark that the solution is the same as for a solid screw; the constancy of the gliding angle renders it analogous to the angle of friction.
4. An erratum published in Volume 2 has been applied: "P. 300, line 6 from top, for "${\tan(\theta -\gamma )}$" read "${\tan(\theta +\gamma )}$," comp. p. 299, line 3. " (Wikisource contributor note)
5. A frequent cause of error in the work of writers of some forty or fifty years ago.
6. Compare §§ 214–216.
7. § 206.
8. Trials of destroyer Daring.
9. The author has, for example, observed such air-core vortices from the after-deck of twin-screw S.S. New York.
10. It is understood that the proportions of the aerodrome are varied with changes of velocity to comply always with the conditions of least resistance, §§ 165, 176, 185.
11. In §§ 181 and 189 it has been pointed out that the $\gamma$ value, independently of body resistance, is in practice greater than can at present be deduced from pure theory.