'/')' BLiJ^Si t;l%Aii tiji^A (Cornell IntograitH SItbratg 3tlfara, Sfem ^axk BOUGHT WITH THE INCOME OF THE SAGE ENDOWMENT FUND THE GIFT OF HENRY W. SAGE 1891 INGINEERING LIBRARY Cornell University Library VM 751.T23 Resistance of ships and screw propulsion ■'3""l924 004 998 625 The original of tiiis book is in tine Cornell University Library. There are no known copyright restrictions in the United States on the use of the text. http://www.archive.org/details/cu31924004998625 RESISTANCE OF SHIPS SCREW PROPULSION .m^ RESISTANCE OF SHIPS SCREW PROPULSION BY D. W. TAYLOR NAVAL CONSTRUCTOR, UNITED STATES NAVY MACMILLAN AND CO. AND LONDON 1893 All rights reserved Copyright, 1893, By MACMILLAN AND CO. J. S. Gushing & Co. — Berwick & Smith. Boston, Mass., U.S.A. PREFACE. In his professional work the writer has often felt the need of a short treatise upon the resistance and propulsion of ships, con- taining data, formulae, and tables for use in making estimates. Such a treatise he has endeavoured to produce. In handling the subject the papers read at various times before the Institution of Naval Architects by the late Mr. William Froude, and by Mr. R. E. Froude, his son, have been made much use of. This was necessarily the case, since the theories set forth in these papers are now generally accepted, and the experimental results - given in them stand alone for accuracy and completeness. The book contains, however, a good deal of original matter which will, it is hoped, be found of value. The writer has endeavoured throughout to discuss ships as they are, not floating bodies in general ; to set forth methods and deduce results as simple as the nature of the subject will allow, and sufEciently accurate for everyday use. The proof-sheets have been revised by Professor W. F. Durand of Sibley College^ Cornell University. His assistance has been of the greatest value, not only as regards correction of the proof, but also in the discovery and correction of discrepancies in the orig- inal text. D. W. TAYLOR. U. S. Navy Yard, Mare Island, Cal., July 22, 1893. CONTENTS. CHAPTER I. RESISTANCE. Skct. Page 1. Preliminary and Definitions i 2. Symbols used 2 3. The Knot 4 4. Frictionless Submerged Solid 6 5. Various Kinds of Ships' Resistance 9 6. Eddy Resistance of Plate 12 7. Skin Resistance of Planes 16 8. Rankine's Method 20 9. Froude's Method 25 10. The Law of Comparison 28 11. Froude's Method (Completed) 36 12. Phenomena of Waves produced by Ships 38 13. Properties of Trochoidal Waves 43 14. Deduction of Law of Wave Resistance 45 15. Laws of Variation of Wave Resistance 48 16. Results of Experiments on Wave Resistance and Approximate Laws 52 17. Simplification of Wave-Resistance Formula 56 18. Increase of Resistance in Shallow Water 58 19. Squat and Change of Trim 59 20. Formula for Total Resistance 60 vii viii CONTENTS. CHAPTER II. THE PROPELLER. Sect. Page 1. Preliminary and Definitions 62 2. Element of Face of Blade treated as a Plane ... 67 3. Extension of Formula for Plane to Propeller ... 76 4. Values of a, f, and the Characteristics 80 5. Efficiency and Power of Various Shapes of Blades. . 85 CHAPTER III. mutual reactions between propeller and ship. 1. Action of Propeller attached to Vessel 95 2. The Wake loi 3. Thrust Deduction 104 CHAPTER IV. analysis of trials and average results. 1. Value of Trials 106 2. Components or Absorbents of the Indicated Horse- Power 108 3. Yorktown Trial Analysis 116 4. Distribution of Power 133 5. Coefficients and Constants for Practical Use . . .136 6. Thrust Deduction and Wake Factor 139 CHAPTER V. the power of ships. 1. Preliminary 14c 2. Admiralty Coefficient Method 142 3. Kirk's Analysis .... 147 4. Extended Law of Comparison 151 5. Standard Curves of Power 155 CONTENTS. IX Sect. Page 6. Model Tank Method i8o 7. The Independent Estimate Method 181 8. Comparison of Methods 182 9. Effect of Rough Weather and Foulness 183 CHAPTER VI. PROPELLER DESIGN. 1. Influence of Shape of Section and Variation of Pitch, 186 2. Standard Blade 194 3. Standard Slip 198 4. Design of a Propeller 201 5. Strength of Propeller Blades 208 TABLES 222 CHAPTER I. RESISTANCE. § I. Preliminary and Definitions. If a ship is towed at a steady speed through still water, the Tow-rope pull on the tow-rope must evidently be equal and opposite to ^^^'^^^^s- the resistance of the ship, which opposes its motion through the water. The resistance of the ship in this simplest case is often called the tow-rope resistance. When a ship is propelled through still water by her own Resistance machinery, there appear in the water various disturbances p^opeued." caused by the propelling machinery which are superposed upon those caused by the ship's hull. Under these circum- stances the resistance of the ship generally differs from the tow-rope resistance appropriate to the speed. I shall deal in this chapter with tow-rope resistance, and discuss in a later chapter the effect of the propelling machin- ery. Used without qualification the term resistance will be understood to mean tow-rope resistance. If a ship, instead of being towed through still water, sMpat were held by a tow-line in water flowing uniformly and steady steadily past her, it is evident that, the relative motion stream, being unchanged, the pull on the tow-line for a given speed of the water would be the same as the tow-rope resistance for the same speed of the ship through still water. In dis- cussing resistance I shall consider the ship moving through still water, or fixed in a uniform s'tream, as is most con- venient. 1 2 RESISTANCE OF SHIPS. § 2. § 2. Symbols Used. There will be found below the meanings attached to cer- tain symbols which will be frequently used. length. The length of a ship in feet is denoted by L. Since the stem and stern post are not usually vertical, the exact value of L depends upon the level at which the length is taken. For purposes of resistance we need only consider the portion of the hull below water. As a rule it is best to measure L at the level of the water Une, making it what is called length on water line. Where there are ram spurs or other projections below the water, a reasonable addition should be made to the length on water Hne to obtain the proper value of L. It would perhaps be a somewhat more nearly exact definition of the symbol L to call it the mean immersed length of hull. Breadth. -pjjg greatest breadth of the immersed hull in feet is denoted by the symbol B. This greatest breadth is usually found at the water line, but in some ships the point of maximum breadth is below water. It should be noted that B is not the molded breadth, or breadth to outside of frames, but should be measured to the outside of the plating or planking where they are flush. In the case of ships plated on the raised and sunken sys- tem, B should be measured to a line \\ times the thickness of the plating outside the frames ; for this line is a fair mean of the irregular contour, being at a distance equal to \ the thickness of the plating inside the surface of the outer strakes, and the same distance outside the surface of the inner strakes. Mean The mean draught in feet at the centre of length is denoted by the symbol H. This draught is exclusive of keel or other projection from the surface. Thus, in the case of a ship with bar keel 1 2 inches deep, plated on the raised and Draught. § 2. SYMBOLS USED. 3 sunken system, with bottom plating | inch thick, and drawing 23 feet amidships to bottom of keel, the value of H would be 23'- 12"+ iJ-x f" = 22'.0937S. The surface of the hull and its appendages below water, wettea commonly called the wetted surface or skin, is expressed in ®"^*°^- square feet, and denoted by the symbol 5. Distinction is and should be made between the wetted surface of the hull proper and that of bar or bilge keels, rudders, shaft tubes, struts, etc. The latter constitute the appendages. The displacement in tons is denoted by the symbol D. Dispiace- Of course at a given draught and trim the displacement depends upon the density of the water. It is customary to consider 35 cubic feet of salt and 36 cubic feet of fresh water as weighing one ton. These are convenient round numbers, and sufficiently close approximations. There are various ratios or coefficients which may well be coeffi- cients. defined and explained here, though I shall have occasion to use them but little. If D denote salt-water displacement, then Z'X3S denotes the volume displaced in cubic feet. The ratio {D y. i^) -r- {L y. B % H) is called the block coeffi- cient. It is the ratio between the volume of the hull below water and that of a rectangular block of the same extreme dimensions. The ratio (Area of Midship Section) -=- (.5 x //^) is called the Midship Section Coefficient. It is the ratio between the area of the midship section and that of a rectangle of the same extreme dimensions. It should be remarked that the mid- ship section is the section of maximum area below water, and is not necessarily situated at the centre of length, though it is usually there or thereabouts in ships of the present day. The ratio (Area of Water Plane) -?- (^ x i^) is called the Water Plane Coefficient, or Water Line Coefficient. 4 RESISTANCE OF SHIPS. § 3- The ratio (Z^x 35)-f-(Area of Midship Section xZ) is called the Cylindrical Coefficient. It is the ratio between the volume of the hull below water and that of a cylinder with a section the same as the midship section, and of the same length as the ship. Resist- Resistance in pounds is denoted by the symbol R. R used alone will usually denote the total resistance, while used with subscripts, as R„ R^, it will denote various components of the total resistance, the subscript indicating the nature of the special resistance denoted. Note. — It will be convenient in other chapters to use R to denote other things than resistance. I shall not hesitate to do so where there is no chance of confusion being caused, as I consider this course preferable to the use of the Greek or German letter R. Speed. Speed in knots is denoted by V, and speed in feet per second by v. § 3. The Knot. Log Line The knot, properly speaking, is a unit of speed, and not of length. The expression came into use from the method uni- versally used in the day of sails (and still much employed) for determining the speed of ships at sea. A nearly triangular log chip is so weighted and connected to a log line that it floats nearly stationary when thrown overboard, and draws the log line off the reel upon which it is wound. Knots are tied in the line at suitable intervals, and by observing the number of knots which pass overboard while a sand-glass is running down, the speed of the ship is determined at once. A common length between knots is 47 feet 3 inches, corre- sponding to a 28-second sand-glass. A speed of one knot, then, would mean a speed per hour of 47-25 X36oo ^g^^g j^^^_ 28 if the log chip did not move at all. The distance between knots is probably made purposely somewhat less than that §3- THE KNOT. 5 corresponding exactly to the nautical mile because of the slight drag of the log chip. The nautical mile or unit used for measuring distances at Nautical sea is not the satne in all countries. Since, upon the ocean, DMerent latitude is more easily and accurately determined than longi- countries, tude, we might naturally expect to find the nautical mile and the minute of latitude identical. There is an intimate con- nection between them. But the arc of a meridian which subtends an angle of one minute at the centre of the earth varies slightly in length from the equator to the poles on account of the fact that the earth is not a perfect sphere. Its average length, according to the astronomer Bessel, is 1852.01 metres. Accordingly, the nautical mile used in France, Germany, and Austria is 1852 metres, or 6076.23 feet. In England the nautical mile corresponding to the "Ad- miralty knot " is 6080 feet. In the United States the nautical mile as fixed by the Navy Department is 6080.27 feet, being "equal to the one- sixtieth part of the length of a degree on a great circle of a sphere, the surface of which was considered equal to the surface of the earth." The "geographical mile" is the length of the arc sub- Geograpu- tending one minute of longitude at the equator, and is 6086.5 feet long. It appears to be an entirely superfluous -unit. The "statute mile" used in land measurements is 5280 statute Mils ^ feet. It is commonly used in navigating river and lake boats, notably on the Great Lakes of America. 'Such vessels never take astronomical observations to determine their posi- tions, and the statute mile is better suited to their needs. While, as stated above, the word knot is properly restricted usage of . . , ■, WordKnot. to denote a unit of speed, the expression nautical mile or sea mile is rather clumsy and tends to produce confusion. Accordingly, there is observed a growing tendency to use the word knot in the sense of nautical mile. RESISTANCE OF SHIPS. §4- This usage is convenient, and though strenuously opposed as a solecism by the grammatical purist and the amateur sailor, it appears probable that in time it will prevail. Solid in Perfect Fluid. Stream Lines. Plane Stream Lines past Cylindrical Solid. § 4. Frictionless Submerged Solid. It is advisable to lead up to the somewhat complex prac- tical case of' the resistance of a ship by a short discussion of a simple ideal case of no resistance. Suppose we have held deeply submerged, in a perfect and incompressible fluid, a small, smooth, or frictionless solid with sharp head and tail, and of fair and easy lines all over. For our purposes it suffices to define a perfect fluid as one destitute of viscosity. Imagine the whole body of fluid to be flowing steadily and horizontally in one direction, and the solid to grow gradually from nothing, so to speak, being kept at rest by a force which does not interfere with the motion of the fluid. We know from the mechanics of fluids that after the solid has attained its full growth the surrounding fluid will continue to flow steadily past it. The direction and magnitude of the velocity at each point of the fluid will not change with time, though they will differ from point to point. The paths followed by the particles of fluid under the above conditions are called the " stream lines " past the solid. Their shape depends upon the shape of the solid, and does not change with change of speed of flow of the fluid. Figure i shows some stream lines past a cylindrical solid, a quarter-section of which is shown in the figure. It is sup- posed to be placed vertically in the fluid, and while of small dimensions relative to the fluid to have its ends at such a distance from the section shown in Figure i, that the flow past this section goes on as if the solid were indefinitely long. Figure i shows the stream lines in one quadrant only, but they are symmetrical in the remaining three. §4. FRICTIONLESS SUBMERGED SOLID. At every point along a stream line what is called the steady steady motion formula holds good, or, Fomiuu. W 2g In this formula p = pressure per unit area ; w = weight of Unit volume of the fluid ; V = velocity of flow ; g = acceleration due to gravity ; c = height above a fixed level ; A = a constant for each stream line, being called the head. If, then, the motion is horizontal, it is evident that along changes of a stream line increase of velocity must accompany decrease anapres- of pressure, and vice versa. sure along I/ine. 150 i4o lie 1^0 ilo 100 90 s'o Vo e'o ^0 ia a'o 20 io 6 rig. 1. — STREAM LINES PAST A CYLINDRICAL SOLID. Consider a stream line passing close to our submerged solid. At a great distance ahead the pressure and velocity are constant, having what may be called their undisturbed 8 RESISTANCE OF SHIPS. §4- or normal values. As the solid is approached, the pressure increases and the velocity falls off. We reach a point, how- ever, somewhere near the head of the solid, where the velocity begins to increase again. EXCESS. 300 Note. — TH&tanoes are measured out from, the centre line of Fig, L except for Curve showing pressure on surface of solid where dis- tances are measured from the forward end of the solid. .j^iO ClTY ALONG _4 _^ DEFECT. Fig. 2. CURVES OF PRESSURE AND VELOCITY AT SECTIONS SHOWN IN FIG. I. It continues to increase until abreast the centre of the solid we have a velocity above the normal accompanied by a pressure below the normal. This is necessarily the case ; for abreast the centre the stream lines are parallel to their directions at a great distance ahead (where the pressure and velocity are normal), but the area available for flow is dimin- ished by the sectional area of the solid. §5- VARIOUS KINDS OF SHIPS' RESISTANCE. 9 After passing the centre of the solid the changes of velocity just described are repeated in reverse order, until at a great distance astern the flow again becomes normal. It appears, then, that at what may be called the bow and stern of the submerged solid the pressure in the fluid is greater than if the solid were not there, while amidships it is less. Figure 2 shows the changes of velocity and pressure set up at the sections indicated in Figure i. The numerical values were obtained by supposing the solid to be 180 feet long, immersed in a fluid weighing 64.32 pounds per cubic foot, and flowing at the rate of 10 knots. The densest salt water weighs nearly 64.32 pounds per cubic foot, and that number was adopted for convenience because it is twice 32. 16, the value of ^, the acceleration due to gravity. Figure 2 also contains a curve showing the pressure on the Net Resist- surface of the solid at each point, or rather the change in ^y^'g pressure from its value for fluid at rest. It is evident that Notung- the net result of the changes in pressure is nil. Since by supposition the fluid is frictionless, and since it exerts no resultant pressure ahead or astern on the solid, it is evident that in the ideal case supposed the solid offers no resistance. We know very well that ships offer a great deal of resist- ance, and in the next section I shall point out how the various resistances are brought about as we proceed from the ideal solid to the actual ship. Note. — The theory of stream lines is beyond the scope of this work, so I have contented myself in the above with simple statements of facts. In any modern treatise on hydrodynamics the reader will find the steady motion formula demon- strated, and the theory of stream lines set forth. § 5. Various Kinds of Skips' Resistance: I propose in this section to enumerate the points of dif- ference between the ideal solid in a frictionless fluid and a ship in water, pointing out the various kinds of resistance arising from these differences. 10 RESISTANCE OF SHIPS. §5- Limits of Stream- Line Motion. Eddy Re- sistance. Skin Re- sistfince. We have seert that for a stream Hne ^-\ = a constant = 2 w 2g _ — + — , where /o and v^ are the normal pressure and velocity, W 2g or the pressure and velocity at a great distance. Then w 2g Suppose v = nvo, n is a quantity depending entirely Upon the location of the point at which the velocity is v. Then /=/o- 2 §9- FROUDE'S METHOD. 27 eddy resistance was usually about 8 per cent of the skin resistance. For well-formed ships of the present day it probably sel- dom exceeds this value, and usually falls below it, so that 5 or 6 per cent would be a fair average allowance. Now for full-sized ships the values of /, the coefficient of friction given by Tideman, are about 5 per cent greater than those used by Froude, so that if we are calculating resistance, it is preferable to use Tideman's Table, and not make a sep- arate calculation for eddy resistance. It is interesting in this connection to compare the resist- compan- ances of eddy-making area and frictional surface. Making and Fric- tional Sur- Joessel's formula for a plane of area A moving perpen- dicular to itself in sea-water becomes, for speed in knots, face. 7?e=4.62 A V^. From this the eddy resistance of one square foot at 20 knots would be 1848 pounds. This is equivalent to the skin resistance at the same speed (for a ship 300 feet long) of about 770 square feet of wetted surface. While the eddy- making appendages of ships do not usually, and should never approximate to planes moving perpendicular to themselves, the necessity for so shaping them as to reduce eddy resist- ance to a minimum is obvious. This is a matter largely under control of the designer, and there is seldom any good excuse for exaggerated eddy resistance. In this connection it should be said that if the lines of a Excessive ship aft are too full for her speed, large, unstable eddies are ^"aifect^" liable to appear, shifting from one counter to the other, and steering, appearing and disappearing in a capricious and irregular way. These suddenly formed and shifting eddies cause sudden and unsymmetrical changes of resistance, and render such a ship very difficult to steer. A case in point is that of the short and fuU-sterned armoured ship Aj'ax of Her Majesty's Navy, which steered very wild at full speed until her run was lengthened and made finer. 28 RESISTANCE OF SHIPS. § 10. Why justly called Froude's Law. § lO. The Law of Comparison. Before completing the description of Froude's Method, I shall deduce and discuss what is commonly called Froude's Law, or the Law of Comparison. The formula of mechanics which is the most general expression of Froude's Law was discovered long before Froude. Its successful application to the comparison of the resistances of ships and their models is, however, due entirely to the late Mr. William Froude, and indeed Froude appears to have attained this result without any knowledge of the general formula alluded to above. rig. 8. ■FOR LAW OF COMPARISON. Dednction of Principle of Law of Compari- son. Referring to Figure 8, let BB be a fixed horizontal plane, and aaa, a stream line in a steady stream of perfect fluid. Suppose the upper surface of the fluid exposed to a constant pressure — as, for instance, that of the atmosphere — which denote by it. Let p denote pressure due to fluid alone ; that is, the actual pressure, less the constant 'w. § lo. THE LAW OF COMPARISON. 29 Let V denote velocity of flow in feet per second. , Let z denote elevation above BB. Let w denote weight per unit volume of fluid. Then, at any point along the stream line, we have, from the steady motion formula, ^ + ^-+2=3. constant =^+—+^0, (O w 2g w 2g ^ ' where /o. ^0. and zo refer to the point of the stream line imme- diately above the origin O. Suppose, now, the fluid removed and replaced by a second, whose pressure, velocity, etc., are denoted by large letters. Then for a stream line AAA in the second fluid, #/ + — + ^=S+ — + ^»- (2) W 2g W 2g ^ ' Reducing (i) and (2), fi=h+ti:J^+,.,-o, (3) W 2g ^ '■ P—P V'^—V^ ^-^+i--^+Z-Z„=o. (4) W 2g Now let AAA be simply an enlargement of aaa, say n times as great. Since the motions are similar, v:v,= V:V,. Whence, Multiplying (3) by n and rewriting, «ZizA+«?^Yi _!iV«(^_^„)=o, (6) W 2g\ IT J and from (4), P-P^ W 2g\ 30 RESISTANCE OF SHIPS. § lo. Now, whatever the value of n, Z=nz; Za=nz^. We are still free to give n a value in terms of v and V. Let n=^ = ^. Subtracting (6) from (7), and cancelling terms which have become equal on the above supposition, we have If, now, the stream lines are around some solid, they will at a great distance from the solid be parallel straight lines. The removal of the origin to such a distant point in the case of the first fluid, and to a point n times as far in the case of the second, does not change any of the previous results. Suppose this done ; then — - and i^ will simply denote W w depths below the surface. Since the second fluid is on n times the scale of the first, undisturbed depths in the second will be n times undisturbed depths in the first. Therefore ^= n^- (9) Combining (8) and (9), 1J? = ^ (10) AppUcation Apply now this result to the case of a ship and its to Ship and , , ■, n ■ ■ r Model. model, supposed floatmg m streams of fluids of different densities. Let L, B, H, the length, breadth, and draught of the ship, be n times /, b, h, the corresponding dimensions of the model. Let D' and d' denote the volumes displaced by ship and model respectively. Then D' = «V, since the volumes of § 10. THE LAW OF COMPARISON. 31 similar solids are as the cubes of their linear dimensions. Let Fand v denote the normal velocities of the streams past ship and model, connected by the relation V=v Vn. Now if the model and ship, which are similar in every respect, produce similar stream lines, Equation io above (— =«— J applies to the pressures at corresponding points of the similar stream lines. If ds denote an element of surface of the model, and dS the corresponding element of the ship, evidently dS=tP'ds. Let d denote the inclination of the normals at ds and dS to the fore and aft vertical plane. Let J?„ r„ denote the part of the ship and model resistances respectively due to the stream-line pressures. Evidently rr= \ p cos 6ds, Rr= \P cos QdS, the integration extending over the whole immersed surface in such case. Now since P = n — /, and dS=n^ds, w R=J^ (p cos en'ds=n'^ Cp cos dds=n'^'; or, r, wt^ w d' Now, ^^2?'= weight displaced by ship, wd' = weight displaced by model; R, displacement of ship _D whence, — - ^jigpiacement of model d ' These ship and model resistances are to each other in the corre- ratio of the displacements, not at the same speed, but at speeds, speeds connected by the relation V^=nv\ where n is the ratio between linear dimensions of ship and model. These 32 RESISTANCE OF SHIPS. § lo. speeds V and v are called the Corresponding Speeds of ship and model. Now, V^ = nv^, n — —-, If L = nl, n = — ; whence, -7= -7 5 or, — -=^=^2, say. v' I LI Speed At corresponding speeds, then, the value of c is the same Ratio. for both ship and model. The quantity denoted by c is called the Speed Length Constant or Speed Length Ratio, the latter name being preferable since c is not a constant. Assump- In the deduction of the law of comparison two principal tions of Froude's assumptions were made. Law. J That we were dealing with a perfect fluid. 2. That the stream lines past models and their correspond- ing ships were similar — differing only in scale, justifloa- We know that water is so nearly a perfect fluid as to justify tion of . , Froude's the first assumption. The second assumption requires some Law. evidence to justify it. The elder Froude made numerous experiments upon models, similar, but differing in size, and found that so far as- careful observation could establish, the wave surfaces, and hence the stream lines, were similar at corresponding speeds. He found also that the Law of Com- parison applied in such cases. Froude recognised, however, that experiments with models, even though one were double the size of another, could hardly be considered conclusive in extending the law to full- sized ships. Accordingly with the assistance of the Admiralty he carried out towing experiments to determine an actual curve of resistance for the Greyhound — a ship 172 feet long and of more than 1000 tons' displacement. Figure 9 shows a comparison between the curve of resist- § 10. THE LAW OF COMPARISON. 33 ance obtained by towing the ship and that obtained by calculating the skin resistance and deducing the residuary resistance by the Law of Comparison from that of a model I of feet long. Figure 9 speaks for itself. So far as I am aware the Greyhound experiments have not been repeated with any other ship, but the wave patterns of many men-of-war have been compared with those of their models at corresponding speeds by the Froudes, and appear to have been similar, so far as could be established by close observation. Pig. 9. 700 860 960 1000 1100 1200 1300 SPEED IN EEET PER MINUTE. Full line — Actual resistance. Dotted Line — Resistance as deduced from that of model. — ACTUAL RESISTANCE OF GREYHOUND AND RESIST- ANCE AS DEDUCED FROM THAT OF MODEL. All things considered, we appear fully justified in accepting the Law of Comparison. It will be well at this point to try and learn something of Laws of the law followed by a resistance which satisfies the Law of ^Mch sat- Comparison. if*^, , ^ Froude's Suppose we have a resistance of such a simple nature that Law. for similar ships of different sizes, at varying speeds, it is always expressed by the formula b ■ LfV", where b, x, and y are constants. Let R' denote this resistance for a full-sized ship, and r' the same resistance for a model. 34 RESISTANCE OF SHIPS. §10 Then R' = bI>V', Now if Fand v are corresponding speeds, i-w-w- since I \d. R' D also at corresponding speeds— r = — r a Hence, R' d)\v) d' {©*}' whence. Now — is not equal to i, so ( — p^"^ can equal i only if d \dJ = 1. x+^—i=o. Hence if a resistance, expressed by the formula bD'Y", sat- isfies the Law of Comparison, we. must have x+-^=i. It is interesting to note the possible cases* for integral values of ^. They are shown below. Value oiy. Value oiK. Law of Resistance. Value oiy. Value of X. Law of Resistance. o I bD 4 I bD^V^ I 1 hiyv 5 i bD^V^ 2 3 bjD*F' 6 bV^ § 10. THE LAW OF COMPARISON. 35 y cannot be =o, for we should then have a resistance which did not increase with the speed ; nor can it be negative, for that would involve a resistance decreasing with the speed. Also y cannot be 6, involving a resistance which does not increase with size; nor can it be greater than 6, for that would involve a resistance diminishing with the size for unchanged speed. We know that eddy resistance varies as the square of the speed; and since it follows the Law of Comparison, it must follow the law, This result could also be demonstrated independently of the Law of Comparison. We shall see later that the wave resistance does not follow so simple a law as to be exactly capable of expression by such a formula as An approximate expression in this form is possible, however, which is applicable to many cases. Since wetted surface varies as the square of the linear dimensions, or as D^, the skin friction for similar ships of any size follows the law. This does not satisfy the Law of Comparison, and this is the reason why in Froude's Method the skin resistance must be calculated. The fact that the skin resistance does not satisfy Froude's Law is of importance in the extension of Froude's Method to the powering of ships. Its bearing will be considered in the ' proper place. 36 RESISTANCE OF SHIPS. § ii. § II. Froudes Method {completed). Apparatus J am now in a position to complete the description of Necessary. Froude's Method. For a full description of the apparatus and methods used by Froude, the reader should consult his papers read before the Institution of Naval Architects. Without going into technical details it may be said that the models used are usually made of paraffin, being cast to approximate shape, and finished to exactly reproduce the ship's form. They are from 9 to 25 feet long. A good and very common working length is 12 feet. When of paraffin the shell is an inch or so thick. The models are ballasted by weights or shot to any desired water line. The tank in which the models are towed should be 300 or 400 feet long, about 30 wide, and at least 10 or 12 deep. The towing and recording apparatus used in resistance work is fitted to tow the model steadily at any desired speed, and to record the speed and the corresponding resistance. There are many practical difficulties in the way of obtaining accurate results, and to secure reliable data there is needed refined apparatus and great care and skill in handling it. Example of Supposing all difficulties overcome, we can plot the results Froude's , . . Method. of towmg experiments upon a model in the shape of a curve such as AAA in Figure 10, showing the resistances of the model plotted upon speeds as abscissas. This curve repre- sents the- total resistance, made up, as we know, of skin resistance, eddy resistance, and wave resistance. We also know that the two latter alone — constituting the residuary resistance — follow the Law of Comparison. The first step, then, is to deduct from th« total resistance the skin friction, which is calculated (as, for instance, by the use of Table IV.). Setting down the skin friction from the curve AAA in Figure 10, we obtain the curve BBB, representing the residuary resistance of the model. §"• FROUDE'S METHOD. 37 Now we know from the Law of Comparison that this curve also represents the residuary resistance of the ship, pro- vided the scales of speed and resistance are suitably changed. In the case shown by Figure lo the model was JL the size 1 D of the ship ; hence corresponding speeds of ship and model are in the ratio Vi6: 1=4:1 and residuary resistances at cor- responding speeds are in the ratio 16^:1 =4096 : i. mo UO 160 180 300 220 StiO 260 280 300 , SPEED OF MODEL, FEET PER MINUTE _l 1 1 1— -H 1 1 1 1- 180 S60 640 720 800 880 960 1040 1120 SPEED OF.SHIP.,FEET- PER MINUTE "7 8 9 10 SPEED OF SHIP IN'KNOTS >I8 Fig. 10. Drawing in the scales for the ship as shown, the curve BBB represents the residuary resistance of the ship in either fresh or salt water, according to the scale used. (Salt-water resistance) = 1.026 (fresh-water resistance). It is now necessary to calculate the skin resistance of the ship, and set it up above BBB to obtain the curve CCC, which gpeciai ez- represents the total resistance of the ship. LT^ln*' The curves of -Figure 10 are those for H. M. S. Grej/- Resistance hound, given by Froude. Figure 9 shows the close agree- work!"^ ^ 38 RESISTANCE OF SHIPS. § 12. ment between the curve CCC, determined as just described, and the actual curve, determined by towing experiments upon the ship. While I have for clearness of description assumed Tables IV. and V. applicable to the case, the skin-resistance constants, used by Froude in this case, were obtained by special experiments. In accurate tank work the skin resistance of a model should always be deduced from direct experiments upon plane surfaces of the same length and area and nature as the model's surface. This is because an error, small in the case of the model, becomes much amplified in proceeding to the full-sized ship. Value oi Froude's Method is at present by far the most accurate Froude *s Method. and reliable known for determining the resistance of a ship. It should be pointed out, however, that it is not, does not pre- tend to be, and cannot be, exact. The skin resistance is calculated, and the coefficients used in calculations must often be different from the actual coefficients. Our knowledge of the exact frictional value of painted surfaces is hmited, and the standard, but twenty-year old experiments of Froude need supplementing by exhaustive experiments confined to painted iron surfaces in various con- ditions, and extending, if possible, to a length of plane greater than 50 feet. Such experiments would, I believe, show that Froude's and Tideman's coefficients are thoroughly reliable for working approximations, but would enable a somewhat closer approximation to be made in many cases. § 12. Phenomena of Waves produced by Ships. Need of It has been seen that the skin resistance can be calculated Formula for .,, ._ . ^ . , ,^, Wave Re- With an accuracy sufficient for practical purposes. The combined wave and eddy resistance can be very closely determined by Froude's Method. In the absence of tank facilities a satisfactory allowance can be made for eddy sistance. § 12. PHENOMENA OF WAVES PRODUCED BY SHIPS. 39 resistance, which should never be more than a very small percentage of the total. Since but few of us are favoured with facilities for model tank experiments, an approximate method for calculating the wave resistance in a given case is much needed. Of the nature, laws, and phenomena of wave resistance we can learn something from pure theory, and something from results of experiments given by the Froudes in various papers before the Institution of Naval Architects. We have seen that in fluid streaming past a submerged cause and , , ,• - , Genesis of body there are changes from the normal pressure and waves, velocity, the pressure in the stream being greater than the normal near the bow and stern, and less than the normal amidships. Tendency toward change of pressure must, of course, exist in a stream past a floating ship. But the pres- sure at the surface must remain constant — being the pres- sure of the atmosphere. Changes in velocity of flow appear ; but at the surface, instead of changes of pressure, changes in the level of the water make their appearance ; or, in other words, waves are produced. If, as imagined by Froude, we could surround a ship by a thin sheet of rigid ice on the surface of the stream, extend- ing to a great distance in all directions, there would be no changes in the level of the surface. There would be changes in pressure and velocity of the stream caused by the ship, but they would be confined to the vicinity of the vessel. When waves are produced upon a free surface, however, the case is different. The waves follow their natural tendency to spread, and much or all of the energy required to produce them cannot be returned to the ship. Consider now a ship with a long parallel middle body. Twosepa- The stream-line pressures at the bow will cause waves. The tems Jen- disturbance of the surface will spread away from the ship, g^p**"''* and if the parallel middle body is long enough there will be 40 RESISTANCE OF SHIPS. §12. at the stern no remaining disturbance due to the bow. BUt as the stream lines close in aft, there will be new variations of pressure which must cause new changes of water level, pro- ducing a new wave system. It is evident, then, that a ship tends to originate two sepa- rate wave systems, one forward and one afi- Pig. 11. — BOW WAVE SYSTEM. I shall distinguish them as the "natural" bow wave or bow wave system, and the natural stern wave or stern wave system. Bow Wave. Figure 1 1 shows the features of a bow wave system. For clearness, vertical heights have been exaggerated. Note. — From a paper by Froude, read before the I. N. A. We observe a diverging series of pronounced crests spread- ing away from the ship. The diverging crests are parallel to each other. The inclination of each crest to the line of advance of the ship is about double that of the "line of divergence," or the line separating the disturbed from the undisturbed water. It is to be noted that the line of divergence is straight or nearly so. § 12. PHENOMENA OF WAVES PRODUCED BY SHIPS. 41 It is also seen by inspection that each of the diverging crests appears to form the end of a "transverse" wave whose crest is nearly perpendicular to the line of advance, and shows in the figure against the side. The natural stern wave system, which tends to form at the stem stern, is similar in character to the bow wave. As a rule, however, we have at the stern more or less disturbance due to the bow wave, and this modifies the natural stern wave in a manner which I shall discuss later. It should be explained. 18 KNOTS. N.B. Position of Wave Crests indicated ty shading. Pig. 12a. PLAN OF WAVE SYSTEM MADE BY 83 FEET LAUNCH AT VARIOUS SPEEDS. too, that in ships with little or no parallel middle body the whole of the fore body comes into play in generating the bow wave, and the whole of the after body in generating the stern wave. As a further illustration of the features of the waves pro- wave sys- ^ , _,. , , tems in duced by a ship, I may refer to Figures \2a and \2b. practice. Note. — From a paper, read by the younger Froude before the I. N. A. These show in plan various wave systems. 42 RESISTANCE OF SHIPS. § 12. In this connection it should be remarked that, except at high speeds or with very bluff vessels, the wave systems are seldom so clearly defined in practice as shown by Figures I2«, etc. 83.FT. LAUNCH. 333 FOOT SKIP. N.B. Position of Wave Crests indicated by shading. Pig. 12b. PLANS OF WAVE SYSTEMS MADE BY DIFFER- ENT VESSELS AT 1 8 KNOTS SPEED. The transverse crests are ijot well marked, and the bow wave especially often appears at low speeds to consist entirely of the diverging crests. At high speeds, however, when the § 13- PROPERTIES OF TROCHOIDAL WAVES. 43 wave resistance becomes important, the features which have been described show plainly near the ship, and it is only at some distance astern that the transverse crests apparently disappear. The "diverging wave system" and "transverse wave sys- tem " are sometimes spoken of as if they were separate and disconnected, but this appears mechanically and physically impossible. They are essentially one. The diverging crests appear always to form the end of the transverse waves, though in some cases the transverse wave is all end, so to speak, extending within the line of the diverging crests but a very small distance toward the ship. The most beautiful large-scale wave systems of my ac- quaintance are those which spread aft from the paddle wheels of a powerful paddle steamer. Their transverse and diverging features are plainly marked, the former being pro- portionately more strongly accented than in the case of bow and stern wave systems. § 13. Properties of Trochoidal Waves. In this section I shall state certain properties of trochoidal waves, which may be found demonstrated by Rankine or other writers upon the subject. Rankine's trochoidal wave theory is founded upon , the Genesis oi Trochoidal assumption that in deep-water waves the particles of water waves, affected by the wave revolve uniformly in circular orbits. The theory is practically exact as applied to a system of uniform regular waves, indefinite in number, and advancing steadily in a direction perpendicular- to their crest lines. Let w denote the weight per cubic foot of the water. Let / denote the length of the wave (j.e. the distance from one crest to the next), and H the height of the wave from crest to hollow, both / and H being measured in feet. Consider a solid mass of water, extending to the bottom. 44 RESISTANCE OF SHIPS. §13- Kinetic Energy. Potential Energy. Transmis- sion of Energy. Compound Waves. Speed of Advance. , of breadth in the direction of the crests = b, and of length in the direction of propagation of the waves = /. Then we have the following properties : 1. The kinetic energy of the mass, due to the rotation of its particles, remains constant while the wave passes, and ' is equal in foot pounds to -^^ wblH^. 2. The mean centre of gravity is raised above the position it would occupy if the water were at rest, so that the mass has a certain amount of potential energy. This potential energy remains the same during the- passage of the wave, and is also equal in foot pounds to -^^wblH'^. 3. The potential energy remains constant because, while the mass of water considered is constantly receiving energy, from the water behind, it delivers energy at the same rate to the water in front. During the passage of one wave {i.e. during one wave period) the amount of energy so transmitted is equal to the constant potential energy of the mass =^wblH'^. 4. Let Hx, Hi, be the heights of two trochoidal wave series of the same length =27ri?, where R is the radius of the rolling or generating circle. Then if one series be super- posed upon the other with an interval between crests = a, we shall have resulting a single series of trochoidal waves of a height H such that /f 2 = ff2 + H,^ + 2 H^H, cos -, R and having the same speed of advance as each of the components. 5. If V denote the speed of advance in feet per second of a trochoidal wave, and R the radius of the rolling or generat- ing circle, then /, the length of the wave, = 2 -irR and. -gR 2ir § 14- DEDUCTION OF LAW OF WAVE RESISTANCE. 45 § 14. Deduction of Law of Wave Resistance. It is found that the distance fore and aft between succes- Trochoidai sive crests of the bow wave system is, as nearly as possible, fo^Bow ** the same as that between successive crests of a trochoidai '^^'^^ ^y=- tern. system, with speed of advance equal to the speed of the ship. Also at high speeds, when the wave resistance becomes a large portion of the total, the transverse waves which appear strongly resemble short lengths of trochoidai waves. We may then for convenience take the energy of the bow wave system as concentrated in the transverse waves, sup- posed to be trochoidai. Advancing into still water, such a system carries its poten- tial energy with it ; but the kinetic energy of the new parti- cles, constantly set in motion, must be derived from work done by the wave resistance of the ship. Let b denote the breadth at a given point from the line of Resisunce 1 1 • 1 1 .1 ^"6 to Bow divergence on the starboard side to that on the port side, wavesys- Let H denote the height at that point of the uniform tro- choidai wave, which is supposed to replace the somewhat irregular actual wave. Let R„ denote the wave resistance at speed V. Then R„ V denotes the work done agajnst wave resistance in unit time, and must equal the kinetic energy generated in the water per unit time. Now the time required to traverse the distance /( = one wave length) is -^- Then in the time required to traverse one wave length, tern. Work done by ship=7?„ Fx -j^RJ; Kinetic energy generated in via.tev=^gWblH\ Whence, RJ= ^V '^^^^^ or R^=^ wbH\ b changes but little with the speed ; hence we may reasonably conclude that as the speed changes the wave resistance varies as//2_ 46 RESISTANCE OF SHIPS. § 14. Now H is the height of the imaginary trochoidal wave, supposed to replace the actual wave, and cannot differ much from the mean height of the actual wave. The exact deter- mination of //" in a given case is impracticable, but, as will be seen, it is not necessary for the purpose in hand, stem The preceding applies directly to the bow wave, and would apply to the natural stern wave. But the actual stern wave is the resultant of the natural stern wave and of part of the bow wave. Although the positions of the first crests of the bow and stern wave systems of a given ship change with the speed, there will at a given speed be a fixed distance between them. Denote this distance, called the wave-making length, by s, and suppose s — mL, where »« is a coefficient varying slightly as the speed changes. Let R denote the radius of the rolling circle of the trochoi- dal bow wave. Then its length = 2 ttT? ; and since v^=gR, its length = g Now the distance from the natural position of the first stern wave crest to the bow wave crest next ahead of it is evidently the remainder after dividing s ; the distance from first bow crest to first stern crest, by , the length from crest to crest of the bow wave. Referring to the compound wave formula, it is seen that this remainder corresponds to a in that formula. Now let s ={n-\-q) , where « is a whole number, and q a fraction. Then a in the compound wave formula =qt """ ; and since R = -, p = 2 7r^, g R § 14. DEDUCTION OF LAW OF WAVE RESISTANCE. 47 cos — = cos 2 TT^ = cos (2 TT^ + 2 ttm) ^v" 2ir{n+q)— 2irR{n + q) = cos 5 — ^ = cos J mL = cos -7. = cos — 5- zr If g S = cos^-5-, where v is in feet per second, ='°'^(iij'^^^''^ ^'^ ^" ''"°*'- 172 Now — - is the speed length ratio squared denoted by c^. Substituting for g its value, and changing from circular or absolute measure to degrees, we have cos— = cos — x 646°. R c Now let H-^ denote the height of the bow wave at a given Resistance breadth b, H^ the height of the natural stern wave at the gt^,^" same breadth, and kH-^, the height of the bow wave when it ''^*'''- has passed aft to the point where the natural stern wave has the breadth b. Let H'^ denote the height of the actual compound stern wave at the breadth b. Then from the preceding H'i=k'H^+Hi+2kH^H,cos ^ 646°. The resistance due to the actual stern wave will be pro- portional to H'i, and the resistance due to the portion of the bow wave, not compounded with the stern wave, will be pro- portional to H^-Js'Hl 48 RESISTANCE OF SHIPS. §15- Totaiwave Then will the total wave resistance be proportional to Resistance. m-ji'm+H'i m -. H^ - k'Hi + l^H^ +H,'+2 kH,H, cos ^ 646° = H^' + i7/ + 2 kH^H^ cos ^646°. This formula is not in shape for practical use, because of uncertainty as to the values of Hi, H^, k, and ;« in a given case ; but it expresses, I believe, the true law of wave resist- ance, and useful practical conclusions, confirmed by experi- ment, can be drawn as to the characteristics and mode of variation of the unknown quantities. It was first given in a slightly different shape by the younger Froude. Height of Waves as aifected by Speed. § 15. Laws of Variation of Wave Resistance. Let us inquire first into the connection between H^ and H^ and the speed. We know that in perfect stream motion the excess pressure, at a point near the bow, for instance, will vary as the square of the speed. In the imperfect stream motion which exists, part of the excess pressure will be devoted as before to the production of stream line acceleration. The remainder will leak away, as it were, and be absorbed in raising the level of the water in the vicinity. Now if the resultant rise of level — or //i the height of the bow wave — were always proportional to the excess stream line pressure, we should have H^ varying as F^, and H^ as V^. The same reasoning applies to Hi. It appears entirely probable that at moderate speeds when the water has time to obey the stream line pressures, so to speak, and the vertical motions are small, the heights of the natural bow and stern waves do vary somewhat closely as the square of the speed. As the speed increases, involving greater wave height and vertical motion of the water, it § IS- LAWS OF VARIATION OF WAVE RESISTANCE. 49 appears that the wave height should not increase so fast as the square of the speed. Now Hx and Hi are proportional to the natural bow and stern wave resistances respectively. If the above reasoning is sound, it follows that the natural bow and stern wave resistance will vary at low and moderate speeds, as the fourth power of the speed, but that as the speed increases, the index will steadijy fall off. Consider next the coefficient k in the expression for wave variation resistance. It is a matter of common experience that at low speeds, when the length of the bow wave is small in comparison with the length of the ship, the bow wave has practically subsided close to the ship by the time the stern is reached, and hence can produce little or no effect upon the natural stern wave. As, however, with increasing speed, the length of the bow wave increases, so that fewer crests appear in the length of the ship, more and more of its energy is found in the vicinity of the stern — available as a component of the actual stern wave. The general nature of variation of k is then obvious, k must be equal to zero at low speeds, and increases with the speed. The .theoretical limit of k is unity, but it does not appear likely that for actual ships of the customary form k can ever exceed the value 0.5. There is little to be said about m. It appears that the dis- vaiueofm. tance between the first bow wave crest and the first stern wave crest is, in all cases, somewhat greater than the length of the ship. Also, as is natural, this distance appears to increase slightly with the speed. For ships of ordinary form and speeds a fairly safe value for m appears to be about i.io. It is to be regretted that the limited amount of data avail- able prevents the deduction of a close approximation to m, but it is not a matter of serious practical importance. Having considered the component factors, we can now form some idea of the nature of a curve of wave resistance. 50 RESISTANCE OF SHIPS. § 15. The wave resistance was taken as proportional to m+Hi+2kH^H^ cos ^ 646°. Wave Denote the resistance due to the natural bow wave by Fonnuia. A^V^, and to the natural stern wave by B^V*'. Then from what has been said, if i?„ denote the wave resistance, R^ = V" (A' + B' + 2kABcos'-^ 646°). c Features of What would be the nature of a curve of wave resistance sistance Calculated from this formula and plotted on speeds as Curves. abscissEc .? At low speeds k=o, n-^arly, and A and B are nearly con- stant. At such speeds, then, the wave resistance would vary as the fourth power of the speed, and would be expressed by F^ X a constant. As the speed increases, A and B fall off, while^y^ ceases to be negligible and steadily increases. Now the term 2^^^ cos ^646= r is sometimes negative and sometimes positive. If, then, the mean wave resistance equals V^{A^-\-B'^), the actual curve following the law, R^ = V (A' + B' + 2 kAB cos ~ 646°), c will sometimes rise above and sometimes fall below the mean curve. This will give rise to "humps" and "hollows" so called in the curve of wave resistance. As the maximum and minimum values of cos ^646° cor- responding to ^2x360° and «x (360°+ 180°) (where n is an integer) are not spaced at equal intervals of speed, the inter- vals between successive " humps " will be greater and greater as the speed increases. § IS LAWS OF VARIATION OF WAVE RESISTANCE. 51 As an illustration I refer to Figure 13, which shows a curve of values of cos ~ 646° plotted on speed. It is as- sumed that the ship is 300 feet long and that in this case m=i.os. .It should be remembered that c=-^c^=—- VZ L Fig. 13. Figure 14 shows two curves of values of cos — 646° plotted on values of c as abscissae. For the full curve m=i.oo, and for the dotted curve m= 1. 10. These are values of m likely to be met with in practice. m Pig. 14. — CURVES OF COS -r 646' The rapid fluctuations of the curves as c diminishes do not indicate features which would appear on a curve of wave 52 RESISTANCE OF SHIPS. § i6. resistance. For low speeds — say for c less than .6 — the coefficient k is practically zero, so it makes no difference what may be the value of cos — 646°. The first maximum value of cos — 646° at which the value of k cannot be ignored is that found about c = i.oo. Since at about this speed k is increasing, and A and B have usually not fallen off much, the "hump" found in this vicinity is a notable one. Few vessels, except torpedo boats, travel at a speed sufficient to surmount this hump. Figure 13 also shows a curve showing for a ship 300 feet long the length of the wave produced plotted on values of v. We have seen that, for speed in feet per second, if / denote the length of a wave of speed v, 27r g g \i6ooJ for speed in knots, and g= 32. 16. Whence g \3600J "^-^ = 167. 19 c^ for Z = 300. § 16. Results of Experiments on Wave Resistance and Approximate Laws. Having set forth in a general way the theory of wave resistance and deduced the corresponding formulas, I propose now to quote certain experimental results bearing upon the Froude's matter. Curves illustrating Figures 15-17 show various curves of residuary resistance Interfer- ence. obtained by Froude from model experiments. The residuary § i6. RESULTS OF EXPERIMENTS. S3 resistance includes both eddy and wave resistance, but in the cases in question the eddy resistance must be so small that the curves may be said to represent practically the >wave resistance alone. Consider first Figure 15. This shows curves of residuary resistance for various speeds, plotted on length of parallel middle body. The lengths of entrance and of run and their shapes are identical, being always 80 feet, while the length of parallel middle body varies from zero to 340 feet. 400 TOTAL LENGTH 300 OF SHIP 3^0 sio 2k sto sio 3^ 1 2^ 1^0 iiio ik lio ilm m LENGTH OF PARALLEL MIDDLE BODY nr Fig. 15. EFFECT OF ADDING PARALLEL MIDDLE BODY UPON RESIDUARY RESISTANCE. These curves are worthy of close study. The fluctuation of wave resistance due to the term in the formula 2 MB cos ^ 646° is marked. It is to be noted that as the speed increases for the same length, the fluctuation increases — due to the increasing value of k. Also as the length is increased for a given speed, the fluctuation decreases, due to the fact that k falls off under these circumstances. This is of course because when the ship is lengthened the stern wave system is initiated at a greater distance aft of the bow wave and there is less interference between the two. If the length of parallel middle body were sufficiently great, the 54 RESISTANCE OF SHIPS. § i6. h* - o 1 p 1 i 1 \ ,1 H- 1 - 1 LfNE RE6IDU ANC to 1 1 „ 3J o> i 2 •< fn M i i CO l\ 1 ^ m =0 w > 11. M o O O o o fe m 2 « 5 Jb- m o O o o o o u\ui £ m w a. 3 M fe P TO 00 05 00 CO CO CO CO BREADTH EXTREME FEET. cn \\\ b 03 to to to b M 1— CD JD OS 1— 00 CO I-" CO CO MEAM DRAUGI EXCLUS OFKEE FEET. CD TJ W W m m \\ \\ Ol Or OI cn cr< Ol ■- S ^ o w \ * — M Ol cn ^ H^ ot cn o -1 S o> Z -^ M \ CO CO Q rt^ CO CO o -^ -:i CD 00 CO CO m H ^ q o O o O o O m "m \^ ■' 00 t; ^ s. CD \S. tA ^^V. > o >.^ "^ ■5^ V \Q, ^ ^S^^r^*" b3 V\1p ^. ->- ^^o^„ M Mi' \ s ^ie ^. x \ \ \->' \ \ \ \o \ t!0 \ \ \% \ to \ \ \ \ _ x \ s\ F- i-\ I-' \ i-t. to 1^ OS to E\ 03 \ o D \ § § g o \ 8 \ § o \ § \ o bO_ o ° o ° \ ° \ ° ° 1 o RESISTANCE IN' POUNDS Fig. 16. CURVES OF RESIDUARY RESISTANCE. §i6. RESULTS OF EXPERIMENTS. SS curve of wave resistance at a given speed, plotted on length of middle body, would finally cease to fluctuate, and become a level line. Consider now Figure i6. It shows curves of residuary curves of resistance of ships of the dimensions stated up to a speed of Res-gtrMe 23 knots. These curves show plainly the "hunips" and X 476000 400000 ^ X / ?- S75000 L^ =4 OC ) ^ ^ -^ 350000 H =38.2 =14.65 /• / 325000 =4 Oi to ris / / 60000 300000 / 55000 275000 ^1 / / / 50000 250000 t / '^ =e / / 45000 225000 0-. 1 / . -^ 20000 100000 ^ / / 7-/ y i% f^ > 15000 75000 /• / y ■)* f ? 10000 50000 / / / / »e ■'■^ j3 ^ s> 5000 25000 X / ^ ^ ep oc i»! , ' ^ ^ — -- ^ ^ ._? r '] s 13 141 [51 6 1 71 8 9! X)S 1i ■8S »! a &i «s 7S «s 93 OS 13 12s AS 4! 153 «: [?S 8! 9d 44 1 J Sti ii 4:4 SiU. SPEED-KNOTS Pig. 17. CURVES OF RESIDUARY RESISTANCE, ETC. " hollows " which we have seen should characterise a curve of wave resistance. These features are more clearly marked in the curves of Figure 16, which are deduced from model experiments, than 56 RESISTANCE OF SHIPS. § 17, they would be in the case of actual ships properly designed to be driven at the speeds to which the curves extend. Residuary Figure 17 shows one of the curves of Figure 16 extended Resistance; o / o Extreme to a Speed of 46 knots. The curve of horse-power corre- sponding to the residuary resistance is also shown. The wetted surface o{ a ship of the dimensions and displacement of that in Figure 17 would be about 20,000 square feet. The horse-power absorbed by the skin resistance of this amount of wetted surface is shown in Figure 1 7. The paramount influence of skin resistance at low speeds and of wave resistance at high speeds is obvious. While no ship of the size specified in Figure 17 has ever moved at anything like the extreme speeds to which the curves extend, it should be borne in mind that a speed of 40 knots for the 400-foot ship corresponds to that of 20 knots for the 100-foot torpedo boat, though the latter can attain the speed of 20 knots only by the adoption of a shape and proportions very different from that of the model of the 400- foot vessel. A feature of Figure 17 worthy of note is that the " humps " and " hollows " of the curve of resistance are appreciably toned down in the curve of the horse-power necessary to overcome that resistance. § 17. Simplification of Wave Resistance Formula. Complete Let us return now to the wave resistance formula Formula. ' This involves too many coefficients of uncertain and vari- able values for everyday use. Omission of Obviously the first step in the direction of simplification Last Term. .,,,,,, Will be to drop the last term, for the reason that k is quite small for the speeds attained in practice by ships of any size. This leaves us R^= V*(A'+B'). § 17- WAVE RESISTANCE FORMULA. 57 Curve. Now we have seen that A and B are not constants, but Approxi- must diminish with the speed. "^** ^'^' So we cannot be sure in advance that this diminution is not so rapid that the best approximate formula for the wave resistance will be i?«, = V^xa constant *" instead of E„=V*xa constant. I am of the opinion that the best working approximation is given by the formula R„=bV^, where ^ is a constant. To test the justness of this conclusion we may assume that conclusions the residuary resistance of Figure 17 is expressed by Experi- mental or R„=bV\ or R.=b^V^- Then knowing the residuary resistance at each point from the curve, it is easy to plot curves of b-^, b, and b^. Figure 18 shows these curves. It is noted that all three curves show a marked lump at about 20 knots. It should be remembered, however, that the form and proportions of the ship were not suited to a speed so high as 20 knots, and that with a form suited to that speed, the excrescence would have been much less marked. Below 12 knots the wave resistance is of little relative importance, and we need not concern ourselves at present about what goes on at speeds above 30 knots. It is evident from Figure 18, speaking broadly (and bearing in mind that the lump about 20 knots is abnormal), that from 12 to 30 knots the curve of b-^ falls off steadily, and that of ^2 increases steadily, while that of b remains fairly con- stant. 58 RESISTANCE OF SHIPS. §i8. Evidently the formula R„ = dV* is best adapted to this case, and so far as my experience goes the wave resistance is Vt 13 U IS 16 17 18 19 20 21 22 23 21 25 26 27 28 29 30 31 32 33 31 35 36 37 33 39 W U 12 SCALE FOR SPEED IN KNOTS. Fig. 18. CURVES FOR LAW OF RESIDUARY RESISTANCE. expressed with sufficient approximation by this formula up to speeds such that the speeds length ratio does not exceed 1.2. § 1 8. Increase of Resistance in Shallow Water. EHect of Shallow It is well known that the form of water waves changes in Water upon shoal Water. According to the trochoidal theory the orbits Waves pro- q£ ^^ particles cease to be circular in shoal water and be- duced by ^ Ship. come elliptical, the excentricity of the ellipse increasing with the shoalness of the water. Other changes in period, etc., § 19- SQUAT AND CHANGE OF TRIM. 59 take place as shown in Table VI., taken from a lecture on water waves by the late Richard Gatewood, naval con- structor, U.S.N. The character of the waves produced by a ship changes in a marked manner in shallow water, and their height is aug- mented. The result is increased resistance. This is an important point in considering the location of a measured course to try the speed of ships. It is reasonable to suppose that at depths where a trochoidal wave of length such as to travel at a given speed is practically unchanged from the deep water wave at the same speed the resistance of a ship at that speed will not be afEected. Using Table VI., we have the following results ; Minimum depth for No Depth of Ship Knots. Change in Resistance. Water FEET. Appropriate 10 28 to Various Speeds. 12 40 14 55 16 71 18 90 20 III 22 135 24 160 26 188 28 218 30 250 The increase of resistance due to shoal water appears liable to exaggeration. Mr. R. E. Froude has stated (Transactions I. N. A., 1892) that for a 5000-ton ship at speeds no greater than 1 7 knots in water of 7 fathoms' depth the increase of resistance above that in deep water is but some 3 or 4 per cent. § 19. Squat and Change of Trim. The line of flotation of a ship moving through the water is different from her line of flotation in still water. We know that in perfect stream motion the pressures at squat. 6o RESISTANCE OF SHIPS. § 20. Change of Trim. bow and stern are in excess, and the pressure amidships in defect. Since the sections amidships are fuller than those forward and aft, it would seem that the ship in motion should show a tendency to sink bodily in the water. This effect is produced to a slight extent, and is called "squat." Its result is to increase the wetted surface, and hence the skin resist- ance, but the increase is very small. The changes of pressure referred to above correspond to perfect stream motion, and being distributed nearly symmet- rically forward and aft, cause little if any change of trim. But the stream line motion not being perfect and the excess pressure at the bow not being balanced by a similar pressure aft, a ship under way will nearly always change her trim more or less by the stern. The change is inappreciable in most cases, but with small high-speed boats is sometimes very marked. Being an effect, and not a cause of resistance (except as it changes the shape, etc., of the under-water body), change of trim is not a matter of much importance in considering resistance. § 20. Formula for Total Resistance. Before attacking the question of propulsion it will be well to summarise the results arrived at as to resistance. Formula Combining the expressions for skin, eddy, and wave resist- Resistance. ance, it appears that a comprehensive formula for total resist- ance would be R=.fSV^-^^ + KV'^+ VnA'^+E^+2kAB cos 646°). Eddy Resistance. The second term on the right-hand side should be made small by careful design. Also it appears that by adopting values of f in Tideman's Table (which are nearly 5 % larger than Froude's values), we make ample allowance for it. So exit, eddy resistance, and we write R=fSV'^^^+ V*fA^+B^+2kAB cos -^ 646°). § 20. FORMULA FOR TOTAL RESISTANCE. 6l The second term now represents wave resistance. We know that for low speeds — and by "low" I mean speeds for which the speed length ratio is not more than .5 to .6 — the wave resistance is seldom more than 10 % of the whole. It is allowable then, when dealing with such speeds, to ^^ calculate simply the skin resistance, and approximate to the total resistance by adding a reasonable amount to allpw for wave resistance. As the speed increases, this method becomes unsafe, and Practical Speeds, indeed is scarcely to be commended for any speed. But up to speeds for which the speed length ratio is not above 1.2, it is allowable to denote the wave resistance by d V\ This gives us R=fSV^-^^+6V\ This is the practical working formula which I propose to adopt. The question of the value of ^ in a given case will be dealt with later. At speeds for which the speed length ratio is greater than 1.2, we are driven to the complete formula, R=fSV^-^+ V^fA^+B^+2kAB cos J 646°)- The values of A, B, and k could, as things stand at pres- Extreme ent, be determined closely only by model experiments, and any one so situated as to be able to make model experiments need not struggle with approximate formulae, but can adopt Froude's Method in toto. Fortunately for the vast majority who have not access to model tanks, speeds for which the speed length ratio is greater than 1.2 are exceptional, having been reached only in one or two instances by vessels other than torpedo boats. Speeds. CHAPTER II. THE PROPELLER. § I. Preliminary and Definitions. Propeller considered apart from Sbip. Theories in Two Classes Disc Theory. In discussing resistance I considered the ship alone — apart from its means of propulsion. I propose in the pres- ent chapter to discuss the propeller alone, — apart from the ship, — taking up later the modifications in the action of the propeller due to its connection with the ship. The problems to be solved are simplified by attacking them in detail. There have been innumerable theories of the action of the propeller. They can be divided into two classes. To the first class may be assigned the theories, which consider the effect of the propeller upon the water, and from the motion of the water deduce the reaction upon the propeller. The " disc theory," so called, of Rankine is a notable example of the first class. To the second class belong the theories which consider only the action of the water upon the propeller, using semi- empirical or experimental methods for dealing with it. The " blade theory " of Froude is a type of this class. At first sight the first class of theories would seem to have the advantage. Given a certain amount of water having a- certain change of velocity impressed upon it, the reaction resulting can apparently be calculated at once from the known density of water. This method would possess a beautiful simplicity if we knew the exact effect of a pro- peller upon the water which it passes through, and if the pro- peller blades were frictionless. Rankine assumed that a 62 § I. PRELIMINARY AND DEFINITIONS. 65 screw propeller gave to a column of water, having a sectional area equal to the " disc " swept by the propeller, a sternward velocity corresponding to the slip. That this is impossible is manifest to any one who has seen a propeller as ordinarily fitted, and considered its working. Useful results may be obtained by supposing only a certain fraction of the disc area column to be given the sternward velocity of the slip. That fraction, however, can be determined only by experi- ment or semi-empirical methods. The friction of the pro- peller will still remain to be dealt with. A curious feature about most of the theories of the first class is that they consider only the change of velocity of the water acted on, while the change of pressure is a matter of equal importance. It is for this reason that theorists of the first class so often give the maximum theoretical efiBciency of a propeller as I — ( ^-^ ) instead of (i— slip), the true theoretical maxi- mum, when neglecting friction. Rankine gave the value (i— slip) for ordinary propellers, but concluded that a form of propeller which would work without "shock" was capable theoretically of an efficiency /^i_?iiP\ How a propeller working in "solid water" could administer to it this mysterious " shock " Rankine did not explain. In Froude's theory, typical of the second class, the face of Blade a propeller blade is treated as if it were made up of a number of small inclined planes advancing through the water. This theory appears logical, and is, I believe, the most sim- ple in practical application. The elder Froude in bringing forward his theory before the Institution of Naval Architects, in 1876, confined himself principally to the discussion of a single small plane element of the face of a blade. From this and other causes the idea arose that Froude's theory, followed out, would give results at variance with common and good 64 RESISTANCE OF SHIPS. §1. Right and Left handed Propellers. Face. Back. Leading Edge. Following Edge. practice. I hope to demonstrate that this is not the case. Before proceeding further with the subject, however, it will be well to give some brief definitions and state the notation that will be used. Figure 19 shows a four-bladed right-handed propeller. A right-handed propeller, viewed from aft, turns with the hands of a watch when driving its ship ahead. Under similar circumstances a left-handed propeller turns against the hands of a watch. Fig. 19. — FOUR-BLADED RIGHT-HANDED SCREW PROPELLER. The "face" of a blade is the rear face^ the side which drives the water aft when the ship is going ahead. The "back " of a blade is the side opposite the face. Care must be taken to avoid confusion, from the fact that the face of a blade is aft, and the back is forward. The "leading edge" of a blade is the edge which leads; i.e. cuts the water first when the screw is turning ahead. The "following edge" is opposite the leading edge. Tip. § I. PRELIMINARY AND DEFINITIONS. 65 The "diameter" of a screw is the diameter of the circle Diameter, described by the tips of the blades. In symmetrical two and four bladed screws it is simply the distance from the "tip' or outermost part of one blade to that of the opposite blade. The " pitch," at a given point of the face, is the distance Pitch, in the direction of the axis of the shaft which an elementary area of the face at the point, if attached by a rigid radius to the axis, would move during one revolution, if working in a solid fixed nut. The pitch may be different at every point of the face. If it is the same at all points, we say the pitch is "uniform." If it is greater along the following than the leading edge, increasing we say the pitch "increases axially." If it groiys greater as we leave the centre, we say the pitch "increases radially." The term " increasing pitch," used without qualification, always refers to axially increasing pitch. The " area " or " developed area " of a blade is the surface Blade Area, of its face, and the "blade area" of a screw, sometimes called its " helicoidal area," is the area of all its blades. The " disc area " of a screw is the area of the circle described by the tips of its blades. The " boss " or " hub " of a screw is the cylindrical or Boss, spherical centre to which the blades are attached, and which takes hold of the shaft. .The "pitch angle" at any point of a screw is the angle Ktch between a tangent plane at that point and a thwartship plane. Thus, Figure 20, if LNP is a section of the tangent plane at O, OD being the fore and aft line, and OM a section of the thwartship plane, the angle POM is the pitch angle. When a propeller is working with " slip," it advances during siip. each revolution a distance less than the pitch, the difference „. . , Slip Angle. between its actual advance and the pitch indicating the amount of "shp." When working with slip, an element such as LL, in Figure 20, does not advance parallel to itself along Disc Area. 66 RESISTANCE OF SHIPS. Fitch and Diameter Ratios. §1- The angle OP, but along a line such as OS, inclined to OP SOP is called the "shp angle." The "pitch ratio" of a screw is the ratio Pitch -5- Diameter, Conversely, the diameter ratio is the ratio Diameter h- Pitch, 1 ' 1 / \ \ 1 / ' / 1 / 1 1 Fig. 20. — MOTION OF ELEMENT OF SCREW FACE. It is necessary in dealing with screws to consider the whole blade from the tip in. Now taking pitch constant, as we come in toward the centre, the pitch ratio increases inversely with the diameter, while the diameter ratio decreases directly § 2. ELEMENT OF FACE OF BLADE AS A PLANE. dy as the diameter. Consider a screw of 8 feet diameter and lo feet pitch, and suppose the hub i foot in diameter. Then the pitch ratio ranges from 1.25 at the tips to 10 at the hub. The diameter ratio ranges from .8 to .1. Used without qualification, pitch ratio and diameter ratio usually refer to the extreme diameter of the screw. In what follows — d denotes diameter in feet ; symtois. ;' denotes radius in feet ; p denotes pitch in feet ; A denotes surface in square feet ; R denotes revolutions per minute ; V denotes speed of ship in knots ; F' denotes speed of propeller in knots. By " speed of propeller " is meant the speed corresponding to no slip ; thus if speed were measured in feet per minute, the speed of . the propeller would always be denoted by pR. s denotes slip expressed as a fraction of the speed of the propeller; i.e. V -V s=- y denotes diameter |^ratio = d § 2. Element of Face of Blade treated as a Plane. I propose now to discuss the action of a small isolated plane area when it is given a motion similar to that of a small element of the face of a propeller blade. Referring to Figure 20, let LL denote a small plane set at Nature of the angle 6 with an axis O which revolves while it advances. pi°^'°° "* Then supposing first that there is no slip, during one Element, revolution, LL, moving along a spiral path shown in plan by 68 RESISTANCE OF SHIPS. § 2. OCCD, will advance to D. If slip is present so that the advance instead of being OD is some lesser quantity, say OD], the spiral path of advance will be denoted in plan by Let r denote the length of the radius from the centre O to middle of the element LL. Then lay off 0M=2 7r^= circumference of circle described by LL when looked at from- aft. Set up MP=OD=th.Q pitch of the element. Let J/^S = sin ^+7^^ ^^ ((9p)2 cos a-p^'rrY+{i-sf\s.'jry -.R^dA _Vi + TT^y VTT^y + (I _ j)2 Vi + Try + /y (i + Try) iry Vi + Try =yie^^^r^.g^ ^-^'+('-")V /Trr vrr^^l L I+Try -^ J ] § 2. ELEMENT OF FACE OF BLADE AS A PLANE. n Determine now the useful work, i.e. the work done in the useful direction of thrust, denoted by dU, and the gross work deliv- ^ork. ered to the element, denoted by dN. Evidently (Figures 20 and 21) dU=dTxMS=dTxpR{i-s) --fRHA as{\.—s) 7ryV7ry+(l-j)2 i+7ry -/(I. V ■)Vi2+7r2. dN=dM-x. irdxR^dM- 7^7 -pR There is no good result obtained by writing these compli- cated expressions at full length. So denote Then Vi + Try by F; TT^j/^Vi + Try by Z. dU= fR\\-s)(flsX'--fY) dA, dN=fR\asX +fZ)dA. The efiSciency of the element, denoted by e, is the ratio Efficiency, between the useful work done by it and the gross work delivered to it. T^, dU I \ asXx —fY Then e= = (i —s) — ^-— • dN ^ ' asX^'+fZ The quantities X^,. Y, and Z can be readily calculated. The first, X^, depends upon the diameter ratio y, and the slip s, while Y and Z involve diameter ratio and known constants only. Table VII. gives values of X^ for various diaimeter ratios and slips, and Table VIII. gives values of Fand Z for various diameter ratios. XS r, and Z. 72 RESISTANCE OF SHIPS. § 2. Curves of It is to be observed that the value of the efficiency depends ciency. ^p^^^ ^.j^^ relative value of a and /, the thrust constant and the coefficient of friction. The expression for the efficiency may be written .= (!-./ -sX^-Y "■ sX^+Z f Froude, as a result of experiments made before 1876, assigned to a the value 1.7, and to/ the value .0085, the unit of speed being one foot per second, and the unit of force one pound. The value of / is double what may be called 'its natural value, to allow us to deal with the area of the face of a blade only, and yet to express the friction of the equal area of the back. For a—i.y and/ = .oo8s —=200. The expression for efficiency then becomes 200 sX^-Y e={i-s) 200 sX^ + Z Figure 22 shows in full lines the curves of efficiency of elementary areas of the diameter ratios indicated, plotted on slips as abscissas. They were obtained from the above formula by the use of Tables VII. and VIII. The dotted curves in Figure 22 show very closely the efficiency curves of actual small screw propellers — the data being taken from papers read by the elder and younger Froude before the Institution of Naval Architects in 1883 and 1886 respectively. It seems impossible for any one to observe the essentially similar nature of the full and dotted curves of Figure 22 without feeling the force of the words with which the elder Froude ended his paper on the subject before the Institution of Naval Architects in 1878: "No theoretical treatment of §2. ELEMENT OF FACE OF BLADE AS A PLANE, 73 the action of an actual screw can be sound which does not incorporate and mainly rest on the principles embodied in the treatment of the problem of the plane, and indeed the char- acter of the results must, in their most essential features, be the same in both cases." Fig. 22. EFFICIENCY OF ELEMENTS AND OF MODEL PRO- PELLERS. To determine X\ we have the expression X,= •7rVV7rV+(i-j)'' I + TT V Substitu- tion of X torXK A little reflection upon this formula and an inspection of Table VII. will make it evident that the difficulties of tabulat- 74 RESISTANCE OF SHIPS. § 2. ing and handling X^ are largely increased by the presence of (i— j)^ under the radical sign in the numerator. Now the slips with which propellers work in practice vary from .15 to .30, as a rule. I have accordingly inserted in Table VIII. a column giving X, which is the value of X^, for a slip of .2. Table IX. gives comparative efficiencies for ele- ments of various diameter ratios obtained by using X^ and X. It is seen that for slips occurring in practice the efficiencies are practically identical. The useful and gross work obtained by using X will be slightly less than if X^ were used for slips below .2, and slightly greater for slips above .2. We shall see later, how- ever, that we thus probably obtained a closer approximation to the working of actual propellers. So henceforth I shall discard X^ and use X, as being sim- pler and better in every way. Then the quantities X, Y, and Z of Table VIII. are functions only of the diameter ratio and determinate constants. Maximum In concluding the discussion of the plane element, I- pro- of ETement. pose to deal with its maximum efficiency under various con- ditions. Our expression for efficiency is e=(i-s)^ s-x-y Denote -Xhyc; f Xhv^: s-X+Z Then ^=(i-j)^^— — , cs-\-Z or e(cs-\-Z)=cs —Y —cs'^-\-sY. (i) Differentiating with respect to s, —- {cs + Z)+ce=c—2cs + Y. (2) as § 2. ELEMENT OF FACE OF BLADE AS A PLANE. 75 At the slip corresponding to maximum efficiency, the effi- de ciency curve is horizontal, or — =o. ds Then if e^ and s^ denote maximum efficiency and slip cor- responding, we have from (2) ce^=c-2cs^-^Y, (3) and from (i) eJfs^^Z)=cs^-Y-cs^+s„Y. (4) Solving (3) and (4) for e^ and j„, we have c s^=-\ -Z+^{Z+c)(Z+V)]. c Also if ^„ denote the angle of slip corresponding to s„. Vi + Try Vtt^/ + .64 Figure 23 shows graphically the maximum efficiencies attainable by elements of various diameter ratios on three suppositions as to a-i-f. The slip angles corresponding are also shown. It is to be observed that the absolute maximum efficiency in each case is found at about a diameter ratio = •35- This applies only to a single element and does not indicate that an actual propeller will develop maximum efficiency if its diameter ratio (corresponding to the extreme diameter) is .35- Figure 23 if rightly interpreted affords conclusive evidence that the diameter ratio of the actual propeller must be greater than .35 for maximum efficiency. For suppose it were but .35. Then the tips of the blades would be working with high efficiency, but the bulk of the blade (within the 76 RESISTANCE OF SHIPS. §3- tips, where the diameter ratio would be less than .35) would be working with much less efficiency. The extreme diameter ratio should evidently be such that the bulk of the work is done under the most efficient conditions. 8° .-8 .4 -B DIAMETER RATIO. Fig. 23. Difference between Small Plane and Element of Face of Propeller. § 3. Extension of Formula for Plane to Propeller. The results obtained so far apply, it must be remembered, to an imaginary isolated small plane given the same motion as an elementary area on the face of an actual propeller blade. What are the points 'of difference between the small plane and the element of a propeller blade face .' I. The plane is isolated; the blade element is surrounded by other elements. §3- EXTENSION OF FORMULA. ■^j 2. The plane is of negligible thickness ; the blade element has the thickness of the blade behind it. 3. The plane is supposed to move in still water ; the blade element moves in water already set in motion by the action of the propeller. The exact effect of the differences enumerated above cannot be predicated on theoretical grounds. It appears probable, however, that their effect is not great, and it is exceedingly probable a priori that it is simply equiv- alent to a modification of the thrust constant a and the coefficient of friction f, so that results obtained from experiments on planes will not be exactly applicable to actual propellers. Without taking up at present the question of the exact values of a and / for a given case, let us proceed to extend the methods already explained to the full-sized blade. I shall take a blade of uniform pitch. Figure 24 shows an expanded propeller blade. To properly Developed develop a blade, it should be first expanded or twisted until "aBiade*' it is all in one plane, and then a new contour line drawn through the extremities of Imes perpendicular to the centre ^^^ single line of the blade and of the length of corresponding circular ^•^*^- arcs to the contour of the twisted blade. To deduce a blade from a given developed shape, the above process should be reversed. Referring to Figure 24, let / denote the length of the strip at radius r, as shown, the width of the strip being dr. Then Idr is an elementary area corresponding to dA in the formula for the plane. Then for the elementary strip, dU=fRHdr (i - s) (asX-fY), dN=fRHdr {asX+fZ). Let U and N denote the useful and gross work of the whole blade. 78 Then RESISTANCE OF SHIPS. U=^dU=fR?^{\ - s) (cisX-fY) l-dr; N=j'dN=p^R^^{asX+fZ) /• dr. §3- I. l^^W^V\v.V\^S\v\VV WAS.^^ c ^>>>>>>>>>>>>>>>>>>>>>>y>>>y f<'i.lmwi\V.<<<<'N.i .<<'\'vm"ys^m 31 < < < CO u « Hi o H H 3 s i ffi < g li- D 5 ° Ul tn fs a UJI 1 0-1 1 °\ / jl / ul\ / >\ / m / c U- M LINE OF HUB / < I to U- o CO A ^ , c A Fig. 24. — EXPANDED PROPELLER BLADE. It is convenient to replace / and dr by the ratios - and — , d d d denoting the extreme diameter of the propeller. Making these substitutions and noting that s is constant for the assumed uniform pitch, we may rewrite our expressions, N=p^R^d^\as Cx^^+/Cz I ^] L '^ d d ^ d d } the integration in each case extending over the whole blade. § 3- EXTENSION OF FORMULA 79 / Now the quantities The char- acteristics. depend on the diameter ratio, the proportions, and the shape of the blade, but not on the size. Their value in a given case may be said to be " characteristic " of the blade. Then J 7 7 ^--^ hy X^, called the X characteristic ; / dr Y- — by Y^, called the Y characteristic : a a ■^■--^by Z,, called the Z characteristic. a a We have then U=p^R^d^ [as{i-s)X,-f{x-s) FJ, N^p^R^d"^ [asX,+/Z,]. As will be seen later, the calculation of X„ Y„ and Z^ from the developed plan of a blade is a simple process. So far I have considered only a single blade, and taken no account of the units of speed and power used. The numerical expressions for the values of a and / will Numiierof • TkT ■ ■ • Blades and depend upon these units. Now it is convenient to retain units used, nearly the same numerical values for a and f as if the knot were the unit of speed, and the pound the unit of force. But we have introduced horse-power, and it is customary and convenient to express pitch and diameter in feet, and revolutions by the number made in one minute. Evidently, then, a factor must be introduced. Expressing pitch in feet and revolutions by the number made in one minute, the unit of speed is one foot per minute. Now one foot per minute = gg^p knots. Hence to denote thrust or force in pounds we should multiply the expressions above by (gffo)^- \3 lOOOOOOOOO I 80 RESISTANCE OF SHIPS. § 4- Now thrust x/i?= work done in foot-pciunds per minute = 33)OOOXthe horse-power developed. Then the factor to be introduced is 33 0"0 ^ (eOff?) ~T3^858666- This is an unwieldy quantity. If for 338858666 we write 333333333^, our results will be much simplified without any real error. The only effect will be that a and / will be expressed by numbers about one and a half per cent smaller than if the change had not been made. The quantity {pR)^ appearing in our expressions will be an unwieldy one in practical cases, since pR is seldom less than 1000 or greater than 3000. So let us substitute for (pR)^ the equal quantity ' VIOOO/ If n denote the number of blades of a propeller, its thrust, horse-power, etc., will be n times those of a single one of its blades. Fomuiaefor So we have finally for a propeller Complete Propeller. lOOOOOOOOO / pR \3 „ Similarly, ^=3 n [j^ d^ [asX,+fZ;\. These expressions are rigorously deduced from the funda- mental Assumptions which were discussed and justified as they were made. § 4. Values of a, f, and the Characteristics. Method of The question now arises, how are we to assign values to Determin- ingaand/. the Constants a and/ for a given propeller.'' Suppose we determined by experiment, with a known propeller, working at a known number of revolutions with § 4- VALUES OF a, /, AND THE CHARACTERISTICS. 8 1 a known slip, the values of U and N. We should then have two equations to determine the values ot a and/. Determi- nation of U and N for the same propeller, at a different num- ber of revolutions, and working with a different slip, would enable us to determine a second set of values of a and /. So by sufficiently extending our experiments we could deter- mine values of a and /for the propeller in hand throughout the range of slip likely to occur in practice. There have never been such experiments made, so far as Ftoude's I am aware, upon propellers of any size unattached to an ^^^^[ actual vessel. The most valuable experimental results avail- Propeller- able will be found published in a paper read by Mr. R. E. Froude before the Institution of Naval Architects in 1886. The experiments were made upon four-bladed model pro- pellers 0.68 of a foot in diameter. The blades were elliptical in shape, and each blade devel- oped would have been (but for the boss) a perfect ellipse touching the axis of the shaft. The extreme width, or minor axis of the ellipse, was equal to .4 of the radius. The elliptical form was departed from slightly in the vicinity of the boss, which was in diameter about ^ the diameter of the propeller. Figure 25 shows very approximately Froude's elliptical character- blade twisted into one plane. Before proceeding further I Froude''s shall show in detail the calculation of X„ Y„ and Z„ in this ^'^^^• case. For the blade shown the extreme diameter ratio is .8, corresponding to a pitch ratio of 1.25. The radius from the tip to the boss is divided into equal parts at points corre- sponding to diameter ratio .1, .2, .3, and so on. At each point of division is determined the value of /, the width of the blade, and the value of -. The diameter d is of course the d extreme diameter, not the diameter to the point where the width is measured. From Table VIII. the values of X corresponding to the 82 RESISTANCE OF SHIPS. §4- diameter ratios at the several points of division are picked out and multiplied into the corresponding values of — . d This gives the data necessary to enable us to draw a curve of X -, as shown in Figure 25, upon the radius, as base. a The area of this curve is readily determined, preferably by an ordinary planimeter. BASE LINE FOR CURVES Pig. 25.- - SHAPE AND CHARACTERISTICS OF FROUDE EXPERI- MENTAL BLADE. The value of X^ bears a ratio to the area of the curve of X — , depending upon the scales used. a Thus if A^ denote the area of the curve in square inches, while the curve was drawn on the scale i inch = n, and the blade diagram on the scale i inch=_o- feet = — of the radius, we shall have X^= A^. 2 m J J The curves of F - and Z — are also shown in Figure d d 2$, and the values of F„ and Z, are readily determined from § 4- VALUES OF a, /, AND THE CHARACTERISTICS. 83 them, following exactly the same process as in determin- ing X^ The values of the characteristics deduced from Figure 25 are X,= .o7S, Y,=.i27, Z,= .322. These are for the Froude elliptical blade of diameter curves ot ratio = .8. The characteristics of the same shape of blade i^tics. .6 / / .5 / --- 0-* / N ^;^ 2- -« H -8 Q.15 .4^ 8 1 ^^ Cy' -^ X .2 ir.lO . UJ CUR\ t OF YC / • e--1^ ^ .1 0.OE y ji^ oJ>>^ ^ - — :=:^ =^ 1 i 3 t 5 3 r 3 3 1 SCALE OF MAXIMUM DIAMETER RATIO Fig. 26. — FROUDE EXPERIMENTAL PROPELLER. CURVES OF CHARACTERISTICS. with other diameter ratios may be readily calculated, and from a sufficient number of such determinations curves may be drawn, showing the three characteristics plotted on the values of maximum diameter ratio as abscissae. Figure 26 shows the characteristics of Froude's elliptical blade up to a diameter ratio of i.o. 84 RESISTANCE OF SHIPS. §4- Constants of Froude's Propeller. ' Value of f . Value of a. Having determined the characteristics of the blade used in Froude's experiments, we can from the curves of useful horse- power and eflficiency published in the I. N. A. paper of 1886, deduce values of a and / in the manner outlined in the first part of this section. The curves from the experimental results, as given by Froude, are not in the shape best for the determination of a and f. It is found, however, that the values of a and f deduced from them are fairly consistent. The average value of / is about .045. This corresponds to a coefficient of friction of .0225, which, while double that for a smooth plane surface, seems not unreasonable for a propeller blade possessing a certain thickness and edge resistance. Upon attempting to determine a, it was found to vary with the diameter ratio of the propeller. This is entirely natural ; for the less the diameter ratio, the thicker the slices of water between successive blades, and the less the interference of the blades with each other. The value of a depends also' upon the number of blades. Froude states that the thrusts of four, three, and two bladed propellers of identical blades are as i. 000:. 865 :.6s instead of as 1. 000 : 75 : 50, as would be the case if a were constant. The- increase of the thrust per blade as the number of blades is diminished is probably due to the diminution of interfer- ence with the fewer blades. TJie values of-« as finally determined from the analysis of' Froude's results may be expressed as below. If m denote the extreme diameter ratio — For four-bladed propellers, a = ?>.^—\.om. = 7-tf-(^ux.uj:^ For three-bladed propellers, «= 9.4— 1.2 »e. a ?-,i3( J!f )' For two-bladed propellers, «= 10.4-^ 1.4 w. 9: j,oi{^ ^ )' While these values of a apply directly only to the experi- mental' propeller used by Froude, they appear to be close §S' EFFICIENCY AND POWER OF BLADES. 85 approximations to the exact values for full-sized composition or manganese bronze propellers, such as are ordinarily fitted to tnen-of-war and fast passenger steamers. For the thicker and blunter cast iron blades customary in the merchant marine the values of a are less than those given by the above.- I shall say more about the values of a and/ in a later chapter. In any propeller the " mean width of blade " area of blade Width Ratio. radius of blade — radius "of boss For the experimental propeller just discussed, the ratio between the mean width of blade and the diameter of the propeller was .166. This ratio is a useful quantity, and it is well to give it a name. I shall call it the " mean width ratio." The mean width ratio of propellers as usually fitted is seldom so low as .166. § 5. Efficiency and Power of Various Shapes of Blades. I propose in this section to discuss the effect of shape of Data,oi blade upon the efficiency and power of a propeller. In ghlpes. Figure 27 are shown developed five shapes of blade, all of the same area, radius, and radius of boss. Their- principal dimensions are given below. Shape No. I. No. 2. No. 3. No. 4. No. 5. Diameter of boss \d %d \d ^d %d - Maximum width .2023 d .16861/ .Id .2023 d .2023 d Minimum width •1349"^ .1686 d .1349 d ■1349 < Width ratio .1686 .1686 .1686 .1686 .1686 It is seen that the five blades differ only in shape. No. 3 is identical with Froude's experimental blade except as- to 86 RESISTANCE OF SHIPS. §S- boss. The boss has been taken in each case as | of the diameter. This is a fair average of the sizes of boss custom- ary in good practice. No. 1.1 No. 2. Pig. 27. DEVELOPED SHAPES OF BLADE. Charac- teristics. The X, Y, and Z characteristics of the five shapes have been calculated up to an extreme diameter ratio of i.o, and are shown graphically by Figures 28, 29, and 30. It is to be observed that above diameter ratio of .5 blade No. 5 has always the greatest characteristic, and in each case the remainder follow in the order No. 2, No. 4, No. i, No. 3. §s- EFFICIENCY AND POWER OF BLADES. 87 7>»-^*-^ .0 .1 .2 .3 .4 .5: ,6 j/ ".15^.3 / H X"^ # <." 8 .14 .2 / >: Co 'RESP 5n5w^ ^^ ,13 .1 / .12 / .2 .3 .4 .5 .6 .7 .8 SCALE FOR EXTREME DIAMETER RATIO. XQ Pig.- 31. AVERAGE MAXIMUM EFFICIENCY AND SLIP FOR FIVE BLADES. The expression for the useful horse-power U delivered by a propeller is (page 8o) U=ln pR lOOO. d'^\_as{i-s)X-f{i-s)Y:i. Take all five propellers as three-bladed lo feet in diameter, of 14.286 feet pitch (diameter ratio = .7), and making 117.6 revolutions per minute. EFFICIENCY AND POWER OF BLADES. 91 Take for a and / their values as deduced from Froude's experiments, namely / = .045, fl = 9.4— 1.2 x. 7 = 8.54. Then our formula becomes y=3X3^^4-286xii7.6)3^^^^^3^^_^^ 1000 x{8.s6j;ir;-.o4s y,\ =900 X 4.741632 X 8.56 X„ (I -^) j.f-^4gj = 36S3oX„(i-.r) j .r-. 00526^ j • The characteristics of the various blades as obtained from Figures 28, 29, and 30 for a diameter ratio of .7 are as below: X, Y, Y,-^X, No. I. .0630 .1088 1.726 No. 2. .0670 .iiiS 1.668 No. 3. .0613 .1080 1-763 No. 4. .0663 .1108 1.672 No. 5. .0700 •I 143 1.632 Substituting the various characteristics in the general. Results for " Revolutions formula, we have v^ constant. For No. I, C/=230i (i-j) (J-.0091). For No. 2, U= 2448 (I - s) (s - .0088). For No. 3, Z7=2237 (i-j) (J-.0093). For No. 4, U= 2420 (i-j) (J-.0088). For No. s, ^7=2557 (i-j) (j-.oo86). Figure 32 shows curves of useful horse-power plotted from the above five equations upon slips as abscissae. The revolu- tions having been assumed constant at 11 7.6, increase of 92 RESISTANCE OF SHIPS. §5- Results for Speed of Advance Constant. slip must be obtained by decreasing the speed of advance of the propeller. Let us now determine similar curves when the speed of advance remains constant, and increase of slip is obtained by increasing the revolutions. 600 500 400 < 300 o 200 100 ^ ^ ^ ;tf3^ ^ w A V / y so ^LE FOR i iLIP .05 .10 .15 .20 .25 .30 .35 Fig. 32. HORSE-POWER AT CONSTANT REVOLUTIONS. Assume in the above case that the constant speed of advance is 1680 feet per minute, i.e. that at 117.6 revolu- tions the slip is nil. Then the only change necessary in the original formula is the substitution of itZ:^ for i—s 1 17.6. S- EFFICIENCY AND POWER OF BLADES. This gives the following five final equations : For No. I, U^22,oi •^~°°9i. {i-sf 2100 For No. 2, U= 2448 J -.0088 (1-^)2 ■ For No. 3, U=22i7 '~-°°l^ - For No. 4, U= 2420 For No. 5, 6''=25S7 J -.0088 5- — .0086 93 2000 ■ P 1900 1800 r 1700 1600 1500 Wr gl400 |l300 y// I!; 1200 ////> to g-1100 / J 1000 / '/// £ 900 ^4 '/ Ml -J 800 ///// S 700 'V/ 600 A y 500 y ^^ 400 y^ ^ 300 <^^ 200 '' , 100 ^ y ^ .05 .10 .15 .20 .25 SCALE FOR SLIP .30 .35 .40 Pig. 33. HORSE-POWER JDELIVERED AT CONSTANT SPEED OF ADVANCE. 94 RESISTANCE OF SHIPS. §5. Figure 33 shows the curves of useful horse-power obtained from the above, plotted on slip as before. Their shape is essentially the same as that of curves of useful horse-power of actual model propellers obtained by experiment. When I come to discuss the question of propeller design I shall have something more to say upon the theory of the pro- peller acting alone. At present, however, I shall leave this subject and discuss the mutual reactions of ship and pro- peller when the latter is driving the former. CHAPTER III. MUTUAL REACTIONS BETWEEN PROPELLER AND SHIP. § I. Action of Propeller attached to Vessel. A little consideration will make it evident that a screw attached to a ship^ and driving it through the water, is work- ing under conditions decidedly different from a screw advanc- ing through undisturbed water in the direction of the axis of its shaft. The resistance of the ship also must be affected by the working of the screw. I shall consider first the case of the screw. Even if the screw of a ship is of correct and uniform pitch, screw its shaft is seldom exactly parallel to the centre line of the advances ship, and to the undisturbed surface of the water. Hence nse*f*'*° when the screw is propelling the ship in smooth water, it is not in general advancing parallel to its shaft. The result is that the slip angle varies from point to point of a revolution. The variation (supposing the water undisturbed) follows the following simple law. Let (^Q denote the mean slip angle of an element, and 2 ^^ the total amount of variation of the slip angle during a revolution. Let i/r denote the angle made by a fixed radius of the shaft with a fixed direction in space, and ^ the actual slip angle at the point of the revolution determined by i/r. Then ^ = <^o -f- 0i sin i/r. If ^1 is not large in comparison with ^q, the slip will vary with sufiScient approximation directly as the slip angle, or j=jg-|-jj sini/r, where s, s^, and s^ apply not only to a single element, but to the whole propeller. 95 96 RESISTANCE OF SHIPS. §1- Disturb- ances of Water of Screw Race. Non-paral- lelism of Stream Lines. Stream Line Wake. Friction2il Wake. Wave Motion Wake. So much for deviation of the shaft from the true level and fore-and-aft line. Now the water in which the screw works is disturbed by the action upon it of the ship. The disturbance is complex and has a powerful influence upon the screw's action. This complex disturbance may be regarded as made up of four comparatively simple components. a. The water closing in around the stern tends to flow parallel, or nearly so, to the surface of the hull, and hence to flow at an angle to the fore-and-aft vertical plane. This man- ner of flow will have but little effect upon the propeller of a single-screw vessel, since in the centre line the water tends to flow parallel to the fore-and-aft plane. With twin-screw ships the result of the water flowing parallel to the hull is to produce a virtual deviation of the shaft. b. Owing to stream line action the velocity of the water in the neighbourhood of the stern is less than the normal speed. Figure 2 illustrates this fact, and shows also that the velocity will not be the same throughout the screw's disc. For a ship advancing through still water this stream line action tends to produce a following current or wake, which is not uniform over the disc of the screw. I shall call it the "stream line wake." c. Owing to the " frictional wake," or the following current set in motion by the friction of the wetted skin, the uniform- ity of the race or column of water acted on by the screw is still further disturbed. The " frictional wake " is not a uniform current, but is strongest close to the hull, and near the surface, diminishing outwards and downwards. d. The presence of waves implies motion of the water, and hence disturbance of the race. The disturbance due to a wave is the greater the nearer the surface, and varies in direction according as the propeller falls beneath a crest or a hollow. A crest implies forward motion of the water; a hollow. § I. ACTION OF PROPELLER. 97 sternward motion. This disturbance I shall call the " wave motion wake." Summing up, it is evident — 1. That the slip, of the blade of a propeller driving a ship summing necessarily varies from point to point of a revolution. 2. That the. apparent slip or (speed of screw — speed of ship) -^ (speed of screw) can be the same as the real slip only by chance. In practice, the. real slip is always greater than the apparent slip. The frictional and stream line wakes always tend to diminish the apparent slip, and even if the wave motion wake is opposed to the two above, it is not sufficiently strong to neutralise them. All our theoretical conclusions concerning the screw have been based upon- the assumption that it advanced parallel to the axis of the shaft into undisturbed water. We see that the actual working conditions are very different. If, then, we are to make any practical use of previous results, it is necessary to bridge the gap -between theory and practice by a reasonable working hypothesis. While at a given point of a revolution the slip varies from Expression of ... forSlipat point to point over a blade, the variation, with the amounts of Any Point, slip met with in practice, must be small, and it seems entirely allowable to assume a mean slip applicable to the entire blade at a given point of its revolution. Now for the screw working in the disturbed wake, the slip of a blade must be at a maximum at some point of the revolu- tion, and at a minimum at some other point. Also from the nature of the motions these points must be nearly opposite one another, ie. about 180° apart. Furthermore, at points half-way between the point of maximum slip and the point of minimum slip, the slip cannot be far from the mean between the maximum and minimum. Let •v^ denote the angular distance of a blade from the position corresponding to the mean slip. Let s denote the 98 RESISTANCE OF SHIPS. § I. slip corresponding to i^, Sq the mean slip, and 2 s^ the differ- ence between the maximum and minimum slip. Then it fol- lows from the above that with fair approximation j = jg-|-jj sini^. Gross and Referring to page 80, it is evident that we may denote NetWorkin t> f t, > j Turbulent N, the gross work delivered to a blade in one revolution at slip s, by K{asX^-\-fZ^, and the useful work f ' by K{\-s){asX-fY:). While turning through a small angle d-^ at slip s, the N'd-^ elementary useful work= , or dN' = — (asX +/2'o) df, 2 TT dl7' = ~{i -s){asX,-fY,)d-^. 2 TT Substituting in the above the value of s in constants and terms involving i/r, namely, we have dN' = — {as^X^ +fZc+as-^^Xc sin i/r) di^, rrj,_^{ {T--SQ){asQX,-fY,) + {i-s^as^X,sm'y^ 1 ~ 2 TT I -s^ sin ^^r {as^X,-fY,) -as^ sin^ -^X, \ ^' Integrating for one revolution between the limits o and 2 TT, we have N' = K(as,X+fZX [/' = K^{i-s,){as,X-fZ) -^ s/j. Hence the gross work absorbed by the blade is that corresponding to the uniform slip Sq, while the useful work is diminished because of the negative term — - s-^- ACTION OF PROPELLER. 99 The diminution is small, however, unless s^ is large com- pared with Sq. This question of the effect of non-uniformity of wake, or turbulence of wake, upon the action of the propeller is a most important one. For that reason a second method of dealing with it may well be introduced. Suppose that of the total work W sent to the propeller, an Efaciency by Second amount w^ is expended at the slip s^, with an efficiency e^, an Method in amount w^ at slip s^, with efficiency e^, and so on. wjuce.*"* Then the useful work The efficiency of the whole transaction _ U _ w-^e-^-{- w^e^ + ^£^3^3 + W 'w-^^ + w^ + w^ + Also the mean slip s^ _ te/, J, + w^s^ + w^s^ + Suppose next that the curve of efficiency of the propeller plotted on slip is a straight line, or that ei = a+cSi, e^ = a-\-cs^, and so on. Then U _ p _ w^(a-\-cs-^-\-'w^(a-Vcs^-\-'w^(a-\-cs^ + W~ ~ 'Zi)i + zv^ + w^ + W-^ + W^ + W^ Wi + W^ + Ws = a+csQ. That is to say, the final efficiency of transfer corresponds to the mean slip if the efficiency curve is a straight line. Now, unless the slip variations are large, the efficiency curve within the limits of variation is practically a straight line. In such a qgse the efficiency is that corresponding to the mean slip. 100 RESISTANCE OF SHIPS. § i. conciu- The above theoretical work appears to justify the follow- eions. . , mg conclusions : I. The variable wake currents around a propeller are equivalent to a single uniform wake. Fronde's 2. The efficiency of a propeller is not appreciably affected mental by the turbulence or variation of the wake unless the varia- conciu- |.Jqjj jg excessive. eions. These conclusions fully agree with the results of model experiments made by the Froudes upon small propellers working behind models of ships. R. E. Froude has stated that in such experiments the turning moment and thrust correspond to the mean slip, and that the experiments appeared to indicate, if anything, a very slight gain of efficiency due to turbulence of wake. While moderate variation in the wake does not affect the turning moment, thrust, and efficiency of the propeller as a whole, it should be remembered that it causes great vari- ations in the thrust of a given blade during a complete Virtual and revolution. Actual Deflection In practice, the wake is always greatest near the surface. ° °' For a twin-screw ship it is also greater in the part of the disc close to the ship than in the part farthest removed. Hence, in a twin-screw ship with propellers turning outward {i.e. with the blades moving away from the ship when in their highest position), the wake is equivalent to a virtual deflection of the shaft (starting from the engine) outward and upward. If the propellers turn inward, the virtual deflection is inward and downward. Care should be taken, in design- ing, not to give the shaft an actual deflection of such a nature as to add to the virtual deflection due to the wake. On the contrary, the virtual deflection should, in the case of twin-screw vessels, be more or less neutralised by a sHght actual counteracting deflection. § 2. THE WAKE. lOI § 2. The Wake. Having concluded that the propeller behind a ship is practically working in a uniform following current or wake, it is necessary to inquire into the effect of this wake upon some of our previous results and formulae. It is convenient to express the speed of the wake as a wake fraction of the speed of the ship. Let us denote ithy wV, ^**^*"' where w is called the "wake fraction," or "wake factor." Real and Retaining the symbol s to denote true slip, it is convenient gyp"^"* to denote apparent slip by the symbol s'. The relation betweeft s and s' must involve w. Now, speed of screw in knots = ^ ^ , 6080 speed of ship in knots = V, speed of wake in knots =wV. PRX6 y A ^ r , 608 608 F Apparent slip = j' = — - — =1 ^^ ^ pRy.6 epR 608 pR X 6 . .j^ -<-— — {i — w)V „ ,. 608 608 r, . True slip = j = t- — =1 — - — —(i — w). ^ pRx6 epR^ ' 608 Whence, (i-.r')=^^^. 1, t ^, , ,, V , s — w s — s' or, s = w-t-s'(i — w,) s'— , w = -• i — w i—s' What is the effect of the wake upon the gross and useful work .'' The thrust and the work delivered to a screw depend Thrust and solely upon the revolutions and the speed of advance ; i.e. siaering upon the true slip. The useful work, however, depends ^^''^• upon the velocity with which the thrust overcomes the 102 RESISTANCE OF SHIPS. §2. resistance opposing it ; and hence, upon the speed of the ship. Referring to page 71, it is evident that, in deducing the useful work done by a propeller attached to a ship, we must multiply the thrust by the speed of the ship, that is, by pR{i—s'), instead of by the speed of advance of the propeller, which is denoted hy pR{i—s). The only change in the final formulae is in the expression for useful work, which becomes ^==3« (^J^Hi-^')(«-f^.-/a while for the gross work N we still have Effloiency Consider next the effect of the wake upon efficiency. Consider- . . ing Wake. The expression for efficiency from page 74 is , .cs— y, CS+ Zc a consideration of the deduction of this expression will show that when the effect of the wake is included, it becomes but I— i-'= . i — w so we can also write the expression for efficiency when wake is considered, ., i-s ^cs-Y„ i — w c^-t^c where s is as usual the true slip. Now e' is an apparent, not a real, efficiency. It is the ratio between the work delivered to the ship by the propeller and the work delivered to the propeller by the shaft, but a certain amount of the work delivered to the ship by the § 2. THE WAKE, 103 propeller is obtained from the water of the wake, instead of from the shaft. A simple ideal case may help to make clear this somewhat difficult point. Imagine a propeller working in a canal, and driving a carriage on rails above the canal. Suppose the carriage so fitted with adjustable brakes that its resistance, and hence the thrust required to drive it, is con- stant for a wide range of speed, say from 5 to 20 knots. Suppose now such a turning moment applied to the propeller that it drives the carriage at 5 knots, the water in the canal having no motion. Keeping the same turning moment, suppose the water of the canal to be flowing at the rate of 5 knots. The thrust of the propeller will be the same as before, and the carriage will now move at 10 knots. The power applied to the pro- peller will be the same, and the portion of this power utilised for driving the carriage will be unchanged ; but an additional power equal to the latter is delivered to the propeller by the water in the canal. Under these conditions the " negative slip " may be large, and the " apparent efficiency " much above unity. This brings me to the much-vexed question of " negative Negative slip" in the case of actual propellers. '"" It is absurd to discuss the possibility of real negative slip. A propeller cannot exert thrust without slip. But suppose we had a propeller, which, with a speed f -^ .. ) of 10 knots and a slip of 10% (making a speed of \6o8/ advance of 9 knots), exerts in still water sufficient thrust to overcome the resistance of a certain ship at 1 1 knots. Suppose the ship has a wake of 2 knots, and that the pro- peller is put in the ship and driven by power sufficient to sive it the same number of revolutions as before. The speed of the propeller, as deduced from the pitch multiplied by revolutions,- will be 10 knots, the thrust will be that neces- sary to drive the ship at 1 1 knots, and she will undoubtedly I04 RESISTANCE OF SHIPS. §3- show that speed, involving an apparent negative slip of io%. The possibility of negative slip depends, then, entirely upon the possibility of a wake of a speed greater than the actual speed of slip. The existence of such an amount of wake in many cases of fuU-sterned ships, where a large stream line wake is added to the frictional wake, is incontestable, and there appears to be no reason why reports of apparent nega- tive slip should be received with discredit and contumely. Apparent negative slip is, however, very objectionable, as it is always a sign of low actual slip, and hence of low propeller efificiency. Cause of Thiust Deduction. Influences causing Variation of Thrust Deduction. § 3. Thrust Deduction. When a propeller works with slip, the faces of the blades exert pressure upon the water, and hence increase the pressure aft of the propeller. Also the backs of the blades exert suction upon the water, and hence decrease its pressure forward of the propeller. If the propeller is attached to a ship, this decrease of pressure extends to the water in contact with the hull, decreases the pressure upon the after-body, and hence increases the resistance. This increase of resistance was called by the elder Froude the augment of resistance, but has been more happily termed the thrust deduction by his son. It does not appear possible that in any case the thrust deduction can exceed that part of the thrust of a screw due to the suction of the backs of the blades, and it must in most cases be much less. The amount of the thrust deduction for a given ship must vary with the position of the screw relative to the hull. The thrust deduction may be expected to change but little with change in size, proportions, or slip of the screw, provided its thrust remains unchanged. § 3- THRUST DEDUCTION. 105 Our knowledge upon the question of thrust deduction is due almost entirely to the researches of the Froudes. It is, however, exceedingly limited. I shall give subsequently some of the results published by the Froudes, which are found of use in considering practical cases. CHAPTER IV. ANALYSIS OF TRIALS AND AVERAGE RESULTS. § I. Value of Trials. Indicated and Effec- tive Horse- Power Differ. Mercantile Value of Trials. Graphic Method used in Analysis. The power indicated by the engines of a ship and the power required to overcome the tow rope resistance are very different. The latter absorbs in practice only from 40 to 60 per cent of the former, the remainder being accounted for in various ways. Evidently the analysis or separation of the indicated horse-power into its several components is very desirable, and may in fact be said to be necessary if the results of trials of actual ships are to be fully utilised in preparing designs for new ships. A careful analysis of trial results of a new ship is of value also from another point of view. It enables the owner to decide whether or not the performance of his ship is what it should be, and may point out defects that can easily be made good. In the long run the results of such defects would, in the case of a merchant vessel, show on the owner's books, but the books would seldom indicate the remedy. Before taking up the question of trial analysis I shall ex- plain a graphic method which I shall have occasion to use. In dealing with trial analysis, where the quantities which we must handle are at best approximate, graphic or semi-graphic methods present distinct advantages over arithmetical or algebraical processes. Figure 34 shows 10 lines radiating from a common point or focus. 106 §1- VALUE OF TRIALS. 107 Now if the positions of these lines had to be obtained by experiment or observation, we should find that they would not all pass through a single point. We might know that if our observations or experiments were exact, the lines would have a focus. But it is not possible for fallible man to entirely avoid errors of observation. Refined methods and apparatus combined with skill may, and often do, so bring it Fig. 34. LINES RADIATING FROM A FOCUS. about that errors of observation may be neglected for any practical purpose. They cannot, however, be eliminated entirely. In Figure 35 the lines of Figure 34 are shown with small errors in position assigned at random. The focal point of Figure 34 is indicated in Figure 35 by a small circle. While few of the lines pass directly through this focal point, it is evident that any one who knew that all the lines should pass through a focus, could not go far wrong in "spotting" it upon io8 RESISTANCE OF SHIPS. fa- Figure 35. If the errors of the lines of the-diagram were not abnormal (a fact which the diagram itself would show), the focal point so spotted would be quite sufficiently close to the actual focus for all practical purposes. The reader will be enabled to form his own conclusions 'as to the accuracy of this graphic method from examples of its application which I shall introduce presently. Fig. 35. LINES RADIATING FROM APPROXIMATE FOCUS. Results of Progressive Speed Trials. § 2. Components or Absorbents of the Indicated Horse-Power. A discussion of the manner of conducting a " progressive trial " of a steamship is beyond the scope of this work. Suffice it to say that in such a trial the indicated horse-power and the revolutions of the engines are determined for various speeds ranging from four or five knots up to the highest speed of which the ship is capable. It is best in dealing with the" results of trials to lay down the revolutions as abscissse set §2- COMPONEiNTS OR ABSORBENTS. 109 up ordinates representing on suitable scales the corresponding speed in knots and indicated horse-power, and draw through the extremities of the ordinates fair curves of speed and indicated horse-power. For twin-screw vessels the revolu- tions are taken as the average of the two screws. For triple- screw vessels, where the middle screw, even if of the same 3100 17 (PLOTTED ON REVOLUTIONS.) h y 3200 10 A / SOOO 15 / 1 2800 U / / 2600 13 / I 1 £2400 12 / •^ 2200 h 11 / 1 g 2000 1^ 10 1 ~ f 1 S 1800 § 9 i;^ a 1600 g 8 ^j V '^) 1 i £ g 1400 "i 7 f ^/ 3 1200" 6 / oy f "1000 6 / / -^ y 80O i / / f COO 3 ,- / / 400 2 / / 200 1 / ly / . , — - '^ 90 100 110 120 130 140 ISO 100 170 ISC 10 20 30 40 60 60 70 SCALE FOR REVOL'OTIONS PER.MINUTE. Fig. 36. CURVES OF SPEED AND I. H. P. OF U. S. S. YORKTOWN. size as the other two, is working under different conditions, it will be necessary to draw separate curves for the middle screw. Figure 36 shows curves of speed and indicated horse- power, plotted as indicated above from the trial results of a twin-screw vessel, the United States gunboat Yorktown. Evidently from Figure 36 we can determine the revolutions and horse-power corresponding to any speed. no RESISTANCE OF SHIPS. §2- Reduced Mean Effective Pressure. Expression for Indi- cated Horse- Power. Initial Friction. Let US trace the processes by which a greater or less proportion of the horse-power indicated in the cylinders of the engines is used to overcome the resistance of the ship. Knowing the cylinder dimensions of the engine, we can reduce the mean effective pressures obtained in the several cylinders to an equivalent mean effective pressure upon the low- pressure piston only. Thus in the case of the Yorktown the high-pressure cylinder of each engine is 22 inches in diameter, the intermediate 31 inches, and the low-pressure 50 inches. Then to reduce an effective pressure shown by an indicator diagram in the high-pressure cylinder to an equivalent effective pressure in the low-pressure cylinder, we must multiply it by the factor (ff)^ = .i936. Similarly, for the intermediate cylinder the factor of reduction is (|-J)^ = .3844. Let /„ denote the total equivalent mean effective pressure, reduced to the low-pressure cylinder. Let R denote the revolutions. Let / denote the indicated horse-power. Let C denote the horse-power constant for the low-pressure cylinder. Then I=C-p^-R. C depends upon the cylinder dimensions. If A denote the effective cylinder area in square inches, and s the stroke 2As in feet. C: 33000 From the above relation between /, /„, and R, it is evident that if either R or/„ is equal to zero, /will become equal to zero. Now at the origin, R = o, and hence /=o. It does not follow that/„ = o. On the contrary, /„ must have a value. Owing to the tightness of glands and bearings, pistons and valves, a certain amount of effective pressure is necessary to start the engine against its own internal resistance — to overcome the "initial friction " or "dead friction." This §2. COMPONENTS OR. ABSORBENTS. m initial friction or internal resistance of the engine is over- come at all rates of revolution. While it probably varies slightly with the power and revolutions, we have no reason to believe the variation great. It is then safe to suppose this resistance constant, and hence the power absorbed by it varying as the rev- olutions. If /o denote the mean effective pressure necessary to over- come the initial friction, and If the corresponding horse-power absorbed at R revolutions, we have If = Cp^. At the origin /„ and/^ coincide, and hence the curve of I coincides with the curve of If. This is because with the exception of I, all the resistances which absorb / diminish indefinitely with the speed, and become zero when the speed and revolutions become zero. If we know p^, we can calculate If for any value of R, and hence for any speed. To determine p^, we proceed as follows : It is supposed that Detennina- in properly conducted speed trials," spots " on the power curve tiaiFnc- are determined at close intervals down to the lowest rate *''"'■ of revolutions which can be steadily maintained by the engines. These spots being set off in both the first and third quadrants, the curve of I. H. P. is plotted through the origin, as shown in Figure 37. From this curve we plot a derived curve of (I. H. P.)-;- (Revolutions), as shown in the figure. Spots for this curve are obtained by dividing the , I. H. P.'s, corresponding to various rates of revolutions by the revolutions. The spots of this curve should be laid off in both the -first and second quadrants, as the curve is horizon- tal immediately above the origin and symmetrical on either side of the vertical axis. Of course we cannot calculate the ordinate for the spot where the vertical axis is cut, since there /=o, R = o. We can, however, obtain spots so close to it on either side that by sweeping a fair curve through — horizontal where it cuts the axis 112 RESISTANCE OF SHIPS. §2. — the value of — for 7?= o is very accurately determined. From the curve of I-i-R a curve of /„ is readily drawn, since whence A =7=. 4 f I. H. P. Fig. 37._CURVES OF I ^^^^ PRESSURE, j I. H. P. (^ REVOLUTIONS. PLOTTED ON REVOLUTIONS. For the Yorktown the low-pressure cylinder area is 1963.5 inches, and the stroke 2.5 feet ; whence ^^ 2x1963.5x2.5 ^ 33000 ^=3.361. ■2975' §2. COMPONENTS OR ABSORBENTS. 113 From the curve, the value of — for ^=0 is .95. Then (remembering that there are two low-pressure cylinders), /„=-x.95x 3.361 = 1.6. 2 Figure 37 shows a curve of /„, including /q. If it is more convenient, this curve may be drawn at once instead of the curve of I-r-R. But while apparently a more round-about process, it is in my opinion, simpler and better to determine /„ from the auxiliary curve of I-r-R. We thus determine directly the quantity Cp used to calculate 1/ by the formula 1/ = Cp^R. Having calculated the values of If, we denote 1—1/ by I„, and call it the net horse-power. It may be remarked here that the value of the initial friction, in the case of Yorktown, as determined above, is very low for new engines. At full speed the initial friction absorbs only about 4J per cent of the total indicated horse-power. In new tripje-expansion engines of the present day, If usually absorbs from 6 to 10 per cent of the total power. An amount of initial friction greater than 10 per cent can always be traced to avoidable causes — such as bad fitting or poor alignment. In cases where feed, air, circulating or bilge pumps are driven by the main engines, the power required to work them should be deter- mined and classed with the initial friction. At the present day, in few large ships do we find any pumps except the air- pump worked as above. I shall give later an estimate by a competent authority of the average amounts of power absorbed by the various pumps — when worked off the main engines. Besides the loss through "initial friction," there is a Load second frictional absorbent of power. This is the "load ^'^''t"'' 114 RESISTANCE OF SHIPS. §2. friction," being the friction of the load upon the bearings-and the thrust block. It necessarily varies with the coefficient of friction of the rubbing surfaces, and hence depends some- what upon the lubrication. There are scarcely any reliable experimental results from which the load friction of marine engines under various conditions can be certainly predicated. It appears to be nearly always about 7 per cent of the net horse- power, /„( = /— ^) seldom falhng below 5 per cent. All things considered, an allowance of 7 per cent of the net horse-power for load friction appears to be reasonable, and I shall adopt it in future. Mr. Isherwood, the distin- guished American engineer, considers y^ per cent the proper allowance. Mr. Blechynden, of Barrow, gives 6^ to y^ per cent for triple-expansion engines of the present day. The horse-power absorbed in load friction I shall denote by /,. Power and We have now traced the powcr to the propeller. Denoting ofPropeiiM. ^^^ power delivered to the propeller by P, we have Of P a certain amount is wasted by the slip of the propel- ler, and a certain amount is wasted by the friction (including head resistance) of the propeller blades. Denoting the sum of these two losses hy I^, we conclude that an amount of power derived from P equal to P — P, is delivered to the thrust block — the true efficiency of transfer P — P bemg p ' ■ Wake Gain. vVe have seen also that a certain amount of power is delivered to the propeller by the wake and transferred to the thrust block, so that the actual amount of power absorbed by thrust is greater than P — P,. Denote the actual power absorbed by thrust by T\ we have the apparent efficiency of the propeller equal to — • §2. COMPONENTS OR ABSORBENTS. 115 Now we have seen (page 102) that if e' denote the apparent . efficiency of a propeller, with wake factor =z£; and e denote the true efficiency, i — w Now e' = --^ e = P P-P, . P ' P-P. whence T= \—w I shall denote the power absorbed by the propeller from the wake by P„. Then we have T=P — P, + P^. Of T, the power absorbed by the thrust, a certain amount Thrust is used to overcome the thrust deduction or suction, exerted ^^""^ "*" by the propeller upon the rear of the hull. Let us denote the thrust deduction by T,. Then if t denote the thrust deduction factor, we have T, = tT. The remaining portion of T is employed to overcome the Effective Horse- natural or tow rope resistance of the ship. This portion of power. the power alone is used to produce the effect for which the power is needed. Hence it is well called the " effective horse-power," and denoted by E. As we have distinguished two components of the resistance skin and {R), namely, R„ the skin resistance, and i?„ the wave resist- sistance ance, it is necessary to divide the effective horse-power into two corresponding portions. We will denote them by E, the skin resistance power, and E„ the wave resistance power. Recapitulating in symbols, we have, / = //+/,+/', P+P„=P.+ T, T = T,+E, E=E,+E„. Power. 1 16 RESISTANCE OF SHIPS. § 2. With a satisfactory method of trial analysis, all of the quantities denoted above can be determined with sufficient aproximation from known data concerning the ship and engines, and the results of careful progressive trials. § 3. Yorktown Trial Analysis. We have seen already how to determine I,'\x\.z. given case, and also that with sufficient approximation, /, = .07 (/-/,). I can best make clear the methods pursued in determining the other components of the indicated horse-power, by analysing in detail the trial results of the Yorktown — shown in Figure 36. Data of The trials of the Yorktown were carefully conducted by a andFormu- board of naval officers, and the accuracy of the results can ise needed, ^g relied on. The trials were made in 1889, shortly after the commissioning of the vessel. Very complete particulars of the ship and machinery and of the trials have been published in the Journal of the American Society of Naval Engineers, and official reports of the U.S. Navy Department'. Those needed for the purpose in hand are given below : Length between perpendiculars, 226'. Length on trial water line, 230'. Mean immersed length, 226'. The vessel has a slight partially immersed overhang, but it did not appear to be large enough to warrant making the mean immersed length greater than the length between perpendiculars. §3- YORKTOWN TRIAL ANALYSIS. 117 Beam extreme on trial water line. 36'. Draught on trial forward, 12' 6"J. Draught on trial aft, 15' o"f. Draught on trial mean, 13' qILSs." /?• ^*3 (This is skin draught.) Displacement on trial, 1680 tons. The displacement and trim on trial were practically identi- cal with the displacement and trim at designed load water line. Wetted surface (S) on trial, 10,840 square feet. Area of midship section {M) on trial, 432 square feet. The propelling machinery is of the twin-screw, horizontal, triple-expansion, three-cylinder type, for boiler pressure of 160 pounds above the atmosphere. The cylinders are 22, 31, and 50 inches in diameter by 30 inches stroke. The propellers are three-bladed, the blades being of the oval or modified Griffiths shape. The pitch (uniform), 12' 6". Diameter of propeller, 10' 6". Diameter of boss, 2' 6". The blade area of each propeller is 25.4 square feet. The mean width ratio = .20i. From the shapes of the blades the characteristics are .found to be as follows : - ^.=.0954. I^»=^i439- Z„=.4469. From Table V. we obtain .0094 as the appropriate skin resistance coefficient for the Yorktown. Il8 RESISTANCE OF SHIPS. § 3- Then 72.= .0094 x 10840 V^-^. R, is in pounds. Now a resistance of one pound at a speed of one knot absorbs .0030707 horse-power. Then ^.=.0030707 VxR, = .0094 X 10840 F^-^^ X .0030707 V = .3129 V^-^^. Denoting the wave resistance R^ by b V*, we have E^ = bV*x .0030707 V = .0030707 b V^. We have seen that the expression for the power /'absorbed by a screw is \IOOO/ In the case of the Yorktown with three-bladed twin screws, n- = 6, /=i2.s, ^=10.5, .X„=.0954, ^^=.4469. Taking the standard value .045 for f, we obtain, upon simplifying the expression for P, 7^= 369.58 ('-^)V+.2i I). The first step of the analysis is the determination of horse- power wasted per revolution by the dead friction. The method to be followed has already been explained, Figure 37, referring to the Yorktown, and giving us the result Ij = .(^t) R. Note. — The diagram and figures used in trial analysis are drawn in practice upon a much larger scale than those in the accompanying plates. The latter have been reduced from the large-sized figures, which must be used if accurate results are desired. Table. Turning now to Table X., it is seen that on the right, under the heading " Remarks and Explanations," are entered the necessary data and formulas, together with results of analysis. § 3- YORKTOWN TRIAL ANALYSIS. i ig In line No. i we enter as headings for the columns the speeds. Lines 2 and 3 contain the corresponding values of the revolutions R, and the indicated horse-power /. They are obtained from the curves of Figure 36. While it would have been possible to extend the table to speeds below five knots, it would be unnecessary labour — the initial friction having been already determined. Knowing that If = .95 R, we fill up line 4 by entering the products of the values of R in line 2 by .95. Line 5 gives the values of I—Ip which are obtained by subtracting the quantities in line 4 from those in line 3. With the standard loss of 7 per cent for load friction, the power P delivered to the propeller is (i.oo-.o7)(/-//) = .93 (/-//). The values of P so obtained are entered in line 6. wake Fac- tor and We have seen also that Thrust >2,li Coefficient. />= 369.58 (^ ) («^ + .I«W)- f R\^ Line 7 gives the values of f 1 obtained from line 2. Then line 8 is obtained by the use of the formula R\^ flj + . 211=/" -^ 369.58 MOO. Deducting .211 from the quantities in line 8, we have in line 9 the values of as. Here s is the true slip which we do not know, because we do not know the wake factor. Also we do not know the value of a. In line 10 is entered the speed of the screw in knots =^-^=.i234i? for the I2'.5 ^ 6080 ^ pitch of the Yorktown propellers. In line 11 is entered the apparent slip obtained by the formula • 1234-^ I20 RESISTANCE OF SHIPS. § 3- The apparent slip s' is also the true slip s if the wake factor w is equal to zero. For a wake of .10 we must add to the apparent slip at speed V .10 V .12347? The quantities to be thus added are entered in line 12, and adding 11 and 12 we obtain in line 13 the true slip for a wake of .10. Again adding lines 12 and 13, we obtain in line 14 the values of the true sHp on the supposition of .20 wake. Now in line 9 we had the values of as, while lines 11, 13, and 14 give the values of s on the three suppositions of (i) zero wake, (2) .10 wake, (3) .20 wake; whence we enter in lines 15, 16, and 17 the corresponding values of a. Thus we obtain for each speed three spots on a curve of values of a, plotted upon values of wake factor. Since a and w are constant, or nearly so, the curves for the various speeds must either all coincide or else pass through a common focal point, corresponding to the constant values of a and w. Figure 38 shows the curves in question. They do not tend to coincide, but clearly tend to pass through a common focus. Naturally we do not find a perfect focus, and it is necessary to "spot" the proper focal point. This is done as indicated in the figure where «=7.8, w=.o8:i. Closeness of Up to this point but two approximate assumptions have tion"^"^' entered into our work ; namely, that 7 per cent of the net horse-power was expended in friction of load, and that the frictional coefficient for the propeller was .0225. While these have been selected as suitable standard values, and are probably fair approximations, it is proper to consider what effect would be produced in the values of a and w, determined §3- YORKTOWN TRIAL ANALYSIS. 121 as above by possible difference between the actual values of / and load friction, and the standard values adopted. By giving to / and to the allowance for load friction values different from the standard values, it is found that for an .025 .050 .075 .100 .125 .150 .1T5 .200 Pig. 38. FOCAL DIAGRAM FOR a AND W. increase in either quantity, the curves of Figure 38 are lowered bodily, and for a decrease are raised bodily. There is little or no change in the focal value of w. While the focal value of a is changed, the change is small, 122 RESISTANCE OF SHIPS. § 3- for any probable variation of / or of the allowance for load friction from the standard values. An error of 30 per cent in estimating the load friction causes an error of only about 2 per cent in the value of a. An error of 25 per cent in/, the coefficient for screw friction, results in an error of only about 3 per cent in the value of a. All things considered, I think it safe to conclude that the method of Table X. and Figure 38 enables us, given accurate trial results, to determine the wake factor w very closely, and the thrust coefiicient a with ample accuracy. Knowing the wake, and hence the true and apparent slip, we are in a position to determine the true and apparent propeller efficiency, and the thrust horse-power T. We have seen that the true efficiency e is given by the formula e=(\-s) "■^^-f^'' . Rewriting, , , aX, For the Yorktown / = .04S, a = 7. 8, Jf;=.0954, F,=.i439, Z.=.4469. Substituting and reducing, i- + .027 Similarly, for the apparent efficiency e' we have = (i-/) -.009 J + .027 Returning now to Table X., line ii„, repeats line 11, the apparent sHp. Line 18 gives the addition for the wake of .083. Since line 12 gives the addition for a wake of .10, line 18 contains the quantities of line 12 multiplied by .83. Line §3- YORKTOWN TRIAL ANALYSIS. 123 19, the sum of ii^and 18, gives the values of the true slip=j. Lines 20 and 21 contain respectively the values of s—.oog and J +.027. Then line 22 gives the apparent propeller efficiency, obtained from lines ii„, 20, and 21 by the formula J + .027 The thrust horse-power T, entered in line 23, is simply the product of the apparent efficiency of line 22 by the propeller power of line 6. Now we have seen that a part of the thrust horse-power T Thrust Deduction . is absorbed by T„ the thrust deduction, the remainder bemg the effective horse-power E. Let t denote the thrust deduc- tion factor ; then, 7^^=Txi; also, T= T,+E= Tx t+E, or, T{i-i) = E. Now E is the sum of the skin resistance power E, and the wave resistance power E„. The former we know, it being equal to .3129 V^-^ The latter is unknown, being denoted by .0030707 d V\ where d is an unknown semi-constant. Then (i — ^)r=.3i29 F^-^^ +. 0030707 <5F^. Line 24 gives values of ^, or .3129 V^^ calculated with the aid of Table XI. Line 25 is taken from Table XI., being values of .0030707 F^ denoted byj. Let ^=c, ^=X. y y Then from the equation {:}.-t)T=E,^- .0030707 b V^ X{i-t)=c+b. 124 RESISTANCE OF SHIPS. § 3- Now t is practically constant, but we know that b varies more or less. If b and t were both constant, our task would be easy. We would have as many equations between them of the form X{i — t) = c-\-b as there are speeds entered in the table, and their value could be determined graphically with great accuracy by plotting the focal diagram iox the lines represented by the equations. Unfortunately the case is not so simple. Remember that we are working with approximate quantities. Now at low speeds the wave-making resistance is so small, proportionally, that a small error in c (depending upon E^, or in X (depend- ing upon T), win render the equation {i. — t)X^c-\-b useless ' for a reliable determination of b. Small errors are neces- sarily made in such work as the present. Again, at high speeds, E„ is substantial in comparison with E^ and T, but it is so large in comparison with Tt that the fluctuations, which we know take place in b, render the equation {i — t)X=c-\-b unreliable for the determination of t. So the determination of b and t must be made in some other way. Average We may, however, determine an average equation between for^fiand t. ^ ^"^ * ^^ determining average values of c and X for a suit- able range of speeds. The upper speeds are preferable for this purpose. As noted in the table (under remarks, etc.), the average value of c for speeds from ii to 17 knots inclusive, is .3563 and of X .6908. The corresponding equa- tion is .6908(1— ^)=. 3 563 +3. If, then, we can in some way determine a fair average value of the fluctuating quantity b, we can from the above equation determine the value of the constant t, and hence the actual values at the various speeds of the fluctuating quantity b. Let us then attempt the determination of b. Returning to Table X., it is seen that line 28 contains the thrust efificiency T-i-I. §3- YORKTOWN TRIAL ANALYSIS. 125 If e denote the efficiency of propulsion, we have e= — Now E = {i-t)T. Then e= ^ — y- — = (i — ^) x thrust efficiency. Then (i — ^) being constant, the curve of true efficiency of propulsion is simply the curve of thrust efficiency on a smaller vertical scale. It follows that the ratio between the effici- encies of propulsion for any two speeds will be equal to the ratio between the thrust efficiencies for the same speeds. Line 29 gives the ratio between the efficiency of propul- sion at each tabular speed, and that at 16 knots, the highest even speed in knots actually attained by the Yorktown. The highest actual speed was about 16.7 knots, the quantities for 17 knots being obtained by extending the curves to that speed. Returning to the expression for efficiency of propulsion, we have - .3129 F^-^^ .0030707?^^^ ~ / / = m + bn, say. Line 30 gives values of m, being obtained by dividing line 24 by line 3. Similarly, line 31 gives values of n, being obtained by dividing line 25 by line 3. If b=o the quantities of line 30 are the true efficiencies. Dividing them by the quantities of Hne 29, we get the values of e^Q (or efficiency of propulsion at 16 knots) from each speed on the supposition that b=o. Line 33 gives the additions to be made to the values of e-^^ if ^ = .5. The quantities entered in it are obtained by multiplying line 31 by .5 (for b = .s) and dividing by the ratios in line ^9. 126 RESISTANCE OF SHIPS. § 3- Now from lines 32 and 34 we have a linear relation between ^ig and b at each speed. Thus for 5 knots we have ^i6 = -534for b=o; = .619 for sjtssj XIBLg.-- ,t^? «!^ bov^i f.»i-*= #«; ITY OF ROME (\0* .55 .60 .65 SCALE FOR BLOCK COEFFICIENT .75 Pig. 45. WAKE FACTORS PLOTTED ON BLOCK COEFFICIENTS. Wake Fac- tors for Design Work. The increase in wake with increase of block coefficient is due to the increase of stream-line wake due to the fulness aft accompanying increase of block coefficient. The abnor- mal wake of the Inflexible, for instance, is undoubtedly due to her full form aft. For purposes of design it is always necessary to estimate the wake. This is best done from results of trials of similar ships. §6. THRUST DEDUCTION AND WAKE FACTOR. 141 Those to whom such results are not available cannot do better than use the mean lines of wake of Figure 45. It is not necessary to know the wake very exactly, and the mean lines of Figure 45 will give results sufficiently near the truth in the vast majority of cases. We shall see later that it is probably better to over- estimate the wake than to underestimate it, but that in either case we can adjust the blades of the propeller to suit the actual wake without any appreciable, diminution of the efficiency. CHAPTER V. THE POWER OF SHIPS. § I. Prelimifiary. How are we to estimate with sufficient approximation the indicated horse-power which must be developed by the engines of a given ship in order to drive her at the intended speed ? Evidently this indicated horse-power will depend not only upon the effective horse-power, but also upon the efficiency of propulsion, or ratio between effective and indicated horse-power. Methods We may, in making estimates, deal with the indicated into Two horse-power directly, or consider separately the quantities Classes. upon which it depends. Accordingly the methods used fall naturally into two classes. 1. To the first class belong methods which deal at once with the indicated horse-power, without considering its com- ponents. 2. To the second class belong the methods which deal separately with the effective horse-power and the efficiency of propulsion. Let us examine the leading methods in each class. § 2. Admiralty Coefficient Method. Assump- tions and Suppose that the resistance consists of skin resistance only, of Method, taken as varying as the square of the speed. 142 § 2. ADMIRALTY COEFFICIENT METHOD. 143 Then we may write R-- constant If the efficiency of propulsion were constant, we should have I=:RVx -0030707 ^ SV^ constant efficiency constant For similar ships of varying sizes, 5 varies as the square of the linear dimensions. Now Ds is proportional to the square of the linear dimensions, and so is the area of the midship section, which we denote by M. So for similar ships with the same efficiency of propulsion we have /= — — — or/= ) where C\ and C^ are constant. These formulas may be taken to apply approximately to sTiips which are not similar, and whose efficiencies of 'pro- pulsion differ. C^ and C« are then what are called the admiralty coeffi- Two coeffi- cients, cients, C\ being the " displacement coefficient," and C^ the "midship section" coefficient. Unfortunately the indicated horse-power of a ship does not variation vary as the cube of the speed, and hence these coefficients cients with are not constant, even for the same ship at various speeds. ^p^^^- Figure 46 shows curves of the " coefficients " for the Yorktown, plotted upon speed. The variation is seen to be great, the values of the coefficients being least at the lowest and the highest speed, and reaching a maximum at about 10 knots. The two curves are essentially the same, their ordinates bearing a constant ratio to each other. 144 For whence RESISTANCE OF SHIPS C,- c,- M constant. §2- 300 o =200 ri5ot so 800 700 600 500 400 300 200 100 .,M0SH!P_8I£TT0|V ^^^^232etA2™^N]Lco££2g^^^^^ SPEED IN KNOTS "s 9 15 tE~ H iS iil iS i5 W Pig. 46. CURVES OF ADMIRALTY COEFFICIENTS FOR YORKTOWN. The shape of curves of the coefficients shown in Figure 46 is typical for a fast ship. The reason why the shape should be as shown is not far to seek. Cattse of Variation with Speed. We have and E ffSV^f^±bV^ r=— =.0020707 (■ § 2. ADMIRALTY COEFFICIENT METHOD. 145 Substituting and reducing C =-~^— F3 ^ .0030707^/5^2-83 + ^^5^^ where k is constant for a given ship. Now at low speeds of a ship, bV^ is comparatively small, and the expression — ^^^„„„ — — -T does not change much with change of speed. fSV^-^ + bV^ But the value oi e, the efficiency of propulsion, is increas- ing rapidly with the speed. So the value of Cj increases at low speeds. As the speed increases, we reach a point where has a maximum value, and then, owing to the rapid increase of the term bV^ in the denominator, it begins to fall off, decreasing continually with the speed. Meanwhile e increases less rapidly as the speed increases, and at the highest speeds is usually nearly constant for several knots. The result is that C^ must reach a maximum value at some speed, and then constantly decrease with increase of speed. Evidently, then, the admiralty coefficients for a fast ship and a slow ship may be expected to be very different, and a coefficient deduced from the performance of a slow ship will be unreliable for use in connection with a fast ship. Also the coefficients may be expected to differ if the variation proportions of a ship be changed. This is because the wetted "jg^"' ^j^i, surface does not vary as D^ except for precisely similar Propo'- ships. Thus, suppose we had a ship of 1000 tons' displacement and 6000 feet wetted surface ; then Z?^= 100 and 5= 60 D^. Now, retaining the same lines and the same midship sec- tion, let us double the length, and hence the displacement. 146 RESISTANCE OF SHIPS. §2. Limita- tions of Method. Present Status of Method. The wetted surface would be very nearly doubled, becoming 12,000. D being now 2000, Z'3= 158.7. So' we have now 5=^6 D^ nearly, a change of about 25 per cent in the ratio between S and D^. With the midship section coefficient, Cg- the case would be even worse. The midship section of the 2000-ton ship being the same as that of the 1000-ton ship, the ratio -— for the 2000-ton ship would be double its value for the 1000-ton ship. Evidently, then, the admiralty coefficients of ships which differ much in their proportions should be expected to be different. So when using this method to approximate to the power of a new design, we can rely only upon coefficients obtained from the performance of ships of similar proportions at similar speeds. When the admiralty coefficients were introduced many years ago, there were few high-powered ships, and the wave resistance was seldom so great that a curve of coefficients would fall much below its maximum at the highest speed of the ship. The introduction of high powers and the multiplication of types were accompanied by such variations in the admiralty coefficients from ship to ship, that this method was soon dis- credited, and is now little used in England. The French engineers still use the reciprocal of the mid- ship-section coefficient with success ; but they fully under- stand its restrictions, and in applying it to a new design are guided by results from ships as nearly as possible similar to the new design. Their success is due to the possession of data from careful trials of numerous types of vessels, and to their own skill and experience in handling what is essentially a treacherous and untrustworthy method. §3- KIRK'S ANALYSIS. 147 § 3. Kirk's Analysis. The fundamental idea underlying the admiralty coefficient same method is that the resistance consists almost entirely of skin ^im^^s" resistance, varying as the square of the speed. Preceding. We saw that one weakness of the method was due to the fact that the wetted surface does not vary as D^, nor as M, except for similar ships. We may then naturally expect to find a method based upon the same fundamental idea, but using the wetted surface itself, or a reliable approximation to it. This is the method brought forward by Dr. A. C. Kirk. The method is usually called Kirk's Analysis, and may be divided into two parts. ELEVATION SECTION PLAN ■ 7 ^rT-" — " ' i^ C -* 1 ^^ '■'' ' ~ ]I" Pig. 47. BLOCK MODEL OF YORKTOWN. SCALE: l" — 80'. The first part of Kirk's analysis is a method for approxi- mating to the wetted surface. The actual hull below the water line is represented by a " block model," or model with plane sides. Let L denote the length of a ship, B the beam, H the Block mean draught, D the displacement in tons of 35 cubic feet, and Mthe area of the midship section. The block model. Figure 47, has the same length as the ship, and a uniform draught equal to the mean draught of 1^8 RESISTANCE OF SHIPS. §3. the ship. The beam of the block model, denoted by B', is such that B'H=M. Referring to Figure 47, let / denote the length CD, or AB, the length of the triangular entrance and run. CD and AB, or /, are made of such a length that the displacement of the block model is the same as that of the ship. Then we have the following relations between the known dimensions, etc., of the ship and the dimensions of the block model : B'Hy.BC-V2y.\B'Hy.CD=DY.i^,oxM{L-2l)+Ml=ii,D; M{L-l) = iSD; l=L- 3SB> M Surface of The surface of the block model is evidently nearly the Model. same as that of the corresponding ship. We have Surface of bottom = 5'(Z-/) =^, 11^= 31^. ^ ' H M H Surface of parallel sides =2 H{L — 2l). Surface of sides of ends ^A^Hy^i — j +/2- The total block model surface is readily calculated from the above. It is as a rule greater than the actual wetted sur- face. Kirk estimates the excess as follows : For very full ships the excess is about 2 per cent. For ordinary nierchant steamers the excess is about 3 per cent. For fine steamers, but not with hollow water lines, the excess is about 5 per cent. For very fine steamers with hollow water lines, the excess is about 8 per cent. Formula of Having determined the wetted surface S, as above, with Analysis. Sufficient approximation, Kirk assumed that the resistance § 3- KIRK'S ANALYSIS. 149 varied directly as the surface and as the square of the speed, and that the indicated horse-power would vary as the cube of the speed. Then he adopted the formula 1 00000' where /^ is a coefficient. According to Kirk, the appropriate values of k under various conditions are as below : 1. For fine ships with smooth and clean bottom and high values of ~, . , ... Constant. efficiency 01 propulsion, ^ = 4. 2. For merchant ships of ordinary proportions and efficiency, k=^. 3. For short, broad ships, k = 6. The above formula was proposed for and intended to be Limita- applied to ships of moderate speed, — not above 12 knots Errors of or so, — being more especially adapted to merchant vessels '*^«t'"'*- of 10 knots' speed. It has one advantage over the admiralty coefficient method, in that it uses a close approximation to the wetted surface itself instead of quantities whose ratio to the wetted surface varies with variation in the proportions of ships. Hence k may be expected to change less from ship to ship than C^ and C^, ante. But owing to the fact that the formula takes no account of the change in efficiency of propulsion in passing from low to high speeds, and no adequate account of the wave resistance, it is found that for a given ship, k varies greatly with the speed. This fact is exemplified by Figure 48, which is a curve of values of k for the Yorktown. It may be noted that the values of k and of the admiralty connection coefficients are connected by a reciprocal relation. i and „ Admiralty , D^V^ kSV^- Coeffl- Thus, /=— r; — = --——:' cients. 61 I 00000 whence, ^1/^= ^°°°°° =a constant for a given ship. ISO RESISTANCE OF SHIPS. 13- Let us see what the values of k given by Kirk imply as regards efficiency. Efficiencies For a steamer, say 350 feet long, the coefficient of skin iSrk's^n- friction is, from Table V., .00916. From Table XI. the value stants. of lO^-^^ is 67.61. Hence the skin resistance at lo knots of lOO square feet of wetted surface of such a steamer is loo x .00916 x 67.61 =62 pounds nearly. 8 T_ 6 _5 (c_i o ii_ ij 3. < o "2 1 SPEED IN KNOTS J T § 9 15 5 ii ii HI IS iS iT Fig. 48. — CURVE OF KIRk's CONSTANT k FOR YORKTOWN. The skin resistance power corresponding is .0030707 X 10x62=1.9. The corresponding indicated horse-power by Kirk's analysis . kx 100 X 1000 7 IS = k. I 00000 If k=4, the efficiency of propulsion, considering skin resistance alone, is -^=.475. 4 §4- EXTENDED LAW OF COMPARISON. 151 Similarly, the efficiencies corresponding to k=s and k=6 are .38 and .317 respectively. These numbers agree very well with what might have been anticipated from our previous work. While Kirk's analysis is preferable to the admiralty coeffi- vaiu? of cient method, it is based upon the same assumptions, which, Analysis, though fairly close approximations at low speeds, are seriously in error at high speeds. Hence the method should always be used with caution. In applying it to a new design, values of the constant k should be used which have been deduced from careful trials of actual ships similar in type and speed to the new design. § 4. Extended Law of Comparison. We have seen that, using the law of comparison, if we know for a given range of speed of a model or small ship the resist- ance following the law, we can determine the corresponding resistance of a full-sized ship, or a similar larger ship, through- out the corresponding range of speed. The question naturally arises whether the Law of Compari- Deduction son cannot be extended to indicated horse-power, even if only Extended in an approximate manner. ^^^' Froude has shown that his law can readily be applied to power as well as resistance, and a method of estimating indi- cated horse-power based upon the law of comparison is already much used and is rapidly growing in favour. Suppose that all the resistance of ships followed the Law of Comparison. Let R, V, D, Rp V^, D^ denote the resistances, corresponding speeds, and displacements of two similar ships. Then A = ^ by the law, ^1 A ^""^ ^=Vf,=V(:f)'=©* 152 RESISTANCE OF SHIPS. § 4- "^"^^' fyrl''(t)*^(0 Now R, V and R-^, V^ are proportional to E and E-i, the effective horse-powers. „,, E RV fD\l Whence, Now E=cl, Ei=e-Jy e and e-^ being the efficiencies of pro- pulsion. If e and e-^ are the same, which will be approximately the case in good work, we shall have /i E^ \dJ ' and ^i=(f7^ Hence, knowing /, V, and D for a given ship, we can readily calculate /j and Vi for a similar ship of known displacement A- So from a speed curve of a ship of a given size we can deduce a corresponding curve for a similar ship of any size. Example of Figure 49 shows the actual speed curve of the Yorktown Power at i68o tons extending to 17 knots, and the corresponding ""^^^" curves deduced by the above method for similar ships of 1000 tons and 2500 tons, extending to 15.6 and 18.2 knots respectively. Errors of There are four principal sources of error in using the Extended - _, Law. Extended Law of Comparison. In the iirst place, as we know, the skin resistance does not exactly follow the law. Take for instance the case of the Yorktown. §4- EXTENDED LAW OF COMPARISON. IS3 The skin resistance power of the Yorktown at i6 knots skin Re- was 800. f *""* does not For a io,ocx>ton vessel similar to the Yorktown '"""'^ Law. (■^j =(^^1 = 1-346 (the speed factor), while the power factor =Jj'yy^x 1.346=8.013. SPEED IN KNOTS. Fig. 49. — POWER CURVE OF YORKTOWN AT 168O TONS. REDUCED TO lOOO AND 25OO TONS. Then if the law applied, the skin resistance power of the 10,000-ton ship at (16 X 1.346) = 21.54 knots would be {800x8.013) = 6410. Now the actual wetted surface of the Yorktown at 1680 tons was 10,840 square feet. 154 RESISTANCE OF SHIPS. §4. Efficiency of Propul- sion varies for Given Ship. The wetted surface of a similar io,ooo-ton vessel would be 10,840 (JY D u O < O iz; < H Pi o in Pk £ tn o tn O in Iz; M _ -a (li ft u . ^^ ^ in t~. N N q H ft "^ c " r^ t^ vd a>- t^ t^ O (0 J< w M M vo t^ t^ 00 t^ o Meai Draug H. Feet w t^ r^ m rp q H d 4 "^ vd in t^ o" N N N N N « s .-s ro i-^ W 00 r~ 0\ i S n £ \d N 4 4 N 00 n fc lO VO r^ VO t^ vo JS Lengt L. Feet, N O "* M t^ Ov H. in in •* N lO ■* tn n ro m ft'S . S lO fT) q ■^ vq ov lO M vd M 00 M vd vd M M O u C3 0\ o ro o t^ N o\ in ro 00 Tj- !>. S O S lO "? in in in in H n o -3 z pi] ' g S «- o to O o O t^ t^ 1.1 Pi N ■!t- vo O CO ^■ hH tn t^ 00 M vo .S 6 H 1/1 ■* t^ ■* 00 Ov S5 a M (3 Mean Draugh H. Feet. M ■* M o in f^ .-I N q l^ ro t^ vo 1 M M d CO CO vd < E -si q N O Ch. q o fll « *> iri 00 00 ro 00 00 ffl f^ ■* •* VO t^ vo vo J3 o m m O m m 1^1 M IH N o N M ^ ro M '^ CO CO g ^ 1? Xi 1 ' 2 Sd 3 o o rt^ -2 a oj a 13 .S ro O ^ so « OS lO M H a>| IT) u^ O •:|- N lo so 00 so t^ so m so m « 1 s g »-i CTs 00 ro N O ro n r^ ■+ N OS CO r^ OS SO 00 00 lO fO 00 ^ m r^ OS so lO IT) u-1 so ■* in in LO lO in in in so H « O u ■ * CA o o oo t^ ro O o o CO r^ o O t^ r^ CSI sn r^ m 'i- so o t^ CO ■fl- M u o Tl- M 00 lO 00 to r^ oo N m so n C^ N lO OS 00 rO ■* t^ M 00 r^ OS so HI HI Q c In *: ro ^ ro o fO ^ M o in O t^ so IH -J 00 N ro N 00 O t^ ro r^ OS so in ■+ u 3 X u s s £ Q ^ sn 1-1 M 00 OS N O CO ■* so N SO M M N « M w N fO N w C4 N M s o o lO N (N o OS o O o o t-- ro I^ •* SO 00 oo ro 00 M 00 00 Os ro ■^ \n •* ■* ■* so x^ SO so so in SO £ ro n n in M to in O in lO in O N u ■ -g M M n ^ ■>t l-l N O c^ HI cq t^ N ,^-£ ■ to ■* LO ■* ro ro ro 'J- CO CO CO 0) CO H '^ > a; • J3 o s t/i (U B -1 s ni O 2 J3 O XI o o 1 as Ci u Q o 00 r^ o ON »^ 00 VO rx t^ 00 r* VO m " M M M w e 2 lO VO o o r^ CO r^ 00 I^ N O ON O If^l N N ^ ON i-^ !>• t^ lO CO CO o VO ■* ' H ro o M CO "t lO •* VO in 1^ r^ VO o" M c^ c< C^ N N N t>» c« « M E . i lO CO CO tH t^ N Oi 00 r^ Ov 00 VO O ^ S n u N VO t^ VO N M ■* ■^ N vn no 00 t^ n II. lO LO lO IT) VO VO t^ VO r^ VO VO VO VO 1^1 ■* N N t^ O ■* ■* ^ f^ vn 0\ Ov ON i-i M o U-) ON lO irt ■5t- M- r< M o \n to lO ^ ■ o\ O o lO CO ■u-, o o o CO r^ o o K M C . " t^ N VO w "d- CO M VO o 1^ CO M- PI tE°° ■* M 00 ^ lO N t^ 00 C4 m vn o N « lO Ov N ■* CO t^ ■!)■ 00 r^ Ov VO M en 5 M Q S 2 h ro M CO CO ^ VO M o m o t^ vn M ^ 00 CO qv 4 o as vq CO o CO Ov> vq vn in VO ^ ^ M ^ M M M H C^ CO N o N N c< g ^ o o o o M o O Ov O O o O r^ S n S m lO t^ lo 00 N 00 CO 00 M 00 00 Ov ro ■* lO CO ■* ■* VO r^ VO VO VO m VO 5 ^ ro O o m in o lO O in in in N 1^1 N w o 00 1^ C^l O w M C4 r^ « CD tT m N CO N CO ^ CO CO CO N CO ^ s ■< ^ > cn OP 1 1 Cli a s o 1 o IS o 1 i O O ■1 3 1) 1 s 1— 1 i ill e o (U o u D P!^ U pq N J U U Pn •aaawn>i M N rO ■* lO VO t~ 00 ON o tH tH M H CO M 1 62 RESISTANCE OF SHIPS. §S- All things considered, it appears that a suitable standard power for the case in hand is 22,500. The power factor 12000 11000 10000 9000 8000 7000 6000 6000 Fig. 53. — STANDARD CURVES. l6 TO I7 KNOTS. from Table XII. being 5.040, the actual power for the 2500- ton ship at 17 knots, as deduced by this method, will be 22500 _ 5.040 = 4465- § §5- STANDARD CURVES OF POWER. 163 en H O > u Q a a-S . J3 Ov ■* « 10 in 00 r- C4 N 00 0\ 00 r^ o\ t^ " 00 ^ t^ r~. 00 vo t^ vo 2 f2 Q "1 ^ n t^ 10 CO r^ 00 r^ N n OV d ^ CO 4 ■* m 10 4 m vd CO CO in t^ CO 4 q vo r^ vd 8 N C) c< « N N N (N c^ w w e^ cq M 0, E J ui to CO w t^ w « 00 t^ Ov 00 00 vo S m Si N VO t^ vo N Ov IH ^ Tf N vo vo 00 00 < B fc- U-) ^n u-i 10 VO vo VO ir^ vo t^ vo vo vo vo j: ■* C^ M t^ o\ ^ ^ M t^ VO t^ Ov Ov 1'^ M HH u-> M ON vn m ^ ^ CO C< tH 10 U-) 10 ■; ir> 10 m \n N 10 MD 00 vo r^ vo vo m •3(ES HI ON 00 r^ Tf ro i~- -^ l^ N ov t^ vo vo u^ 00 vo ro 00 ■^ in CO r^ Ov sou u-1 u~, 10 LO ■* lO m m vo in m a -3 * * f^ , IS 0^ 10 CO N in CO m r- tij J~- N VO N Tl- M CO vo t^ -* CO ■'t 1 •* hH 00 •+ 10 >o M i>- 00 « in r^ vo N VO CT\ C^ ■* CO r^ ■* 00 r^ o> ON vo p-< Q - • Q ' sL« . CO t^ Ov W vo in ^1 "^ "S- yD )_, ■* Ov o\ r^ « CO ■* ■* vo N H l-t CJ IH N CO N N M C4 M . vo vo vo vo m J! fO iri IT) CJ 1J~) in in in in N t-l 00 in I^ N C» M CO « i~. CO ■* 10 CJ CO Ht N fO ■* CO CO CO CO w u 5 I? J3 s 1 a 2 IS 1 13 4-> t ■4-> 13 ■ V i s So .a s s U £ 1 h) ■ 2 1 § U ^ ^ pti u U m W i-l CL, M- vo t* ON M N CO ■d- m •aaawnN w tH W M W M II64 RESISTANCE OF SHIPS. §5- 14000 13000 12000 11000 10000 Q. i 9000 8000 7000 6000 10 (fell. Fig. 54. — STANDARD CURVES. 1 7 TO 1 8 KNOTS. §5- STANDARD CURVES OF POWER. I6S m H O z 00 o > U Q « Q < o tn o tn Z o tn Z u a'S .-2 o\ On N M m ov ■* o in 00 o t^ 00 r^ O t^ Ov t^ cq M H-t 00 00 en Z o t-l *^ M N HI '^ M CM CM c» M M lO in C\ o o o 00 00 m t-- CO N VO « ^ ■^ ■+ OV ov o * CO t^ CO H o ro HH 1-1 CO CO VO m m VO in t-« o~ « N N N CM CM « CM w N CM CM E . -K o U-) 00 CO CO M 00 N o cq CM Ov u CQ u d N M r^ 00 VO w m Ov in W VO < O fe lO lO lO \n ■* m m VO VO VO VO VO t - N •* VO « Ov r^ Ov o Ov CM M- VO ^•| N M O o r- in CO t-t O o Ov 'i- lO in m in ■-' Q If^l VO ro N CO r* CO 00 ■* m O VO O .-1 ^ 00 ^ CO M ov in 00 •"I- in VO Ov < tH ■* 00 M m Tl- VO CO ov t^ x^ ■* BS C4 M IH rq M M CM M )H C4 ° E ^ o o o O o O o o VO CO o o n fa N to ■* r^ CM in M VO in CO cq M m ro ■* in m CO m CO N ■* ^ VO £ fO m o o in m N VO CM r^ m M . S Tj- N ro o 00 CO CM in VO t^ M s -^^ 10 CO ■* in m N ■* CM M M CM CO u '^ o (A (4-1 o g > rrt .3 ni i O s a c n tn s x^ f3 "ni ID tn o s ^ 2 o u CJ O t3 Pi <; >* U > m •aaawriN w N ro ^ m VO t~ 00 ov a M M i66 RESISTANCE OF SHIPS. §S- 16000 8. 15000 cS jMi-- ^ ^ fi. 3. 5. 14000 -Fm^ i^^**'' ^ -^ ,^ 7. 18000 - — -b5J ^ -■\5is' ^-- yoji T2i!!i. ^ 9. ^ 12000 ^ ^J^i^ VH^ -^ 4. 11000 ^^ -^ ^tfl^ ^ t^ ,^ ^^ 10 2. 10000 _^ ^ == ■— "Ot Sb^ ^ 1. 9000 ^ =^^ jHi5!=5 a^^ ^ 8000 -^ 7000 18.0 .1 .8 .3 .4 .5 .6 .7 .8 .9 19.0 Pig. 55. STANDARD CURVES. 1 8 TO IQ KNOTS. STANDARD CURVES OF POWER. 167 a-S .S On ri in in N ■* ■1 00 d f^ Ov W t^ f^ M M M tn •H N N M M c^ « w N Cfl 10 m ■!■ vd vd N 0) M N N N c^ N N t» Q 8 0" E . -i >9 to Ov t-i 't r^ N q N vo S n ii ffl d. cJ t^ ti vd in ^ m d\ m o5 ^ 10 u-> VO in VO VO \o vo vo r^ J3 Tj- N Tt- t^ m in OS t< 00 !3 M r^ in in to t-H t- Ui C kJ 1) IT) Ln 'I- ■!■ ■*■ ■* ■* •* to 0) g u s; g q M 6 in w 10 y3 00 M ^d M oo_ t< M in q t^ dv CO iSiES M 00 r^ 0\ 00 00 to r^ t^ vo tn r~- \o in VO in 00 N 00 Ov H irj VO ■* in 'J- ^ in ^ in -^ > c^ mu-D ' ' ' ' ' * (J S- N in m N to Ed -y M r^ vo 00 w 00 ON 00 M N CJ .2 H Q Tj- 00 to ■* r^ M vo m o\ a 3 5 N o\ M N to t^ m n _1J ■t 1 g 1> s a ■ M ^ N ,iA M ^ C3 0} s ■4-* ii CO •c 1— 1 fH ^ ^ -^ j3 • rH q s 1) J« "3 U p s P^ U > u > ■aaawriN M N to •* m ^ tr* 00 ov i68 RESISTANCE OF SHIPS. §5. 20000 19000 ^ 6 4 2 3 18000 ^ ^^^^ ^ ^ ^ 17000 ^ ^ '-j^ ^L- ^ 16000 ^ ^ so^ -0> ^ 7 15000 P^ "^^ ^ V ^ ■^>-^ 14000 ^ ^ 8 1. 13000 ^ ^^ ^t ^ ON* <^i>- "^ 12000 ^^^ i 11000 ^^ 10000 19.0 .1 .2 .8 .4 .5 .6 .7 .8 .9 20.0 Pig. 56. STANDARD CURVES. IQ TO 20 KNOTS. STANDARD CURVES OF POWER. 169 a'S .a N 10 N ■. t^ q 'f CO -^ H M to d 4 10 m NO NO « N N N c^ M C) eq 8 0" E ^ ro ON -^ t^ c^ q e^ NO ^ S n S r^ N 10 vd 10 ON 10 in n fe 10 VO VO NO NO NO NO t^ H a - N ^ 10 in On 00 1^ in ro tH r- 10 ■^ '^f "*• ^ ^ -*■ CO . 00 00 m r^ t^ VO If) VO IT) m 00 N 00 ON > j^- 5 5S 10 ^ 1" ■* u^ •^ "? "^ (J s jrf N in N CO H M3 CO 00 On 00 M N u 3 00 ro t^ « NO 10 On Q On M CO w PI ir> 1^ s Q 3 .2 "rt "rt p s u ►i5 >« U > ■aaarenivi M N fO ^ 10 NO t^ 00 I/O RESISTANCE OF SHIPS. §5- 26000 25000 / 8. 24000 / /^ ^ 23000 ^ 22000 c^' \^^ ^^ 6. 1. 21000 ^ ^ ::::i^ 7. 20000 •^ ^ ^ ^ -^ ^ 4. 9. 19000 ^ :^-i5< }1phin, ^^ j^ i9^ ^ 5. 18000 '^ ^^ "^ S^ ooS- joS-^ ^ ^ 10 17000 ^ P^ ^ 16000 ^ 6^ ^ -^ 15000 ^' 14000 ^ 13000 12000 20.0 .1 .2 .3 .4 .5 .6 .7 .8 .9 21.0 Pig. 57. — STANDARD CURVES. 20 TO 21 KNOTS. §s- STANDARD CURVES OF POWER. 171 a'S . S 10 N ir> q q ■* "it q in fO « N d 4 in N N M Ht M N N N w N N « N « N Mean Draught H. Feet. 10 VO \o VO Tj- 00 in r^ vo t^ J>. t^ HI M 'r °. 'f CO •>!}■ 0" in IN 6 in ■^ in in vd vd E ^ OS ^ ■* ■<1- r-. « q cj VO S BiS n (I. cJ in d 06 00 vd in dv m tn ^ VO vo M3 m m V3 vo vo vo *^ s § a - ■* IT) vo VO m Ov W 00 l-^J r^ irj IT) ■* ■* to f r^ ■* ^ ■* ■* ■* •^ ■* ■St- ■^ CO N H ■a 10 0, U *J « IT) q q t^ vo t^ 00 in q S > vd 4 m t^ r^ CO vd N t^ dv ra Si E^ S5-S ■f ^ 10 in in ^ in ■^ in ■>!■ Z Isplace- ment D. Tons. « 10 0\ OS « CO u H 00 CO N m in 00 M N 2 t^ r^ ID N N w vo m Ov Q t-* M HI CO M N m < Q fe fi Q Mean raught H. Feet. a\ IT) 00 00 00 ■* m .J « q !>• in in q 00 ■f m ■ i < 3 ro M M t^ ro dv HI C4 Ul Q P!i ," eq M N N N ro f) M N CO § J "^ tn Z S5 z u g 5 (4 S a ■ t<3 .S 3 § c ■| E .H cs '4 d S »-r a tn iH M ^ r3 cS .a OJ ni ri m u Q t^ U A > u >.o IH M fO -f m vo r^ 00 Ov ■aaawni^ HI 172 RESISTANCE OF SHIPS. §S- 28000 27000 26000 ^^ 25000 \^ ^ -''^ 24000 ^-^ -^ ^ ^ 23000 ^ ^ ^ ^^ ^ :^ 22000 ^ ^ ^ cpS! )05^^ 21000 ■^ ^ ^ -^"^ 20000 ^ ^ 19000 ^ ^^ 18000 ( «2S2^ t^-- -^ ITOOO ^ 16000 "^ 15000 14000 21.0 .1 .2 .3 .4 .5 .6 .7 .8 .9 22.0 Fig. 58. STANDARD CURVES. 21 TO 22 KNOTS. STANDARD CURVES OF POWER. i;3 m H O eg eg o H eg xn > o n c (« o Pk tn O O c-S .2 M U^ W o o ■<*■ ^ to ^1%^ lO w N ■^ to N N M (2 N « N N w w « CI Q M m O VO VO Tf 00 r^ 4 d q to CO vd N - N N ^ S n ii Tf w lO 00 00 VO to to m to VO VO VO VO to VO VO VO ro Tt- lo vn vn to n N l-'l / 0\ r^ m ^ rf PO M o ^ •t ^ 't •* ^ ^ ■* . -O n o\ w lO I^ VO t^ to o u «»■ O t^ o ^ 4^ r^ 00 VO t^ 00 r^ 00 o o 00 ro r^ to in IT) o o 00 N O ■* ■* ^ lO to ^t to to E- m o o • g S l|di VO N lO ON Ov o o CO P4 o oo 00 in to Ov 00 C4 1 CO rO t^ N N N VO Ov .2 2 H t-l M w M ro M « Si a Q lt»^l o\ o\ o 00 00 00 'r o J N w to to O 00 to 5 N f^ »H to fO r^ CO r^ Q E ^ \n to o N w M o CO « to M M 00 ON OV VO VO CO to r^ w N w t»- CO ^ 1^1 O u-> lO ro ro o VO t^ lO -^ Ov N N o N VO N N N N CO M N ^ a tn ■c 13 8 1 1 2 3 rt o o ^ pq u fe U > > ■saawnM M N ro ■* to VO t^ 00 174 RESISTANCE OF SHIPS. 81000 22, Pig. 59, .2 .3 A .5 .6 .7 .8 .9 23.0 ■STANDARD CURVES. 22 TO 23 KNOTS. §s- STANDARD CURVES OF POWER. i;s tn H O z fed m o H N (M o.'S . 2 „ N 0\ O ro ■* to o S > o lO N N ■* tn N N N f2 N N N N N N N c< sf^ti N o rO VO \o N 00 o N r^ O M t^ O r^ ^f £ ■* O ^ r^ r^ xn lo l^ N w N N N « N w o" E ^ M ■* w ■* ^ o « t^ ^ S n S ■* lO o\ 00 00 ■^ in N n h •o VO VO m U-) kO VO NO J3 n to M VO VO r~ O to o\ ir> lO ■* ^ o ■* ■H w •raawnM - N ro •* lO NO t^ 00 176 RESISTANCE OF SHIPS. 36000 35000 y 34000 y / 33000 ^ 32000 \ ^ y ^ 81000 ^ ^ ^ 30000 ^ 0^ ^ ^ y qT ^ ^ ^ ^ / 28000 ^ ^ t P^ y^ 27000 ^ ^ 26000 ^ y^ ^ 25000 ^ / ■0- JN^ ^ 24000 ^ ^ 23000 ^ ^ 22000 4. 1. 23.0 .1 .2 .3 A .5 .6 .7 .8 .9 24.0 Fig. 60. — STANDARD CURVES. 23 TO 24 KNOTS. §s- §s- STANDARD CURVES OF POWER. i;7 a'S . £ M q q q w3 1 o u *^ o IT) Mean H. Feet. N VO VO o M N § n o o" E . *J M M- Tt- Tf i S n S -=(- 00 oo' oo' in PQ Ik VO m u-> VO i-i O J3 fO VO VO N pi O •!f ■* ■* ■* ■<1- ■* ■* (N . O Top Speed V. Knots, q VO H M (-1 M M CO o ic *; o u c 00 o O ^ W5 lO o o VO > ^ S o K ■ 1 •aaanriN w N ro ^ 178 RESISTANCE OF SHIPS. §s. 43000 42000 41000 40000 39000 p 38000 / 37000 c 4^ ^ 2. 86000 A /- 85000 ^ y ^ 34000 c u / 38000 ^ y / 32000 ^ y ^ 1. 31000 ^ ^ 30000 y .^ f^ 29000 ^ ^ 28000 ^ 2TO00 24.0 .1 .2 .3 .4 .5 .6 .7 .8 .9 25.0 Pig. 61. — STANDARD CURVES. 24 TO 25 KNOTS. §s- STANDARD CURVES OF POWER. 179 m H O Z \^ 10 t> u O Pi VO « a\ ■* •* "* ■*■ ■^ ft'S . i c> r^ vq II >> M * M Hock ents. 00 M 10 VO ^ lO ■* ^ W M ' g §? .g VO Ov td IT) fO ^ a|Q g CO M N .2 g & M M M Q Q Mean Iraught H. Feet. o\ 00 q w 10 w t w 5 B E ^ 10 PI q S P9 S N dv 4 (S fe CO N CO 5 ^ ro r^ IT) Estimating to know, in order to determme the values 01 the 1. H. r.. Efficiency the efficiency of propulsion, depending upon the hull efficiency °[„^"'"''"' ^3, the propeller efficiency e^j and the engine efficiency e-^. These are best determined from analysis of trials of simi- lar ships. In the absence of sufficient information thus Wetted Surface. 1 82 RESISTANCE OF SHIPS. Common Features of All Methods. Admiralty Coefficient Method. Kirk's Analysis. obtained, suitable approximate values may be adopted as indi- cated in Chapter IV. for the engine efficiency. The hull efficiency may be taken equal to unity. The propeller efficiency in actual ships often falls below .6. An efficiency of .7 is, however, possible, and if care is bestowed upon the design of the propeller, we should always obtain an efficiency of .65. Then we have ^ = .65 X i X e^^. Since e^ varies, as we have seen, from .80 to .89, the value of e to be anticipated in designing will vary from .52 to .58, nearly. If the design contains abnormal features, they should be allowed for in making estimates of the efficiency to be expected. § 8. Comparison of Methods. It will have been made evident by what has gone before that any method for estimating indicated horse-power must be based upon the results of trials of actual ships. Any one of the methods which I have described will give good results in skilful hands, provided it is used in connection with a sufficient amount of data regarding actual trials of ships of diversified types. But the methods described are not all of the same intrinsic value and reliability. The weak points of each method were pointed out after describing it. Summing up the subject, it may be said that the admiralty coefficient method is the least reliable of those described. It is based upon erroneous assumptions, and is essentially untrustworthy. Kirk's analysis has all the faults of the admiralty coefficient method except one. While of value for a special purpose, — the determination of the power necessary for the 'lo-knot cargo steamer, — its range is too limited to allow it to be used with confidence for other types of vessels. §9- EFFECT OF ROUGH WEATHER. 183 The model tank method, though the most accurate of all Model Tank those described, is not available for every-day use. The extended law of comparison and the independent Extended estimate method are both sufficiently accurate in theory and inaepend- practice for practical purposes. They must, of course, be ^^t^^*'' used understandingly ; but it does not seem possible to devise Method, a method which will give reliable results without a certain amount of intelligence and judgment on the part of the person using it. The independent estimate method requires more skill and judgment than the extended law. Given the skill, however, it appears to be an essentially more reliable method, especially for low-speed vessels. § 9. Effect of Rough Weather and Foulness. Since any method of estimating the I. H. P. required by a seagoing given ship for a given speed is based upon trial results, and conditions since trials are almost invariably made in smooth water, and °'**'" with the ship's bottom clean, the power for a given speed is naturally estimated under these conditions. It should not be forgotten, however, that ordinary seagoing conditions differ from trial conditions. It is found, and very naturally too, that more power is developed by a ship during short measured mile trials than can be maintained for a long voyage. But apart from this, with a given development of power, Efieotof . 1. • -n 1 1 Rough a ship under ordinary seagoing conditions will not show the water, same speed as on measured mile trials — made in smooth water. Supposing her bottom clean, in average moderate weather at sea a ship will do from ^ to f of a knot less than in dead smooth water, at the same displacement, and for the same power. In rough and stormy weather the speed of a small, low freeboard vessel may be reduced to almost nothing. But, l84 RESISTANCE OF SHIPS. § 9. providing her propellers do not race, necessitating reduced power, it must be stormy weather indeed when the lofty- sided Atlantic "greyhound" loses more than a knot and a half, or thereabouts, through stress of weather. Absolute size is a great advantage in maintaining speed through rough water, and in the Atlantic liners large displace- ment is combined with a small proportion of beam to length, the latter being also a feature conducive to the maintenance of speed in a seaway. Effect of The difference between speed on the measured mile and Fouling. the speed shown in service for the same power is principally due to foulness of bottom. The exact effect of foulness upon the coefificient of skin friction is a somewhat obscure matter, but marked reduction in speed due to fouling is a matter of common experience. It appears safe to conclude that an amount of fouling is fre- quently encountered in practice which doubles the coefificient of skin friction, while ships do at times become so covered with barnacles, grass, etc., that the skin friction is four or five times its value for a clean bottom. Figure 62 shows the effect upon the Yorktown power curve of doubling the skin resistance power, making the justifiable assumption that the efficiency of propulsion at a given speed is not appreciably affected by the increased resistance. It is seen from Figure 62 that in this case the effect of the amount of fouling supposed is to cause always a reduction of two knots, more or less, in the speed shown for a given power. Indirect It should be pointed out that foulness has an indirect effect Foining. upon the speed as well as a direct one. Owing to the in- creased resistance, it is impossible, with a given effective pressure, to obtain the same number of revolutions with a foul bottom as when the bottom is clean and the resistance less. It follows that unless the adjustment of cut-off, etc., when working at full power with a clean bottom are such that the §9- EFFECT OF ROUGH WEATHER. 185 mean effective pressure can be increased when fouling ensues, the revolutions and I. H. P. will fall off when the ship becomes foul, and there will be a double loss of speed. 4000 SPEED-KNOTS Pig. 62. CURVES OF I. H. P. FOR YORKTOWN, WITH CLEAN AND WITH FOUL BOTTOM. There would be some much-needed light thrown upon this Need of Progressive question of fouling by a few progressive trials, when foul, of Thais of ships whose trial results when clean were known. foui ships. By analysing the trials, the effect of the fouling upon the wake and the slip of the propeller could be determined, as well as the amount of increase in the skin resistance. CHAPTER VI. PROPELLER DESIGN. Sections of Actual Blade. Lines of Advance. § I. Influence of Shape of Section and Variation of Pitch. Before taking up in detail the question of the design of a propeller to suit given conditions, I shall discuss a question of the greatest practical importance. A propeller blade, like many other things in this world, has two sides, a fact which should never be lost sight of. Figure 63 shows six sections of an actual propeller blade whose expanded outline is shown. The blade is somewhat narrow, and being cast-iron, is rather thick. The diameter of the propeller was 14 feet, and the pitch ig feet. The backs of the sections are circular arcs — a very common shape, especially in merchant work. The sections are taken at radii corresponding to diameter ratios of .2, .3, and so on up to .7. From Table XIV. I have taken the slip angles for each section corresponding to a slip of 20 per cent, and drawn the corresponding " lines of advance " which show the directions in which the sections advance into still water with the above slip. Owing to the fact that through interference of the blades the water has a certain sternward velocity when the edge cuts into it, the actual slip angles would be somewhat smaller than shown in the figure. It is seen that in every section of Figure 63 the back of the blade meets the line of advance at a much greater angle than the front of the blade. 186 § I. INFLUENCE OF SHAPE OF SECTION, ETC. 187 It necessarily follows that the backward pressure on the Backward leading part of the back of the blade is greater than the Thras't?'"' forward pressure on the leading part of the face of the blade. o I- o g < CENTRE LINE OF SHAFT Fig. 63. — SCALE l"=l'8". Some distance from the leading edge the water at the back of the blade will break into eddies, and from that point to the following edge the back of the blade will exert suction, and so help to increase the thrust. But the actual effective thrust will be the difference between the gross thrust on the blade and the backward i88 RESISTANCE OF SHIPS. §1- Variation of t and a. thrust on the leading part of the back. Evidently this back- ward thrust is a source of double loss. A good deal of power is required to generate it, and once called into existence it neutralises an equal amount of the forward thrust. Referring now to Figure 64, Section A shows on a larger scale the section of Figure 63 at the diameter ratio of .6. Section A. LINES Section B. .OFjDVANCE_F OR a SLIP. Fig. 64. On Section A are drawn lines of advance, corresponding to slips of 10, 20, 30, and 40 per cent. The following conclusions appear obvious : 1. The backward thrust will decrease with increase of slip, which is as much as to say that the thrust coefficient a will increase with increase of slip. 2. As the slip increases, there will be more eddying and less friction on the back of the blade. 3. As the slip increases, the angle at which the face of the blade meets the water will increase, involving diminished friction on the face. (See page 16.) Then the ratio - is not a constant quantity, but falls off with increase of slip. The elder Froude noted the diminution of friction with increase of slip, and stated in 1878 that at a slip of 30 per cent the friction at the back of the blade appeared to be diminished by half. § I. INFLUENCE OF SHAPE OF SECTION, ETC. 189 It will be necessary to consider the result of the variation / of -, but before so doing I wish to call attention to Section B of Figure 64. Section B is simply Section A distorted in such a manner improved that the leading portion of the back of the section is a straight Una tangent to the line of advance for 20 per cent slip. The result is increased pitch of the leading portion of the face, and transfer of the backward thrust to the side of use- ful thrust. This gain far outweighs the slight loss due to the fact that the leading portion of the face will work with greater slip than before. We have seen that the slip must increase a good deal before the efficiency is seriously diminished. Owing to the short length and great thickness of Section A, the efficiency of the distorted Section B, while above that of A, will still be less than it would have been if A had not had such a large angle between the face and the back of the blade at the leading edge. The advantages of thin blades, and of leading edges so thin that the line of advance for a fair working slip (which I take as 20 per cent) will clear the profile of the back with little or no increase of pitch along the leading edge of the face, are obvious. With such blades the value of a will be greater than with the ordinary shape of section, and the change in a as the slip increases will be slight, involving less variation in -. To fix our ideas, when considering the effect of the varia- Effect of f Variation tion of -, let uSitake the case of the Yorktown propeller. of /ana a « uponEftt- Here we took /= .045, and deduced a = y.S for slips in the ciency. neighbourhood of 20 per cent. Then, ^=184, {='OOS77- igo RESISTANCE OF SHIPS. § i. For efficiency of the propeller we have Z, a For the Yorktown propeller -¥.=.0954, F,= .i439, ^=.4469. Whence, ^=.322(1—5-) -■ .2\%s — •^ a f Taking - constant and equal to .00577, we have the curve of efficiency for - constant, shown in Figure 65. Suppose, now, - is variable, changing steadily from .00750 at zero slip to .00350 at 30 per cent slip. The resulting curve of efficiency is shown in Figure 65, f marked for - variable. a The curve of efficiency of one of Froude's model propellers is also shown in Figure 65. Practical What are the conclusions to be drawn from Figure 65 .-' Conclu- ^ 1. The curve of efficiency with - variable resembles the ■'a J. curve of the model propeller more closely than when - is "taken as constant. 2. The maximum efficiency corresponds to a greater slip for - variable. a 3. The maximum efficiency is not much changed, and in the vicinity of practical working slips, say from 15 per / cent to 25 per cent, the assumption that ~ is constant is sufficiently near the truth for practical purposes. § I. INFLUENCE OF SHAPE OF SECTION, ETC. 191 tion of Ex- perience. There are some facts of experience that accord with the conobora- theory set forth above of the influence of the shape of the blade section. I. It is generally admitted that for the same slip the cargo boat requires an essentially larger propeller than the fast twin- screw vessel, and that the propeller of the cargo boat should work with greater slip than that of the fast twin-screw vessel. S2.4- .2- .0- .05 .10 .15 — I — .20 IT — I — .80 — 1 — .35 .40 SCALE FOR SLIP Fig. 65. EFFICIENCIES ON VARIOUS SUPPOSITIONS. When I say "larger propeller," I mean, of course, one more than equivalent to the two of the twin-screw vessel. Now the cargct boat is nearly always fitted with heavy cast- iron propeller blades, while the twin-screw ship will have sharp brass or bronze blades. Hence the cargo boat will have a lower value of a, requiring more blade area to get the thrust with a given slip. 192 RESISTANCE OF SHIPS. §1- Azially Increasing Pitch a Source of Loss. Also the variation in - will be greater in the case of the cargo boat and extend further, so that the maximum effi- ciency will be found at a higher slip. 2. When a cast-iron propeller is replaced by a brass or bronze one, exactly similar except that the blades are thinner and sharper, the engines will not show the same maximum number of revolutions as before. This is because the value of a is increased by the change, more than counterbalancing any slight diminution oif, so that the turning moment neces- sary to obtain a given number of revolutions is greater than before. Hence, unless the maximum effective pressure of the engines can be increased, the same maximum revolutions as before cannot be reached. The question of increasing pitch should naturally be con- sidered in connection with the shape of section. There can be no doubt that axially increasing pitch is a source of dead loss. It is usually adopted with the idea of making the leading portion of the face of the blade tangent, or nearly so, to the line of advance. With a blade of any thickness this involves serious loss, because of the large backward thrust resulting upon the leading portion of the back of the blade. Even if the blade had, no thickness, axially increasing pitch would probably involve loss of efficiency. If constant over the whole blade, loss of efficiency would cer- tainly ensue. For, suppose we took two blades A and B of the same size and shape and delivering the same useful work. Suppose A of uniform pitch, and B of axially increasing pitch. The slip of every part of A is the same, and can be made the slip of maximum efficiency. The leading portion of B will have less slip than the a 1 were tion of Ex- perience. § I. INFLUENCE OF SHAPE OF SECTION, ETC. 193 uniform slip of A, and hence will do its share of work with less efficiency than the corresponding part of A. Part of B about the centre will have the same slip and efficiency as the corresponding part of A. But the rear portion of B must work with greater slip, and hence less efficiency than the corresponding part of A, the total thrust of A and B being the same. While — is probably not constant over the whole blade, it is, without doubt, so nearly constant that the above applies to the case. Axially increasing pitch is then, without doubt, a source conflrma- of loss, though the loss is probably not very great. This conclusion is confirmed in a negative way by common experience. Though Rankine and others have recom- mended increasing pitch, and though it has been tried very largely, and is still much used, it has never prevailed over the uniform pitch, for the simple reason that numerous trials have not shown any appreciable advantage from its use. While axial increase of pitch is not advantageous, it is Pitch probable that a very slight gain would result from a pitch Radiauy which varied radially in such a manner that each portion of the blade would work at the slip corresponding to its maximum efficienc)' The gain would be so slight, however, that it is not worth while attempting to make it. The only point to be noted in connection with this is that increase of pitch obtained by twisting a blade bodily, as is frequently done with adjustable blades, causes a variation of pitch rather favourable to effi- ciency than otherwise, while if the pitch be decreased by twisting, there will result a slight tendency toward lessened efficiency. Hence in designing such propellers we should be careful to estimate the pitch too small rather than too large, in order 194 RESISTANCE OF SHIPS. § 2. that change of adjustment, if found necessary, may be made without lessened efficiency. Table XV. will be found of interest in this connection. It gives the fraction of itself by which the pitch is changed for elements of various diameter ratios from . i to i.o, and amounts of twist varying from i to 6 degrees. The pitch of a twisted blade as usually stated is that of the tip. It would be preferable to take as the true new pitch that at the centre of effort of the blade. The centre of effort may be taken as at about f of the radius for ordinary shapes of blade. Summing Summing up what has gone before, it appears that the up. pitch of the face of a propeller blade should be uniform, unless it is necessary to increase the pitch along the leading edge in order to avoid backward thrust. Even this variation should be avoided, if possible, by making the angle between the face and the back of the blade equal to, or less than, the expected angle of slip. This, however, can seldom be done with blades of cast iron, which are necessarily somewhat thick. § 2. Standard Blade. Curved We frequently find propellers with blades bent backward along a straight line or a curve. This is with a view to avoid centrifugal action, which is supposed to involve increase of thrust deduction. The gain can be very slight, and is doubtless nearly or entirely neutralised in practice by increased area, etc., of blade upon the same diameter. So it appears that straight blades will be as good as any other kind, although there are cases when, on account of the nature of the supports or the struc- ture of the ship, the blades of a propeller should be bent backward. § 2. STANDARD BLADE. 195 The final question to be passed upon is that of the shape of shape of blade which should be used. We saw in Chapter II. that the shape of blade was not a very important matter as regards efficiency. Now a blade must have a good deal of length at the root in order to secure strength where it merges into the hub, and at the same time the length here, being limited by the size of hub, is usually necessarily less than the greatest length of blade. Also a blade should have well-rounded ends, with no cor- ners that are sharp or rounded with a small radius. This is because such outlying and semi-isolated parts of a blade contribute their full share to friction, but not to thrust. All things considered, a symmetrical blade with moderate curvature of leading and following edges and a well-rounded end appears to be about the best. Also it is convenient to adopt a blade such that we can express mathematically the curves of its outline. The standard shape which I propose to use is shown developed in Figure 66. The maximum length occurs at half-radius. Between this point and the hub the outline is that of an ellipse which, if continued, would touch the axis. Beyond the half-radius the outline is not elliptical, but represented by a cubic equation of the form — -f--^ = i- Of course only the part of the curve represented by the above which lies in the first quadrant is used for the blade. To draw this blade for a given length and radius it is not Method of necessary to have recourse to formulae. In Figure 66 there tion. is noted the total length of the blade at every twentieth of the radius, the lengths being expressed as fractions of the maximum length at half-radius. Hence, having determined the maximum length and the 196 RESISTANCE OF SHIPS. §2. radius, the developed shape of blade is easily determined by- using the fractions of Figure 66. ^^^1 .... .647 Z2\ / 18/ .787 1 \ .869 1 \ / le/ .922 \ \ \ xy__ ._.. .956 \ \ /20 14/ .978; /lo .991? /20 ~ — — 12/ -..^ __^-.— - - .997! > __n/ \ V _ . I /20 .995! /20 lL __. .980! 1 jL .954 7 \ s/ .•J17 ! / V.._...l.„ .866! / \ M HUB=%RADIUS / CENTRE LINE OF SHAFT Pig. 66. STANDARD BLADE. §2. STANDARD BLADE. 197 The diameter of hub is taken as -| of d, the diameter of the propeller. The size of hub may be slightly changed, or the standard shape modified near the hub without appreciably changing the characteristics of the blade. For the standard blade above, if / denote the maximum width or length, we have Area of blade =.35 a?/. Mean width =.9/. = Whence mean width ratio = .9— , , d — being the maximum width ratio. d It is, of course, necessary to be able to determine the characteristics of the standard blade without detailed calcu- lations for every case. We have seen that in order to determine the X charac- teristic we have ■ I dr X, Cyldr -J^d^' the integration extending over the blade. Evidently, if we double the width of a blade throughout, we double the characteristic. Then denoting the mean width ratio by k, we can write X^ = hXf, where Xf may be called the '.'function for X characteristic." Idr Then ^^.= ^»=J'^^f' y I Cyl dr ^'^'O^'di: Table XVI. gives values of X,, Y^ and Z, for the standard blade, whence the characteristics for any desired maximum or mean width ratio are readily deduced. igS RESISTANCE OF SHIPS. §3. § 3. Standard Slip. It is obvious that the problem of propeller design in the ordinary run of cases would be materially simplified if we knew in advance the best slip at which to work. By "best slip " I do not mean the slip which corresponds to the abso- lute maximum efficiency of the propeller, but the slip which will give the best all-round results. Assuming - to be constant, we have for the efficiency 'asX,+fZ, Since the functions of Table XVI. are all proportional ro the characteristics, they may be substituted for them in the above. That is, we may write ^ ^ asXf+fZ, Maximum Following the methods of Chapter II. in order to determine Efficiency. . . the maximum efficiency e^, and the corresponding slip s„, and denoting - X, by c, we have While e={i-s) cs-Yf cs+Zf Xp Yp and Zj are readily determined from Table XVI. We have for /the standard value .045, and for a I shall adopt the values obtained in Chapter II. from analysis of the trial results for Froude's experimental propeller. These values are below. §3- STANDARD SLIP. 199 For three-bladed propellers a=g.4— 1.2 m. For f our-bladed propellers « = 8. 4 — i .0 wz. Where m is the extreme diameter ratio. These values are slightly higher than will be found with propellers of the ordinary blade section, but probably not far from the true values for the standard blade section recom- mended in this chapter. .6- m .5- a z _MAXlMUM_EFFIClENcy ■H .3- .2- 3ir SLIP CORRESPONDING TO \ SCALE FOR EXTREME DIAMETER RATIO .5 .7 .9 7^ Pig. 67. EFFICIENCY AND SLIP. THREE-BLADED PRO- FELLERS. The calculation of the data necessary to plot Figures 6y and 68 is now easy. Figure 6y refers to three-bladed, and Figure 68 to four-bladed, propellers. 200 RESISTANCE OF SHIPS. §3- Each figure shows a curve of maximum efficiency and corresponding sHp and of efficiency at 20 per cent slip, all plotted upon values of extreme diameter ratio. Si.5- <.4- !.3- .2- .1- MAXIMUM EFFICj ENrv SLIP CORRESPONDING TO MAXIMUM EFFIOIENgL. SOALE FOR EXTREME DIAMETER RATIO i 1 .7 -^ "T" .5 T" A .5 .6 Fig. 68. EFFICIENCY AND SLIP. FOUR-BLADED PROPELLERS. These curves are upon the supposition that - is constant. We have seen that the effect of the variation in - is to increase the slip corresponding to maximum efficiency. Twenty per All things Considered, then, it appears reasonable to aim standaia. always at a "standard slip" of 20 per cent in designing. In Slip. § 4- DESIGN OF A PROPELLER. 201 the best propellers the maximum efficiency will probably be found at a somewhat smaller slip, but the falling off at 20 per cent will be immaterial. If we were designing propellers for ships which ran always considera- in smooth water with clean bottoms, it would be better to inflii::icing adopt a slightly higher standard slip with a view to securing the greater engine economy which results from quick -running machinery. The final economy as regards coal consumed by the ship in steaming a given distance would not be changed, but the advantages of smaller, lighter, and cheaper engines would be secured. The propeller, however, must be capable of working at the increased slip found in rough weather, or when the bottom is foul, without a marked loss of efficiency. For this reason a standard slip of 20 per cent seems best for all-round pur- poses, but the reasons for its adoption should be clearly understood, as a different slip should often be adopted to meet special circumstances. § 4. Design of a Propeller. Havins: disposed of the question of slip, we are now in a Pointstoiie ° ^ ^ '^ Settled in position to undertake the design of a propeller to suit a given Designmga ship ; that is, to settle the diameter, pitch, width ratio, and ^"p^"*'- revolutions, unless the latter are already fixed. To fix our ideas, let us suppose that we wish to design a propeller for a 20-knot twin-screw ship of 10,000 indicated horse-power. Suppose that we have a right to anticipate an engine efifi- ciency of 88 per cent,, and estimate by the methods previ- ously given that the wake factor will be 10 per cent. Then each screw must absorb 4400 horse-power. The speed of ship minus speed of a 10 per cent wake will be 18 knots, or 1824 feet per minute. 202 RESISTANCE OF SHIPS. § 4- The fundamental formula needed is The standard slip being .2, we have .8/7?= 1824. Whence, pR = 22So; f^Y= 11.852. Viooo/ Then P=iS-SS^nd\.2aX,+fZ^ = 7.111 nd^{aX,+ SfZ:i- Hevoiu- The method of procedure from this point depends upon Kxed?" whether or no the revolutions are fixed. Suppose first that they are not fixed. In that case we may assume the mean width ratio. Fair maximum working values appear to be .18 for four-bladed propellers, and .2 for three-bladed propellers. Let us investigate first four-bladed propellers. Since Jf„=.i8-J^ and so on, we have = S.i2^V^/+S/Z/). Let the extreme diameter ratio = .5. Then ^=8.4— i.ox.s = 7.9. .Z, (Table XVI.) = .236. f standard = . 045 . Z/ (Table XVI.) =.65. y^ (Table XVI.) = .557. P (given) = 4400. M- — 5.12x2.010. d= = 20.68. d P = •5- P = 2d=^l .36, pR = -- 2280. R = 2280 55.12. §4- DESIGN OF A PROPELLER. 203 Then 4400=5.12^2(1 354^ i^g^ Whence, d^ = = 42 7. 5, Now, Whence, and Whence, The efficiency from Figure 68 would be about 69 per cent. Assuming other values of extreme diameter ratio, and hence deducing other corresponding values of d, etc., we obtain data to plot Figure 69, which shows in full lines curves of pitch, revolutions, and efficiency, plotted upon diameter. This is for four-bladed propellers of mean width ratio = . 18. If we try three-bladed propellers of mean width ratio=.20, we have /' = 7. 1 1 1 X 3 X ^2 X .2(flX/+ 5/Z/), = 4.267 d\aX,+ sfZ,). Proceeding as before, we obtain the data for the dotted lines of Figure 69. From the curves of Figure 69, suitable values of diameter, pitch, and revolutions may be selected. If, for any reason, it is not possible to adopt suitable values of the above quantities from Figure 69, the circumstances of the case will usually indicate whether we must adopt a slip different from the standard .2, or change the values of mean width ratio used in Figure 69. 204 RESISTANCE OF SHIPS. §4- Revoiu- In practice, the revolutions at which the propeller is to work are usually fixed beforeh proportion the propeller to suit. i^Adv^^ct* work are usually fixed beforehand, and it is necessary to Fig. 69. — FOUR-BLADED PROPELLER, FULL LINES. THREE- BLADED PROPELLER, DOTTED LINES. Thus, suppose in the preceding, the revolutions had been fixed beforehand at 120 per minute. Retaining the standard slip of .2, we still have J>R = 22io; and if R = i20, p=i(). §4- DESIGN OF A PROPELLER. 20S If h denote the mean width ratio, we have P=4400= 7. 1 1 1 nd^k{aXf+ 5/Z/). Xf and Z, are functions of the diameter ratio, and hence not known in advance. Assume a diameter ratio of .8. Then for four-bladed propellers n =4; d =.8x 19=15.2; d^ =231, nearly; a =7.6; X/=.46i ; Z,= 2.04. Everything being now known except h, we find on solving, h =.169. And from Figure 68 the efficiency for .8 diameter ratio = .675. Proceeding thus for other assumed values of diameter ratio, we obtain the data needed to plot the full curves of Figure 70, which show, plotted upon diameters, the necessary mean width ratios, and resulting efficiencies at every diameter. The dotted lines refer to three-bladed propellers, and are obtained in precisely the same manner as the full lines. In choosing a suitable diameter from . Figure 70, it is consiaera- important to remember that the mean width ratio should not encing exceed .18 for four-bladed propellers, and .20 for three- width, bladed propellers. While wider blades are often used, there can be no doubt that wide blades greatly aggravate the evil of interference, and that up to a certain point narrow blades are per se more efficient. But wide blades mean small diame- ter, and hence in many cases increased efficiency. 206 RESISTANCE OF SHIPS. §4- Values Cbosen. In the case shown in Figure 70 it is desirable, from the point of view of efficiency, to keep the diameter as small as possible. Accordingly, adopting a four-bladed propeller and the limiting mean width ratio of .18, we have a diameter corre- sponding of 14.93 feet, or 14' 11", very nearly. .7 — X ^5y?yES_0F EFFICIENCY 0.4 2.2 cc — o SCALE FOR DIAMETER, FEET 11 Fig. 70. li li li "X 16 IT IF li CURVES OF EFFICIENCY AND WIDTH RATIO, FUI,L LINES FOR 4 BLADES, DOTTED FOR 3 BLADES. Coefficients and Con- stants need not be Exact. The mean width ratio being. 1 8, the mean width = .i8x 14.93 = 2.7, nearly. 2' 7 The extreme width then /=— ii-=?' and the outline of the •9 blade can be readily plotted from the fractions given in Figure 66. It should be remembered that throughout the above we are working with approximate quantities. Fortunately our approximations may be appreciably in error without seriously impairing the value of the conclusions reached. For any slip between 15 and 25 per cent the efficiency of a propeller is practically unchanged from the maximum. §4- DESIGN OF A PROPELLER. 207 The approximate quantities used in the preceding will sel- dom or never be so much in error that the variation of slip will be more than 2 or 3 per cent above or below the standard 20 per cent. If it happens that the estimate of wake or horse-power is so much in error that the engine at full power works off the steam too fast, or not fast enough, it is very easy, by adjust- ing the pitch of the propeller, to hold the revolutions at the proper number. We have seen that this twisting of the blades will not materially affect the efficiency. It is well known that the propellers of cargo steamers comparison , • , , , . , ,. of Propel- usually work with rather a high slip. lers as It would seem, then, that the adoption of a standard slip ^Hlg^aw of .2 would result in larger propellers than are now generally Kttea. fitted. But if the section recommended in the early part of this chapter be used, it will be found that the sizes of propeller will be but little, if any, increased. This is because with the segmental section commonly used the value of a is often reduced to six, or thereabouts, while with the section I propose the values of a deduced from Froude's experiments will be found very close to the truth. It will be well to tabulate these values of a for three and four bladed propellers, and the efficiencies at the standard slip of 20 per cent. They are given below for the range of diameter ratio likely to occur in practice. In design work the above values of a are only to be used when reliable constants from trials of propellers, similar to the one being designed, are not available. There is some reason to believe that, owing to minor ^^^^ influences which I have not considered, the maximum effi- Efficiency, ciency possible corresponds to a somewhat higher diameter ratio than indicated by the above table, probably about .65 to 208 RESISTANCE OF SHIPS. §4- .7. This is a matter of no practical importance, however, since from .5 to .8 the efficiency is practically constant. It should be noted that this is independent of the values of a and/. Possible changes in a and / would change the effi- ciency for each diameter ratio, but the variation of efficiency with variation of diameter ratio would not appreciably change. Extreme Diameter Ratio. Three Blades. Four Blades. Value of Efficiency Value of Efficiency a. at .2 Slip. ct. at .2 Slip. •4 8.92 .688 8.00 .676 ■5 8.80 .702 7.90 .692 .6 8.68 .704 7.80 .694 ■7 8.56 .698 7.70 .687 .8 8.44 .686 7.60 ■67s •9 8.32 .672 7.50 , .660 I.O 8.20 •657 7.40 •643 § 5. Strength of Propeller Blades. Elementary Thrust, Moment, etc. It is, of course, essential that the thickness of a propeller blade should be such that the blade will not break. At the same time it is very desirable that the blade should be no thicker than required for strength. I propose, then, by the methods already used, to investigate the stress upon a propeller blade, and the strength required. To do this it is necessary to return to first principles. We have seen (page 70) that the elementary thrust upon a small area, dA of a blade, situated at radius r, is given by dT=fR^ as 7r2j/2V7rV+(l —^T l + iry ■2i,2 ^/Vi + TrYd dA, § S- STRENGTH OF PROPELLER BLADES. 2O9 or using the notation subsequently adopted, dT=fR\asX-fY)ldr, where / is the width of the blade at radius r. The moment of this elementary thrust about a line through the axis, parallel to the middle of the eXevatnt = dTy.r, very nearly. Now r=^py, where j is the diameter ratio at radius r. Then moment of thrust = 9i .606 .6 .692 .644 1.0 .696 .600 •7 .684 .625 §5- STRENGTH OF PROPELLER BLADES. 213 This table and Figure 71 afford the information necessary working to enable us to determine the positions of the centres of lor Thrust thrust and of transverse force in any case. andTrans- -' verse Force. .7- .5- A- .3- EXTREME DIAMETER RATIO J \ L ■'A .7 .1.0 Fig. 71. VALUES OF Kj AND Kg The values of the transverse force and thrust can be deduced with sufficient approximation from the propeller power P- Let A denote the transverse force in pounds exerted against a single blade of a propeller of n blades. Then A-n-k^—2 7ri?=work delivered to the propeller in one minute in foot-pounds = 33,000 P. Whence 33000 /• A = n • k^divR 214 RESISTANCE OF SHIPS. § S- Let B denote the thrust in pounds upon one blade. Then B ■ n -pii — j)7? = useful work done per minute in foot- pounds =33,000 U. 33000 U Then B=- .p.{i-s)R The standard slip being .2, we have i — j = .8. Also it is best in this work to assume a constant efficiency of propeller of .7. This is slightly above the truth, and allows a slight margin of safety. Making the necessary substitutions, we have 33000 X .7 /* _ 28875 P .8 npR np • R . ,. ,. From the above we can deduce the transverse and longi- Application ° to Practical tudinal bending moments upon the root section of the blade, which is the most strained. The modus operandi will be best understood from a practical example. For the four-bladed propeller at 120 revolutions of § 4, we have f=440o; d= 14.93 ; i?=i2o; i^=4; Jo=-^=786. By interpolation in Table XVIIL, or directly from Fig- ure 71, k^=.6^T, /&i^=s'.i3; ^2=-6iS; /^2 2=4'- 59- §5- STRENGTH OF PROPELLER BLADES. 215 From the above we deduce A = 10,485 ; i5= 13,930. The most strained section is that at the root. Now the radius of hub=^ x 14.93 = 1.66. Then transverse moment at root = 10,485 (4.59— 1.66) X 12 = 10,485 X 2.93 X 12 in inch-pound units = 368,600. Fore and aft moment at root =5C,^i--i.66')infeet = i3>930 (5-13 — 1-66) 12, in inch-pound units. i3>930X 3.47x12 = 580,100. These moments are shown graphically in Figure 72. By resolving them into two other moments, perpendicular and parallel to the face of the blade at root, we obtain the bending moments upon the root section which are perpendic- ular and parallel to the face. These latter moments can also be calculated.' Referring to Figure 72, ^^ =/=i9'; CB =2 7rXi.66=io'.47; tan ^=-^=1.815; 10.47 ^=6i"09'; sin (9=. 8759; cos ^=.4825 ; 2l6 RESISTANCE OF SHIPS. §S- MT denotes the transverse moment, and MS the fore-and- aft moment. A Fig. 72. BENDING MOMENTS ON ROOT SECTION. '5- STRENGTH OF PROPELLER BLADES. 217 Draw 77e, SQ parallel to the face of the blade to meet a line perpendicular to the face. Then moment perpendicular to face of blade = MS cos e + MT cos (90° - 0) = MS cos 9 +MT sine = 279,900 + 322,900 = 602,800, denoted by M^. Moment parallel to face of blade =MV=SQ-TR =MS sin e-MT cos d = 508,150-177,850 = 330,300, denoted by M^. Having arrived at the bending moments upon the root sec- Moments of tion, it behooves us next to consider the moments of resist- ^^^^\ ^ etc., 01 Root ance of the latter. section. The root section is usually made a segment of a circle. This is so close to a parabolic segment for the proportions of depth and width found in practice that I shall call it para- bolic, since the moments of inertia of a parabolic segment can be expressed very simply. Fig. 73. PARABOLIC SEGMENT. Figure 73 shows a parabolic segment of width or length of base AB denoted by /, and of depth or thickness CD denoted by k. Section. 2l8 RESISTANCE OF SHIPS. § 5- The properties of this segment which we shall need are as follows : Kx&z.=\AB-KCD = \lh. DG, or distance of centre of gravity of the section from the face, =\CD=\h. Moment of inertia of section about a line through G par- allel to the face =^-^W : denote this by Z^. Moment of inertia of section about a line through G per- pendicular to the face = -^ P/t : denote this by I^. Now Mj^ denotes the bending moment perpendicular to the face of the blade, and M^ the moment parallel to the face, stresses on Then by the well-known formulae of applied mechanics for the strength of beams, we have Fr9m M^, Tension at A and B =-hy. -^ 5 A „ . ^ ^ 3 , Ifj 105 i^i Compression at c =- h y. -~ = —^ -rk- ^ 5/18 IB From M^, Tension at ^ = j^ = ,„, ^ . 2 /g /% Compression at ^ = j^ = ^ '■ ■ ^ J. n L fir In all practical cases the maximum tension, which I shall denote by^^, is at A, and the maximum compression, denoted by/2, at C. Then. /,=3S Mi.^llM^, ■'^ 8 /A2 ■ / is fixed by the width of the propeller near the boss, so we can from the above determine the requisite thickness for a given maximum tension or compression. §5- STRENGTH OF PROPELLER BLADES. 219 The amount of these maximum strains allowable depends Maximum Stresses upon the material. An ample margin of safety should be AUowawe. allowed in all cases, and the fact that these are fluctuating stresses borne in mind. All things considered, the following seem allowable work ing stresses: Table XIX. Material. Working Stress Allowed. Pounds per Sq. In. In Tension. In Compression. 2000 5000 30C30 5000 6000 lOOOO 4oeo 6000 Cast Steel Composition Manganese and phosphor bronze .... Let us determine the root thickness necessary for the Thickness . , . . at Root for propeller discussed m the preceding part of this section, various supposing cast iron the material used. Materials. We have J/i=6o2,8oo; « ^2=330,300; /i=20oo; /2=6ooo. The maximum width of blade is 3 feet, or 36 inches. Then the width at root (Fig. 66) = 36x.86 = 3i". From the equation for maximum tension we have ,000=35^602800^ 15x330300 or, whence. 4 31^^ >^2_2.s78>^=8s.o9; ^ = 10". 5, about 961 h 220 RESISTANCE OF SHIPS. From the equation for maximum compression, , 105,602800. 6000=— ^x j^; whence, k = 6".6. Of course, the greatest value of k must be used, since the blade must be strong enough to stand both tension and compression. I have calculated the values of h for the above propeller blade for the various materials and stresses of Table XIX. The results are of interest, and are given below in Table XX. Table XX. Jiooi Thickness of Blades of Various Materials. Thickness of Root from Con- Material of Blades. sideration of Tension. Compression. Cast iron ... io".5 6".6 Cast steel 6".4 5".i 8". I Composition 8".4 Manganese or phosphor bronze . . . 6".4 6".6 Having settled the thickness at root, the thickness near the tip is made just enough to allow the blade to be cast easily, and the back of the blade is given a uniform taper. By this method, if the root is thick enough, the thickness will be sufficient throughout. The formulae given above for the segmental section apply with ample approximation to the modified section, shaped to avoid backward thrust, which I have recommended. The extra thickness of blade necessitated when cast iron is §5- STRENGTH OF PROPELLER BLADES. 221 used is very objectionable. Manganese and phosphor bronze are usually used for propellers at present in high-class work. It appears probable that in time cast iron will cease to be used except for special cases. For a tug, or ship exposed frequently to danger of damage to its propeller, cast iron is preferable. Its brittleness now becomes a virtue. When a cast-iron propeller strikes an obstruction there is seldom more damage done than is involved in the loss of a piece or the whole of a blade. If a propeller of stronger and tougher material strikes an obstruc- tion, there is danger of serious injury to the shaft or engines. 222 RESISTANCE OF SHIPS. Table III. Froude's Frictional Constants for Salt - Water, Paraffin, or Smoothly- Painted Surfaces. u •S-2 £ •s a u . u Power according to which Fric- tion varies. u lA u m u >^ MO .J ^0 Ek *o g 'g.2 ♦• Power according to which Fric. tion varies. f. •wcr v^ f. xn^ /VT. 8 .01197 1.825 80 .00933 1.825 9 .01177 i( 90 .00928 ti 10 .01161 (( 100 .00923 it 12 .01131 iC 120 .00916 it 14 .01106 tt 140 .00911 tt 16 .01086 ti 160 .00907 it 18 .01069 ce 180 .00904 tt 20 •01055 ic 200 .00902 it 25 .01029 a 250 .00897 ti 30 .01010 it 300 .00892 it 35 •00993 (C 350 .00889 if 40 .00981 ti 400 .00886 it 45 .00971 tf 450 .00883 it SO .00963 it 500 .00880 tt 60 .00950 {{ 550 .00877 tt 70 .00940 tt 600 .00874 tt TABLES. 223 Table IV. Surface Friction Constants for Paraffin Models in Fresh Water. £ s s •s g li -H U Povrer according to which Fric tion varies. ti u a "5 •a t a u u b, ■0 a V U Power according to which Fric- tion varies. f. J31. -v^ t. Uf. ■y- 2.0 .01176 1.94 12.0 .00908 1.94 3-0 .01123 *« 12.5 .00901 (C 4.0 .01083 u 13.0 .00895 cc 5-0 .01050 li 13-5 .00889 It 6.0 .01022 ti 14.0 .00883 it 7.0 .00997 it 14-5 .00878 tt 8.0 .00973 ti 15.0 .00873 tt 9.0 •00953 if 16.0 .00864 it lO.O .00937 it 17.0 .00855 tt lo.s .00928 it 18.0 .00847 ti II.O .00920 tt 19.0 .00840 It "•5 .00914 li 20.0 .00834 tt 224 RESISTANCE OF SHIPS. Table V. Surface Friction Constants for Ships in Salt Water of 1.026 Density. Q. Is "s s M.S g Iron Bottom, Clean and Well Painted. Copper or Z nc Sheathec Sheathing Smooth and in Good Condition. Sheathing Rough and in Bad Condition. f. -SB. f. n^ f. 10 .01124 1-8530 O.I 000 1-9175 .01400 1.8700 20 .01075 1.8490 .00990 1.9000 •01350 I.86IO 30 .01018 1.8440 .00903 1.8650 .01310 1^8530 40 .00998 1-8397 .00978 1.8400 •0127s 1.8470 5° .00991 1-8357 .00976 1.8300 .01250 1.8430 100 .00970 1.8290 .00966 1.8270 .01200 1.8430 150 .00957 1.8290 •00953 1.8270 .01183 1.8430 200 .00944 1.8290 .00943 1.8270 .01170 1.8430 250 .00933 1.8290 .00936 1.8270 .01160 1.8430 300 .00923 1.8290 .00930 1.8270 .01152 1.8430 350 .00916 1.8290 .00927 1.8270 .01145 1.8430 400 .00910 1.8290 .00926 1.8270 .01140 1.8430 450 .00906 1.8290 .00926 1.8270 .01137 1.8430 500 .00904 1.8290 .00926 1.8270 .01136 1.8430 TABLES. 225 Table VI. Comparison between Waves in Shallow and Deep Water. Depth of Water as a Fraction of the Length of 'Wave. Ratio between Quantities for Shallow Water and Corre- sponding Quantities for Deep Water. Length and Velocity for Given Period. Length for Given Velocity. Velocity for Given Length. .01 .063 15.90 .251 .02 .124 8.08 •352 •03 .186 5^376 •431 .04 .246 4.065 .496 •05 •304 3.289 •552 •075 •439 2.277 .663 .10 •557 1.796 .746 •IS •736 1-358 .858 .20 .847 1. 180 .920 .25 .917 1. 09 1 •958 •30 •955 1.047 •977 •35 •975 1.026 .987 .40 .987 I.013 •993 •45 •993 1.007 .996 .50 .996 1.004 .998 •55 .998 1.002 •999 .60 .999 1. 00 1 •9999 •75 •9999 1. 000 1 , ^99999 1. 00 .99999 I.OOOOI .999999 226 RESISTANCE OF SHIPS. > -^ a s " I o r^ vo •* r^ N ro q 00 vo lO ■* ro M M o\ OV o\ OV 0^ o\ OV N « « N M N w vo 00 lO N O o\ OV lO ro N M O 00 r^ 0) V3 vo VO VO VO m lO N N N N N w CM o lO M t^ Tt- IM M 00 N o Ov r^ VO lO ■5t- ro ro N M N N N N N N N N N N vo o\ ^ Ov lO N o I- 00 vo m ro N M o M M q\ M ■ On M qv M qv M 00 o ■* 00 ro Ov VO (0 ■<*■ ^0 M Ov 00 vo lO vo M vq M lO in M vo 00 M ^ OS 't o in o 00 f^ m rO IM M H ' M M ■* •* vo o\ N VO , H( ■+ VO ■* N M OV 00 VO Ov Ov qv Ov 00 00 00 m O ro t^ W t^ N Ov ro )H o\ 00 vo VO ro vq vo in lO in VO vo OJ O M Os 00 tr^ VO vo N H a\ 00 t^ vo lO ro ro N N N w « O vo M t^ rr> Ov lO 0\ 00 00 t^ t^ VO vo q q q q q q o fe >. to B ° .5 13 X II lO o iri o lO o << n ^ X .2 t^ Q o r- 0\ 0\ O O N Tj- o^ O O 00 lo lo lo ^ *o lo in Ti- OS O ■* o «3 O o ■o to '^ r^ M to VO ro vo VO VO VO to 00 00 M VO OS VO to O r^ VO VO IT) HH r^ T)- t^ a\ VO to OS o VO VO irj VO M to w SO to so so 00 o OS so to OS in so r^ so so so IT) SO o 00 "1 so so ■* 00 t>* 00 ■* o so SO so 00 so so SO OS O OS SO SO lO 00 to OS so VO o -* o N OS so to ' OS o so so in o to TABLES. Table XI. Powers of Speeds needed in Calculalions. 229 Speed V. • V3.17. V2.83, V6. .0030707 v. I 1. 00 I.O 1.0 I 0.0 2 3-56 4-5 7-1 32 0.1 3 7-47 10.8 22.4 243 0.7 4 12.64 20.2 50.6 1024 3-1 5 19.02 32-9 95-1 3125 9.6 6 26.55 48.8 159.2 7776 23-9 7 35-20 68.2 246.4 16807 51.6 8 44.94 91. 1 3S9-5 32768 100.6 9 55-75 117.1 501.9 59049 181.3 10 67,61 147.9 676.1 I 00000 307.1 II 80.49 181.9 885.4 16IO5I 494-5 12 94-38 219.7 1132.6 248832 764.1 13 109.27 261.3 1420.7 371293 1140.1 14 125.14 307.0 1752.0 537824 1651-5 IS 141.99 356.6 2129.8 359375 2331.8 16 159-79 410.1 2556.6 1048576 3219.9 17 . 178-53 467.8 3035-0 1419857 4360.0 18 198.22 529-4 3568.0 1889568 5802.3 19 218.84 595-4 4158.2 2476099 7603.4 20 240.37 665.6 4807.5 3200000 9826.3 21 262.82 740.1 5519-2 4084101 12541.1 22 286.18 818.5 6296.2 , 5153632 15825-3 23 310-43 901.4 7139-9 6436343 19764.1 24 335-57 988.5 8053.7 7962624 24450.9 25 361.60 1080.4 9040.0 9765625 29987.3 26 388.51 1176.3 10101.2 11881376 36484.2 27 416.29 1276.6 11239.6 14348907 44061.3 28 444-94 1381.5 12458.3 17210368 52848.0 29 474-45 1490.8 13759-0 205 1 1 149 62983.7 30 504.82 1604.6 15144.4 24300000 74618.2 230 RESISTANCE OF SHIPS. Table XII. Factors for Reduction of Speed and Power to Standard Displacement of 10,000 Tons. Displace- ment. Factor fcr Displace- ment. Factor for Displace- ment. Factor for Speed. Power. Speed. Power. Speed. Power. 100 2.155 215.450 5500 1-105 2.009 10800 .987 .914 200 1.919 95.970 5600 1. 102 1.967 10900 .986 .904 300 1-794 59-798 5700 1.098 '■ifz 1 1000 .984 ■895 400 1. 710 42.750 5800 1-095 1.888 1 1 100 -983 .885 500 1.654 33-079 5900 1.092 1. 851 1 1 200 .981 .876 600 1.598 26.638 6000 1.089 1. 815 1 1 300 .980 .867 700 1-558 22.313 6100 1.086 1.780 1 1400 .978 .858 800 1-523 19.043 6200 1.083 ^•747 1 1 500 ■977 .850 900 1.494 16.598 6300 1.080 1. 714 I1600 •975 .841 1000 1.468 14.678 6400 1.077 1.683 H700 •974 .833 1 100 1.445 13-133 6500 1.074 1-653 11800 •973 .824 1200 1.424 11.866 6600 1.072 1.624 I1900 .971 .816 1300 1.405 io.8o8 6700 1.069 1.596 12000 •970 .808 1400 1.388 9-913 6800 1.066 1.568 12100 .969 .801 1500 1-372 9.146 6900 1.064 1.542 12200 •967 •793 1600 1-357 8.483 7000 1. 061 1. 516 1 2300 .966 ■785 1700 1-344 7-903 7100 1-059 1. 491 12400 •965 .778 1800 I-331 7-394 7200 1.056 1.467 12500 ■963 ■77' 1900 1-319 6.942 7300 1.054 1.444 12600 .962 ■764 2000 1.308 6.538 7400 1. 05 1 1. 421 12700 .961 •757 2100 1.297 6.177 7500 1.049 1.403 12800 .960 ■750 2200 1.287 5-850 7600 1.047 1-377 12900 ■958 •743 2300 1.278 5-555 7700 1.045 1-357 13000 •957 •736 2400 1.269 5.286 7800 1.042 1-336 13100 •956 •730 2500 1.260 5.040 7900 1.040 1-317 13200 -955 -723 2600 1.252 4.814 8000 1.038 1.297 13300 -954 .717 2700 1.244 4.607 8100 1.036 1.279 13400 •952 .711 2800 1.236 4.416 8200 1.034 1. 261 13500 -95" •70s 2900 1.229 4-238 8300 1.032 1.243 13600 -950 .699 3000 1.222 4-074 8400 1.030 1.226 13700 •949 •693 3100 1. 216 3.921 8500 1.028 1.209 13800 .948 .687 3200 1.209 3-779 8600 1.026 1.192 13900 .946 .681 33°o 1.203 3-645 ?r° 1.024 1. 176 14000 •945 •675 3400 1. 197 3-521 8800 1.022 1.161 14100 •944 .670 3500 1. 191 3-403 8900 I.OZO 1.146 14200 •943 .664 3600 1. 186 3-293 9000 1. 018 1.131 14300 .942 .658 3700 1. 180 3.190 9100 I.0I6 1.116 14400 .941 •653 3800 1. 175 3.092 9200 I.0I4 I.IOZ 14500 .940 .648 3900 1.170 3.000 9300 1. 012 1.088 14600 -939 •643 4000 1. 165 2.912 9400 I.OIO '•075 14700 -938 • 638 4100 1. 160 2.830 9500 i.ooS 1.062 14800 -937 •633 4200 1. 156 2-751 9600 1.007 1.049 14900 -936 .628 4300 1. 151 2.677 9700 1.005 1.036 15000 •935 .623 4400 1. 146 2.606 9800 1.003 1.024 15100 •934 .618 4500 1. 142 2.539 9900 1. 001 I.012 15200 •933 .613 4600 1.138 2.474 lOOOO 1. 000 1. 000 15300 -932 .609 4700 I-I34 2.413 lOIOO .998 .988 15400 -931 .604 4800 1. 130 2-354 10200 ■997 -977 15500 ■930 .600 4900 1. 126 2.298 10300 •995 .966 15600 .929 •595 5000 1. 122 2.245 10400 •993 •955 15700 .928 •590 5100 1. 118 2.194 10500 .992 •945 15800 ■927 .586 5200 1.115 2-145 10600 .990 •934 15900 .926 .582 5300 i.iii 2.098 10700 .989 .924 16000 •925 .578 5400 1. 108 2.052 TABLES. 231 Table XIII. Factors for Reduction of Dimensions to Standard Displacement of jo,ooo Tons. i Q 1^ 1^ i I u 1 = i, .2 fi .2 E Q u I 100 4.642 3300 1.447 6500 1-154 9700 1.010 12900 •919 200 3-684 3400 1-433 6600 1. 149 9800 1.007 13000 .916 300 3.218 3500 1-419 6700 I- 143 9900 1.003 13100 .914 400 2.924 3600 1.406 6800 I-I37 lOOOO 1. 000 13200 .912 500 2.714 3700 1-393 6900 1. 132 lOlOO -997 13300 •909 600 2.SS4 3800 1-381 7000 1. 126 10200 •993 13400 .907 700 2.426 3900 1-369 7100 1. 121 10300 .990 13500 .905 800 2.321 4000 '-3S7 7200 i.ii6 10400 .987 13600 •903 900 2.231 4100 1.346 7300 I. no 10500 .984. 13700 .900 ICXX3 2. 1 54 4200 I-33S 7400 1. 105 10600 .981 13800 .898 IIOO 2.087 4300 1-325 7500 1. 100 10700 .978 13900 .896 I2CX) 2.027 4400 1-315 7600 1.096 10800 -975 14000 .894 1300 1.983 4500 i-3°5 7700 1. 091 10900 .972 14100 .892' 1400 1.926 4600 .1.295 7800 1.086 1 1000 .969 14200 .890 1500 1.882 4700 1.286 7900 1.082 I'l 100 .966 14300 .888 i6cx) 1.842 4800 1.277 8000 1.077 II200 -963 14400 .886 1700 1.805 4900 1.268 8100 1-073 1 1300 .960 14500 .883 1800 1.771 5000 1.260 8200 1.068 1 1400 -957 14600 .881 1900 1-739 5100 1.252 8300 1.064 11500 -955 14700 .879 2000 1. 710 5200 1.244 8400 1.060 11600 .952 14800 .877 2100 1.682 5300 1.236 8500 1.056 11700 ■949 14900 .876 22CX] 1.656 5400 1.228 8600 1.052 11800 -946 15000 .874 2300 1.632 5500 1. 221 8700 1.048 11900 •944 15100 .872 2400 1.609 5600 1. 213 8800 1.044 12000 .941 15200 .870 2500 1.587 . 5700 1.206 8900 1.039 12100 -938 15300 .868 a6oo 1.567 5800 1. 199 9000 1.036 12200 -936 ,15400 .866 2700 1-547 5900 1. 192 9100 1.032 12300 -933 15500 .864 2800 1.528 6000 1. 186 9200 1.028 12400 -93' 15600 .862 2900 1.511 6100 1. 179 9300 1.025 12500 .928 15700 .860 ' 3000 1.494 6200 I-I73 9400 1. 021 12600 .926 15800 .859 ' 3100 1-477 6300 1. 167 9500 1. 017 12700 •923 15900 -857 3200 1.462 6400 1. 160 9600 I.OI3 12800 ' .921 16000 •855 232 RESISTANCE OF SHIPS. 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