UC-NRLF Irlllllill C 2 fill 57fl A STUDY OF TI3E FUNDAMENTAL PRINCIPLES OF CURRENT METERS A study of the fundamental principles of current meters By Joseph Ridgeway Gunn, Jr. B.S. (University of Texas) 1921 THESIS Submitted in partial satisfaction of the requirements for the degree of MASTER OF SCIENCE in Mechanics in the GRADUATE DIVISION of the UNIVERSITY OF CALIFORNIA Approved Instructor in Charge Deposited in the University Library /t+ f^ /ff 12- Date Librarian PREFACE. The steady development of water resources for hydraulic power and irrigation purposes, and a consequently increasing de- mand for current measuring instruments more accurate than any which have heretofore been designed has suggested that an inves- tigation be made of the defects existing in the present meters. A study of the problem reveals an interesting fact, i. e., that all basic principles seem to have been given insufficient con- sideration by the designer. The effects of eddy currents ap- parently have never been considered; stream lines have always as- sumed parallel; the. requirements for the very low velocities en- countered in irrigation investigations have been ignored. Although the courts of this country recognize no stan- dard, they usually accept the gagings determined by a turbine type meter because of its popularity. The absence of literature on the subject of fundamentals themselves makes the acutenes? of the situation more easily realized. The results of extensive investigations of the principles, in so far as the writer has been able to determine, have never been recorded by American engineers. Schmidt, Sandstrom, Bateau and Spper have investigated many phases of the problem; but the Euro- pean requirements are essentially different from the American. 3ver, very little of these authors' works were available, but those few articles which could be obtained were invaluable to the writer. The preparation of this thesis is a step into a new field, and like all things new it will likely receive a great deal of criticism all of which the writer welcomes. Besides an investigation of some of the fundamentals of stream flow, a theory of resistance or interference as advanced by fir. E. J. Hoff has been set forth. Mr. Hoff has been connected with the Department of Agriculture, Bureau of Public Rod-s, Di- vision of Rural Engineering, Irrigation Investigations, for over 16 years during which time he has been intensely interested in the current meter problem and has contributed much to its advancement. An absence of comparative tests between the turbine type of meter and the propeller type of meter will be particularly con- spicuous in this paper. The writer first thought of making tests in the weir channel here in the hydraulic labratory of the Univer- sity of California, but after more thought on the subject the idea was abandoned because the results obtained in a single channel would be of little value 4 A series of tests of this kind, to be of any worth, would require a long period of time, and should be conducted under government supervision. Tests should be made in channel? of different cross sectional areas and proportions, with varying velocities and depths of water, and it would be highly desirable to determine a factor which would show a definite relation between the type and the factors listed. The huge volume of data thus ob- tained would then warrant a critical comparison of types. The writer is deeply indebted to the Berkeley office of the Department of Agriculture, and especially to Mr. Hoff, for its encouragement in the preparation of this paper. Thanks are also due the Spring Valley ~7ater Company, San Francisco, for the use of their Belmont rating station. J. R. G. Berkeley, California, April, 1922. CONTENTS. . . Page Preface iii I. INTRODUCTION. 1. Brief History of Current Meter Development.. 1 2. Principle of Meter Method of Measuring Stream Flow Is Wrong 3 3. Purpose of Thesis 4 4. Only Scientifically Designed Meter: Ott 4 5. More Figid Demands Placed Upon Meter When Used for Irrigation Work 6 II. FUNDAMENTAL PRINCIPLES OF CURRENT METERS. A. Fluid Motion and Obstacles. 6. Obstacles 6 7 . Eddy Effects 9 8. Example of Rod in Stream 13 B. Hoff Theory of Resistance. 9. Explanation 17 10. Vertical Shaft Meters 17 11. Explanation of Turbine Rotation 24 12. Other Characteristic Curves for Vertical Shaft Instruments 27 13 . Horizontal Shaft Instruments 31 14 . Summary 40 C. Supplementary Tests. 15. Purpose 42 16. Effects of '.Vires, Rods, Yokes and Cables on Current Meter Operation 42 17. Tests of Price Meters as Horizontal Shaft Meters 50 III. REQUIREMENTS FOR IDEAL CURRENT METER MEASUREMENTS. 16. Requirements 52 19. possibility of Realizing Ideal Requirements. 53 IV. THE VERTICAL SHAFT TURBINE 1ETER VS. THE HORIZONTAL SHAFT PROPELLER METER. 20. Advantages of the Vertical Shaft Turbine Type Meter 58 21. Disadvantages cf the Vertical Shaft Turbine Type Meter 59 22. Advantages of the Hoff Horizontal Shaft Propeller Type lleter 61 23. Equation for the Rotation at Various Values of the Angle $ 71 24. Disadvantages of the Hoff Horizontal Shaft Propeller Type Meter 82 V. CONCLUSIONS. 25. Conclusions Bibliography 85 Index 87 A STUDY OF THE FUNDAMJTrTU, PRINCIPLES OF CURRENT METERS I. INTRODUCTION. 1. Brief History of Current Meter Development . Instru- ments of various forms for the measurement of stream flow have been in general use for one hundred and fifty years or more. The first instruments employed were simple wooden or cork floats; later these were replaced by semi-floating rods of different types. These rods were of wood, cork, or metal tubing weighted at the bottom so as to float vertically. As the science of hydrology developed, and eco- nomic conditions demanded the conservation and industrial utiliza- tion of hydraulic power, more accurate instruments were in demand. One of the first meters of modern design was invented by Revy after considerable experience in gaging South American rivers. No thought "was given the fundamentals in the design of this instru- ment; since the propeller rotated in a moving stream, no farther study of the theory seemed necessary. This meter was equipped with a 6 in. propeller of the Griffith screw type as used on ships. The blades were mounted around a hollow metal boss, the idea being to minimize instrumental friction by reducing the relative weight of the rotating parts. All of the early meters, like the Revy meter, were of the horizontal shaft type, very bulky in design, and bur- ^In this paper, a distinction will be made between "turbine" and propeller". A turbine may be defined as a number of cups mounted concentrically upon a vertical shaft. A propeller may be defined as a number of blades mounted radially upon a horizontal shaft. (1) (1) Price Meter, Model 623, as cable meter. Meter equipped with Hoff improved 1:1 and 5:1 contact chamber, and turbine spindle. (2) Price Meter, Model 618, with eccentric base. (3) Price Meter, Model 618, with concentric base. Fig. 1. dened with a mass of non-revolving parts. These instruments, in spite of floating bosses and other similar devices, showed a very large friction factor. This general type of instrument in a very materially improved form is on the market today under the names of Haskell, and Ott. In 1885, W. G. Price, Assistant Engineer of the Corps of Engineers, United States Army, was engaged in measuring the flow of the Ohio River. At this particular time the river was at a very high stage, and quite muddy. Mr. Price experienced a great deal of difficulty with the horizontal shaft type of meter, and he designed the original Price meter which is nothing more nor less that a modi- fied type of anemometer. As far as it was possible to estimate, this new meter gave fair results. As can be seen from this very brief sketch of the history of current meters given above, the instruments have never been de- signed from a basis of fundamentals. The designer seems to have always been satisfied with the fact that his meter rotated in the stream, and has gone no further. 2. Meter Method of Measuring Stream Flow. In the first place, the meter method of measuring stream flow will never pro- duce accurate results. As will be shown later, if any body, no matter how saall, be immersed in water, it will set up eddy The use of this word is likely to confuse the reader. To avoid this, the writer will give his definition: An eddy current is a current which has suffered a perceptible change of direction. The term is usually applied to a current whose direction has been com- pletely reversed. Eddy forces refer to the new forces exerted by the eddy currents . < ' " <. , I O- . (. 7 .Pi- : . f drees and cross currents. This alteration of the original forces, and direction of stream filaments introduces an error from the very beginning. A perfect meter is then out of the question. 5. Purpose of Thesis. It is the purpose of this thesis, therefore, to point out certain fundamental principles of current meter design, and to show that a careful study of the existing evils, and a combination of the lesser evils, modified as far as possible, will result in a meter whose observations will show greater precision. 4. Only Scientifically Designed Meter: Ott. The only scientifically designed meter with which the writer is familiar is the Ott meter manufactured by A. Ott, Kempten, Bavaria. The blades of the propeller were designed after a mathematical development, and should theoretically offer no interference to parallel stream filaments. Under actual conditions, however, the instrument offers a small interference, though it is ce'rtainly a minimum. In low velocities, the Ott meter shows a somewhat high friction factor, which is not to be desired, but for the higher velocities, taking all factors into consideration, it is one of the best instruments to be had. The meter was designed upon the assumption that all stream filaments were parallel. If this were the case, this instrument to- gether with many other types of meter manufactured today would be very nearly perfect; but this is not the case, for, as will be shown later, parallel filaments exist only in theory. Since parallel currents do not exist, would it not be bet- ter in the 'design of current meters to choose between evils, and se- , . . U ^IJ , (1) Hoff Meter, Model 21, as rod meter. (2) Hoff Meter, Experimental Model, as cable meter. (3) Hoff Meter, Model 22, as rod meter. Fig. 2. lect the lesser ones? In other words, if a meter could be designed which would automatically utilize only those components of stream velocity perpendicular to the base plane of measurement, even though this instrument offered somewhat more interference to stream flow, would it not be the better plan to design the meter after the former principle? 5. More Rigid Demands Placed upon Meter when Used for Irrigation Work. As it becomes necessary to cultivate lands located in sections which receive only seasonal rainfall, irrigation is com- ing more into practice. This phase of hydraulic engineering has placed more exacting demands upon the current meter. In the first place, the low velocities usually encountered in irrigation canals- the average of which is below 2 ft. per second demand a meter which will accurately indicate these velocities. This obviously necessi- tates the reduction of the friction factor. In the second place, the very small cross section of the irrigation canal requires a meter whose propeller or turbine is of a smaller diameter than that of any meter being manufactured today. The Hoff meter, with which this paper will deal at considerable length, was designed primarily to meet the new and more rigid requirements, though the latest style meter, model 22, will function equally well in the higher velocities. II. FUNDAMENTAL PRINCIPLES OF CURRENT METERS. A. Fluid Motion and Obstacles. 6. Obstacles .--If any object is immersed in a moving stream of perfect fluid, which fluid may be defined as one without viscosity, each filament will flow around that object in perfectly ". . -- .89. t , smooth stream lines, and at every point will be tangent to the sur- face of the object. No fluid, however, is entirely devoid of vis- cosity, and the existence of this characteristic causes the flow to be sinuous or eddying. Thus, if some object be immersed in water, the stream lines will not be smooth. No matter how small or large the object, whether it be a pebble or a huge boulder, eddying effects and cross currents will be produced to a greater or lesser degree, and the resulting stream lines present a problem infinitely more difficult. IVith the exception of the simplest cases which Prof. Hele Shaw has worked out very successfully, the problem of stream line flow absolutely defies mathematical analysis. It is necessary, there- fore, to make all the experiments possible, and draw such conclusions as seem reasonable. Since any object, regardless of its size, will produce eddy currents and cross currents, it is obviously impossible for parallel filaments to exist under actual field conditions; yet it is under this assumption that practically all current meters have been de- signed. Aside from foreign obstacles themselves, the viscosity of the water serves to produce non-parallel filaments. Velocity con- tours and vertical velocity curves both substantiate this statement. From these curves, it is seen that the velocity varies from a mini- mum at the bed to a maximum somewhere near the surface, the curve being theoretically a parabola. Contour curves show similar results: i. e., the velocity varies from a minimum at the sides and bottom of the channel to a maximum at a point near the center of the surface. Since the velocity of a stream varies throughout the cross section, the viscous dragging effect existing between filaments al- ev-n. , rfflXH W< u> ters the direction of the filanents in a very uncertain way. Or, to put it in the words of another, "resistance is not caused by the liquid rubbing against their beds: it is almost entirely caused by the liquid changing shape, and so forming 'eddies 1 . 11 ' 7. Eddy Effects. --Consider the rectangular plate, AB, im- mersed in a moving stream as shown in Fig. 3 (a). The water adheres to the surface of the plate, and consequently this film is not in motion. The stream filament which just misses the edge of the plate would be expected to follow the course AC 1 on account of its inertia, "its proximity to the body and the reduction in motion caused by the retardation of the layers of the fluid in its immediate neighborhood, however, cause the particles to tend to move along a path of smaller 2 radius of curvature and the stream line bends inward along AC." ' In the same way, each neighboring filament is affected. Consequently, behind the plate two whirls or eddies will be observed, one at each edge. These eddies are produced periodically, and finally at some distance behind the plate they merge with the other filaments. Often times another type of eddy is formed as shown in Fig. 3 (b). This is different from the first only in the time at which the whirls appear, the whirls at one edge seeming to lag be- hind those at the other edge. Many experiments have been conducted for the purpose of ascertaining just when to expect the second type of eddy, but no definite conclusions have as yet been drawn. How- ever, it seems reasonable to assume that the latter type would "Motion of Liquids", by R. de Villamil. 2 "Aeronautics in Theory and Experiment", by Cowley and Levy. ' . eecf oVBri a. only be produced when some non-symmetrical disturbance had entered. It is not possible to determine the magnitude of these eddy forces, nor can it be said in just what manner they resolve themselves. However, their effect is felt in the force required to hold the plate in the stream, and their importance cannot be over- estimated, because, as will be shown later, the rotation of the Price meter turbine depends largely upon their existence. Upon approaching the plate, the stream filaments are slowed down; but in passing around the edges and moving into the whirls, they are again accelerated. n . . . the total energy per unit volume at a given point in a stream filament is composed of potential energy, or pressure, and kinetic energy, which is measured by the velocity: and the sum of these two is a constant at all points of the stream filament. Expressed generally, per unit volume of the stream filament, p p + gmv a constant potential energy kinetic energy = constant where p = pressure, v * velocity, and m = the density of the liquid; p and this p + igmv must retain the same value at all points of the same stream line provided, of course, that it always lies in the same horizontal plane. . . . Potential energy may be converted into kinetic energy, or vice versa, but the sum of the two must re- main constant." 1 Consequently, as the velocity of these filaments 1 "Motion of Liquids", by R. de Villamil. 11 increases, the kinetic energy increases, while the potential energy decreases by an equal amount, because the sum of the kinetic and potential energy must at all times be a constant. In other words, the whirls or vortices are characterized by a pressure which is lower than that of the neighboring fluid, but the eddies thus formed exert a very appreciable "reactive" force, or force oppo- site in direction to that of the original velocity force. The pressure on the plate is made up of two components: one is the "velocity pressure", and the other is the "static pres- sure". If the liquid were stationary, the static pressures on each side of the plate would balance themselves, and in the ab- sence of any velocity pressures, no force would be required to hold the plate up; but as soon as the stream begins to move, an entirely different condition has to be contended with. The static pressure on the anterior face is increased very slightly due to a piling up of the water. The static pressure on the posterior face is consider- ably reduced, as explained above, and partly on account of this re- duction, eddy currents are formed which exert a velocity pressure on the posterior face opposite in direction to the velocity pres- sure exerted on the anterior face. Thus, it has been shown again that due to eddy currents set up in the rear of the plate the force required to hold the plate in position against the velocity pressure of the oncoming stream is somewhat smaller than the force of the velocity itself. 12 Fig. 4. 8. Example of Rod in Stream. If a rod is immersed in a moving stream, a side elevation of the above phenomena may be ob- served. Fig. 4 sho-ws a rod meter held in a stream, with the direc- tion of the stream filaments diagrammatically represented. At a distance _c in front of the rod, the stream filaments begin to slow down and pile up, and immediately in front of the rod the water is raised a distance a above the normal level of the stream. At this point in the hump, the filaments momentarily come to rest on account of a transposition of pressures: i. e., due to the presence of the rod as an obstacle, the velocity pressure or energy is all consumed in lifting the water to the height a. In other words, the velocity pressure is reduced to zero while the static pressure is increased to a maximum. Between the up-stream and down-stream sides of the rod there is an immediate and quite pronounced drop in the water level, as shown. Behind the rod there is a concave surface or hol- low into which the water rushes from its slightly higher elevation in front. The depth of the hollow reaches a maximum value indicated by b, and at a distance d behind the rod the filaments reach the nor- mal level of the stream where the disturbance is finally absorbed by the more smoothly moving filaments . The formation of whirls or vortices explains the presence of this hollow. In the same way that eddy currents were formed behind the plate, as described in Section 7, they are formed behind the rod in this case, and as part of the static pressure is changed into velocity pressure in the vicinity of the eddies, the pressure in the rear of the rod is re- duced, with the consequent formation of a slightly vacuous space or hollow. . vel wol "> *( aur I , 14 If the stream is moving slowly, the slowing down of the filaments can be detected at e. considerable distance in front of the rod, but the water is lifted to only a very slight elevation. In the same way, b_ is small, and d_ is large. Vice versa, if the stream is moving rapidly, _c and d_ are small, but ia and are proportionately larger. In other words, c = k X 1 V a = k 2 X V b = k_ X V o c = k X 1 4 V where, Ic , k , k_ and 1^ are constants, and V is the velocity of the stream. '.Then an obstacle is placed in a moving stream, the filaments split in front of the obstacle on what might be termed a "liquid prow", and proceed around the edge as shown in Fig. 3. This same phenomena is present in the case of the rod in the stream. The edfe of the liquid prow is at a distance c 1 (not shown) c in front of the rod. The stream divides at this point, flows around the sides of the rod, and finally merges with the smoothly flowing filaments at a distance d' (not shown) = d behind the rod. If the stream is moving slowly, c 1 and d 1 , and the distance out from the sides of the rod to which the disturbance can be felt are relatively large. Vice versa, if the stream is moving rapidly, these distances are much shorter. The stream filaments directly in front of the rod, a, b, c and d are not drawn to scale. 15 after having suffered a change in internal energy- -having gained an amount of potential energy at the expense of the kinetic energy- proceed around the front of the rod at a lower velocity than that of the unaltered filaments. As soon as they reach the edge of the rod, their direction is changed, and in moving into the vacuous space behind the rod they give up part of their potential energy which is changed immediately into kinetic energy, and due to the viscosity this increased velocity forms very rapidly moving whirls or eddies. However, the point to be emphasized is that the stream filaments directly in front of the rod turn from their original course with about the same amount of velocity energy in all cases, independent of the original velocity; but the greater the original velocity, the higher the distance a to which the water is piled. In other words, all "surplus" velocity energy is transformed into potential energy in the piling up process, and the filaments depart from their original course with a constant velocity energy. The greater the velocity the more difficult it is for the filaments whose pressures have been rearranged to push aside the undisturbed filaments, and, in general, it may be stated that the area of the surface of the stream in which the disturbance is felt is inversely proportional to the velocity of the stream. It is this rod phenomena which explains the surface curve of a current meter. larfc SO.';' r T 33 Q) t> 1- U 3 3 o f -i . A * V f-H n I. D o 17 B. Hoff Theory of Rosistance . 9. Explanation. The Hoff theory of resistance, or theory of interference as it might well be called, deals with rating curves for the various types of current meters from a standpoint of the in- terference offered oncoming stream filaments by the rotating member itself. It explains the irregularities ordinarily found in rating curves, and establishes a definite relation or connecting link between the test curve and the theoretical or ideal curve. Incidentally the theory offers an explanation for the rotation of a turbine; and it might be said here that in none of the scant amount of literature on the subject of current meters which the writer has been able to obtain has he seen any explanation of this action put forth. 10. Vertica^l Shaft Meters. Fig. 5 shows the normal characteristic rating curve of a vertical shaft meter, such as the Gurley (or Price), Lietz, and Lallie. The "starting velocity curve" 2 begins at 0.05 ft. per second zero revolutions per second and Figs. 5, 9, 10, 11, 12 and 15 are general curves, not applicable to any one particular meter, but simply cEar&cteristic of the type as a whole. They have been reproduced from L'r. Hoff's "Cur- rent Keter Studies". Certain sharp breaks will be observed in these curves. These are exaggerated in order that the irregularity at the particular point will be plainly evident. In all of the rating curves included in this paper, the heavy line indicates the test curve, while the light line indicates the theoretical curve which passes through the origin. Although the starting velocity curve begins at 0.05 ft. per second, the meter will not rotate until about 0.16 ft. per second. Veloci- ties above 0.5 ft. per second are considered reliable by Messrs. TIT. & L. E. Gurley in their "Manual of Gurley Hydraulic Engineering In- struments". Velocities as low as 0.2 ft. per second are not un- common in the rice fields. , 8r - ' ' ' en - e>i oJ ids t OK j I Mil on ri^ Bwori i -- ariJ - rO" e' i, J. r^" . -nl gniiaf nje > 19 Table 1. Rating of Price Meter, Model 142. June 21, 191S. Time in Revo- Revolutions Ft. per seconds, lutions. per second, second. Course: 120 ft. 16.5 53 3.21 7.28 18 52 2.89 6.667 20.75 52.5 2.534 5.783 21 52 2.476 5.714 26 52 2 4.615 26.5 52 1.962 4.528 33.61 52 1.562 3.603 36.5 52 1.425 3.288 38.5 52 1.35 3.117 48 52 1.083 2.5 62 51+ 2.25 1.935 75 51+ .69 1.6 Course: 60 ft. 45 25+ .55 1.333 50.5 25+ .5 1.188 91 23.75 .261 .659 76 24 .317 .789 115 23 .2 .522 138 22.5 .164 .435 151.5 22 .145 .396 188.5 21 .111 .316 ends at 2.2 ft. per second 1 revolution per second. Above the veloci- ty of 2.2 ft. per second lies the "higher velocity curve" which coin- cides with the ideal curve through the origin. An explanation of these new terms would not be out of place at this time. The ideal curve for a vertical shaft instrument is a straight line drawn through the origin and as many observed points as possible. The term "starting velocity curvo" is used to designate the lower section of the rating curve, whic h is a curve, but which, for practical purposes, is usually approximated with a straight line. The upper section of the curve which is a straight line will be known as the "higher velocity curve". Field observations are only reliable when they are above the starting velocity. What causes the break in this rating curve? Heretofore, it has been generally accepted that instrumental friction was en- tirely responsible for this irregularity, but this is not the case. Although instrumental friction does exist, it is almost negligible. The important factor, however, is the interference of the turbine with the oncoming stream filaments. Fig. 6 shows an actual rating curve of a Price meter. However, the curve is not different from the characteristic curve just described, so no discussion is necessary. Figs. 7 and 8 represent stream filaments flowing past a Price meter turbine. Fig. 7 is a characteristic picture of the filaments when their velocity is below 2.2 ft. per second. A close observation of the stream lines and their relation to the Fig. 7. turbine is invitad. Consider the lines approaching the tur- bine as being parallel. Upon meeting the turbine their course is altered to accommodate the rotating obstacle. From zero to 1 revolution per second, or from zero to 6 cups per second, the water has time to enter the space within the cups, and it is this interference of the filaments with the rotation, plus a small friction factor, that causes the starting veloci- 22 Fig. 8. ty section of the rating curve to lie below the ideal curve. 10 of the filaments flow around the outside of the turbine while others flow in between the cups producing certain eddy currents and whirls. Now observe the filaments as pictures in Fig. 8. Here, their velocity is 2.2 ft. per second, or more. What is the difference between the filaments in thi s picture and Ihose in Fig. 7? First, none of the filaments are able to 23 flow in between the cups, because at this velocity the turbine is moving with a velocity of 1 revolution per second, which is equivalent to 6 cups per second, and this is sufficient to shut out the oncoming filaments. Second, an eccentric, solid, whirling mass is observed within the central space. Third, the lines approaching and departing from the tur- bine appear to be more radically disturbed at this higher rotation. It is this factor which accounts for the break in the rating curve. From this point up, the meter is rotating so rapidly that the filaments cannot enter the region between the cups. In fact the obstacle presented to the moving fila- ments is now entirely different from that for the lower ve- locities. It is a solid mass, the center being filled with water which rotates at the speed of the turbine. A side elevation of the turbine at this time would be a mass with parallel top and bottom and rounded ends. So, the rota- tion above 2.2 ft. per second may be considered as normal, because at this point the interference factor is a minimum, and as the shape of the obstacle remains unchanged at the higher velocities, it is constant; and with a constant and almost negligible friction factor the higher velocity curve coincides with the ideal curve. 24 11. Explanation of Turbine Rotation. At first glance one would think that a turbine such as the one shown in Figs. 7 and 8 would rotate in a clockwise direction, re- ceiving the torque components on the slanting sides of the cup like a propeller or windmill. Upon placing the turbine in a moving stream, however, it will be found that it ro- tates in the opposite direction. The explanation of this is comparatively simple after having previously considered the effects of eddy currents. On the right, or positive half of the turbine, two torque components exist. First, a veloci- ty force is exerted against the open face of the cup which may be imagined as being filled with water and thus presen- ting a flat surface to the approaching filaments. Second, the viscous dragging effect existing between the moving filaments and the film of water with which the cups are coated exerts a pull. On the left, or negative half of the turbine, three torque components exist, one of which is a counter- clockwise or positive component. Here, a velocity force is exerted against the slanting sides of the cups. Theoretical- ly this resulting torque component should be somewhat smaller than the component produced from the same kind of force on the right. The pull due to the viscosity or adhesion exists here also. Then, from what source does the turbine receive its ac- tuating torque? The real force which turns the turbine is produced by the eddies which are formed behind the cups on the left side due to a relative motion, or difference in motion between these cups and the filaments, and it is important to note that this force is a positive one. Neither is this component of small consequence, be- cause it requires an appreciable force to rotate a turbine at the speed at which a flowing stream does, and since the direct veloci- ty forces and the viscous dragging forces practically, or very nearly balance each other on the two sides, it remains for the eddy forces to do the turning. Above 2.2 ft. per second, a slightly different condition exists. The same forces act as before, but less cup surface or area is exposed to the moving filaments since they cannot enter the inner space, which means that the number of active filaments is reduced in proportion. However, the resulting effect of these fewer filaments is not materially reduced, because the cups are not required in this instance to buck the filaments which pre- viously worked their way in among them (the cups). On the left side of the turbine, there exists a relative motion between the cups and the moving filaments with the resultant production of eddy cur- rents and vacuous spaces in the rsar of the cups, just as before. Since the turbine is moving too rapidly for the filaments to en- ter, that dead mass of water already within the cups tends to move itself over to the left and rush into the vacuous spaces, thus exerting a counter-clockwise force. This movement to the left sets the mass eccentrically, as shown. The more radical disturbance of the filaments approaching and departing from the turbine is simply due to the fact that thuy are sucked in on the approach and hurled off at the departure on ac- count of this higher velocity. Thus, it has been shown that the existence of two separate and distinct rating curves one, the starting velocity curve, and the other, the higher velocity curve is not entirely due to instrumen- tal friction, but is largely accounted for by the interference of the turbine itself. 12. Other Characteristic Curves for Vertical Shaft In- struments. Fig. 9 shows the surface curve for a Price meter: i. e., a rating whose data was observed with the top edge of the turbine only 0.25 in. below the surface of the water. Up to a velocity of 1.2 ft. per second, the curve is not unlike the ordinary starting velocity curve for this type of meter. However, at this point, or at this speed, a piling up effect begins to appear at the surface above the turbine just as in the case of the plate and the rod. Fron this point to 2.2 ft. per second, the slope of the curve is less than that of the ideal curve. This is explained by the fact that the higher the velocity, the greater the height to which the water is piled, and this piling up effect absorbs part of the kinetic en- ergy which is taken from the active filaments. Consequently, this means a proportionately slower rotation with the resultant lowering of the curve. At 2.2 ft. per second, however, the meter first ap- pears partly freed from the water, and from this point on the water is lifted from above the cups at an increasing rate until at the 28 p O M n o - 00 t-> o 6^ P O "-" 33 M O f-i > M O t- V. 3 .H ^ 3 o to 3 o C c -P l .P >> 03 4) K -P +3 i-l O - O > (D rH ID O C p C C >r > as Hi - 1 1 C] <0 H r-l 5 s K ^ ^ ^* w s s 13 1 1 &- c* H H< Si rH ^ d o t tri ^ 01 m *^ *"^ r 31 X-axis indicates a negative friction factor which is, of course, im- possible . 15. Horizontal Shaft Instruments. In Fig. 11 there is shown a characteristic rating curve for a horizontal shaft instru- ment, such as the Haskell, and Ott. This curve is very much the same as a normal curve for a vertical shaft instrument in that it is made up of two sections, i. e., the starting velocity curve and the high- er velocity curve; but unlike the previous type curve, this one does not reach the ideal . One advantage of this type meter over the turbine type, which is immediately noticed, is a shortening of the starting velocity curve an advantage, because as long as the veloci- ties are within the zone of the starting velocity, the observations are not reliable on account of the varying friction and interfer- ence factors. But why does the higher velocity curve not coincide with the ideal curve as it does in the case of the turbine type? It will be remembered that the only reason why the turbine meter curve coincided with the ideal curve was because the higher veloci- ties shut out the filaments from within the turbine itself, and thus reduced the interference factor to a constant minimum. In the case of the propeller, or horizontal shaft type meter, however, this shutting out effect can obviously not be accomplished. No matter how fast the propeller rotated, these filaments would always find their way through between the blades. Hence, the vertical dis- tance between the starting velocity curve and an extension of the The ideal curve for a horizontal shaft type of meter is drawn through the origin and parallel to the higher velocity curve. lev +em nl - eri.l J-e - blx/ow eJrtsn 32 C E-* C I t? 0) s- curv o n urve -', C o c o *+-! ^ id r-, c c 4-> a j>^ EC K tc o "x 4' |~1 j^ C d H * . I O -^J .r" 1 *~3 a) . o C' 1-1 cc P l a a .^ H > i r~* H r- 1 r-t ^ 1-1 r-i T-i Ul 1 s q-i tj ->' S* o hi 'o !_' c/- trt CM "g Q V O 33 higher velocity curve so as to cross the X-axis is indicative of instrumental friction, while the vertical distance between the high- er velocity curve and the ideal curve is indicative of interference which the propeller offers to the moving stream filaments. Fig. 12 shows a characteristic rating curve for the airme- ter type of current meter, such as the Fteley-Stearn, and Eckman. The curve really shows no signs of a single redeeming feature for this type. It is simply another case where the designer has com- pletely ignored all fundamental principles of stream flow, interfer- ence, etc. The reader should first be reminded of some of the physi- cal features of these meters. Both instruments have a multi-bladed propeller mounted on a horizontal shaft, the circumference of the propeller being bounded by a metal band. The shaft is pivoted in a yoke whose up-stream edge is sharpened in stream line fashion. Bearing these facts in mind, an understanding of this type curve will be more easily arrived at. The starting velocity curve extends to 2.5 ft. per second. It will be noticed that this curve crosses the ideal curve at a veloci- ty of 1.125 ft. per second. This crossing is due to the effect of the proximity of the yoke to the front of the propeller. The stream filaments splitting on this yoke are greatly accelerated as they move around the sides, just as in the case of the flat plate and the rod, and they strike the blades with this increased velocity thus causing a higher propeller rotation than they would if the yoke were not present. At 2.5 ft. per second, the propeller is swallowing a max- imum amount of water: the rim around the blades prevents any more ausei - -if ataiiw eejBO v cmei d erf.} "io eone-: tarle 3*1 eoJI maei, ' . 3V J J-JB e/VIIJO l Jo ell e eri m&slje erfT t : as be,' bru IAI B al)*Id er a 5 **** ZlSSttl * a> !' * : i < -to '& K OH Table 2. Table 5. Rating of Hoff Meter, Model 21, Ko. 47. Jan. 5, 1922. Course: 200 ft. Rating of Hoff Meter, Model 21, No. 48. Jan. 23, 1922. Course: 200 ft. Time in Revo- Revolutions Ft. per Time in Revo- Revolutions Ft. per seconds lutions . per second. second. second. lutions . per second. second. 124 150 1.21 1.61 73.2 159 2.732 2.17 83.5 153 1.83 2.39 64.5 159 3.1 2.465 75.2 154 2.04 2.66 51.5 159 3.88 3.088 84 154 1.83 2.38 38 159 5.26 4.18 68.8 154 2.24 2.91 74.2 159 2.691 2.145 62.5 154 2.46 3.19 75 159 2.665 2.12 55 154 2.8 3.64 85 159 2.35 1.87 44.8 155 3.46 4.46 114 158 1.76 1.39 38.5 155 4.02 5.19 111.2 159 1.8 1.43 38 155 4.07 5.26 151.5 156 1.32 1.03 95 153 1.61 2.11 154 156 1.3 1.01 97.5 153 1.57 2.05 120 157 1.67 1.31 111 152 1.37 1.8 131.2 158 1.52 1.2 122.8 150 1.22 1.63 142.3 158 1.41 1.11 122.6 150 1.22 1.63 148.4 157 1.35 1.06' 110 152 1.38 1.82 153.1 157 1.31 1.02 111 152 1.37 1.8 160.5 156 1.25 .972 106 152 1.43 1.89 170. 7 156 1.17 .915 111 152 1.37 1.8 177.3 157 1.13 .885 148 150 1.01 1.35 186.3 157 1.07 .843 104 152 1.46 1.92 225.7 154 .887 .685 161 150 .93 1.24 197.8 155 1.01 .783 174.5 150 .86 1.15 188.7 156 1.06 .827 198 148.5 .75 1.01 251.5 154 .796 .613 200 148.5 .74 1. 258.1 153 .775 .592 120 153 1.27 1.67 369.7 137 .541 .371 119 152 1.28 1.68 369.9 141 .54 .381 116 154 1.33 1.73 346.7 142 .577 .41 111 151 1.36 1.8 524.4 126 .382 .24 111 153 1.38 1.8 525 115 .381 .219 435 134 .459 .308 326.5 139 .613 .426 260.3 146 .769 .561 161.2 154 1.241 .955 260.2 146 .769 .561 372 137 .537 .368 384.6 137 .52 .356 439.6 135 .455 .307 353.8 139 .565 .393 380.2 138 .526 .363 296.8 140 .674 .471 472.3 129 .423 .273 37 38 water from passing through. So, the higher velocity curve drops, and crosses the ideal curve at a velocity of 3.333 ft. per second. Fig. 13 shows the rating curve for a Hoff meter, model 21, No. 47. This curve is not unlike the characteristic curve for the horizontal shaft type of meters shown in Fig. 11; consequently, no explanation need be made. The velocity of the car cculd not be carried below 1 ft. per second at the time the rating was made, so the curve does not show what takes place at the lower velocities. After making a slight change in the spindle (and changing the num- ber of the meter to No. 48), another rating was made. The re- sultant curve is shown in Fig. 14. One notable advantage over the Ott and Haskell meters which this curve indicates is a marked de- crease in the starting velocity. It is this improvement which makes the Hoff instrument particularly well adapted to the low velocities encountered in irrigation investigations. Fig. 15 shows the surface curve for one of the Hoff trial meters . The data for this curve, like that for the curve in Fig. 9, was observed with the top edge of the meter only 0.25 in. below the surface of the water. At the point where the starting veloci- ty curve ends, a piling up of the v,-ater at the surface over the me- ter is apparent. As before, a certain amount of the velocity ener- gy is consumed in lifting the water; and as a result, less velocity energy is imparted to the propeller blades, and the curve begins to drop down from the ideal curve. In this case, the meter is at no time entirely free from the water, and consequently the curve does This meter was one of Mr. Hoff's experimental meters of the hori- zontal shaft type which he used in the development of his present meter. . . iVb* ~ 1 . Sft ese a 39 40 not rise again as it does in the case of the Price meter at the sur- face. 14, Summary. Before leaving this subject, the outstanding features should again be pointed out. (1) Every normal rating curve is made up of two separate and distinct curves the starting velocity curve, and the higher velocity curve each of which has its own independent equation. (2) TWO factors affect the characteristics of a rating curve: the instrumental friction factor, and the interference factor. The interference factor causes the test curve to lie below the ideal curve; and the instrumental friction factor causes the starting velocity curve to break from the higher velocity curve at the velocity at which that factor ceases to be a constant. To be more specific: in a vertical shaft turbine meter, the starting velocity curve rises from a point at the right of the origin on the X-axis to the ideal curve. When the turbine just be- gins to rotate, the almost negligible friction factor is a maximum. At 2.2 ft. per second, it is a minimum and a constant. Also, at the start, the very appreciable interference factor is a maximum; and at 2.2 ft. per second, it has been reduced to an almost negligible minimum which is a constant. The sum of the instru- mental friction and interference factors makes the starting velocity curve lie below the ideal at the start. At 2.2 ft. per second, both factors are practically negligible, and are constant; hence } above this velocity the higher velocity curve and the ideal curve are one and the same thing. 1 a,; 41 In the case of the horizontal shaft propeller meter, the starting velocity curve rises from a point at the right of the ori- gin on the X-axis to a point where it merges with the higher veloci- ty curve, this point being below the ideal curve. '(Then the propel- ler first begins to rotate, the almost negligible friction factor is a maximum. At 1 ft. per second, it is a minimum and a constant. It is this friction factor which is responsible for the two separate curves. The interference factor, in this case, is the factor which holds the test curve below the ideal curve, because at no time is it negligible, but, above 1 ft. per second, it is always a small con- stant factor. (3) The reliability of observations made in the zone of the starting velocity is to be questioned, because of the varying friction and interference factors. (4) It is important that the meter be immersed to a depth at which the static pressure on the filaments approaching the turbine or propeller is sufficient to present the piling up of the water above. In general, the depth of submergence is proportional to the volume of the meter. Mr. E. C. Murphy states that 0.5 ft. 2 is sufficient for the small Price meter] Mr. F. C. Scobey states that 0.3 ft. is sufficient. (5) If a rating curve at any point crosses the ideal curve, it is an indication of either a faulty rating, or a poorly designed instrument. Engineer, U. S. Geological Survey. Irrigation Engineer, Office of Experiment Stations. C. Supplementary Tests^ 15. Purpose. The results of four sets of tests will be set forth in this section. The results of the first three sets are included so as to show that what has been said about the for- mation of eddies about obstacles and their effects is directly applicable to current meters. Owing to the general nature of the tests, and to the uncertainty of the behavior of eddy currents, no rigid conclusions can be drawn; but general statements may be made, which the writer trusts will be of some value. The fourth test was designed to prove that the Hoff theory of interference is correct. 16. Effects of Wires, Rods, Yokes and Cables on Current Meter Operation. As shown in Sections 7 and 8, the velocity of the water flowing around any obstacle is at some points increased, while at others it is decreased. A corresponding transposition of the forms of internal energy was also noted. These statements are substantiated by some of the following tests. If a wire C.06 in. in diameter is held 6 in. in front of a Price meter, the rotation is reduced 3 per cent. A rod 0.5 in. in diameter suspended 9 in. in front of a turbine meter in water moving at a velocity of 4 ft. per second will reduce the rotation about 16 per cent; at 2.5 ft. per second, about 13 per cent. The same rod 9 in. in rear of a turbine at 4 ft. per second will cause a decrease in rotation of about 3 per cent; at 2.333 ft. per second, about 9 per cent. Reversing the Price meter so that the yoke points up- stream will cause an increase of about 10 per cent in the turbine rotation. Table 4. Table of current meter ratings showing the relation between cable and rod suspension. Ratings made at the Hydraulic Labratory, Colorado Experiment Station, Ft. Collins, Colorado. Rating No. Date ' Suspension Weight Equation Mean Equation. Gurley Meter No. 1728 471 6-21-17 Rod V 2 .191R + .030 518 8-16-17 it V = 2 .188R + .023 V = 2.19R + 0.025 502 8-14-17 4ft. cable Single V = 2 .249R + c .016 503 n 5ft. c&ble " V = 2 .247R 4- .035 504 * 6ft. cable V = 2 .277R 4- .022 505 7ft. cable n V = 2 .256R + .014 506 n 8ft. cable it V = 2 .274R 4- .004 507 n 9ft. cable n V = 2 .245R + .030 508 8-15-17 10ft. cable n V 2 .261R 4- c .012 509 n lift, cable it V = 2 .258R 4- .013 510 n lift, cable Double y 3S 2 .294R 4- .038 511 it 8ft. cable n V = 2 .268R 4- .042 512 n 5ft. cable it V = 2 .286R 4- .012 V = 2.26R -t- 0.025 Gurley Meter No. 51 347 3-31-16 Rod V _ 2 .29 R + c .04 453 3-30-17 n V = 2 .295R + .055 567 10-8-17 it V = 2 .299R 4- .035 V - 2.30R +0.04 643 10-29-18 Cable Double V = 2 .401R 4- c .043 V = 2.40R +0.04 Gurley Meter No. 1314 109 9-21-15 Rod V = 2 .193R + .040 112 9-24-15 n V = 2 .188R 4- .015 261 11-8-15 it V = 2 .196R 4- .033 468 6-21-17 it V 2 .184R 4- c .023 V = 2.19R + 0.03 113 9-24-15 Cable Single V = 2 .262R 4- c .015 V = 2.26R + 0.02 Lietz Meter No. 8374 107 9-20-15 Rod V _ 2 .362R 4- .025 264 11-9-15 n V = 2 .346R 4- .045 266 " it V = 2 .352R -"- .060 V = 2.35R + 0.05 108 9-20-15 Cable Single V = 2 .356R 4- .049 V = 2.36R + 0.05 Lai lie Meter No. 300 296 11-15-15 Rod V = 2 .241R 4- .041 V = 2.24R + 0.04 306 11-16-15 Cable Single V = 2 .29 R 4- .06 V = 2.29R + 0.06 Gurley Meter No. 1640 358 5-8-16 Rod V - 2 .20 R 4- .02 V = 2.20R + 0.02 359 it Cable Single V = 2 .22 R 4- .02 V = 2.22R + 0.02 44 Table 5. Zahlenta.fel 8. Xo. Means of ! , Meter v o. 289/11, Ott catalog VII, propeller b: heavy, cylindrical contour. 1. Flat iron bar, 7 x 50 . . Cable, 9 mm. di^r.eter. (;' Top of railroad rail, m. loi crted by two cables. (! 4. Ire-. (242 - . n r * C.92P 0.934 O.S44 ' n VVVifcm ntaf-1 9. Meter Xo. r <: , -'Inr of conicnl form, 22 cm. diameter, 1. ir of ova] x 5.4 cm. 0.679 2. Cable with b 0.685 3. Iron pipe, 4.5 cm. diameter. 7 00 , 7 . 5 c r . . 706 Zahentafel 10. Large I r ice meter, I'D. 136, cup .turbir.* . 1. Free suspension by cable. 0. 2. Iron rod, 2 cm. diameter. 0.95 3. Iron rod, 4 cm. diameter. 0.990 45 Table 4 is a summary of a series of tests made at the Hydraulic Labratory, Colorado Experimental Station, Ft. Collins, Colorado. These tests were carried out in an attempt to find some definite relation between the rating of a cable meter and the rating of the same meter suspended by a rod} however, they will serve another purpose quite as well. These ratings were made before the advent of the Hoff theory, and consequently disregard the fact that a rating curve is divided up into two separate sections; however, this erroneous idea will not materially affect the conclusions to be drawn in this in- stance, since all the curves were plotted in the same manner, i. e., a straight line through as many points as possible. With the exceptions of the test on the Gurley meter No. 1314 and the test on the Lallie meter No. 300, the only deviation between the mean rod and cable equations appears in the slope fac- tor. In the two exceptions noted, there is a difference of 0.01 and 0.02 in the friction factor, respectively; but since these differences are in the opposite directions, it would be quite pos- sible to attribute them to personal errors. Table 5 is a reproduction of Tables 8, 9 and 10 from Dr. Schmidt's article . These tables were compiled from the ratings of current meters supported by various means, some by cable, and others by rods of different cross sections. With the same meter, a wide variation in the coefficient k, the slope factor, is noted. The coefficients should not differ, since the same meter is being used, but it is the change in the interference factor the change in the shape of the non-rotating parts due to the change in the See bibliography. 46 means of support that causes the resultant variation. What general conclusions can be made now after having studiid these tables carefully? In the first place, it has been shown that by the introduction of obstacles, the resulting distur- bance of the filaments and the alteration of their forces material- ly affects the rotation of the meter. In other words, any obstacle in a stream alters the internal energy of the flowing filaments, just as in the case of the flat plate and the rod, and if these affected filaments are the ones causing rotation, the standard rating equation is not applicable, since it was determined after a rating in still water where apparently undisturbed parallel . filaments exist. Second, these results, though rather loosely related to ore another, are responsible for this statement: The slope of a rating curve depends not only upon the pitch of the propeller or turbine, if such a term may be applied to a turbine, but also upon the inter- ference which the complete meter offers the oncoming filaments including all protecting devices such as wires, rods, etc., yokes, tails, weights, suspension rods and cables, spindle housings, sup- ports, binding posts, contact wires, etc., etc. The importance of this statement cannot be overestimated. In reply to this objection, one will say that these factors are all covered in the rating. True enough, but field conditions are so radically different from the ideal rating conditions that an operator never knows how close his results are going to be, because in the former instance parallel stream filaments never exist, while in the latter, it is not un- reasonable to assume that they exist at all times, and a change in the direction of a filament flowing around an obstacle alters the s p 1 t f * EC "3 10 3 ^* r-H 0) U y rH t-. O' pJ tf'-P I i-sirs -3^ I a* u i CO , P *5 rH C 1 03 rj *-, N fe ^ Tf t) 00 \1 'H " i. -0 *> ^ PC o 3 * " -^5 as ID .p H .H H \ >kj o ^ g m - - . ^: C C 3 M .H o ^ * ** S X CO Xv ; ) ^ r-l r-H 3 '" u ^ g v> o \- K m O, ^-> 03 fe K (X J - > - O i-l ^ ( * A. O> CC ^i. *^ Njv + t 4 4 * 4< X tffc V & > X p. I \ x \ to CO X co t^- to co X CO O r-l Ol t-1 tf CO O CO UJ >a w : o m o cj cv f> > fl> H * Sj^. ?i r O ^v *^ 03 " r= d P 5 C7* J- \V S O M r-l K CT JO W. 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Tl -P O 3 JT * O ,*d * \ o T? 4) ~4 .-) .-I r 1 ^ 1^ O a u rvj pu ** & 00 ** jc r S L-- , ^i -J ^J P M * * 3 ^ W rH i > ^-1 V. r-l r3 pj -j o ^^* O f4 C5 ^^ I -;$ N a < w \ : i-l S3 Mir; H t % I fci .1. -H +5 \ >~ -a jn u o JS CC O C C 1 JC "M " * \ ' F ? 1 ;.; x: js H -i 1 * & ; 5 \ C E c -i -:--^ -JJ}- Jj V (jjCJ tf'lJJ^^-'Vf ._ O t) "T^ ^ J "* ? T3 J- ^ V CJ i- C SM P K tl K t> g N. <: > c^ o \ u: t-" ^ * fr. r. PM r,' (L \ \ i r> i to v PM t< \ to in 4 a> t- oi o C" C^- ' CM ' tC -S 1 CO CJ ca r-l O O I c w -< ^ 4, H H ''j. ?- (; P O tr r-. cr 42 ^ o Cii U- 4^ {* C 1< o ro 10 >u r-l X -> 4> i ?: U ^v 3 4> 1C : r-l V> NX 62 BJ ul f: ' fc. ^vN r^ H I- O 63 shown in Fig. 16. There is nothing to get out of repair. If the contacts neod an adjustment, the points are tightened or loosened, as the case may demand; if the plate is removed beforehand, this adjustment can be mado more quickly and accurately, since the opera- tor can see just what he is doing. (2) This meter is characterized by a very small friction factor, as can be seen from the rating curves in Figs. 13, 14 and 19. This advantage has been gained by the employment of an ingenious set of bearings. The end thrust is absorbed by a jewel bearing in the rear socket. The shaft bearing that bearing which carries the weight of the spindle and propeller is simply a guide ring into which the shaft fits rather loosely. By making this bearing loose- ly fitting, any additional looseness caused by wear is not likely to affect the rating. The propeller is very small, being only 4 in. in diameter. A drawing of the propeller to full scale is shown in Fig. 20. Its weight is materially reduced by making it of hard rubber the specific gravity of which is 1.2. Besides reducing the friction factor, a rubber propeller has the advantage of being able to withstand a great deal of abuse without any danger of changing its pitch. (3) Reducing the friction factor is naturally productive of a shortening in the starting velocity curve. The starting veloci- ty curve of all of the Hoff meters terminates at approximately 1 ft. per second. This is quite an advantage over the turbine type of meter, because with this type the starting velocity curve invariably extends to 2 or 2.25 ft. per second. Since the reliability of ob- servations made within the zone of the starting velocity is to be questioned, this meter is particularly well adapted for irri- 1J8VJ oee 64 gation investigations. (4) The Hoff meter is not burdened with any unnecessary non-rotating parts. It has no yoke; it has neither protecting wires nor rods for the propeller. The simplicity of design has eliminated all these dispensable parts which are usually found in the construc- tion of a current meter. This meter causes less disturbance in the stream flow, and less eddies are produced. Consequently more accu- rate results should be obtainable with its use. (5) Another point in favor of the meter is its small pro- peller, this meaning, of course, that the rotating part itself will cause less disturbance in the flowing filaments. The small propel- ler is especially advantageous in gaging -very small channels be- cause observations can be made nearer the side walls and bottom. (6) The Hoff meter rotates from 78 to 80 revolutions per 100 ft. With only a 1 to 1 counting device, it would be quite difficult to accurately record the higher velocities. This instru- ment, as described before, has two counting devices: the 5 to 1 ratio used for the higher velocities, and the 1 to 1 ratio used for the lower velocities. This makes the meter equally well adaptable to both high and low velocities. (7) Suppose a horizontal shaft meter is lowered into a stream so that its axis is parallel to the stream filaments, and the propeller rotates at the rate of 0.5 revolutions per second. Now turn the meter about the supporting rod as an axis so that the cen- ter line of the shaft makes an angle tf of, say, 60 with the stream filaments. The meter should now rotate at 0.5 X Cos 60 or 0.25 revolutions per second. In other words, the discharge of a stream, 65 Q - av where a is the cross sectional area of the stream, end v is the veloci- ty of the stream perpendicular to the plane of area ji. Thus, if the base plane in which the gaging is being done is not taken perpen- dicular to the stream (assuming for the moment that parallel fila- ments do exist), but makes an angle (90 - $} with the stream, the discharge, Q a'v Cos $ where a 1 is the new crosn sectional area. If a meter can be designed which, when held perpendicular to the above base plane, will indicate a velocity of TT Cos j6 in- stead of v, the designer will have made another long stride toward the ideal. A meter fulfilling this requirement is said to follow the cosine law. Obviously, it can only be a propeller type meter, the propaller having flat face*. A met. e v whi^h follows the cosine law will accurately gage any stream regardless of the direction of the filaments. It will indicate only that component of velocity which is perpendicular to the base plane, provided the instrument is rigidly supported, and the angle between the base plane and the propeller axis is 90 . When $ = 90 , the rotation should be zero. Consequently, a meter of this type would be particularly well adapted to the integration method of gaging, because any vertical movement would not tend to rotate the propeller. A test of the Hoff meter with various values of the angle shows that this instrument is not far from the ideal in this respect. Tables 10 and 11 show the data obtained. If the meter were rotated about the supporting rod as an axis in a clockwise , . Tftfc * 66 S a; tu O 3 o 67 W a cs to U > ... a u 0; cox i CS <-l t > O coco o o 1/2 o in o 68 Theoretical Curve ,Vith Arbitrary Correction Applied THEORETICAL AND EXPERIMENTAL COHVES SHOWING! : VELOCITIES AS -ORBilSATES- AGAINST ; VALUES OF AS '/iBBG IS<>AS . Fig. 23, 69 Table 10. Table 11. Rating of Hoff Meter, Model 21, No. 48. Jan. 25, 1922. '.Then j6 is positive. Time in Revo- Revolutions Ft. per seconds, lutions. per second, second. Course: 200 ft. 101.7 157 1.54 1.97 287.8 153 .532 .695 10 176.2 146 .828 1.14 10 216 149 .69 .926 10 196.4 149 .759 1.02 Course: 51.5 ft. 21.8 41 1.88 2.36 40.5 40.7 1.005 1.27 o 68.8 41 V .596 .748 5 22 40 1.82 2.34 f.0 39.5 39.9 1.01 1.3 62.7 39.2 .625 .823 10 o 40 38.2 .955 1.29 10 o 65.2 37 .569 .791 10 o 21.8 39 1.79 2.36 15 o 22 37.3 1.69 2.34 15 o 41.2 37.2 .904 1.25 15 o 64 36.6 .572 .805 20 o 24.2 34 1.4 2.12 20 c 41.8 32.5 .777 1.23 20 o 52.4 33 .63 .983 30 o 23.7 26 1.1 2.17 30 o 24.2 27 1.12 2.13 30 c 41.9 26.2 .625 1.22 40 o 24 18 .75 2.14 40 o 41.6 17.9 .43 1.23 40 Q 60.8 16.8 .276 .846 4 ^o 21.7 12.6 .58 2.37 45 o 40.6 12.4 .307 1.27 61.8 11.8 .191 .834 50 21.1 8.1 .382 2.43 50 39.8 7.5 .189 1.3 50 62 6.2 .103 .83 55 22 4.2 .191 2.34 55 40.5 3.2 .079 1.27 55 60.2 2.2 .037 .856 60 No rotation. 65 n n 70 n it 75 it u 80 C 21.8 -5.5 -.252 2.36 85 21.8 -9.1 -.417 2.36 80 21.8 -10.5 -.482 2.36 iVhen $ is negative. Time in Revo- Revolutions Ft. per seconds, lutions. per second, second. 41.7 64.9 22 40.2 65 22 41 65.2 21.8 4C.9 63.7 21.8 41.8 62.5 21.9 39.9 62.6 22 39.6 63.5 22.3 40 60.7 22 39.7 39.9 39.8 39.5 39 37 33.8 36.8 35.1 33.5 33.8 30.6 29.7 26.3 25 23.3 16 15.1 13.7 11.9 10.8 8.7 8. 7.2 6.05 4. 2.3 No rotation. it it 1.81 .953 .609 1.77 .92 .52 1.67 .856 .514 1.55 .748 .475 1.21 .598 .373 .73 .378 .219 . 41 .273 .137 .359 .18 .0997 .182 .058 2.34 1.24 .794 2.34 1.28 .793 2.34 1.26 .791 2.36 1.26 .808 2.36 1.23 .823 2.35 1.29 .823 2.34 1.3 .811 ? Z w W 1.29 .849 2.34 1.3 21.7 21.8 22 -5.5 -9. -10.8 -.253 -.413 -.491 2.37 2.36 2.34 70 direction, $ was marked plus; in a counter-clockwise direction, minus. The corresponding curves are plotted in Figs . 21 and 22. In Fig. 23, ft. per second are plotted as ordinates against $ as abscissas, where the normal velocity was 1.5 ft. per second. The dsta for this curve, marked "test curve" is taken from Figs. 21 and 22, at the velocity of 1.5 ft. per second, and listed in Table 12, columns (2) and (3). The average of the positive and negative values, column (4), are the ones plotted. The ideal Table 12. (1) (2) (3) (4) (5) (6) Revolutions Revolutions per second per second Average for positive for negative revolutions Ft. per Cos $ X 1.5 ft. values of $ values of $ per second second per second 1.58 1.58 1.58 1.5 1.5 5 1.56 1.54 1.55 1.47 1.49 10 1.51 1.48 1.5 1.42 1.48 15 1.44 1.39 1.42 1.35 1.45 20 1.32 1.26 1.29 1.22 1.41 30 1.03 1.01 1.02 .965 1.3 40 .7 .61 .66 .625 1.15 45 .49 .44 .47 .445 1.06 50 .29 .31 .3 .285 .964 55 .13 .14 .13 .123 .86 60 .75 65 .635 70 .513 75 .389 80 -.21 -.21 -.21 -.199 .261 85 -.35 -.35 -.35 -.332 .13 90 -.42 -.42 -.42 -.399 curve is marked "cosine $" . The values, 1.5 ft. per second X Cos 0, are shown in column (6). There is an appreciable difference between the ideal curve and the test curve. Still, the Hoff meter is the only one which even approximates the ideal, and this is certainly o another point in its favor. V, r hen jp 90 , the rotation is negative, 71 (o) uatlo . - an end sect: a Hoff iler V'in.st ' ? Lve faces A, , no force . n for this r- 72 resentation will be disclosed later. Face A_ makes an angle a with a line perpendicular to the center line of the propeller shaft; B, an angle b; C, an angle c. Fig. 24 (b) shows a filament acting against the front face, A. What torque component does this filament exert? Let dF the force due to impact of the filament. Then, dF makes an angle (90 - a) with the face A. Upon striking the face, dF is resolved into two components: ds, the slippage component parallel to A, and de, the effective com- ponent perpendicular to A. de is then resolved into two components: dt, the thrust component parallel to the spindle, and dq, the torque component perpendicular to the spindle. de - dF Cos a dq = de Sin & . . dq dF Cos a Sin a where dq is the component causing rotation. Suppose, however, that the filament is not parallel to the spindle, but makes an angle with it, as in Fig. 24 (c). Then, what is dq in terms of a and j6l de - dF 1 Cos (a + 0) dq = de Sin a . . dq dF 1 Cos (a + $) Sin a Let & angle of displacement from the position o shown, where -6- = ) . In other words, when the propeller has revolved one-fourth revolu- tion, -6- = 90; one-half revolution, -6-= 180; etc. \7ith the fila- ment approaching the propeller at the angle $, what is the torque pJb sieriw 73 component when $ = 90? At -9- = 180 At -6- = 270 dq Q dF 1 Cos a Sin a Cos j6 A 7W 180 dq 2 o = dF 1 Cos a Sin a Cos fi At -6- - 360, or dq = dF 1 Cos (a + $) Sin a In other words, the torque component varies from a minimum at &-- to a maximum at -d- * 180 , and back to a minimum at = 360 , or 0, the curve being two straight lines. If the face A were the only one active, the propeller would theoretically follow the cosine law; but this is not the case. When reaches 25 , face B, at -9- = 0, is parallel to the filament, and any further increase in $ brings B into play. In a like manner, when $ reaches 40 , face , at -9- * , is parallel to the filament, and any further increase in $ brings C into play. So, it is necessary that each face be considered separately. Disregard the existence of face A for the present, and assume that the filaments impinge upon faces ]3 and , as shewn. An analysis of the way in which the force dF* is resolved gives t. e following equations similar to the previous ones: dq n o = dF 1 Cos (b + f} Sin b O V dq - dF 1 Cos b Sin b Cos j6 B 90 dq o z dF' Cos (b - $) Sin b B 180 dq = dF 1 Cos b Sin b Cos B dc lr. no = dF 1 Cos (c + $) Sin c v U dq o * dF f Cos c Sin c Cos c yo Tt ."o rio < ' ari , x anb-toi 77 T> tO tO to LO LO CO CO .. ^ CO to *d* CO CO ^ to to o tOC^CO O>O> O^OSCOD^ fj o . x I 1 OO ir I 1 SP O i II S-O (-, ^ ooo ooooooc "l ooo ooooooo CM 10 C! tO T}< tO CM i 1 rH CM tO ^J ! ^ 1 1 1 1 r II pj rH ' -P o 5 o to CM to to CM to to tOlOtO rH rHtOtOtOC- rt rH o J O 1 1 1 1 I 3 n C O 1 1 u a! ,- O a. ooo ooooooo 000 OOOOOOO o rH rH rH rH rH i to^c-toa>totototototo ,0 o &2 t** O ^O i ! O JO O> vJJ *J O o -* ^ LO C*- C CO Oi O* OS Oi U * QJ c CO r, rH r-H V II a ni -. (^ k i OOOOOOOOOOO to S-*O i LO LO LO O LO LO LO LO LO LO LO r-H O id Q CO LO ^* ^ tO CXJ i 1 1 c 4 OJ ^H CO _^ 1 1 C' n II ,5. C ,0 Q tooit- c-cnto^co C 1 rH H- EJJ ?R ^"^ Lu rt ^^ Iv^ r^ ^J O o ^* CVi O C) C^ ^* LO C*- CO O* 3 o 1 R O O 1 1 1 1 1 1 1 CJ> rH O f ooooooooooo LO LO LO C3 tO LO LO LO LO LO LO '^, CO ^^ OO Oi O5 C5 f~~^ C\J tO ^^ LO i-i r-t <-H H i-! f-l 7 CO CO CO LO LO CO CO ^ CO CO *vj* CO 00 ^* CO CO ii o c: co ^- oo *j~> o^ O5 0^ oo t o CO r/ r-i rH rH O to O s n 4 , ^ (j) r^i ooo ooooooo ^Q | ooo ooooooo o o t LO ^ tO CVl rH rH CM tO ^ CM m tiii n n OS B _J ^ ^ t7 tO CM tO tO CM tO CO 49 o o ri j< ^ t~ t- ^ 'i 1 to (OUJtO rH rHtOLOtOC- a II V. O 1 1 1 1 I rH I o f7 ooo ooooooo + ooo ooooooo ci lO to t*- CO O^ O i 1 CM CO ^t* rH rH rH rH rH OO OOOOOOOOO OOOiOOOOOOOO I I 78 180 dq o = dF 1 Cos c Sin c Cos $ C I U A glance at Figs. 25, 26 and 27 will make the problem clearer. In Fig. 25, the function of the torque component for the face A is plotted for values of $ from to 90 . These values are plotted for only one-half revolution of the propeller because the curves are identical for the two halves. Also, the curves were located by plotting only the values at -Q- = , and $ = 180, because they are all straight lines. In Fig. 26, similar curves for face B are plotted; in Fig. 27, for face . Referring to Fig. 25, what do the negative components mean? Take $ 50 , for an example. From -&- = to -0- 16, face A is not exposed to the approaching stream; it is hidden behind the face B. These components which are negative, in this case, are not active . In Fig. 26, the curves are somewhat different. Here, the positive components indicate nothing; where the curves show positive values, the blade B is inactive. The negative components indicate that the face B is exposed to the oncoming filaments, and that this torque tends to reverse the rotation. Fig. 27 presents a still different situation. Here again, the negative components mean nothing; face C is hidden by faces A and B. When the components are positive, the face C is exposed to the oncoming filaments, and the resultant torque tends to turn the propeller in the normal direction; it assists the torque exerted on face A. This analysis is a difficult one to make clear to the reader, because it is a three dimensional problem. Possibly Ta- Table 14. 79 (1) Average value of a ct i ve c omp one nt from curve for face- A B C (2) (3) (4) H H X! CM (5) X! > c H X! V~N to (6) o X! fl H X! **"N CT) Total theoretical CD ^torque component X K * 3-1 LO 1 r-t \ e> -tf g T( O 4) tf) K (9) 10 .643 .633 .444 .437 .444 .437 1.5 1.477 20 .605 .418 .418 1.413 25 .583 .408 .403 1.363 30 .557 -.043 .385 -.039 .346 1.17 40 .494 -.129 .341 -.117 .224 .756 50 .5 -.212 .087 .345 -.192 .02 .173 .584 60 .492 -.287 .171 .339 -.26 .039 .118 .399 70 .47 -.353 .25 .324 -.321 .057 .06 .203 80 .433 -.409 .322 .299 -.371 .074 .002 .007 90 .383 -.453 .383 .264 -.411 .088 -.059 .-.199 80 bles 13 and 14 will be of some benefit. Table 13 ^hows the calculations from which the curves in Figs. 25, 26 and 27 were plotted. Table 14 shows the average torque components as taken from these curves. The average torque exerted on face A for one revolution for a particular value of $ is proportional to the mean ordinate of that portion of the curve corresponding to that particular value of $ above the X-axis. These are positive components; the values are tabulated in column (2). The average torque exerted on face B for one revolution for a particular value of $ is proportional to the mean ordinate of that portion of the curve corresponding to that particular value of $ below the X-axis. These are negative components; the values are tabulated in column (3). The average torque exerted on face C for one revolution for a particular value of $ is proportional to the mean ordinate of that portion of the curve corresponding to that particular value of j6 above the X-axis. These are positive components; the values are tabulated in column (4). Since the height of the blade is the same for all faces, and since the three faces appear in each of the four blades, the torque on any face for the four blades is proportional to the values taken from the- curve multiplied by the width of the face. Let w = width of face in in. Then, w =0.9 A w = 1.0 B w = 0.3 C . . the total torque on the face A for the four blades T A = k X M X w A A A where k is a constant including the length of the blade, the number 81 of blades, the friction factor, adhesion, and any other factors which tend to oppose the rotation of the propeller; M is the mean value A of the torque function taken from the curve . On the face B, T = k X M X w - B B B On the face C, T = k X M X w ~~ c c c Values for M. are shown in column (2); for K , in column A B (3); for Mp, in column (4). Values for T. are shown in column (5); for Tg, in column (6); for T., in column (7). T + T + T = total theoretical torque X K ABC and the values are shown in column (8). These values are reduced to ft. per second by multiplying them by the constant 1.500; this 0.444 does not change the normal velocity of 1.5 ft. per second, con- sequently, the resultant curve is comparable with the test curve shown in Fig. 23. Referring to Fig. 23 again, it will be noted that the curve just developed, marked "theoretical curve as developed", . o crosses the X-axis when is about 80 . The test curve shows that the propeller does not rotate between $ - 60 and $ = 75 , and beyond the latter value of $ the rotation is negative. Much to the writer's disappointment, the two curves show little similarity. The theoretical curve shows two decided breaks. At $ * 25 the curve begins to fall rapidly due to the retarding influence of the face Bj at $ = 40 , the slope is decreased slightly due to the propelling influence of the face C. A decided break, such as these are, would naturally not appear in the test curve, though it seams that a slight depression might be expected. If an arbitrary constant is applied to the theoretical curve so that it will also cro-?* the X-axis at $ = 60 , a resultant 82 curve, marked "theoretical curve with arbitrary correction applied", is obtained. In attempting to apply a standard equation to the test curve, it was found that Cos 3 j6 fits the positive values very ~Z~ closely. It appears to the writer to be nothing more than a co- incidence, though there may be some reason for it. It might be said here that a development of the torque exerted on a very thin blade having only two flat, parallel sides, or on a blade whose cross section is a closed triangle will result in a cosine curve. '.Vhether or not a meter with this type of pro- peller would actually give a cosine curve, or even a close approxi- mation could only be determined by experiment. 24. Disadvantages of the Hoff Horizontal Shaft Propeller Type Meter .--Compared with other current meters, the Hoff instrument has no disadvantage. The rating curve always lies below the ideal curve on account of the small interference factor. This may appear to some as a disadvantage, though the writer believes that this type of curve--a straight line slightly below the ideal curve is ju-rt as well. Of course the Hoff meter is not perfect. Ag was stated in the beginning, a perfect meter cannot be made. This meter offers small interference to the stream filaments; it possesses a small a- mount of friction; it does not exactly follow the cosine law. The first two objections cannot be eliminated. Whether or not the third objection can be overcome, the writer cannot say; but the Hoff curve approaches the .copine curve, and this can be said of no other meter. So, taking all factors into account, and considering both the con- clusions drawn from the theoretical analysis and the actual tests, the writer believes that the Hoff instrument is nearer the ideal 83 than any other raster which has yet, been made. V. CONCLUSIONS, 25. Conclusions .--In conclusion, it might be well to re- view the most important points as set forth in this thesis. 1. The current meter method of measuring stream flow .fill never produce accurate results, because when any obstacle, suce as a currant meter is, is placed in a moving stream, a cer- tain retardation of the fluid is effected, and eddy currents and cross currents are produced from the non-parallel filaments which introduces errors of uncertain magnitude . 2. The vacuous spaces formed behind any obstacle material- ly affect the force required to hold that obstacle in the moving stream. It is to this phenomena that the turbine meter is largely responsible for its rotation. 3. A disturbance is more widely felt when the velocities are low, while the height to which the water is piled in front of an obstacle and the depth to which it falls in the rear is greater when the velocities are high. This makes the accurate gaging of small channels very difficult, because the stream filaments are deranged throughout the entire cross section of the ditch when a meter is lowered into it. 4. The slops of a rating curve depends not only upon the characteristics of the rotating element itself, but also upon the interference offered by all of the non-rotating parts: the supporting rod or cable, tail, yoke, any protecting devices for the propeller or turbine, etc. The interference factor of the propeller or turbine causes the higher velocity curve to lie below the ideal 84 curve. The instrumental friction factor gives the starting veloci- .ty curve a steeper slope than that of the higher velocity curve, and, on this account, the two curves intersect at a point below which the observations cease to be reliable. 5. All current meters have their advantages and disad- vantages. By a careful study of the nature of fluid flow around obstacles, and close attention to the minute details, the designer can eliminate many of the disadvantages, and improve those which are indispensable. The horizontal shaft propeller meter is correct in principle. Its precision cannot be excelled by any other type. 85 Bibliography. Bateau, M. : "Experiments and Theory on the Pitot Tube and on the Current Meter of ifoltmann." (Annales de Mines Memoires, Vol. 18, 1898, and 10th series, Vol. 2, 1902.) (Manuscript in the Office of Irrigation Investigations, Berkeley, California.) Ccwley and Levy: "Aeronautics in Theory and Experiment". de Villamil, R. : otion of Liquids". Epper, M. : 1 Hydrometric Experiments . Foote, A. D.: ">7ater Meter for Irrigation (A)". (Trans. A.S.C.E., Vol. XVI, 1887.) Groat, 8. F.: "Characteristics of Cup and Screw Current Meters". (Trans. A.S.C.E., Vol. LXXVI, 1913.) "Current Meters". (Part V, "Cheni-Kydroinetry and its Ap- plication to the Precise Testing of Hydro-Electric Generators". Trans. A.S.C.2., Vol. LXXX, 1916.) Gurley, V, r . & L. E. : "Manual of Gurley Hydraulic Engineering Instruments". Hele Shaw, H. S. : "Experiments on the Nature of the Surface Resistance in Pipes and on Ships". (Trans. Inst . Naval Arch., 1897.) "investigation of Stream Line Motion Under Certain Experimen- tal Conditions". (Trans. Inst. Naval Arch., 1898.) "Stream Line Flow of a Viscous Fluid". (Report of British Assn., 1898.) "The Distribution of Pressure Due to Flow Round Submerged Sur- faces". (Trans. Inst. Naval Arch., 1900.) "The Motion of a Perfect Fluid" 1 . Hoff, E. J.: "Current Meter Studies". Hoyt, W. G.: "The Effects of Ice on Stream Flow". ('iTater Supply Paper, No. 337, U. S. Geological Survey.) Hoyt and Grover: "River Discharge". Hydraulic Labratory, Colorado Experiment Station, Ft. Collins, Colo. Tables, Tests, Rating Data, etc. Not available. 86 Lyon, G. J.: "Equipment for Current Meter Gaging Stations". (Water Supply Paper, No. 371, U. S. Geological Survey.) Murphy, E. C.: "Accuracy of Stream Measurements". (.Tater Supply Paper, No. 95, D. S. Geological Survey.) "Current Meter and " r eir Discharge Comparison". (Trans. A.S.C.E., Vol. XVI, 1887.) Noble, T. A.: "Current Meters". (Trans. A.S.C.E., Vol. XLI-2, 1899.) Rensselaer Polytechnic Institute, Troy, N. Y. : Thesis on Current Keters . Sandstrom, J. ".. . : "Hydrometric Experiments". (Stockholm, 1912.) (Manuscript in the Office of Irrigation Investigations, Berkeley, Calif.) Schmidt, 1.1. : "Untersuchungen uber die Umlaufbewegung hydrometrischer Flugel". (ilitte i lunge n uber Forschungsarbeiten auf dem Gebiete des Inge- nieurwesens, 1903.) Scobey, F. C. : "Behavior of Cup Current Meters Under Conditions Not Covered by Standard Ratings", (journal of Agricultural Research, Vol. II, No. 2, 1914.) Stearns, F. P.: "On the Current Meter Together With a Reason IThy the Maximum Velocity of Water Flowing in Open Channels is Below the Sur- face". (Trans. A.S.C.E., Vol. XII, 1883.) Unwin, "f. C.: "Hydraulics". (Encyclopaedia Britannica.) "Hydromechanics". (Enclyclopaedia Britannica.) Fortier & Hoff: "Defects in Current Meters and a New Design . (Engineering News-Record, Vol. 85, No. 20, 1920.) 1 Wot available. 87 I8DEX. Reference to pages. Contour curves, 7. Current meters, demands on for irrigation work, 6, 54. fundamentals, 6. historical, 1. horizontal shaft, 31'. air meter type, 31, 32. ideal curve, 31. rating curve, 30. Hoff, 5, 16. advantages, 61. disadvantages, 82. equation, 71. rating curve, 34, 35, 55, 68. surface curve, 37, 38. ideal measurements, requirements, 52. possibilities of realization, 53. method of measuring stream flow, 3. only scientifically designed, 4. Price, 2, 20. as horizontal shaft, 50. rating curve, 18. surface curve, 29. vertical shaft, 17. advantages, 58. disadvantages, 59. ideal curve, 20. rating curve, 16, 28. without rigid support, 29. Eddy currents, definition, 3. effects, 9. reactive force, 11. Higher velocity curve, definition, 20. Interference, see Resistance. Obstacles, 6, 42. Parallel filaments, 4, 7. Propeller, 62. definition, 1. mass, 54. Rating curves, factors affecting, see Interference. Resistance, Hoff theory, 17. explanation, 17. proof, 50. 88 Rod in strean, 13. Starting velocity curve, definition, 20. Stream filaments, energy, 10. parallel, 4, 7. Turbine, definition, 1. explanation of rotation, 17. Vacuous spaces, 13. Vertical velocity curves, 7. Viscosity, effect, 7. UNIVERSITY OF CALIFORNIA LIBRARY BERKELEY Return to desk from which borrowed, This book is DUE on the last date stamped below. ? 6 1961 LD 21-100m-ll,'49(B7146sl6)476 NON-CIRCULATING BOOK U.C.BERKELEY LIBRARIES 476547 f UNIVERSITY OF CALIFORNIA LIBRARY