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THE NEW YORK
POBUC LIBRARY
Q
O it
O
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I
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WATER POWER ENGINEERING
THE THEORY. INVESTIGATION AND DEVELOPMENT OF WATER POWERS.
BY
Daniel w. mead.
Member American Society Civil Enginan
Consul/ f fig" Engineer
Professor of Hydraulic and Sanitary Engineering
University of Wisconsin
First E'lition.
NEW YORK
McGraw-Hill Book Co, 19O8
THE NEW YORK
PUBLIC LIBRARY
63248^
•iTO« LC«tUR AMD
Copyrighted 1907-1908
BY
Daniel W. Mead
:': ••• •. .•
*• •*•*! •*.
Btatm JouaiTAL yRonam CourAgrt Ma»i«on, WiaooMnr
PREFACE
In the development of a water power project the engineer is fre- qently called upon to do more than design and construct the power plant. He may be required to report on.the adequacy of the supply, the head and power available and the probable variations in the same, the plan for development, the cost of construction and opera- tion, and the advisability of the investment. A study of the entire project, therefore, becomes essential, and each factor must be care- fully considered in detail to assure ultimate success. Each of the features of the development is of equal importance to the commer- cial success of the project. The majority of the failures in water power development have occurred from causes other than structural defects, and a knowledge of these other important and controlling factors is therefore quite as essential as a knowledge of design and construction. It must be said, however, that in respect to some of these controlling factors current practice has not been what it should l>e. This has resulted in many over-developments and illy advised installations, from which the power generated has not been equal to that anticipated, and in many poor financial investments amount- ing frequently to practical failures. The engineer has given much attention to design and construction but too little attention to the other fundamental considerations mentioned above on which the success of the project depends to an equal extent.
In the preparation of this book the author has endeavored to con- sider, briefly at least, all fundamental principles and to point out the basis on which successful .^'Mer power-, cqvelopment depends. The method of study and inyestig^tipn outlined herein was developed by the author during tweuty^five year'$ of professional practice and in his efforts to illustrate the. principles underlying the subject in his lectures to the senior ^•class^iji.wateJ*. power engineering at the Uni- versity of Wisconsin. A somewhat extended acquaintance with the literature relating to water power engineering leads the author to believe that in a number of features the principles and methods de- scribed in this book are ^mewhat in advance of present practice.
VI
Preface,
In current practice, the hydraulic engineer, to determine the ex- tent of a proposed hydraulic development, commonly depends on a study of the monthly averages of stream flow and of observed maxi- mum and minimum Oows. He usually assumes from his previous knowledge and study that the development should be based on a certain minimum or average stream discharge per square mile of drainage area. The value of this method depends on the breadth of the engineer's local knowledge of rainfall and run-oflf relations. With a sufficient knowledge of these conditions, this method may form a safe basis for water power development but it fails to give the complete information which is essential for a full comprehension of the subject. In other cases the development is predicted on a single, or on a very few, measurements of what is believed, or as- sumed to be, the low water flow of the stream. This method, evettfl when accompanied by careful study of rainfall records, is a danger* ous one to employ as many over-developed water power projects demonstrate. Neither of these methods compares favorably with the more exact method of studying flow by actual or comparative^ hydrographs as is described in Chaps. IV, V, VI 11 and IX. "
In current practice the head available is usually determined for average conditions, or, perhaps, occasionally for low, average and high water conditions, and no detailed study of the daily e fleet on power is attempted* In Chaps. IV and V this subject is presented in detail and a method of the investigation of this important subject, under all conditions of flow and all conditions of use, is outlined.
On the basis of the kno%vledge gained from the study of flow and head, the study of the power that can be developed for each day ini the year and during each year for which actual or comparative hy-J drographs are available, is outlined. An outline of a method for the consideration of possible variations in flow during periods for] which no measurements arc available based on the available rain-J fall records, is also given*, ii>f:haj]^,-VJ.:VM •and VIIL A study of the effect of pondage OTv'pfiier' k mc^tMmpc>rtant matter, though not always carefully considcrel^, pr ^{^J-er^sftcd, is also discussed in — considerable detail in Chaps. lV,:V^"na'^XxVl. |
In the selection of turbines,4fi)j:a'\vs(6en power project* the current practice has been for the erfgtntir; Whilfe rtfawing certain conclu- sions from the tables of manufacturers' catalogues, to present to the manufacturer the conditions under which the power is to be devel-^ oped and to rely largely or entirely on the manufacturer for advice]
Preface. vii
as to machinery to be used. In such cases he is dependent for re- sults on guarantees which are usually quite indefinite in character and seldom verified by actual tests, under working conditions, be- fore the wheels are accepted and paid for. This has resulted in many cases in the installation of wheels which are entirely unsuited to the particular conditions under which they are installed.
Practical turbine analysis has not been treated except in the most general way in any publications except the various German treatises on the turbine in which the subject is discussed from the basis of turbine design. The author has developed the method of turbine analysis and selection, outlined in Chapters XIV and XVI. which applies to all wheels when tests of wheels of the series or class considered are available. These methods are based on the practical operating conditions of actual tests and are both theoreti- cally and practically correct. The engineer should be able to intel- ligently select the turbines needed for the particular conditions of his installation and to determine, with a considerable degree of accuracy, the results on which he can depend during all conditions of head and flow.
It is believed that this treatment of the subject is sufficiently complete to place the selection of turbines on a better footing and that, when adopted, it will lead to the selection of better and more improved designs and assure more satisfactory results.
The subject of turbine governing has, for electrical reasons, be- come an important one. While a number of important papers have appeared on this subject, there is, so far as the author knows, no discussion in English which offers the engineer a basis for a com- plete consideration of this subject. Chap. XVIII, on the principles of turbine governing together with appendixes A, B and C, offer, it is believed, suggestions for the consideration of this subject which may prove of value to water power engineers.
The report on a water power project should involve a careful and complete investigation of the entire subject, and should be based on the broadest considerations of the project in all its rela- tions. Many reports which have come to the author's attention bave been too limited in scope and have included only general opin- ions which have not. to his mind, been sufficiently specific or based on sufficient information to warrant approval without extended in- vestigations. In Chap. XXVIII the author has outlined his idea
viii Preface.
of the extent and scope of such investigation and report, which h believes is essential for an intelligent investigation and a reliabl opinion on this subject.
ACKNOWLEDGMENTS.
There can be little which is strictly new or original in any technics work, and in offering this book to the profession, the author wishes t acknowledge his indebtedness to the large number of technical ai tides that have already appeared on various phases of the subjecl Many references to such literature have been given at the end of th various chapters.
Many illustrations have been taken, with more or less chang from Engineering News, Engineering Record, Cassier's Magazin and Electrical World and Engineer. Various manufacturers hav furnished photographs and, in some cases, cuts of their wheels, go\ ernors and apparatus, in connection with which their names appeal
Tlie author has been greatly aided by his assistants, both of hi own private office and of the University staff. He wishes especiall to acknowledge the assistance of Mr. L. F. Harza to whoi Chap. XVIII on The Speed Regulation of Turbine Water Wheel and appendixes A, B and C are largely due. Mr. Harza has als been of much assistance in the editorial work of publication. Ej pecial acknowledgment is also due to Professor G. J. Davis, Ji for the preparation of the diagrams of friction of water in pipes an of Bazin's and Kutter's coefficients, etc. Mr. Robert Ewald assistc in the selection of material for illustrations, in the investigation < German literature, and the preparation of various graphical diagram including the first development of the characteristic curve.
The author also desires to acknowledge his indebtedness to h principal assistant, Mr. C. V. Seastone, for advice and assistance i the arrangement of many of the chapters in this work and assis ance in the editorial work of publication.
The sources of various other tables, illustrations, etc., are a knowledged in their proper places. D. W. M.
Madison, Oct. i, 1908.
CONTENTS
CHAPTER I.
Introduction.
The Hwtory of Water Power Development— Every Development of Water Power — ^The EiarlieBt Type of Water Wheel — ^The Undershot Wheel — The Overshot and Breast Water Wheel — ^The Development of the Turhine— Fundamental Ideas of the Turhlne— The Modem Turhin&— The American or Francla Turbine — ^Modern Changes in Turbine Practice — Historical Notes on Water Power Development — Development of Water Power in the United States — ^Literature. ••• 1
CHAPTER IL Power.
The Development of Potential Energy — Definition of Energy — Solar Energy the Ultimate Source — ^No Waste of Energy in Nature— Laws of Energy Conservation — Efficiency — Natural Limit to Efficiency — • Practical Limits to Efficiency — Efficiency of a Combined Plant- Capacity of Each Part of a System not Identical — The Analysis of Losses — The Losses In a Hydro-Electric Plant— Units of Energy'— Conversion of Energy Units — ^Kinetic Energy — Uniform Motion — Uniform Varied Motion — Compound Motiour-Qraphlcal Representa- tion of the Laws of Motion — ^Transformation — ^Literature 19
CHAPTER IIL
Hydraulics.
Basis of Hydraulics — Mathematical Expression for Energy— Velocity Head*— Entrance Head — Submerged Orifices — Friction Head — Kut- ter's Formula — Bazin's Formula — ^Efficiency of Section — ^Determina- tion of Canal Cross-Section — The Back Water Curve — Flow of Water in Pipes — The Flow of Water Through Orifices — Flow over Weirs — ^Literature 40
CHAPTER IV.
Wateb Powis.
The Study of the Power of a Stream as Affected by Flow— Source of Water Power — Factors of Stream Flow — Broad Knowledge of
Contents.
Stream Plow Neceasary— The Hydrograph— The Use of Local Hydrography— Use of Comparative Hydrographs— Reliability of Comparauve Hydrographfl— When no Hydrographs are Available— The Hydrograph as a Power Curve 7S^
CHAPTER V.
Wateb Poweb (Continued)
The Study of the Power of a Stream as Affected by Head— Variations in Head — The Rating or Discharge Curve— The Tail Water Curve — The Head Water Curve— Graphic Representation of Head — Effects of Design of Dam on Head — Effect of Head on the Power of the Plant — Graphical Representation of the Relations of Power, Head and Plow — Graphical Study of Power at Kilbourn — Power of the Kilbourn Wheels Under Variations in Flow— Effects of Low Water Flow — Effects of Number of Wheels on Head and Power • 9 J
CHAPTER VL
Rainfall.
Importance of Rainfall Study — Distribution of Rainfall — The Rainfall Must be Studied in Detail — ^Local Variation in Annual Rainfalls— Local Variations in Periodical Distribution of Annual Rainfall — . Accuracy of Rainfall Maps and Records — Rainfall and Altitude — Value of Extended Rainfall Records — Accuracy in Rainfall Obser- vation*—District Rainfall — Study of Rainfall as Affecting Run-off — Literature.. ..c 111-
CHAPTER VII.
The Disposal of the Rainfall.
Pactors of Disposal — The Rate or Intensity of Rainfall— Condition of Receiving Surfaces and Geological Strata — Effects of Wind — Effects of Vegetation — Percolation— Evaporation — Evaporation Relations — Practical Consideration of Losses — Literature 133-
CHAPTER VIII.
Run-off.
Ron-oCf — Influence of Various Factors — Relations of Annual Rainfall and Run-off of Water Year — Relation of Periodic Rainfall to Run- off— ^Monthly Keiatlon of Rainfall and Run-off — ^Maximum iStream Flow— Estimate of Stream Flow 146;
Contents. zi
CHAPTER IX.
RuN-OFF» ( Continued )
Relation of Run-off to Topographical Conditions — Tweets of Geological Condition on the Run-off— The Influence of Storage on the Distri- bution of Run-off— Effects of Area on the Run-off — ^The Study of a Stream from Its Hydrographs — Comparative Run-off and Compara- tive Hydrographs — Comparative Hydrographs from Different Hydrologlcal Divisions of the United Statest— Literature 17S
CHAPTER X.
Stream Flow.
Plow in Open Channels — Changes in Value of B^tors with Changes in Flow — Effects of Variable Flow on the Hydraulic Gradient — Effects of a* Rising or a Falling Stream on Gradient — Effects of Channel Condition on Gradient — Effect of Change in Grade and of Obstructions — Relation of Gauge Heights to Flow — Variations in Velocity in the Cross-Section of a Stream — Effects of Ice-Covering on the Distribution of Velocities 1»«.
CHAPTER XI.
The Measurement of Stream Flow.
Necessity for Stream Flow Measurements — ^Methods for the Estimate or Determination of Flow in Open Channels--Estimates from Cross-Section and Slope — Weir Measurement — Measurement of Flow by the Determination of Velocity — The Use of the Current Meter — Current Meter Ob?ervatons and Com putatlon'— Float Measurements — The Application of Stream Gaugings — Literature. 21 Jl:
CHAPTER XII.
Water Whescls.
Classification of Water Wheels — Gravity Wheels— Reaction Wheels — Impulse Wheels — ^Use of Water Wheels — Classification of Tvtr- bines— Conditions of Operation — Relative Advantage of Reaction and Impulse Turbines — Relative Turbine Efficiencies — Turbine De- velopment in the United States — The American Fourneyron Tur- bine—The American Jonval Turbine — The American Type of Re- action Turbine — The Double LefTel Turbine — Other American Wheels — ^Early Development of Impulse Wheels — American Im- pulse Wheels — Turbine Development In Europe 23 T
:xxi
Contenta,
CHAPTER Xtll,
Th§ Runner — Its Material and Manufacture — Diameter af tte Run* jier— The Detalli oC the Runiter — Vertical Turblae Bearlnga^— Hoii* fontal Turbine Bearings — Thrusts-Bearing In Snoqualmle Fall* Turblus— The Chute Case — Turbine Gates— The Draft Tube. 2S4
tJHAPTER XIV.
Htdbacxics of the TtniBirffi.
"practical Hydraulics of the Turbine^NoniencIature Used la Chapter— First Principies^lmpuise and Keactlon— The impulse Wheel— irffect of Angle of Discharge on Efficiency^ — Reaction Wheel- Graphical Relation of Energy and Velocity In Reaction Turbine* Turbine Relations— Relation of Turbine Speed to Diameter and Head— Graphical Expression of Speed Relations— Relations of fp and Efficiency — Discharge of Turbine at Fixed Gats Openings Power of a Turbine — The Relation of Diacbarge to the Diameter of a Turbine — The Relation of Power to the Diameter of a Turbine — Relation of Bpeed to Di:?charge of Turbines.— Relations of Speed to Power or Turbines^ Value of Turbine Constants— Literature,,.,
CHAPTER XT.
TunaiKE Testinq.
The Importanrc of Testing Machinery — The Testing of Water Wheeli^— Smeatoa's Experiments — The Barly Testing of Turbine Water Wheels — The Testing of Turbines hy James Emerson— The Kolyoke Testing Flume — The Value of Tests — Purpose of Turbine Testings P^tors that Influence the Results of a Test — Measurement of Dl»> Charge — Measurement of Head — Measurement of Spetd of Rota- tion— Measurement of Power — Efficiency — Illustration of Methods and Apparatus for Testing Water Wheela— Tests of Wheels In Place— Literature ,,.* , ..^ ».., Sfl
CHAPTER XVL
The Seiactioh of the Tuhbie^i^
Eflert of Condttons of Operation — Baals for the Selection of the Tur- bines-Selection of the Turbine for Uniform Head and Power— Ths Selection of a Turbine for a Given Speed and Power to Work under a Given Fixed Head— To Estimate the Operating Results of a Tur- bine under one Head from Test Results Secured at Another Head — To Estimate the Operating Results of a Turbine of one Diameter from Test Results of Another Diameter of the Same Series— To Estimate the Operating Results of a Turbine under Variable
Contents. xiii
Heads from a Test Made under a Fixed Head — A More Exact Graphical Method fi)r Calculation^— The Construction of the Char* acteristlc Curves of a Turhlne — The Consideration of the Turbine from its Characterletlc Curve — Other Characteristic Curves — Graphical Analysis as Proposed by Mr. W A. Waters 884
CHAPTER XVII.
The Load Cubvb' and Load Factoes, and Theib Influence on thb Design or
THE Power Plant.
Variation In Load — Load Curves of Light and Power Plants.— Factory Load Curves — ^Load Curve of London Hydraulic Supply Company — Railway Load Curves — ^Load Conditions for Maximum Returns*— The Load Curve in Relation to Machine Selection — Influence of Manage- ment on Load Curve — Relation of Load Curve to Stream Flow and Auxiliary Power — Literature • 420
CHAPTER XVIIL
The Spied Regxtlation of Turbine Water Wheels.
The Relation of Resistance and Speed — Self-Regulation in a Plant with Variable Speed and Resistance — ^The Relations Necessary for Con- stant Speed — ^The Ideal Governor — Present Status — Value of Uni- form Speed — ^The Problem — ^Energy Required to Change the Pen- stock Velocity — Hunting or Racing — ^Nomenclature — Shock of Water Hammer Due to Sudden Changes In Velocity — ^Permissible Rates of Gate Movement — ^Regulation of Impulse Wheels — Influences Opposing Speed Regulatiour- Change of Penstock Velocity — Effect of Slow Acceleration on Water Supplied to Wheel — Value of Racing or Gate Over-Run — Energy Required to Change the Penstock Velo- city—Effect of Sensitiveness and Rapidity of Governor — The Fly- wheel— ^The Stand-Pipe — ^The Air Chamber — Predetermination of Speed Regulation for Wheel set In open Penstocks — Predetermina- tion of Speed Regulation, Plant with Closed Penstock, — Predeter- mination of Speed Regulation, Plant with Standpipe — Application of Method. Closed Penstock — ^Application of Method, Open Penstock —Application of Method, Plant with Standpipe— Literature 440
CHAPTER XIX.
The Wateb Wheel Go\'ebnob.
Types of Water Wheel Governors — Simple Mechanical Governors — ^Anti- racing Mechanical Governors — Details and Applications of Wood- ward Govemora— The Lombard-Replogle Mechanical Governors — Essential Features of an Hydraulic Governor— Details of Lombard Hydraulic Govemor^Operatlng Results with Lombard Governor — The Sturgess Hydraulic Governor — Test Results with Sturgess Gov-
xiv . Contents.
ernor — Control from Swltchboarfl — Connection of Governors to Gatea— Relief Valves — Lombard Hydraulic Relief Valves — Sturgess Relief Valves ...c 470
CHAPTER XX.
ABBAI7GEMENT OF THE REACTION WheEL.
General Conditions — Necessary Submergence of Reaction Wheels^— Ar- rangement of Vertical Shaft Turbine — ^Arrangement of Horizontal Turbines— Classification of Wheels — ^Vertical Wheels and Their Con- nection— Some Installations of Vertical Water Wheels — Some In- stallations of Vertical Wheels In Series — Some Installations of Horizontal Water Wheels — Some Installations of Multiple Tandem Horizontal Wheels — ^Unbalanced Wheels 500
CHAPTER XXI.
The Selection of Machinery and Design of Plant.
Plant Capacity — Influence of Choice of Machinery on Total Capacity- Effect of Size of Units on Cost — Overload — ^E#conomy In Operation- Possibilities in Prime Movers — Capacity of Prime Movers — The In- stallation of Tandem Water Wheels — Power Connection — ^Various Methods of Connection in Use— Use of Shafting— The Wheel Pit — Turbine Support— Trash Racks 525
CHAPTER XXII. Examples of Watee Power Plants.
Sterling Plant— Plant of York-Haven Water Power Company — Plant of South Bend Electric Company — Spier Falls Plant of the Hudson River Power Transmission Company — Plant of Columbus Power Company — Plant of the Dolgevllle Electric Light and Power Co. — Plant of the Shawlnigan Water and Power Company — Plant of the Concord Electric Company — Plant of Winnipeg Electric Railway Co. — Plant of Nevada Power, Mining, and Milling Co. — Literature. . 637
CHAPTER XXIII.
The Relation of Dam and Poweb Station.
Ceneral Consideration — Classification of Types of Development — Con- centrated Fall — Examples of the Distribution of Water at Various Plants — Head Races only— Plants Located in Dam— High Head De- velopments • 661
Contents. xv
CaEIAPTER XXIV,
PsmoiFLiB or CoNBTBucnoN or Dams.
Object of Construction— Dams for Water Power Purposes^-Helght of Dam — ^Ayallable Head — 'Vhe Principles of Oonstmction of Damfr^ The Foundations of Dams — Strength of Dams — Flood Flowsr^Im? pervious Ctonstructionr-The Stability of Masonry Dam»— Calcula^ tions for Stability — Further Considerations— Types and Details of Dams— Literature 579
CHAPTER XXV.
Appendages to Dams.
Movable Dams — Flood Gates — Flash Boards — Head Gates and Gate
Hoists — Flshways — ^Logways — ^Literature 603
CHAPTER XXVI.
Pondage and Storage.
Effect of Pondage on Power — ^Effect of Limited Pondage on the Power Curve — ^Power Hydrograph at Sterling, Illinois — Effect of Pondage on other Powers— Effect of Limited Storage — ^Effect of Large Stor- age— Effect of Auxiliary Power — ^EJffect of Maximum Storage — Cal- culation for Storage — ^Method of Storage Calculation — ^Analytical Method— Literature 624
CHAPTER XXVII.
Cost, Value and Sale of Poweik.
¥*inancia] Consideration — Purpose of Development — Cost of Water Pow- er— Depreciation — ^Annual Cost of Developed Power— Cost of Distri- bution—Effect of Partial Loads on Cost of Power— Cost of Auxil- iary Power or Power Generated from other than Water Power Sources — Market Price of Water Pow^r — Sale of Power — ^An Equi- table Basis for the Sale of Power— Value of Improvements Intended to Bffect Economy— Value of a Water Power Property— Literature. 646
CHAPTER XXVIII.
The Investigation of Water Power Projects.
The ESztent of the Investigationr— Preliminary Investigation and Re- port—Study of Kun-off- Study of Rainfall— Study of Topographi- cal and Geological Conditions — Study of Flood-flow — Study of Back Water Curve— Study of Head— Study of Storage and Pond- age— Study of Probable Load Curve — Study of Power Development Study of Auxiliary Power— Study of Site of Dam and Power Sta- tion— Study of Plant Designr— The Estimate of Cost — The Report. . 675
I
xvi Contents.
APPENDICES.
A. Water Hammer — B. Speed Regulation, a more Detailed Analysis than in Chapter XVIII— C. The Stand-Pipe — D. Test Data of Turbine Water Wheels— E. Elffect of an Umbrella upon Formation of Vor- tices—P, EJvaporatlon Tables— G. Two New Water Wheel Governors — H. Miscellaneous Tables Including: Equivalent Measures and Weights of Water— Equivalent Units of EJnergy— Velocities in Feet per Second Due to Heads from 0 to 50 Feet— Three Halves Powers of Numbers, 0 to 100 — Five Halves Powers of Numbers, 0 to 50 — ^Re- lation of mean Rainfall to Maximum and Minimum Discharge of Various Rivers — Rainfall, Run-oft and Evaporation for Storage. Growing and Replenishing Periods or 12 Streams of the United States *.. 685-757
WATER POWER ENGINEERING.
CHAPTER L
INTRODUCTION.
THE HISTORY OF WATER POWER DEVELOPMENT.
I. Early Development of Water Power. — Most methods of power generation can be traced to an origin at no very remote period. Their development has been within historic times. The first development of water power, however, antedates history. Its origin is lost in remote antiquity.
Air and water, both physical agents most essential to life, have ever been the most obvious sources of potential energy and have each been utilized for power purposes since the earliest times. Beside the Nile, the Euphrates, and the Yellow Rivers, thou- sands of years ago the primitive hydraulic engineer planned and constructed his simple forms of current wheels and utilized the energy of the river current to raise its waters and irrigate the otherwise arid wastes into fertility. Such primitive wheels were also utilized for the grinding of corn and other simple power purposes. From these simple forms and primitive applications have gradually been developed the modern water power installa- tions of to-day.
2. The Earliest Type of Water Wheel— The crude float wheel driven directly by the river current developed but a small por- tion of the energy of the passing stream. The Chinese Nora, built of bamboo with woven paddles, is still in use in the east (see Fig. i), and was probably the early form of development of ^his type of wheel. The type is by no means obsolete for it is yet used for minor irrigation purposes in all countries. These ^vheels, while inefficient, served their purpose and were exten- sively developed and widely utilized. One of the greatest de- velopments of which there is record was the float wheel installa-
Introduction.
Pig. 1, — Chinese Nora, or
Float Wlieel Used Present.
From Earliest Times to
lion used to operate the pumps at London Bridge for the first water supply system of the city of London, and constructed about 1581 (see Fig. 2). In all such wheels the paddles dip into the unconfined current which, when impeded by the wheel, heads up and passes around the sides of the wheel and thus allows^ only a small part of the current energy to be utilized. H
3. The Undershot Wheal, — The introduction of a channel con- fining the water and conducting it to a point where it could be applied directly to the undershot wheel, was an improvement that permitted the utilization of about thirty per cent, of the theo-J
rig. E. — Float Wbeel Opemttng; Fiunps for Water Supply ot London 1S8 (From Matthews' Hydraulia Loud. 1835.)
The Overshot and Breast Water Wheel. 3
retical power of the water. This form of water wheel was most widely used for power development until the latter half of the eighteenth century.
In the float and undershot wheels the energy of water is ex- erted through the impact due to its velocity. The heading up of the water, caused by the interference of the wheel, results also iii the exertion of pressure due to the weight of the water, but this action has only a minor effect. The conditions of the application of the energy of water through its momentum is not favorable to the high efficiency of this type of wheels and the determination of this fact by Smeaton's experiments undoubt- edly was an important factor in the introduction and adoption of the overshot water wheel.
.s^i^^i
Fig. S.— Breast Wheel Used From About 1780 to About 1870.
4. The Overshot and Breast Water Wheel. — In the overshot water wheel the energy of water is applied directly through its weight by the action of gravity, to which application the design of the wheel is readily adapted. Such wheels when well con- structed have given efficiencies practically equal to the best modem turbine, but on account of their large size and the serious effects of back-water and ice conditions, they are unsatisfactory for modern power plants (see Fig. 11).
Following the work of Smeaton, the breast wheel (see Fig. 3) was developed in England largely through the work of Fairbairn ^^^ Rennie. The latter in 1784 erected a large wheel of this ^ype to which he applied the sliding gate from which the water flowed upon the wheel instead of issuing through a sluice as formerly. About this time the fly-ball governor, which had been ^^igned and adapted as a governor for steam engines by Watt, ^^ applied to the governing of these wheels and by means of these governors the speed of the wheel under varying loads was
iDLroduclioo.
Fig, 4.— Breast Wheel About 1790 Showing Early Application of Governor,
(After Glynn.)
kept sufficiently constant for the purpose to which they were then applied, (See Fig* 4*)
Another mode of applying water to wheels under low falls was introduced by M. Poncelet, (See Fig< 5.) Various changes and improvements in the form of buckets, in their ventilation so as to permit of complete filling and prompt emptying, and in their structure, tcxjk place from time to time, and until far into the middle of the nineteenth century these forms of wheels were widely used for water power purposes.
Fig. 5.— Poticelet's Wheel
5. The Development of the Turbine. — The invention of any important machine or device is rarely the work of a single mind. In general such inventions are the result of years of experience of many men which may be simply correlated by some designer.
Fundamental Idea of the Turbine.
to 'whom often undue credit is g^ven* To the man who has gathered together past experiences and embodied them in a new and useful invention and perhaps through whose energy practical applications are made of such inventions, the credit is frequently assigned for ideas which have been lying dormant, perhaps through centuries of time. Every inventor or promotor of val- uable improvements in old methods and old construction is en- titled to due credttj but the fact should nevertheless be recalled that even in the greatest inventions very few radical changes are embodied, but old ideas are utilized and rearranged and a new and frequently much more satisfactory combination results. Im- provements in old ideas are the improvements which are the most substantial. Inventions which are radically new and strictly original are apt to be faulty and of little practical value*
I
^FH5. 6, — Anctent Indian Water WheeK (After Glynn J ContalnEng FuB^ dameutal Suggest ion of Both Turbine and Impulse Wlieela.
6, Fundamental Ideas of the Turbine. — ^The embryo turbine may be distinguished in the ancient Indian water mill (see Fig. 6). A similar early type of vertical wheel used in Europe in the six- teenth century, the illustration of which was taken from an an- cient print (see Sci. Am. Sup* Feb. 17, '06) is shown in Fig- J. Barkers mill in its original form or in the form improved by M* Mathon de Cour, embodied the principal idea of the pressure
6 Introduction*
turbine, and was used to a considerable extent for mill purposes. In 1845 James Whitlaw suggested an improved form which was used in both England and Gennany early in the nineteenth cen- tury. (See Fig. 8.) Many elements of the modern turbine were conceived by Benjamin Tyler, who received letters patent for what he termed the "Wry Fly" wheel in 1804. T!ie description of this wheel as contained in the patent specifications is as follows :
Fig. 7,— Early Vertical Wheel.
Containing fundamental auggeatioii of tli» Turbine.
'The Wr>' Fly is a wheel which, built upon the lower end of a perpendiciilar shaft in a circular form, resembles that of a tub. It is made fast by the insertion of two or more short cones, which, passing through the shaft, extend to the outer side of the wheel. The outside of the wheel is made of plank, jointed and fitted to each other, doweled at top and bottom, and hooped by three bands of iron, so as to make it water-tight ; the top must be about one-fifth part larger than the bottom in order to drive
4
Barker's MiU. 7
the hoops, but this proportion may be varied, or even reversed, according to the situation of place, proportion of the wheel, and quantity of water. The buckets are made of winding timber, and placed inside of the wheel, made fast by strong wooden pins drove in an oblique direction ; they are fitted to the inside of the tttb or wheel, in such a manner as to form an acute angle from the wheel, the inner edge of the bucket inclining towards the w^ter, which is poured upon the top, or upper end of it about twelve and a half degrees ; instead of their standing perpendicular with the shaft of the wheel they are placed in the form of a screw, the lower ends inclining towards the water, and against the course of the stream, after the rate of forty-five degrees ; this, however, may be likewise varied, according to the circumstances of the place, quantity of water, and size of the wheel."
Elevation.
Plan and Partial Section.
Fig. & — Early Vertical Wheel. Containinjir Fundamental Suggestion of the Tnrbine. (After Glynn. )
Inlroduction*
Fig. 9. — ^Roue A* CurveB (After Glimii).
From the description it will be noted that, with the exception
of the chuteSp the principal features of the modern turbine were here anticipated. The "Wry Fly" wheel was an improvement on the "tub" wheel which was then in use to a considerable extend in the country.
These various early efforts received their first practical con- summation and modern solution ihrough various French in- ventors early in the nineteenth century. The "Roue a Ciives*' (Fig. 9) and the **Roue Volant" {Fig. 10) had long been used in France, and were the subject of extensive tests by MM* Pio- bert and Tardy at Toulouse. Those various wheels received the water tangentially through an opening or spout, being practically an improvement on the old Indian mill by the addition of a rim and the modification of the form of buckets.
7. The Modem Turbine, — The next improvement in the United States consisted in the addition of a spiral or scroll case to the wheel, by means of which the water was applied equally to all parts of the circumference passing inward and downward through the wheel. To the French inventors, Koechlin, Foumeyron and Jonval, is largely due the design of the turbine in a more modern and practical form. By the middle of the nineteenth century these wheels had met with wide application in France and been
■
I
4
The Modern Turbine.
I
Fig. 10. — Roue Volant (After Glynn).
adopted and considerably improved by American and German engineers, but were scarcely known in England. (See "Power of Water," by Jos. Glynn, 1852.) The turbine was introduced into the United States about 1843 ^Y Elwood Morris, of Penn- sylvania, but was developed and brought to public attention more largely through the inventions of Uriah A. Boyden, who in 1844 designed a seventy-five horse-power turbine for use at Lowell, Mass, (See Fig. 132, page 251.) The great advantage of the turbine over the old style water wheel may be summarized as fol- lows: (See Figs. 11 and 12). First: Turbines occupy a much smaller space. Second: On account of their comparatively high speed they "CJin'frequently be used for power purposes without gearing and with a consequent saving in power. Third: They will work submerged.
Fourth: They may be utilized under any head or fall of water. (Turbines are in use under heads as low as sixteen inches and as high as .several hundred feet.)
Fifth: Their efficiency, when the wheel is properly constructed, « comparatively high.
Sxth: They permit a greater variation in velocity without ma- terial change in efficiency.
to
Imroduclion.
The Francis Turbine.
zx
Seventh: They are more readily protected from ice interfer- ence.
8, The American or Francis Turbine. — ^Through the efforts of Uriah A. Boyden and James B. Francis (1849), ^^e Fouraeyron turbine became the leading wheel in New England for many years.
In 1838 Samuel B. Howd of Geneva, New York, patented the "inward flow" wheel, in which the action of the Fourneyron tur- bine was reversed. This seems to have been the origin of the American type of turbine, and the Howd wheel was followed by a large number of variations of the same general design on which American practice has been based for many years. About ^849, James B. Francis designed an inward flow turbine of the same general t3rpe as the Howd wheel. Two of these wheels
IS. — Inward Flow Wheel by S. B. Howd t After Francis).
^'cre constructed by the Lowell Machine Sliop for the Boott Cotton Mills. In the Lowell hydraulic experiments (page 61) ^Jr. Francis refers to the previous patent of Howd and says : "Under this patent a large number of wheels have been con- structed and a great many of them are now running in diflferent
I?
Introduction.
parts of the country. They are known in some places as the^ Howd wheels in others as the United States wheel. They have uniformly been constructed in a very simple and cheap manner in order to meet the demands of the numerous classes of millers and manufacturers who must have cheap wheels if they have any." M
Fig. 13 shows a plan and vertical section of the Howd wheels as constructed by the owners of the patent rights for a portion of the New England states. In this cut g indicates the wooden
Fig, 14, — Original Francis Turblna
guides by which the water is directed on to the buckets; W ifi dicates the wheel which is composed of buckets of cast iroi! fastened to the upper and lower crowns of the wheel by bolts. The upright crown is connected with the vertical shaft S by arms. The regulating gate is placed outside of the guides and is made of wood. The upright shaft S runs on a step at the bottom (noi shown in the cut). The projections on one side of the buckets. it was claimed, increased the efficiency of the wheel by diminish^ ing the waste of the water. f
The wheel designed by Francis was on more scientific lines, of lietter meclianical construction (see Fig. 14) and is regarded bi
Modem Changes in Turbine Practice.
13
many as the origin of the American turbine. The credit of this design is freely awarded to Francis by German engineers, this type of wheel being known in Germany as the Francis Turbine. The Francis wheel was followed by other inward flow wheels of a more or less similar type. The Swain wheel was designed by A. M. Swain in 1855. The American turbine of Stout, Mills and Temple (1859), ^^^ Leffel wheel, designed by James Leflfel in i860, and the Hercules wheel, designed by John B. McCormick in 1876, are among the best known and earliest of the wheels of this class.
9. Modem Changes in Turbine Practice. — A radical change has taken place in later years in the design of turbines by the adop- tion of deeper, wider and fewer buckets which has resulted in a great increase of power as shown by the following table from a paper by Samuel Webber (Transactions of Am. Soc. M. E. Vol. XVII) :
T1811 h— Showing Size, Capacity and Power of Varimis Txirbinee Under a ee-foot Head.
Inches Diameter.
Cubic Feet
Water per
Second.
Horse Power.
Boyden-Fourneyron . .
Ri«lon
Bisdon "L. C."
B»don"L. D."
LeHel, Standard
Wfel, Special
Tyler..
SviiiL
Hunt, "Swain bucket'
Hnnt, New Style
lAl, ••Samson"
"Htttsoles"
'TieU»'»
^ 8wtin
36
:^6
36 36 35 36 36 36 36 35 36 25 86
22.95 35.45 48.27 80. 40.46 60. 40.7 58.2 48.8 98. 109.1 107.6 108.8 89.5
55
89 121 199
96 148
95.8 140 121
289.74 264 253.5 266 215
By 1870 the turbine had largely superseded the water wheel for manufacturing purposes at the principal water power plants in this country. The old time water wheel has since become of comparatively small importance, but it is still used in many iso- '^ places where it is constructed by local talent, and adapted to local conditions and necessities.
14 Introduction.
The current wheel is still widely used for irrigation purposes and in many instances is a useful and valuable machine.
10. Historical Notes on Water Power Development. — ^Water mills were introduced at Rome about seventy years B. C. (see Strabo Lib. XII), and were first erected on the Tiber. Vitruvius describes their construction as similar in principle to the Egyp- tian Tympanum. To their circumference were fixed floats or paddles which when acted upon by the current of the stream drove the wheel around. Attached to this axis was another ver- tical wheel provided with cogs or teeth. A large horizontal wheel toothed to correspond with it worked on an axis, the upper head of which was attached to the mill stone. The use of such water wheels became very common in Italy and in other countries sub- ject to Roman rule.
Some of the early applications of water power are of interest. In 1 581 a pump operated by a float wheel was established at London Bridge to supply the city of London with water. In 1675 ^ri elaborate pumping plant driven by water wheels was established on the Seine river near Saint Germain. For this plant a dam was constructed across the river and chutes were arranged to conduct the water to the undershot water wheels. Thcse were twelve .pr more in number, each operating a pump that raised the waters of the Seine into certain reservoirs and aqueducts for distribution.
The pumping of water for agricultural irrigation and drainage, domestic supplies and mine drainage, was undoubtedly the first application of water power, and still constitutes an important application of water. Fig. 15, from an article by W. F. Dupfec, published in Cassier's Magazine of March, 1899, illustrates a primitive application of the water wheel to the pumping of water from mines. The frontispiece also shows the great Laxy over- shot water wheel in the Isle of Man which is still used for mine drainage. The wheel is about seventy feet in diameter and the water is brought froin the hills a considerable distance for power purposes.
11. Development of Water Power in the United States. — ^In this country one of the first applications of water power was the old tidal mill on Mill Creek near Boston, constructed in 1631, which was followed by the extensive developments of small powers wherever settlements were made and water power was
Development of Water Power.
IS
available. Often availability of water power determined the location of the early settlement.
About 1725 the first power plant was established along the Niagara River. This was a water-driven saw-mill constructed
Ckronologieal Development of Water Power of the United States to 1898.
Year.
Lowell, Mass
Nwhoa, N. H
<5ohoee,N. Y
Norwich, Conn
ADgQBta,Me. ,
Mmchester.N. H ,
Hooksett, N. H
Liwienoe, Mass.
Aopirta,Ga
Holyoke, Mass
Uvnston, Me.
Oolomboa, Ga
Bocbeeter, N. Y
St. Anthony Falls, Minn. . Kiagara,N. Y. (Hy. canal)
Turner's Falls. Conn
FoxRiver, Wis
KiminghaiD, Conn
Bingor, Me.
Augusta, Ga
timer's Falls. N. Y
Mechanicsville, N. Y
^ Cload, Minn
little Falls. Minn
Spokane, Wash
Howland, Me
^wtt Falls, Mont
Aiatln, Texas.
gwhSte. Marie, Ont
Johom, Cal
^id,N.H
JWenajMont
Junneapolis, Minn
Mechanicsville, N. Y
1822 1823 1826 1828 1834 1835 1841 1845 1847 1848 1849 1850 185(> 1857 18()1 1866 1866 1870 1876 1876 1882 1882 1885 1887 1888 1888 18<»0 1891 1S91 1891 1894 1894 1896 1897 1897 :898
Fall Ft.
36
36 104 16 17 52 14 30 50 50 50 25 236 50 90 35 185 22 9 50 30 20 14 14 70 22 42 60 18 55 13 170 446 32 18 18
Minimum Horse Power.
11,845
1,200
9,450
700
3,5U0
12,000
1,81'0
11,000
8,500
14,000
11,900
10,000
8,000
15,500
15,000
10,000
1,000
1,767
8,500
1,125
3,636
4,500
4,000
18,000
6,000
16,000
10,000
10,000
6,200
5,000
50,000
2,U40
10,000
6,000
3,270
Drainage
AreaSq.
Miles.
4,088 516 3,490 1,240 5,907 2,839 2,791 4,625 8,830 8,000 3,200
14,900 2,474
19,736 271,000 6,000 6,449 2,000 7,200 6,830 2,650 4,476
13,250
11,084 4,180
22,000 40,000 51,600
2,350
271,000
360
14,900
19,737
4,478
^7 the French to furnish lumber for Fort Niagara. Mr. J. T. Fanning gives the following list of the dates of establishing some ^ the principal water powers of the United States :
The last few years have witnessed a still more rapid develop- ment. The increase in manufacturing industries and other de-
i6
Inti'oduclJoo
mands for power and energy, I lie increased cosi of coal, am improvement in electrical methods of generation and tram sion have all united to accelerate the development of water p plants. Water powers once valueless on account of their tance from centers of manufacturing and population are accessible and such powers are rapidly being developed and energy brought into the market.
rig.
IS.— Earl f Application of Undershot Water Wheel to Mtne Date Unknown (from C&ssiers Mag. March, 1S99J
Dri
LITERATURE.
I
AppletoTi*a Cy doped la of Applied Merhanlcs* Modem Me Chan la
S, pp* 891-901. Description of the development of the Spon*s Dictionary of Knglneerlng. Barker's Mill, pp. 230-23&. do. Float Water Wheels (includtng undershot wheels), pp. 15 do, Overshot Water Whet^ls, p, 2557.
do. PoDcelet*s Water Wheels, p 2G(J0. ^H
do. Turbine Water Wheels, pp. 3014-3023, ^|
Knights Mechanical Dictionary, Vol. 3, Water Wheels, p. 27*
bines, pp. 2C56-2C^8.
i
Literature. i7
4. Emerson, James. Hydrodynamics. Published by author. Willimansett,
Mass. 1892. Describes several types of American turbines.
5. Matthews, William. Hydraulia. London, 1836. (Description of London
Bridge Water Wheels, p. 28.)
6. Palrbairn. William. Machinery and Mill work. Description of undershot
water wheel, pp. 145-150; description of earlier types (^ tur- bines, pp. 151-173.
7. Francis, James B. Lowell Hydraulic Experiments, pp. 1-70. Descrip-
tion and tests of Boyden-Fpurneyron Tremond Turbines; also
the Boyden-Francls "Center-Vent" Turbine, in which the Flow
was Radially Inward. New York, D. Van Nostrand, 1883. & Welsbach, P. J. Mechanics of Engineering, vol. XL Hydraulics and
Hydraulic Motors. Translated by A. J. DuBois. New York.
J. Wiley & Sons. 9. Morin, Arthur. Experiments on Water Wheels having a Vertical Axis,
Called Turbines, 1838. Translated by EUwood Morris in Jour.
Franklin Inst, 3d ser.. vol. 6, 1843. pp. 234-246, 289-302. 370-377.
370-377.
10. Morris, ESlwood. Remarks on Reaction Water Wheels Used in the
United States and on the Turbine of M. Foumeyron. Jour. Franklin Inst, 3d ser.. Vol. 4, 1842, pp. 219-227, 289-304.
11. Morris, EUwood. Experiments on the Useful Effect of Turbines in the
United States. Jour. Franklin Inst., 3d ser.. Vol. 6, 1843, pp. 377-384.
12. Wbitelaw, James. Observations of Mr. EUwood Morris's Remarks on
Water Wheels. Jour. Franklin Inst, 3d ser.. Vol. 8, 1844. pp. 73-80.
13. Franklin Institute. The Koechlin Turbine. Jour. Franklin Inst, 3d
ser.. Vol. 20, 1850, pp. 189-191. (Report of experiments made by members of the institute at the request of Emile Qeyelin, who introduced the Koechlin turbine at Dupont's powder mill.)
H. Ewbank, Thos. Hydraulic and Other Machines for Raising Water. New York, 1847.
15. Qeyelin, Emile. Experiments on Two Hydraulic Motors, Showing the Comparative Power Between an Overshot Wheel and a Jonval Turbine made for Troy, N. Y. Jour. Franklin Inst. 3d ser.. Vol. 22, 1851, pp. 418, 419.
16 Glynn, Joseph. Power of Water. London, 1850. pp. 39-97. Weales Scientific Series.
17. Webber, Samuel. Ancient and Modem Water Wheels. Eng. Mag., Vol. 1,
1891, pp. 324-331.
18. Frlzell, J. P. The Old-Time Water Wheela of America. Trans. Am. Soc
C. E., Vol. 28, 1893, pp. 237-249.
^5- Aldrlch, H. L. Water Wheels. Description of Various Types of Ameri- can Wheels. Power, Vol. 19, No. 11, 1894.
20. Francis, James. Water Power in New England. Eng. Rec, Vol. SS» 1896. pp. 418, 419. 1
1 8 Introduction.
21. Geyelin, Emile. First Pair of Horizontal Turbines ever Built Working
on a Ck>nimon Axis. Proc. Eng. Club, Philadelphia, Vol. 12,
1895. pp. 213, 214.
22. Francis, James. Water Power in New England. Eng. Rec. Vol. 33,
1896, pp. 418, 419.
23. Webber, Samuel. Water Power, its Generation and Transmission. Trans.
Am. Soc. Mech. Eng., Vol. 17, 1896, pp. 41-57.
24. Tyler, W. W. The BJvolution of the American Type of Water WhoeL
Jour. West Soc. Eng., Chicago, Vol. 3, 1898, pp. 879-901.
25. Johnson, W. C. Power Development at Niagara. Jour. Asso. Eng. Soc,
July, 1899, pp. 78-90. Hist of early development of power at Niagara.
26. Christie W. W Some Old-Time Water Wheels. Description of Various
old wheels in Eastern U. S. Eng. News, Vol. 42, 1899, pp. 394-395.
27. Ruchel, E. Turbines at the World's Fair, Paris, 1900. Review of Tur-
bine development in various countries. Zeitschr. d ver Deutsch, Ing. p. 657. 1900.
28. Foster, H. A. The Water Power at Holyoke. Jour. Asso. Eng. Soc., Vol.
25, 1900, pp. 67-34.
29. Thomas. R. Development of Turbine Construction. Zeitschr. d ver
Deutsch. Ing. p. 409, 1901.
30. Rice, A. C. Notes on the History of Turbine Development in America.
Eng. News, Vol. 48, 1902, pp. 208-209.
31. Fanning, J. T. History of the Development of American Water Powers.
Rept 22d Ann. Meeting, Am. Paper and Pulp Asso., 1898, pp. 16-24. Progress in Hydraulic Power Development Eng. Rec- ord, Vol. 47, 1903, pp. 24-25.
32. Fanning, J. T. Progress in Hydraulic Power Development BJng. Rec-
ord, Jan. 3d, 1903.
33. Slckman, A. F. The Water Power at Holyoke. Jour. N. E. W. W. Afi80.»
Vol. 18, 1904, pp. 337-351. Historical.
CHAPTER II.
POWER.
12. The Development of Potential Energy. — ^The development of natural sources of potential energy, the transformation of such energy into forms which can be utilized for power, and its trans- mission to points where it can be utilized for commercial pur- poses, constitutes a large portion of the work of the engineer. The water power engineer primarily deals with energy in the form of flowing or falling water, but his knowledge must extend much further for he encounters other forms of energy at every turn. Much of the energy available from the potential source will be lost by friction in bringing the water to and taking it from the wheel. Much is lost in hydraulic and mechanical fric- tion in the wheel ; additional losses are sustained in every trans- formation, and, if electric or other forms of transmission are used or auxiliary power is necessary for maintaining continuous operation, the engineer will be brought in contact with energy in many other forms.
13. Definition of Energy. — Energy is the active principle of nature. It is the basis of all life, all action, and all physical phenomena. It is the ability to exert force, to overcome resist- ance, to do work. All physical and chemical phenomena are but "manifestations of energy transformations, and all nature would DC rendered inactive and inanimate without these changes.
14. Solar Energy the Ultimate Source. — A brief consideration <^f the various sources of potential energy makes the fact mani- fest that solar energy is the ultimate source from which all other forms are directly or indirectly derived. The variations in solar heat on the earth's surface produces atmospheric currents often ^f tremendous power. This form of energy may be utilized, in *ts more moderate form, to drive the sailing vessel and the wind- n^JI, and in other ways to be of service to man. The energy of fuel is directly traceable to solar action. Through present and past ages it has been the active cause of chemical and organic
20 Power
change and growth. From this has resulted fuel supplies avail- able in the original form of wood, or in the altered forms, from ancient vegetation to the forms of coal, oil and gas, and from which a large portion of the energy utilized commercially is derived.
A brief study of meteorological conditions shows that through the agency of solar heat, and the resulting atmospheric move- ment, a constant circulation of water is produced on and near the earth's surface. Hundreds of tons of water are daily evapor- ated from the seas, lakes, rivers and moist land surface, rise as vapor into the atmosphere, circulate with the winds, and, under favorable conditions, are dropped again upon the earth's surface in the rainfall. Those portions of the rain that fall upon the land tend to flow toward the lower places in the earth's crust, where lie the seas and oceans, and such portions of these waters as are not absorbed by the strata, evaporated from the surface or utilized in plant gfrowth, ultimately find their way to theSe bodies of water to again pass through this cycle of changes which is constantly in progress. Thus we find water always in motion, and always an active agent in nature's processes. Due to its peculiar physical properties and chemical relations, it is one of the essential requisites of life, and is also of great importance in nature's processes through the energy of which it is the vehicle.
15. No Waste of Energy in Nature. — Active continuous en- ergy transformation is^a most important natural phenomenon. Changes from one form to another are constantly in progress. In nature's transformations energy is always fully utilized. As the running stream plunges over the fall, the potential energy, due to its superior elevation, is transformed into the kinetic en- ergy of matter in motion, and through the shock or impact the kinetic energy is transformed into thermal energy due to a higher temperature, which again may be partially changed in form by radiation or vaporization. Thus the quantity of energy is con- tinually maintained, while its quality or conditions constantly vary. There is, and can be, no waste or loss of energy as far as nature itself is concerned. Wasted or lost energy are terms that apply only to energy as utilized in the service of man. Nature itself never seems to utilize the entire quantity of energy from one source for the development of energy of a single form, but always differentiates from one form into a number of other forms. When the engineer therefore attempts to utilize any source of
Laws of Energy Conservation. 2i
potential energy for a single purpose, he at once encounters this natural law of differentiation and finds it impossible to utilize more than a portion of the energy used in the manner in which he desires to utilize it. Much of this loss may be due to the form of energy available, much to the medium of transformation and transmission, and much to physical difficulties which it is im- possible to overcome.
i6. Laws of Energy Conservation. — Primarily it should be fully understood and clearly appreciated that matter and energy can neither be created nor destroyed. Both may be changed in form or they may be dissipated or lost so far as their utilization for commercial needs is concerned. But in one form or another they exist, and their total amount in universal existence is al- ways the same. In any development for the utilization, trans- formation or transmission of energy, the following fundamental axioms must be thoroughly understood and appreciated:
First : That the amount of energy which can be actually utilized in any machine or system can never be greater than the amount available from the potential source.
Second: That the amount of energy which can be utilized in any such system can never be greater than the difference be- tween the amount entering the system and the amount passing from the system as waste in the working medium.
17. Efficiency. — Efficiency is the ratio or percentage of energy utilized to energy applied in any system, part of a system, ma- chine or in any combination of machines.
The efficiency df a given machine or mechanism, or the per- centage of available energy which can be obtained from a given system of generation and transmission therefore can never be greater than represented by the equation :
E E'
Efficiency or amount of available energy = — =; — in which
E equals the energy in the working medium entering the machine
E' equals the energy in the working medium passing from the machine.
18. Natural limit to efficiency. — The total energy in a workins; medium such as water, steam, air, etc., is the energy measured from the basis of the absolute zero for the medium which is being considered. For example, the average surface of I_^ke Michigan is 580 feet above sea level ; each pound of water, there- fore, at lake level contains 580 foot pounds of potential energy. This amount of energy must therefore be expended in some man-
22 Power.
ner by each pound of water passing from the lake level to the ocean level, which may be regarded as the absolute zero refer- ence plane for water power. This energy cannot be utilized at Chicago for there no fall is available. A small portion of this energy is now utilized in the power plants at the falls of Niagara. Some energy will be ultimately utilized on the Chicago Drainage Canal, where a fall of some thirty-four feet is available from the controlling works to Joliet. Perhaps ultimately in its entire course one hundred and seventy feet of fall may be utilized by the waters of the drainage canal, in which case the absolute avail- able energy of each pound of water cannot be greater than shown by the following equation :
Available energy = ^ — = ^^ = .2931, or 29.31 per cent.
With any other form of energy the same conditions also pre- vail. Consider a pound of air at 760 degrees absolute tempera- ture Fahr., and at 75 pounds absolute pressure. The number of heat units contained will be given by the equation :
Heat units = temperature X weight X specific heat.
B. T. U. = 760 degrees XIX .1«^> = 128.
To Utilize all of the energy in this air, it would be necessary to expand it down to a temperature of absolute zero and exhaust it against zero pressure. In any machine for utilizing com- pressed air, it will be necessary to exhaust it against atmospheric pressure. This will expand the air 3.10 times, and if expanded adiabatically it will have a final temperature of 474 degrees. The heat units in the exhaust will therefore be as follows :
B. T. U. = 474 degrees X 1 X .169 = 80,
and the available energy will be as follows :
228 80 48
Available energy = — = -^ = .375, or 37.5 percent.
In this case also the temperatures vary directly as the heat units, and are therefore a measure of available energy:
A -1 ui 760 — 474 rt-»r o- m
Available energy = ^r-r-r — = .375 or 3/. 5 per cent.
/oU
In the ideally perfect furnace the efficiency is somewhat higher. The fuel may be consumed at a temperature of about 4,000 Fahr. absolute, and the gas may be cooled before escaping to about 600 Fahr. In this case the possible efficiency or available energy is:
Practical Limits to Efficiency, 23
4000 660
Available energy = -^^ — = .832 or 83.2 per cent.
The above examples show, therefore, the limits which nature itself places on the proportion of energy which it is theoretically possible to utilize. For such losses the engineer is not account- able except for the selection of the best methods for utilizing such energy. The problem for his solution is, what amount of this available energy can be utilized by efficient machines and scientific methods.
19, Practical Limits to Efficiency. — The preceding equations are the equations of ideally perfect machines. Of this available energy only a portion can be made actually available. In practice we are met with losses at every turn. Some energy will be lost in friction, as radiated heat, some in the slip by pistons, or as leakage from defective joints. In many other ways the energy applied may be dissipated and lost. From this it follows :
The amount of energy which can be utilized can never be greater than the difference between the amount supplied to any given machine or mechanism, and the amount lost or consumed in such machines by friction, radiation or in other ways. Hence it follows that the efficiency of a given machine, or the percent- age of energy available, or which can be obtained from the ma- chine, can never be greater than the following:
!,« . E — fE' +E' + E"-fE"etc.). , . . Efficiency = ^ • ^ ' in which
B 8 total energy available
E* E' E" etc. = the energy lost in friction and in various other ways, in the machine or system, and rejected in the exhaust from the same.
Every transmission or transformation of energy entails a loss, hence, starting with a given quantity of energy, it gradually dis- appears by the various losses involved in the mechanism or ma- chines used. Other things being equal, the simpler the trans- niission or transformation, the greater the quantity of the orig- inal amount of energy that can be utilized.
The term efficiency as here applied represents always the ratio "Ctween the energy obtainable from the mechanism or machine and the actual energy applied to it.
Therefore the efficiency of a pumping engine is the ratio be- tween the energy of the water leaving the pump and the energy ^ the steam applied to the engine.
24
Power.
The efficiency of a hydro-electric plant is the ratio between the energy in the electric current delivered at the switch board and the energy in the water entering the water wheel.
The efficiency of the dynamo in the same plant is the ratio be- tween the energy furnished by the dynamo and the energy ap- plied to it.
If a shaft receives from an engine lOO horse power and de- livers 90, ten horse power being lost in friction, etc., the efficiency of the shaft transmission is 90 per cent.
If a steam engine receives 1,000,000 heat units from the steam it uses, and is able to deliver only the equivalent of 10,000 heat units; i. e., 7,780,000 foot pounds of work, the efficiency of the engine is only one per cent.
20. Efficiency of a Combined Plant. — In any plant or connected arrangement of mechanisms and machines for the transforma- tion or transmission of energy the efficiency of the plant is the product of the efficiency of each of its parts.
Hence, to estimate total efficiencies, the efficiency of each part may be estimated, and the combined efficiency then obtained. From the same calculation, the necessary relations between the input and the output of energy can be obtained. Thus, if a boiler has an efficiency of 50 per cent., and an engine has an efficiency of 10 per cent., the combined efficiency will be .50X.10 =.05 or five per cent.
In the following examples the loss and efficiency of the unit and the combined efficiency of the various units in the system are shown.
FIRST EXAMPLE. Example of Energy Loss in Well-Designed Steam Power Plant.
Per Cent
Lost.
Per Cent Efficiency
Furnace
Ik)iler
Steam Pipe
Enjijine
Belt
Shafting, Belts and Counter Shafts
Lathes or other Machine Tools
Percentage of original energy utilized useful work
20 15
5 94
5
40 60
80
85
95
0
95 (K) 50
Net Effi- ciency from Potential Source.
80
68
64.5 3.87 3.67 2.2 1.1
1 1
Efficiency of a Combined Plant.
25
SECOND EXAMPLE. Exampie of Energy Lo9B in Ilydfnulie Plant for Electric Lighting,
|
Per Cent Lost. |
Percent Efficiency |
Net Effi- ciency from Potential Source. |
|
|
H4^ And Tnil Raree . . |
5 20 15 6 5 8 10 20 80 |
95 80 85 95 95 92 90 80 20 |
95 |
|
Turbine. |
76 |
||
|
Gearing |
64 6 |
||
|
Shaft .'.; |
60 37 |
||
|
Belt |
57 35 |
||
|
Generator |
52.76 |
||
|
Line Loss |
47 48 |
||
|
Tranftformer .• |
87.98 |
||
|
JjAinp ...c ..*.x |
7 00 |
||
|
Percentage of original energy utilized in oaef ul work .* |
7.60 |
||
THIRD EXAMPLE. Example of Energy Lost in Steam and Electric Pumping Plant
Per Cent Loet.
Per Cent Efficiency
Net Effi- ciency from Potential Source.
Boiler and Furnaco.
bteam Pipe
Eneine
Belt
Generator.
Line
Motor
Pomp.
Suction and Discharge Pipe
PercentRge of original energy utilized in oseful work
30 5
90 5 20 10 10 26 20
70 95 10 95 80 90 90 75 80
70
66.6 6.65 6.82 5.05 4.55 4.09 3.06 2.45
2.45
21. Capacity of Each Part of a System Not IdenticaL — In each of the transmission systems outlined above a much larger amount of energy enters the first unit of the system than is de- livered by the last. Each unit in the system receives a decreas- ing amount of energy.
In consequence, the first units in the system must be of greater proportional capacity, and in practice each unit must be selected of a size or capacity suited for its position in the system. Thus in the first example, for each 100 units of energy rereived by the furnace, the engine receives but 64.5, and the shafting but 4.
26 Power,
aa. The Analysis of Losses. — In estimating power losses the loss in each step from the generation to the utilization of the power should be carefully examined. Four steps may ordinarily be considered in any system :
1. Generation of power from potential source.
2. Conversion of power into form for transmission.
3. Transmission of power.
4. Utilization of power.
An analysis of the first three items is shown in Table 11. In Table III is shown the ordinary maximum and minimum ef- ficiencies obtained from various motors and machines in prac- tical work. Higher efficiencies are sometimes obtained under test conditions where great attention is g^ven to secure favorable conditions, and, in many places where careless work is permitted, neglect and unsatisfactory conditions will result in much lower efficiencies than the minimum shown.
as. The Losses in a Hydro-electric Plant — ^To emphasize and point out in greater detail the various losses encountered in the generation and transmission of energy, especially as applied to hydro-electric plants, attention is called to Fig. 16. In this diagram is traced the losses from the potential energy of the water in the head race of the power plant to the power avail- able at the point where it is used. In each case considered it is assumed that 1,000 horse-power of energy is applied to the par- ticular work considered.
First, consider the transmission of power for traction pur- poses. If a certain head is available when no water is flowing in the raceways, that head becomes reduced at once when the wheels begin to operate. A certain amount of head is also lost in order to overcome the friction of flow through raceways, racks and gateways. In the problem here considered it is assumed that the above losses are five per cent, of the total energy avail- able in the head-race, and that this loss occurs before the water reaches the turbines : hence, 95 per cent, of the potential energy is available at the turbine. The turbine loss is here assumed to be about 20 per cent. First-class turbines under three-quarter to full load conditions, will commonly give 80 per cent, efficiency, or a little better.
Professor Unwin, in his "Development and Transmission of Power," page 104, gives the following percentage of loss in tur- bines :
The Losses in a Hydro-Electric Plant.
27
Shafting, friction and leakage 3 to 5 per cent
Unutilized energy 8 to 7 per cent
Friction in shaft, guides and passages 10 to 15 per cent.
Total loss of energy IG to 27 per cent
TABLE II.
Method of Generation.
z
J* 0
z
0
•-i
X
< H
z
<^
z
0
<
z
Fuel.
'Internal Coinhustion Engine
Gas — Oil Engine losses.
( Direct (Vacuum Pump) C Furnace. Steam ] i Boiler.
( Indirect I Piping.
(Direct (Ram) Ram losses. Indirect (Wheels) ) ^WoL^.'-'
h
a H
2
''I
a
o
I
i
i
<
h O
c o
' i
Water Power.
Minor Sources.
Electric (Primary Batteries) .
Wind (Mills)
Waves (Motors)
^Sun Heat (Solar Engines) . . .
" Various mechani- cal and other losses due to method used.
" Internal Combustion Engine Included in engine
Steam.
Electrical .
Engine and con- nection losses.
Dynamos and wire losses.
Hydraulic Pump 1
Pneumatic Compressor losses.
f Direct connected,— Shaft f
Mechan- i Cables, Ropes, Ch:iins ) Various losses due
ical I Electric ] to method need.
tCombination (,
(Entrance head. Pipe friction. Mmor losses. Connections.
Electrical .
Pnenmatic .
fTran former losses. j Wire losses. I Motor losses. [Connections.
(Pipe friction. Air cooling. Motor lossep. Connections.
The Losses in a Hydro-electric Plant. 29
The next loss shown on the diagram is the loss in transmitting the energy through the bevel gear and the shafting to the gen- erator. The loss in gearing, shafting, etc, is shown as 10 per cent., which is probably much less than actually takes place in most plants of this kind, but may be considered as representing the results of good practice.
The loss in the transformation of power in the generator is given as 8 per cent. The generator is an alternator, and the cur- rent generated would be at about 2,300 volts. This current must be raised to a higher voltage, by means of transformers, for long distance transmission. These transformers would g^ve an efficiency of about 96 per cent. The line loss is dependent on the size of the copper used, but would probably not exceed 10 per cent. At the distributing point, where the energy is to be used, the high voltage current must be transformed again into suit- able voltage for distribution. The same energy loss is estimated, for these transformers. If the current is to be used for traction purposes, it will be necessary to convert it into direct current by means of a rotary converter, the efficiency of which is esti- mated at 92 per cent. The voltage from the general distribution system would probably be too high for direct use in the rotary converter, and would have to be transformed to a lower voltage before passing into the converter. A loss of about 6 per cent., therefore, should be allowed for this transformation.
The current from the rotary converter is subject to a line loss which may be again assumed at 10 per cent. The loss in the car vQxAor may be estimated at 7 per cent. The percentage of loss and the percentage of efficiency for each unit in this generation and transmission system is based, of course, on the actual energy supplied by the unit next previous to it in the system, so that the percentages mentioned are not based on the total potential power available in the head-race but on the power actually reach- ing the machine.
In the solution of any actual problems of this character it is necessary to determine the efficiencies of the various units of 4c plant under the condition of actual service. The efficiency will be found to vary under various conditions of load. It may therefore be desirable to determine the probable losses under various working conditions.
In the selection of the various machines which are to form a part of such a system of transmission, the choice should be
30
Power,
based on an effort to establish a plant which will give the maxi- mum economy when all conditions of loading are considered. The losses in the transmission of power for traction purposes, as shown on the diagram, may be traced through in tabular form as follows:
Total Energy Available.
Per Cent Lose.
Per Cent Efficiency
1,000 HOBSK
Power.
LoBsin horsepower
Head race
Turbine
Shaft and gearing
Generator
TransformerB.
Transmission line.
.Step-down Transformers. Secondary Transformers.
Rotary Converters
Line
Traction Motor
5
20
10 « 4
10 4 6 8
10 7
95 80 90 92 96 90 96 94 92 90 93
50
UK)
76
64.7 25.2 60.4 21.7 31.3 39.3 45.1 28.4
Power utilized for operating the cars, or 37J per cent of the original energy 374 .5 Horse Power.
In the generation and transmission of power for lighting pur- poses, the losses will be similar to those above mentioned, up to and including the step-down transformers at the point of dis- tribution. In this case, however, no secondary transformers or rotary converters would be necessary. The only loss between the step-down transformers and the light will be the line loss assumed at 5 per cent. The loss in the individual transformer for the light will be about 8 per cent., leaving the available en- ergy for actual use in the lamp at about 456.2 horse power, or a little less than 46 per cent, of the total energy in the head-race.
In the case of the utilization of this energy for manufacturing purposes, the loss would be the same up to and including the step-down transformers at the point of distribution. The line loss in the distribution from the transformer house to the manu- facturing establishment may be assumed at 5 per cent. The motor, if properly selected, may be run at the line voltage, and no transformer losses need be considered. The motor efficiency is here shown at 92 per cent., although in most cases the per- centage of efficiency would be considerably less.
The belt loss in transmitting the power from the motor to the line shafting is estimated at 5 per cent.
Efficiency of Generators and Motors. Tablb m. — Ordinarif Ejfloiency of Oenerators and Motors,
31
Glass or Maceinsbt.
Cent at Fcll Load,
mum.
Mini- mum«
Water Wbeds.
CoDdesaiii^ - - * - { gleam Engines . { **""
Kon-CondeneSng ) Steam Engines.. }**""
BefttEngmei. i> *#««
Gteom Air Compreadon .
^llotor
Bectrical Macbintry . , .
Tnmamitting Mechaix-
TmuDusaioD Methods.
f Overaliot WJieels. ♦ -
Bn^flSi Wheels' .
Undershot Wheels .
Tarbhiw
Impulse Wheela. > . .
i Boilers ^ - « . < \ Steam Pipe
f Triple Expansion Corlij^ Compfitind GorUBS .Simple CorliBa Compound High Speed * .
f Cbtnponnd Corllea
1 Simple CorlisB Compound tliurh Speed, Simple Uif^h^peed. . ., , Simple Slide Val?e . . . . .
iGas or Oil Engines . Diesel Motor
'Compound Con. Corliss*
Simple Con. Corliss .
Simple Corliss*,***. ***, High Pressure
^SmallStraijfht Line. ....
5 Air^ cold
( Air J reheated.
r Dynamos .... ! Motor, large.. \ MotoFi email , [Trans I or user..
fBelt .•.,..
Hope . *
Cable. .- ...,,
Direct connection
Shafting **
Gearing -
Bevel Gefiring
Pneumatic r per mile , Hydraulici |^r mile * ElectriCr usual
75
iS5 40 S5 85
75
18 15 12 12
12
7 7
20 30
12 2
3
60 7U
92 90 85
95
^7 tm ln>
S5 75
97
ys
»5
65 60
25 60 75
60
75
15 12
10 10
10
7 7 6 5
10 25
10 7 5 3 IS
SO 60
SO 80 75 50
85 90
75 95 70 50 60
9^ 90 85
32 Power.
The shafting necessary for the general distribution of power through the factory is estimated at 75 per cent, efficiency.
The belt loss from the shaft to the individual machine is esti- mated at an additional 5 per cent., leaving the total energy avail- able for use in the machine at 308.8 horse power, or about 31 per cent, of the original energy in the head-race.
It should be noted that in each of the three transmission sys- tems mentioned above, the actual power utilized at the point of application is less than half of the energy available in the head- race. It is the function of the engineer to see that these losses are reduced to the greatest practicable extent. These losses must be limited in both directions. They must not be too great, nor too small. Tliey must be adjusted at the point where true economy would dictate. This limit is the point where the cap- italized value of the annual power lost is equal to the capitalized cost of effecting further saving. In other words, true economy means the construction of a plant that will save all the power or energy which it is financially desirable to save, and will per- mit such waste of energy as true economy directs.
24. Units of Energy. — Energy is known by many names and exists in many forms which seem more or less independent. The principal forms of energy are measured by various units. Those most commonly considered in power development and trans- mission are as follows:
Work is energy applied to particular purposes. In general it is energy overcoming resistance, mechanically it is .the exertion of force through space.
Power is the rate of work, or the relative amount of work done in a given space of time.
The unit of work is the foot pound, or the amount of work required to raise one pound one foot. One pound raised one foot, one-tenth pound raised ten feet, ten pounds raised one- tenth of a foot, or any other sub-division of pounds and feet whose product will equal one requires one foot-pound of work to perform it.
The unit of power is based on the unit of work, and is called "horse power.'* It is work performed at the rate of 550 foot pounds per second, or 33,000 foot pounds per minute.
Units of Heat. The unit of heat is the amount of heat which will raise one pound of water from 39 degrees Fahr. to 40 degrees Fahr. at atmospheric pressure. It is called the British Thermal Unit, and is indicated by the initials B. T. U.
Conversion of Energy Units. 33
Electric Unit. The unit of quantity of electricity is the coulomb. One coulomb per second is called an ampere, and one ampere un- der a volt pressure is equal to a watt, the unit of electric power.
Water Power. Water power is the power obtained from a weight of water moving through a certain space. In water power the unit of quantity may be the gallon or the cubic foot ; the unit of head may be the foot; and the unit of time may be the second or minute. The weight of water, unless highly mineralized, at ordi- nary temperature, varies from 62.3 to 62.5 pounds per cubic foot. As these weights vary from each other less than one-third of one per cent., the difference is insignificant in practical problems where the errors and uncertainties are often large. In the further discus- sion of this subject, therefore, the weight of 62.5 pounds is used as the most convenient in calculation.
Steam Power. The unit of steam power in ordinary use is the pound of steam, its pressure, and rate of use. It is, however, based on the heat unit, and must be so considered for detailed examina- tion.
Definite quantities of work are also designated by the **horse power hour," equivalent to 1,980,000 foot pounds, and the "kilowatt hour," equivalent to 2,654,150 foot pounds.
The pound of steam may be considered as containing an aver- age of 1,000 British thermal units, which may be utilized for power. This is equivalent to 778,000 foot pounds.
35. Conversion of Energy Units. — The various forms of energy as expressed by the units named are convertible one into another in certain definite ratios which have been determined by the most careful laboratory methods. In considering these ratios, however, it must be remembered that, as shown in the preceding examples, in the transformation from one form of energy into another the ratios given cannot be attained in practice on account of losses wliich can not be practically obviated. Such losses must be, in good practice, reduced to a minimum, and the ratios given are, therefore, the end or aim toward which good practice strives to at- tain as nearly as practicable when all conditions and facts are duly considered.
Energy must be considered in two conditions as well as in the above named forms, viz.: passive and active or potential and kinetic
Potential energy is energy stored and does not necessarily in- volve the idea of work. Kinetic energy is energy in action and
34 Power.
involves the idea of work done or power exerted and for its meas- urement must be considered in relation to time.
The most common units of potential energy and their equiva- lents are as follows: The footpound (one pound raised one foot).
=1/62.5 or .016 foot cubic foot (of water), =1/8.34 or .12 foot gallon (of water). =1/2655.4 or .0003766 volt coulombs. =1/778 or .001285 British thermal units. The foot cubic foot (one cubic foot of water raised one foot). =62.5 foot pounds. :=7.48 foot gallons. =.08 British thermal units. =.02353 volt coulombs. The foot gallon (one gallon of water raised one foot) s=8.34 ^^^^ pounds. =.01072 British thermal units =.00314 volt coulombs. =.1334 foot cubic feet. The volt coulomb
=2655.4 foot pounds. =42.486 foot cubic feet. =318.39 foot gallons. =3.414 British thermal units. The British thermal unit =778 foot pounds. =12.448 foot cubic feet. ==93.28 foot gallons. =.2929 volt coulombs. Quantities of energy available, used or to be used, and eithe«" potential or kinetic may be measured in the above units.
When the rate of expenditure is also stated these units express units of power. Some of the equivalent values of power are as fol- lows, those most commonly used being printed in black-face type : The horse power
=1980000 foot potinds per hour. =33000 foot pounds per minute. =550 foot pounds per second. =31680 foot cubic feet per hour. =528 foot cubic feet per minute.
Conversion of Energy Units, 35
=8.8 foot cubic feet per second. =237600 foot gallons per hour. =3960 foot gallons per minute. ==66 foot gallons per second. ^=74^ watts.
=2545 British thermal units per hour.
. ^=42.41 British thermal units per minute.
=.707 British thermal units per second.
The foot pound per minute
=1/33000 or .0000303 horse power.
=1/778 or .00129 British thermal units per minute;
=.0226 watts.
=i/8.34=.i2 foot gallons per minute.
=i/62.5=.oi6 foot cubic feet per second.
The foot cubic foot per minute =62.5 foot lbs. per minute. =i/528=.ooi89 horse power. =1412 watts.
=748 foot gallons per minute. =.0803 British thermal units per minute.
The foot cubic foot per second
=3750 foot lbs. per minute.
=62.5 foot lbs. per second.
=i/8.8=.ii36 horse power.
=^48.8 foot gallons per minute.
=7.48 foot gallons per second.
=4.820 British thermal units per minute.
=.0803 British thermal units per second. Th* watt
=44.24 ft. lbs. per minute.
=.00134 horse power.
=.0568 British thermal units per minute.
=5.308 gallons feet per minute.
•=.7089 ft cu. ft. per minute.
Thf British thermal units per minute ^78 ft. lbs. per minute. ^=.02357 horse power. =17.58 watts. =93.28 ft gal. per minute. =12.48 ft. cu. ft per minute.
36 Power.
26. Motion in General — In moving a body against a given force or resistance the work done in foot pounds is the product of the space passed through (in feet) and the resistance (in pounds). Thus in raising a ten-pound weight 100 feet high, 1,000 foot-pounds of work is performed. But this is not the only work performed. To pro- duce motion in a body or to bring a body to a state of rest neces- sitates a transfer of energy. For all moving bodies are endowed with kinetic energy — the energy of motion — and this energy must be given to them to produce motion, and must be taken from them to produce a state of rest.
Hence, Newton's laws of motion:
1. "Every body continues in a state of rest, or of uniform mo-
tion in a straight line except in so far as it may be com- pelled by impressed forces to change that state."
2. "Change of motion is proportional to the impressed force
and takes place in the direction of the straight line in which the force acts."
3. "To every action there is always an equal and contrary reac-
tion."
The acceleration of gravity is the acceleration due to the weight of a body acting on its mass.
The weight of a body W (on account of centrifugal effect of the earth's revolution) varies, being least at the equator and greatest at the poles. From Newton's second law it follows that the accel- eration in motion designated by g and caused by the weight of any body acting on its mass will be proportional to its weight, i. e., g^= constant X W, and hence the weight of a body divided by the ac- celeration will always be constant. This constant quotent desig- nated by the letter M is termed the mass of the body.
(.)M=:^
Let W=The weight of a body. M=Mass. g^=Acceleration due to gravity=velocity of a falling body at
end of first second, and is ordinarily taken as 32.2 ft.
per sec. per sec. A=Acceleration of moving body=velocity of body at end
of first second. W'^=Weight acting. W"=Weight acted on.
Kinetic Energy. 37
V=Velocity at end of time L
Va=Average velocity.
t?=Time force has acted.
S=Space passed through.
h=Height passed through by falling body.
V'=Initial velocity.
S'=Initial space passed through.
27. Uniform Motion. — In uniform motion the moving body passes through equal spaces in any equal divisions of time.
Hence by definition :
The space passed through (S) equals the product of the velocity (V) and the time (t).
(2) S=Vt
(3) V=-|
28. Uniformly Varied Motion. — If the velocity of a body is in- creased or diminished uniformly, the motion is termed uniformly varied motion and is termed uniformly accelerated motion in the first case and uniformly retarded motion in the latter case. In all such cases the following relations hold:
(4) A=^g.
(5) V=At=^g t (6)Va=4i (7)S=Vat=--=- (8) V=VTXs.
With falling bodies:
S=h. A=g. From which equation (8) becomes
(9) V=V 2gh, ^hc well known basis of hydraulic calcu- lations.
(10) Work==W h=W VV2g=-=M VV2.
>9. Compotmd Motion. — ^\Vhen bodies are already in motion and additional force is applied, the following relations hold :
(11) V=V'+At.
(12) S=S'+V't+^
38 Power.
30. Graphical Representation of the Laws of Motion. — In each
case—
The vertical ordinates represent velocity
Abscissas represent time.
Areas represent space passed through.
|
SPACE |
||||||
WNiroitM Monof*
|
^.^-.-'-^ |
a»Ace |
||||
|
^^^^t*"*^ |
|||||
|
*\.^\ |
TIM« MNirOAM ACCBUCRATCO MOTION
^
00*4^0UN0 MOTION - UMI^tMI-lkV ACCCkCnATCO
>Htr** IMITIAU VCLOCiTV
^T =s constant S = Vt
V = At = ,^ gt
W
v. = 4^
S = V^t =
V=t/2AS"
At«
2 " lA
V = V + At
At*
8= S' + V t + ^
Pig. 17. — Graphical Representation of the Laws of Motive.
31. Transformation. — ^The transformation of potential to kinetic energy is well illustrated by water acting upon a water wheel. The energy in a body is always constant whatever its form, except as said energy be given up to other bodies or lost and wasted in vari- ous ways. Consequently the sum of the potential and kinetic en- ergies in any body is a constant quantity unless the difference be accounted for by energy loss or transfer as above noted.
Water that has fallen to sea level has lost all the energy it may have once possessed, its energy having been expended in perform- ing some kind of work.
If, in a hydraulic plant, we have an available fall of 8.8 ft. every cubic foot of water falling each second should produce 350 ft. lbs. of work per second or one horse power. After the water has passed through a well designed turbine it flows sluggishly away, having used up nearly all its energy in the turbine to which
Literature. 39
It has transferred its energy. If, however, on account of bad de- sign the water flows away at a rapid rate, say at lo feet per second, the head lost, fc=vV2g i. e. h=ioV644=i-55 ft. of vertical fall. Under these conditions the energy due to this fall still remains in the water, after it has left the wheel, and is lost, the loss being 17.8 per cenL of the original energy.
LITERATURE.
1. Thurston, Robert H. Conversion Tables of Weights and Measures. NeW
York. J. Wiley ft Sons. 1883.
2. Oldberg, Oscar. A Manual of Weights and Measures. Chicago. O. J.
Johnson. 1887. 8. Everett, J. D. Illustrations of the C. G. S. System of Units. New York: MacMillan ft Co. 1891.
4. Anderson, William. On the Conversion of Heat into Work. Discussion
of energy conversion. London. Whittaker & Co. 1893.
5. Unwin, W. C. On the Development and Transmission of Power. Long-
man ft Co. London. 1894.
6. Oswald, Wilhelnu Manual of Physics, — Chemical Measurements. New
York. The MacMillan Co. 1894.
7. Peabody, Cecil H. Tables of the Properties of Saturated Steam. New
York. J. Wiley ft Sons. 1895.
8. Richards, Frank. Compressed Air. New York. J. Wiley ft Sons. 1895.
9. Bolton, Reginald. Motive Powers and Their PracticaJ Selection. New
York. Longmans, Green & Co. 1895.
10. Holman, Silas W. Matter, Energy, Force and Work. New York*. The
MacMillan Co. 1898. IL Kent, Wm. Notes of the Definition of Some Mechanical Units. Am.
Asso. Adv. of Sci. 1898. See also Eng. News, Vol. 40, p. 348. U Mead, Daniel W. Commercial Transformation of Energy. Trans. 111.
Soc. Eng. 14th report, 1899. U. Reeve, Sidney A. The Steam Table. New York. The MacMillan Co.
1903. 11 Kohlrausch, F. An Introduction to Physical Measurements. New York.
D. Appleton & Co. 1903. 15. Carpenter. R. C. E3xperimental Engineering. New York. John Wiley
ft Sons. 1903. 11 Herwig, Carl. Conversion Factors. New York. J. Wiley ft Sons. 1904. 17. Smithsonian Institution. Physical Tables. 3d Edition. 1904. 11 American Institute of Electrical Engineering. Report of Committee on
Standardization. 1907. Proc. Am. Inst. E. E. Vol. 26, pp. 107&-
llOC
CHAPTER IIL
HYDRAULICS.
32. Basis of Hydraulics. — ^The science of hydraulics is an empir- ical, not an exact science, but is based on the exact sciences of hydrostatics and dynamics. Its principal laws are therefore founded on theory, but on account of the multitude of modifying influences and of our necessarily imperfect theoretical knowledge of their varying characters and extent, the formulas used must be derived .from or at least modified by observation and experience and can- not be founded solely on theoretical considerations. The condi- tions under which hydraulic laws must be applied are so varied in both number and kind that the application of the laws must be modified to suit those various conditions and for this reason their successful application depends largely on the practical experience of the engineer.
In the following discussion the letters used will have the signifi- cance shown below :
E=Energy (abstract).
P=Horse power.
W=Total weight of water.
h=The total available head in feet
hi=The velocity head.
h2=The entrance head or influx head.
hs=The friction head.
q=The quantity of water (in cubic feet per second).
w=The weight of each unit of water (cu. ft.=62.5 lbs.).
a=Area (in square inches) against which pressure is ex- erted.
s=The space (in lineal feet) through which the area moves under pressure.
v=The velocity of flow (in feet per second).
gi=Acceleration due to gravity (32.2 feet per second per sec- ond.)
t=The time in seconds.
33. Mathematical Expression for Energy. — Mechanically, energy is the exertion of force through space. The amount of available
Mathematical Expression for Energy. 41
energy of water that may be theoretically utilized is measured by its weight (the force available) multiplied by the available head (the space through which the force is to be exerted), 1. e., (i) E=: Wh. From this it will be noted that the energy of water is in direct proportion to both the head and quantity. Tliis energy may be exerted in three ways which may be regarded as more or less distinct but which are usually exercised, to some extent at least, in common. The exertion of this energy in the three ways men- tioned, expressed in terms of horse power, are as follows :
First: By its weight which is exerted when a definite quantity of water passes from a higher to a lower position essentially with- out velocity. This method of utilization is represented by the equation
^ ' 560
Second: By the pressure of the water column on a given area exerted through a definite space. This method of utilization is rep- resented by the equation ^
^'^ ^ 650r"
Third: By the momentum of the water exerted under the full velocity due to the head. The energy of a moving body is repre- sented by the formula :
Wv»
(4) E = ^
The equation for the horse power of water under motion is there- fore represented by the equation :
^ ' 560 X 2g
An analysis of these formulas will show that under any given conditions the theoretical power exerted will be the same in each case.
34. Velocity Head (hj). — It has already been pointed out (chap- ter II) that energy must be expended in order to produce motion in any body and that the head (hj necessary to produce a ve- locity (v) is
(«) K = S
This proportion (hj/h) of the available head h has to be ex- pended to produce and keep in motion the flow of water. This teid (hi) is not necessarily lost (it has simply been converted into
42 Hydraulics.
kinetic energy, and it may be re-cohverted into potential energy by correct design or it may be utilized in some other way, as, for example, by pressure or impact in hydraulic motors).
Whatever head (hx) is necessary to maintain the velocity (v)^ with which the water leaves the plant, will be lost to the plant. It is, therefore, desirable to keep v at this point as low as may be found practicable when other conditions are considered.
Sudden enlargements or contractions in pipes or passages may wholly or partially destroy the velocity and cause the permanent loss of the corresponding head (hj).
In this case an additional amount of the available head (h^) must be used to again generate the velocity (v) required to convey the water through the remainder of its course. Gradual change in the cross-section of all channel conduits or passages is, therefore, de- sirable in order that the transformation from kinetic to potential energy, and the reverse, shall be made without material loss.
Not only the head (hj) but still other portions of the total avail- able head (h) may be lost in the channels and passages of a ma« chine or plant by improper design.
35- Entrance Head. — The loss of head (hg) which occurs at en- trance into a raceway, pipe or passage may be called the "influac head." The amount of this loss differs considerably with the shape- and arrangement of the end of the pipe or passage. In general, the- influx head may be determined by the formula:
(7) h, =1^- — 1 |-2^(Merriman*8 Hydraulics, Art. 66)
In this formula the coefficient can be obtained from table IV, lit which the variations of the constant under various conditions, with reference to a pipe inlet, are shown, and from which it will be noted that its magnitude depends on the shape and arrangement of the inlet,
TABLE IV. Arrangements of a pipe uUet with corresponding coefficients.
|
Arrangement of Pipe. |
c |
^- |
|
A. Proiectinflf into reservoir |
.716 .825 .950 .990 |
.956 |
|
B. Mouth flush with side of reservoir |
.469 .106 |
|
|
C. Bell shaped month ' ;j^°™ |
.020 |
|
Submerged Orifices.
43
To find the value of h^, the value of -i- — i corresponding to the given conditions, is to be selected from Table IV and substituted in formula (7). The ordinary arrangement of suction pipes is for
a square ended pipe to project di- rectly into the suction pit. In res- ervoirs the pipe may be flush with ry..'.,,'..'ym. thi^ masonry or project as in the
,, , ^ ^^g^ ^£ suction pipes. With condi- tion (A) formula (7) becomes
^\}$f:\/ik\iykttji^^
(8)
h, = .956
2«
|
0m |
||
|
^-^ - -^ ^^ ^ |
•**>* |
|
|
' — -^— _-^ir_ ^T-- "-i: |
||
|
^2 |
||
The value of h, can be readily obtained from equation (8), as it will be 95.6 per cent, of the veloc- 3 ity head.
With the mouth of the pipe flush, with the side of the reservoir the loss would be 46.9 per cent, of the velocity head, and with a bell mouth pipe the loss would be de- creased to from two per cent. to. ^10.8 per cent, accoi'ding to the de- sign of the bell mouth entrance.
The arrangements of inlet pipes as referred to in Table IV are ^^" ^^- shown in Fig. 18.
36. Submerged Orifices. — A similar loss is sustained in the flow through gates or submerged openings or in the flow past any form of obstruction which may be encountered by the water in its flow through channels, pipes or other forms of passages. Openings or obstructions with square edges may cause a serious loss of head which may, however, be reduced.
First: By increasing the opening, thus causing a reduction in velocity and consequently a saving in head, or
Second : By rounding the corners of the opening or obstruction,, thus causing a gradual change in velocity and a partial recovery of any head necessarily used for creating greater velocity through such passage or past such obstruction.
But few experiments have been made on submerged orifices and tubes. These indicate a coefficient of about .62 for complete con- traction which increases to .98 or even .99 with the contraction
44
Hydraulics,
completely suppressed. Certain experitucnts have recently been made at the hydraulic laboratory of the University of Wisconsin, on the discharge through orifices and tubes four feet square and of ■ various thicknesses or lengths and with various conditions of con- traction. The values of the coefficients as determined in these ex- periments with various losses of head and various conditions of entrance, are shown in Table V.*
The FormM of Entrance and Outlet Used for the Tubes in tM &Fperimeni
were as follows:'
A- Entrfincej all corner 90**
OutleL; tube projecting into wftt«r on down stream side of bolkbesuL a Entrance; contraction eupptet^sed on bottom.
Outlet; ttibe projecting; into water on down stream side of bulkhead. b Entrance; contractioD Buppres^ied on bottoii and one side. ^
Outlet; tube projecting into wat«r on down stream aide of bulkhead* ^M C Eiiinmce; contraction sup pressed on bottom and two sides.
Outlet; tube projecting into water on down etream side of bulkhead. d' Entrance; contract] on euppreseed on bottom and two eidm*
Outlet: square cornet^ with bulkhead to sides of channel presenting J the return current alon^ the aides of the tube. d Entrance; contraction suppre^sBed on bottom, two sides and top*
Outlet; tube projecting into water on down itream side of bulkhead.
I
From this tabic it will be noted that a partial suppression of con- traction does not always improve results, and that by complete sup- pression, the coefficient is greatly increased with a corresponding decrease in head lost. fl
37, Friction Head (h^) — In raceways and short pipes the velocity head (hj) and the influx head (h^) are frequently the sources of the /i^eatest losses of head. In canals and pipes of considerable length the friction of flow may become the most serious sotirces of energy loss>
The principles of flow in such channels may be considered as follows :
First Principle: In any fnctionless pipe, conduit^ channel or pas-^ sage of any length the flow may be expressed by the formula;
(»)
lll = ^ or T ■* V2gh
In practice, however, we find friction is always present and a friction factor must be introduced in the above formula in order to
i
♦Prom experiments by Mr, C, B. tilt University of Wisconsin*
Stewart at tlie Hrdraulle Laboratory ot
J
|
^^^ |
Friction Head. 45 ■ |
|||||||
|
1 represent the actual conditions of practice. (9) therefore becomes: ^| |
||||||||
|
(10 |
hj=q' Z_ or ▼ » c VSgh ^^^B |
|||||||
|
TABLE V. ^^H |
||||||||
|
Value of the Co^Usieni 0/ Di^^arg^ for flow through horUontal mibmergoi ^^H |
||||||||
|
lufie; 4 f^^t square, for vanous lengths, lanes of head artd forme of enfraitee ^^^| |
||||||||
|
and ouikL |
^H |
|||||||
|
Lo«of |
Forms of En- |
Length of tube, in feet ^| |
||||||
|
■ |
||||||||
|
bead^b» |
tt%UQ& |
O.Sl |
0.62 |
1.25 |
2,60 |
6.00 |
10.0 |
14.0 ■ |
|
in feet* |
and Ontbt |
■ |
||||||
|
Valne of the coefficieat, c. ^| |
||||||||
|
.0§*. «*•»•<• |
A |
.650 |
.672 |
.769 .742 |
.807 .810 |
.621 |
.838 H .848 ■ |
|
|
b |
,740 |
.7139 |
.S32 |
.862 H |
||||
|
c |
,H34 |
.7139 |
.875 |
.690 H |
||||
|
c' |
,87& ■ |
|||||||
|
d |
.948 |
,943 |
.940 |
.927 |
,931 H |
|||
|
: .10.-, |
A |
.611 |
.631 |
.647 |
.718 |
,783 |
,780 |
.79^ H |
|
a |
.636 |
.698 |
.771 |
.801 ■ |
||||
|
b |
.685 |
.718 |
.791 |
.813 ■ |
||||
|
0 |
.772 |
.718 |
.828 |
.841 V |
||||
|
c' |
M^ ^ |
|||||||
|
d |
M2 |
.911 |
.899 |
.892 |
M% |
|||
|
.IS,*,*..... |
A a |
,609 .630 |
,628 |
.644 |
.70S .689 |
,75a .767 |
,779 |
,794 .803 |
|
b |
.677 |
.708 |
.767 |
,814 |
||||
|
c |
.765 |
.708 |
.828 |
.839 |
||||
|
c' |
.82^ |
|||||||
|
d |
.936 |
,010 |
,899 |
.893 |
.894 |
|||
|
M |
A a |
.609 |
.630 |
.647 |
.711 .694 |
.788 .777 |
.794 |
,809 .814^ |
|
b |
,678 |
,711 |
.796 |
.8:^3 |
||||
|
c |
.771 |
.711 |
.838 |
.85a |
||||
|
c' |
,84^ |
|||||||
|
d |
1 .048 |
.923 |
,911 |
.906 |
.905 |
|||
|
1 J5 ,, |
A a |
.610 .634 |
,631 |
.662 |
.720 .705 |
.782 .790 |
.812 |
.828 |
|
b |
.683 |
.720 |
.809 |
|||||
|
e |
.779 |
.720 |
.854 |
|||||
|
d |
,966 |
.938 |
.028 |
|||||
|
[,» — ^ |
A b e d |
.014 .639 .689 .788 .9i$4 |
.639 |
.660 |
.731 |
.796 |
,832 |
,66a |
|
^^^ |
46
Hydraulics.
The formulas (9) and (10) represent one of the important funda- mental principles from which many hydraulic formulas arc de- veloped.
Second Principle: In any pipe, conduit, channel or passage we may fairly assume:
First: From axiomatic considerations the resistance to the flow of water may be regarded as directly proportional to the area of The surface in contact with the water.
Second : From observed conditions the resistance is found to be directly proportional to the square of the velocity of flow.
Third: Experience leads to the conclusion that the resistance to flow is inversely proportional to the cross-section of the stream.
These conclusions may be expressed by the following equation:
P . __ (Yelocity)*X area of cont-act
"" area ot section
Fig. 19.
Tlie area of the surface of a channel is the product of the wetted section or wetted perimeter (p) times the length of the section, or» to p X 1. (See Fig. 19.) The velocity is represented by v and the cross-section by a. Hence, from the above considerations, we may write for the friction head :
(11) hg = ^^-^ and by transposition v* = -— ^
That is to say, the square of the velocity is in direct proportion to the area of the section and to the friction head and inversely proportional to the wetted perimeter and to the length of the sec- tion.
In practice it is found that there are numerous factors which
Kutter's Formula. 47
affect the theoretical conditions, as above set forth, which must therefore be modified in accordance with the conditions which ob- tain. In formula (11) therefore it is necessary to apply a coeffi- cient (c') which represents the summation of such other influences. The form in which this last equation is ordinarily written is
Ordinarily this form is somewhat abbreviated by substituting for a/p the hydraulic radius which represents this ratio. That is to say,
area of cross section __ a _ wetted perimeter ~ p ~
The "hydraulic radius" is also sometimes termed the "mean <icpth" or the "mean radius." For the ratio of the resistance head to the length of section the equivalent slope s is substituted. That is to say:
Resistance head __ h, _ Length of section "* 1 ""
With these substitutions the formula (12) assumes the final form of:
(13) V = ci/rs"
In the use of this formula three factors must be determined by measurement or estimate in order to derive the fourth, v, r and s <^ be determined experimentally or measured directly. The factor c is the most difficult to ascertain as it depends upon a very ^cat variety of conditions which can only be known and appre- ciated by a thorough knowledge of the conditions under considera- tion, and by comparison of such conditions with similar observed conditions. Various attempts have been made to derive a formula which would give the value of c in accordance with the varying conditions. The principal formulas for the values of c are those of Ganguillet and Kutter and of Bazin. Ganguillet and Kutter's form- ula for the value of c is as follows :
38. Kutter's Formula. —
4i.fl + l:^ + 2:«^l
a*) c = "
,+(..e+<L«^)_,L-
From this formula it will be seen that Ganguillet and Kutter as- sume c to vary with the slope, with the square root of the hydraulic ^dius and with a new factor "n" which is termed the coefficient
48
Hydraulics.
VELOCITY "^V -iM FEET PER SECOND
-J
Fig. 20.
Kutter's Formula*
49
Fig. 21.
so Hydraulics.
of roughness. The value of this coefficient as determined by these
experiments is as follows: For large pipe with the following characteristics:
Exceptionally smooth cast iron pipe n= .Oil
Ordinary new cast iron or wooden pipe .0125
New riveted pipes and pipes in use .014
Pipes in long use .019
For open channels of uniform sections :
For planed timber sides and bottom n= .009
For neat cement or glazed pipe .01
For unplaned timber xyi2
For brick work .013
For rubble masonry joiy
For irregular channels of fine gravel X)2
For canals and rivers of good section .025
For canals and rivers with stones and weeds . . . .030
For canals and rivers in bad order .035
The relation of the above factors may be determined by the dia- grams, Figs. 20 and 21. If with a known slope and a known value of n (for example, let n=o.i5 and s=.ooo2, as at A, Fig. 20), a straight line be drawn on this diagram to the scales of the hydraulic radius (at B) it will show at the intersection with the scale for the coefficient (c) the relative value of this coefficient for these condi- tions, or with a known c and the known hydraulic radius and the given slope the value of n of a channel may be likewise determined. After a line has once been drawn connecting these four known values the velocity can be determined by drawing a line from the hydraulic radius scale (B) to the proper point on the scale of slope or hydraulic gradient at x, and then from the point of intersection of the line A B with the coefficient scale at x' drawing a line par- allel with xB which will intersect the velocity scale at the point B', giving the velocity (in this case equal to 1.34 ft. per second). These formulas only apply with accuracy where the channels or passages are uniform and if applied to channels or passages which are not uniform the sections .selected must be fairly representative. If the sections selected are not fairly representative the value of c or n determined from observations and experiments may vary consid- erably from the values which would otherwise be anticipated. That is to say, the calculations based on c and n will take into account irregularities in channels and other unknown or unrecognized con- ditions, including curves, bends and obstructions which may not
Bazin's Formula.
S-
|
^^'■~~" |
/ |
1 |
T-iei |
||||||||||
|
Baiin'fi ForiDulaior the T«tiie of c in the foriiuil& T=ci^rs iSf in Eiigjish |
/ |
/ |
1 |
||||||||||
|
/ |
/ |
1 |
|||||||||||
|
/ |
/ |
/ |
1 |
||||||||||
|
/ |
/ |
/ |
J |
||||||||||
|
/ |
/ |
/ |
/ |
" ** |
|||||||||
|
S7 |
/ |
/ |
/- |
— - |
|||||||||
|
0 = |
rr^ |
I |
1 |
f |
/ |
||||||||
|
.5.2 + ^- B]=0,06forimooth plank or matched boards. niM^,16 for plauka and brick* m=0.4G for nmflonrjr, m=0.85 for r^ular eanh beds. m^L30 for canaU in good order. |
/ |
J |
/ |
||||||||||
|
/ |
I |
/ |
|||||||||||
|
/ |
/ |
/ |
1 |
_^ |
1 ■ |
'5 |
|||||||
|
/ |
/ |
||||||||||||
|
/ |
/ |
J |
1 |
||||||||||
|
/ |
/- |
-/ |
/ |
f |
H |
||||||||
|
■ ' |
/ |
f |
|||||||||||
|
/ |
-/ |
/ |
/ |
||||||||||
|
I |
/ |
/ |
1 |
3 J |
|||||||||
|
/ J |
/ |
/ |
1 |
*t |
|||||||||
|
iQ^L75 ia very bad order* |
^ */ |
— |
':: |
a 5 < OE ft u li |
|||||||||
|
r^ |
i |
f |
\ |
91 |
dI |
||||||||
|
J |
fj |
1 |
1 |
||||||||||
|
/ |
/ |
/ |
y |
iVk |
|||||||||
|
i^ |
0 |
/ |
/ |
y |
r |
/ |
/ "^ |
1 |
|||||
|
/ |
/ |
/ |
r |
/ |
i |
f |
-v. 1/ |
||||||
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^^ |
i*^ |
/ |
J- |
/ |
1 |
1/. |
|||||||
|
- >/' |
/ |
/ |
^ |
7^~ |
/ |
/ — |
/ |
1% - J4 — — fk |
|||||
|
/ |
.^ |
y |
^ |
X |
z |
||||||||
|
^ wy |
^ |
^ |
^ |
ip^^ |
- |
^ |
^ |
||||||
|
* e e c G |
1 1 OEr |
■ 1 f 4 ricii |
3 < [NT |
3 < |
IN |
a 1 ■ ( roH |
a 1 ■ 1 MULi |
3 \ V |
n '=C1 |
a 1 /rs |
» 4 |
9 0 S 12 |
5 |
Pig. 22. — ^Diagram For Solution of Baz!n's Formula.
GRAPHICAL 5DLUTI
V-VCLDCITY IN FCCT PCH ICCONQ.
C ^EOEFFIUIENT.
R- HYDRAULIC RADIUS LR PCCT = -^,
■ -BINE DT BLDPC
- Jl
V = c V
P- h-i
VALU 1
u
n
.§ .7 a 9 iQ
V = VELD[:iTIES
V
F CHEZYS FORMULA
= cv¥T
p I
ym BQ rccT or channel bcctidn
rtED PCnrMCTER DF CHANNEL BCCTION IN LINEAL FEtT.
I. IN rCET BETWEEN FDINT8 CONBIDERCO.
IBTH OR DISTANCE. BETWEEN POINTS CDNBIOCHED. IN LINEAL FECT.
FEET PER SECDNO
54 Hydraulics.
have been considered at the time the original observations were made.
39. Bazin's Formula. — It has been questioned by many observers whether the slope of the channel has any material influence on the value of the coefficient c. Bazin has derived a formula based on his examination of this subject in which he assumes that c does not vary with the slope. His formula, which is intended for the calcula- tion of flow in open channels is shown, together with a gfraphical table based thereon, in Fig. 22. This figure illustrates the law of variation of c and is applicable in principle in a general way to all channels and passages.
The graphical diagram. Fig. 23, which was prepared by the writer in connection with Mr. J. W. Alvord, affords a ready method of solving Chezy's formula (13).
40. Efficiency of Section. — From equations (12) and (13)
(16) q « velocity X area = va
or q = ca|/r8~= ca^5»
With c and s constant q varies as a|/r or as^/? —
\p
From this the conclusion may be drawn that other things being equal the maximum quantify of water will pass through any sec- tion of any river or other channel in which the hydraulic radius is a maximum or the wetted perimeter a minimum. Where a choice exists as to the class of material with which the channel is to be lined c becomes a variable and q will vary as
ca y'r or as c ^5 —
That is to say, under circumstances where different characters of lining may be used the maximum quantity will pass a given sec- tion with c and r maximum or with c a maximum and p a minimum for given a.
41. Determination of Canal Cross-section. — ^The velocity of the water in any artificial channel must be limited by the class of ma- terial used in its construction and the head which it is found prac- ticable to use. As noted above the efficiency of a section is greatest with the value of p minimum. Therefore, the semi-circular sec- tion is the most advantageous cross-section that can be used in a channel where resistance alone is considered and when the canal
Determination of Canal Cross-section.
55
IS to be lined with material which can be readily shaped into this form. If the canal is to be lined with stone masonry it is fre- quently more advantageous to make the face perpendicular and to place the batter of the wall at the back. Where the canal is cut from stone or shales which will not readily disintegrate in contact with the water, a slope of 90"* to 40** may be sometimes used. Quite steep slopes can also be used with dry masonry walls. In material which can be handled with pick and shovel, slopes may be used from i to 1.25 to i to 1.50. With artificial banks of dirt and gravel a less slope angle is necessary and the slope must frequently be made as low as one to two.
Table VI, which is taken partially from "Uber Wasserkraft und Wasser Versorgungsanlagen," by Ferdinand Schlotthauer, is of considerable value in determining the most advantageous cross- section in various sections which may be adopted in the construc- tion of a canal. As seen in the discussion above, the most advan- tageous cross-section, other things being equal, is that in which the
Fig. 24.
wetted perimeter is a minimum or the hydraulic radius is a maxi- mum. The following general discussion of the relations is based on Fig. 24. From this figure it will be seen that
(16) a = bd + d'cota
(17) p = b -f- 2d cosec a The transposition of (17) gives
(18) b = p — 2d coeec a Substituting (18) in (16)
(19) « ss dp — 2d* coseca -f- d'cotor The above equation now contains the area, depth, wetted peri- meter and functions of the slope angle, in this case a constant. 'Hie conditions of maximum efficiency of a canal section require
56 Hydraulics.
that the wetted perimeter be a minimum or what amounts to t\ same thing with a given wetted perimeter the area a must becon a maximum. The value of d which makes a the maximum is d
termined by putting ^i^ = o
(20) ^^ = p — 4d cosec a + 2d cota
(21) 0 = p — 4d cosec a + 2d cota
(22) d = 5
4 coseca — 2 cota
Substituting for p its value in (17)
,oQx J _ b -f 2d cosec a
""4 cosec a — 2 cota Equation (16) transposed reads
(24) b = ^-^y^
d Substituting this value in (23) we have
-pk— d cota -f 2d cosec a
(25) d = -5—^ 5—-
4 cosec a — 2 cota
Clearing:
(26) 4d«co8ec a — 2d«cota = a — d*cota + 2d*co8eoa
Transposing :
(27) d« =
2coseca — cota Transforming trigonometric functions
(28) d« = 2
-: cos a cosec a
Bin a
(29) = 2 — sin a cos a cosec a
sin a
(30) Finally.
a sin a 2 — cos a
(31) d = .JI^
• coaa Equation (24) may be written
(32) b = -g- — dcota
Table VI is calculated from the formulas:
(31) d = J/«"^^
^ \2 — COS a
Determination of Canal Cross-section.
57
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•— 1-^ ^i i-H ^ ,ph , g
58 Hydraulics.
(32) b = -j^ — dcota
(33) B = b + 2dcota
(34) p = b+^ ^ ' *^ ' sin a
In the above, a=cross-section area ; d=depth of water in channel ; b=bottom width ; B=width at water level ; p=wetted perimeter ;
c=the length of slope which is equal to -; —
In Table VI the relation of these functions, for the slopes ordi- narily used in practice have been calculated as well as for the semi- circular section. The use of the table may be illustrated as fol- lows: The quantity of water which it is desired to deliver is de- termined by the conditions of the problem or by measurement The velocity to be maintained in the channel is determined by the ex- isting slope, the nature of material encountered, or the friction head which it is found desirable to maintain. The area of the cross-section required to carry the quantity q with velocity v is a=-3- After the slope angle has been selected, for the material in which the channel is to be constructed, the corresponding values may be taken out of the table from their respective columns and multiplied by the square root of a. The result thus obtained gives the desired dimensions. If, for example, we desire to carry loo cu. ft. of water per second in a canal at a velocity of 2 1/2 ft. per second at which velocity small pebbles are unaffected, and with a side slope of 1.5 to i, which is suitable for loose earth, has been decided upon, the required area of cross-section will be 100/2.5 =40 sq. ft. The square root of 40 is 6.33. The required dimensions- of canal as taken from the table are
Depth d=.689 x 6.33=4.36 ft.
Bottom width b=.4i8 x 6.33=2.65 ft.
Top width 6=2.485 X 6.33=15.73 ft. and
The wetted perimeter p=2.904 x 6.33=18.38 ft Computation of the area from the above dimensions gives 40 sq. ft Hence the work has been checked.
42. The Back Water Curve. — One of the problems which be- comes very important in many water power installations is the effect on the elevations of the stream produced by the erection of a dam or other obstruction therein. The back water curve can best be determined by the use of the simple formula of flow, equa- tion (13).
Flow of Water in Pipes, 5p
From this, as shown in equation (15) From this equation can be derived
(35) h. = 2^=a^xi
With ^^ constant, h, : h', ::^ : JBl, therefore (36) h/.^ h»P'<^' ^ h»>'r
That is to say, with the quantity of water and length of section constant, if the coefficient remains constant the head due to any obstruction will vary in accordance with equation (36).
Where the water is greatly deepened in proportion to its orig- inal depth the value of c will not remain constant but will vary. Where such is the case and where q*l is constant, under which condition
The difficulties in the determination of the value of c are, of course, obvious, but it is believed that the back water curve can be closely calculated by this simple formula in which the new value of c is the only factor to be estimated, and where the other elements of the problem can be determined by actual measure- ments. In using this formula the original value of c under exist- ing condition of flow can be determined by calculation based on actual observation of flow under different conditions of water and tjie conditions of the channel under the new regimen can be closely estimated. New values of c can be very closely estimated on the basis of the values known to exist under other similar cir- cumstances. This method will permit of a more practical solution of the problem than by the use of formulas based on entirely the- oretical consideration of conditions which can never be approxi- mated in practice.
43. Flow of Water in Pipes. — Mathematical expressions for the flow of water in pipes may be derived from either of the funda- mental hydraulic formulas
v = ci/ra or V = c^/^ba
Starting with the former equation, in the case of a pipe flowing
6o Hydraulics.
full the hydraulic radius p=-^- where d is the diameter of the pipe and for s we may substitute --i We then have
(38) '=*'A^*
In a pipe of unit length and unit diameter without friction the flow would be expressed by the formula
— v»
V = i/2gli or h = ^
To modify this for friction a friction factor f is introduced and the equation then reads:
The friction varies directly as the length and is assumed to vary inversely as the diameter. Hence, for any pipe of length 1 and diameter d the complete equation is :
Placing (38) and (39) equal it will be found that
16.04
so that the equations can be made equivalent by the proper modi- fications of friction factors. An extensively used formula for the determination of c in equation (38) is that of Darcy. It reads :
For new pipe a = .00007726 and fi = .00009647. For old pipe a = .0001543 and fl = .00001291.
These coefficients were determined from experiments on small pipes and therefore in the case of large pipes with high velocities the velocities computed by this formula are too small.
Various modifications of the Chezy formula, having the general form
(41) v = cr°8»
have been proposed or derived from experiments. Lampes and Flamant's are the best known of this type. Lampes reads
(42) v = 77.68 d0.6M gO.US and Flamant's
(43) v = cd* 8*
in which c:=76.28 for old cast iron pipe and 86.3 for new pipe.
Flow of Water in Pipes.
6i
63
Hydraulics*
The value of c in the formula v^^cV^s may vary from 75 to 15c for large cast iron pipe. For riveted steel pipe the coefficient varies but little with velocity and diameter and at ordinary velocities ranges from 100 to 115, A- L, Adams gives values of c for wood stave pipe ranging from 100 to 170. Experiments on the Ogden pipe line showed average values of about 120. ^
An examination of the various formulas proposed for calculating' the flow of water in pipes will show a very wide range of results For example, for calculating the head lost in a four-foot new cast iron pipe, some of the principal formulas offered and the graphical solution of the same are shown by Fig, 25, From these results it will be seen that the data from which the formulas were derived are evidently obtained under widely varying conditions and that in the relation of such formulas for use on important work, they must be chosen after a careful consideration of all the elements of the problem, and that it is usually much better, when possible, to utilize the original data and obsenation along similar lines when such can be obtained, and derive the formula to be used instead of accepting one whose basis may be obscure or unknown.
In construction %vhere pipes are short and comparatively unim- portant, a formula may be selected which seems to agree with the
Flow of Water in Pipes.
63;
9.Ct «J0
VBLoeirv iM rccT pir sccono
Fig. 27.
1.0 *.o
vcLoeiTv IN rccT wtn second
Fig. 28.
I
Hydraulics,
elements of the problem. The formulas offered by Tutton seem to agree well with the actual results of expferiments and several diagrams based thereon are shown in the following pages* In two of these diagrams (Figs, 26 and 27) the limiting values are shown and the results obtained from any pipe of the character represented therein should lie between these limits depending on its condition. 44- The Flow of Water Through Orifices.— It is found that water flowing through an orifice in the side of a vessel acquires, a velocity practically equal to that which would be acquired by ^ falling body in passing through a space equal to the head above the center of the opening, i, e.j
(44) v= i/2iir= 8.025/E
in which
v=veIocity of spouting jet* g=acceleration of gravity=32,a- h^=head on opening. The discharge through the opening would therefore be (45) q=^* va^^aV^gh or practically (46) q^caV^gh where c is a coefficient varying with the size and shape of the orifice and with various other factors.
A more accurate determination of the theory of flow through a^ given orifice is derived as follows: ^|
If a thin opening is considered at a depth y be- low the surface the discharge through the ele-^ mentary section Idy would be
(47) dq = Idjy 2^
Integrating this equation between the limit
h^ and h^ we obtain the following;
(49)
t = IKht*— h|*)i/2g or practically
m being the coefflcient of practical modification due to condition of the orifice.
45. Flow Over Weirs. — In a weir h|=o. Hence equation (49) becomes
(bQ) q=^ m(|)ll/5ih*
in which h is the head on the crest of the weir. That is, the ver- tical distance from the water level above to the crest of Uie weir.
4
Flow Over Weirs.
65
^- For practical use the coefficieiit m together with the constpnts ^E- and 2g are combined as follows:
^H e = m |;/2g =: M ^2g afid equation (50) beooEnea
■ (51) q = c Iht
^m The value of m and consequently of c varies with the shape of Hhe weir and with other factors and must be determined experi* Hnentally. This has been done with weirs of many forms, both by ^Eajiin in France and by Rafter and Williams at the Cornell hydrau- lic laboratory. The results of these experimental determinations Hpe given by Figs, 30 to 34, inclusive. These figures are reduced ^^irectly from the diagrams of Mr Rafter in the Report of the Board of Engineers of Deep Waterways, 1900. In practice many weir formulas are in use, based on various ex- ^periments and observations. The formula of Francis', equation- ^f(S2). is probably the best known in this country. It is best adapted lo long, sharp crested weirs without end contractions.
q = BM Ih*
Fig, 30.— Wetr Catfficlenta for Weirs of Various Shapes.
Fiow Over Weirs,
67
Fijr. 35.— Weir Ooeffidentt for We! n of Tanons 91iap««»
•66
Hydraulics.
|
Heod Flf r jl' IF itr' |
on CntsT of Weir in Tecr |
i3 |
|
^5^._^-44^^ q: |
__ .^._^^.. J^^ |
- ^ - - 1 |
|
|3^f^l|t|gi^riTp-if|j^| *P| I'H \\\ ]'] \ j | |
-3 , |
|
|
^M,dzz -^ ^~~ii i^ r- ;:: |
tefj^ feUwi iN^^I mI-1 \-\ l-U-R-H |
zz: |
|
W«f - ' ,^^::f-i ^11 |
IJEEhEEEE^EEEEE^^iEEEEEEE |
|
|
1 *^ .1^ ^^^,--i_z.i-,^3H |
i^W^^ffl |
E!s±? |
|
lai -"z--''"z-i = = --z ^"H |
H ' IbHtttmiTf 1 |
:: - |
|
^30 z ^_z---: |
-It |
|
|
"5^*h+j i + W'HMihHH''^L-tr(''W^ |
||
|
ntr^^^y-'lthi-.i^rrzyz-^ |
||
|
E^rEEEEEEEEEEEEEEEEEEE: zziz=zzziz3is;PBBJJlJ3 = C; |
!?||! = ?Ee55e!EeEEEE:EEEEE |
' |
|
f [Ifljlm^ |
EE;iEipipEE;iE::^::EE^EE |
^,. |
|
?s-=i==ii=ii=EEH!! 0»' :""-E""Ei====±-"- |
■ 1 ■ -^ ir-i |
.., |
|
i«9---pl-^~— ----1— — — :^^-::: |
— t |
|
|
*7.zz-:_-Lij^44: - Ji_.i3aB J3W.Z--I i = i = — = ---^5^:^: |
|?!?:|||? = = *?*:?5:" = = = i |
|
|
^aMp^^^nu^ |
^i |
|
|
^s^» -z----'i?^!iiii-i=izzi: r^^liJJjinlllllll4t^ |
M M n r[m"H Jon Crast of Weir in Fo«r |
^. |
Fig. 31, — Weir CopfReienla for Welis of Various shapea.
Flow Ov<fr Weira.
67
Head on Great df Weir in fiset
g^fl y> 4fl
Hood on Creor of Weir rn Fe«r
Pi|E. .^,--Weir OoefRc!eTi*i for Weire of Vatinni 3hap«e*
70
Hydraulics,
fir- 38— Weir CoelBcients for Wein of Vailoua Shai>M.
Flow Over Weirs.
71
Head on Cresf 0/ Wetr in F««t,
Cneat of W«r m fc^
fig. 34.— Weir CoemclentJ, xor Weirs ot Vaj"loufl Sllai^es^
I-
lb oceompqny Report an 5|K
1899
US.0QARO or Engineers on deep waterwavs
WATER SUPPLY DIVISION
Diagram showing Ofacborqe over v/elrs wffh irrcqular Crests. os per Boiirv's Experirr^t r?l8 . in compariSQn svith Dischorqe a& per Ff5knciS'& Formula fiif a stiorp crested woir
"/;
r' r'ff ■ ■gli?''^"ra -^
^ -— ^fe — vis — rt — KB — fk >Ar-Tn! — rit — rst — cAi — iJo— ■
Dl^hang* in Cubic f «ei- per s
SBCfuna of AKpvnmeflM Mm^
|
1 >»^ f"^-^*^ |
^^ |
130 ITO |
||||
|
fr Hoirr f«e 173 |
135 |
|||||
|
on5Bov»rw*(r,«9foloni^ffrpnaFbot,(flQJjic Fcef par 5(scona^Appin;sqma^ |
||||||
|
wHQttr |
t«e |
173 |
03 |
130 |
f/O |
9%w«e>**«Hh |
|
T ■ |
ao |
a A |
*« |
a^ |
' "^olT" |
|
|
■a |
I4t |
Lift |
r»> |
t.tt |
lEO |
1 »& |
|
^m |
4aa |
as* |
3/0 |
A9« |
3^ |
3H. |
|
ijd |
T>0 |
ftflA |
TtO |
TtO |
T«C |
«rt |
|
I9S |
II » |
IO>« |
H*9 |
leoo |
teoe |
94t |
|
03 |
IS«B |
iy4a |
■■30 |
iroo |
ir jc |
19. » |
|
a» |
VQ» |
e»OA |
ie» |
Has |
irso |
|
|
hDO |
M49 |
t5.«e |
eats |
M«o |
e»iQ |
2l9(» |
|
>.AJi |
JAM |
Vl »1 |
a* 7* |
Uil3 |
3daa |
^£4^ |
h »rb — wi^ — licj u'a ■ le^o n'o — sse 190 — sm "»r5 sja — »p aJa sJg — Wt
Dt 0f Cf"«6+
n a ^tr-iiirttr Jit
74
Hydraulics.
A number of different tormulas for the flow over weirs are given on Fig. 35 and the flow as calculated by these formulas is showi> on the diagram. L in these formulas represents the length of the weir crest which in the dimension above is represented by 1.
Figure 36 shows graphically the results of the application of the value of c as given on Figs. 30 to 34 as compared with Francis formula.
In small weirs the effect of end contraction and of the velocity of approach becomes important and corrections to the formulas must be applied in order to allow for those influences.
If n==the number of end contractions and the effect of each is to reduce the effective length of the weir by one-tenth the head on the weir, equation (51) will become
'>'
(63) q = c(l.
The effect of the velocity of approach, for a given quantity, is tc reduce the head on the weir by the velocity head. This reductioD is given by the formula:
(64)
in which v'=velocity of approach and h'==velocity head.
TABLE VII. Coefficient of discharge C for use with Hamilton Smith, Jr.'s formula (56) for flow of water over sharp crested weirs having full contraction, I = length of weir.
|
Effective h6«d=h |
.66 |
i(?) |
0 |
2.6 |
3 |
4 |
5 |
7 |
10 |
15 |
19 |
|
.1 |
.632 |
.639 |
.646 |
.650 |
.052 |
.053 |
.653 |
.654 |
.655 |
.655 |
.656 |
|
.16 |
.619 |
.625 |
.634 |
.637 |
.0:^ |
.K39 |
.640 |
.040 |
.641 |
.642 |
.642 |
|
.2 |
.611 |
.618 |
.<>26 |
.629 |
.6:10 |
.631 |
.631 |
.632 |
.633 |
.634 |
.634 |
|
.26 |
.605 |
.612 |
.621 |
.623 |
.624 |
.625 |
.62t> |
.627 |
.628 |
.628 |
.629 |
|
.:^ |
.601 |
.608 |
.(il() |
.618 |
.619 |
.621 |
.621 |
.623 |
.624 |
.624 |
.625 |
|
.4 |
.595 |
.601 |
.609 |
.612 |
.613 |
.014 |
.615 |
.617 |
.618 |
.619 |
.620 |
|
.5 |
.5<.K) |
.596 |
.605 |
.607 |
.608 |
.610 |
.611 |
.613 |
.615 |
.616 |
.617 |
|
6 |
.587 |
..V.)3 |
.601 |
.604 |
.(J05 |
.607 |
.608 |
.611 |
.613 |
.«14 |
.615 |
|
7 |
.58.') |
.590 |
.598 |
.601 |
.603 |
.604 |
.tM)6 |
.609 |
.012 |
.613 .614 |
|
|
8 |
.'>95 |
.598 |
.600 |
.602 |
.604 |
.607 |
.611 |
.612 |
.618 |
||
|
9 |
.592 |
.596 |
.598 |
.600 |
.603 |
.606 |
.60i» |
.611 |
.612 |
||
|
1 0 |
.5JH) |
.593 |
.5^5 |
.598 |
.601 |
.(504 |
.60S |
.610 |
.611 |
||
|
1 1 |
.587 |
.591 |
.593 |
.596 |
.599 |
.603 |
.606 |
.(K)9 |
.610 |
||
|
1 2 |
.585 |
.589 |
.591 |
.5^4 |
.5M7 |
.601 |
.605 |
.608 |
.610 |
||
|
1 H |
.5S2 .580 |
.58(5 .584 .5*<2 |
.589 .587 .585 |
.692 .5fX) .589 |
.596 .594 .592 |
.599 .598 .5*»6 |
.604 .002 .601 |
.607 .606 .605 |
.609 |
||
|
1 4 |
.609 |
||||||||||
|
1 5 |
.606 |
||||||||||
|
1 (> |
.580 |
.582 |
.587 |
.591 |
.595 |
.600 |
.604 |
.607 |
|||
|
1.7 2.0 |
|
.594 |
.699 |
.603 |
.607 |
||||||
|
::::::i:::::: |
::::::i.;:... |
•••• |
Literature. 75
To allow for the influence of velocity of approach h' must be added to h and equation (53; becomes
m q = c(l~n^)(h4-hM'
Experimental results at the hydraulic laboratory of the Uni- versity of Wisconsin show- that for small sharp crested weirs, with end contraction, the formula (56) of Hamilton Smith, Jr., is prac- tically correct :
(56) q = c 1 1^2^ Ihf
In this formula
c?=coefficient of discharge (to be taken from Table VII), h=observed head on crest (H) plus correction due to velocity of approach.
Variations in the forms of the crest of weirs and in the arrange- ment of sides and bottom of the channel of approach cause con- siderable variation in their discharging capacity. It is therefore apparent that unless the conditions closely agree with those on which experimental data is available that the error of calculation may be considerable.
LITERATURE.
BEFEBXNOES ON GENERAL HYDRAULICS.
1. Francis, Jas. B. Lowell Hydraulic Experiments. New York. D. Van-
Nostrand. 1883. t Panning, J. T. Hydraulic and Water Supply Engineering. New York.
D. Van Nostrand & Ck). 1886.
3. Smith, Hamilton, Jr. The Flow of Water Through Orifices, Dver Weirs,
and through Open Conduits and Pipes. New York. Wiley A Sons. 1886. 3a. Church, Irving P. A treatise on Hydraulics. New York, Wiley ft Sons.
4. Welsbadi, P. J. Hydraulics and Hydraulic Motors. Translated by A,
Jay Dubois. New York, Wiley ft Sons. 1891.
5. Carpenter, L. G. Measurement and Division of Water. Bulletin No. 27.
Colo. Agric. Expt. Sta.. Ft. Collins, Colo. 1894. €. Boyey, Henry T. A Treatise on Hydraulics. New York. Wiley ft Sons.
1895. 7. Merriman. Mansfield. Treatise on Hydraulics. New York. Wiley 6
Sons. 1903. *• Hydrographic Manual, Water Supply and Irrigation Paper No. 94. U. S.
G. S. 1904. ^- Hoskins, L. M. Hydraulics. New York, Henry Holt ft Co. 1907.
BEFEBENCES ON FLOW OF WATER IN CANALS.
^^ Hill, A. Flow of Water in Rivers and Canals. Van. Nost Bng. Mag. Vol. 8, p. 118. 1870.
Hydraulics.
11* Ganpiniet, E. Unlfomi Motloa la CansLls and Rivera. Vau. NosL Eng. Mag* Vo!. 2, p. 211. 1870.
12. Searles, W> H< Slope of Water Surface in tlie Brie Canal* Trans. Am.
Soc. a E., Vol. C, pp. 290-296, 1S77
13. Ellis, Tlieo, G. Flow ol Water, Eng, News, Nov. 26, 1881, Vol. 8,
478-9.
14. Cunnlnghaio, Allan. General DlBcnssion of Flow In Canals* Proo. In
Clr. Eng, 18S2-E3, pp. 1-95. IG. Fteley, A. and Stearns, F. P. Flow of Water In Conduits. Trans. Soc. C. E, Vol. 12 (1883), p. 114.
16. Mmn, P. J. Irrigation Canals and Otbar Irrigation Works and Flow
Water In Irrigation Canals. Denver, Colo. 1892.
17, Adamai, A. L. Diagram for Calculating Velocities, Grades and Mean
Radii for Flumes and Ditches. Eng. News, Feb. 13, 1892. p. 157.
18* GanguUlet, E. and Kutter, W. R. A General Formula for the Uniform
Flow of Water In Rivers and Other Channels. Trans, by Ru
[ dolph Herring and John Trautwine. New York, Wiley & Sons.
1893.
19. Bou&sinesq, H. The Gradual V&rlatloas in th© Flow of Water la Chan-
nels of Large Section. Comptes Rendus. May 31, 1897.
20. Bouflfilnesq, J. Expertmental Verification of the Theory of Gradually I Varied Flow in Open Channels. Comptes Rend us. June 14.
1897. 2L The New Formula of Bazln. Genie Civil, March 5, 1S9S,
22. A New Formula by Bazin for Computing Flow of Water in Open Chan^
nels. Eng. News, July 14, 1893*
23. Bazln's New Formula for Flow in Open Channels. Eng. News, 1898, Vo
2, p. 26.
24. A Study of a New Formula for Calculating the Discharge of Open Chan-
nels. Ann ales des Fonts et Chaussees. 2 Trimegtre, 1898.
25. Determination of Flow in Rivers and Canals. Zeltsclir. d Oesterr. Ing. u
Arch. Ver., Vol 50. pp, 533^34. 1898.
26. Swan, Chas. H. and Horton, Theo. M, Hydraulic Diagrams for the Dis
charge of Con du its and Canals, New York, Eng. News Pub Co. 1899.
27. Croathwaite, Ponsby Moore. Two Graphic Methods Applied to HydraulicI
Calculations. EngineeHng. Loudon. July 15, lS98t 1
38, Concerning the Conception of a Hydraulic Moment of Conduit Cross Sec^
tlon. ZeitscJir, fur Arch, u Ing. Vol. 4G, 1900, Heft-Ausgabe.
Col, 402-417. 29, Siedek, Richard. Studies of a New Formula for Estimating the VeloclTy
of Water In Brooks and Small Channels. Zeltschr, d Oeaterr.
Ing. und Arch. Ver, Vol. 55, pp. 98-106. 1903. J
BErEBKNCES ON FLOW OF WATEB THROUGH Fn*ES,
30. Francis, Jas. B. Flow Through Pipes, Trans, Am, Soc C. E, Vol, 2, p. 45. 1872,
31. Danach, a G. Flow of Water In Pipes under Pressure. Trans. Ahl So< C. E. Vol. 7, p. 114. 1878.
32. TVehage, H, Fnction Resistance in Pipes. Dingler's Polytechnlsrhei Journal. 1884, p. 89.
I
Literature. 77
33. Steams» F. P. Flow of Water Throat a 48^ Pipe. Trans. Am 8oc C.
R, Vol 14, p. 1. 1886.
34. Mair, J. G. Flow Through Pipes at Different Temperatures. Proc. Inst
C. E. Vol. 84, p. 424. 1886.
35. Duane, James. Effect of Tuberculatlon on Delivery of a 48^^ Water Main.
Tnuus. Am. Soc. C. B. 1893, p. 26.
36. Tuttle, Geo. W. Economic Velocity of Transmission of Wlater Through
Pipes. Eng. Rec. Sept 7, 1895.
37. Coffin, Freeman C. The Friction in Several Pumping Mains. Eng. News,
Feb. 20, 1896.
38. Hawks, A. McL. Flowage Test of 14"^ Riveted Steel Main at New Wes^
minster, B. C. Eng. News, July 30, 1896. 3S. Flow of Water in Wrought and Cast Iron Pipe. Am. Soc. Mech. Eng.
Dec. 1897. 40. Herschel, Clemens. 116 Experiments on the Carrying Capacity of Large
Riveted Metal Conduits. New York. John Wiley & Sons. 1897. 4t Gould, E. Sherman. The Flow of Water in Pipes. Am. Mach. Mar. 8,
1898. 42. Hawks, A. McL. Friction Coefficient for Riveted Steel Pipes. Proc. Am.
Soc. C. E. Aug. 1899. 4S. Palton, C. H. Flow of Water in Pipes. Jour. Ass'n Eng. Soc. Oct. 1899. 4i Marx, C. D., Wing, Chas. B., and Hosklns, L. M. Experiments on the
Flow of Water in the Six Foot Steel and Wood Pipe Line of
the Pioneer Electric Power Company. Proc Am. Soc C. E.
Feb., 1900; April, 1900; May, 1900.
45. Gregory, John H. Diagram Giving Discharge of Pipes by Kutter's For-
mula. Eng. Rec. Nov. 3, 1900.
46. Pbnnulas for Flow In Pipe. Eng. News, 1901. Vol. II, pp. 98, 118, 332.
476.
47. Noble^ T. A. Flow of Water in Wood Pipes. Trans. Am. Soc C B. Vol.
49, 1902. ^S- Sapb, A. V. and Schoder. E. W. Experimental Study of the Resistance of the Flow of Water in Pipes. Proc. Am. Soc. C. B. Maj, 1903; Oct, 1908.
BETEBENCES ON FLOW OF WATEB OVEB WEIBS.
49. Pteky, A. and Steams, F. P. Flow of Water over Weirs. Trans. Am.
Soc C. B. Vol. 12, p. 1. 1883.
50. Francis, J. R Experiments on Submerged Weirs. Trans. Am. Soc C. B. Vol. 13, p. 303. 1884.
51 Henchel, Clemens. Problem of the Submerged Weir. Trans. Am. Soc
a B. Vol. 14, p. 189. 1885. 52. hrestlgations on the Flow over Submerged Weirs. Zeltschr. des Ver.
Deutsch. Ing. 1886, p. 47. W. Hind, R. H. Flow over Submerged Dams. Proc. Inst C. B. VoL 86, p.
307. 1886. W. Kaberstroh, Chas. B. Epxerlments on the Flow of Water Through Large
Gates and over a Wide Crest Jour. Ass'n Bng. Soc Jan., 1890,
p. 1.
5
f8
Hydraulics.
S5. 66.
67* 53.
eo.
6t
ea.
$5. 66.
67
6S.
€9.
70. 71.
72.
73,
74.
75. 76.
77.
The Floir of Water orer Dams and Spillways, Bug. Rea Jun« f , 1900. Flow of Water over Sliarp Greeted Weirs, Annales des Ponta et Chaua^
sees. Jan, 1, 18&0; Nov., 1S91: Feb., 1894. Also Proc. Eng. Club
of Philadelphia, Jan., IBM; July. 1S02; OcL, 1892; Apr., 1893. Flow over a Weir of Curved Proflle. Keltschr. d Oestarr, Ing. n AicIl
Ver. June 2, 1906. Flymi, A, D. and Dyer, C. W* D. The ClppcletU Trapezoidal Weir.
Trans. Am. Soc C. B. July, 1894, Warenaklold, N. Flow of Water over Rounded Crest Eng. Newi, J;
ai. 1895* Vol. 83, p. 75. PrUzel, J. P. and Herachel, Clemena, Flow over Wide Horizontal Top
Welrt, Eag. News, 1892. Vol 11, pp, 290, 440, 446: 1895, Vol. 1,
p. 75. John son, T. T. and Cooley, B. S. New Experimental Data for Flow ot©? a
Broad Crest Dam. Jour, W, Soc Engrs. Jan., 1896* Wide Cr^t Weirs, Bazln'e Formula. Eng. News^ 1890. Vol. I, p. 16^
Vol, ir, p. 577: 1896, Vol, I, p. 26. E:cperinient3 on Flow OTor Dams, Eng. News, 1900, P< 207* Hafter, Geo. W. The Flow of Water over Dama, Proc Am* Soa C- IL
Mar,, 1900. Heyno H. Study of Hydraulic Coemcienta, 2eltschr. d Oesterr In^. n
Arch. Ver Dec. 6 1900. Dery, Victor A, E. D, Experiments on the Measurement of Water otw
Weira Proa Inst, G. E, Vol. 114, p. 333, 1893.
K^
^
BEFZBEKCES Olf BACK WATI3 AKP IN THEFEBIKOI.
d
Wood, De Volson, Back Water la Streams as Produced by Dams. Trans,
Am. Soc. C, E, Vol. 2, pp. 255-26L 1873, Hutton, W. H, Back Water Caused hy Contractions, Transu Am* Soc. C*
m Vol. 11, pp. 212-240. 1882. Olllmore, Q, A. Ohsrt ruction to River Discharge by Bridge Plera, Van.
Host, Eng. Mag, Vol. 2$, p. 441. 1882. J
Back Water from Dams. Eng, Rec, July 9, 1892, ^
Ferrlday, Robert Measurements of Back Water* Eng. Newa, 1896, VoL
n, p. 28.
Frescolm, S. W* Back Water Caused by Bridge Piers and otber Obf^tniG'
tlons. Jour Eng. Soc, Lehigh Univ. Feb., 1899. The Estimation of Damages to Power Plants from Back Water. Eag.
Rec April 26, 1902. Harria, E, G., Taylor, W. D.. Ladshaw, T. B. Back Water from Dams.
The E^ect on Meadow Lands, Eng, NewSf 1902* YoL II«
142 and 311 Tables for Computation of Swell on Open Water Courses* Zeltachr.
Ardi. und Ing. Vol. 49, Cola. 268-274* 1903. Fllegoer, A. A New Method of Computing the Back Water Curva
SchwelzerlBChe Bauaeltting. Aug. 22. 1903. Tolmaa, BreiUIav. The Computation of Back Water Curves. Oesterr,
Wocbensohn f d Oeffent Baudienst July 1, 8, 1905.
ms.
I
CHAPTER IV.
WATER POWER,
THE STUDY OF THE POWER OF A STREAM AS AFFECTED BY FLOW.
46. Source of Water Power. — ^Water power depends primarily on the flow of the stream that is being considered for power pur- poseSy and on the head that can be developed and utilized at the site proposed for the power plant. Both head and flow are essen- tial for the development of water power, but both are variable quantities which are seldom constant for two consecutive days at any point in any stream. The variations in head and flow radically affect the power that can be generated by a plant installed fdr power purposes. These variations also greatly affect the power that can be economically developed from a stream at any locality. The accurate determination of both head and flow therefore be- comes very important in considering water power installations and hence should receive the careful consideration of the engineer. The neglect of a proper consideration of either or both of these factors has frequently been fatal to the most complete success of water power projects.
47. Factors of Stream Flow. — ^The quantity of water flowing in a stream at any time, which is more briefly termed "stream flow" or "nm-off," depends primarily upon the rainfall. It is, however, mfluenced by many other elements and conditions. It depends not only upon the total quantity of the yearly rainfall on the drainage area, but also on the intensity and distribution of the rainfall throughout the year. In addition to these factors the geological structure of the drainage area, the topographical features, the sur- face area of the catchment basin, the temperature, the barometric condition, all influence and modify the run-off. Sufficient data is not available for a full understanding of this subject, but enough » available so that the general principles involved can be intelli- gently discussed knd the problems considered in such a way as to ?ivc a fairly satisfactory basis for practical work. A knowledge 0^ the importance of the factors above mentioned and the extent to which they modify, influence or control stream flow, is essential
Be-
Water Power.
to a broad knowledg^e of water power engineering. These factor? are discussed in more detail in chapters VI, VII and VIIL
48, Broad Knowledge of Stream Flow Necessary. — The flow of a stream is constantly changing and any single measurement of that flow will not furnish sufficient data on which to base an in- telligent estimate of the extent of its possible or even probable economical power development* A knowledge of the economical possibilities of such development must be based upon a much broader knowledge of the variations that take place in the flow of the stream. In order to fully appreciate the power value of a stream, the character and extent of its daily fluctuations must be known or estimated. Averages for the year, monthly averages, and estimates of average power have been ordinarily taken as a basis for water power estimates, but they are more or less misleading, unsatisfactory and uncertain for the reason that such averages in- clude extremes, the maximum of which are often unavailable for water power purposes without more extensive pondage than is usually practicable. These maximum and minimum flows which affect the power of a stream not only through the quantity flowing but also through the head as well, as will be hereafter discussed* arc of the utmost importance for a broad consideration of water power. So also is a knowled^ife of the various stages of flow and the length of time that each will prevaiL Such knowledge demands daily observations or estimates of daily flow which can be repre- sented in graphical form by the hydrograph,
49, The Hydrograph, — ^The hydrograph, constructed for the study of stream flow and its influence on water power, may be drawn by representing the daily flow in cubic feet per second at the point of observation by the ordinates of the diagram and the element of tame by the abscissas, (See Fig. 37.) The result is a graphic diagram which shows the character and extent of the daily fluctua tions in the flow of a stream at the point of observation during thi period for which the hydrograph has been prepared,
A single observation of the flow of a stream represents a totally inadequate and unsatisfactory criterion for water power consid- eration. By reference to Fig, 37 it will be seen that, if the dis- charge of the Wisconsin River at Necedah had been measured only on August $t 1904, the conclusion would have been reached that the discharge of the river was about 2,100 cubic feet per second. If the measurement had been taken only on August 15, 1904, the flow would have been determined at about 5,850 cubic feet per second, or almost three times as great as on the first date, Thfl
The Hydrograph.
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Water Powtr<
difference between the dates might be even greater, and no slng!^ measurement nor any series of measurements for a single week or month would ^ve a fair criterion from which the normal flow of the river could be judged.
The hydrograph of the daily flow of a river for a single year gives a knowledge of the variation in flow for that year only under the peculiar conditions of the rainfall, the evaporation, and the other physical factors that modify the same and that obtain for that particular year. Such infonnationj while important, is noi altogether sufficient for the purpose of a thorough understanding of the availability of the stream flow for power purposes. Observa- tions show that stream flow varies greatly from year to yeafp and while, with a careful study of the influences of the various factors on stream flow, together with a knowledge of the past variations in such factors, the hydrograph for a single year may give a fairly clear knowledge of the variations to be expected in other years where conditions differ considerably, still it is desirable that the observations be extended for as long a period as possible. Such long time observations may remove the estimates of flow entirely irom the domains of speculation and place them on the solid ground of observed facts. Hydrographs of a river that cover the full range of conditions of rainfall, temperature, etc., which are liable to pre- vail on its drainage area, give a very complete knowledge of the flow of the stream for the purpose of the consideration of water power.
It is rare, however, that observations of stream flow for a lon^ term of years are available at the immediate site of a proposed power plant. Such observations are ordinarily made only at loca- tions where power has been developed and where water power oi similar interests have been centered for a long period of time, Oc casionally, however, the future value of potential powers is rccog* nized and appreciated^ and local observations are maintained for < series of years by interested parties, having a sufficient knowledge of the subject to recognize the value and importance of such in- formation. The variation of flow for some considerable time pre- vious to construction is thus available upon which to base the desigra. . In considering new installations, one of four conditions obtains - First: Hydrographs are available at the immediate site proposed Second ; Hydrographs are available at some other point on tN^* river above or below the proposed installation.
The Use of Local Hydrographs. 83
Third: Hydrog^phs are not available on the river in question but are available on other rivers where essentially similar condi- tions of rainfall and stream flow prevail.
Fourth: No hydrographs, either on the river in question or on other rivers of a similar character and in the immediate vicinity, are available.
50. The Use of Local Hydrographs. — ^When hydrographs, con- structed from observations taken at the immediate site of the pro- posed water power installation, are obtainable, for a considerable number of years, the most satisfactory character of information is available for the consideration of a water power project. Under such conditions the engineer is not obliged to consider the rela- tion of rainfall to run-off or to speculate as to the relative value of the stream in question compared with other adjacent streams, or as to tl)e effects of the physical conditions of drainage area, evap- oration, temperature and other factors on stream flow. The actual daily flow of the stream from day to day, perhaps through all ranges of rainfall, temperature, evaporation and other physical con- ditions, is known and the principal points which must be consid- ered are : First, the head available ; Second, the effects of the varia- tions of flow on the variations in head; and Third, the extent to which the flow can be economically developed or utilized. Gen- erally, however, even where local hydrographs are available, they arc not sufficiently extended to cover all the variations in river flow which must be anticipated, and it is ordinarily desirable to com- pare the available data with the flow at other points on the stream in question or with other streams in the immediate vicinity.
51. Use of Comparative Hydrog^phs. — Hydrographs taken at other points on the same river, or on other adjacent rivers where conditions are reasonably similar, are of great value in considering the local stream flow, — ^provided all modifying conditions are under- stood and carefully considered. Hydrographs are ordinarily pre- pared to show the cubic feet per second of actual flow at the point at which observations are made. If the observations (and the hydrographs based thereon) made at some other point on a stream, or on some other streams, are to be used for the considera- tion of the flow at a point where a water power plant is to be installed or considered, the relation of the flows at the several points must be determined.
I As a basis for such comparison of stream flow, it may be as-
Water Power.
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Use of Comparative Hydrographs. 85
stream, or at points on different streams under similar circum- stances, is essentially the same. This is not strictly true, or per- haps it may be more truly said that the apparent similarity of condi- tions is only approximate and hence differences in results must necessarily follow. For a satisfactory consideration of the subject of comparative hydrographs, the variations from this assumption, as discussed in another chapter, must be understood and appre- ciated. For practical purposes, however, the assumption is often essentially correct and forms a basis for an intelligent considera- tion of stream flow where local hydrographs are not available. Fig. 37 is a hydrograph constructed from observations made on the Wisconsin River at Necedah, Wisconsin, by the U. S. Geological Survey for the water year, 1904, and shows the daily rate of dis- charge of the Wisconsin River at that point for the year named. The area of the Wisconsin River (see Fig. 38) above Necedah is 5,800 square miles. If, therefore, we draw a horizontal line from the point representing 5,800 cubic feet per second on the discharge scale (see Fig. 37), the line so drawn will represent a discharge at Necedah of one cubic foot per second per square mile of drainage area, and a similar line drawn from the 11,600 cubic foot point on the vertical scale will represent a discharge of two cubic feet per second per square mile, and so on. These lines may be fairly regarded not only as indicating the flow per unit of area of the river at Necedah, but also the relative flow per unit of area of the Wisconsin River at points not greatly distant therefrom. At Kil- l>oum, (see Fig. 38) located on the same river about forty miles below Necedah, the flow may be assumed to be similar and pro- portionate to the flow at Necedah. Above Kilboum the drainage 2rea is 7,900 square miles, and with similar flow the discharge would be proportionately greater. The fact must be recognized, ^d acknowledged, that the hydrograph is strictly applicable only to the point at which^ it is taken, and that certain errors will arise in considering its application to other points, yet observations and comparisons show that, while such errors exist, they are not nearly so important as the errors which arise from the consideration of averages, either annually or monthly.
Consider, therefore, on this basis the Necedah hydrograph as ^hown in Fig. 37. On this diagram a flow of one cubic foot per second per square mile at Necedah, representing an actual flow of 5»8oo cubic feet per second at that point, would, by proportion, present a flow of 7,900 cubic feet per second at Kilbourn and,
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with a suitable change in scale, the diagram may be redrawn to rep- resent the flow at Kilboum as shown in Fig. 39. This same method can be applied to any point on the same river or to comparative points on different rivers.
S3* Reliability of Comparative Hydrographs. — It must be clearly understood that comparisons as above described hold good only as the conditions are essentially similar at the various points com- pared.
Stream flow at the best is very irregular and varies greatly from year to year. The actual departure from the truth can best be imderstood and appreciated from an actual comparison of flows on adjacent drainage areas where observations have actually been made for a term of years. From such an investigation, which can be made as extended as desirable, the true weight to be given to the comparative hydrograph can best be judged. It is not believed that the actual variations from the truth, as shown by carefully selected comparative hydrographs, will be any greater than the flow variations which actually take place from a drainage area from year to year under the varying conditions of rainfall and climate. This method, therefore, is believed to be a scientific and systematic one for the consideration and discussion of probable variations in stream flow at any given point, if its limitations and the modifying in- fluences known to exist on different drainage areas and under liferent geographical, geological and meteorological conditions are known and appreciated-
53, When no Hydrographs are Available. — In a new country where no observations are available either on the drainage area under consideration or on other areas adjacent thereto, the study of comparative hydrographs is impossible and a different method ol consideration must be used. If no data are available, time must be taken to acquire a reasonable amount of local information which should include not less than one year s observation. In addition to such observation a study as thorough as practicable should be made of the geology, topography, and other physical conditions that prevail on the water shed. Rainfall data is commonly avail- able for a much greater range of time than the observations of stream flow. The relations of rainfall to run-off are hereafter dis- cussed and approximate fixed relations are shown to exist between them. From such relations, and from a single year s observations, conclusions may be drawn as to the probable variations from the observed flow which will occur during the years where the rainfall
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The Hydrograph as a Power Curve, 89
varies greatly from that of the year during which observations are available. Such conclusions are necessarily unsatisfactory, or at least much less satisfactory than conclusions based on actual stream flow. The consideration of the best information available on any project is the basis on which the engineer should always rest his conclusions, and all relations which will throw light on the actual conditions should be g^ven careful attention. If a water power plant must be immediately constructed upon a stream con- cerning which little or no information is available, then the risk is proportionately greater, and safety is obtained only by building in such a conservative manner that success will be assured for the plant installed and on plans that will permit of future extensions should the conditions that afterward develop warrant an extension of the same.
54. The Hydrograph as a Power Curve. — ^The hydrograph, by a simple change in the vertical scale similar to that already consid- ered, may also be made to show graphically the variations in the power of the stream. If, for example, at Kilbourn, a constant fall of seventeen feet be assumed, then a flow of one cubic foot per second per square mile represents a total flow of 7,900 cubic feet per second, and this flow, under 17 foot head, will give a theoretical hydraulic horse power as follows :
H.P. = :?520X17.^ 15281
Now if a hydrograph be constructed on such a scale that the line of flow of one cubic foot per second per square mile will also repre- sent 15,261 horse power, the result will be a power hydrograph (sec Fig. 40), which represents the continuous (24 hours per day) theoretical power of the river under the conditions named.
On account of losses in the development of power the full theoret- ical power of a stream cannot be developed, and hence the actual power that can be realized is always less than the theoretical power of the stream. If it is desired to consider the actual power of the stream on the basis of developing the same with turbines of 80 per cent efliciency, the line representing the flow of one cubic foot per second per square mile will represent the actual horse power to an amount determined as follows :
A rxT> _7900X17X .80 7900X17 -,,,^ ^^•■^- 878 = 11 = ^^^
A hydrograph platted so that the line of one cubic foot per square mile will represent this amount, will represent the actual
90
Water Power
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horse power of the river at Kilbourn with the wheels working with the efficiency and under the head named. Such a hydrograph is shown by Fig. 41, referred to by the left-hand scale (A). Powcr^ however, is not always used continuously for twenty-four hours. If pondage is available the night flow may be stored and utilized during the day. If the flow of twelve hours at night is impounded and used during the day under the seventeen foot head, the power will be double that shown on scale A, and can be represented by another change in scale as shown by Fig. 41, referred to scale B. If the flow for the fourteen hours of night is stored and utilized in the ten hours of day, then the hydrograph can be made by another change in scale to represent the ten hours power as shown by Fig. 42.
The total horse power hours which are available from a stream for each day may be represented (either theoretically or actually) by multiplying the scale of continuous power by 24. The actual horse power available at Kilbourn under the conditions named is represented by scale C in Fig. 41. It will be noted that by pointing off one place in the figures of scale C, Fig. 41, the hydrograph will represent the same condition as shown in Fig. 4a.
CHAPTER V,
WATER POWER (Continued.)
THE STUDY OF THE POWER OF A STREAM AS AFFECTED BY HEAD.
55. Variations in Head. — In the previous chapter the graphical representation of stream flow has been considered. A method for the expression of the power resulting from the fluctuations of stream flow and under a constant head has also been shown. Ex- perience shows, however, that such a condition seldom if ever occurs. In some cases where the available head is a very large element of the possible power, the fluctuations may be so small as to be of little or no importance. In many other cases where the available heads are considerable, the importance of the fluctuation in head is comparatively small, under which condition the diagrams already discussed are essentially correct and are satisfactory for the consideration of the varying power of the stream. In power developments under the low heads available in many rivers, the fluctuation in head is almost or quite as influential on the con- tinuous power that may be economically developed from a stream wthe minimum flow of the stream itself.
The hydraulic gradient of a stream varies with the quantity of Wer flowing. At times of low water the fall available in almost every portion of its course is greater than is necessary to assure ^he flow between given points and frequent rapids result (see R. ^ %• 43) which are commonly the basis for water power develop-
rieed rrow.
M«dium Wiatttr Loiv Wafmr •tr«oni Bad.
Fig. 43. — ^Hydraulic Gradients of a Stream Under VarlOYiB Conditions
of Flow. • 0
94
Water Power.
ments. As the flow increases, however, a higher gradient anc greater stream section is necessary in order to pass the greater quantity of water, and the rapids and small falls gradually become obscured (as shown by the medium water lines, Fig. 43) or dis- appear entirely under the larger flows (as shown by the higher water linei Fig. 43) • Water power dams concentrate the fall of the
Ftg. 44. — Hydraulic Gradients of the Same Stream After the ConBtnietloii Dam and Under Various Conditions of Flow.
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river that is unnecessary to produce flow during conditions of lo' and moderate water (as shown in Fig, 44), and when the gradient of the water surface and the cross section of the stream are tn- ^ creased to accommodate the larger flow, the fall at such dams is frequently greatly reduced (as shown by the medium water line In Fig. 44) or, during high water, the fall is largely or completely de- stroyed (as shown by the high water lines in the Figtire), or at least is so reduced as to be of little or no avail under practical water power conditions. M
The cross section of the river bed, its physical character ana longitudinal slope, are the factors which determine the hydraulic gradient of a stream under different flows* They are so variable in character and their detail condition is so difiicult of determina- tion that sufficient know^ledge is seldom available, except possibly in the case of some artificial channels, to determine, with reason- able accuracy, the change of the surface gradient and cross section of the water under various conditions of flow. Where a power pi is to be installed, it is important to ascertain the relation of floi to head in order that the available power may be accurately detei mined. Where a river is in such condition as to make the &i termination of a discharge rating curve possible, either by din river measurement at the point in question or by a comparison wi the flow over weirs at some other point, such determination shoul be carefully made, as such knowledge is of the utmost importain in considering the problem of continuous power.
The Rating Curve,
95
S6. The Rating or Discharge Curve, — The rating curve, which will be discussed in some detail in a later chapter, is a hydrograph that represents the relation of the elevation of the v;^ater surface in a channel to the quantity of water passing a given cross section. The form of this curve varies with the various conditions of the cross section both at the immediate point and for a considerable distance above and below the location considered and can usually be de- termined only by detail observations. The rating curve is a uni- form curve only for channels in which no radical change in form of cross section occurs with the increase of fiow. (See A Fig. 45.) If, on account of o%^erflow conditions, or sudden enlargements of the cross section, that cross section varies radically in form at a given height, then at this elevation a radical change in the slope of the rating curve is likely to occur. (See B and C Fig, 45,)
m
Ftg, 45. — Tte Influence of the Stream Cross Section on the Rating Curve.
Any change in the bed of the stream may, and frequently does, modify to a considerable extent the rating curve, which must be expected to vary under such conditions to an extent that depends on the variations that take place in the cross section and elevation of the stream bed. Such variations, however, are not, as a rule, of great magnitude and consequently will not usually affect the head materially at a given point.
k-^
96
Water Power*
In Fig, 46, which shows the rating curve of the Wisconsin Rin at Necedah, Wis., as determined at different times during the years 1903 and 1904, an extreme change of head of about six inches will be noted for ordinary flows. When tlie change in head is of s
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Discharge in Ciu Ft. Per Second.
Fig. 46. — Eating CurTes^ Wisconsin River at Necedah, Wla*, Showing Ch; in Head Due to Changes In Cross Section.
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ficient importance to warrant the expense, the river channel may b^ so dredged out as to restore the original head when the reduction in head is occasioned by the filling of the section* ^
57. The Tail Water Curve. — It will be readily seen that while the rating curve sliows the relation between stream flow and river height prior to the construction of a dam, it will still represent the condition of flow below the dam after construction is completed. The water flowing over the dam will create a disturbed condition immediately below. If the velocity of the flow is partially checked , or entirely destroyed, a heading-up of the water may result beloi^fl the dam suflicicnt to give the velocity required to produce the iow^ in the river below, but it will soon reach a normal condition similar^ to that which existed previous to the construction of the dam*
58* The Head Water Curve, — In Chapter III is shown (see Fi| 35 and 36) the discharge curves over weirs of various forms and lh( formulas representing them are also quite fully discussed, Froi
The Graphic Representation of Head.
97
these formulas or diagrams a discharge curve can be readily cal- culated, with reasonable exactness, for a dam with a certain form and length of crest. Such a curve will show the height of the head waters above the dam and under any assumed conditions of flow. From the rating curve of the river at the point considered, and the discharge curve of the weir proposed, the relative positions of head and tail waters under varying conditions of discharge can be readily and accurately determined, and if a weir is to be built to a certain fixed height, it will be seen that the head under any given conditions of flow may be thus determined.
59. Graphic Representation of Head. — Fig. 47 shows the rating curve of the Wisconsin River (see lower curve marked "Tail Water
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OISCHARBC IN CUBIC rtlT PER SeOONO
Tig. 47.--Showing Head at the Kilboum Dam Under Various Conditions of
Flow.
Water Power,
I Curve") at Kilboum. On this diagram has also been platted scv-
■ eral discharge curves^ two being for a weir of 300 feet in lengtb I and two for a weir of 350 feet in length* Both weir curves in the I upper set are based on the assumption that the entire flow of water — I is passing over the weir. The crest of the dam is shown as raise^|
■ to gauge 19, and the distance between the rating curve, which now m represents the height of the tail water, and the weir discharge
■ curves, which represent the height of the head water (with two dif-
■ ferent lengths of weir) under different conditions of flow, wilt show I the heads that obtain at all times under these assumptions. I The entire discharge of the stream, however, will not pass over I the dam except when the plant is entirely shut down, which wouhl I seldom be the case. The essential information which is desired I therefore is the available head when the plant is in active operation.
■ At the Kilbourn plant the discharge of the turbines to be installed
■ under full head will be 7,000 cubic feet per second, hence, with the I plant in full operation, this quantity of water will be passing I through the wheels. Therefore in determining the relation between I head water and tail water it must be considered that with a flow of I 7»ooo cubic feet per second, the water surface above the dam will I be at the elevation of its crest, no flow occurring over the spillway,
■ and that only the flows greater than this amount will pass over I the dam. Another curve for each weir has therefore been added I to the diagram in which the zero of the weir curves is platted I from the point where the line representing the height of the dam I (elevation 19) intersects the line representing a discharge of 7,000 I cubic feet per second. From this diagram (Fig. 47) it will be seen I that other heads, shown in Table VIII, will obtain under variou I conditions of flow. I It will readily be seen that the line representing the height
■ the dam is not essential and that the curves may be platted relative I to each other, leaving the height of the dam out of the question
entirely and indeterminate. A curve constructed on this basis but otherwise drawn in the same manner as in Fig. 47, is shown in Fi^_ 48. In Fig. 48, wherever the weir or head water curves pass abovfl the tail water curve, it shows that an increase in the head will re-
Lsult under the corresponding condition of flow and wherever they pass below such curve, it shows that a decrease in the head will result under the corresponding condition of flow, the amount of which is clearly shown by the scale of the diagram- Consequently, :
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J
The Graphic Representation of Head.
99
of, no discharge, the head available under any other condition can be immediately determined from the diagram.
From this diagram the changes in head (as shown in table IX) can be determined and these, with a 17 foot dam, will give the total
TABLE VIII.
Oauge heightM and heada available at Kilboum Dam under varioue conditions
of flow, teith a length of ttpillway ofSOO and SSOfeet
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Hkad Water |
Tail Water. |
Head with |
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Flow in cabic feei per second. |
300 ft. dam. |
a^o ft. dam. |
300 ft. dam. |
350 ft. dam. |
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7000 |
10 23.9 25.2 27 28.5 30.2 31.5 32.7 |
19 22.3 24.6 26.2 27.7 29.3 30.4 31.6 |
2 6.1 8 10.3 12.2 13.6 14.7 15.6 |
17 17.8 17.2 16.7 16.5 16.6 16.8 17.1 |
17 |
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14000 |
17 2 |
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21000 |
16.6 |
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28000 |
15.9 |
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35000 |
15.5 |
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42000 |
16.7 |
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15.7 |
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56000 |
15.8 |
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heads available under various conditions of flow as shown in the last two columns. These heads will be seen to correspond with the heads given in table VIII.
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•II8MAI6C fr WlfCOMIN RIVCR AT KILIOORN —IN CUIIC FT. PCR 8CC. FIc- 48. — Showing Change in Head at Kilboum Dam Under Various Condi- tions of Flow.
6:J24;j?
lOO
Water Power.
TABLE IX.
Vhangev in heafi at Kilbourn D^m trifh lengths of crest ^/ SDO and S50 feet owrf under isarious conditionM of flow vnth result in ff total available head with 17 ft. dam.
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CflAXtifift IN |
HiiAD WITH |
TotAL Head with |
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Flow in cubic feet per eecoQd. |
300 (t. dam. |
350 ft. dam. |
300 ft. dam. |
350 ft. dam. , |
|
7000 , |
0 + ,8 + .2 — .3 — .5 — ,4 — ,2 + .1 |
0 + .2 — A — l.l —1.6 —1.3 —1-3 —1.2 |
17 17.8 17.2 16.7 16.6 16.6 16,8 17.1 |
17 |
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14000 |
17.2 |
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210t)0 |
16.6 |
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15*9 |
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350UO, ..* , |
16 6 |
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42000 **.***. |
15.7 i |
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49000...... |
15,7 ' |
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66000 - |
15.8 |
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I
60. Effects of Design of Dam on Head. — It should be noted in both of the last diagrams that the height of the water above the
dam is readily controlled by a change in the form and length of the weir; that a contraction in the weir length produces a corre* sponding rise in the head waters as the flow increaseSp while the lengthening of the weir will reduce the height of the head water under all conditions of flow. The physical conditions relative to overflow above the dam will control the point to which the head waters may be permitted to rise and will modify the length and the construction of the dam. Where the overflow must be limited, the waters^ during flood times, must he controlled either by a suffi- cient length of spillway or by a temporary or permanent reduction in the height of the dam such as the removal of flash boards, the opening of gates, or by some form of movable dam, ■
- Having determined the head available at all conditions of river flow, the hydrograph, as previously shown, may be modified to show the actual power of the river under the varying conditions of flow- The vertical scalei in this case, instead of being uniform must be variable as the head varies. Fig. 49 shows graphically the variation in the continuous theoretical power of the river taking into con- sideration the variation in head which wtll actually occur* Com- pare this hydrograph with Fig. 40 in which no variation in head IS considered. ^|
61. Effect of Head on the Power of the Plant — ^It is important^ at this point to take into consideration the effect of head and fiow^ on the actual power of the plant. In most rivers^ under flood coj
EfiEects of Design of Dam on Head.
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Water Power,
tions, the power theoretically available is largely increased, forJH while the head may diminish, the flow becomes so much greater that the effect of head on the theoretical power is more than off- set thereby. Practically; however, the conditions of head under which a given water wheel will operate satisfactorily (L e. at afl flxed speed) are limited, and, while the theoretical power of the river may radically increase, the power of the plant installed under such conditions will often seriously decrease, and under extreme conditions may cease entirely. The discharging capacity of any opening is directly proportional to the square root of the head, andl the water wheel, or water wheels, simply offers a particular fornlfl of opening, or openings, and operates essentially under this general law» With a fixed efficiency, therefore, the power which may be dereloped by a water wheel is in direct proportion to its discharging capacity and to the available head. Hence, the power of the wheel decreases as the product of these two factors, and therefore the power available under conditions of high flow and small head are much less than where the head is large and the total flow of the river is less. The only way, therefore, to take advantage of the large increase in theoretical power during the high water condi- tions is to install a surplus of power for the condition of average water. This may sometimes be done to advantage, but its extent soon reaches a practical limitation on account of the expense. ItH often becomes desirable to take care of such extraordinary condition by the use of supplemental or auxiliary power. Such power can^ usually also be applied during conditions of low water flow whe the power is limited by the other extreme of insufficient water undc maximum head.
In considering the effect of head on the power of a plants it is necessary to understand that water wheels are almost invariably selected to run at a certain definite speed for a given power plant and cannot be used satisfactorily unless this speed can be main- tained. Also that any wheel will give its best efRciency at a fixed speed only under limited changes in head. If the head change
L radically, the efficiency changes as well and this fact become more serious imder a reduction in head. As the head is reduced/ the discharging capacity of the wheel and its efficiency is also^ rapidly reduced so that the power of the wheel decreases moii rapidly than the reduction in the diseharj^^in^ capacity would indicate. When the reduction of head reaches a certain point thi wheel is able to simply maintain its speed without developinf
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