Water-power. An outline of the development and application of the energy of flowing water

Front matter

BOUGHT WITH THE INCOME FROM THE

SAGE ENDOWMENT FUND

THE GIFT OF

Preface

Water-power, which was formerly the chief reliance of mankind in industry, has been greatly overshadowed in recent times by the cheapness of coal and the development of the steam-engine. This condition, while permanent to all immediate practical intents, is nevertheless, upon a broad view of the subject, merely temporary. Two hundred and fifty years ago, when unbroken forests stretched from the Penobscot to the Wabash, the idea of a scarcity of wood and timber appeared grotesque. At the present day, the idea of the exhaustion of existing deposits of coal appears equally so. Nevertheless that time will certainly come. Our deposits of coal are finite ; every ton taken from the mines leaves a ton less to be mined. Coal does not have the power of reproduction even to the extent that wood and timber have. It does not, like water, have the power of rising in the form of vapor after having developed energy by descending. Coal deposits are simply an enormous store of fuel in rapid process of destruction by fire. Water, on the other hand, will continue to stand mankind in good stead long after the reluctant earth has yielded up its last ton of coal ; at least, long after coal has become too inaccessible and consequently too expensive to be used for power.

Recent remarkable developments in electricity and other modes of transmitting mechanical energy have recalled water-power to something like its former position in industrial economy. Whereas formerly it was necessary that the in-

IV PREFA CE.

dustry should be located at the waterfall, it is now possible to make large use of water-power for industries located many miles distant. These changed conditions have given a new interest to water-power and directed the attention of investors to sources which formerly appeared entirely outside the range of practical consideration. They have also led to some noteworthy improvements in utilization, development, and transmission of power.

These improvements are embodied in records of engineering societies, in the pages of engineering journals, in the plans of skillful engineers, in catalogues and advertisements of machinists. It has appeared to the writer no unworthy task to review these sources of information, to select what appears worthy of preservation, or necessary for the illustration of the subject and put it in an accessible and preservable form. In other words, to bring the entire subject of water-power up to date.

This is in no sense an elementary work. It does not deal in axioms, definitions, or the elements of science. It assumes the reader to be possessed of elementary notions of mechanics and mathematics. It discusses some subjects which cannot be made clear without recourse to the higher mathematics, and in these cases the writer does not hesitate to plunge boldly into , these branches of analysis. He does not conceive it necessary to offer any apology for this proceeding. It is true that there are successful engineers — successful at least in the sense of obtaining and retaining lucrative employments — who affect to despise mathematical knowledge, and make a boast of their ignorance of these branches. As well might the lawyer boast his ignorance of the common law. Mathematics is the common law of engineering. It is to engineering what the common law is to the legal profession. Granting that in practice no scientific principle can be applied, the truth of which has not been avouched by experiment, the fact remains that experiments cannot be intelligently conducted until their nature and scope have been indicated by the results of mathematical

PREFACE. V

inquiry ; neither can their results be intelligently coordinated and understood without recourse to the same potent instrument of investigation.

The work, however, is by no means valueless to readers of limited mathematical knowledge. These will be obliged to omit certain demonstrations, and content themselves with the results of the inquiry. They will find the main part of the work, entirely intelligible.

The work does not attempt to go much into constructive details. It directs the reader's attention to the principles involved in the design, and the results to be aimed at, without invading the province of the machinist, the carpenter, or the mason. The illustrations are conformable to this view, being generally no more elaborate than necessary to elucidate the principles to which they relate ; and these principles are often illustrated by sketches of very little value in a constructive point of view. Photographs, except to show general situations, have been avoided. A photograph of a finished machine is generally no more instructive to the student of mechanism than the photograph of a man in an overcoat to the student of anatomy.

Preface To The Third Edition

The passing of this book to a third edition appears to approve the author's judgment in attempting the work. In the interval a fact has transpired well calculated to awaken reflections as to the nature of the agency now chiefly relied on for mechanical energy. We have seen a dispute between coal-mine owners and their employees carry the prices of all grades of coal to more than double their ordinary level, and nearly a year elapse without witnessing a return to the same level. This fact appears to show an abnormal sensitiveness in the coal market to any dis-

VI PREFA CE.

turbance of the supply, and has an ominous significance as showing what the coal consumer has in future to expect from combinations of mine-owners or of mine workers.

In this edition some typographical and numerical errors which, despite great care, had crept into the former, have been eliminated. The subject of dams on soft bottom has been considerably extended, and a section added on dams in tidal waters. The length of wasteways in relation to the appurtenant drainage area has been more fully considered. The subject of tidal power has been outlined, and extended attention has been given to storage reservoirs. The subject of leases and rentals of water-power and land appurtenant thereto has also been more fully elaborated.

Boston, April, 1903.

Contents

XXVI.

XXVII.

rACB

Chapter I. Natural Watercourses

Physicists are wont to assert that the sun is the source of all power upon the earth. It creates fuel by separating carbon from the carbonic acid of the atmosphere and storing it in the substance of plants. In former times, when this constituent of the atmosphere was much more abundant than at present, it thus originated the vast accumulations of vegetable growth which now constitute the coal-mines. By rarefying parts of the atmosphere and disturbing the barometric equilibrium, it creates the winds, which involve enormous power. A current of air one mile wide, ioo feet high, and moving with the moderate velocity of 30 miles an hour represents an expenditure of more than 1 00 000 horse-power. The sun creates the food which nourishes animals and gives them the strength to perform labor. It also sets in motion the agencies which give rise to water-power.

Watercourses. — About three-fourths of the earth's surface is water, from which evaporation is constantly going on. Water in the form of vapor is constantly rising from the seas. It is distributed by the winds in very unequal measure throughout the land, falls in the form of rain or of snow which resumes the liquid form, and, gathering into streams, pursues its course toward the sea. Rivers owe their existence to this ceaseless play of natural forces, and to the further fact that the contraction of the earth's crust has tended to form its surface in a series of ridges and valleys.

Flow of Streams. — The agencies which sustain the flow of streams are of extremely variable and intermittent character, whence arises great variation in the flow of streams. No feature of streams is more striking than this. Rivers as large as the Connecticut at Hartford, with a drainage-ground of IOOOO square miles, carry forty times as much water at one time as another. The maximum flow of the Merrimac at Lowell, where its drainage-area is some 4000 square miles, is sixty or seventy times the minimum. The Kanawha River at Charleston, W. Va. , with a drainage-area of 9000 square miles, has varied in the ratio of one to one hundred. The Minnesota River at Fort Snelling, Minn., draining some 16 000 square miles, has been known to carry 8 or 10 cubic feet per second for every square mile of its drainage-area; at other times, not over 1 cubic foot per second to every 30 square miles. In small streams draining 10 square miles and under, the variations are almost without limit. The extent of the drainage-area together with the rainfall of the region forms the most reliable indication of the flow of a stream both in flood and low water.

The Slope of a stream or fall per mile, together with the quantity of water to be relied on, determines, its value for water-power. Nearly all the streams emptying into the Atlantic and the Gulf, except the Mississippi and its lower tributaries, have some points of resemblance in respect to slope. From the point where they are large enough to be regarded as rivers to where the influence of the tide becomes apparent, they have a fall of from 2 to 10 feet per mile. The slope is greatest in the upper reaches of the stream, and diminishes toward tide-water. It is by no means uniform. Shoals and rapids occur, often many miles in extent, with a fall much above the average, and stretches of quiet water with a fall much below.

STREAMS IN DIFFERENT SECTIONS. 3

Streams in Different Sections of the United States. — In

other respects there is a material difference between the streams of the Northern and those of the Southern States. This difference arises from three causes, viz. : i . The effect of glacial action, occasioning great deposits of drift over the Northern and especially the Northeastern section, which are wanting at the §outh. 2. The greater effects of volcanic action in the former. 3. The difference in climate. From the first of these causes it results that Northern rivers flow in beds of gravel and boulders, with valleys sloping gently upward from the stream, while Southern rivers flow mainly in canyons or trenches worn out of the original rock with nearly vertical escarpments, and often partly filled with alluvial deposits. To glacial action is also due the immense number of lakes which dot the surface of the Northern section, and exert a marked influence on the flow of streams, a feature entirely wanting at the South. To the second cause is due the prevalence of metamorphic rocks in the Northern section which by their upheaval have created so many falls and rapids, and by their great hardness defy the abrasive action of the water. The Southern river-beds usually lie in sedimentary rocks of nearly horizontal stratification and often very fragile structure. In the third place the heat is more intense in the Southern States, the hot weather of longer duration, and in the Southwestern part the rainfall is less. Little snow falls and no ice accumulates. The rains, when they do come, are more violent. These causes, together with the absence of any regulating effect of lakes or great bodies of drift, tend decidedly to increase the fluctuations of streams, diminishing the low-water flow and increasing the floods. In some streams of the extreme Southwest the variations of flow are literally infinite, their beds being dry at times and roaring torrents at others.

In regions where snow accumulates, streams flowing east or west, like the Ohio, the Missouri, or the Minnesota, are liable to severer floods than streams flowing south, like the Connecticut, the Hudson, and the Upper Mississippi. In the first, the

NATURAL WATERCOURSES.

melting takes place simultaneously along its course, and the water is all precipitated into the channel. In the latter, the melting extends slowly up the stream, the result of one day's melting being well on its way before the next day's contribution enters the channel. Streams flowing in a northerly direction, like the Red River of the North and some of the tributaries of the St. Lawrence, are subject to great floods during the breaking up of the ice, which commences at the head of the stream and moves northward, constantly encountering firmer ice. The ice accumulates in immense masses, jams and dams the water, which is all the time increasing from the melting of snow. On the Red River of the North it is no uncommon thing to see driftwood among the branches of trees 40 feet above the ground. Such conditions are very trying to any artificial structure placed in the river.

In the Northeastern section, in the latitude of Boston, the total volume of water carried annually by streams is distributed throughout the year very much as follows :

January. . .

February .

March . . .

May

June

10 per cent.

July.

2 per cent.

August 3

September. . 3 October. ... 5 November. . 6 December. . . 8

In the Mississippi valley, in the same latitude, this proportion is changed by a different distribution of the rainfall. In the extreme North, the lowest water occurs in the winter, the temperature remaining below freezing, often for months in succession, during which time no water enters the streams except from springs. The extremely low stage of the Minnesota above referred to occurred in the winter, after a long period of extremely cold weather. In the South Atlantic and Gulf States there is no snow to affect the spring flow. The streams are usually high during the winter and become very low during the latter part of the summer and autumn. In Texas and New

RELATION OF SLOPE, DEPTH, AND VELOCITY. $

Mexico the normal flow of the streams is maintained bysprings, the result of rainfall in previous years. The drainageareas being vastly greater than for streams of equal flow in other sections, a concurrence of rains in all the tributaries gives rise to an enormous flood, which comes down with the velocity of a bursting reservoir or a tidal bore. It subsides rapidly and the river very soon returns to its normal condition.

Nearly one-half the territory of the United States, called the Arid Region, comprising the entire area west of the iooth meridian except the narrow margin between the Pacific Ocean and the Coast Range, is a region deficient in water, the rainfall ranging from 6 inches per annum in Nevada to 30 inches in other parts. The entire State of Nevada and a part of Utah lie in a basin which has no drainage to the sea, all the water that falls upon it being removed by evaporation. In many parts of the Arid Region no agricultural production is possible without artificial irrigation, and in no part of it can agriculture be carried on with full success without this aid. The interests of irrigation on all the streams are held to be paramount to those of water-power, and the latter thus becomes of quite secondary importance throughout this part of the country.

Table 1 gives some facts of interest in regard to the flow of streams in different parts of the United States. It is taken from vol. 16 of the 10th United States Census Report, and from the 18th Annual Report of the U. S. Geological Survey.

Relation of Slope, Depth, and Velocity in Streams.— If we carefully determine, by levelling, the fall in a given length of a running stream, and divide the fall expressed in feet by the length in feet, this quotient is called the slope and is represented by the letter S. Suppose the length to be a mile, and

2 the fall 2 feet, the slope is — ^ = 0.000379. Let FiS- l

represent the cross-section of a river-channel, ab the surface, acdb the bottom. Measure the depth at a number of equidistant points; the average of these measurements is the mean depth. Measure also the velocity at a number of equidistant

NATURAL WATERCOURSES.

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RELATION OF SLOPE, DEPTH, AND VELOCITY. "J

points by methods to be hereafter explained, and thus determine the mean velocity, which is designated by v. Determine the length acdb of the line representing the bottom. This

Fig. i.

length is called the "wetted perimeter." The mean depth multiplied by the width ab is the area of the cross-section. This area, divided by the wetted perimeter, is called the " hydraulic mean depth " or the " mean radius," and is represented by R. It is obvious that in a natural river-channel the wetted perimeter is not materially different from the width, and that no great error can result from taking the one equal to the other. In an artificial canal with vertical side walls, Fig. 2, the wetted perimeter may differ very materially from

Fig. 2.

the width. The relation between the slope, depth, and velocity, in a stream, is ordinarily expressed by the formula

v = cVRS (i)

Though this is the best formula that modern hydraulic science has to offer for the case in hand, it is still very imperfect. The factor c varies between such wide limits that no general value can be assigned to it. It is not only different for different streams and for different localities on the same stream, but it varies for the same locality at different stages of the stream, which shows that it does not truly express the law of the phenomenon. Nevertheless, in default of a better expression, we have to make such use as we can of it. The only rational use that the writer is able to make of it is by the aid of a table of data contained in the work of Ganguillet and Kutter, translated by Trautwine and Hering. This contains results of measurements by different experimenters on a great number of streams and channels from which it is usually possible to select the value of c suited to the case we have in view.

For. rivers and canals with fairly regular channels in earth, with values of R less than 0.5 c may be 30 ;

from O.

to

45;

" 1

1 1

55;

" 2

i I

" 65;

" 3

1 i

80;

" 4

1 i

" 100.

On great rivers, where/? is from 30 to 75 c may run from 125 to 275.

Flowage Occasioned by Dams. — Questions as to the extent to which the surface of a stream is raised by a dam are of very frequent occurrence. Such questions as to an existing dam can be best determined by levelling. It is often required, however, to find how far the flowage of a proposed dam will extend, which can only be determined by calculation. Even in that case a survey of the river-channel would be made and numerous cross-sections taken of the stream and valley. In endeavoring to elucidate the principles of the case, with no existing data, we must proceed upon mere supposition as to the contour of the ground.

We will assume a stream 100 feet wide, a maximum depth of 6 feet, and a mean depth of 4 feet, the area of cross-section being 400 square feet. Suppose the fall to be 2 feet per mile and the mean velocity 3 feet per second, which would make s = 0.000379 and would imply a value of yj for c, the relation of slope, velocity, etc., being expressed by the formula

v = 77 VRS.

FLO WAGE OCCASIONED BY DAMS. 9

We will assume a parabolic cross-section for the valley in

which the river flows, a supposition which ordinarily represents

the facts reasonably well. The equation of the parabola is

yz — 2Px, where x and y are the coordinates of any point in

the curve, x being, in this case, the maximum depth, and y the

semi-width. We have the relation 50 — 2P X 6, so that

2500 zP = , = 417, and when the water rises, the relation

between depth and width will be expressed by

(semi- width)' = 417 X max. depth.

That is to say, for purposes of computation we may put the width of the stream as raised by a dam equal to

2 ^417 X max. depth = 40.8 ^max. depth.

We neglect the small error that results from assuming the width equal to the wetted perimeter, which allows us to put R equal to two-thirds the maximum depth.

Now suppose the surface of the stream to be raised 12 feet by a dam. If the water in the pond stood at an exact level, the flowage would extend 6 miles up the stream. This point where the natural surface of the stream is at the same level as the water at the dam is called the hydrostatic limit of the flowage, and its distance from the dam is called the hydrostatic amplitude. The surface of water, however, cannot be at an exact level while it is in motion. The water in the pond is moving, very slowly near the dam, more rapidly further up. Some head is required to maintain the movement, and the stream will be somewhat raised at the hydrostatic limit. The velocity increases and the depth diminishes as we go up the stream, so that no formula can apply to the entire reach under consideration. The best we can do is to divide the channel into lengths such that the velocity and depth can, without serious error, be regarded as constant in each, and compute the slope for each length. For an approximate computation,

IO NATURAL WATERCOURSES.

we will take lengths of a mile each, and will consider the

cross-section and velocity at the middle of each mile section.

At a distance of half a mile up-stream from the dam, we

should have the maximum depth 6 -\- 12 — 1 = 17 feet.

The mean depth we take at two -thirds the maximum, viz.,

11.333, and this is approximately equal to R. The width is

40.8 V17 = 168.3. F°r the increased depth we must adopt a

higher value of c, say c = 105. The cross-section is 11.333

We have therefore for the fall in the first mile

(0.629)2

For the second mile we have the depth i£ miles above the

, . 1200

cross-section = 1580, velocity = — - = 0.760. We may

take c = 100. We have therefore for the fall in the second mile

„ (°76)8

For the third mile we have the max. depth = 12 -(- 6 5

= 13, R = 8.67, width = 40.8 Vi 3 = 147. 11, cross-section = 1275. In this section the slope of the surface sensibly affects the cross-section.

We have for the fall in the first mile 0.0167

Second mile 0.0305

Estimated in half the third mile 0.0300

Correction for the cross-section 0.0772 X 147. 1 1 = 11.36. Making the corrected cross-section 1286.36,

. . 1200

velocity = i2g6 -6 = 0.933. We take c = 95.

FLO WAGE OCCASIONED BY DAMS. II

(o.cnO2 Fall in third mile = 5280. \a „ ^ = 0.0587.

For the fourth mile we have max. depth at 3J miles

12 + 6—7= 11.0000

Add fall in first, second, and third miles and half of

fourth, estimated at 0.06 1659

Corrected max. depth 11. 1659

»0(

We may take c = 90.

Fall in fourth mile = 52807 — ^ = 0.1225.

For the fifth mile we have for the max. depth

124-6 — 9= 9.0000

Add fall in first, second, third, fourth, and half of fall

in fifth, estimated at 0.1450 0.3734

Corrected max. depth 9. 3734

Fall in fifth mile = 5280^, x ^ = 0.2700.

For the sixth mile max. depth = 7.8188, R = 5.213, width = 40. 8 Vy. 81 88 = 114.09, cross-section = 594.73, velocity 1200

594-73

2.018, c may be taken at 81.

Collecting these several quantities we have:

Fall in first mile 0.0167

At the hydrostatic limit the water is raised . . 1 . 1 269

To pursue the computations above the hydrostatic limit we adopt a method of trial and error which is in many hydraulic problems the only available method. We assume a value for the swell and execute the computations on that assumption. The result will show how the assumption should be amended. Guided by preliminary computations which need not be introduced, I adopt 0.732 as the rise of the surface at middle of the seventh mile. This makes the max. depth 6.732, mean depth 4.488, width 40.84/6.732 = 105.86, cross-section 475.10, velocity = 2.5258. We here take c = 79.

Fall in the seventh mile = 5280^.3 — - j^g = 1.2026.

This result makes the surface at the middle of the above what we may call the pond-level. Our assumption would make it 0.732 4- 1 foot fall in original surface 1.7320

At the end of the seventh mile the surface of the above pond-level. The natural surface was 2.

Swell of stream 1 mile above the hydrostatic limit. . 0.3295

In like manner, guided by preliminary computations, we assume the swell at the middle of the eighth mile o. 1 89, max. depth 6 . 1 89, mean 4.126, width 101.50, cross-section =418.79, velocity = 2.8654. Take c = 78.

FLOWAGE OCCASIONED BY DAMS. 13

This result makes the swell at the middle of the

Our assumption makes it o. 1 890

At the end of the eighth mile the swell is

That is, at 2 miles above the hydrostatic limit the swell is about two-thirds of an inch.

In applying these methods to an important case, the engineer would take half-mile and quarter-mile sections near the hydrostatic limit of the flowage. He would draw the cross-sections, compute the areas, measure the wetted perimeter, and thence compute the values of R. He would also be guided by the nature of the bed and banks of the stream in finding the values of c.

It is worth while to remark that before the building of the dam the entire power of the stream was expended in giving motion to the water. After the water is raised by the dam a very slight fall suffices for the movement of the water, leaving the greater part available for power.

It is also to be remembered that the above computation of the flowage takes account only of the immediate and usually temporary effect of the dam. In the course of time other agencies come into operation which may have the effect of extending the flowage much farther up the stream. The stream, in high water, is constantly bringing down sediment, either earthy matters held in suspension, or gravel and pebbles rolling on the bottom. The heavier matters are arrested at the head of the pond where the velocity begins to slacken ; the lighter are deposited farther down. Both, by diminishing the waterway, tend to extend the swell of the dam. In the course of years these deposits consolidate and harden and do not yield readily to an increased velocity of the current. Some dams are carried away with considerable regularity at every

J J NATURAL WATERCOURSES.

recurrence of extreme high water, an event which carries with it the removal of the accumulated deposits. In this case, the evil referred to only becomes apparent upon the construction of a permanent dam. In cases of litigation between millowners and parties claiming to be aggrieved by flowage, the dam, in order to demonstrate that it does not create the flowage complained of, is virtually removed, by opening waste-gates and drawing down the pond till the fall is practically obliterated. It is urged that the flowage cannot be created by the dam, since it does not disappear upon removal of the same. This is an inconclusive test. It is necessarily made at a low stage of the stream and lasts but a few hours, during which time the current is entirely inadequate to the removal of the deposits which have accumulated in the course of years at the head of the pond. The fact remains that these deposits were occasioned by the dam and would disappear upon its permanent removal.