Encyclopaedia Britannica [14, 11 ed.]

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THE

ENCYCLOP .tED lA BRITANNICA ELEVENTH

FIRST SECOND THIRD FOURTH FIFTH SIXTH SEVENTH EIGHTH NINTH TENTH ELEVENTH

EDITION

e dition , published in three " " " " " " " " "

"

ten

volumes,

eighteen twenty

twenty " twenty " twenty-one " twenty-two " twenty-fi ve " ninth edition and eleven

" " , " " " " "

1 768-1771. 1777-1 784. 1788-I797• 18or-r8ro. r8rs-r8r7. !823-1824. I830-I842. I853-186o, r87s-1889.

supplementary volumes,

1902-19(>3•

published in twenty-nine volumes,

I9IQ-I9II•

COPYRIGHT in

all countries subscribing to the Bern Convention

by

THE CHANCELLOR, MASTERS AND SCHOLARS of the UNIVERSITY OF CAMBRIDGE

.All n'ghts rmrv;ed

THE

ENCYCLOPlED lA BRITANNICA A

DICTIONARY ARTS,

SCIENCES,

OF

LITERATURE

INFORMATION

ELEVENTH

AND

EDITION

VOLUME XIV

HUSBAND

to

ITALIC

New York

Encyclopcedia Britannica, Inc. 342 Madison Avenue

GENERAL

Copyright, in the United States of America, 1911, by

The Encyclopredia Britannica Company.

INITIALS USED IN VOLUME XIV. TO IDENTIFY INDIVIDUAL CONTRIBUTORS,1 WITH THE HEADINGS OF THE ARTICLES IN THIS VOLUME SO SIGNED.

A. Ba.

ADOLFO BARTOLI ( 1 833-r8q4).

A. Bo.*

AuGuSTE BouorNHON, D . D . , D .C.L.

Forn:erly Professor of Literature at the lstituto di studi superiori at Florence. Author of Sto i1� della letteratura Italiana ; &c. Profcssor of Canon Law at the Catholic University of Paris. Paris. Editor of the Canoniste contemporain.

A. Cy. A. C.

ARTHUR ERNEST CowLEY, M.A . , LrTT.D.

Sub-Librarian of the Bodleian Library, Oxford.

Honorary Canon of

{

{

and Fishes in the British Museum ; Reptiles of British India ; Fishes of Zanzibar ; Reports �n the " Chaltenger " Fishes ; &c. REV. ALFRED ERNEST GARVIE, M.A . D . D .

A.E.H.L.

AuGusTus EDWARD HouGH LovE, M.A., D.Sc., F.R.S.

,

A. F.C.

{-

Principal of New College, Hampstead. Member of the Board of Theology and the Board of Philosophy, London University. Author of Studies in the inner Life of Jesus ; &c.

Hon. Sedleian Professor of ?\atural Philosophy in the University of Oxford. Fellow of Queen's College, OxfGrd; formerly Fellow of StJohn's College, Cambridge. Secretary to the London Mathematical Society.

ALEXANDER FRANCIS CHAMI!ERLAIN, A.M., PH.D.

MAJOR ARTHUR GEORGE FREDERICK GRIFFITHS (d. rgo8). H.M. Inspector of Prisons, I8?8-I8g6. Author of The Chronicles of Newgate ; Secrets of the Prison House ; &c.

A. Ge.

SIR ARCHIBALD GEIKIE, LL.D .

See the biographical article, GEI KIE ,

REV. ALEXANDER GoRDON, M. A

A. G. G.

SrR ALFRED GEORGE GREENHILL, M.A., F.R.S.

Formerly Professor of Mathematics in the Ordnance College, Woolwich. Author ; Notes on of Differential and Integral Calwlus with A pplications ; Hvdrostatics · Dynamics ; &c.

A.M. C.

AGNES MARY CLERKE.

General in the Persian Army.

Author of

{ .

A.N.

ALFRED NEWTON, F.R.S.

A. So.

ALBRECHT SociN, P H D . (r844-r8gq).

JHutton' l

{

A.W.H.*

.

.

{{

.

Isfahan

.

(in part). .

Huygens, Chnst1aan.

{ Ib"

.

the Geolog1cal Society, London.

'

Hydromechamcs.

Formerly Professor of Semitic Philology in the Universities of Leipzig and Tiibingen. Author of A rabische Grammatik ; &c.

ARTHUR SMITH WooowA�D, LL.D , F.R.S. : Keep�r of _Geol_ogr, Natural H1story Museum,

James

Illummar1•

{

Eastern Persian Irak.

See the biographical article, N EWTON, ALFRED.

Wo.

.

ldent1ficatron.

See the biographical article, CLERKE, A. M.

w. R.

Infinitesimal Calculus'

�

.

SIR A. HOUTUM-SCHINDL ER, C.I.E.

A.

Immortality; Inspiration.

{

Lecturer on Church History in the University of Manchester.

A. H.- S.

.

.

Semitic.

(in part) .

Ichthyology

{

SIR A.

A. Go.*

A.W Po

��

bitorum·' Infallibility ·

Assistant Professor of Anthropology, Clark University, Worcester, Massachusetts. Indians' North American. Member of American Antiquarian Society; Hon. Member of American Folk-lore 1 l Society. Author of The Child and Childhood in Folk Thou�ht.

A. G.

A.S.

(in part).

Index Llbromm Prohi-

J Ibn

Keeper of Zoological Department, British Museum, I8?S-I89S- Gold Medallist, Royal Society, 1878. Author of Catalogues of Colubrine Snakes, Batrachia Salientia ,

A. E. G.*

.

a irol; l lnscnptiOns:

Fellow of Magdalen College.

ALBERT CHARLES LEWIS GoTTHILF GUNTHER, M . A., M.D., PH.D., F.R.S.

G.

Italian Literature

South Kensington.

ARTHUR -WILLIAM HOLLAND.

Formerly Scholar of St John's College, Oxford. 1900.

Secretary of

,

Bacon Scholar of Gray s Inn,

ALFRED WILLIAM POLLARD, M.A.

Assistant Keeper of Printed Books, British Museum. Fellow of King's College, London. Hon. Secretary Bibliographical Society. Editor of Books about Books and Bibliographira. Joint-editor of The Library. Chief Editor of the " Globe "

Chaucer. ALEXANDER WooD RENTON, M.A., LL. B .

Puisne Judge of the Supreme Court of Ceylon.

Laws of England.

Editor of

{

IS; ICterus.

f Ichthyosaurus; -l Iguanodon

{

{

•

Imperial Cities; Instrument of Governmen•... ·

Incunabula.

r Inebriety

Encyclopaedia of the I Insanity:' Law. �

J ;\ complete list, showing all individual contributora. appears in the final volume. v

(in part) .

Irak-Arabi

Law

of•"

c.

{

INITIALS AND HEADINGS OF ARTICLES

n

P.A.

CHARLEs FRANCIS ATKINSON.

Formerly Scholar of Queen's College, Oxford. Captain, rst City of London (Royal Fusiliers). Author of The Wilderness and Cold Harbour.

COLONEL CHARLES GRANT.

C.. G.

Formerly Inspector of Military Education in India.

C. H. Ha.

CARLTON HUNTLEY HAYES, A.M., PH.D.

C. Ll. M.

CoNWAY LLOYD MoRGAN, LL.D., F.R.S.

C. R. B.

c. S.* C. T. L.

C. We. D. B.Ma.

D. G. H.

Assistant Professor of History at Columbia University, New York City. of the American Historical Association.

Member

�

Pn;fessor of Psychology at the U ni v�r si t y of Bristol. P�incipal of niversity College , Bnstol, r887-·1909. Author of A nzmal Ltfe and lntellzgence; Habtt and lnstmct.

{

{

CH ARLES R.-w�ro:-;D BEAZLEY, ::\LA., D.LITT . , F.R.G.S., F.R.HIST.S. Professor of :\lodern History in t he Cniversity of Bir min gham . Formerly

Felln are or otherwise, any real and personal property, in the same manner existing or not at the dissolution of the marriage. On the death of as if she were a feme sole, without the intervention of any trustee. the husband, his children take one-third (called legitim) , the widow The property of a woman married after the beginning of the takes' one-third (jus relictae) , and the remaining one-third (the dead act, whether belonging to her at the time of marriage or acquired part) goes according to his will or to his next of kin. I f there be no children, the jus relictae and the dead's part are each one-half. If after marriage, is held by her as a feme sole. The same is the case the wife die before the husband, her representatives, whether children with property acquired after the beginning of the act by a woman or not, are creditors for the value of her share. The statute above­ married before t he act. After marriage a \Voman remains liable mentioned, however, enacts that " where a wife shall predecease her for antenuptial debts and liabilities, ancl as between her and her husband, the next of kin, executors or other representatives of such wife, whether testate or intestate, shall have no right to any share of husband, in the absence of contract to the contrary, her separate the goods in communion ; nor shall any legacy or bequest or testa­ property is deemed primarily liable. The husband is only mentary disposition thereof by such wife, affect or attach to the said liable to the extent of property acquired from or through his goods or any portion thereof." It also abolishes the rule by which wife. The act also contained provisions as to stock, investment, the shares revert if the marriage does not subsist for a year and a day, Several later acts apply to Scotland some of the principles of the insurance, evidence and other matters. The effect of the act English Married Women's Property Acts. These arc the Married was to render obsolete the law as to what created a separate Women's Property (Scotland) Act 1877, which protects the earnings, use or a reduction into possession of choses in action, as to equity &c., of wives, and limits the husband's liability for antenuptial debts to a settlement, as to fraud on the husband's marital rights, of the wife, the Married Women's Policies of Assurance (Scotland) Act r 88o, which enables a woman to contract for a policy of assurance and as to the inability of one of two married persons to give for her separate use, and the Married Women's Property (Scotland) a gift to the other. Also, in the case of a gift to a husband and Act r 88 I, which abolished the jus mariti. A wife's heritable property docs not pass to the husband on wife in terms which would make them joint tenants if unmarried, they no longer take as one person but as two. The act contained marriage, but he acquires a right to the administration and profits. His courtesy, as in English law, is also rccogHized. On the oth.cr a special saving of existing and future settlements ; a settlement hand, a widow has a terce or life-rent of a third part of the husband 's being still necessary where it is desired to secure only the enjoy­ heritable estate, u nless she has accepted a conventional provision. Continental Eu rope . Since 1 882 English legislation in the matter ment of the income to the wife and to provide for children. The act by itself would enable the wife, without regard to family of married women's property has progressed from perhaps the most backward to the foremost place in Europe. By a curious contrast, claims, instantly to part with the whole of any property which the only two European countries where, in the absence of a settle­ might come to her. Restraint on anticipation was preserved ment to the contrary, i ndependence of the wife's property was recog­ by the act, subject to the liability of such property for antenuptial nized , were Russia and Italy. But there is now a marked tendency debts, and tc the power given by the Conveyancing Act r88r towards contractual emancipation. Sweden adopted a law on this subject in 1874, Denmark in r 88o, Norway in 1 888. Germany to bind a married woman's interest notwithstanding a clause followed, the Civil Code which came into operation in 1 900 (Art. of restraint . The Married Women's Property Act of 1893 1367) providing that the wife's wages or earnings shall form part of repealed two clauses in the act of r 88 2 , the exact bearing of her Vorbehaltsgut or separate property, which a previous article -

4

HUSHI-H USS

( 1 365) placed beyond the h usband's control. As regards property accruing to the wife in Germany by succession, will or gift inter vivos, it is only separate property where the donor has delit;erately stipulated exclusion of the husband's right. In France it seemed as if the system of community of property was ingrained in the institutions of the country. But a law of 1907 has brought France into line with other countries. This law gives a married woman sole control over earnings from her personal work and savings therefrom. She can with such money acquire personalty or realty, over the former of which she has absolute control. But if she abuses her rights by squandering her money or administering her property badly or imprudently the husband may apply to the court to have her freedom restricted. A merican Law.-In the United States, the revolt against the common law theory of husband and wife was carried farther than in England, and legislation early tended in the direction of absolute equality between the sexes. Each state has, however, taken its own way and selected its own time for introducing modifications of the existing law, so that the legislation on this subject is now exceedingly complicated and difficult. James Schou\er (Law of Domestic Relations) gives an account of the general result in the different states to which reference may be made. The peculiar system of Homestead Laws in many of the states (see HOMESTEAD and ExEM PTION LAW!ii ) constitutes an inalienable provision for the wife and family of the householder.

HUSHI (Rumanian Hufi) , the capital of the department of Falciu, Rumania ; on a branch of the Jassy-Galatz railway, 9 m. W. of the river Pruth and the Russian frontier. Pop. ( 1 900) I 5,404, about one-fourth being Jews. Hushi is an episcopa: see. The cathedral was built in I49I by Stephen the Great of Moldavia. There are no important manufactures, but a large fair is hdd annually in September for the sale of live-stock, and wine is produced in considerable quantities. Hushi is said to have been founded in the I 5th century by a colony of Hussites, from whom its name is derived. The treaty of the Pruth between Russia and Turkey was signed here in I 7 I r . HUSKISSON, WILLIAM (I 7 7o-I83o) , English statesman and financier, was descended from an old Staffordshire family of moderate fortune, and was born at Birch Moreton, Worcester­ shire, on the uth of March I 7 7o. Having been placed in his fourteenth year under the charge of his maternal great-uncle Dr Gem, physician to the English embassy at Paris, in I 783 he passed his early years amidst a political fermentation which led him to take a deep interest in politics. Though he approved of the French Revolution, his sympathies were with the more moderate party, and he became a member of the " club of I 789," instituted to support the new form of constitutional monarchy in opposition to the anarchical attempts of the Jacobins. He early displayed his mastery of the principles of finance by a Disc ours delivered in August I 790 before this society, in regard to the issue of assignats by the government. The Discours gained him considerable reputation, but as it failed in its purpose he withdrew from the society. In January I 793 he was appointed by Dundas to an office created to direct the execution of the Aliens Act ; and in the discharge of his delicate duties he mani­ fested such ability that in I 795 he was appointed under-secretary at war. In the following year he entered parliament as member for 1\Iorpeth, but for a considerable period he took scarcely any part in the debates. In I 8oo he inherited a fortune from Dr Gem . On the retirement of Pitt in I8oi he resigned office, and after contesting Dover unsuccessfully he withdrew for a time into private life. Having in I 804 been chosen to represent Liskeard, he was on the restoration of the Pitt ministry appointed secretary of the treasury, holding office till the dissolution of the ministry after the death of Pitt in January I8o6. After being elected for Harwich in I 8o7, he accepted the same office under the duke of Portland, but he withdrew from the ministry along with Canning in I 809. In the following year he published a pamphlet on the currency system, which confirmed his reputation as the ablest financier of his time; but his free-trade principles did not accord with those of his party. In I 8 I 2 he was returned for Chichester. When in I814 he re-entered the public service, it was only as chief commissioner of woods and forests, but his influence was from this time very great in the commercial and financial legislation of the country. He took a prominent part in the corn-law debates of 1 8 14 and I 81 5 ; and in 1 8 1 9 he presented a memorandum to Lord Liverpool advocating a large

reduction in the unfunded debt, and explaining a method fOf the resumption of cash payments, which was embodied in the act passed the same year. In 1 82I he was a member of the committee appointed to inquire into the causes of the agricultural distress then prevailing, and the proposed relaxation of the corn laws embodied in the report was understood to have been chiefly due to his strenuous advocacy. In I 823 he was appointed president of the board of trade and treasurer of the navy, and shortly afterwards he received a seat in the cabinet. In the same year he was returned for Liverpool as successor to Canning, and as the only man who could reconcile the Tory merchants to a free trade policy. Among the more important legislative changes with which he was principally connected were a reform of the Navigation Acts, admitting other nations to a full equality and reciprocity of shipping duties; the repeal of the labour laws; the introduction of a new sinking fund; the reduction of the duties on manufactures and on the importation of foreign goods, and the repeal of the quarantine duties. In accordance with his suggestion Canning in 1 827 introduced a measure on the corn laws proposing the adoption of a sliding scale to regulate �he amount of duty. A misapprehension between Huskisson and the duke of Wellington led to the duke proposing an amend­ ment, the success of which caused the abandonment of the measure by the government. After the death of Canning in the same year Huskisson accepted the secretaryship of the colonies under Lord Goderich, an office which he continued to hold in the new cabinet formed by the duke of Wellington in the following year. After succeeding with great difficulty in inducing the cabinet to agree to a compromise on the corn laws, Huskisson finally resigned office in May r 829 on account of a difference with his colleagues in regard to the disfranchisement of East Retford. On the I 5th of September of the following year he was accidentally killed by a locomotive engine while present at the opening of the Liverpool and Manchester railway. See the Life of Huskisson, by J. Wright (London 183 1 ) . HUSS (or Hus) , JOHN (c. 13 73-14I 5) , Bohemian reformer and martyr, was born at Hussinecz,l a market village at the foot of the Bohmerwald, and not far from the Bavarian frontier, between I 3 73 and 1 3 7 5, the exact date being uncertain. His parents appear to have been well-to-do Czechs of the peasant class. Of his early life nothing is recorded except that, notwithstanding the early loss of his father, he obtained a good elementary education, first at Hussinecz, and afterwards at the neighbouring town of Prachaticz. At, or only a very little beyond, the usual age he entered the recently (1348) founded university of Prague, where he became bachelor of arts in 1393, bachelor of theology in I 394, and master of arts in 1396. In 139S he was chosen by the Bohemian " nation " of the university to an examinership for the bachelor's degree; in the same year he began to lecture also, and there is reason to believe that the philosophical writings of Wycliffe, with which he had been for some years acquainted, were his text-books. In October 1401 he was made dean of the philosophical faculty, and for the half-yearly period from October 1402 to April 1403 he held the office of rector of the university. In 1402 also he was made rector or curate (capellarius) of the Bethlehem chapel, which had in 1 391 been erected and endowed by some zealous citizens of Prague for the purpose of providing good popular preaching in the Bohemian tongue. This appoinment had a deep influence on the already vigorous religious life of Russ himself; and oo C!25 ?(46ro+o)_; . the weight, inertia, pressure, &c., which produce the visible t e ��eJl\�ct� � {t �� ��� �e�����c�f';he 1�r�:��;:.twnal to the absolute mot ions i � is th e �bject of hydraulics to estimate. On the other p a re h d , wh 1e a fl d pas: es cas!_ 1y from one form to another, it an The value of f(�r water at 7 i ° Fa hr. is, according to H . von . Helmholtz and G. P1otrowski, opposes considerable resistance to change of volume. p. = o·o?o or8 8, . . It is easily deduced from the absence or smallness of the t�e u!nts bemg the same as belore. For water decreases rapidly tangential stress that contiguous portions of fluid act on each w1th mcrease of temperature. . . . other w1th a pressure whtch 1s exactly or very nearly normal § 4· ·when a fluid flows in a very regular manner, as for instance to the interface which separates them. The stress must be a

I

1

K

•

JJ.

p.

1

Ul

JJ.

f.l.

·

pressure, not a tension, or the parts would separate.

Further,

at any point in a fluid the pressure in all directions must be the

same; or, in other words. the pressure on any small element of surfa ce is independent of the orientation of the surface. § 2 . Fluids are divided into liquids, o r incompressible fluids, and gases, or compressible fluids. Very great changes of pressure change the volume of liquids only by a small amount, and if the pressure on them is reducer! to zero they do not sensibly dilate. In gases or compressible fluids the volume alters sensibly for small changes of pressure, and if the pressure is indefinitely diminished they dilate without limit. . In ordinary hydraulics, liquids are treated as absolutely incompressible. In dealing with gases the changes of volume which accompany changes of pressure must be taken into · account. 3 \"iscous fluids are those in which change of form under a § · cont inued stress proceeds gradually and increases indefinitely. A very viscous opposes great resistance to change of form in a short time, and yet may be deformed considerably by a block of pitch is more small stress acting for a long period. easily splintered than indented by a hammer, but under the action of the mere weight of its parts acting for a long enough t ime it 11attens and flows like a liquid. .\ll actual t1uids are viscous. They oppose a resistance tv the relative motion of their parts. This resistance diminishes with t he velocity of the relative motion , and becomes zero in a fluid the parts of which arc relatively at rest. When the motion of different parts of a fluid is small, the viscosity may be neglected without introducing important errors. On the other hand, where there is considerable relative motion , the viscosity may be ex"" pccted to have an influence ' a /a ' b ::::::]. / '6' I too great to be neglected. I

fluid

A

out

relative ___

___

T ' I '

I I

I I

I I I I

c �.w'.a-0�;.:�;@ &rz;zwJa%

I I I I I I I I

}.fr.asurement of Viscosity. C o effi c i e n t of Viscosity.­ ab, fig. :

Suppose the plane of area to move with the velocity V relat1velv to the FIG. I . surface The cd and paraflel to it. �et the spac;c b-- �-:-� , , � "'

-

.

------- _ _ _ _ _

• .•

1 I1

-- ----

- ----

-----

----··

- · · · - - -- - - � - · ·· ---.... . ,,.• ';.'.•.a AI. : . '

J1 ')'bl))

j

\ �... .lJ - - - - - - - - - - - - ---=-·· ·"'

A

FIG. 6.

been conceived that the variations of direction and magnitude of the velocity are periodic, and that, if for each point of the stream the mean velocity and direction of motion were substituted for the actual more or less varying motions, the motion of the stream might be treated as steady stream line or steady laminar mot ion . § 1 3 . T'olume of Flow.-Let A (fig. 6) be any ideal pln of motion, and let V

A :l

A_.

--�- """'--- ..,...,..,

I

f)

FIG. 8.

(j

'"

"8

account. If the motion is steady the weight of fluid between two cross sections of a stream must remain constant. Hence the weight flowing in must be the same as the weight flowing out. Let P1, p. be the pressures, v�o v2 the velocities, G1, G2 the weight per cubic foot of fluid, at cross sections of a stream of areas A1, A2 • The volumes of inflow and outflow are A1v1 and A2v2, and, if the weights of these are the same, G1A1v1 = G2A2v2 ; and hence, from (Sa) § 9, if the temperature is constant, p1A1v1 = p2A2v2• (3 )

HYDRAULICS § 1 5· Stream Lines.-The characteristic of a perfect fluid, that is, fluid free from viscosity, is that the pressure between any two parts into which it is divided by a plane must be normal to the plane. One consequence of this is that the particles can have no rotation impressed upon them, and the motion of such a fluid is irrotational. A stream line is the line, straight or curved, traced by a particle in a current of fluid in irrotational movement. In a steady current a

[DISCHARGE OF LIQUIDS

shipshape body such as a screwshaft strut. The arrow shows the direction of motion of the fluid. Fig. IO shows the stream lines for a very thin glycerine sheet passing a non-shipshape body, the stream lines being practically perfect. Fig. I I shows one of the earlier air-bubble experiments with a thicker sheet of water. I n this case the stream lines break up behind the obstruction, forming an eddying wake. Fig. I2 shows the stream lines of a fluid passing a sudden contraction or sudden enlargement of a pipe. Lastly, fig. 13 shows the stream lines of a current passing an oblique plane. H. S. Hele Shaw, " Experiments on the Nature of the Surface Re­ sistance in Pipes and on Ships," Trans. Inst. Naval A rch. ( I 897). " I nvestigation of Stream Line !\lotion under certain Experimental Conditions," Trans. Inst. Naval A rch. ( r 8g8) ; " Stream Line Motion of a Viscous Fluid," Report of British A ssociation ( r 8g8). I I I . PHENOME NA OF THE DISCHARGE OF LIQUIDS FROM ORIFICES AS ASCERTAINABLE BY EXPERIMENTS § I6. When a liquid issues vertically from a small orifice, it forms a jet which rises nearly to the level of the free surface of the liquid in the vessel from which it flows. The difference of level h, (fig. I4) is so small that it may be at once suspected to be due either to air resistance on the surface of the jet or to the viscosity of the liquid or to friction against the sides of the orifice. Neglecting for the moment this small quantity, we may infer, from the eleva­ tion of the jet, that each molecule on leaving the orifice possessed the velo­ city required to lift it I against gravity to the I I height h. From ordinary I j j dynamics, the relation between the velocity and height of projection is given by the equation

l FI G. 9·

each stream line preserves its figure and posttlon unchanged, and marks the track of a stream of particles forming a fluid filament or elementary stream. A current in steady irrotational movement may be conceived to be divided by insensibly thin partitions follow­ ing the course of the stream lines into a number of elementary streams. If the positions of these partitions are so adjusted that the volumes of flow in all the elementary streams are equal, they represent to the mind the velocity as well as the direction of mo�i?n of the particles in different parts of the current, for the veloetttes

. I I

______

.

FIG . I I. FIG. I 2 . are inversely proportional t o the cross sections o f the elementary streams. No actual fluid is devoid of viscosity, and the effect of viscosity is to render the motion of a fluid sinuous, or rotational or eddying under most ordinary conditions. At very low velocities in a tube of moderate size the motion of water may be nearly pure stream line motion. But at some velocity, smaller as the diameter of the tube is greater, the motion suddenly becomes tumultuous. The laws of simple stream line motion have hitherto been investi­ gated theoretically, and from mathematical difficulties have only been determined for certain simple cases. Professor H. S. Hele Shaw has found means of exhibiting stream line motion in a number of very interesting cases experimentally. Generally in these experiments a thin sheet of fluid is caused to flow between two parallel plates of glass. I n the earlier experiments streams of very small air bubbles introduced into the water current rendered visible the motions of the water. By the use of a lantern the image of a portion of the current can be shown on a screen or photo­ graphed. In later experiments streams of coloured liquid at regular distances were intro­ duced into the sheet and these much more clearly marked out the forms of the stream lines. With a fluid sheet o · 02 in. thick, the stream lines were found to be stable at almost any required velocity. For certain simple FIG. 13. cases Professor Hele Shaw has shown that the experimental stream lines of a viscous fluid are so far as can be measured identical with the calculated stream lines of a perfect fluid. Sir G. G. Stokes pointed out that in this case, either from the thinness of the stream between its glass walls, or the slowness of the motion, or the high viscosity of the liquid, or from a combination of all these, the flow is regular, and the effects of inertia disappear, the viscosity dominating everything. Glycerine gi,·es the stream lines very satisfactorily. FIG. 9 shows the stream lines of a sheet of fluid passing a fairly FIG. IO.

!

·

v = .,f 2gh.

___

_ _

(I)

As this velocity is nearly reached in the flow from FIG. I 4 . well-formed orifices, it is sometimes called the theoretical velocity of discharge. This relat10n was first obtained by Torricelli. If the orifice is of a suitable conoidal form, the water issues in filaments normal to the plane of the orifice. Let w be the area of the orifice, then the discharge per second must be, from eq. (I), Q = wv = w,/zgh nearly. (2) This is sometimes quite improperly called the theoretical dis­ charge for any kind of orifice. Except for a well-formed conoidal orifice the result is not approximate even, so that if it is supposed to be based on a theory the theory is a false one. Use of the term Head in llydraulics.-The term head is an old millwright's term, and meant primarily the height through which a mass of water descended in actuating a hydraulic machine. Since the water in fig. 14 descends through a height h to the orifice, we may say there are h ft. of head above the orifice. Still more generally any mass of liquid h ft. above a horizontal plane may be said to have h ft. of elevation head relatively to that datum plane. Further, since the pressure p at the orifice which produces outflow is connected with h by the refation pjG = h, the quantity pfG may be termed the pressure head at the orifice. Lastly, the velocity v is connected· with h by the relation v2/2g = h, so that v'fzg may be termed the head due to the velocity v. § I 7 . Coefficients of Velocity and Resistance.-As the actual velocity of discharge differs from v 2gh by a small quantity, let the actual velocity where

Cv

= Va = Cv../2gh,

(3)

is a coefficient to be determined by experiment, called the coefficient of velocity. This coefficient is found to be tolerably c_:on­ stant for different heads with well-formed simple orifices, and it very often has the value 0·97. The difference between the velocity of discharge and the velocity due to the head may be reckoned in another way. The total height h causing outflow consists of two parts-one part h, expended effectively in producing the velocity of outflow, another hr in over­ coming the resistances due to viscosity and friction. Let where

h, = c, h"

is a coefficient determined by experiment, and called the coefficient of resistance of the orifice. It is tolerably �;onstant for different heads with well-formed orifices. Then Cr

t•a = v 2gh, = v \2ghf(r +cr) ].

(4)

DISCHARGE OF LIQUIDS]

and c,. for any orifice is easily found :­ �·" = c,-./ 2gh = -./ (2J;h/(I +c.)} c, = -./ ( 1 /(r +c.)} . (5 ) c. = r /c,2 - I . (Sa) Thus if c, = 0 · 97, then Cr =o·o62S. That is, for such an orifice about 6l % of the head is expended in overcoming frictional rf water, in which the stream lines are concentric circles, and i n which --;=== =-

I

I

P3

q I

�

......

��

=[\

:

I I - -: - v

: l

I jJ, ,G I I I I I I ',

iiiiiliiii ----1 � r

p iG = a2r'/2g+constant.

(9)

Let p,, r,, v1 be the pressure, radius and velocity of one cylindrical section, P2, r2, v2 those of anut�1cr ; t:1en

I I I

' • iii liiii liii liii J . _ _ _ _ _ _ _ _ _ J_ i_ !_ I I I

4 .5

HYDRAULICS

STEADY MOTION OF FLUIDS]

I

/

....

-­

p J (G-o.2r1 2 /2g = p, jC-a2 r22/2g; (p2-p i )fC = a'( rl-r,')/2!!, = (z'i'-v,2)/zg.

----

That is, the pressure i ncreases from within outwards \ - �f I I

I I I I I

-

____, -d - - - .J.

� __

m

(Io) a curve

;:1\ ;I \

I

P.

� I I

: I A I

� iii Iii;; liiii �·��-�� - 1 -1- - - - - - H ;_ -1\1- -\ - !- - - _J_ - -:1- ... I I

�

I

I

I

I

I I

FIG. 36.

I

the total head for each stream line is the same. Hence, if by any slow radial motion portions of the water strayed from one stream line to another, they would take freely the velocities propel to their· new positions under the action of the existing fluid pressures only . . For such a current, the motion being horizontal, we have for all · ' the circular elementary streams H = P/G +v'/2g = constant ;

:.dH = dp/G +vdv/g = o.

Consider two stream lines at radii r and r +dr (fig. 36). (2), § 3 3 , p = r and ds = dr, v2dr/gr+vdv/g = o, dv/zo = -dr/r,

(7)

Then i n

V ., I jr, (8) precisely as in a radiating current ; and hence the distribution of pressure is the same, and formulae 5 and 6 are applicable to this case. Free Spiral Vortex.-A s in a radiating and circular current the equation' of motion are the s a m e , they will also apply to a vortex in which the motion is compounded of these motions in any pro­ portions, provided the radial component of the motion varies in­ versely as the radius as in a radial current, and the tangential component varies inversely as the radius as in a free vortex. Then the whole n· locity at any point will be inversely proportional to the rad ius of the poin t , and the fluid will describe stream line; having a constant inclination to the radius drawn to the axis of ti.e current. That is, the stream lines will be logarithmic spirals. \\'hen water is delivered from the circu mference of a centrifugal pump or turbine into a chamber, it forms a free vortex of thi;; kinrl. The water flows spirally outwards, its velocity diminishing and its

I

1 FIG. 37· which in radial sections is a parabola, and surfaces of equal pressure are paraboloids of revolution (/ 2gH velocity of approach is negligible. Let 00 he the surface level in the = 5 · 35c(l -o · InH1H}. reservoir, and let be at a height h" below 00, and h' above a'. This is Francis's formula, in which the coefficient of discharge c is Let h be the distance from 00 to the weir crest and e the thickness much more nearly constant for different values of l and h than in of the stream upon it. Neglecting atmospheric pressure, which has the ordinary formula. Francis found for c the mean value o-622, no influence, the pressure at a is Gh" ; at a' it is Gz. If be the the weir being sharp-edsed. velocity at § 43 · Triangular Notch (fig. 46).-Consider a lamina issuing be­ v2/2g = h' +h" -z = h - e ; tween the depths h and h +dh. Its area, ne:,;lecting contraction, will Q = bd 2g(h -e). be bdh, and the velocity at that depth is ..; (2gh). Hence the dis­ Theory does not fu:-nish a value for but Q =o for e =o and for charge for this lamina is e = h. Q has therefore a maximum for a value of e between o and h, obtained by equating dQ/de to zero. This gives e �h, and, inserting b� 2gh dh. this value, But B/b = H/(I-1 - h) ; b = B (H -h)/H. Q =0 ·385 bh � zgh, Hence discharge of lamina as maximum value of the di3charge with the conditions assigned. = B (H - h) V (2gh)dhjH ; Experiment shows that the actual discharge is very approximately and total discharge of notch H equal to this maximum, and the formula is more legitimately ap­ J = Q = B v (2g) (H - h) h�dh/H plicable to the discharge over broad-crested weirs and to cases such = 15 B � (2g)I-!L as the discharge with free upper surface through large masonry ,

I

---

--------

--1

r

- - - - · - - ---· - - ··-· · - -

�

a5

_ _

I

"'

0

a

v

' a ,

e,

=

a

o

Coefficients for the D ischarge over Weirs, derived from the E-cperiments of T. E. Blackwell. same head, and the results were pretty uniform, the resulting coefficients are marked 2s very strongly marked.

I 1

Head; in inches measured from still 'Water in Reservoir.

Sharp Edge .

3 ft. long.

· 677 ·675 ·630 ·f17 ·602 " 593

Planks 2 in. thick, square on Crest .

3 ft. long.

6 ft . long.

I

Crests 3 ft . wide.

ro ft . long, wing-boards making a n angle of 6o0•

I o ft . long.

3 ft. long, level.

3 ft. long, 3 ft . long. fall I in I 8 . fall r in 1 2 .

1

•467 · S4S •754 •459 "4S2 " 435 •509* ·561 ·675 •546 "482 · s b 5* · S37 ·563* · S97* ·569* "441 '431 •419 ·602* · S49 ·656 •57 S 4 ·588 ·516 ·67! ·60! * ·609* "479 s ·501 * · S93* ·6o8* •576* 6 .. . •513 •488 ·617* ·6o8* •576* .. 7 . ·6o6* •590* •548* "470 .. "491 I 89 ·6oo ·569* ·558 * "476 . "492* .. . . ·614 * •539 .. . ·S34 * I 1012 . .. •525 . "534* . . .. .. . ·S49* .. .. . l 14 The discharge per second varied from ·461 to ·665 cub. ft. in two experiments. The coefficient 1 2 3

.

.

.

. .

1

r o ft . long.

I

When more than one experiment was made with the with an (*). The effect of the converging w ing-boards

·809 ·803 ·642* ·656 ·65o* .. -581 "530 ..

. .

. .

. .

. .

.

. .

'

.

.

.

.

.

.

·467 •533 "S39 •45S .

.

"53 I

·527 . . "498

6 ft . hng, level.

.

.

" 492* " 497* · S07 " 497 . •48')* •465* •467* . .

.

IO ft. bng, level.

·381 •479* . .

.

.

·518 · SI3 . •468 •486 " 455 .

I o ft . long, fall I in 1 8

•467 '495* .. ·SIS . "S43 . · S07 . .

.

I ·435 is derived from the mean value. . . .

.

. .

.

.

.

.

. .

.

.

. .

.

.

.

HYDRAULICS 49 Q, = �cb� :ig{h! - h,!j t sluice openings than the ordinary weir formula for sharp-edged (3) weirs. It should be remembered, however, that the friction on Q2 = cb(h, -h)� zgh f · the sides and crest of the weir has been neglected, and that this In the case of a rectangular notch or weir, h, = o. Inserting this tends to reduce a little the discharge. The formula is equivalent value, and adding_ the two portions of the discharge together, we get to the ordinary weir formula with o- S77 for a drowned wetr Q = cb� 2gh(h, - h/3), (4) SPECIAL CASES OF DISCHARGE FROM ORIFICES § 4 S- Cases in which the Velocity of Approach needs to be taken :-vhere h is the difference of level of the head and tail water, and h2 into Account. Rectangular Orifices and Notches.-In finding the 1s the head from the free surface above the weir to the weir crest so). wlocity at the orifice in the preceding investigations, it has been (fig.From some experiments by Messrs A. Fteley and F. P. Stearns assumed that the head h has been measured from the free surface ( Trans. Am. Soc. C.E., 1 883, p. 102) some values of the coefficient of still water above the orifice. In many cases which occur in can be reduced practice the channel of approach to an orifice or notch is not so large, relatively to the stream through the orifice or notch, that the h,jh, c' h,jh, c n�locity in it can be disregarded. o- 629 0· 1 · 0•7 O S 78 Let ht. Jv, (fig. 48) be the heads measured from the free surface to o- 8 0·6I4 O· S83 the top and bottom edges of a rectangular orifice, at a point in the 0· 600 O·S96 o- 607 O·S90 o-s8z 0·628 O·S78 If velocity of approach is_t_aken into account, let � be the head due to that velocity ; then, addmg 9 to each of the heads in the equations (3), and reducing, we get for a weir Q = chi 2g[(h,+9) (h + 9 ) i - Hiz + 9 ) ! - � 9l] ; ( S) an equation which may be useful in estimating flood discharges. Bndr,e Pzers and oth�r Obstructions in Streams.-When the piers of a bndge are erected m a stream they create an obstruction to the flow of the stream, which causes a difference of surface­ level above and below the FIG. 48 pier S I). If it is neces­ channel of approach where the velocity is u. It is obvious that a sary (fig. to estimate this differ­ fall of the free surface, ence of level, the flow between the piers may be � = u2/zg as if it occurred over has been somewhere expended in producing the velocity u, ami treated drowned weir. But the hence the true heads measured in still water would have been h, + f) avalue of in this case is and h, +b. Consequently the discharge, allowing for the velocity imperfectly FIG. so. known. of approach, is § 47 · Bazin's Researches on Q = �cbl2g{ (Jv, + 9 ) � - (h, +Wl(r) Weirs.-H. Bazin has executed a long series of researches on the And for a rectangular notch for which h , = o, the discharge is flow over weirs, so systematic and complete that they almost supersede other observations. The account of them is contained Q = �cb�2g{ ( h2 + () ) l - (Jil(z) a series of papers in the A nnales des Pants et Chaussees In cases where u can be directly determined, these formulae give the in(October January I89o, November I891, February I894, discharge quite simply. \Vhen, however, u is only known as a December I888, 2nd trimestre I898). Only a very abbreviated function of the section of the stream in the channel of approach, they account canI896, be given here. The general plan of the experiments become complicated. Let n be the sectional area of 2the channel was to establish first the coefficients of discharge for a standard where h, and h._ are measured. Then u = Q/ n and 9 = Q /zg Q2 without end contractions; next to establish weirs of other This value introduced in the equations above would render them weir in series with the standard weir on a channel with steady excessively cumbrous. In cases therefore where n only is known, types flow, to compare the observed heads on the different weirs and it is best to proceed by approximation. Calculate an approximate to determine their coefficients from the discharge computed at value Q' of Q by the equation the standard weir. A channel was constructed parallel to the Canal de Bourgogne, taking water from it through three sluices Q' = �cb� zg{h2� -h1lj. X 1 ·0 metres. The water enters a masonry chamber I S metres Then f) = Q'2/zgQ2 nearly. This value of 9 introduced in the equations 0·3 by 4 metres wide where it is stilled and passes into the canal above will give a second and much more approximate value of Q. long § 46. Partially Submerged Rectangular Orifices and Notches.­ at the end of which is the standard weir. The canal has a length \\'hen the tail water is above the lower but below the upper edge of I S metres, a width of 2 metres and a depth of I ·6 metres. From of the orifice, the flow in the two parts of the orifice, into which it is d i v ided by the surface of the tail water, takes place under different conditions. :\ filament M,m, (fig. 49) in the uppl,r part of the orifice issues with a hmd h' which may have any value between DISCHARGE FROM ORIFICES]

c=

c

-�-� � �

c

�-- -�?��-��-:�-�--;�;-����0:� -

�M4V/£bdj���'

•-

_

·

---t---,-----------·

;.L

' '

1

k

I

S I .l this extends a channel zoo metres in length with a slope of I mm. per metre. The channel is 2 metres wide with vertical sides. The channels were constructed of concrete rendered with cement. The water levels were taken in chambers constructed near the canal, by floats actuating an index on a diaL Hook gauges were used in ........L determining the heads on the weirs. Standard Wcir.-The weir crest was 3·72 ft. above the bottom of the canal and formed by a plate t in. thick. It was sharp-edged with free overfalL It was as wide as the canal so that end con­ tractions were suppressed, and enlargements were formed below FIG. 49 · the crest to admit air under the water sheet. The channel below h, . and h. But I { r +p./Gh} ;

iJ

n h,

'

;E '

, z

:E

H . ..

FIG. 63.

B is the reservoir of water to be pumped ; C is the reservoir into which the water is pumped. D I SCHARGE WITH VARYING HEAD § 53 · Flow from a Vessel when the Effective Head varies with the Time . Various useful problems arise relating to the time of empty­ ing and filling vessels, reservoirs, lock chambers, &c. , where the flow is dependent on a head which increases or diminishes during the operation. The simplest of these problems is the case of filling or emptying a vessel of constant horizontal section. -

Time of Emptying or Filling a Vertical-sided Lock Chamber.­

Suppose the lock chamber, which has a water surface of !] square ft. , is emptied through a sluice in the tail gates, of area w, placed below the tail-water level. Then the effective head producing flow through the sluice is the difference of level in the chamber and tail bay. Let H (fig. 64) be the initial difference of level, h the difference _ _ _ .J:_ _ _ _ - ...._ H 1-dh - -

· - - - - - :- - - ,.- - - - - - - - - - - - - - - - -l - - - .._ _ I

�-=--=:-_ -==:;_ - - - - - -

or, putting

p.(G = 3 4 ft. , if !]/w � Y { ( h+34)/h}.

In practice there will be an interruption of the full bore flow with less ratio of !:1/w, because of the disengagement of air from the water. But, supposing this does not occur, the maximum discharge of a mouthptece of this kind is Q = wy ( zg(h+p.(G )l ; that is, the discharge is the same as for a well-bellmouthed mouth­ piece of area w, and without the expanding part, discharging into a vacuum. § j2. Jet Pump.-A divergent mouthpiece may be arranged to act a,; a pump, as shown in fig. 62. The water which supplies the energy

a

n

Q = !:1v1 = c,ny (2gh).

Hence the discharge de­ pends on the area of the stream at EF, and not at QoF all on that at C D , and the latter may be made as small as we please without affecting the amount of FIG. 60. water discharged. There is, however, a limit to this . . As the velocity at CD is greater than at EF the pressure is less, and therefore less than atmospheric pressure, if the discharge is into the air. If CD is so contracted that p = o, the continuity of flow is impossible. In fact the stream disengages itself from the mouthpiece for some value of p greater than o (fig. 6 r ) . From the equations,

F I G. 6 1 .

D

FIG.

64.

o f level after t seconds. Let - dh be the fall of level i n the chamber during an interval dt. Then in the time dt the volume in the chamber is altered by the amount - ndh, and the outflow from the sluice in the same time is cw.Y (2gh)dt. Hence the differential equation con­ necting h and t is .cw'o/ (zgh)dt+nh = o.

53

HY D RA U LICS

DISCHARGE FROM ORIFICES]

Forvalue the time t , during which the initial head H diminishes to any not be too small (say half a foot) to decrease the effects of errors of other h, measurement. The section of the jet over the weir should not exceed one-fifth the will section streaminto behind the weir, or the velocity approach needoftothebe taken account. A triangular notch - {r!./(cw" 2g) l f�dh/V h = J:dt. ofis very suitable for measurements of this kind . If the flow is variable, the head h must be recorded at equidistant . · .t = 20(" H - " h)/{cw" (2g) l intervals of time, say twice daily, and then for each I 2 hour pe iod = (rl/cw){" (2H/g) - " (2h/g)l. For the whole time of emptying, during which h diminishes from Scale Wetr H to o, T = (r!./cw)" (2H/g). Comparing this with the equation for flow under a constant head, it will be seen that the time is double that required for the discharge of an equal volume under a constant head. The time of filling the lock through a sluice in the head gates is exactly the same, if the sluice is below the tail-water level. But if the the sluice is above the tail-water level, then the head is constant till level of the sluice is reached, and afterwards it diminishes with the time. PRACTICAL UsE OF ORIFICES IN GAUGING WATER § S-+· If the water to be measured is passed through a known orifice under an arrangement by which the constancy of the head is ensured, the amount which passes in a given time can be ascertained by the formulae already given. It will obviously be best to make the orifices of the forms for which the coefficients are most accurately determined; hence sharp-edged orifices or notches are most com­ moniy used. Water In ch . For measuring small quantities of water circular sha.-p-edged orifices have been used. The discharge from a circular orifice one French inch in diameter, with a head of one line above the FIG. 66. top edge, was termed by the older hydraulic writers a water-inch. common estimate of its value was 14 pints per minute, or 677 A the discharge must be calculated for the mean of the heads at the English cub. ft. in 24 hours. An experiment by C. Bossut gave beginning and end of the time. As this involves a good deal of 634 cub. ft. in 24 hours (see Navier's edition of Belidor's A rch. troublesome calculation, E. Sang proposed to use a scale so graduated llydr., p. 2I2). as to read off the discharge cubic feet per second. The lengths of . L. J. \Veisbach points out that measurements of this kind would be principal the ?fin such a �cale easily calculated by· made more accurately with a greater head over the orifice, and he puttmg Q = I graduations 3 • 2• · · ions the ordmary areformulae notches proposes that the head should be equal to the diameter of the orifice. the int.e�mediate graduat_ may be taken ac'-1) = {plviY/('Y - I ) ){ I (v{'f-1 - I /v/ -1) = { p,v,f('Y - I) l ( I - (v1jv2)Y -1 } . (2 ) Gt . G2

(Pt /GI) ( I /('Y - I ) ) ( I - (G,fGI )">'-1 ) , = P1Vt{ I ( (-y - I ) ) ( I - (p2/p1)CY-I)/Y) .

(2a) (2b) § Modification of the Theorem of Bernoulli for the Case of a Compressible Fluid.-

m

:!.. -

A

iect, p per

w

pw

D

w

w

B

w

A,

w

w

P w

z�o z2

!

W

VI. STEADY FLOW OF COMPRESSIBLE FLUIDS.

§ 6r . External Work during the Expansion of Air.-1£

wz,

FIG.

v2 , G ,

j�pdv

t

G1w1v1t

=

G,w2v2t.

�

t

- f · · · · · - - · ·1':f· · · · · · · · · · · · . · -V . . . . .;,.;

f-------j.....

�

P1w1V1l -p,w,v,t, = CP1/Gt -pz(G,) Wt. Wt lb

is

(2g) (v22 -v,2 )t.

�:

dv = (Wj2g) (v,2 -v12) t ; vl : . z1+P1/G1+v12/2g = zz +P'/Gz+vz2 /2g pdv. (I)

v2•

p

(v,2 -v12 )/2g = (pt/G1)

(fig. 76) .

P1

v v1

v + dv v2 , 2 11 pdv . VJ

pdv ,

(2)

(2 a ) (p1(p,). § 6I ) Z1 +P.fG, +v12/2g = z2+P2(G,+v,2/2g - (pt(G,){ I /('Y -'-- I )) 1 l I - (Pz/Pt ) n- l 1r ) ; (3)

v �o

p,

v

t

G1w1v1t(z1 - z2) = W(z1 - z,)t,

Ia, § 6 r ) z1 +P1/G1 +vt'j2g = z2 +p,fG, +v,2(2g - (pt!G1) (G1/G,). P1/G1 = p,(G, ; (p1fp,) , :. z1+v12/2g = z2+v22/2g - (pt (G1)

I lb

I

77.

B

W(z1 -z2)t + (ptfGi - p,(G,)Wt+Wt

· · · ·V1 · · · -

B'

B:

w

P.

p2 ,

B

t.

P�o w�o V�o G1

B.

A'

It

2a,

(v,2 -v12)j2g = (pt/G1)[I + I/('Y - I ) ( I - (p,jp1)CY-1 ) /y)] - p,(G,. pt(p,

(3a)

§ 63. Discharge of Air from an Orifice.-

z, +P1/G1 +vt'/2g = z, +Pz(G, +v,2/zg - CP1/G1)( I ((-y - I ) ) [ I _ (p,fp1)CI-1)/y},

HYDRAULICS 57 the expansion being adiabatic, because in the flow of the streams of It is easily deduced from Weisbaeh's theory that, if the pressure air through an orifice no sensible amount of heat can be communi­ external to an orifice is gradually diminished, the weight of air dis­ charged per second increases to a maximum for a value of the ratio cated from outside. Suppose the air flows from a vessel, where the pressure is p, and p,fp, {2/(-r + I ) } Y - r /y the velocity sensibly zero, through an orifice, into a space where the = 0·527 for air pressure is p2• Let v2 be the velocity of the jet at a point where the = o· 58 for dry steam. convergence of the streams has ceased, ·so that the pressure in the For a further decrease of external the discharge diminishes, jet is also p, . As air is light, the work of gravity will be small -a result no doubt improbable.pressure new view of Weisbach's compared with that of the pressures and expansion, so that z,z, formula is that from the point whereThe the maximum is reached, or may be neglected. Putting these values in the equation abovenot greatly differing from it, the pressure at the contracted section ceases to diminish. p J/G, = P / G + v 2/2g - (pJ(G,) ( I /(-y - I ) } f I - (p,jp, ) < Y - r)/y ; A. F. Fliegner showed (Civilingenieur xx., I874) that for air flow­ Vz2/2g = pJ(G, - p,(Gz + (pJ/G,) { I/(-y - I ) l { I - (p,;p,) (y- r)/y l ing from well-rounded mouthpieces there is no discontinuity of the = (pJ /G1) { / ( - I ) - ( P2fp1) - r /y/ (-y - I ) } -P,/Gz. law of flow, as Napier's hypothesis implies, but the curve of flow But pJ /G1"� = p2 (G2"� :. p2/G2 = (pJ/G1) (p2/p1)(y - r) /y bends so sharply that Napier's rule may be taken to be a good approximation to the true law. The limiting value of the ratio v22/2g = (PdG,){-y/(-y - I) } { I - (p2/p1)-n, and n is taken 1 · 795, then values of m at each temperature are praGtically constantm. Temp. F. m. Temp. F. 57 70 So 90

I ·72 I • 75 I · 8S 1 ·8 7 1 •95 2 ·0 2·0

The vanation of each of these coefficients is within a comparatively narrow range, and the selection of the proper coefficient for any given case presents no difficulty, if the character of the surface of the pipe is known. It only remains to give the values of these coefficients when the quantities are expressed in English feet. For English measures the following are the values of the coefficients:Kind of Pipe.

I

Tin plate Wrought iron Asphalted iron Riveted wrought iron K ew cast iron Cleaned cast iron Incrusted cast iron

m

·0265 ·0226 ·0254 ·0260 ·02 1 5 ·0243 · OHO

X

n

I · IO 1·21 1 ·127 1 • 390 1 · 168 I · 1 68 I · I 6o

1 • 72 I •75 1 · 85 1 · 87 1 •95 2·0 2·0

I

Dtstribution of Velocity in the Cross Sectwn of a Ptpe. Darcy made experiments with a Pitot tube in I 850 on the velocity at different points in the cross section of a pipe. He deduced the relation § 78.

-

\' - v = 1 I ·3(r�jR) � i,

where V is the velocity at the centre and v the velocity at radius r in a pipe of radius R with a hydraulic gradient i. Later Bazin repeated the experiments and extended them (A1em . de l'Academie des Sciences, xxxii . ?\o . 6). The most important result was the ratio of mean to central velocity. Let b = Ri/C2, where U is the mean velocity in the pipe ; then V/U = I +9·03 v b. A very useful result for practical purposes is that at 0·74 of the radius of the pipe the velocity is equal to the mean velocity. Fig. 84 gives the velocities at different radii as determined by Bazin. § 79 · Influence of Temperature on the Flow through Pipes . Very careful experiments on the flow through a pipe o· I 236 ft. in diameter -

FIG. 84.

0•000276 0•000263 o·ooo257 0·000250

I OO I IO I 20 1 30

0·000244 0•000235 0·000229 0·000225 0·000206

I6o where again a regular decrease of the coefficient occurs as the temperature rises. I n experiments on the friction of disks at different temperatures Professor \V. C. Unwin found that the re­ sistance was proportional to constant X (r-o·002It) and the values of m given above are expressed almost exactly by the relation m =0·0003 I I ( I -0·002 1 5 t). I n tank experiments on ship models for small ordinary variations of temperature, it is usual to allow a decrease of 3 % of resistance for I 0 ° F. increase of temperature. § So. Influence of Deposits in Pipes on the Discharge. Scraping Water Mains.-The influence of the condition of the surface of a pipe on the friction is shown by various facts known to the engineers of waterworks. Jn pipes which convey certain kinds of water, oxidation proceeds rapidly and the discharge is considerably diminished. A main laid at Torquay in 1 858, 1 4 m. in length, consists of ro-in., 9-in. and 8-in. pipes. It was not protected from corrosion by any coating. But it was found to the surprise of the engineer that in eight years the discharge had diminished to 5I % of the original discharge. J . G. Appold suggested an apparatus for scraping the interior of the pipe, and this was constructed and used under the direction of William Froude (see " Incrustation of Iron Pipes," by W. Ingham, Proc. Inst. Mech. Eng., 1 899). lt was found that by scraping the interior of the pipe the discharge was increased 56 %. The scraping requires to be repeated at intervals. After each scraping the dis­ charge diminishes rather rapidly to 1 0 % and afterwards more slowly, the diminution in a year being about 25 %. Fig. 85 shows a scraper for water mains, similar to Appold's but modified in details, as constructed by the Glenfield Company, at Kilmarnock. A is a longitudinal section of the pipe, showing the scraper in place ; B is an end view of the plungers, and C, D sections of the boxes placed at intervals on the main for introducing or with­ drawing the scraper. The apparatus consists of two plungers, packed with leather so as to fit the main pretty closely. On the spindle of these plungers are fixed eight steel scraping blades, with curved scraping edges fittmg the surface of the main. The apparatus is placed in the main by removing the cover from one of the boxes shown at C, D. The cover is then replaced, water pressure is ad­ mitted behind the plungers, and the apparatus driven through the

FIG. 85. Scale .J5 •

and 25 ft. long, with water at different temperatures, have been made by J. G. i\lair (Proc. Inst. Civ. Eng. lxxxiv.) . The Joss of head was measured from a point I ft. from the inlet, so that the loss at entry was eliminated. The I ! in. pipe was made smooth inside and to gauge, by drawing a mandril through it. Plotting the results logarithmically, it was found that the resistance for all temperatures varied very exactly as v1 '795, the index being less than 2 as in other experiments with very smooth surfaces. Taking the ordinary equation of flow h = !(4L/D) (v2/2g), then for heads varying from I ft. to nearly 4 ft. , and velocities in the pipe varying from 4 ft. to 9 ft. per second, the v3.lues of ! were as follows :Temp. F. Temp. F. r ! ·0039 to •0042 57 ·0044 to ·0052 , 100 ·0037 to ·0041 ·0042 to ·0045 I 10 70 ·0037 to ·004I 8o ·004I t o ·0045 1 20 90 ·0040 to ·0045 130 ·0035 to ·0039 ·OOJ-5 fo ·0038 I 6o I

/

\

main. At Lancaster after twice scraping the discharge was increased 56! %. at Oswestry 54! %. The increased discharge is due to the diminution of the friction of the pipe by removing the roughnesses due to oxidation. The scrape� can be easily followed when the majns are about 3 ft. deep by the nmse it makes. The average speed of the scraper at Torquay is 2! ni. per hour. At Torquay 49 % of the deposit is iron rust, the rest being silica, lime and organic matter. I n the opinion of some engineers it is inadvisable to use the scraper. The incrustation is only temporarily removed, and if the use of the scraper is continued the life of the pipe is reduced. The only treatment effective in preventing or retarding the incrustation due to corrosion is to coat the pipes when hot with a smooth and perfect layer of pitch. With certain waters such as those derived from the chalk the incrustation is of a different character, consisting of nearly pure calcium carbonate. A deposit of another character which has led to trouble in some mains is a black slime containing a good deal of iron not derived from the pipes. It appears to be a n

[STEADY FLOW I N P IPES

HYDRAU LICS

organic growth. Filtration of the water appears to prevent the growth of the slime, and its temporary removal may be effected by a kind of brush scraper devised by G. F. Deacon (see " Deposits in Pipes, " by Professor J . C. Campbell Brown, Proc. Inst. Civ. Eng.,

I903-I904) . § 8 1 . Fl ow of Water through Fire Hose.-The hose pipes used for

fire purposes are of very varied character, and the roughness of the surface varies. Very careful experiments have been made by ]. R. Freeman· (Am. Soc. Civ. Eng. xxi., I 889). It was noted that under pressure the diameter of the hose increased sufficiently to have a marked influence on the discharge. In reducing the results the true diameter has been taken. Let v = mean velocity in ft. per sec. ; r = hydraulic mean radius or one-fourth the diameter in feet ; i = hydraulic gradient. Then v = n.,j (ri) . Diameter in Inches.

l l l

Solid r u b b er hose \Voven cotton, rubber lined Woven cotton, rubber lined Knit cotton, I l rubber lined cotton, I Knit I rubber lined ' \Voven cotton , I rubber lined 1 Woven cotton, rubber lined U n lined l i nen hose

2 ·65

2I5 344 200 299 200 3I9 I 32 299 204 3I9 I 54 240 54• 8 298 57•9 33I

"

2•47 "

2•49 "

l

2·68

" 2·69 ..

{ {

Gallons (United States) per min.

-

2· I 2

" 2·53 "

2·60 "

v

· I 863 •47 I 4 •2464 ·5269 •2427 •5708 · o8 og "393 I ·2357 ·5I65 "3448 • 7673 ·026 I ·8264 •04I4 I · I624

I 2 · 50 20·00 I 3•40 20·00 I 3·20 2 I ·OO 7•50 1 7 •00 I I ·50 I 8·00 J,4•00 2I · 8 I 3•50 I9·00 3•50 20•00

vdvo

I 23•3 1 24•0 l l9 · I I 2 I .5 I I 7· 7 I 22· I I I I ·6 I I4·8 IOO· I I 05·8 l l 3•4 1 I 8·4 94"3 9 I ·O 73•9 79·6

=

(do(dl)'.

+

r------ -lJ t

A

•

i

i

;,. B

!l

_____ -

--

-·--

-+----- --lz - - - - - · '

+

:Iz

- - -

·

.,.�� •

-----

-ls · - - - - .��.:1

l

t)

•

- - - - - - - - - ---l - - - - - - - - - - - - - - - - - - - - -.----�

--------�·d--

I

FIG. 86.

In fig. 86 let A be the main of variable diameter, and B the equivalent uniform main. In the given main of variable diameter A, let l�o l,. . . be the lengths, d�o d,... the diameters, v�, v,. . . the velocities, t1 , x,. . . the slopes, for the successive portions, and let l, d, v and i be corresponding quantities for the equivalent uniform main B. The total loss of head in A due to friction is

h = id1 +i,Z, + . . . = ) (v12·4ld2gd1) + )(v,'·4/,fzgth) + . . .

and in the uniform main

' ''

II

FIG. 87.

�. = (vo - VJ)2/2g = (w1/wo - I )2vN2g = { (dJ/do) 2 - I }2Vt2/2g �. = ),v12/2g,

if ), is put for the expression in brackets. w

dwo = dddo = ), =

r.r

r.2

r.s

r.7 r.B

1.9

2.0 2.5 s.o 3.5 4.0

s.o

6.o

(I )

7.0 S.o

1.05 I.IO 1 . 22 I.30 1.34 1.38 I.4I 1.58 1.73 1.87 2.00 2.24 2-45 2.65 2.83 .or

.04 . 25

.49 .64

.8r r.oo 2.25 4.oo 6.25 9.00 r6.oo 25.00 36.0 49·0

A brupt Contraction of Section .-When water passes from a larger to a smaller section, as in figs. 88, 89, a contraction is formed, and the contracted stream abruptly expands to fill the section of the pipe.

§ 82. Reduction of a Long Pipe of Varying Diameter to an Equivalent Pipe of Unij�Jrm Diameter. Dupuit's Equation.-Water mains for

the supply of towns often consist of a series of lengths, the diameter being the same for each length, but differing from length to length In approximate calculations of the head lost in such mains, it is generally accurate enough to neglect the smaller losses of head and to have regard to the pipe friction only, and then the calcula­ tions may be facilitated by reducing the main to a main of uniform diameter, in which there would be the same loss of head. Such a uniform main will be termed an equivalent main.

II

d;. �m�

The head lost at the abrupt change of velocity has already been shown to be the head due to the relative velocity of the two parts of the stream. Hence head lost or

n

i

§ 83. Other Losses of Head in Pipes. -M ost of the losses of head in pipes, other than that due to surface friction against the pipe, are due to abrupt changes in the velocity of the stream producing eddies. The kinetic energy of these is deducted from the general energy of translation, and practically wasted. Sudden Enlar!J.ement of Section.-Suppose a pipe enlarges in section from an area w0 to an area w1 (fig. 87) ; then V1/Vo = wo/WI ; or, if the section is circular, v.; t J,

FIG. 89.

Let w be the section and v the velocity of the stream at bb. At aa the section wi!J be c,w, and the velocity (w/c,w)v = v/Cto where c'. is the coefficient of contraction. Then the head lost is

�m = (vjc , -v) 2/2g = ( I/c , - I )2v2j2g;

and, if c, is taken 0·64,

(2)

The value of the coefficient of contraction for this case is, however not well ascertained, and the result is somewhat modified by friction : For water entering a cylindrical, not bell-mouthed, pipe from a reservoir of indefinitely large size, experiment gives

�. = o·505 v2j2g.

(3)

= )czN2g

(4)

If there is a diaphragm at the mouth of the pipe as in fig. 89 � let w1 be the area of this orifice. Then the area of the contracted tream is Ccwl, and the head lost is �. = { (w/c ,w1) - I }2v2j2g

if r. is put for { (w/c,w,) - I }'. Weisbach has found experimentally the following values of the coefficient, when the stream approaching the orifice was considerably larger than the orifice :wdw =

Cc =

r. =

O.I

0.2

O.J

0-4

o.s

o.6

.6r6

.614

.612

.6ro

.6!7

.6os

2JI-7

50·99 I9.78 g.6I2 5-256 3-077 I.876 r r6g 0-734 0.480

0.7

.6o3

o.8

0.9

I.O

.6or

·598

·596

When a diaphragm was placed in a tube of uniform section (fig. go)

il = )(v2·4//2gd) .

If the mains are equivalent, as defined above,

)(v' ·4l/2gd) = )(v12· {/d2gd1) + )(v,2·4l,j2gth) + . . .

But, since the discharge is the same for all portions, l,.d"'v = t,.d12V1 = l,.d,'v, = . . . v1 = vd2/d12 ; v, = vd2/d22 • • •

Also suppose that ) may be treated as constant for all the pipes. Then lfd = (d'/d1') (ldd1) + Cd'fd,') (Md,) + .. .

l = (d5/dl5)ll + (d5fd,5)t,+ ...

which gives the length of the equivalent uniform main which would have the same total loss of head for any given discharge.

FIG. 90.

the following values were obtained, w, being the area of the orifice and w that of the pipe :w1/w =

o.I

0.2

0.3

o.:t

o.s

o.6

0.7

o.8

0.9

c, =

.624

.632

.643

.659

.68r

.7r2

· 155

.8r3

.892

�. =

225-9 47-77 J0.83 7.801 1.753 ' · 796

· 797

.290

.Q6o

I.o

r.oo

.000

STEADY FLOW IN PIPES]

HYDRAULICS

Elbows.-\\'eisbach considers the loss of head at elbows (fig.9 r ) t o b e ciue t o a contraction formed b y the stream. From experiments with a pipe r l in. diameter, he found the loss of head �. = t,v2/2g ;

(S)

;. : .:. .·: . I ::: :::. I ::.1 :: : I :�: I:·�. I:·: . I::. r. = 0•9457 sin2�4> +2 ·047 sin 4 �q,.

Hence at a right-angled elbow the whole head due to the velocity n�ry nearly is lost. Bends.-\Yeisbach traces the loss of head at curved bends to a similar cause to that at elbows, but the coeffi­ cients for bends are not very satisfactorily ascer­ tained. Weisbach ob­ tained for the loss of head at a bend in a pipe of circular section �b = t&v2/2g ;

(6)

)b = o· r 3 r + r · 847 (d/2p)l ,

() =

)v =

45°

50°

55°

60°

650

r 8· 7

32·6

s8·8

I I8

256

1

I

700 900 751

Ci)

§ 84. Practical Calculations on the Flow of Water in Pipes.-ln the following explanations it will be assumed that the pipe is of so great a length that only the loss of head in friction against the surface of the pipe needs to be considered. In general . .. · 8 it is one of the four quantities � "'' ::. L d, i, v or Q which requires �--I to be determined. For since the loss of head h is given by the relation h = il, this need not be separately considered. FrG. 95 · There are then three equations (see eq. 4, § 72, and 9a, § 76) for the solution of such problems as arise :--

--

-- -_

____ __ __

7/II//Uii/ ��· y,�,,,.,p, ,i/f,,/, w .�.

______ _

:.�,.,� '-:1//i/IV//, /I

! = a ( r + r/r2d) ;

where a = o·oos for new and =o·OI for incrusted pipes. jv2/2g = idi.

(r)

(2)

Q = i1rd2v. (3) where d is the diameter Problem r . Given the diameter of the pipe and its virtual slope, of the pipe and p the radius of curvature of t::> find the discharge and velocity of flow. Here d and i are given, FIG. 9 1 . the bend. The resistance and Q and v are required. Find ! from ( r ) ; then v from (2) ; lastly Q from (3). This case presents no difficulty. at bends is small and at present very ill determined. By combining equations ( r ) and (2) , v is obtained directly :Valt•es, Cocks and Sluices.-These produce a contraction of the water-stream, similar to that for an abrupt v = ..J (gdi/2!) = ..J (g/2a) ..J [di/{ I + r /r2d} ] . (4) diminution of section already discussed. The For new pipes ..J (g/ 2 a.) = s6· i 2 loss of head may be taken as before to be . = 40· 1 3 For incrusted pipes (7) �. = tvv2/2g ; For pipes not less than I , o r more than 4 ft. in diameter, the where v is the velocity in the pipe beyond the valve . and !v a coefficient determined Ly experiment. The mean values of t are For new pipes . . 0·00526 following are Weisbach's results. . o· or os2 . For incrusted pipes Sluice in Pipe of Rectangular Section (fig. 92) . F IG. 92. Section a t sluice = w1 in pipe =w. Using these values we get the very simple expressions·

bl

I I I

\

wdw =

l''

I ·O 0·9 O·OO

lv =

.

•09

0·8

0• 7

o·6

0 •4

o·s

•

0 •3

0 2 O·I

' 95 2 · 0 8 4•02 8 · 1 2 1 7 • 8 44" 5

' 39

1 93

Sluice in Cylindrical Pipe (fig. 93). "

'•'•"'

''}

PCf!ing to diameter f pipe

wt/w = lv =

I.O

1 . 00

o.oo

t

!

i

II =

Ratio of

.609

·466

.

0.07

o.Sr

2 . 06

5-5 2

17. 0

0.26

cross sections r. =

97·8

Within the limits stated, these are accurate enough for practical purposes, especially as the precise value of the coefficient r cannot be known for each special case. Problem 2. Given the diameter of a pipe and the velocity of flow, to find the virtual slope and discharge. The discharge is given by (3) ; the proper value of r by ( r ) ; and the virtual slope by (2) . This also presents no special difficulty. Problem 3· Given the diameter of the pipe and the discharge, to find the virtual slope and velocity. Find v from (3) ; t from ( r ) ; lastly i from (2) . If we combine ( r ) and (2) we get i = t(v2/2g) (4/d) = 2 a{ r + r fr 2d}v2/gd ;

40 0

45°

·38 5

•315

•250

1 7 •3

3 1 ·2

52 6

�\?(.,

·

I

I

1

F IG . 94· Angle through which cock 20°

25 0

300

35°

·692

·613

• 535

•458

I · S6 I 3 • 1 0

5•47

9·68

60°

650

82°

• 137

·091

0

2n6

486

55° · i:')O roC

l

Throttle Valve in a Cylindrical Pipe (fig. 95) 0=

i

· '59

(4a )

(5)

and, taking the mean values of l for pipes from I to 4 ft. diameter, given above, the approximate formulae are i = o·ooo3268 v2/d for new pipes t ( S a) =o·ooo6536 v2/d for incrusted pipes I · Problem 4 · Given the virtual slope and the velocity, to find the diameter of the pipe and the discharge. The diameter is obtained from equations (2) and ( r ) , which give the quadratic expression

j

}

i

JIS

-740

FIG. 93· Cock in a Cylindrical Pipe (fig. 94). is turned =II. 10° 50 ISO 0= Ratio of 1 • 772 · Bso c r o s s '· •926 sections ·29 I • 75 · OS r,: =

i

t

0.948 .Ss6

v = ss·3I..J (di) for new pipes ( = 39· u..J (di) for incrusted pipes l ·

Ci)

d2 - d(2av2fgi) - av2/6gi =o . (6) . ' . d = av2/gi +..J { (av2/gi) (av2/gi+ r f6) } . For practical purposes, the approximate equations (6a) d = 2a.v2/gi + r / r 2 = o • oo031 v2/i + ·o83 for new pipes = o·ooo62 v2/i+ ·o83 for incrusted pipes are sufficiently accurate. Problem 5 · Given the virtual slope and the discharge, to find the diameter of the pipe and velocity of flow. This case, which often occurs in designing, is the one which is least easy of direct solution. From equations (2) and (3) we getd" = 3ztQ2fg7r•i. (7) If now the value of r in ( r ) is introduced, the equation becomes very cumbrous. Various approximate methods of meeting the difficulty may be used. (a) Taking the mean values of r given above for pipes of I to 4 ft. diameter we get d = V (32!/g1r2)'{/ (Q2/i) (8) = 0·22 16 './ (Q2/i) for new pipes = 0·2541 V (Q2/i) for incrusted pipe s ; equations which are interesting a s showing that wheR t h e value of ! is doubled the diameter of pipe for a given discharge is only in­ creased by 13 %.

AThisecsovalndumee isthod is to obtain a rough value of by assuming isfhoulthed avebe rcagelculdeatmedandfor iso galgallonslonsperpeherahed peadr pedayr .day, the maies . Whe n t h e pl a n of t h e ar r a nge m e n t of mai n s i s de t e r m i n e d Then a vcr) approximate value of is upon, and t h e s u ppl y t o e a c l o c l i t y and h e pr e s u r e r e q ui r e d i s as c e r t a i n e d , i t r e m ai n s de t e r m i n e h e di a me t r s of t h pi p e s . Le t s h ow an l e v at i o n of a mai ABCD . . be i n g t h e r e r v oi r fi g . ?nd a revised value of not sensibly differing from the exact value, flterhvomeldatiwhingucmopeh tlihrnaeet,siouHrnsppl, beyandiisngdeHatrhiv,eedhe. i.gLehtthteofhetighbehte swatthofe edatrthseuumrmaifalcinee iabove ofn tthhee forExpandi m Equatngiotnhe tmayerm bein brputackinettsh,e eglectingthetermsaftcorthesecond,. and fofroirnnecruwstpiedppiespes. ussausppluealrvlyiecdesurbyppleseirgrevdoiabyvir,tagrwhiTowntiaovicnhtafmairtnioomtunfsranaromiestorage reservoir or by pumping reservoir from the same datum. Set up next heights . fdayssrhooulm'dasbeulpplosweuycrhorltehvateinl.watimTheeporr istsadeentrlviviceearserdesatemucravoiprhessmorhuoulr e.dofcatIotntsleaaeislnetvabouttathrioen fsouprpplmesyaenofltoinweegatrchleimlmioictnalimtoyum.theTheprleisneurA1ofe heBvi1Cirg,thtDual1 . sc.leoispea.rlyinfTheeorwhitnheciadefh hesqhiuatoulghtdes . . ar e t a ke n r e p r s e n t i n g t h e ac t u al l o s e s of he a d i n e a c h t o t h e hi g he s t par t of t h di s t r i c t . The gr e a t e s t pr e s u r e i n l e n gt h of t h rr . a i , Ao B o C o wi l be t h l i n e of vi r t u al i s us u al l y about f t . , t h e pr e s u r e f o r whi c h or d i n ar y pl;:>ts and fit ings are designed. Henc if the dist ict supplie has prsreoqepesui,ruerandde.isForitdewifianycliebentpoi,obviandnt oausindiattffhewhatrlentgtcpoihoiofnctesthofseucmaidih aasmen, weterandhaveof maiE0, nhies Pr e s u r e he i g ht Whe r e no ot h e r c i r c u ms tfofro flamogiw.frvacntGeulrennegtbyrahlhydryoftmaihaeulvenic,tlaosncchcioctoeynskliimnledaie:wtdsrsaattheiorenamailooflsimntofishtealheitesisoafndbeeoftytowtofbeehetnashveesliomaigcneanditdny pealprorengfts.uetcrpeheosnd,rmaiesveecandnond.asitnhfOctehydr. dicpeascsraihulsarnaleicgoelnd.macditmhheiUsninveiseuhrlaloeycsl,iywortsyothasikeintpiveogplroeundeescdiucrteyraecdisehonmemeorsinmwisoushathfets. atftohre thtooeT.wnsaenlFannild'ossofupploftnhhegeymaigia:-dvewhins. theisfolliaoblwientgo rveenlodecirtitehseaspressuiutrabline uinffipiciepnets VeA,DiB,alomeCci(tfiyegri. infeinbetchpeetshrrseec.res.ervoirs connected by the arrangement eotf . grlsoeweneaitrcevarprreisaetuirvoensoi. rofFiandgle.vaelrashngemusowsoftabeprdiesditrsvucirdteeofdinitnwthoezloonewesrofediachisthrgiwihec trfhrandoitms lpitknown ehnpegtemaishs, handndiown,apimepth-et.e/di,rSuppos dischsarchgaresgtrheeqa/diuin2,drmedven. sloi cnsityandin/3,tposhe itihornse ofporbethtienognspiphoeesf galAmelotrnsoicape, trhefhet.saudpplThepeyritsday.octolnssiudButpplerayblinyr manygrquieatdert,iosbutwnsin ,alEnglandso ianndesmanypeaboutcialcyasiens heliniIegfshtaofprvierstumeualreacsluoorlpeuinmn.g tIhifetiprhnterfsorduceusreersevuatdroifatarcAeandlseuvtpplehleatiwates BeirsandwiabovelwiC,rlisbeeandtothiaef .�-------·---- . iI.sebequeabovelonwtly thAereandare tBhresucpplaseys:Cons l IToIIII.. detR elbeervmleoilwnwie twh1h ch case has to be . . debysupposalat swie!Ut�tche. ip1n.Theptehefnrogitmhveerne �condiaB sc;rlmiosplnsede, I I maicond1atn.,t�1and0ncanthbee hedeitgehtrmofinedfr.e Foru ftahcis . .. . wheofHentrchee twios tporhe ticonsmmonof tdiheschpiarpge. tmaihgoodreonughpopuls tdehteahaflletuofibrocnttauhncatseusihpplounmaipplieydnt.hiseoflButodeastmdibyiandsntrldeeibamuskageutteirnmtgbeionsfyialntshgtleotmwemaiheisdncfasopac.lrc.uTheliIattyeissdofuusfpplrtuohalmye fopetwirhoelmncalianwhilsgoewticnflheohandswluficrsoeimsebeacB,staiwlsyande Iobtn. tahIifeandnecda.isBe liIewifscltahtsalheenanlIowI.fliosIopewfgrtenoawarintegrdtthsheB,anslcuandaicsee to take maximum deman(t the average demand. Hence is case I ., and is already complet ly solv d. 66

1

HYDRAULICS

(b)

d

.1 = " ·

d' = '.J (32Q2/g1r2i)'.J ct = 0 ·63 1 9 '.J (Q2/i)'V ct. .I

25

§ 86. Determination of the Diameters of Different Parts of a Water Main -

( = ct( I + r/ I 2d') ;

97

d,

IS

[STEADY FLOW IN PIPES

NN

,R

Hb

(c)

7

d = '.J ( 32 nQ2/g1r2i) 'V (I + r / I 2d) . (9)

\' ( I + I/I 2d) = I + I j6od - I /I 800dl- . . :\ d = i/ ( 32 n/g1r2)V (Q2/i).{ I + I /6od}

= V (32 a.(gr.2 )V ( Q2/i) + o 1 667 ;(9 a) ·

'V (32a. lgr.2) = 0 · 2 1 9 = 0·252 § 85. A rrangement of Water Mains

for Towns' Supply.-

AAw BB 1 . . .

100 lt . the m�ins

�

200

••

�b, � . . la, h, l....

Do

z

!

= H, - H. - (�a +�b + . . . �.).

r! ro

4!

20

2od

I i I

-·

' � ·4 � �

l.

I'

· - · - · - · - � ""- - - · - . _

,. .J.

. - · - · - ·-

FIG. 96.

96

roo

' '.

�0

ISO'

·

1

-

J.

.

it

200

�'"""

�-:::,. i:

..�,--=-

:.

25

�

""=:-.-.....

-

R

�

t

Hr

; : 1 l · ...._ � '--:::::-..� · ·� ·�::;_'ffi'flh.:: .. .i ...,1 :A� ."'�"'- . l 1 -..... !"b: "- -......:. : te �--� �-":':� .L: i '· �--�:A:

.

�-:-�� �

1

'

)Y__ · · - · - ·---- - .

;

:

• a:·

_____

:1

: : ; ;

iJa::

N

._ _ _ )'(;¥ •• ·-· • ··-·

, ·-:

"

'·'-c!, ·

: C1, -\ !' ·

�: �: �! i. .;. ....\ . .l &--. .· :.=· *j ..........Z�---·-t ---·Ji,. ...i · . . . . . .. * '* - � ... . . . . :.......

. ....

::

. �

··

· ·•·-

a

i

X,

i i

i

1

i

l

i

:

E.

•

·

. ." .

1:

!:

at twice

R

X,

a R , Rb, Rc

R

! !.

C.

R

b ; Q, = Q, +Q,.

R

b ; Q� + Q2 = Q3•

b ; Q, = Q, ; Q, =o

I

-----· ··-"----•·---··-··'····

N

X

Is

h'

·

\F

b,

b,

!

i -�� .01 _ ....q,..?. · : ................_ I F. ;n.--�:---...:.�

h'

t£e

4 2·5

XR,

FIG. 97 ·

of

. .

.

, - f ---··---·�·-··-··-··---·------- --.-·········--·· · · · · -

.

__

8 12 IS 24 30 36 3·0 3·5 4·5 5 · 3 6·2 7·0 § 87. Branched Pipe connecting Reservoirs at Different Levels .-L 98 ) d, Q1, v1 ; d2 , Q2, v2 ; d3, Q3, v3

- · - ·4

...

X

ha - h' = !(v,2/2g)(4l,/d,) = 32!Q'2/J/g7r2d,5 ; h' - h. = t(v,2/2g )( 4lsfd,) = 3 2 !Q'2l3/gr.2d,5 ; Q' (ha - h')/(h' - he) =Mlfl,d,5, h'

h'

hb,

h' = hb ,

hb.

HYDRA U LICS

COMPRESSIBLE FLU IDS IN PIPES] The true value o f h must lie between h' and hb. value of h, and recalculate Q,, Q2, Qa. Then if or the value chosen for If

Q, > Qz +Qa i n case 1., �.h + Q2 > Qa in case I I I . ,

h

I

time dt the mass of air between AoA1 comes to A'0A'1 so t hat A0.'\ 'o = udt and A1A\ = (tt +du)dt,. Let n be the section. and m the hydraulic mean radius of the pipe, and W the weight oi air flowing through the pipe per second. From the steadiness of the motion the weight of air between the sections AoA'o, and A,A', i s the same. That is,

1

i s too small, and a new value must be chosen.

mr

or

Choose a new

Wdt = Giludt = Gn(u+du)dt.

Q, < Q. +Qz in case 1 . , Q, + Q2 < Qz i n case I I I . ,

By a�alogy with liquids the head lost in friction is, for the length dl (see § 72, eq. 3 ) , l(u2/zg) (dljm). Let H = u"j2g. Then the head lost is I(H/m)dl ; and, since Wdt tb of air flow through the pipe in the time considered, the work expended in friction is - I(H/m)Wdl dt. The change of kinetic energy in dt seconds is the difference of the kinetic energy of AoA'o and A1A'1, that is, (W/g)dt ( (u+du) 2 - u2)/2 = (W/g) u du dt = \\ dHdt.

the value o f h i s too great. Since the limits between whic-h h can vary are in practical cases not very distant, it is easy to a pproximate to values sufficiently accurate. § 88. H1ater Ha m e . I f in a pipe through which water is flowing a sluice i:; suddenly closed so as to arrest t he forward movement uf the water, there is a rise of pressure which in some cases is serious enough to burst the pipe. This action is termed water hammer or water ram. The fluctuation of pressure is an oscillating one and gradually dies out. Care is usually taken that sluices should only be closed gradually and then the effect is inappreciable. Very careful experiments on water hammer were made by N. J . J oukowsky at :\ J oscow in I 8g8 (Stoss in Wasserleitungen, St Petersburg, I 900) , and the results are generally confirmed by experiments made by E. B. Weston and R. C. Carpenter in America. Joukowsky used pipes, 7 , 4 and 6 i n . diameter, from rooo to 2500 ft. in length. The sluice closed i n 0·03 second, and the fluctuations of pressure were auto­ matically registered. The maximum excess pressure due to water­ hammer action was as follows :-

-,

Pipe 4-in. diameter. Velocit,, ft. per sec.

O· S 2·9 4• I 9• 2

1....--....::..

I

I

I I

3' I 68 2 32 519

I

Excess Pressure. tb per sq. in.

Velocity ft. per sec.

I

l

I

0·6 J ·O s· 6 7·S

43 I 73

369 426

------------�--·-----------�

I

In sanae cases, in fixing the thickness of water mains , I oo lb per sq. i n . excess pressure is allowed to cover the effect of water hammer. \Vith the velocities usual in water mains, especially as no valves can be quite suddenly closed, this appears to be a reasonable allowance ( see a 1 so C arpenter, .fi• tn. �oc. ")Jec h . Eng. , I 8 9 3 ) . "

IX.

FLO\V OF C0:\1 PRESS I B L E FLU I DS I N PIPES

§ 89. Ftow of A ;r in Long

Pipe>.-'v\'hen air flows through a l on g pipe, by far the grcat.?r part of the \\ o rk expended is used in overcoming frictional resistances due to the surface of the pipe. The work expended i n friction generates heat, which for the most part must be developed in and given back to the air. Some heat may be transmitted through the sides of the pipe to surrounding materials, but 111 experiments hitherto made the amount so conducted away appears to be very small, and if no heat is transmitted the air in the tube must remain sensibly at t h e same temperature during expansion. I n other words, the expansion may be regarded as isothermal expansion, the hear generated by friction exactly neutralizing the cooling due to the work done. Experiments on the pneumatic tubes u sed for the transmission of messages, by R. S. Culley and R. Sabine (Proc. Ins!. Cic'. Eng. xliii. ) , show that the change of temperature of the air flowing along the tube is much less than it would be in adiabatic expan�ion. . . , . . . . , § 90. Differcn!t�l Equatton ?f the .Sfeady �fotton of Au flowmg tn a Long Pzpe of Lmform Sectwn.-�\'hcn mr expands at a constal!t absolute temperature r, the relatwl! bctwecyt the press!lre P 1� pou_nds per square fo;>1 and the dens1ty or we1ght per cub1c foot (, IS g1ven by the equatwn PfG = cr, (I) where c = 53 · I S . 6o° Fahr.,

Taking r = 52 I, corresponding to

a

temperature of

(2) cr = 27690 foot-pounds. The equation of continuity, which expresses the condition that in steady motion the same weight of fluid, W, must pass through each cross section of the stream in the unit of rime, is

•iiii..iiiiiiiiiiiiiiiiii!iiiiiiii[i !' '

'��----L--t.U ----------i

"'wliii ii! i A:o :�iiiiiiii i ii i -. ii .A: ziiiiii i ' iiil!! � AQ l :

j

�

0

;

G nu = \V = constant,

where n is the section of the pipP, and u the velocity of the air. Combining ( I ) and

(J )

'

and the work of the pressures is zero. Adding together the quantities of work, and equating them to the change of kinetic energy,

WdHdt = - (crW /p)dp dt - I(H/m)\V dldt d H + (cr/p)dp + nH/m)dl = o, dH/H + (cr/Hp)dp + r;dl/m = o (-t) u = crW/Ilp, But 2 /2g12 •r2\\2 and c u2/zg = 2p , H= :. dH/H + (zg£l2P/crW') dP + Idl/m = o. (4a) For tubes of uniform section m is constant ; for steady motion \V is constant ; and for isothermal expansion 7 is constant. I n tf"grating, log H +gn'p'/W'cr + ll/m = constant ; (5 ) for l = o, let H = Ho, and P = p0 ; and for l = l, let H = H, , and p = p1 • log (HdH o ) + (gi22/W2cr) (PI' -Pu') + tl/m = o. (sa) where Po is the greater pressure and p, the less, and the flow is from Ao towards A,. By replacing W and H , I (6) 1og (p0jp1 ) + (gcr / u02p0')( p12 - p0' ) + 1l/m = o. Hence the initial velocity in the pipe is

I

'1

uo = v [\ gcr( Po' - p,')}/\ Po' ( l'li m +log(PofP, ) } ] . 'When l is great, log Po/P1 is comparatively small, and then

(7)

Uo = v [ (gcrm /ll) { (p02 - p1") /Po' l ] . ( ? a) a very simple and easily used expres,ion. For pipes of circular section m = d/4, where d is the diameter :Uo = ...; [ (gcrd /4 1l) { (p.,Z - p,' ) /Po'll ; (? b) or approximately Uo = ( I · I J I 9 - 0· 7264PI /Po) v (gcrd/.J,Il) .

(7c)

§ 9 1 . Coefficient of Friction for Air.-A

discussion by Professor Unwin of the experiments by Culley and Sabine on the rate of transmission of light carriers through pneumatic tubes, in which there is steady flow of air not sensibly affected by any resistances other than surface friction, furnished the value I= ·007. The pipes were lead pipes, slightly moist, 2{ in. (0· 187 ft . ) in diameter, and in lengths of zooo to nearly 6ooo ft. In some experiments on the flow of air through cast-iron pipes A. Arson found the coefficient of friction to vary with the velocity and diameter of the pipe. Putting

I

!

1

- - --------

l = o.fz+fl, he obtained the following values-

l

Diameter of Pi p in feet.

(3)

nup/W = cr = constant. (3a) .A.'1 Since the work done b y gravity on the air during it� FIG. 99 · flow through a pipe due 1 o variations of its level is generally small com pared with the work done by changes of pressure, the former may in many cases be neglected. Consider a short length dl of th� pipe limited bv sections Au, A, at a distance dl (fig. 99) . Let p, u be the pressure iwd velocity at Ao, and u +du those at A1• Further, suppose that in a very short

P+dp

du(dp = - crW/12p2•

And the work done by expansion is - (crW/p)dp dt. The work done by gravity on the mass berween Ao and A1 is zero if the pipe is horizontal, and may in other cases be neglected without great error. The work of the prtcssures at the sections AoA1 is pnudt - (p+dp)n(u +du)dt = - (pdu + udp)ndt B u t from (3a) p u = constant, pdu+ udp = o ,

Pipe 6-in. diameter.

Excess Pressure. tb per sq. in.

l

'

The work of -expansion when nudt cub. ft. of air at a pressure p expand to n(u +du)dt cub. ft. is npdu dt. But from (3a) u = crW/r!p, and therefore

I

II o

1

I

·-

I ·64 I ·07 ·83 . 3 38

· 266 · I 64

.

1

�

·00 29 ·00972 • 0 1 525 ·03604 ·03790 ·045 1 8

I

{1

•00483 ·00640 ·00]04 ·0094 I ·00959 ·OI I67

I for 100 ft. per second.

·oo6so

· 00484 ·007 1 9 ·00977 · 00997 ·OI2I2

.

\

lI

(8)

It is worth while to try if these numbers can be expressed in the form proposed by Darcy for water. For a velocity of IOO ft. per second, and without much en or for higher velocities, these n umbers agree fairly with the formula !Qi 1 = 0 · 005 ( 1 + 3 / I od) , which only differs from Darcy's value for water in that the seeond term, which is always small except for very small pipes, is larger.

imentl=o·sl,onagroao28(e veberryt+3/elarwigIOed)thsc.tahle, valbyue Stockalper twiahlantihghtetaosfyeteltcleucgarvraephsie;r,byth eemelerectsirsiatcgeaitncys.earofThee madewhitucbehinsntoarthaeerloatulidbeandundes inplrLoragcreo.dundonin atSomeThethesSte lpiaGottpeerstehwexarperderTunne \tneogneVlehengcltetchtehthworevarvarkiadoneitaitoprinoonofbybablo erlxyepansvsleeulsrofeirooin,tsughhveeandpirtyhptansehm.enalArForl,soitnh'isat. nocalsoengewermaysafe andimatst onlprecorypeiantillsiyozntuousalh. etAecra,icnrandugirteht.nehteflIofnocwaaimosrofrieftroairsrcary. sdcteiimnnttsorotduch ecetudrbeorenorrteofmdroveaaiwnrdistwihsrtoehughaoutdy Put t i n g f o r t h e t i m e of t r a ns i t f r o m o t o bevepreilnoandccgisiettylhyeienqtlhuiethveatalprpiieoennspsteu.tofroeThist,hesetwqeandquatouateinoidsonnftofohmayretdehebenpiflsioptwusie seofa,dbandowatfvoer eatrnh,tyehedatflandomewum,aofn From neglecting andrputting . cFroalo§mgaseq.uation it results that the pres ure at ft. from that From (r) and r ewhindcofh itshofe pithpee fwheormre th pre srureris i fFioacrgeany. levegil vsfhoeowsnr tpihepceapilwicpuetlhaistgi,edtvhecnrueendrfvoersepr, ofaesparprueraesbolsu. raeThewifotrhctuhorwrvoeizofofontfSabirael axisnuers'­. But = experiments, in one of whi h th pres ure was grea er than atmo- Iwhif ch5gi21°ve,sctohreets=ipmondie ofntgratonsmis ion in ter)ms of the .initial and final pres ures and the dimensio of theThetube.mean velocity is or, for The fol owing table Ab�gi=0·ve0s7lus0to8eme-.j results :Prepes rurseqs. in . Mean Velenlogtcihtiens foertTubes . of VacPresmkumg:mre l ro 67·2 4707·3 38·8 Wor FI G . r o o. sTheprpheesruobsicreprserwievsetduhroutpre, eandsbruarceiksnearttsh.e giTheotvhentpilipensebrwastahcanktehtates mpneandosupmathetheriiccaprtluceubeslautbered­ Working 84· 7 tThew enprFees nucrhesurarche Stgivre nt iandn incthesCeofntme.rarlLecStutray.ion, beydsthe. inprleensgturhe then i n t h e l a s t e q uat i o n t h e r e be put =o, andvelocveityloatcityanyat ota hgiervesnecsteioctn.ionFro!o=cmtheonsqpiuattpae;nt;ion (3a) the pr s ur and end,whepipersea, astrhesfuolvert orlwhiocifiitcyhesappar i s i n de p e n de n t of t h e pr e s u r e at t h e ot h e r e n t l y mus t be abs u r d . Pr o babl y f o r l o ng ,rtwhiherchisthaelfimorimultoatihseapplatiocaofblet.he initia and so that, for any given uniform pipe, t e r m i n al pr e s u r e s f o which hasgivesalrtehaedyvelbeoceitny deat eanyrminsedc.tioFin gi.n termgisvofes the prveeslocuirtey, § FLOW I N RIVERS AND CANALS. hen water a, thpie pseecytt.hioenWhensdeectpioeinntdsflatoonwsanytinhpoieanvenopetloicsnitdeychtdueanneermtinloewitdhtbyeh dynami teheuppefocrmarl ofscfluoorndiwstfhaectieinboundar f r o ns . Suppos e grtchaeangidualincnrlg.yeasfiwateButl ,ofetrhciferadmiotsshecsteinoecfldntoiwtandnotandano vethunfieldiocmliateinalyndutatciaatonalneiatof.cshheThevepoialdoncctihstyannegrcoatansduallteaawintclyh, poipeverlonmctianetaty tnattinreaceafgitmheepoiiasnetismtarableaischloiemnsdi.ta. ntThe, andcetfherwmotardiotnhei sescteiaody,n andor Istantf whe,fotrhealmotltsheecitomotinonsis,ouniandn ifsosrmties.acdyoBynstthahypot ent;sectthioernsseifso,ofrtehtehmusienflsottrwbeeamcoiarnss ectaoalntn­l e q ual al s o r m s e c t i o n t o s e c t i o n. The c s e i s t h n one of uni f o r m s t e dy FI G . mot i o n. I n mos ar t i fi a l c h anne l s t h e f o r m of s e c t i o n i s c o ns a ni , cprveurelovsceuistryfeoircnutcrhveeastwehavesoceoxnspealidreimaradyeblntybes toeofnwarCuldrdalewn.sythandatlteSabinwidlonfbeet,hfsoerpinwhiptehcatwheh three andtIhfewhedethpentbehstdiesahascdyonsamottuniantifo,onandrmis setlsohtpeeabls. tirsIehnaemd attshuercfassece ttihsoensparmotaral eunequal uni f o r m l t o t h e bed. , tchhea§rprgeeds peurre siesclondast.is (equation 3a)- The weight of air dis­ smotfleocwtioin.s imodiOrsdteifinaeardyd ybymotrivewerosniarrswieoritnh tobshvaristyrcuoincndigtiovetnsiol.n,ocieShortsypefcrtioamlunobsyswheecttirouenctthhetdoee From equation = for a pipe of circular section and diameter lveernlIgtoonrc,hialttihelofesac,meathruaiosalvnesretmaynecrteiaormsnbetinhtetheesaudiltrefffndagtecasrehnandofbet ifluninugcifdepunrfittmrlaefomesremovicnthiteosnacnhavegwituftalahsouttdiseerffctegrtirohensannatt. tflmayhooswebeofnesneatrgelatemscteed,botreandstoonmthaandathypotalsl itdhehesefi.sliasThemethnattorstmaydhinsarYarbey itfaroteriaomtneuldofasevefhaviolroctinhyge >\pproximately mmonhveesilso,caitply aenequallayetor me(fign. velobecitywofe n hsecsitornseam.normOnal has been found cheaper tParo tirsa,nsBemrltinme, London, s ages inandpneotuhmater tiocwnstube, its athcisohypot 68

E.

r

Z() - z, - po/Go - p,f G, - I(v�/2 g) (1/m) = o ,

p,

Po

[FLOW I N RIVERS

HYDRAULICS

G1

Go

Z()

(10)

Time of Transit through the Tube.t l, 1 t = dl(u, o di-I/H, m = d/4, (4a) dl = gdfl2pdp/2 s-W2cr

.z,

v

(3)

92. Distribution of Pressure in a Pipe in which A ir is Flowing.­ (7a) l p, Po , (n) P = Po" { - luo2/mgcT } ;

t = J��gdfl'p2dp(2!;\Vsc2T2,

= gdfl' (pos-p,s) /61\V3c2T2. W = Pouoll(CT ; :. t gdcT(Po3-Pt3) /6sPo3Uo3, = !;!l�(Po'-p,') /6 (gcTd k (Po'-p12)� 60° F., t ·OOI 4I :2!;lll (Po3 - p,3) /dl (Po2 - p,2) J ;

P = " (al +b)

roo

T=

� :

"·f-4-19

P,.e...�su,.�: J.O·O:: ·'19·7. " "

Vl

•

u

-"'

. .!d

..!!: Cl "" §� _a ·c u P< -" � � 0 0 � O o! s :;;

(/)�

- � " " " 0

..n �

;j � 6� -" ..!l b!J.O

,.o o :l

��

�" ..c.

"�

�I �

·C ('j � g § QJ (J;i:! �>. "� >=< ..=.S "il ...£l .,o "" � .� � -�

�6

5� � = ;: � a ""B > � ]

-- -- -- -- -- -- -- -- -- -- -- --

•25 ·s •75 I ·O I ·s 2·0 2·5 3•0 3•5 4•0 4'5 s ·o s· s 6·0 6·5 7•0 7•5 S·O

125 I35 I 39 I41 I43 144 I45 I 45 I 46 I46 146 I46 I46 147 I 47 1 47 I47 I47

95 I IO 1 16 I I9 I22 I 24 126 I 26 I27 I 2S 12S I2S I 29 I 29 I 29 1 29 I29 I 30

57 26 72 36 S I 42 S7 4S 94 56 9S 62 IOI 67 I04 70 IOS 73 1 06 76 IO? 7S lOS So I 09 S2 I IO S4 I I O ss 1 1 0 S6 I I I S7 I I I ss

r S ·s 25·6 30·S 34'9 4I•2 46·0 .. 53 .. ss .. 62 .. 65 .

.

67 .

.

69

s-s 9•0 9·5 I O·O II I2 I3 I4 IS I6 I? IS 20 25 30 40 00

.so

I 47 I 47 1 47 I 47 147 I 47 I47 I 47 I 47 1 47 I 47 I4 7 I 47 qS I4S qS I 4S I 4S

I 30 I 30 I 30 I 30 I 30 I 30 I 30 1 30 I 30 I 30 I 30 I 30 I3I I3I I3I I3I I3I I3I

I I2 I I2 I I2 I I2 113 I I3 I I3 I I3 114 I I4 I I4 I I4 114 I IS IIS I I6 I I6 II?

S9 90 90 9I 92 93 94 95 96 97 97 9S 9S roo I02 1 03 I04 lOS

. .

?I .. 72 .. 74 .. .. 77 .. .. .. So .. S3 ss S6

�

\ from§ 99the · Ganguillet and Kutter's Modified Darcy Formula . Starting genera} �xpression v = c -./ mi, qanguillet and Kutter l

-

. . exammed the vanatwns of c for a Wider vanety of cases than those discussed by Darcy and Bazin. Darcy and Bazin's experiments were confined to channels of moderate section, and �o a llrr:ited variation of slope. Ganguillet and Kutter brougnt into the dis­ cussion two very distinct and imoortant additional series of results. The gaugings of the Mississippi by A. A. Humphreys and H. L. Abbot afforcl data of discharge for the case of a stream nf exceot:

HYDRAULICS negative, and the stream is diminishing in depth in the direction of flow. In fig. I23 let BoB1 be the stream bed as before ; CoC1 a line drawn parallel to BoB 1 at a height above it equal to H. By hypo­ thesis the surface AoA1 of the stream is below CoCt, and the depth has ·j ust been shown to Bo from diminish Cn Going toward'S B1• up stream h ap­ Ct proaches the limit H, and dhfds tends to the limit zero. That is, up stream AoAt is asymptotic to CoC. Going down stream h diminishes FIG. I23 . and u increases ; the inequality h > u'/g diminishes ; the denominator of the frac­ tion (I - (U2/2gzh)/(I-u2/gh) tends to the limit zero and con­ ' sequently dlz/ds tends to oo . That is, down stream A0A1 tends t.o !l direction perpendicular to the bed. Before, however, this lumt was reached the assumptions on which the general equation is based would cease to be even approximately true, and the equation would cease to be applicable. The filaments would have a relative motion, which would make the influence of internal friction in the fluid too important to be neglected. A stream surface of this form may be pro­ duced if there is an abrupt fall in the bed of the stream (fig. I 24). On the Ganges canal, as orig­ c o n­ inally structed, there were abrupt precisely falls FIG. I24. this kind, of and it appears . . th N2.

I f such a stream i s interfered with by the construction of a weir which raises its level, so that its depth at the weir becomes h1 > u2;g, then for a portion of the stream the depth h will satisfy the con­ ditions h < u2/g and h > H , which are not the same as those assumed in the two previous cases. At some point of the stream above the weir the depth h becomes equal to u2/g, and at that point dh/ds becomes infinite, or the surface of the stream is normal to the bed. It is obvwus that at that point the influence of internal friction will be too great to be neglected, and the general equation will cease to represent the true conditions of the motion of the water. It is known that, in cases such as this, there occurs an abrupt rise of the free surface of the stream , or a standing wave is formed, the conditions of motion in which will be examined presently. It appears that the condition necessary to give rise to a standing wave is that i> !/2. Now ! depends for different channels on the roughness of the channel and its hydraulic mean depth. Dazin calculated the values of ! for channels of different degrees of rough­ ness and different depths given in the following table, and the corre­ sponding minimum values of i for which the exceptional case of the production of a standing wave may occur.

N aturc of Bed of Stream.

which a Stand­ ing Wave is

Slope below

Standing Wave Formed.

impossible in

Slope in feet I Least Depth in feet . per foot .

feet per foot.

Very smooth cemented surface

O·OOI47

Ashlar or brickwork

o·ooi86

Rubble masonry

0·00235

Earth

0·00275

{ { { {

0·262 •098 ·o6s '394 · I97 •098 I · I 8I •525 ·262 3 ' 478 I •S42 '91 9

0·002 0·003 0 •004 0·003 0·004 0·006 0·004 0·006 O·OIO 0·006 O·OIO 0·015

STANDING \VAVES The formation of a standing wave was first observed b y Bidone. Into a small rectangular masonry channel, having a slope of 0·023 ft. per foot, he admitted water till it flowed uniformly with a depth of 0·2 ft. He then placed a plank across the stream which raised the level j u st above the obstruction to 0·95 ft. H e found that the stream above the obstruction was sensibly unaffected u p to a point I S ft. from it. At that point the depth suddenly increased from 0·2 ft. to o· 56 ft. The velocity of the stream in the part un­ affected by the obstruction was 5·54 ft. per second. Above the point where the abrupt change of depth occurred u2 = s· 542 = 30·7, and gh = 32·2 Xo·2 = 6·44 ; hence u2 was> gh. J ust below the abrupt change of depth u = s·s4 X 0·2/o·s6 = I ·97 ; u2 = 3· 88 ; and gh = 32·2 X0·56 = I 8 ·03 ; hence at this point u2 < gh. Between these two points, therefore, u2 = gh ; and the condition for the production of a standing wave occurred. The change of level at a standing wave may be found thus. Let fig. I 26 represent the longitudinal section of a stream and ab, cd

§ I2I.

c

c

FIG. I 2 5. lo represent some miles' length of the canal bed above the fall

AA parallel to the canal bed is the level corresponding to uniform

motion with the intended velocity of the canal. In consequence of the presence of the ogee fall, however, the water surface would take some such form as B B , corresponding to Case 2 above, and the 1·elociry would be greater than the intended velocity, nearly in the inverse ratio of the actual to the intended depth. By constructing a weir on the crest of the fall, as shown by dotted lines, a new water su:fa� > .:_;;r

/,\-�"';� ------�

c

�\

\ ""'-

I

;

L

I J 1�

� - 17 �

:J

A

-

·-;:��"'-=- '{ ��� ---'- \� l -=-___::.:r �T :M

--

-

stream and to check oscillations of the water column. Let the difference of level of a pair of tubes A and B (fig. I45) be taken to be h = kv2/2g , then k may be taken to be a corrective coefficient whose value in well-shaped instruments is very nearly unity. By placing his instrument in front of a boat towed through water Darcy found k = I ·034 ; by placing the instrument in a stream the velocity of which had been ascertained by floats, he found k = I ·Oo6 ; by readings taken in different parts of the section of a canal in which a known volume of water wab flowing, he found k = 0·993. He believed the first value to be too high in consequence of the disturbance caused .D by the boat. The mean of the other � two values is almost exactly unity ("lY (Recherches hydmuliques, Darcy and Bazin, I 865, p. 63) . W. B. Gregory used somewhat differently formed lm._ \:.��/ Pitot tubes for which the k = I (Am. I I Soc. Meek. Eng., I903, 25). T. E. 7 Jt Stanton used a Pitot tube in deter­ mining the velocity of an air current, and for his instrument he found k = I ·030 to k = I ·032 ( On the Re­ sistance of Plane Surfaces in a Current of Air," Proc. Inst. Civ. Eng., I904, I 56). One objection to the Pitot tube in its original form was the great difficulty and inconvenience of reading the height h in the imme­ diate neighbourhood of the stream surface. This is obviated in the Darcy gauge, which can be removed from the stream to be read. Fig. I 46 shows a Darcy g;wge. It consists of two Pitot tubes having their mouths at right angles. In the instrument shown, the two tubes, formed of copper in the lower part, are united into one for strer,gth, and the mouths of the tubes open vertically and horizon­ tally. The upper part of the tubes is of glass, and they are provided with a brass scale and two verniers b, b. The whole instrument is sup­ ported on a vertical rod or small pile AA, the fixing at B permitting the instrument to be adjusted to any height on the rod, and at the same time allowing free rotation, so that it can be held parallel to the current. At c is a two-way cock, which can be opened or closed by cords. If this is shut, the instrument can be lifted out of the stream for reading. The glass tubes are connected at top by a brass fixing, with a stop cock a, and a flexible tube and mouthpiece m. The use of this is as follows. If the velocity is re­ quired at a point near the surface of the stream , one at least of the water columns would be below the level at which it could be read. It would be in the copper part of the i1�strument. Suppose then a little air is sucked out by the tube m, and the cock a closed, the two columns will be forced up an amount correspond­ ing to the difference between atmospheric pressure and that in the tubes. But the difference of level will remain u naltered. When the velocities to be measured are not very small, this instru­ ment is an admirable one. It requires observation only of a single linear quantity, and docs not require any time observation. The law connecting the velocity and the observed height is a rational one, and it is not absolutely necessary to make any experiments on the coefficient of the instrument. If we take v = k..J (2gh), then it appears from Darcy's experiments that for a well-formed instrument k does not sensibly differ from unity. It gives the velocity at a definite point in the stream. The chief difficulty arises from the fact that at any given point i n a stream the velocity is not absolutely constant, but varies a little from moment to moment. Darcy in some of his experiments took several readings, and deduced the velocity from the mean of the highest and lowest. § I45 · Perrodil Hydrodynamometer.-This consists of a frame abed (fig. 147) placed vertically in the stream, and of a height not less than the stream's depth. The two vertical members of thi: frame are connected by cross bars, and united above water by a circular bar, situated in the vertical plane and carrying a horizonta� graduated circle ef. This whole system is movable round its axis, being suspended on a pivot at g connected with the fixed support mn. Other horizontal arms serve as guides. The central vertical rod gr forms a torsion rod, being fixed at r to the frame abed, and, , passing freelv upwards through the guides, it catties a horizontal

��

I

�

"'-..!

�

-::::::, · l--=:.j-::='Y-��"Y1'8?/��'r.\.�b�"":_; ?5-=§l-::t� FIG. 144·

which the resistance of ship models is ascertained. I n that case the data are found with exceptional accuracy. § I44. Darcy Gauge or modified Pilot Tube.-A very old instru­ ment for measuring velocities, invented by Henri Pitot in I 730 (IJistoire de l'A cadcmie des Sciences, I 732, p. 376), consisted simply uf a vertical glass tube with a right-angled bend, placed so that its mouth was normal to the direction of flow (fig. 1 45). The impact of the stream on the mouth of the tube balances a column in the tube, the height of which is approximately h = v2j2g, where v is the velocity at the depth x. Placed with its mouth parallel to the stream the water inside the tube is nearly at the same level as the surface of the stream , and turned with the mouth down stream, the fl u i d si n k s a d e p t h h ' =v2/2g nearly, though the tube in that case interferes with the free A c flow of the liquid and B somewhat modifies the FIG. 1 45 · result. Pitot expanded the mouth of the tube so as to form a funnel or bell mouth. Jn that case he found by experiment h = I · s�·'/2g. But there is more disturbance of the stream. Darcy preferred to mak.., the mouth of the tube very small to avoid interference with the

STREAMS

"

HYDRAULICS

AND RIVERS]

needle moving over the graduated circle ef. The support g, which carries the apparatus, also receives in a tubular guide the end of the torsion rod gr and a set screw for fixing the upper end of the torsion rod when necessary. The impulse of the stream of water is received on a circular disk x, in the plane of the torsion rod and the frame abed. To raise and lower the a paratus easily, it is not fixed directly to the rod mn, but to a tube k . sliding on mn. f'iuppose the apparat us arranged so that the disk x is at that level &he stream where the velocity is to be determined. The plane

f

will be given by the distance of the needle from zero. Observation shows that the velocity of the water at a given point is not constant. It varies between limits more or less wide. 'When the apparatus is nearly in its right position, the set screw at g is made to clamp the torsion spring. Then the needle is fixed, and the apparatus carrying the graduated circle oscillates. It is not, then, difficult to note the mean angle marked by the needle. Let r be the radius of the torsion rod, l its length from the needle over ef to r, and a the observed torsion angle. Then the moment of the couple due to the molecular forces in the torsion rod is

M = E , Ia/l ; where E, is the modulus of elas­ ticity for torsion, and I the polar moment of inertia of the section of the rod. If the rod is of circular section, I = !11"r4• Let R be the radius of the disk, and b its leverage, or the distance of its centre from the axis of the torsion rod. The moment of the pressure of the water on the disk is

.-_-_::-_:_ :::-� =- =- ::-_-�=--:. - -- -- - - - -· - �--=--� =- � =-----�� - - - · -- - - - - ­ . .. - - - - - 1-i--=�.,___-f - - ­ _ _

- - -

- -

- -·

Fb = kb(Gj2g)7rRzv2, where G is the heaviness of water and k an experimental coefficient. Then

E,I a/l = kb(G/2g)11"R2tfl.

n\} '' .' '.

For any given instrument,

v = cv a,

FIG. 1 47 ·

where c is a constant coefficient for the instrument. The instrument as constructed had three disks which could be used at will. Their radii and leverages were in feet Jst disk 2nd , 3rd ,

R=

o·05 2 . O • 1 05 . o·2 IO

b=

o· 1 6 0·32 o·66 For a thin circular plate, the coefficient k = 1 · 12. I n the actual instrument the torsion rod was a brass wire o·o6 in. diameter and 6! ft. long. Supposing a measured in degrees, we get by calculation v = o·335v a ; o· n s.V a ; o·o42,f a. Very careful experiments were made with the instrument. I t was fixed t o a wooden turning bridge. revolving over a circular channel of 2 ft. width , and about 7.6 ft. 'circumferential length. An allowance was made for the slight �urrent produced in the channel. These experiments gave for the coefficient c, in the formula v = cv a,

c ___

1st disk, e = 0 · 3 1 26 for velocities of 3 to 1 6 ft. 2nd , 0· 1 1 7 7 ., , I i to 3i , 3rd , 0·0349 . , , less than I i The instrument is preferable to the current meter in giving the velocity in terms of a single observed quantity, the angle of torsion, while the current meter involves the observation of two q uantities, the number of rotations and the time. The current meter, except in some improved forms, must be withdrawn from the water to read the result of each experiment, and the law connecting the velocity and number of rotations of a current meter is less well-determined than that connecting the pressure on a disk and the torsion of the wire of a hydrodynamometer. The Pitot tube, like the hydrodynamometer , does not require a time observation. But, where the velocity is a varying one, and consequently the columns of water in the Pitot tube are oscillating, there is room for doubt as to whether, at any given moment of closing the cock, the difference of level exactly measures the impulse of the stream at the moment. The Pitot tube also fails to give measur­ able indications of very low velocities.

PROCESSES FOR GAUGING STREAMS § 146. Gauging by Observation of the Maximum Surface Velocity.­

A

FIG. 146. abed is placed parallel to the direction of motion of the water. Then

the disk x (acting as a rudder) will place itself parallel to the stream on the down stream side of the frame. The torsion rod will be un­ strained, and the needle will be at zero on the graduated circle. If, then, the instrument is turned by pressing the needle, till the plane abed of the disk and the zero of the graduated circle is at right angles to the strea m , the torsion rod will be twisted throu �h an angle which mt>asures th ne/(sr.+sforTheinth2eseAeplraotnautes, ofical Socto 2iefty. s q uar , wer e bal a nc e d by i n ge n i o us me c h ani m de s i g ne d by F. H. We n ham and Spe n c r Br o wni g, i n s u a manne r t h at bot h t The pr e s u r e s we r e me a s u r e d by pr e s u r e c o l u mns . Expe r i m e n t s tsheplainarpinecurpserweenirtneand. splqauartchedee.laoppostTheeraliprtfeorscaeublwereahrofset bythroA.ugh wat�Ioeonirn, gaveplates of G.tbutoPiothbeesrqetx. peft.riarmenat,sdrweaandwnre madeverDiticdianiloyna fprstrheopeemsarbluaartfseatlniyvarnismethiueeiadndisgfurroefrmdot.mionaThetofwoode orimentaperlinr.esofquuarlwatts ewiefrototprh eofthsuetrhepr. pleThesaneurf,eosandl ogiwivaenncgoarm­bye trehsrosForeoreughvoiwhiarfaiiofcxhredcprtohplmparecyaenedfoeai.undntivemovily smnalglcuderrpeetnshtu.loft FormorwateseirmnE.ilacaMarrcploradiotancetsemoveefowiundthd partDucheleirobseshoeeminulrofvnse'sgidtrhvuprelnee.sxinpeuTherpounds s l a s t va\ e s ar e obt a i n d by t a ki n g P e on nor m al u r f a c e : Dubuat , e x pe r i m e n t s i n a cur r e n t of wat e r l i k t h os e mesq. nftti.oarneThieda, babove obt a i n e d t h e val u e s f Angl e be t w e n Pl a ne and Di r e c t i o n aul t e x pos e d t o wi n d pr u r e pl a ne s of and 2· 5 of Bl a s . and f o undft o var y f r o m t o t h e me a n val u e beAtinthge NationalaPhysresuilctaagrl Labore ingatweoryl , wiLondon, th DubuatE.. Stanton car ied HorLateizraolntpralespruerseuLre R. . So 20o 2 outsinmalalcurssecraileenstbutofofexwiaipetrhpasimexesncietpsgtonitohnalrtohughelydiacsantcruibraiauttreiotmenruofnk.anspreofThes umerse onawesusruemfonaecntas. Normal pres ure byL2Duc+R2hemin's rule Theis smsealelxrpeeclravitmiv.e, nlys ditofftehrFiefrgc.rmosthsosehceowstialonetofhdyetdihgiesvtcurerinbutirnenitohnatofthpreepls amne·e veonofthatrprtaiconoassql uarutsrheecetonlieopln.awttharee. dwisindtdwarehseoftdihtdeshteandrplicbaeuttnetiroianl steiocntionoofn. tprhSiesmleuilrwaerarlyond stihdeiswioftnhdwarae didisatdgonalriandbu­ tsshieqdce.tficotewas.n.,nwhetreiThenofrealtGhliencitsplaestnaehtseetyweonigthhtvepr2/wiofegsnaudwarrceupebiatdcr ',[SJ :::g ':: , , fcuresadeoixdotcpcerieeprnofntcdshtuelneiaainonrneafrpltgtand.athtapehetievefroetrsdhmeprgeec.eofspr,Onvetueandhrsleeotucplhiirsteeaiytunilsonee().valfwfoForarttrhhumdeee lcaei rcweucarltadrngulorsideasqrwasuarplaeteplateGvs tGv2h!2egjr2eandgs.ultForafntor prof2 etsheupercure sonrqe.nttfhi.enwheplft.artpee rwassiesc.tPheOnvealolcoingty A v grfnarwir etahrotaieswmr, altrSthelcanapltntaangulononesfaoirnundcitplrhcaeurtleeaiasrritsplhtreaunctnk.re.ssulItnantlagrteprraetsesrutsrtehonanwaslathrgosneeraplobsrlyaneersveind Theascasiegdenintofegrmtihmienporatloadiotanncdueofetihtnoe prmanywiensdurpreeabescutiwrr.eaeonl nqueasloflspituiondnsgandand, fosrcuurinfrasvcteaedncinroet,ofhisns, andplaneesxpemoverimdnctisrchaveularlybetehnromadeugh aibyr andHutwatton,erVion caewhi, andrlinThig macbaulhtinone. HYDRAULICS q,

1 70)

a

A

.

B

q,.

a

f.

L

L the

- -- - - .

.

' I , . .

' I I

!

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FIG. I 70.

N

f.

· I3

1 •39 1 ·49 1 • 64 .

0·25 0·63 I ·1 I

.

.

.

( I 734-I 809)

m

q,

1 •24 1 • 43

I •525 .. I ·784

.

3

J.

(1 795-I 88o) , 0·3 f=2·I8 ; a

f = I ·25.

cp((I

n

J= m +n,

(1 793-1 871 )

2·7

=

1'.

'

',

'

a

c

. '

'...,

,'

#,

! '' , ,

169

/ �

/

v

/

P=G

0·48 0·66

6

FIG. I 69.

adb

aeb

afb

. I

d

(Proc. Inst.

v

lb

I8 %

2

lb

= 0·00126 6o %

§ 1 68. Case when the Direction of Motion is oblique to the Plane. ­ is

cp),

I

18

I

= 3·31 ,

.

-Y

ab acb

q,).

q,

1'1!

m = I · I 86 ; n = o·670; I·I7 1 ·568 2 · I 25 ,

1904).

cp

I.

f = 1 ·834, § 1 67. Stanton's Experiments on the Pressure of A ir on Surfaces.­

Civ. Enf!,.

q,)

I 872

j = I ·36, a

I ·856.

cos .

q,).

I

6i

m = I ; n = 0·433 ; /= I ·433 ·

( I 798-I878)

=

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0·4 I ·6 r ·6s 1 ·605

·7 3 3•31 0·6I 1 ·96 1 • 26 3 •3 I 2 ·05 3 ·01 2 ·027 3•276 3•3I

WATER MOTORS

In every system of machinery deriving energy from a natural water-fall there exist the following parts:!. A supply channel or head race, leading the water from the highest accessible level to the site of the machine. This may be an open channel of earth, masonry or wood, laid at as small a slope as is consistent with the delivery of the necessary supply of water, or it may be a closed cast or wrought-iron pipe, laid at the natural slope of the ground, and about 3 ft. below the surface. In some cases part of the head race is an open channel, part a closed pipe. The channel often starts from a small storage reservoir, constructed near the stream supplying the water motor, in which the water accumulates when the motor is not working. There are sluices or penstocks by which the supply can be cut off when necessary. 2. Leading from the motor there is a tail race, culvert, or discharge pipe delivering the water after it has done its work at the lowest convenient level. 3· waste channel, weir, or bye-wash is placed at the origin of the head race, by which surplus water, in floods, escapes. 4· The motor itself, of one of the kinds to be described presently, which either overcomes a useful resistance directly, as in the case of a ram acting on a lift or crane chain, or indirectly by actuating transmissive machinery, as when a turbine drives the shafting, belting and gearing of a mill. With the motor is usually com­ bined regulating machinery for adjusting the power and speed to the work done. This may be controlled in some cases by automatic governing machinery.

92

HYDRAULICS

[WATER MOTORS

§ 169. Water Motors with A rtificial Sources of Energy.-The the hi�hest range of mills. A second canal takes the water which great convenience and simplicity of water motors has led to their has dnven turbines in the highest mills and supplies it to a second series of mills. There is a third canal on a still lower level supplying adoption in certain cases, where no natural source of water the lowest mills. The water then finds its way back to the river. power is available. In these cases, an artificial source of water With the grant of a mill site is also Lased the right to use the water­ power is created by using a steam-engine to pump water to a power. A mill-power is defined as 38 cub. ft. of water per sec. reservoir at a great elevation, or to pump water into a closed during r6 hours per day on a fall of 20 ft. This gives about 6o h.p. effective. The charge for the power water is at the rate of 2os. per reservoir in which there is great pressure. The water flowing h.p. per annum. from the reservoir through hydraulic engines gives back the § I 73· A ction of Water in a Water Motor.-Water motors may energy expended, less so much as has been wasted by friction. be divided into water-pressure engines, water-wheels and Such arrangements are most useful where a continuously acting turbines. steam engine stores up energy by pumping the water, while the Water-pressure engines are machines with a cylinder and piston work done by the hydraulic engines is done intermittently. or ram, in principle identical with the corresponding part of a § 1 70. Energy of a Wa ter-fall. -L et H 1 be the total fall of level from st�am-engine. The water is alternately admitted to and dis­ the point where the water is taken from a natural stream to the point where it is discharged into it again. Of this total fall a portion, charged from the cylinder, causing a reciprocating action of the which can be estimated independently, is expended in overcoming piston or plunger. It is admitted at a high pressure and dis­ the resistances of the head and tail races or the supply and discharge charged at a low one, and consequently work is done on the piston. pipes. Let this portion of head wasted be l),. Then the available The water in these machines never acquires a high velocity, and head to work the motor is H = H, - l),. It is this available head which should be used in all calculations of the proportions of the motor. for the most part the kinetic energy of the water is wasted. Let Q be the supply of water per second. Then GQH foot-pounds The useful work is due to the difference of the pressure of per second is the gross available work of the fall. The power of the admission and discharge, whether that pressure is due to the fall may be utilized in three ways. (a) The GQ pounds of water may weight of a column of water of more or less considerable height, be placed on a machine at the highest level, and descending in con­ tact with it a distance of H ft. , the work done will be (neglecting or is artificially produced in ways to be described presently. \Vater-wheels are large vertical wheels driven by water falling losses from friction or leakage) GQH foot-pounds per second. (b) Or the water may descend in a closed pipe from the higher to the from a higher to a lower level. In most water-wheels, the water lower level, in which case, with the same reservation as before, the acts directly by its weight loading one side of the wheel and so pressure at the foot of the pipe will be p GH pounds per square foot. If the water with this pressure acts on a movable piston like that causing rotation. But in all water-wheels a portion, and in some of a steam engine, it will drive the piston so that the volume described a considerable portion, of the work due to gravity is first em­ is Q cubic feet per second. Then the work done will be pQ = GHQ ployed to generate kinetic energy in the water ; during its foot-pounds per second as before. (c) Or lastly, the water may be action on the water-wheel the velocity of the water diminishes, allowed to acquire the velocity v = v 2gH by its descent. The kinetic and the wheel is therefore in part driven by the impulse due to 2 energy of Q cubic feet will then be �GQv /g = GQH, and if the water Water-wheels are there­ is allowed to impinge on surfaces suitably curved which bring it the change of .the water's momentum. finally to rest, it will impart to these the same energy as in the fore motors on which the water acts, partly by weight, partly by previous cases. :Vlotors which receive energy mainly in the three impulse. ways described in (a) , (b) , (c) may be termed gravity, pressure and Turbines are wheels, generally of small size compared with inertia m'?tors respectively. Generally, if Q ft. per second of water Before act by wetght through a distance h1, at a pressure p due to h2 ft. of water wheels, driven chiefly by the impulse of the water. fall, and with a velocity v due to h 3 ft. of fall, so that h1 + h2 + h3 H , entering the moving part of the turbine, the water is allowed then, apart from energy wasted b y friction o r leakage o r imperfection to acquire a considerable velocity; during its action on the of the machine, the work done will be turbine this velocity is diminished, and the impulse due to the GQh, +PQ + (G/g)Q (v2/2g) = GQH foot pounds, change of momentum drives the turbine. the same as if the water acted simply b y its weight while descen ding In designing or selecting a water motor it is not sufficient to H ft. consider only its efficiency in normal conditions of working. § q r . Site for Water Motor.-\Vherever a stream flows from It is generally quite as important to know how it will act with a higher to a lower level it is possible to erect a water motor. a scanty water supply or a diminished head. The greatest The amount of power obtainable depends on the available head difference in water motors is in their adaptability to varying and the supply of water. In choosing a site the engineer will conditions of working. select a portion of the stream where there is an abrupt natural fall, or at least a considerable slope of the bed. He will have Water-pressure Engines. regard to the facility of constructing the channels which are to § 1 74 . In these the water acts by pressure either due to the convey the water, and will take advantage of any bend in the river which enables him to shorten them. He will have accurate height of the column in a supply pipe descending from a high­ measurements made of the quantity of water flowing in the level reservoir, or created by pumping. Pressure engines were stream, and he will endeavour to ascertain the average quantity first used in mine-pumping on waterfalls of greater height than available throughout the year, the minimum quantity in dry could at that time be utilized by water wheels. Usually they seasons, and the maximum for which bye-wash channels must were single acting, the water-pressure lifting the heavy pump be provided. In many cases the natural fall can be increased rods which then made the return or pumping stroke by their by a dam or weir thrown across the stream. The engineer will own .weight. To avoid losses by fluid friction and shock the also examine to what extent the head will vary in different velocity of the water in the pipes and passages was restricted seasons, and whether it is necessary to sacrifice part of the fall to from 3 to 10 ft. per second, and the mean speed of plunger to and give a steep slope to the tail race to prevent the motor being I ft. per second. The stroke was long and the number of strokes drowned by backwater in flood;;. Streams fed from lakes which 3 to 6 per minute. The pumping lift being constant , such engines form natural reservoirs or fed from glaciers are less variable than worked practically always at full load, and the efficiency was streams depending directly on rainfall, and are therefore advan­ high, about 84 % - But they were cumbrous machines. Tliey are described in Weisbach's Mechanics of Engineering. tageous for water-power purposes. The convenience of distributing energy from a central station § I i2. Water Power at Holyoke, U. S. A . -Ab out 85 m. from the mouth of the Connecticut river there was a fall of about 6o ft. in to scattered working-points by pressure water conveyed in pipes a short distance, forming what were called the Grand Rapids, below -a system invented by Lord Armstrong-has already been which the river turned sharply, forming a kind of peninsula on which the city of Holyoke is built. In 1 845 the magnitude of the water­ mentioned. This system has led to the development of a great power available attracted attention, and it was decided to build a variety of hydraulic pressure engines of very various types. dam across the river. The ordinary flow of the river is 6ooo cub. ft. The cost of pumping the pressure water to some extent restricts per sec. , giving a gross power of :w.ooo h . p. In dry seasons the its use to intermittent operations, such as working lifts and power is 20,000 h . p . , or occasionally less. From above the dam a system of canals takes the water to mills on three levels. The first cranes, punching, shearing and riveting machines, forging and canal starts with a width of 140 ft. and depth of 22 ft., and supplies flanging presses. To keep down the cost of the distributing ==

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HYDRA ULICS

WATER MOTORS]

mains very high pressures are adopted, generally 700 lb per sq. in. or r 6oo ft. of head or more. P worked by water at In a 1arge numb er of hydraulic machmes " high pressure, especially lifting machines, the motor consists of a 95 % direct, single acting ram and cylinder. In a few cases doubletO acting pistons and cylinderS are used ; but they involve a · ·mg of th e piston not easi·1y accessib1e. In some water-t l· gh t pac k" cases pressure engines are used to obtain rotative movement, so %. E. B. and then two double-acting cylinders or three single-acting 8S % cylinders are used, driving a crank shaft. Some double-acting cylinders have a piston rod half the area of the piston. The 8S %, pressure water acts continuously on the annular area in front of the piston. During the forward stroke the pressure on the so % front of the piston balances half the pressure on the back. During 1 1 7 6 . Direct-A cting Hydraulic the return stroke the pressure on the front is unopposed. The 1 7 1) .-This is the water in front of the piston is not exhausted but returns to the Lift (fig. supply pipe. As the frictional losses in a fl�id are independent simplest of all kinds of hydraulic of the pressure, and the work done increases directly as the motor. A cage W is lifted directly pressure, the percentage loss decreases for given velocities of by . water pressure acting in a flow as the pressure increases. Hence for high-pressure machines cylmder C, the length of which is somewhat greater velocities are permitted in the passages than a little greater than the lift. A for low-pressure machines. In supply mains the velocity is ram or . plunger R of the same from 3 to 6 ft. per second, in valve passages 5 to ro ft. per second, length IS attached to the cage. O! in extreme cases 2 0 ft. per second, where there is less object The water-pressure admitted by a in economizing energy. As the water is incompressible, slide cock to the cylinder forces up the valves must have neither lap nor lead, and piston valves are ram, and when the supply valve is preferable to ordinary slide valves. To prevent injurious com- closed and the discharge valve In pression from exhaust valves closing too soon in rotative engines i opened, the ram descends. with a fixed stroke, small self-acting relief valves are fitted to the i this case the ram is 9 in. diameter, cylinder ends, opening outwards against the pressure into the with a stroke of 49 ft. It consists valve chest. Imprisoned water can then escape without over- of lengths of wrought-iron pipe screwed together perfectly waterstraining the machines. In direct single-acting lift machines, in which the stroke is tight, the lower end being closed fixed, and in rotative machines at constant speed it is obvious by a cast-iron plug. The ram that the cylinder must be filled at each stroke irrespective of the works in a cylinder I I in. dia­ amount of work be done. The same amount of water is used meter of 9 ft. lengths of flanged whether much or little work is done, or whether great or small cast-iron pipe. The ram passes weights are lifted. Hence while pressure engines are very water-tight through the cylinder efficient at full load, their efficiency decreases as the load de- cover, which is provided with creases. Various arrangements have been adopted to diminish double hat leathers to prevent this defect in engines working with a variable load. In lift ing leakag� outwards or inwards. As machinery there is sometimes a double ram, a hollow ram the weight of the ram and cage is enclosing a solid ram. B y simple arrangements the solid ram much more than su fficient to cause only is used for small loads, but for large loads the hollow ram is a descent of the cage, part of the locked to the solid ram, and the two act as a ram of larger area. weight is balanced. A chain atIn rotative engines the case is more difficult. In Hastie's and tached to the cage passes over a Rigg's engines the stroke is automatically varied with the load pulley at the top of Level increasing when the load is large and decreasing when it is small: the lift, and carries But such engines are complicated and have not achieved much at its free end a -==­ Disc'(oarge success. Where pressure engines are used simplicity is generally balan.ce _weight . B , workmg m T Iron • a first consideration, and economy is of less importance. • g�ides. Water i � ad­ t. § I 7S· Efficiency of Pressure Engines.-lt l' mitted to the cylmder a • from a 4-in. supply some ' t a. v pipe through a two­ i th; stroke, � way slide worked by "' . r .. a rack, spmdle and a ratio 4 12. I I endless rope. The o s shock be (r - r )'v•f2g I f�. m lift works under 73 :l[ b z -. v2• ft. of head and lifts I ' I a I 3 5° 1U at 2 ft. per rv /2f!.. s e c o n d. T h e e ffi­ f '':1 ciency is from 7 5 to p 8o o/ to·

93

r

Level of Supply

-...-· ., I

•I

I'I. · 't I I

,, I

1

I

to

I

H I I I I I I I I I I I I I I I I

11

tstrohkeeogerhydrenteicraal uhcce from the centre of the outlet surface of the turbine to the tail­ = o·4I4a.2r2/gH = 0·828, water level which is wasted, but which is properly one of the losses about 1 7 % of the energy of the fall being carried away by the water belonging to the turbine itself. In that case the velocities of the discharged. The actual efficiency realized appears to be about 6o %. water in the turbine should l>e calculated for a head H-!>, but the efficiency of the turbine for the head H. so that about 2 1 % of the energy of the fall is lost in friction, i n § r86. Gross Efficiency an.d Hydraulic Efficiency of a Turbine.-Let addition t o the energy carried away b y the water. Td be the useful work done by the turbine, in foot-pounds per § 1 84. General Statement of Hydrodynamical Principles necessary for second, Te the work expended in friction of the turbine shaft, gearing, &c., a quantity which varies with the locaL conditions in the Theory vf Turbines. which the turbine is placed. Then the effective work done by the (a) \Vhen water flows through any pipe-shaped passage, such as water in the turbine is the passage between the vanes of a turbine wheel, the relation be­ T = Td+T1 • tween the changes of pressure and velocity is given by Bernoulli's The gro�s efficiency of the whole arrangement of turbine, races, theorem (§ 29). Suppose that, at a section A of such a passage, ht is the pressure measured in feet of water, Vt the velocity, and z, the and transmissive machinery is 'It = Td/GQH,. (6 ) elevation above any horizontal datum plane, and that at a section And the hydraulic efficiency of the turbine alone is B the same quantities are denoted by hz, Vz, zz. Then 77 = T/GQH. ( 7) (1) ht-hz = (vz'Lvt2)/2g+zz-Zt. It is this last efficiency only with which the theory of turbines is If the flow is horizontal, z. = Zt ; and concerned. ht-hz = (vz2-1!t2)/2g. (ra) From equations (5) and (7) we get (b) \Vhen there is an abrupt change of section of the passage, or 77GQH = (GQ/g) (wtrt-W2r2 ) a ; an abrupt change of section of the stream due to a contraction, then, (8) '1 = (wtr,-w,r,) a.fgH. in applying Bernoulli's equation allowance rriust be made for the loss of head in shock (§ 36) . Let 7)[ , Vz be the velocities before and This is the ft!ndamental equation in the theory of turbines. I n after the abrupt change, then a stream of velocity v, impinges on a general,1 Wt and w,, the tangential components o f the water's stream at a velocity v2, and the relative velocity is Vt-v2• The motion on entering and leaving the wheel, are completely inde­ pendent. That the efficiency may be as great as possible, it is head lost is (v1-v,)2/2g. Then equation (ra) becomes obviously necessary that w2 = o. In that case (2) = /2g-(z•cv,)2 2g v (vt-v

the machine. \Vhen the machine is at rest the water issues from the orifices with the velocity .V (2gH) (friction being neglected) . But when the machine rotates the water i n the arms rotates also, and is in the condition of a forced vortex, all the particles having the same angular velocity. Consequently the pressure in the arms at the orifices is H + a2r2/2 g ft. of water, and the velocity of discharge through the orifices is v = ...[ (2gH + a2r2) . I f the total area o f the orifices is w , the quantity discharged from the wheel per second is

-

h,-h,

(vt'Lv.•z2)

/

= ,

,) (g.

To diminish as much as possible the loss of energy from irregular eddying motions, the change of section in the turbine passages must be very gradual, and the curvaw ture without discontinuity. .4 .1;

:

:

:l.i

(/

I 1 � C

_,.!)· ' ».

/ £ .1.. /

Aa

:.rizontally, and the turbine case is placed en:irely below the tail water. The water is supplied to the turbine by a vertical pipe, over which is a wooden pentrough, containing a strainer, which prevents sticks and other solid bodies getting into tb� turbine. The turbine rests on three foundation stones, and, the pivot fur the vertical shaft being under water, there is a screw and lever arange­ ment for adj usting it as it wears. The vertical shaft gives motion to the machinery driven by a pair of bevel wheels. On the righ• are the worm and wheel for working the guide-blade gear. § I SS. Hydraulic Power at Niagara.-The largest development of hydraulic power is that at N iagara. The N iagara Falls Power Company have constructed two power houses on the United States side, the first with IO turbines of 5000 h.p. each, and the second with IO turbines of 5500 h.p. The effective fall is I 36 to I 40 ft. In the first power house the turbines are twin outward flow reaction turbines with vertical shafts running at 250 revs. per minute and driving the dynamos direct. In the second power house the turbines

FIG. I 9 I . are inward flow turbines with draft tubes o r suction pipes. Fig. I 9 I shows · a section of one o f these turbines. There i s a balancing piston keyed on the shaft, to the under side of which the pressure due to the fall is admitted, so �hat the weight of turbine, vertical shaft and part of the dynamo is water borne. About 70 ,000 h.p. is daily distributed electrically from these two power houses. The Canadian Niagara Power Company are erecting a power house to contain -eleven units of 10,250 h .p. each, the turbines being twin inward flow reaction turbines. The Electrical Development Com­ pany of Ontario are erecting a power house to contain I I units of I 2 ,500 h.p. each. The Ontario Power Company are carrying out another scheme for developing 200,000 h.p. by twin inward flow turbines of 12 ,000 h.p. each. Lastly the Niagara Falls Power and Manufacturing Company on the United States side have a station giving 35,000 h.p. and are constructing another to furnish wo,ooo h:p. The mean flow of the Niagara river is about 222,000 cub. ft. per second with a fall of I 6o ft. The works in progress if completed will utilize 65o,ooo h.p. and require 48,ooo cub. ft. per second or 2 I ! % of the mean flow of the river (Unwin, " The Niagara Falls Power Stations," Proc. Inst. Meek. Eng., 1 906). § I 89. Different Forms of Turbine Wheel.-The wheel of a turbine or part of the machine on which the water acts is an annular space, FIG. 190. furnished with curved vanes dividing it into passages exactly or l, l, l on the outside of the case. A worm wheel on one of the roughly rectangular in cross section. For radial flow turbines the spindles is rotated by a worm d, the motion being thus slow 1 wheel may have the form A or B , fig. 1 92, A being most usual with

. �.... .. . . . ra

�--·

I

I

•

. ..

·····

'£' · · · · · · · · · · · -;

�

FIG. 1 92.

c

J-IYDRA ULICS

TURBINES]

inward, and B with outward flow turbines. I n A the wheel vanes are fixed on each side of a centre plate keyed on the turbine shaft. The vanes are limited by slightly-coned annular cover plates. In B the vanes are fixed on one side of a disk, keyed on the shaft, and limited by a cover plate parallel to the disk. Parallel flow or axial flow turbines have the wheel as in C. The vanes are limited by two concentric cylinders.

Theory of Reaction Turbines. § 190. Velocity of Whirl and Velocity of Flow.-Let acb (fig. 193) be the path of the particles of water in a turbine wheel. That path will be in a plane normal to the axis of rotation in radial flow turbines, and on a cylindrical surface in axial flow t urbines. At any point c of the path the water will have some velocity v, in the direction of a tangent to the path. That velocity may be resolved into two components, a whirling velocity w in the F IG. 193 · d i r e c t i o n of t h e wheel's rotation at the point c , and a component u at right angles to this, radial in radial flow, and parallel to the axis in axial flow turbines. This second component is termed the velocity of flow. Let Vo, w., Uo be the velocity of the water, the whirling velocity and velocity of flow at the outlet surface of the wheel, and v;, w;, v; the same quantities at the inlet surface of the wheel. Let a. and fJ be the angles which the water's direction of motion makes with the direction of motion of the wheel at those surfaces. Then w. =·v. cos{J; u. = v. sin fJ � (10) Wi = Vi Cos a : tti =V� sin a ) · The velocities of flow are easily ascertained independently from the dimensions of the wheel. The velocities of flow at the inlet and outlet surfaces of the wheel are normal to those surfaces. Let n. , f!; be the areas of the outlet and inlet surfaces of the wheel, and V the volume of water passing through the wheel per second ; then V0 = Q/f!0 ; v; = Q/fl;.

(I I)

Csing the notation i n fig. 1 9 1 , we have, for a n inward flow turbine (neglecting the space occupied by the vanes) , (12a) flo = 2 71'r0d0 ; f!; =271'T;d;. Similarly, for an outward flow turbine, ( 1 2b) f!0 = 2 trr0d ; fl; = Ztrr;d ; and, for an axial flow t)lrbine, (12c) flo = fl; = tr (r22-r ,2 ) . Relative a.nd Common Velocity of the Water and Wheel.-There is another way of resolving the velocity of the water. Let ,V be the velocity of the wheel at the point c, fig. 194· Then the veloCity of the water may �e resolved into a component V, which the water has in common with the wheel, and a component v., which is the velocity of the water relatively to the wheel. Velocity of Flow.­ It is obvious that the frictional losses of head in the wheel passages will increase as the velocity of flow is greater, that is, the smaller the wheel is FIG. 194made. But if the wheel works under water, the skin friction of the wheel cover increases as the diameter of the wheel is made greater, and in any case the weight oi the wheel and consequently the journal friction increase as the wheel is made larger. It is therefore desirable to choose, for the velocity of flow, as large a value as is consistent with the condition that the frictional losses in the wheel passages arc a small fraction of the total head. The values most commonly assumed in practice are these :In axial flow turbines, Uo = U; = 0· 1 5 to 0•2,Y (zgH) ; In outward flow turbines, u; =0 · 2S.Y 2g(H - �) i'-"". ..,--7l)"') ; Uo=O·ZI t o 0• 1 7V 2 1!7(H In inward flow turbines, Uo = u; =O·IZSv (zgH). § 191. Speed of the Wheel.-The best speed of the wheel depends partly on the frictional losses, which the ordinary theory of turbines

101

disregards. It is best, therefore, to assume for Vo and V; values which experiment has shown to be most advantageous. In axial flow turbines, the circumferential velocities at the mean radius of the wheel may be taken

Vo = V; = o·6.Y 2gH to o· 66,Y 2gH. In a rad1al outward flow turbine, - ...,,.., (H.,.._ IJ ) v, = o· s6v.-2g = V. = V;rofr; , where ro, r; are the radii of the outlet and inlet surfaces. In a radial inward flow turbine, V; = o · 66 v 2gH, Vo = V;ro/r;. If the wheel were stationary and the water flowed through it, the water would follow paths parallel to the wheel vane curves, at least when the vanes were so close that irregular motion was prevented. Similarly, when the wheel is in motion, the water follows paths rela­ tively to the wheel, which are curves parallel to the wheel vanes. Hence the relative component, Vr, of the water's motion at c is tan­ gential to a wheel vane curve drawn through the point c. Let v., V., Vro be the velocity of the water and its common and relative components at the outlet surface of the wheel, and v; , V;, v,.; be the same quantities at the inlet surface ; and let () and q, be the angles the wheel vanes make with the inlet and outlet surfaces ; then Vo2 = .Y (v,.2 +Vo2 - zVoVro cos ) � ( 1 3) v; = .J (v,;2 +V;2 - zV;v,.; cos 8) ) ' equations which may b e used t o determine q, and 8. § 192. Condition determining the A ngle of the Vanes at the Outlet Surface of the Wheel.-lt has been shown that, when the water leaves the wheel, it should have no tangential velocity, if the effici­ ency is to be as great as possible ; that is, W0 = 0. Hence, from ( 10), cos fJ =o, fl = 90°, Uo = V,, and the direction of the water's motion is normal to the outlet surface of the wheel, radial in radial flow, and axial in axial flow turbines. Drawing Vo or Ito radial or axial as the FIG. 195. case may be, and V. tangential to the direction of motion, Vro can be found by the parallelogram of velocities. From fig. 195, tan t/> =Vo/V. = u.{V. ; (14) but q, is the angle which the wheel vane makes with the outlet surface of the wheel, which is thus determined when thf velocity of flow Uo and velocity of the wheel Vo are known. When q, is thus determined, Correction of the A ngle q, to allow for Thickness of Vanes.-In determining q,, it is most convenient to calculate its value approxi­ mately at first, from a value of Uo obtained by neglecting the thick­ ness of the vanes. As, however, this angle is the most important angle in the turbine, the value should be afterwards correc�ed to allow for the vane thickness. Let q,' = tan-'(u.{V.) = tan-' (Q/fl.Vo) be the first or approximate value of , and let t be the thickness, and n the number of wheel vanes which reach the outlet surface of the wheel. As the vanes cut the outlet surface approximately at the angle ', their width measured on that surface is t cosec q,'. Hence the space occupied by the vanes on the outlet surface is For A, fig. 192, ntdo cosec (15) B , fig. 192, ntd cosec q, C, fig. 192 , nt(r2- rt) cosec q, Call this area occupied by the vanes w. Then the true value of tlte clear discharging outlet of the wheel is flo - w, and the true value of tto is Q/(l!o - w). The. corrected value of the angie of the vanes will be = tan [Q/V.(flo - w)]. (16) § 193. Head producing Velocity with which the Water enters the Wheel.-Consider the variation of pressure in a wheel passage, which satisfies the condition that the sections change so gradually that there is no loss of head in shock. When the flow is in a hori­ zontal plane, there is no work done by gravity on the water passing through the wheel. In the case of an axial flow turbine, in which the flow is vertical, the fall d between the inlet and outlet surfaces should be taken into account.

}

•

HYDRAULICS

1 02

I

V;, Vo be the velocities of the wheel at the inlet and outlet surfaces, Vo the velocities of the water, u; , Uo the velocities of flow, v,;, the relative velocities,

Let

V;,

I

v,.

[TURBINES

This angle can, if necessary, be corrected to allow for the thickness of the guide-blades. Condition determining the A ngle of the Vanes at the Inlet § Surface of the Wheel.-The single condition necessary to be satisfied at the inlet surface of the wheel is that the W- �·; (I +V.�) + ��2 } l

For an inward flow turbine,

v; =c,. �[2g l H - ;�2 ( r+V$) +�� n .

� I94· A nf.le which the Guide-Blades make with the Circumference r;f the Whee .-At the moment the water enters the wheel, the

radial component of the velocity is and the velocity is v;. Hence, if 'Y is the angle between the guide-blades and a tangent to the

u,,

wheel

•.

2g

H),

V;

= V;/211"r; = I ·0579v (H..J H/Q) ; Vo V;r./r; = 0 ·33...; 2gH. =Uo(Vo =0• I25/0· 33 = =2

H - (VN�g) V 2) +tt;2j2g =H{r - · 4356(I +ors6 } =0·5646H. = · g6..J 2g(· 5646H) =0· 72 I..J2gff. Sin 'Y =u;/v; = o· I25/0·72I = o· I 73 ; = I0° nearly. Tangential Velocity of Water entering Wheel. =V;COS =0 ·7 JOIV 2gH. A ngle of Vanes at !�let Surface. Cot 8 = (w; - V;)/u; = (·7IOI - · 66)/-I25 = ·4008 ; 'Y

Wi

'Y

8 = 68 ° nearly. Hydraulic Efficiency of Wheel. T/ w; ; g

= V / H = ·7IOI X ·66 X2 =0•9373 ·

This, however, neglects the friction of wheel covers and leakage. The efficiency from experiment has been found to be to o· 8o.

0• 75

§

I9 7·

Impulse and Partial Admission Turbines. Th e principal defect of most turbines with complete

admission is the imperfection of the arrangements for working with less tha n the normal supply. With many forms of reaction turbine the efficiency is considerably reduced when the regulating

TURBINES)

HYDRAULICS

sluices are partially closed, but it is exactly when the supply of water is deficient that it is most important to get out of it the greatest possible amount of work. The imperfection of the regulating arrangements is therefore, from the practical point of view, a serious defect. All turbine makers have sought by various methods to improve the regulating mechanism. H. Fourneyron, by dividing his wheel by horizontal diaphragms, virtually obtained three or more separate radial flow turbines, which could be successively set in action at their full power, but the arrangement is not altogether successful, because of the spreading of the water in the space between the wheel and Fontaine similarly employed two concentric guide-blades. axial flow turbines formed in the same casing. One was worked at full power, the other regulated. By this arrangement the loss of efficiency due to the action of the regulating sluice affected only half the water power. Many makers have adopted the expedient of erecting two or three separate turbines on the same waterfall. Then one or more could be put out of action and the others worked at full power. All these methods are rather palliatives than remedies. The movable guide-blades of Professor James Thomson meet the difficulty directly, but they are not applicable to every form of turbine. C. Calion, in x 84o, patented an arrangement of sluices for axial or out ward fk>w turbines, which were to be closed success­ ively as the wat• supply diminished. By preference the sluices were closed by pairs, two diametrically opposite sluices forming a pair. The water was thus admitted to . opposite but equal arcs of the wheel, and the forces driving the turbine were sym� metrically placed. As soon as this arrangement was adopted,

103

portion of the sluice, and stopped each time it passes a closed portion of the sluice. It is thus put into motion and stopped twice in each rotation. This gives rise to violent eddying motions and great loss of energy in shock. To prevent this, the turbine wheel with partial admission must be placed above the tail water, and the wheel pas!jages be allowed to clear themselves of water, while passing from one open portion of the sluices to the next. But if the · wheel passages are free of water when they arrive at the open guide passages, then there can be no pressure qther than atmospheric pressure in the clearance space between guides and wheel. The water must issue from the sluices with the whole velocity due to the head ; received on the curved vanes of the wheel, the jets must be gradually deviated and discharged with a small final velocity only, precisely in the same way as when a single jet strikes a curved vane in the free air. Turbines of this kind are therefore ·termed turbines of free deviation. There is no variation of pressure in the jet during the whole time of its action on the wheel, and the whole energy of the jet is im­ parted to the wheel, simply by the impulse due to its gradual change of momentum. It is clear that the water may be admitted in exactly the same way to any fraction of the circumference at pleasure, without altering the efficiency of the wheel. The diameter of the wheel may be made as large as convenient, and the water admitted to a s-mall fraction of the circumference only. Then the number of revolutions is independent of the water velocity, and may be kept down to a manageable value. § 198. General Descripti1m of an Impulse Turbine or Turbine with Free Deuiation.-Fi�. 197 shows a general sectional elevation 'of a 1n turbine, Girard w h i c h t h e fl o w i s T h e w a t e r, · axial. admitted a h o v e a hor i z o n t a l · fl o o r, passes down through wheel the annular containing the guide­ blades G, G, and

, �����. '��ttl•'wW.

a

a

L)7ffi�

The revolving wheel · is fixed to a hollow shaft . sus p)2\ Variation of Pressure in the Pump Disk.-'-Precisely as in the case of turbines, it can be shown that the variation of pressure between the inlet and outlet surfaces of the pump is h. - h; = (V.2 - V;2)/2g - (v,.2 - v,.;2) /2g. Inserting the values of v,., v,; in (4) and (5), we get for normal conditions of working =

J'

[PUMPS

h. -h. = (V.2- V;2) /2g -u.2 cosec2/2g+ (u;2 +V;2)/2g (6) = Vo'(2g - u.' cosec 24>/2 g+u ; 2(2 g. Hydraulic Efficiency of the Pump.-Neglecting disk friction, journal friction, and leakage, the efficiency of the pump can be found ln the same way as that of turbines (§ 186). Let M be the moment of the COUple J'Otating the ump, and a its angular velocity ; W0, To the tangential velocity o the water and radius at the outlet surface ; w;, r; the same quantities at the inlet surface. Q being the discharge per second, thP. change of angular momentum per second is

r

(GQ /_t) (w.r. - w;r;).

Hence M = (G(,J/g) (w.r. -w;r;). I n normal working, w; =o. Also, multiplying by the ang ular velocity, the work done per second is M a = (GQ/g)w0r0a. But the useful work done in pumping is GQH. Therefore the efficiency is (7) '1/ = GQH/M a =gH;w.r.a = gH/w.V.. § 209. Case I. Centrifugal Pump with no Whirlpool Chamber.­ When no special provision is made to utilize the energy of motion of the water leaving the wheel, and the pump discharges directly into a chamber in which the water is flowing to the discharge pipe, nearly the whole of the energy of the water leaving the disk is wasted. The water leaves the disk with the more or less considerable velocity Vc, and impinges on a mass flowing to the discharge pipe at the much slower velocity v,. The radial component of v. is almost necessarily wasted. From the tangential component there is a gain of pressure (w.2 - vs') (2g - (w. - v,)2(2g = v,(w.-v,)(g,

which will be small, if v, is small compared with w.. Its greatest value, if v, =lw., is !w.2(2g, which will always be a small part of the whole head. Suppose this neglected. The whole variation of pressure in the pump disk then balances the lift and the head Ui2(2g necessary to give the initial velocity of flow in the eye of the wheel. u,3(2g+H ""'V .'/2g - u.' cosec 2(2g +u;2/2g, H = V.2/2g - u.' cosec 2cj>(2g I (8) V. = v (2gH +u.2 cosec 2 . \ or and the efficiency of the pu mp is, from (7), 'I = gH /V .w . = gH/\ V (V. - n . cot q,) j , = (V.2 -u.o2 cosec 24>)/{2V.(V. - u. cot l, (9 ). 71= (V.2 - tto2) /2V.2, For t1> = 9o•, which is necl)ssatily less than �. That is, half the work expended in driving the pump 1s wasted. By recurving the vanes, a plan intro­ duced by Appold, the efficiency is increased, because the velocity If 4> is very small, Vo of disi:harge from the pump is diminished. cosec cJ> cot 4> ; and then 71 = (V. +u. cosec cJ>)/2V., which may approach the value I, as q, tends towards o. Equation shows that. Uo cosec 4> · cannot be greater than V.. Putting u. = 0' 25 v (2gH) we get the following numerical values o1 the . efficiency and the circumf"rential velocity of the pump :·

=

(8)

PUMPS] '7

v.

0•47 0·56 0·65 0·73 0·84

HYDRAULICS

1 ·03.V 2gH I •o6 1 • 12 1 •24 1 •75

cannot practically be made less than 2o" ; and, allowing for the frictional losses neglected, the efficiency of a pump in which q, = 20° is found to be about ·6o. § 2 10 . Case 2. Pump with a Whirlpool Chamber, as in fig. 2 10.­ Professor James Thomson first suggested that the energy of the water after leavmg the pump disk might be utilized, if a space were left in which a free vortex could be formed. I n such a free vortex the velocity varies inversely as the radius. The �ain of pressure in the vortex chamber is, putting r., r,. for the radii to the outlet surface of wheel and to outside of free vortex,

q,

(

)

'Vo2 1 - ro2 =vo' ( I - k2) ' r,.2 2g 2g k = r.frw.

if The lift is then, adding this to the lift in the last case, H = {Vl-uo2 cosec2q, +vo'(I - k2) }j2g. vo' = V."--2Vouo cot

But

q,+uo' cosec21> ;

: . H = { (2 - k2)Vo' -2kVouo cot q, - k2uo' cosec2}/2g. (10) Putting this in the expression for the efficiency, we find a con­ siderable increase of efficiency. Thus with = 90° and k = !, '7 � 1 nearly, q, a small angle and k = !, '7 = 1 nearly.

\Vith this arrangement of pump, therefore, the· angle at the outer ends of the vanes is of comparatively little importance. A moderate angle of 30° or 40° may very well be adopted. The following numerical values of the velocity of the circumference of the pump have been obtained by taking k = !, and Uo = 0·25-J (2gH).

900 4 0 5

30"

20°

v.

--

·762.V 2gH ·842 •91 1 1 •023

The quantitv of water to be pumped by a cen.trifugal pump neces­ sarily varies, a�d an adjustment for differ1�nt quantities of water can­ not easily be introduced. Hence it is that the average efficiency of pumps of this kind is in practice less than the efficiencies given above. The �dva_ntage of a vortex c�amber is �lso �enerally neglected. The vcloctty m the supply and dtscharge ptpes ts also often made greater than is consistent with a high degree of efficiency: Velocities of 6 or 7 ft. per second in the discharge and suction pipes, when the lift is small, cause a very sensible waste of energy ; 3 to 6 ft. would be much better. Centrifugal pumps of very large size have been constructed. Easton and Anderson made pumps for the North Sea canal in Holland to deliver each 670 tons of water per minute on a lift of 5 ft. The pump disks are 8 ft. diameter. J . and H. Gwynne constructed some pumps for draining the Ferrarese Marshes, which together deliver 2000 tons per minute. A pump made under Pro­ fessor J . Thomson's direction for drainage works in Barbados had a pump disk 16 ft. in diameter and a whirlpool chamber 32 ft. in The efficiency of centrifugal pumps.when delivering less diameter. or more than the normal quantity of water is discussed in a paper in the Proc. Inst. Civ. Eng. vol. 53 ·

.

§ 2 1 1 High Lift Centrifugal Pumps.-It has long been known that centrifugal pumps could be worked in series, each pump overcoming a part of the lift. This method has been perfected, and centrifugal pumps for very high lifts with great efficiency have been used by Sulzer and others. C. W. Darley (Proc. Inst. Civ. Eng. , supplement to vol. 1 54, p. 1 56) has described some pumps of this new type driven by Parsons steam turbines for the water supply of Sydney, N.S.W. Each pump was designed to deliver rt million gallons per twenty-four hours against a head of 240 ft. at 3300 revs. per minute. Three pumps in series give therefore a lift of 720 ft. The pump consists of a central double­ sided impeller 1 2 in. diameter. The water entering at the bottom divides and enters the runner at each side through a bell-mouthed passage. The shaft is provided with ring and groove glands which on the sucti.on side keep the air out and on the pressure side prevent leakage. Some water from the pressure side leaks through the glands, but beyond the first grooves it passes into a pocket and is returned to the suction side of the pump. For the glands on the suction side water is supplied from a low­ pressure service. No packing is used in the glands. During the trials no water was seen at the gl nds. The following are the results of tests made at l'\ewcastle :- -

Duration of test . . hours Steam pressure lb per sq. in. Weight of steam per water h.p. hour . . . . lb Speed in revs. per min. Height of suction . . ft. Total lift . . . ft. . . Million galls. per day pumped. . By Ventun meter . By orifice Water h.p . .

1 09 I.

II.

III.

I V.

2 57

1 •54 57

1 •2 84

1 ·55 55

27•93 3300

30•67 3330

28·83 3710

27·89 3340

762

744

917

756

1 •573 1 · 623 252

1 • 499 1 ·513 235

1 ·689 1 ·723 326

1 •503 1 ·555 239

II

II

II

II

In trial IV. the steam was superheated 9S° F. From other trials under . the same conditions as trial I. the Parsons turbine uses 1 5 ·6 lb of steam per brake h.p. hour, so that the combined efficiency of turbine and pumps is about 56 %, a remarkably good result. § 2 1 2. A ir-Lift Pumps.-An interesting and simple method of pumping by compressed air, invented by Dr J. Pohle of Arizona, is likely to be very useful in certain cases. Suppose a rising main placed in a deep bote hole in which there is a considerable depth of water. Air compressed to a sufficient pressure is con­ veyed by an air pipe and introduced at the lower end of the rising main. The air rising in the main the diminishes average density of the contents of the main, and Slilnd Pipe their aggregate weight no longer balances the pressure at the lower end of the main due to its sub­ mersion. An up­ ward flow is set up, and if the air supply is suffi­ cient the water in the rising main is lifted to any required height. The higher the lift above the level in the bore hole the deeper must be the point at which air is Fig. injected. 2 1 2 shows an air­ lift pump con­ structed for W. H. Maxwell at the Tunbridge Wells water­ works. There is a two-stage steam air compressor, FIG. 2 12 . compressing airto f ro m go to 100 lb . per sq. in. The bore hole is 3 so ft. deep, lined with steel pipes I S iri� diameter for 200 ft. and with perforated pipes I3!.in. diameter for the lower r so ft. The rest level of the water is 96 ft. from the ground-level, and the level when pumping 32,000 gallons per hour is r 20 ft. from the ground-level. The rising main is 7 in. diameter, and is carried nearly to the bottom of the bore hole and to 20 ft. above the ground-level. The air pipe is 2! in. diameter, In a trial run 3 1 .402 gallons per hour were raised 13,3 ft. above the level in the well. Trials of the efficiency of the system made at San Francisco with varying conditions will be found in a paper by E. A. Rix (Journ. Amer. Assoc. Eng. Soc. vol. 25,

,HYDRA-ZENE;

1 10

calculated on the work of compression only. It is zero for no dis­ I QOO) . Maxwell found the best results when the ratio of immersion charge, and zero also when there is no resistance and all the energy to lift was 3 to x at the start and 2 · 2 to I at the end of the trial. given to the air is carried away as kinetic energy. There is a dis­ In these conditions the efficiency was 3 7 % calculated on: the charge for which this efficiency is a maximum ; it is about half the indicated h.p. of the steam-engine, and 46 % calculated on the discharge which there is when there is no resistance and the delivery indicated work of the compressor. 2 • 7 volumes of free air were pipe is full open. The conditions of speed and discharge correspond­ to the greatest efficiency of compression are those ordinarily used to 1 of water lifted. The system is suitable for temporary mg taken as the best normal conditions of working. The curve marked purposes, especially as the quantity of water raised is much greater than could be pumped by any other system in a bore hole of a given size. It is useful for clearing a boring of sand '" . and may be advantageously used permanently when a boring is in sand or gravel which cannot be kept out of the bore hole. · ·""'"' The initial cost is small. t- '(o��y § 213. Centrifugal Fans.-Centrifugal fans are · constructed similarly to centrifugal pumps, and are used for compressing I/ E:.ftic, "'-= air to pressures not exceeding ra to I S in. of water-column. t--;l }(; . -...;;; "�... With this small variation of pressure the variation of volume r--- � v and density of the air may be neglected without sensible error.. !l:. f-. The conditions of pressure and discharge for fans are gener­ ��

N · N0---7

KS7r> N · N H2 -7

K2S04 + N2H 4.

HYDRA�ONE�HY»ROCBF HALUS P. J. Schestakov (J. R.ltss. Phy.s. Chem. Sec.,� 1:905, 3 7 , - p. · .� are conjugate functions of x and,y, +>lri =J(x+yi), v'Lif; =o, �.p ... o ; or putting

(6)

and,rJ. +Uy = f' · is the stream function of the relative motion of tbe liquid past the cylinder, and similarly if; - Vx for the component velocity V along Oy ; and generally . .P' = f + Uy - Vx (4) is the relative stream-function, constant over a solid boundary mo�ing with components U and V of velocity. If the liquid is stirred up by the rotation R of a cylindrical body, dif;(d$ 9!!ftd ,to. m6!v� so that the cylinder may swim for an instant in the liquid without with the velocity due to the others, the resultant streaJnJunctjpn distortion, with this velocity U1 ; and w in ( 1 ) will give the liquid being motion in the . interspace between . the fixed ,cylinder r ""a and the if; = 'Zm log r = log I Irm ; (9) · concentric cylinder r = b, moving with velocity Ut. \Vhen b = o , Ut = oo ; and when b = oo , U1 = - U, so that at the path of a vortex is obtaiaed by equating tqe yalue of 1{1 at the vortex to a constant, omitting the rm o£ the vortex itself. infinity the liquid is streaming in the direction xO with velocity U. When the liquid is bounded by a cylindrical surface, the motion If the liquid is reduced to rest at infinity by the superposition of of a vortex inside may be determined as due to a series of vortex­ an opposite stream given by w = - Uz, we are left with images, so arranged as to make the flow zero across the boundary. , w "' Va2/z, (6) For a plane boundary the image is the optical reflection of the q, = U (a2/r) cos 8 = Ua2x/(x2 +y2), ( 7) vortex. For example, a pair of equal opposite· vortices, rno!ving on if; = - V(a2(r) sin 0 = -- Ua2yf(x2 +y2) , a line parallel to a plane boundary, will have a corresponding pair (8) of images, formin� a rectangle of vortices, and the path of a vortex of the tylinder r = a with the motion due �o the . passag� giving . . will be the Cotes spiral velocity U through tbe· ongm 0 m the d1rect 1on Ox. r si n 28 = 2a, or x-� +y-2 = a�� ; (10) . If the direction of motion makes an angle (J' with Ox, this HI therefore the path of a single vortex in a right-an�led- corner ; , d.p jd.p 2XY 1. , , x2 -y = tan 2(J , 0 = .o tan IJ = and generally. if the angle of the corner is 1r[n, the path 19 the Cotes' (9) dy/ dx =

( I - �)

·

,

spiral

r sin nfJ = na.

'

(It}

A single vortex i n a circular cylinder of radius ·a at a distani::e c from the centre will move with the velocity due to an equal opposite image at a distance a,2Jc., and so describe a circle with velocity mc/(a'-(:2)in the periodic time 2'1r(a:L.c�)/m. (12) Conjugate functions can be employed also· for the motion 'Of liquid in a thin sheet between two concentric spherical surfaces ; the com­ ponents of velocity along the meridian and parallel in colatitude. 8 ' and longitude >. can be written d.p 1 do/; I d df dfJ = sin 8 fiX.• sin 0 di.. = - d/J'

ahd the velocity is

__t:__ = y (y - c) sin2 1.• 8 ' x2 + y' a' ' 1 2

throuj{h it.-A stream-function Y, conditions

must be determined to satisfy the , . v"Y = o , throughout the liquid ; (1) '' (2 ) .;, "" constant, over any fixed boundary ; dif;/ds =- normal velocity reversed over a solid boundary, (3 ) sc that, if the solid i� moving with velocity U in the direction Ox, df/ds = - Udy/ds, or .Y + U> = constant over the moving cylinder ;

.

sm

8,dD'

2y

-

c d.1_

as = �JS·

(10) ( 1 1)

o n the radius of curvature is !a2/(y - !c.), which skows that the curve is aft Elastica or Lintearia. (}. C. Maxwell, Collected Works, ii. .208 ,). If q,1 denotes the velocity function of the liquid filling the cylinder r = b., and moving bodily with it with velocity U1,

1 = - U1x,

and then

+fi = F(tan !IJ. eAi) . (14) :z8. Uniplanar Motion of a Liquid due to the Passage of a Cylinder

Ustfr2•

. Alortg t\le path of a particle, defined by the Ca of (3) ,

and over the separating surface r = b

�

·

(

(l2)

)

a2 + b2 a' • U � = -u, I + /i2 = a2 - b2'

and this, by § 36, is also the ratio of the kinetic energy in the annular interspace between the two · cylinders to the kinetic energy of the liquid moving bodily inside r = b . Consequently the inertia tc overcome in moving the cylinder r = b, solid or liquid, is its own inertia, increased by the inertia of liquid (a' +b2)/(a2 - lr) times the volume of the cylinder r = b ; this total inertia is called the effective inertia of the cylinder r b, at the instant the two cylinders are concentric. =

HYDRODYNAMICS] With l_iqt•id ?f de�sity p, this gives rise to acceleration d L /dt, g1ven by

;HYBROMECHANICS

a

•a' + b2 dU a' + b' . dU pb 1r a•-bi Tt ""ii'-1Ji M lfi •

kin et ic reaction to

·un!t

if � · denotes the ma:;s of liquid displaced by le �gth of the cyhnder r = b. I n particular, when a = (X) , the extra 1itert1a is M'. Wryen the cylinder r =a is moved with velocity U and r = b with · veloc1ty C, along Ox, .

2

U ' � ' (� + r) cos II - U , b a b2

�·

=

.f

�

= - Ub' a• (

- r) sin I! - U 'h"

� a' (; + �) cos II, 2

�a• (r - �) sin li ;

and similarl y , with velocity components V and V1 along

and

(r+�) sin

V ' �· ' ( � + r) sin II - Vw � ' b a a

¥- =

V ,�' , (� - r ) cos ii +V'bi� • (r - �) cos ll ; b a a

II,

then for the resultant motion

( 1 7) ( r8)

(20 )

and over r = b

(2 1 )

and the difference X-X, is the component' momentum of the liquid in the interspace ; with similar expressions for Y and Y1• Then, if the outside cylinder is free to move

b2 - �

X = -rrpa2Ub2 + a•·

(22)

But if the outside cylinder is moved with velocity U1 , and the i n side cylinder is solid or filled with liquid of density u,

2pb2 • U, X = -,..era. U, U = p(b2 +a') + or < cr. 30. The expression for w in ( r ) § 29 may be i ncrea sed by the addition of the term im log z = -mO + im log r, (r)

representing vortex motion cfrculating round the annulus of liquid. Considered by itself, with the cylinders held fixed , the vortex sets up a circu mferential velocity m/r on a radius r, so that the angular momentum of a circular filament of annular cross section dA is pmdA, and of the whole vortex is pmr (b2 -:-a2) . . Any circular filament can be started from rest by the application of a circumferential impulse 1rpmdr at each end of a diameter ; so that a mechanism attached to the cylinders, which can set up a uniform distributed impulse 1rpm across the two parts of a diameter in the liquid, will generate the vortex motion, and react on the cylinder with an impulse couple - pm1ra2 and pm1l"b2, having re­ sultant pm ... (b• -·a2), and this couple is infinite when b = oo , as the angular momentum of the vortex is infinite. Round the cylinder r =a held fixed in the U current the liquid streams past with velocity q' = 2 U sin 1J +m/a ;

(2)

and the loss of head due to this increase of velocity from U to q' is q'L U2

zg

(2 U sin 11 +m/a)L U2 2g

(6)

(8) = 2pdymUr1(cos 8 -a2r'l cos 31!), and with y = b tan II, r = b sec II, this is 2pmUdl!(i - a2b--2cos 31! cos II) , (9) and integrating between the limits 0 = ± !,.., the resultant, as before, is :i1rpmU. 3 1 . Example 2.-Confocal Elliptic Cylinders.-E mploy the elliptic

coordinates "' t. and r = 'l +ti. such that z = c ch r, x = c ch 'l cci s t, y = c sh 'l sin r ; (r ) then the curves for which 'I and E are constant are confocal ellipses and hyperbolas, and

J = ��::

if OD is

distances,

�� = 2( c

ch211 - cos'�)

= !c2(ch211 - cos2�) = r1 r2 = 0D2,

the semi-diameter conjugate to OP, and r11

r, , r2 = c (ch 'l ± cos t) ; r!! = x2 +y2 = c2 (ch' 'l - sin2�)

r2

(2 )

the focal

= !c2(ch 211+cos 2�). Consider the streaming motion given by

w = m ch (t - 'Y ) , 'Y = a + ,Bi, (5) (6) = m ch ('l - a)cos (t - .8) , Y, = m sh (71 - a)sin(� - ,8) . . Then Y, =co over the ellipse 'I = a, and the hyperbola t = fj, so that these may be taken as fixed boundaries ; and Y, ts a constant on a C4• Over any ellipse 'l, moving with components U and V of velocity, Y,' = f+Uy - Vx = [m sh ('l - a) cos ,B + Uc sh'l] sin t - [m sh (11 - a) sin .B + Vc ch 11l cos � ; (7 ) so that if' = o, if m sh('l - a) . "' m sh('l - a) "' cos .., , V = U = --c· ( 8) � c ch 'l sm .., ,

having a resultant in the direction PO, where P is the intersection of an ellipse 'I with the hyperbola ,8 ; and with this velocity the ellipse 'I can be swimming in the liquid, without distortion for an instant. At infinity

u = - c e-•cos ,8 =

m

'

so that cavitation will take place, llnless the head at a great distance exceeds this loss. The resultant hydrostatic thrust across any diametral plane of the cylinder will be modified, but the only term in the loss of head which exerts a ·resultant thrust on the whole cylinder . is zmU sin l!/ga, and it s thrust is 21rpmU absoll!te units in the direction Cy, to be counteracted by a support at the ceri.tte C ; the liquid i s streaming past r = a with velocity U rEiversed. and the cylinder is surrounded by a vortex. Similarly, the Streaming velocity V reversed will g1ve rise to a thrust 2rpmV in the direction xC. Now if the cylinder is released , and the eomponents U and V are rever,;ed so as to become the velocity of the cylimler with respect

( 7)

Thus with g = o, the cylinder will describe a circle with angular velocity 2pw/( (1 4 ) 4-t = th ('7 -4) over the surface of " ; lsO that parallel to o�. the effective inertia of the cylinder "' displacing M' hquid, is increased by M'th '!/th ('I.;A) , reducing when a. = oo to M' th 11 = M ' (b/a). Similarly, parallel to Oy, the increase of effective inertia is M'/th '7 th(17-a), reducing to M'/th 17 = M ' (ajb), when A = oo , and the liquid extends to infinity. 32. Next consider the motion given by .P = m ch 2 (17- A) sin 2� ,Y = m sh 2 ('1 A) cos 2� ; (1) in which ,Y == o over the ellipse A, and .Y' = .Y + ! R (x• +r) = [ -m sh 2 ('7 - A) + l Rc2Jcos 2�+lRc2 ch 217, (2) wh1ch is constant over the ellipse 'I if lRc2 = m sh 2 ('7 - A) ; (3) so that this ellipse can be rotating with this angular velocity R for an instant without distortion, the ellipse A being fixed. For the liquid filling the interior of a rotating elliptic cylinder of cross section X2/a2+y2/b2 = I , .Y/ = mt (x2/a2 + Jl/b2) ��· = -2R = -2mt ( I/a2 + x /b2�, with !ft = mt (x• ta• + r/b2) - !R(x2 + Jfl) (6) =-! R (xLy2 ) ( aL b• )f(a2 + b2) , 1 = Rxy (a2 - b2)/(a2+b2), Wt = 1/>t +M = - !iR(x +yi)2(a2 - b2)/(a2+b2). The velocity of a liquid particle is thus (a2 - b2) /(a2+b2) of what it would be if the liquid was frozen and rotating bodily with the ellipse ; and so the effective angular inertia of the liquid is (a2 - /;2) 2/ (a2 +b2) 2 of the solid ; and the effective radius of gyration, solid and liquid, is given by k2 = l (a2+ b2) , and l (a2 - b2) 2/ (a2+ b2). (7) For the liquid in the interspace between a and "' m ch 2 (-q-a) sin 2�

= � lRc2 sh 217 sin 2HaL b2)/(a2+b2) = I /th 2 (ra)th 211 ; (8) and the effective k2 of the liquid is reduced to lc2/th 2 ('1!-A) sh 2'1, (9) which becomes ic2/sh 2'1 = Ha2 - b2)/ab, when et = oo , and the liquid surrounds the elli pse 71 to infinity. An angular velocity R, which gives components - Ry, Rx of velocity to a body, can be resolved into two shearing velocities, - R parallel to Ox, c>.nd R parallel to O:r ; and then if; is resolved into .y, +.Y• • such that .Yt +!Rx2 and f• +!Rr is constant over the boundary. Inside a cylinder (1 0) 1/>• +t/tti = - !iR(x +yi) 2a2/ (a2 + b2 ) , (I I ) li>2+t/t•i = iiR(x+yi)2b2/ (a2+ b2), and for the interspace, the ellipse a. being fixed, and a.1 revolving with angular velocity R , +.Yd = - iiRc2sh 2 ('7 - a.+�i) (ch 2a.+ I )/sh 2 (at - cr.), (I2) 2+.Y•i = hRc2sh 2 ('7 - « +ti) (ch 2A- I ) /sh 2 (at - a), (I3 ) satisfying the condition that t/11 and .Y2 are zero over '7 "' a, and over •

-

I

•

,

-

-

t/11 +!Rx2 = l Rc2 (ch 2At + r), t/t• + ! Rr = l Rc2 (ch 2a.1 - I ),

(14) (I S)

constant values. In a similar way the more general state of motion may be analysed, given by w = m ch 2 (s-'Y) , ")' = a.+fJi, (I6) ios giving a homogeneou < strain velocity to the confocal system ; to whtch may be added a circulation, represented by an additional term ms in w.

(HYDRODYNAMICS

Similarly, with

(1 7 )

the function

(I8) .P==Qc shH'I- a.) sin! (I;- {J) will give motion streaming past the fixed cylinder " = a., and dividing along I; = {J ; and then ·

x2 -y2 =c2 sin i; ch 7J, 2xy =c2 cos i; sh 'l. In particular, with sh A= I, the cross-section of '7 = A is

(I9)

x4+6x2y2+y4 = 2c4, or x4+y4 "' c '

(20)

when the axes are turned through 45°. 33· Example J.-Analysing in this way the rotation of a rectangle filled with fiquid into the two components of shear, the stream function t/11 is to be made to satisfy the conditions ( i.) v'.Yt = o, (ii.) ,Y1 + ! Rx2 = !Ra2, or .Yt =o when x = "'= a, (iii. ) t/11 +lRx' = !Ra2, .Yt = !R(aLx2), when y = "'= b Expanded in a Fourier series, � cos(2n+ I H'II'x/a _ -� a2 x• ...3a2 � . (2n + I ) 3 so that a• � cos(2n+ IH'II'x/a . ch (2n + 1 ) !-ry(a y,1 - R� � {2n+ r)a . ch( n+ l ) ! rb/a ' ra . ·n i6 2 � cos(2n + I H 'II'z /a Wt =t +t/ttt =�a. 'll' aa � (2n+ r 3 ch(2n + x )!rb/a' )

-

•

(I)

2

_

•

2 ( ) an elliptic-function Fourier series ; with a similar expression for .Y• with x and y, a and b interchanged ; and thence t/t = .Yt + t/tz.

-

Example 4 . Parabolic cylinder, axial advance, and liquid stream­ ing past. The polar equation of the cross-section being rl cos !8 = al, or r + x =2a, (3) the conditions are satisfied by (4) t/t' = U r sin 8 -2Ua!r! sin !8 =2Url sin !B(rl cos !8 - al), ,Y = 2Uairl sin !8 = - Uv [2a(r -x)], (5)

w ,;,-2Ua!z!,

(6)

and the resistance of the liquid is 21rpaV2j2g. A relative stream line, along which .Y' = Uc, is the quartic curve 2 :- (y - c)• r 4a•y• + (y - c )• (7) = 4a(Y Y-c = n v=

(2)

u ,., -yf+z.,, -z�+xf, W != -�1f+Y�·

Now suppose �he liquid to be melted� and additional c.oml?onent� of angular veloc1ty f!" Qa, Ua commumcated to the elliJ?.SOidal ·case ; the additional velocity communicated to the Jiqui!l, w1U be due to a velocity-function

4> ==

c2 - aJ2sx - o,a''a2 - b2•xy, -n'bb2-c1 +b • +c2Yz-Otc• +a

(3)

as may be verified by considering one term at a time. ' If v , denote the components of the velocity of the liquid relative to the axes,

u', w'

2a1 'o.y- t 2a1 Otz, u' =u+yR-zQ= a•+b c +a• zb2 D.x, 2. b 2 = v +zP - xR = jji +c•f!,z- ar.:F/ii 2c2 20tx-li22c1 w' =w+xQ-yP = c•+a +tzllAy, P=Dt+�. Q '= !lt+lf, R =Da+f. u'!..+v'l.b. +w'!... o, ' v

Thus

c,

a.

(S) (6)

(8)

==

so that a liquid particle remains always on a similar ellipsoid . The hydrodynamical equations with moving axes, taking into account the mutual gravitation. of the liHuid, become

,du. ,du dt - vR +wQ +u,du+ ;;- .fx +4""pAx +du dx v dy +w dz = o, 1

do

where

...

� · · · · �r�)

abed>.. A , B, C , )o{• (a•+>.., IJ2+>.. , c•+>.)P P2 = 4(a2 +X) (112+>.) (&+>-). u, u', w', =

fi, w,

With the values above of of the fgrm

'

v ,

which, as Z is a quadratic function of are non-elliptic i ntegrals; also for ,Jt, where w cos i/1, 'I = - w sin >/;; In a state of steady motion

r•,

�=

so:

n n , ar ae = o, T, ='it '

(27 )

I Oi�+!l21f = flw, dq,

(30)

tj> =;l/t = nt, suppose,

(J or

'a2+c2 w dt = r - a2 -c2 iir' d>/; 2a2 n dt = - a2+c•wr' a2+c2w = 2a2 n 1 - a•-c•o - a•+c•;;;· (�-. aJ-c') ' _:(a2 - c1)(ga2 - c2) 0 ia2+c2 4(a2+c') '

the equations become

( I I) (I 2) ( 1 3)

and a state· of steady motion is impossible wh en

and integrating

pp-1+zrp(Ax2+By2+Ct) . + l (a.x2+ �yt+yz2 + zfyz.+zgzx +2hxy) ,. co»st.,

(3 I )

3a> c >a.

(3 3)

experiment was devised by Lord Kelvin for demonstrating this, in which the difference of steadiness was shown of a copper so that the surfaces of equal pressure are similar quadric surfaces, shell filled with liquid and spun gyroscopically, according as the which, symmetry and dynamic;;tl considerations show, must . be shell was slightly oblate or prolate. According to the theoty coa�tial surfaces ; and J, g, h :vamsq, as follows also by algebra1cal ' above the stability is regained when the length is more than three reductio n ; and diameters, so that a modern projectile with' a cavity more than three diameters long ,should fly steadily when iilled with water ; n� while the' old-faphioned type, not so elongated, would be h�ghly ' ( I 5) n� unsteady ; ·and for the same reason the gas bags of a dirigible balloon should be over rather than under three diameters long. with similar equations· for {3 and 'Y· I f we can make 40. A Liquid Jet.-.:..:.B y the use of the complex variable and its (16) conjugate fp.nctions, an. attempt can be made . to give a mathe­ the surfaces of equal pressure jlre similar to the elCternal case, which matical interpretation of problems such as the efflux of water in a. can then be removed without'affecting the motion, provided {3, "Y jet or of smoke from a chimney, the discharge through a weir, the remain constant . · This is so when the axis of revolution is a principal . axis, saY Oz ; flow of water through the piers of a bridge, or past the side of a ship, the wind blowing on a sail or aeroplane, or against a wall, when f!, = q, or impinging jets of gas or water; cases where a surface of If Q3 = o or 0 3 = f in addition, we obtain the solution of Jacobi's discontinuity is observable, i.nore or less distinct, which separates ellipsoid of liquid of thre{! unequal axe�, rotat:in� bo�ily ab?ut the the running stream from the dead water or air. . ,

a= 4c2(c2-a2) (c•+a•)• 4h2(a2-b') - (a•+b2)2

( 14). '

An

- a2 - ) 2 (c'c•+a·flt '1 a�-b2 (a• +b'o.-r) '

(�11'pA + a)x2 = (�upB+.B)b' = (4rpC + "Y) c2,

a,

Ut =O,

�=o, 7J=O.

( I7)

Maclaunn s solutton 1s obtamed of least axis ; and putting • · the rotating spheroid. In the general motion again of the liquid filling a case, when n, may be replaced by zero, and the equations, hydrodynamical and dynamical, reduce to .

a= b,

a = b,

. zc• n.t, d17t = 1+ 2a� 01f' .ddtf •2c2 ( Ol�-flt'J) (r8) a +c• dt= a2+c• d a c2 �+c2 dQ dB �+� dt = ntr+ a2-c2�!. Tt = - f!, ! - a2 - c2�r; a• (20) r- +'7• = L -czr"· (a•;t=c•)� (2I) nf +n� M + 2C2(a- c2)�' a2+c2� ('; f!,E+OttrN = + (22) 4c

de

=

:i which three integral:> a,re

=

•

.

.

Uniplanar motion alone is so far amenable · to analysis; the velocity function q, and stream function 1/; are given as conjugate functions of the coordin,ates X, � by

w =f(z);

and then

.

where

z=x+yi, w = q,+Y,i,

dw d .d•/1 dz = +�dx = -u+v�;

dx u =q o 9, = q 1'1, i= - Qddz=:.JL_·=g(u+t�l u-m

so that, with

c s

w

v

(2)

the fu nction

sin

�.

•

('1)

.

,

·

,.,

Q

q

(cos

B+i

sin O) ,

(3)

gives f as a vector representing the reciprocal of the velocity 1 in direction and magnitude, in terms of spme standard velocity Q. To determine the motion of a jet which 1ssues from a vessel with plane walls, the vector r ·must be constructed so as to have a constarit

-HYDRO MECHANICS

[HYDRODYNAMICS direction 8 along a plane boundary, and to give a constant skin If (J = « acro the end JJ' of the jet, where u = q= Q velocity over the surface of a jet, where the pressure is constant. h ,.... "b-a' . . .�a-b It is convenient to introduce the function c n .. =/t =O, m; _ d u - dq, d t -:- 1rqu - ,. q u that i f c denotes the ultimate breadth JJ' of the jet, where the velocity may be supposed uniform and equal to the skin velocity Q, ,.AB = "2 du c Jb 'l u m=Qc,. c=m{Q. (b-�')..J(a-u)] 11"c.!!:! . (26) =-{ [�a> b>o>a'> -oo ; (9 ) and then Jr =� = [ · /b-a'. /!!. _ . fa-b . /-a'] tfn' (32) b-a.b-a') dw !!!., dn _]_ ..J( 'la-a 'Jb 'la -a''J b xx' Q (r o) d u - 2n(u-b)..,f (u-a.u-a1)' dzt - ..-u' ss

oo ,

-=----::1 , s

so

,

�.

_

w

u.

�

_

=

oo

oo

oo

so

I

·

n

·

n

n

,.

_

I

•

_

_ _

_ _

the formulas by which the conformal representation is obtained . . For the 0 polygon has a right angle at and a zero angJe at so where fJ changes from o to !-o-/n and n increases by. .

u=a, a',

u = b,

·

li,-Jn; dn A (11) ' where A = ..J (b-a.b-a') 2n du == (u- b)..J (u- a. u- a') the w polygon· has a zero angle a t u o , cCI , where 1/1 changes And from o to m and back again, so that w changes by im;·and dw B -m ( 12) du = u • where B = -;· Along the stream line xBAPJ, ( 13) >/t =o, u =at""' I"'; and over the jet surface J PA, where the skin velocity is Q, � = - q = -Q, u =>ae"sQ/m =ae"stc, denoting the arc AP by s, starting at u =a; lb-a;. fu-a (15) ch nfl = cos n8� '\1 a -a 'lu- b ' . . .. la -b . lu-a', ( 1 6) sh nO= J sm n8= 1 'f a- a''l u-b > u =ae'"''>a, (17 )

ili"

oo ,

=

giving the contraction of the jet compared with the initial breadth of the stream. Along the line of flow x'A'P'J', u = a'e-"4>'"'• and from x' to A', cos nfJ = t , in ch nn = ch log

=

co

and this give& the intrinsic equation of the jet, and then the radius of curvature

i dw = i dw /dn = o1 d.pdfJ = oao oau au . u-a') ' u-b ..J(u-a =;·c 2n-u ..J (a-b.b-a') aot requiring the integration of (11) and (12) ds

p = - iW

.(xS)

>/t=m,

s nB=o,

�D = sh log

sh

Along the jet surface ·

-u' /b -a; 'Jlab -u (Qq ) " = 'la-a (�) n """ ":�:'"b=:', ·

·

o> u>a'.

A'J', q = Q,

-u' . 1� 'J. /ab-u ch nQ=tos nfJ = 'la-a . . , la-b . la'-u' h nfl = f sm n8 = f '\J a- a 'J·b -u

s

(33) (34) (35)

(36) (37) (38)

giving the intrinsic equation. The first problem .of this kind, worked out by H. v. Helm­ holtz, of the effiu:K of a jet between two ed�es A and A1 in an infinite wall, is obtained by the symmetrical duphcation of the above, with =- o, , as in fig. 5, n=

41. 1, b

a' = -oo

"u-a �-a ' -- sh U = (1) u u ' and along the jet APJ, oo >u =aetr•l•>a, "�=ie-i"''", sh O = i sin 0 =i (2) co • -!,•1• -l, f •l c c PM , sin IJ �s Je ds = 1,.e = 1,. sin IJ, (3) ch O =

==

- ·

=;o

HYDRO MECHANICS

HYDRODYNAMICS]

so that PT = cf!,., and the curve AP is the tractrix ; efficient of contraction, or breadth of the jet ,. breadth of the orifice ,.+2 ·

and the co-

43 · When the barrier AA' is held oblique to the current, the stream line xB is curved to the branch point B on AA' (fig. 7 ) , and so must be excluded from the boundary of u ; the conformal re- C presentation is made now with ·

A change of and IJ into 1ill and niJ will give the solution for two walls converging symmetrically to the orifice AA1 at an angle ,.;n. With · n = !, the re-entrant walls are given of Borda's mouthpiece, and the coefficient of contraction becomes !. Generally, by making a' = , the line x'A' may be taken as a straight stream line of infinite length, forming an axis of symmetry ; and then by duplication the result can be ob­ 0 tained, with assigned n, a, A and b, of the efflux from a symmetrical converging

!J

au

=-

---·u ,. -j.u -j"

_ mj'+m'j b - m +m' ' taking u

:jA

(I)

dw m I m' I dU = - -:;;, u -j - -;. u -j ' m +m' u -b

-oo

A

.,; (b -a.b -a') (u -b)-1/ (u -a.u -a1)

dO =

cf> =oo ,

=

00

(2)

at the source where

FIG.

"'

5·

FIG. 6.

B

cf> = -oo ;

ch !O

Over the jet surface

b u

, sh n!J

t/l =m, q = Q,

�

=

�� b

u = -tr'll'fm = -berr•l•,

ch !J = cos n1J

�

= err• !!+ I , sh !J=i sin niJ =i e"�;:� I ' (2) e!rr•l• = tan niJ , � ds = 2n (3) c iiii sin 2niJ"

For a jet impinging normally on an infinite plane, as in fig.

6, n = I ,

e!rr•i• = tan IJ , ch (!,.s/c) sin 21J = I , sh !,.x(c = cot !J, sh �y/c = tan iJ, sh !,.x(c sh !,.y(c = I , e!rr. =

a constant,

(a" +>-)P' tjo P2 =ofta2 +>..) (b2 +i\) (c'+>.),

(I6) (I7)

( I 8)

where M denotes a constant ; so that of- is an elliptic integral of the · second kind. The quiescent ellipsoidafsurface, over whkh the motion is entirely tangential; is the one. far which

2 (42+>..)

�� +.Y = d,

and this is the infinite boundary ellipsoid if we make the upper limit

X1 = oo .

The velocity of the ellipsoid defined by >. = o is then U=

=

-2a•d,Yo (JJ; ->Yo M

abc -:M

with the

so

notation

that in (4)

.":' abc /

j.....,

(I

9

Md>. (a2 +i\)P

(20)

-Ao) ,

A (if AA = � abc dX

)A (a2+}.)P a rod" = - 2ab c'ifi1) "P'' A

M

= iJOcxA =

UxA .

I - Ao' q,,

=

xA:.

I

(21 ) (22)

- Ao ' in ( I ) for an ellipsoid. , The impulse required to set;. up the motion in liquid of density p is the resultant of an impul,sive . pressure p over the surface S of the ellipsoid, and is therefore

.

ffpq,ldS = p,Y0f(xldS = P,Yo (v olume of the eltipsoid) = .YoW',

(23)

where W' d,enotes the weight of liquid displaced . Denoting the effective inertia of the· liq uid parallel to Ox by aW', · the momentum aW'U = .YoW '

(6)

(8)

d.p �'· +xdof­ Ts as "" as"'

x·'-+2 (a!+>.i.Pf-,f.' "' as"' . tfii. lli'

put

(2)

but a rotation will stir up 1;he liquid in the cavity, so that the x's depend on the shape of t:he surface. The ellipsoid was the shape first worked out, by George Green, in his'Resea1'th on the Vibtatio11 of a Pendulum in a Fluid Mediu'fll. ( 1 833) ; the extension to any other surface wilt form an important· step in . this subject. A system of confocal ellipsoids is taken y,2 . x2 y2

+

\II)

>. ,Y ilid>.. , J� � (b2+X . c2+X) .

(a2 +>.. ) 3 "a

so that we may

a fixed surface, and at infinity ; the same for X2 and xa . For a cavity filled with liquid in we interior of the body, since the liquidinside moves bodily for a motion ofltanslation only,

a2 +>-.

��·

aU - a,UI =

(I )

tlie normal ' at the surface

2 ""'-y, cf>a = -z;

. (Io)

l,

and so the boundary cortdltion is . satisfie.d ; !"?reo�er, any ellip�oid!ll surface X may be SUJ?pdsed movmg as tf ngtd With the veloc1ty In : ( I l), without disturbrng 'the liguid motion for the moment. > The continuity is secured If the liquid between two ellipsoids >. and >..� o . �oving with t.he velocity U and U1 of eq�tion ( I I ) , is squeezed'otit or sucked m acros$ the plane x = o a:t a rate equal to the integral 4ow of the velocity 1/1 across the annular ·area a1 - a of the two ellip¥Jids. �2,�.!?.y .x = o ; or if . ·· i

V21 =o, throughout the liquid ;

d , = �l,

4 = � >Y +2(a2 +>-)�f �

so that over the .sui'F�� .of an ellipsoid where }.. and. of- a.re constant • ' the .no.t'n,a.alvelodty',is the same as that of the ellipsoid itself, movizw · as a 591id with;yelodty paraJlel to Ox

where the '8 ajld x's ate fu,nttions of. x,' y, z/depending on the shape of the body ; inte'rpreted dynamically, C - p represents the impulsive pressure required to stop the motion, or C +P to start it · again from rest. The terms of cf> may be determined 01,1e at a t,i me, and this problem is purely kinematical; thus to determine ,, the component U alone is taken to exist, and then l, m, n, denoting the direction cosines of the normal of the surface drawn into the e10terior liquid, the function f/>1 must be determined to satisfy the conditions (i.)

. .,

.

·

a=

(24)

{T I�f;; ;

(25)

=

in this way the air drag was calculated by Green for an ellipsoidal pen�ulum. . Similarly, the inertia parallel to Oy and Oz is

tJW' =

r - Co

(26)

B;... c�. J["" A (bqx, c"+>-)P ; abed>..

=

and For

Bo W' , -yW' = �W' ,

r -Bo

a

A + B +C = abc/!P, Ao +Bo+Co � r .

sphere

·

a = b ,., c, A. -Bu•Co = !, a = {J :o -y = i,

(2Q}

·H YDROM ECHA·NlCS

HYDRODYNAMKS]

so

!:bat the effective inertia of a sphere is increased by half the weight of liquid displaced ; and in frictionless air or liquid the sphere, of weight W, will describe a parabola with vertical acceleration W - W' (3o) W + ! W 'g .

is p. cos P5x,

:J 3 J

PHx

and of the source H aJJ.d line ciak OH is Jl.(ajj).cos

and - (p.ja) (PO - PH) ; so that f=

p.

(cos PSx+ J cos

PH x -.P

O

� P!:!) ,

(4)

and f - JJ , a constant, over the surface of the sphere, so that there Thus a spherical air bubble, in which W(W' is insensible, will begin is no llow across. When the source S is inside the sphere and H outside, the line to rise in water with acceleration 2g. , 45 · When the liquid is bounded externally by the fixed ellipsoid � sink must extend from H to infinity in the itnage system · to realize X = Xt. a slight extension will give the velocity function of the · physically the condition of zero llow across the spher�, an equal liquid in the interspace as the ellipsoid X = o is passing with velocity � sink must be introduced at some other internal point S'. When S and S' lie on the same radius, taken along Ox, the Stokes' U through the confocal position ; must now take the form x(.P + N ) , function can, be written down ; and when S and S' coalesce a doublet and will satisfy the conditions i n the shape is produced, with a doublet' image at H. ' abc h! abcdX + For a doublet at S, of moment m, the Stokes' function is A+B = Ux . C Ux �� .1. (� (I) 2 . d Bo+ C.- B, - C, = 1_ abc abcd X ' 1ft C1J cos PSx (5) X a,b,c, o (a2 + ) P and for its image at 1-J tjle Stokes' funCtion is and any confocal ellipsoid defined by '- • internal or external to d X = X1 , may be supposed to swim with the liquid for an instant, ' m cosPHx = m aa_z_ S; (6) p PH dj Ox along velocity without distortion or rotation, with so that for the combination B A + CA -B , - c. u I (C!:_a I t a3 f3 \ Bu + Co - B, - C1' (7 ) my2 \]J P H 3 - 'PS3 ,P = Since - Ux is the velocity function for the liquid W' filling the ! · the sphere. and this vanishes over· the surface o f · · ellipsoid X =o, and moving bodily with it, the effective inertia · f the There is DO Stokes' function when the axis of the doublet at S liquid in the interspace is ?� not � !frough q; the image syste� will . consist of an Ao+B, +C, mchned doubtet at H , maktflg an equal angle w1th OS as the doublet (2) W' Bo+Co - B. - C, · S, and of a parallel negative line doj!blet, extending from H to 0, of moment varying as the distance from (). If the ellip:;oid is of revolution, with b =c, . A distribution of sources and doublets over a moving surface A +2B, .1. - 1 u x ) (3 will enable an expression to be obtained .for the velocity function Bo - Bt' .,. - , of a pody moving in ,the presence of a fixed $phere, or inside it. and the Stokes' current function ,P can be written down The method of electrical images will enable the stream function f' • B - B, to be inferred from a distriliution of doublets, finite in number .Y, = - 21Uy --- ; B o - B, when · the surface is .comp?sed of two spheres intersecting at au angle 1rjm, where m 1s an mteger (R. A. H erman, Quart. Jour. of reducing, when tlae liquid extends to infinity and B, = o, to Math. xxii. ) . A Thus for m = 2, the spheres are orthogonal, and it can be verified (S) !1> = 11 U xs;;, .1., = -:�1 UJ.tB B. ; that so that in the relative motion pru;t thl' body, as when fixed in the a13 a.a a3 (8 ) ,P' = j.Uy2 I - ,,� - 7 +;:a , current U parallel to xO, 23 =

·

J -j"''

+

·

mlsa;

=

·

·)

= m.r (w-��·

· ·

·

·

•

·

( I +:.). ,Y' = !Uy (1 -fJ

·

(

)

·

where a,, 4ll! , a = a1a2N (a.•+a.2) is the rad:ius of the spheres .and their Cir�le o( intersection, and r,, '•· r the dls�ances of , a point · Changing the origin from the centre to the focus of a prolate I from their centres. The corresponding expression for two ortbogoil!ll cylinders wi\1 be spheroid, then putting lr = pa, >.. = X'a, and proceeding to the limit ! ·a where a = oo , we find for a paraboloi!i of revolution . 1/t; = Uy I- i � �; (9 r1 r2 +� r2) 1__£_ � b . (7) B With a, = oo , these ·reduce to· - -�p +x'· o;' = !Vx

Bo-m·

/tx• = p +>..' - zx,

(6)

;;

(

(8}

with >..' = o over the surface of the paraboloid; and then .P' = !U[y2-p.V (i2+y2f +px] ; (9) ( IQ) .P = -! Up ( v (x2 +y2)-x] ; cp =-lUP log [ � (x2+y2) +xJ. (n) The relative path of a liquid particle is along a stream line 1/1' = }Uc•, a constant , - (y2 - c•)• .V( • = p•y x +yt) - &•+(f - ct)t x ( 1 3) 2p(y2 - c2) ' 2p(y - &)

i for

( I2 )

a particle in space" will be given by dy == - r - x -y = -y22:/>-Y ' · dx y2 - c2 "" a2e--., tP, ( I S) ' 46. Between two concentric spheres, with a2 +X = r2 , lfi +X1 = a12; (I) A = B = C =a,sf3r3, 63 a3 a8 a3 2 1 Tii + a,3 1 r3 a,3• - 2 Ux �, .Y - 2U f � _ , I-/li) 3 12 = (x+ I ) v xl +i(x - I) v x,, X x = :ax4+2ax3 ='" 3 (a +b)x2 + 2bx ::b, W = - 8 (a +b),

(27) (28) (29)

will give a possible state of motion of the axis of the body ; and the motion of the centre may then be inferred from (22).

so. The theory preceding is of practical application in the investigation of the stability of the axial motion of a submarine boat, of the elongated gas bag of an airship, or of a spinning rifled projectile. In the steady motion under no force of such a body in a medium, the centre of gravity describes a helix, while the axis describes a cone round the direction of motion of the centre of gravity, and the couple causing precession is due to the dis­ placement of the medium. In the absence of a medium the inertia of the body to trans­ lation is the same in all directions, and is measured by the

h=

� !i � ({J - ) l +a u2 = a W c, g r +fJ g ·

(7)

5 1 . An elongated shot is made to preserve its axial flight through the air by giving it the spin sufficient for stability, without which it would turn broadside to its advance ; a top in the same way is made to stand upright on the point in the position of equilibrium, unstable statically but dynamically stable if the spin is sufficient ; and the investigation proceeds in the same way for the two problems (see GYROSCOPE). The effective angular inertia of the body in the medium is now required ; denote it by C, about the axis of the figure, and by C2 a,bout a diameter of the mean section. A rotation about the axis of a figure of revolution does not set the medium in -motion, so that C1 is the moment of inertia of the body about the axis, denoted by Wk� . But if Wk� is the moment of inertia of the body about a mean diameter, and w the angular velocity about it generated by an impluse couple M, and M' is the couple required to set the surrounding medium in motion, supposed of effective radius of gyration k',

(I ) - Wkiw "" M - M ' , W'k'2w = M', (Wk � +W!k'2)w = M , (2) · (3 ) C2 "" Wk i +W'k'2 = (W+W'•) k � , in which we have put k'2 esembling the trochoidal curvt;s. which can be .looped, investigated in § l9 for the motion of ·· a cyhnder under grav>ty, when surrounded by a vortex.

1 In some cases hydroids have been l"eared in aquaria from ova of medusae, but· these hydroids h ave not yet been found in the sea (Browne [Io a]).

HYDRO MEDUSAE confinement. The alternative is to fish all stages of the medusa in its growth in the open sea, a slow and laborious method in which the chance of error is very great, unless the series of stages is very complete. At present, therefore, classifications of the Hydromedusae have a more or less tentative character, and are liable to revision with increased knowledge of the life-histories of these organisms. Many groups bear at present two names, the one representing the group as defined by polyp-characters, the other as defined by medusa-characters. It is not even possible in all cases to be certain that the polyp-group corresponds exactly to the m edusa­ group, especially in minor systematic categories, such as families. The following is the main outline of the classification that is Adopted in the present article. Groups founded on polyp­ �haracters are printed in ordinary type, those founded on medusa­ characters in italics. For definitions of the groups see below.

[ORGANIZATION

below the hydranth, or it may extend farther. In general there are two types of exoskeleton, characteristic of the two principal divisions of the Hydroidea. I n the Gymnoblastea the perisarc either stops below the hydranth, or, if continued on to it, forms a closely-fitting investment extending as a thin cuticle as far as the bases of the tentacles (e.g. Eimeria, see G. J. Allman [I],l pl. xii. figs. I and 3). In the Calyptoblastea the pensarc is always continued above the

Sub-class Hydromedusae (Hydrozoa Craspedota) . I . Eleutheroblastea. I I . Hydroidea (Leptolinae). Sub-order I. Gymnoblastea (Anthomedusae) ., 2. Calyptoblastea (Leptomedusae). Order I I I . Hydrocorallinae. IV. Graptolitoidea. V. Trachylinae. Sub-order 1 . Trachomedusae.

Order

•

.,

From Allman's

2. Narcomedusae.

Society.

Order VI. Siphonophora. Sub-order I . Chondrophorida. 2. Calycophorida. 3· Physophorida. 4. Cystophorida. "

FIG.

Gym110blastk Hydroids, by permil;sion of

the Counca of the Ray

2.-Stauridium productum, portion of the colony magnifif'd '· p, polyp ; rh, hydrorhiza.

hydrocaulus, and forms a cup, the hydrangium or hydrotheca (h t) standing off from the body, into which the hydranth can be retra�ted for shelter and protection. The architecture of the hydropolyp, simple though it be furnishes a Organization and Morphology of the Hydromedusae. ion� st;ries of variati