Turbines

Nov., 1886.]

Turbines.

35 r

the metal cannot be distilled. By reducing caustic-soda with the carbide of a metal, or its equivalent, and using only sufficient carbon to carry out the reaction stated above, the gases given off consist of hydrogen, carbonic-oxide and carbonic-acid, which mixture has little or no effect upon the vapors of sodium. T h e crucibles, after treatment, contain a small amount of carbonate of soda and all the iron of the " c a r b i d e " still in a fine state of division, together with a very small percentage of carbon. These residues in the crucibles are treated with warm water, and the solution of soda evaporated to recover the carbonate of soda, while the fine iron is dried, mixed with tar and coked to produce more of the so-called " c a r b i d e equivalent." TURBINES. BY J. LESTER WOODBRIDGE, M. E., Stevens Institute of Technology. [With an Introduction by DE VOLSON WOOD, Professor of Mechanical Engineering.] (INTRODUCTION.) TO THE COMMITTEE ON PUBIACATIONS :

I forward to you, for publication, the graduating thesis of Mr. "Woodbridge, upon " T u r b i n e s . " T h e first well-grounded theory of these wheels was b y M. Poncelet, who read a paper upon this subject before l'Academie des S6ances, Paris, entitled " L a Th6orie des Effets M6caniques de la Turbine-Fourneyron," published in Comptes Rendus, Paris, I838. This solution was so thorough and elegant that it is often referred to as " historical" and " classical." The analysis was founded on the principles of work and energy, and his methods have been followed by m a n y writers since his day. Rankine, in his Steam ~.ngine and other Prime Movers, has given a partial solution of a special case founded on the principle of the Moment of the Momentum, and he has been followed by Unwin, in his article on " Hydro-mechanics," in the Encyclopwdia ]~ritannica, I X edition. It will be seen that Mr. W o o d b r i d g e ' s solution is founded upon the resolution of the pressures of elementary volumes and their moments, and is more general than either'of the preceding, if not the most general solution of the frictionless turbine yet made. Yours truly, DE VOLSON WOOD, Prof. J~[ec]l, End,r.

Hoboken, '.Froze 2% I886.

3 52

TVoodbridge :

,'/

i

~ t ~

l f - - / -" / l ~

EJ. F. I,, ~"

.t

/ ~/~ ~/ /~'/~ /i I- _

/ : 7/, !

)'/' I

I 0 FIG. I.

I propose to discuss the action of turbines in general. Consider first the water after it has entered the wheel and is passing

Nov., i886.]

Im.&ncs.

3 53

along its vanes. Conceive the water to be divided into an infinite n u m b e r of filaments by vanes similar to those of the wheel, but subjected to the condition that, at each point, their width id, Fi~. 1, m e a s u r e d on the arc, whose centre is O, shall subtend at the centre a constant angle d 0. Conceive each filament to be divided into small prisms, whose bases are represented b y the shaded areas a'b'e'd', d ' e ' e " d " and abed, by vertical planes normal to the vanes m a k i n g the divisions ae, el, intercepted on the radius by" circles passing t h r o u g h the consecutive vertices on the same vane a', d ~, d " , etc., equal. L e t p = the radius v e c t o r ; x ~ the h e i g h t of an e l e m e n t a r y prism ; then, &,, - ~ ae, ef, etc. ; p dO dp = . b e d , etc., ~ area of the base

of an infinitesimal prism ; x ,, d 0 d / , --- v o l u m e of an infinitesimal prism ; x a o d 0 d p - - m = the mass of prism, 3 being its d e n s i t y ; F = s a)~ = angle between the normal to the vane at a n y point, and the radius 0 a p r o l o n g e d t h r o u g h that point ; , .... _ velocity of a particle along the vane at p ; ~o := the uniform a n g u l a r velocity of the wheel, and p .~-- the pressure of the water at the point p. F o r the element of time, d t, we will take the time occupied by one of the liquid prisms in passing along the vane t h r o u g h a distance equal to its own length, or, outwardly, a distance d p , thus m a k i n g t a function of p, the latter being considered the independent variable. W e have d t d,o - d p --

d :, _ _ d~! _ . d p - - v.sqn r dt

L e t a a ~ be the distance passed t h r o u g h by this element circumferentially in the time tit, then a a ' = to/, d t =

w ,,, ~/d2 d,o /

and the particle a e will, at the end of time dr, be at a t d. To find the reaction of a particle m o v i n g on a curved path, let the particle m o v e from P , Fig'. 2, along a curve to the consecutive WHOLE NO. VOL. CXXII.--(TH1RD SEP.IES, Vol. xcii.) 23

354

Woodbridge :

[ J. F. I.

point, F , changing its direction by an angle, dF, from P A to P~B. The velocity along P A being v, it will have acquired the

velocity, v d F, perpendicular to_P A, in time d t, and its acceleration in the same direction will be V--,dF and the force ~--- n~v d F dt dt where m is the mass of the particle. T h a t is, the reaction of a particle, the direction of whose motion is being changed, is equal to its momentum multiplied by the rate of the angular change of its motion, and is in a direction opposite to that in which the change takes place. The mass m will have two motions : one along the vane, the other with the wheel perpendicular to the radius. By changing its position successively in each of these directions, both its velocity with the wheel and its velocity along the vane may suffer changes both in amount and direction, as follows : (I.) By moving from a to d , fi'zg. I, in the arc of a circle-(:.) ~o p may be increased or diminished; (2.) o) p may be changed in direction ; (3.) v may be increased or diminished; (4.) v may be changed in direction. (II.) By moving from a' to d r along the v a n e - (5.) ~op may be increased or diminished ; (6.) o) p may be changed in direction ; (7.) v may be increased or diminished ; (8.) v may be changed in direction. These changes give rise to corresponding reactions, as follows: (No. I.) By the conditions imposed, w g' will be constant, and hence the reaction ~--- O. (No. 2.) By moving from a to a p, the velocity ~op is changed in direction from a k to d k' in the time dt. The momentum is o) p, and the rate of angular change is kak t ~o dt dt dt - -

- -

~AJ,

Nov., i886.]

355

Turbines.

and hence the reaction will be m0)2p in a direction radially outward. T h i s is the c e n t r i f u S a [ f o r c e , as designated by most writers. Resolving into two components, we have ~n 0)2 p sin ~" along the vane,

m 0)2p cox r normal to the vane. (No, 3.) According to the conditions imposed, this value o f v remains constant, hence, for this case, the reaction will be zero. (No. 4.) In moving from a to a ' the velocity along the vane, v, is changed in direction from a k to ark r at the rate 0) as in No. 2. The m o m e n t u m is ~n v, and the force will be m v 0), which acts in the direction n a, and being resolved, gives 0 along the vane, -=_ m v 0) normal to the vane. (No. 5.) In passing from a r to d t, the increase of 0) p will be 0) dp, in a time dr, and the reaction will be dp dt'

in a direction tangential to the motion of t h e wheel but backwards, and its components will be m 0) d ~ cos ?- along the vane, d p

-

m ~o~

-

.

sm ?" normal to the vane.

(No. 6.) In passing from a t to d t, 0) p will be changed in direction b y the angle a I 0 d t --

a t

"q

__

d p cot ~",

P

P

and the rate of angular change will be cot r

d p

and the m o m e n t u m will be fa0)p ;

hence the reaction will be d,o

~n 0) cot r ~ [ ,

which acts radially inward, and its components are

Woodbridg~c ."

356

[J. F. I.,

- - "m, ~o cos r d p d~ -

(No. 7.) amount

-

along the vane,

ra ,o cot r eo.~ r d 8 dt

normal to the vane.

By moving from a r to d', v will be increased by an

d p

in the time d t, and the reaction will be d v m d ,o

d p. dl '

which acts backward along the vane, and its components are -

-

~?t

dv d [, -

.

along the vane,

dp d t

0

normal to the vane.

(No. 8.) In passing from a ' to d', v is changed in direction by two amounts : (I.) T h e angle T changes an amount _ _ d 7 d p.

dp and (2.) the radius changes in direction an amount d p cot F P

as in No. 6; hence the total change will be the sum of these, and the rate of change will be the sum divided b y d t, which result, multiplied b y the m o m e n t u m m v, will give the reaction, the components of which will be 0 along the vane, " eot~" d p ,.

d t

dT"

d,o ! normal to the vane.

,t p

dt ]

This completes the reactions. N e x t consider the p r e s s u r e in The intensity of the pressure on the two sides a b and e d differs b y an amount the w h e e l

8 p :

~P- d l'. d [,

T h e area of the face is d e X x : x p d 0 sin r, and the force due to the difference of pressures will be :c [,dO.,.in r d p d [ , .

7?trbines.

Nov., ~886.]

357

If d p is positive, which will be the case when the pressure on d e exceeds that on a b, the force acts backwards, and the preceding expression will be minus along the vane. In regard to the pressure normal to the vane, if a uniform pressure _p existed from one end of the vane V W to the other, the resultant effect would be zero, since the pressure in one direction on V Wwould equal the opposite pressure on X Y , F~$;. I. If, however, in passing from d to a, the pressure increases by an a m o u n t - dp, since V a is longer than X b , the pressure on Va will exceed that on X b by an amount

dp - - d p x X a h -= - - d p . x . ,o d 0 cos r -=- - - x [, cos "[d 0 ~[~ d p. Collecting these several reactions, we have NORMAL TO T H E (2.)

VANE.

ALONG T H E

-]- ~t to 2 p C08 ~'.

-~ m w 2 p

(4.) - - m ~o v.

dp + m ~ocos r 3-[" dp

dp --mwcosr(H.

(6.) - - m ~ocot r cosr ~ .

(8.)

0 +m

--mdP. dt v~

sin, y. O.

dp (5.) - - m ,o sin r j-[.

(7.t

VANE.

I~'ot y -.

i p

dp dt

d ~" d/,

d "f d p] dp d~"

0

dP dtdO. -- xDsin y ~

(9 ) - - x p e ° s y d P

The sum of the quantities in the second column, neglecting friction, will be zero; hence

mw2psin~,

r a dd-t P " da_vv p

x p sin y dd P~od p d O = 0

(1)

Substituting

dP--vsinr, dt

andx[,dOdp

- - ra

and dividing by m sin y, we have ~o2 [, d p _ _ - ~ l d p = o

v d v.

(2)

3 58

Woodbridge :

[ J. F. I.,

Integrating, P__~limit

(~_12 J

The sum of the quantities in the first column gives the pressure normal to the vane, which, multiplied by/) sin ?', gives the moment. This done, we have, after substituting as above,

%p d 2 M -= my sin r

(vP-

,o cos r

• +ve°sr--~

) --P

-

r

o cos ?" d p v3 dp

"

Putting my sin ~ = 23 Q d p d O, where Q is the quantity of water flowing through the wheel per second, and integrating in reference to 0 between 0 and 2 ~z, we have

d5

oosr d~

~v

v3

d

Multiplying (2) by P- cos r, V

we have c°2 P~ cos r d p - - p c ° s t v v 3

d l) d p = ,o eos r d V d p d,o

which substituted above gives d M--.3 Q~

2 w p d p 4 - p c o s r d Vd p + v c o s ~ d p - - p d ~o

vsinr ;

d [~(4)

and integrating, _~ = a Q [ - - ,o F + ,° v c°s r] --

3 Q ,o [oJ , o -

(5)

v cos r ] l~mit. limit.

But w p ~ v cos i" is the circumferential velocity in space of the water at any point, and 3 (2 P [w p - - v cos r] is the moment of the momentum, hence, integrating between limits for inner and outer rims, the m o m e n t exerted by tlze w a t e r on the wlceel equals the difference in its m o m e n t o f m o m e n t u m on entering a n d leaving #~e wheel. Thus we have deduced an expression which some writers

have made the basis of their investigations. Let the values of the variables at the entrance of the wheel be Pl~ J'l, VD P l ' and at exit ~o~,?'z, v2, P2.

Nov., I886.]

359

Turbines.

Equations (3) and (5) become 1 oj2 (p12

2

p55)__pl--p5 --

--

1,

~

~ ~vl - -

2,

v5 ;

M = 3 Q [w (,0;5 - - ,025) - - ,01 v t cos ?'1 -t- P5 v~ cos ?'5] • ** V ~

(6)

3

M ( o ~ 3 Q 40 [(2) (1012~

(7)

i025) ~ i 0 1 v 1 c08 ?'1 ~- P2 v21308 ?'2] (8)

Equation (8) gives the work per second in terms of the known quantities 3, w, Pl, ?', P2, ?'2, and three quantities Q, Vl, % as yet unknown. These three quantities are, however, connected by the condition that the quantity of water flowing through all the sections radially is constant. Calling a, the entire area of all the orifices at the entrance of the wheel ( = 2 z Pl x~), and a 2 those at exit ( = 2 z P2 xs), we have Q = a~ v~ sin ?'t = a2 v5 sin ?'2

(9)

which reduces the number of unknown quantit{es to one.

Equation (6) is the equation of the motion ot the water in the wheel. Besides the velocities v 1 and %, it contains Pl and Ps" L e t p~ = the atmospheric pressure, h = the mean depth o f . t h e wheel below the surface oI tail-race, Pt = 3 9h ~ the pressure due to the flooding of the tail-race; then P 2 = P ~ + P c "

360

Woodbridgc :

[ J. F. I.,

T h e pressure P l , where the water passes from the guide blades into the wheel, is unknown. A n o t h e r condition is necessary, which m a y be found b y considering the passage of the water from the guide plates into the wheel. H e r e it must be observed that the path of a liquid stream, either in space or relative to some moving object, is always a continuous curve, and never contains a sharp angle, the motion being c o n t i n u o u s - - a principle confirmed both b y theory and experience. This granted, we c~n show the fallacy of some statements of Professor Rankine, in regard to the impulse of a stream of water on a vane. T a k e the case given on page I63 of Rankine's S t e a m Engine, Fig. 57, and discussed on page ~69, Case V. ' I copy his diagram and lettering, adding a few lines that he omitted. On page x69, he says, " In this case the easiest method of solution is the following : " Let v' =- D G Fig. 3, be the velocity of the jet relatively to the vane. Let the angle C / ) K, which it makes with the normal to the vane, be denoted b y O. Resolve v' into two components, viz. : D K normal to the vane = v r cos O, K C along the vane -~ v' sin 0. " T h e n , according to the supposition that friction is insensible, / ( C is not affected in magnitude b y the vane, but D K is entirely taken away." Now, as shown, the stream will move in a curve on account of the self-cushion of water, and it is a well-established principle that in a frictionless fluid, the relative velocity o f a particle suffers no loss in p a s s i n g around a curve, if subjected to no other forces than the reaction of the guiding surfaces; or, in other words, if the deviation be free. Therefore, if the relative velocity of the jet was v' before impinging, the velocity along the vane after impinging will also be v', and not v' ~qn O, as Rankine states it. T h a t his solution is erroneous, m a y also be shown b y the fact that his expression for the energy imparted to the wheel added to that lost, does not equal the original energy. Thus, according to his equation (47), the energy exerted on the. wheel is P ~ = D (2 U v' cos 0 cos 3 f!

Nov., I886.]

3 6t

7hrbincs.

T o find the e n e r g y lost: if t-t ,s~/n 0 be the velocity along the vane, and u the velocity with the vane, then will the velocity on leaving the vane be the resultant of these velocities, or 1 v r-'~'in "zO i ~ ( " + 2 ~ v j s i n O ~ i n

3,

and the e n e r g y will be D -Qi v '2 sin "l 0 -'~ .~ + 2 u v' sin 0 sin 2g i '

, 2

which, added to the enerfiy exerted, gives 1)_ Q i v '~ sine O ~ u 2 -:- 2 u v' cos f 2q,

,

(a)

where (f ~ 0 - - 3 ~ the angle between the velocities u and v ~. Now, the actual velocity before striking the vane was the resultant of u and v', or 1/ u 2 ~ H 2 ~- 2 u vt cos ~.

and the total e n e r g y was

D ¢2 : 2 g

,0 ~+ ~,'' + 2 ~ v' cos v

i

(b )

which exceeds equation (a) ; hence, according to R a n k i n e ' s solution, there would be a disappearance of energy, which is c o n t r a r y to the law of the conservation of energy. His error will at once be a p p a r e n t b y m a k i n g the vane normal to the stream, in which case 0 will be zero, and D K ~ v ~, and K C would b e c o m e zero, whereas it should be v r to agree with the statement above. R a n k i n e m a k e s no mention of the fact he has plainly shown in the f i g u r e - - t h a t of a division of the s t r e a m into two streams u p o n striking the vane, but treats it as if it were guided as in a bent tube. O n page I73, Art. I47, he again refers to the subject, and divides the pressure exerted on the vane into two p a r t s : " ( I . ) T h e pressure arising from the c h a n g i n g of t h e direct c o m p o n e n t v cos a of the velocity of the water into the velocity o f the vane. "(2.) T h e term reaction is applied to the additional action d e p e n d i n g on the direction and velocity with which the water glances off from the vane." This latter is the reaction due to the c h a n g e in direction of the relative velocity of the jet. It is the sole cause of the pressure

Woodbridge :

362

[ J. F. I.,

exerted in all cases of free deviation, and I fail to see a n y meaning in his so-called " d i r e c t a c t i o n " as here used. This conclusion agrees with the principles in W o o d ' s Mechanzcs of Flztids. Rankine remarks further, in the same article, that for a flat vane moving normally, as in his Fig. 55 (which I copy), this

direct action is the only action which occurs, but Fig. 4 (his Fig. 55) shows plainly that the direct component of the velocity, v cos a, is not changed into the " v e l o c i t y z~ of the vane," but the stream is simply deviated, and I fail to see any distinction between the above case, which Rankine calls ,, direct action," in which the stream is deviated by a cusp of cushion water, and the case shown in Fzg. 5, in which that cusp is formed of the same material as the rest of the vane. c

/3

In Fzg. 6, let A (7 be the tangent to the guide.plate at its extremity, g the actual velocity of the water on leaving the guideplate, ~o P1 = A / ) , the velocity of the initial rim of the wheel, then will/1_ 23 be the velocity of the stream relative to the wheel. Now, if A B does not coincide with the tangent to the vane at A, the

Nov., 1886.]

Turbines.

363

stream cannot be suddenly made to coincide with the direction of the vane, or float ; neither can we follow Rankine, and assume that the component of A B which is normal to the vane is lost. The water, by cushioning, will make its own angles, as roughly shown

in Fig'. 7. It is impossible, either practically or theoretically, to determine the new angles, and probably they are not constant; neither is it possible to determine the loss of energy due to eddying; we therefore make the hypothesis that the final direction of the guide-plates, the initial direction of the vanes, the angular velocity of the wheel and the velocity of the flow, are so related

that the water on leaving the guide-plates shall coincide in direction with the initial elements of the vanes. _Any three of the four quantities above mentioned being fixed, the fourth becomes known b y this relation. W e will leave the angle of the guide-plates to be determined later.

364

Woodbridffe ."

[ J. F. I.,

Let V = the actual velocity of the water on leaving the guideplates ; v I = the velocity relative to the vane, as before. Then V must be the resultant of ~opt and v,. and we have F@. 8, 172 =

(10)

V12 .2f- /'02 1(,12 __ 2 Vi /'O [(1 008 rl

From Bournilli's Theorem, we have for the flow of water in the supply chamber, the equation, P~

Pl

0-? +

+

V2

(11)

t~ being the mean height of the surface of the water in the reservoir above the wheel. W e thus have two equations (10) and (11), introducing one new unknown quantity. Eliminating V in (10) and (11), we have 2 Pl 2 p, Vl2 -~- /'02 ['12 ~ 2 V1 (0 D1 c08 ~'l = 2 g I~ --- - - 3 - + --3-- (12) Substituting in equation (6) p2 = p~ + g 3 h, we have /,02 iO12 ~

/'02 '('22 __

21h

(~ _~

2~

.a, + 2 g h =

(13)

Vl2 - - v22

Adding (12) and (13), we have 2/'o 2 [,~ - - / ' 0 2 ,%? = 2 g (b - - h) + 2 v 1/'o P1 cos r~ - - v22 (14)

From (9) and (14), H being substituted for l~ - - h , we find

Vl ~

(0

el

i~ a 2 sin r2 a~ sin r~ ~w[, 1 cosy 1 a l sin'[1 + a 1 s i n ~1

00821"1 a l l ~ 8 ~ 2

~1 -~ [02 ( D 2 i l -

2 /'12) -f- 2 g

v2 __ a 1 sin_ ~ Vl _~ (-~o P1 cos ~'1 a 2 sin "(2 + a 2 sin r~ " a 1 sin r l k_

;'02

a~ 2 sin2 ~'2 ,12 e°s2 r l -t- /'°2 ( [ , 2 2 - - 2 [,1 ~) + 2 , H [ ( i 5 a ) a l 2 sin2 Yl Q -~ a 2 sin ~'2 v., =- (~2 sin ~'2 a22 ~in2 ~'2

~0

a 2 sin r2 + P1 cos yl a l sin ?'1

~02 /~12 0052 ~1 ~/12-5,~'~,t2 ;'1 _1 /'02 (p2 2 - -

2 ['1 2) -~- 2 g

H~

(16)

Nov., I886.]

Tztrbim's

365

T h e efficiency will be, from equation (8), E--

1 (t 2 8 ~ Y2 g H ( (021012- - '02'0:2 q- W !"'2 COS j 2 - - al-sin "1-;1 a 2 sin 2"2 a x sin "fl . . . .

co / J l CO8 T1 - P

.............

a,,2 sin 2 ?-o

(/922 ~

!

?,}

2/h 2) + 2 .q H 4- (, t"2eizn2 r": w2 /q '~ cos 2 ~.~

(18)

T o find the a n g u l a r velocity t h a t will give a m a x i m u m effidE ciency, m a k e d w - - 0 in ( 1 8 ) . F o r brevity, m a k e

m =

5 ~

--1Ol

c~1 sin--r1 + ,02 eos r2

% sin r2 Pl cos ?'1 al sin "(1

lU----- P ~ 82 ~

oo8~" 1

[)22 __

(2o)

a: 2 s'in2 ~'2 c°s2 "(1 a12 sin 2 "(a + ,%2 _ _ 2 pl 2 [)12 -

~,~a- - 2 8 i n "(2 / J l 008 "(1.

a a sin "J'l

dE T h e n will ~

(19)

(9,1)

(22)

----- 0, give w s21/ w 2 1 2 + 2 g H = m *o212 + m g H ; g H~s

~-

rs'--m

(23)

2 l~

and this substituted in (18) will give the m a x i m u m efficiency for • a n y turbine, and in (16) will give the q u a n t i t y of water discharged. (To be Conlinued.) ERRORS IN DELICATE WEIGHING.--M. Hennig de Wurtzbourg having

noticed some incomprehensible differences in the weights of equivalent quantities, undertook some minute investigations, which showed that balances of precision are often influenced by the electric state of the glass cage which surrounds them. This electricity influences the small auxiliary slides which are placed upon the beam of the balance in order to make it more sensitive. The error resulting from this influence may amount to 6oo milligrammes when the cage is strongly charged, and two hours afterwards there may still be an error of ten milligrammes.--Cosmos, Oc[. 5, z885.