M ECHANIC9, 
74fr 
In the preceding calculus the pump has been fuppofed 
<jf uniform bore throughout; when this is not the cafe, the 
folution is rendered fomewhat more complex, but not 
difficult. To calculate the effort of the internal air when 
the water has not reached the body of the pump, having 
only attained the height H N, for example, (fig.49.) we 
mul ufe this proportion : As the (pace QOVNMIQ : the 
(pace CDVNMIC :: 33 feet: a fourth term, which, being 
added to the weight of the column of water whofe height 
is NH, ought again to be equal to 33—;', as before. Be- 
fides, when the fucking-pipe F G is of a fmaller diameter 
than the body of the pump, if the conditions which we 
have before fpecified obtain, the pump cannot fail to pro¬ 
duce the proper effect; for the air is dilated with more 
facility in this latter cafe than when the whole is of the 
fame internal diameter. We need only add on this point, 
that, if the length of the ftroke in a uniform pump, which 
is requifiteto render the machine effectual, be greater than 
can conveniently he made, it may be diminWhed by con- 
trailing the diameter of the fucking-pip! in the fubduplicale ratio 
if the diminution of the length of the ftroke. 
As to the effort of which the power ought to be capa¬ 
ble to fultain the water toa determinate height YH (fig. 49.) 
it will he meafured according to what we have laid re- 
fpefting the lifting-pump, hy the weight of a column of 
water whofe bafe is equal to CD, and height that of XY 
above RS. Here, too, we drop the confideration of fric¬ 
tion and the weight of the pifton. 
The velocity of the water flowing from the fucking-pipe 
into the barrel ffiould be equal to the velocity with which 
the pi if on moves. For, if it be greater, lefs work will be 
done than the pump is competent to effect; and, if it be 
lefs, a vacuum will be produced below the pifton, ^ which 
will' therefore he moved upwards with great difficulty. 
If V be the velocity of the water in the fucking-pipe, d the 
diameter of that pipe, D the diameter of the barrel or 
body of the pump, and tithe velocity of the pifton ; then 
4. The centrifugal pump acts upon a principle different 
from the three preceding, and muft therefore he confidered 
feparateiy. It confifts of a vertical tube,and a horizontal arm 
of equal bore, or fometimesof two horizontal arms of lefs 
bore than the vertical pipe; that of which we (hall fpeak 
has only one horizontal arm. In fig. 54, KL may repre- 
fent the vertical tube, the end K of which is immerfed in 
the water of the refervoir, and LI the horizontal arm com¬ 
municating with the former. There is a conical valve 
at K opening upwards, and one at the horizontal arm 
opening outwards. The whole machine being filled with 
water and turned fwiftly round upon pivots at K and L, 
the arm LI retaining its horizontal pofition, the water 
will, when the motion is properly regulated, continue to 
be difcharged from the moveable extremity I of the arm LI 
in an uninterrupted ftr&am. 
In order toinveftigate the tnoft ufeful theorems relative 
to the operation of this machine, put a = the length of the 
arm LI in feet, the length of the leg KL in feet, t— the 
time of a revolution in (econds, g-=z 32^ feet, the meafure 
of the force of gravity, and ir— 3-141593, the circumfer¬ 
ence of a circle whofe diameter is unity. Then, fince the 
centrifugal force is as the velocity it generates in a unit of 
-time, we (hall have the centrifugal force of a particle i- of 
/ j \ 
the fluid at the diftance x from L equal to J 2 -c-a~ 
the fluent of which is 
z<tr‘x i 
d L 
D 2 
will be the velocity of the water in the barrel, 
d* 
—, if the machine be perfefh 
and we muft have »=V 
If h be the height of a column of water whofe weight is 
equivalent to the preffure of the atmofphere, a the alti¬ 
tude of the water in the fucking-pipe, x any other height 
to which it afcends in following the pifton, £=32*- feet; 
then will the moving force C th — x, the quantity of matter 
moved CC*,and, confequently, the accelerating force a ‘ 
12 . —. - -- - = the centrifugal 
motive force of the column x; which, when x = a, be- 
2 TT*” CL ^ 
comes ——— for the whole centrifugal motive force of the 
water in the arm LI. Now the preffure of a columa 
whole length is a will, cat. par. vary as^a; hence it will 
. zn-a 2 zir 2 a 2 , , 
be, ga : a :: —-— : - - the length of a column of 
t 2 gt 2 
water Whofe preffure is equivalent to the centrifugal force. 
If from this we deduct the altitude / of the vertical leg, 
the remainder ^ -/ will be the length of a column 
gt* 
whofe aflion would expel the water from the orifice I with 
the fame force as the whirling motion will occafion; and 
the column moved w ill be a + l, which will alfo denote the 
fpace through which the fluid is accelerated. Confequently 
the circumltances of the cafe before us are the fame as if 
a conftant head of water of the height 
aw a 
gt* 
■ l impelled 
• thx 
Hence we have 
ents, there refults v the 
(A . -hyp. log 
eonlequently the correft fluent will be 
x ' a column of water horizontally at the bottom of the depth 
/-2w s a 2 \ 
and, taking the flu- «+/. Hence the accelerative force <p is ^ -‘y-T’Ca+O* 
velocity of the water 
x) — x. But when 
*) > 
X = a, V = O ; 
and the fpace j is a-f-/; fo that the velocity generated will, by 
the rules of dynamics, bee zzefztpg s—^jzg • a -—I'j 
v — JzgXk • hyp. log. X ~— (x— a). 
It may juft be added here, that the meafure we have all 
along given of the external force is only what is neceffary 
for balancing the preffure of the water in the riling pipe. 
But, in order that the pump may perform work, it muft 
furmount this preffure, and caufe the water to iffue at X 
with fuch a velocity that the required quantity of water 
may be delivered in a given time. This requires force, 
even although there were no oppofing preffure; which 
would be the cafe-if the main were horizontal. The wa¬ 
ter fills it; but it is at reft. In order that a gallon, for 
inftance, may be delivered in a fecond, the whole water 
in the horizontal main muft be put in motion with a cer¬ 
tain velocity. This requires force. We muft therefore 
always diftin.gu.ifh between the (late of equilibrium and 
the Itate of aftual working. It is the equilibrium only 
that we have confidered ; and no more is neceffary forun- 
derffanding the operation of the different fpscies of pumps. 
VOL. XIV. No. loro. 
=3 8 - oi 6 
2 % 2 a 2 
\ 
.st 2 
■l. Or, if we adapt the theorem to 
practical purpofes, the co-efficient 8-ci6 muft (p.715, 16.) 
be multiplied by *62; and the equation will become 
1 2 IS 
: w 
very nearly. 
gt* . . 
In this inveftigation we have paid no regard to the rota-, 
tory motion of the fluid molecules in the vertical tube, but 
have confidered it as though it were indefinitely narrow. 
It is manifeft that / muft never exceed 33 feet, becaufe 
a greater column cannot be fupported by the preffure of 
the atmofphere; fo that this machine cannot raife water 
higher than the common fucking-pump. Befides this, the 
time of a revolution of the arm has limits, between which- 
alone the pump is effedlive, fo that the inveftigation muft 
be carried a little.further. When the centrifugal force is 
barely equal to the weight to be railed, the expreffion under 
the radical vani(hes P and v = o, or the water is ttationary 
9 P m 
