tually variable. If we make z constant, as TABLE OF VALUES FOR LAYING OUT THE RING: would follow from turning the ring of uniform thickness, p would necessarily vary with x and be infinite at the split when the ring is in use. This would indicate no bending of the ring near the split, and hence the tip end would here bear heavily upon the cylinder, with a probable space of no contact for some distance back. The only useful supposition for a is that it shall remain variable. As pointed out above, considerations of convenience make p constant, and hence necessarily equal p1. Hence we have

x

sin

z=sint

x

2

2

2

5°

.087

.197

10

.174

.311

20

.342

.489

30

for the final equation expressing the law of thickness, z, of ring.

It appears from the equation that the relation between z and z, is independent of the radius of the cylinder, and of the pressure of ring against the surface of cylinder. Hence, taking z=1, a table can be computed for the values of z, by which a ring for any cylinder can be Then, when the particular value any practical case, is found to be other than 1, the values of z are to be proportionately modified. Making z,=1, we find the following

Fig.2

By these values the ring is drawn in. Fig. 2 to a scale. The outside is taken, the circle of the cylinder into which the ring is supposed to be compressed when ready for service.

From this drawing it is found by placing one leg of a pair of dividers a distance, a, toward the split of the ring, and swinging the other leg about near the inside edge of ring, that about of the inner line of ring lies almost exactly on the arc of a circle, and with its center at a certain distance, a, from the center of the exterior of ring. The distance of this center from the inner sur face of ring opposite split is r-z,+a. the radius of the inside circle of ring.

At the intermediate point of 90° from worn so as to separate a little at the tips, the split, this center is at a distance the working fluid may enter between the

tips at one side, pass under the ring to the opposite side, thence out and escape. It is evident that for thus lapping the tips, the thickness, to that extent, should be constant, and not vary as in Fig. 2. To adjust the bearing of the lapping part against the cylinder, it is plainly necessary to dress off a trifle of the over-lapping tip outside, to compensate for the increased stiffness due to the deviation from Fig. 2, to a uniformity of thickness. The extent to which the tips may be made to overlap, as in Fig. 3, for a given excess of diameter at which the ring may be turned, will be

OUTWARD PRESSURE OF RING.

If it be assumed that this circle gives the inner form of the ring with a sufficient degree of exactness, for about its extent, the ring can be formed in a turning lathe complete, except for the of inside surface, a half which is on each side of split. This could be considered subsequently. dressed off subsequently. The lathe work would then most conveniently consist of mounting upon a face plate, or in a chuck, acylindric shell long enough to make several rings, with a stay of cross arms. The inside is to be first bored deep enough for several rings, and then the ring offset to the eccentricity a=.206 219 and the outside turned off. The rings are then to be cat off with such breadth as desiried. The eccentricity is observed to be a little over the greatest thickness of ring.

In practice the tip ends of the ring at the split should be lapped a little, in

It is evident that the outward pressure of the ring against the inside surface of cylinder will depend upon the diameter of the ring, its thickness, and the excess in diameter being given it when turned.

Let A be the center of the ring when turned to the radius r,, and B the center when it is compressed into the cylinder of radius r,. Let DC be a small part of that ring, so that

DC=rd0=r, ᏧᎾ' = p dᎾ'

(6)

being the radius of curvature due to

stead of being as shown in Fig. 2. A the bending or compressing of the ring,

TENDENCY OF FLUID TO LIFT THE RING.

In a thin space between the ring and cylinder, the fluid will flow, as shown in Fig. 5, from p, to p,, the frictional resistance of an inelastic fluid like water in the space t, hindering the flow, and giving cause, for pressure upon ring; and in elastic fluids, like air, both the frictional resistance and expansive pressure in the space, t, between ring and cylinder, will have a combined effect to pres 3 the ring back, and to increase the space for escape.

1. Case of Ordinary Water Pump.Here the density does not sensibly change between p, and p

In this case

nearly as the excess of diameter at which the ring is turned.

Fig.5

the velocity of flow will be constant within the space t, and the reduction of This excess in external diameter, of pressure will be in two parts; first, for ring over the cylinder bore, is arbitrary. acceleration at entry to the space, and But in assuming the excess, probably no second, for frictional resistance along the one would take the mean diameter of space. According to an article which I ring, inside and outside, less than that of have lately published in VAN NOSTRAND'S the cylinder bore. A little consideration ENGINEERING MAGAZINE (see p. 372), would probably fix it larger. If we adopt such a relation that of the ring '+h"=h= will be outside the groove when first sprung into it, we have

or

va fsl va
+
2gu a 2g d

Sv3
P1-P=2gu2

rad. of piston, r=r,— }(r,—inside rad).
But inside rad.=r-z,+a=r,-.794 z, for entry, and
.. r=r,—.529 z,

Sfsl v2
a 2g

=

P1-P, P-P2
+
d

(10)

. . (11)

for the space t, in which v is the velocity in the space, u the coefficient of velocity for the orifice of entry, here about 0.85;

fall of pressure due to (10) and (11), we may take their ratio. Hence

P.-P.

& the weight per unit of volume of the
flowing fluid; a, the sectional area, trans
verse to the flow, of so much of the
space t as is considered; s, the perimeter
of the same section; f, the coefficient of
friction of the fluid against the ring or
cylinder; l=b, the length in the direc-in which
tion of motion of the flowing sheet, or

ring

akt s 2k. ·

. (15)

breadth of ring; and p, the absolute since for a given extent, k, along the pressure of fluid just within the entrance to the space t. The pressure will gradually decrease, from p to P, as the fluid passes along the space t. If x be taken for the distance from the p, side of ring, we have

dfsx v2

a 2g

(12)

Hence, the fall of pressure varies only with t and b, the thickness of space and length of flow between ring and cylinder. But it will, of course, be desirable that t be very small, far within the hunit be taken in thousandths of an inch, dredth part of an inch. If, for example, we have for f=.006, b=.5 and u2=.7

where p will now be the pressure of
fluid in the space t at the distance x.
This pressure tends to push the ring
back, and it being variable, the total
amount of it for the whole breadth, b, of t=1 thousandth
ring and for a unit's length along the
ring, will be,

2

66

4

8

1 hundredth

Hence, with a thickness of space t, of

a little over 4 thousandths of an inch, half the fall of pressure from p, to p will be consumed in entering the fluid into the space t. As the thickness of this space thus appears to be an import(14) ant element in determining and conse

(A)

Р

P

quently the thickness z, of ring, the lat-
ter diminishing with P, it becomes desir-
able to determine the admissible limits
of t. This is to be done by aid of the
equation just preceding (10), whence the
velocity of escape is found by

v2=!
P1-P2 2g
δ 1 2/b

2

+ u2

t

(16)

The percentage of the whole piston displacement in the volume escaping past the piston will be

an expression giving in terms of the
pressures of fluid at opposite edges of
packing ring, the ratio of the total press-
ure tending to push the ring away from
the cylinder to the pressure at the ex-
haust side of piston. These pressures
are per square unit of area absolute. A q
noticable fact is that this ratio is inde-Q
pendent of the breadth of ring, of the
density of fluid, and of the coefficient of
friction, except so far as p may depend
upon them.
That p will not coincide
with p, is evident from (10).

where V is the piston speed in ft. per sec. Combining this with (16) by eliminat

To find the relative proportions of ing v, and we obtain,

2

To illustrate further, let the pump be working where

As the maximum admissible leakage of | piston would probably be assumed from

p 6 and of this, gives 4.

P2
Then the table for eq. (14) gives
p

Р

-- =2.5 for =4;

and this, combined with eq. (9), gives Ez,

2.5P48

If p, 1 atmosphere 15 lbs. per sq. inch, and if the ring be made of cast E=20,000,000. Then iron, as probably a good material, when

=.097.

If the pump has a 12" cylinder

2,=.584 inch

so that for the conditions assumed, a 12-inch piston packing ring for a pump will be about of an inch thick at the thickest place.

The eccentricity, a, for use in turning the ring will be

a=.206 z,=.12 inch, or about of an inch.

These results are for a pump where the pressure p,, on the suction side of piston, is one atmosphere; and p,, on the pressure side, six atmospheres.

For similar conditions, except the working pressure p, 2 atmospheres, we have

z,=.496 inch, or about half an inch. a= inch.

It is noticeable, from eq. (9), that z, 1 to 5 per cent. of piston displacement, is proportional to r for a given pressure,. it appears that the greatest admissible so that the thickness of ring for other thickness of space, t, is about 2 thou- sizes than 12" pistons, can readily be sandths of an inch in pumps. obtained.