z=sins 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 2 sin 2 ring is in use. This would indicate no 5° .087 .197 bending of the ring near the split, and 10 • .174 .311 hence the tip end would here bear heavi 20 .312 .489 ly upon the cylinder, with a probable 30 .500 .630 space of no contact for some distance 40 .745 back. The only useful supposition for 50 .766 .837 x is that it shall remain variable. As 60 .908 pointed out above, considerations of con 70 .940 .960 venience make p constant, and hence 80 .990 necessarily equal p. Hence we have 90 .1 .1 By these values the ring is drawn in z=z,' sin 2 (4.) Fig. 2 to a scale. The outside is taken, the circle of the cylinder into which the for the final equation expressing the law ring is supposed to be compressed when of thickness, z, of ring. ready for service. It appears from the equation that the From this drawing it is found by relation between z and 2, is independent placing one leg of a pair of dividers a of the radius of the cylinder, and of the distance, a, toward the split of the ring, pressure of ring against the surface of and swinging the other leg about near cylinder. Hence, taking z=1, a table the inside edge of ring, that about of can be computed for the values of z, by the inner line of ring lies almost exactly which a ring for any cylinder can be on the arc of a circle, and with its cendrawn. Then, when the particular value ter at a certain distance, a, from the cenof 2, for any practical case, is found to ter of the exterior of ring. The disbe other than 1, the values of z are to be tance of this center from the inner sur proportionately modified. Making 2 =1, face of ring opposite split is r-2, +a. we find the following 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 p-z=1°—%, sin 45° to the opposite side, thence out and from the inner surface of ring, which is escape. It is evident that for thus lapalso the radius of the inside circle. Mak- ping the tips, the thickness, to that esing these distances equal, we find tent, should be constant, and not vary r-2, +a=r-2, sini 45° as in Fig. 2. To adjust the bearing of a=2,(1-sini 45°) the lapping part against the cylinder, it is plainly necessary to dress off a trifle a=0.206 z, nearly (5) of the over-lapping tip outside, to comIf it be assumed that this circle gives pensate for the increased stiffness due to the inner form of the ring with a suffi- the deviation from Fig. 2, to a uniformicient degree of exactness, for about f its ty of thickness. The extent to which extent, the ring can be formed in a turn the tips may be made to overlap, as ing lathe complete, except for the } in Fig. 3, for a given excess of diameter of inside surface, a balf which is on at which the ring may be turned, will be 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 It is evident that the outward pressure a chuck, acylindric shell long enough to of the ring against the inside surface of make several rings, with a stay of cross cylinder will depend upon the diameter The inside is to be first bored of the ring, its thickness, and the excess deep enough for several rings, and then in diameter being given it when turned. the ring offset to the eccentricity a=.206 Let A be the center of the ring when %, and the outside turned off. The rings turned to the radius rį, and B the center are then to be cat off with such breadth when it is compressed into the cylinder as desiried. The eccentricity is observed of radius r. Let DC be a small part of to be a little over the greatest thick- that ring, so that ness of ring. DC=rdA=r, do'=pde" (6) In practice the tip ends of the ring at the split should be lapped a little, in- p being the radius of curvature due to stead of being as shown in Fig. 2. A the bending or compressing of the ring, suitable way of doing this is shown in Fig. 3. For a perfect and complete stop OUTWARD PRESSURE OF RING. arms. TENDENCY OF FLUID TO LIFT THE RING. . whence (8) p ni In a thin space between the ring and But as a constant value of p has been cylinder, the Auid will flow, as shown in adopted, p=p, and hence, combining Fig. 5, from p, to p,, the frictional resist. (8) with the equation preceding (3), we ance of an inelastic fluid like water in have the space t, hindering the flow, and =2Pro giving cause, for pressure upon ring; 12 and in elastic fluids, like air, both the whence the amount of normal pressure frictional resistance and expansive of ring per square unit is pressure in the space, t, between ring and cylinder, will have a combined effect to press the ring back, and to increase the space escape. Hence, it appears that the intensity of 1. Case of Ordinary Water Pump:pressure of ring upon cylinder will vary Here the density does not sensibly directly as the cube of the thickness, and change between p, and p. In this case 3 for 1 If we + nearly as the excess of diameter at which the velocity of flow will be constant the ring is turned. 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. va fsl v*_P-PP-P adopt such a relation that of the ring h' +h'=h= + will be outside the groove when first 2gu a 29 d d sprung into it, we have δυο rad. of piston, r=r, - }(r, -inside rad). (10) But insiderad.=r; -2, +a=r, -.794 z, for entry, and .. r=r,-.529 2, ofelva 7; -,_.529 2, P-P,= (11) nearly 2g ni for the space t, in which v is the velocity Hence P= (9) in the space, u the coefficient of velocity or P.-P=2gui . =.5 a E t a 66 o the weight per unit of volume of the fall of pressure due to (10) and (11), we flowing fluid; a, the sectional area, trans. may take their ratio. Hence verse to the flow, of so much of the space t as is considered; s, the perimeter P.-P of the same section; f, the coefficient of . (15) p-pe fsbu2fbu friction of the fluid against the ring or cylinder ; 1=6, the length in the direc- in which tion of motion of the flowing sheet, or 2 breadth of ring; and p, the absolute since for a given extent, k, along the pressure of Auid just within the en. ring trance to the space t. The pressure will a-kt S=2k. gradually decrease, from p to P», as the fluid passes along the space t. If x be Hence, the fall of pressure varies only taken for the distance from the p, side of with t and b, the thickness of space and ring, we have length of flow between ring and cylinotsa va der. But it will, of course, be desirable P-P 29 (12) that t be very small , far within the hun dredth part of an inch. If, for example, where p will now be the pressure of it be taken in thousandths of an inch, fluid in the space t at the distance 2. we have for f=.006, b=.5 and u'=.7 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 P-P: ring and for å unit's length along the 2 .476 (B) ring, will be 4 .952 1.904 1 hundredth 2.381 a 29 Hence, with a thickness of space t, of a little over 4 thousandths of an inch, ofs va b half the fall of pressure from p, to pa a 29 2 +p,b=(p—P.)ż+p. will be consumed in entering the fluid by eq. (11). Hence into the space t. As the thickness of this space thus appears to be an importP P * P (14) ant element in determining and consePA P, and numerically for quently the thickness z, of ring, the lat . P ter diminishing with P, it becomes desir=1.5 able to determine the admissible limits P2 of t. This is to be done by aid of the 4 2.5 6 equation just preceding (10), whence the 3.5 (A) velocity of escape is found by 8 4.5 10 5.5 29 (16) Ô 1 2 16 an expression giving in terms of the + pressures of fluid at opposite edges of น packing ring, the ratio of the total pressure tending to push the ring away from displacement in the volume escaping The percentage of the whole piston the cylinder to the pressure at the ex. past the piston will be haust side of piston. These pressures are per square unit of area absolute. Aq v X area of whole spacetVX 2 art noticable fact is that this ratio is inde- Q piston displacement πη2V pendent of the breadth of ring, of the 2ot density of fluid, and of the coefficient of V friction, except so far as p may depend upon them. That p will not coincide where V is the piston speed in ft. per sec. with p, is evident from (10). Combining this with (16) by eliminatTo find the relative proportions of ing v, and we obtain, P. P = 2 t 2 (8) 2 P=4. + 2, =.584 inch 8gt PI not exceed a third of the total fall from P, to pz. In other words, the fall of (17) pressure, p-pa, in passing the ring, should not be less than two-thirds of the t total difference P, -p, of pressure on opIf it is desired to take account of the posite sides of piston. effect of the motion of piston and of To illustrate further, let the pump be packing ring, upon the actual velocity of working where fluid escape, we find the latter to be, Pi =6 and f of this, gives P2 P. 2 Then the table for eq. (14) gives and then the first member of (17) should Р -=2.5 for =4; be P. PA Ez 2.5 p.= This may be put in a more convenient 48r* form by factoring and tabulating a part. If p=1 atmosphere=15 lbs. per sq. Thus, if d = 2r = diam. of piston, inch, and if the ring be made of cast 1 iron, as probably a good material, when Q -1(Tab.val.) (18) E=20,000,000. Then dvd P2 where, for water pump rings, of a width 2.=.097. b=.5 of an inch, and for t = 1, 2, 5, 10, thousandths of an inch. If the pump has a 12" cylinder Tab. val.=.0057 .014.0480 .1092 (C.) For these values p, is taken at one at- so that for the conditions assumed, a mosphere. 12-inch piston packing ring for a pump To obtain an idea of the practical val- will be about f of an inch thick at the nes of (18), we have, for a piston speed thickest place. The eccentricity, a, for use in turning of pump piston of V=2 ft per sec., the ring will be P. These results are for a pump where the ton, is one atmosphere ; and 9 = 2, =.005 .010 .048 .109 (D) pressure side, six atmospheres. P: Q For similar conditions, except the =8,.... 66 =.015 .027 .127 288 working pressure p, = 2 atmospheres, we 2t have the term a ..496 inch, or about half an inch. obtained. This thickness, compared with the table If we ignore the refinements about of values for eq. (15) indicates that we the orifice of entry, and assume p as may count on identical with p,, we only require formu las (14) and (9) to determine the ring. P, -P =not greater than o to .5. In fact, they may be combined, giving P-P2 24p. or the fall of pressure on entering the 1+ (19) E P, space t, should not exceed a half of the fall due to passing the ring; or, it should Values of %, computed by this for the Pis on the a=t inch. |