begins its stroke. If the piston begins its stroke before the admission of steam begins the valve is said to have negative lead, and its amount is the lap of the edge of the valve over the edge of the port at the instant when the piston stroke begins.

Lap-angle the angle through which the eccentric must be rotated to cause the steam edge to travel from its central position the distance of the lap.

Angular advance of the eccentric lap-angle + lead angle.
Linear advance lap + lead.

=

Effect of Lap, Lead, etc., upon the Steam Distribution.Given valve-travel 234 in., lap 34 in., lead 1/16 in., exhaust-lap 1 in., required crank position for admission, cut-off, release and compression, and greatest port-opening. (Halsey on Slide-valve Gears.) Draw a circle of diameter fh travel of valve. From O the centre set off Oa =

=

lap and ab = lead, erect perpendiculars Oe, ac, bd; then ec is the lap-angle and cd the lead-angie, measured as arcs. Set off fg= cd, the lead-angle, then Og is the position of the crank for steam admission. Set off 2ec+cd from 1 to i; then Oi is the crank-angle for cut-off, and fk+fh is the fraction of stroke completed at cut-off. Set off Ol exhaust-lap and draw Im; em is the exhaust-lap angle. Set off hn ec+cd-em, and On is the position of crank at release. Set off fp =ec + ca+em, and Op is the position of crank for compression, fo+fh is the fraction of stroke completed at release, and hqhf is the fraction of the return stroke completed when compression begins; Oh, the throw of the eccentric, minus Oa the lap, equals ah the maximum port-opening.

If a valve has neither lap nor lead, the line joining the centre of the eccen¿ Cut-off

Release

FIG. 146.

tric disk and the centre of the shaft being at right angles to the line of the crank, the engine would follow full stroke, admission of steam beginning at the beginning of the stroke and ending at the end of the stroke.

Adding lap to the valve enables us to cut off steam before the end of the stroke; the eccentric being advanced on the shaft an amount equal to the lap-angle enables steam to be admitted at the beginning of the stroke, as

before lap was added and advancing it a further amount equal to the lead angle causes steam to be admitted before the beginning of the stroke. Having given lap to the valve, and having advanced the eccentric on the shaft from its central position at right angles to the crank, through the angular advance = lap-angle and lead-angle, the four events, admission, cut-off, release or exhaust-opening, and compression or exhaust-closure, take place as follows: Admission, when the crank lacks the lead-angle of having reached the centre; cut-off, when the crank lacks two lap-angles and one lead-angle of having reached the centre. During the admission of steam the crank turns through a semicircle less twice the lap-angle. The greatest port-opening is equal to half the travel of the valve less the lap. Therefore for a given port-opening the travel of the valve must be increased if the lap is increased. When exhaust-lap is added to the valve it delays the opening of the exhaust and hastens its closing by an angle of rotation equal to the exhaust-lap angle, which is the angle through which the eccentric rotates from its middle position while the exhaust edge of the valve uncovers its lap. Release then takes place when the crank lacks one lap-angle and one lead-angle minus one exhaust-lap angle of having reached the centre, and compression when the crank lacks lap-angle+ lead-angle+ exhaust-lap angle of having reached the centre.

The above discussion of the relative position of the crank, piston, and valve for the different points of the stroke is accurate only with a connecting-rod of infinite length.

For actual connecting-rods the angular position of the rod causes a distortion of the position of the valve, causing the events to take place too late in the forward stroke and too early in the return. The correction of this distortion may be accomplished to some extent by setting the valve so as to give equal lead on both forward and return stroke, and by altering the exhaust-lap on one end so as to equalize the release and compression. F. A. Halsey, in his Slide-valve Gears, describes a method of equalizing the cut-off without at the same time affecting the equality of the lead. In designing slide-valves the effect of angularity of the connecting-rod should be studied on the drawing-board, and preferably by the use of a model. Sweet's Valve-diagram.-To find outside and inside lap of valve for different cut-offs and compressions (see Fig. 147): Draw a circle whose

diameter equals travel of valve. Draw diameter BA and continue to 41, so that the length AA bears the same ratio to XA as the length of connecting-rod does to length of engine-crank. Draw small circle K with a radius equal to lead. Lay off AC so that ratio of AC to AB = cut-off in parts of the stroke. Erect perpendicular CD. Draw DL_tangent to K; draw XS perpendicular to DL; XS is then outside lap of valve.

To find release and compression: If there is no inside lap, draw FE through X parallel to DL. F and E will be position of crank for release and compression. If there is an inside lap, draw a circle about X, in which radius XY equals inside lap. Draw HG tangent to this circle and parallel to DL; then H and G are crank position for release and compression. Draw HN and MG, then AN is piston position at release and AM piston position at compression, AB being considered stroke of engine.

To make compression alike on each stroke it is necessary to increase the inside lap on crank end of valve, and to decrease by the same amount the

inside lap on back end of valve. To determine this amount, through M with a radius MM AA1, draw arc M P, from P draw PT perpendicular to AB, then TM is the amount to be added to inside lap on crank end, and to be deducted from inside lap on back end of valve, inside lap being XY.

For the Bilgram Valve Diagram, see Halsey on Slide-valve Gears. The Zeuner Valve-diagram is given in most of the works on the steam-engine, and in treatises on valve-gears, as Zeuner's, Peabody's, and

Spangler's. The following is condensed from Holmes on the Steam-engine: Describe a circle, with radius OA equal to the half travel of the valve. From O measure off OB equal to the outside lap, and BC equal to the lead. When the crank-pin occupies the dead centre A, the valve has already moved to the right of its central position by the space OB+ BC. From C erect the perpendicular CE and join OE. Then will OE be the position occupied by the line joining the centre of the eccentric with the centre of the crank-shaft at the commencement of the stroke. On the line OE as diameter describe the circle OCE; then any chords, as Oe, OE, Oe', will represent the spaces travelled by the valve from its central position when the crank-pin occupies respectively the positions opposite to D, E, and F. Before the port is opened at all the valve must have moved from its central position by an amount equal to the lap OB. Hence, to obtain the space by which the port is opened, subtract from each of the arcs Oe, OE, etc., a length equal to OB. This is represented graphically by describing from centre 0 a circle with radius equal to the lap OB; then the spaces fe, gE, etc., intercepted between the circumferences of the lap-circle Bfe' and the valve-circle OCE, will give the extent to which the steam-port is opened. At the point k, at which the chor 1 Ok is common to both valve and lap circles, it is evident that the valve has moved to the right by the amount of the lap, and is consequently just on the point of opening the steam-port. Hence the steam is admitted before the commencement of the stroke, when the crank occupies the position OH, and while the portion HA of the revo

lution still remains to be accomplished. When the crank-pin reaches the position A, that is to say, at the mencement of the stroke, the port is already opened by the space OC - OB = BC, called the lead. From this point forward till the crank occupies the position OE the port continues to open, but when the crank is at OE the valve has reached the furthest limit of its travel to the right, and then commences to return, till when in the position OF the edge of the valve just covers the steam-port, as is shown by the chord Oe', being again common to both lap and valve circles. Hence when the crank occupies the position OF the cut-off takes place and the steam commences to expand, and continues to do so till the exhaust opens. For the return stroke the steam-port opens again at H' and closes at F. There remains the exhaust to be considered. When the line joining the centres of the eccentric and crank-shaft occupies the position opposite to OG at right angles to the line of dead centres, the crank is in the line OP at right angles to OE; and as OP does not intersect either valve-circle the valve occupies its central position, and consequently closes the port by the amount of the inside lap. The crank must therefore move through such an angular distance that its line of direction OQ must intercept a chord on the valve-circle OK equal in length to the inside lap before the port can be opened to the exhaust. This point is ascertained precisely in the same manner as for the outside lap, namely, by drawing a circle from centre 0, with a radius equal to the inside lap; this is the small inner circle in the figure. Where this circle intersects the two valve-circles we get four points which show the positions of the crank when the exhaust opens and closes during each revolution. Thus at Q the valve opens the exhaust on the side of the piston which we have been considering, while at R the exhaust closes and compression commences and continues till the fresh steam is readmitted at H.

Thus the diagram enables us to ascertain the exact position of the crank when each critical operation of the valve takes place. Making a résumé of these operations of one side of the piston, we have: Steam admitted before the commencement of the stroke at H. At the dead centre A the valve is already opened by the amount BC. At E the port is fully opened, and valve has reached one end of its travel. At F steam is cut off, consequently admission lasted from H to F. At P valve occupies central position, and ports are closed both to steam and exhaust. At Q exhaust opened, conse quently expansion lasted from F to Q. At K exhaust opened to maximum extent, and valve reached the end of its travel to the left. At R exhaust closed, and compression begins and continues till the fresh steam is admitted at H.

PROBLEM.-The simplest problem which occurs is the following: Given the length of throw, the angle of advance of the eccentric, and the laps of the valve, find the angles of the crank at which the steam is admitted and cut off and the exhaust opened and closed. Draw the line OE, representing the half-travel of the valve or the throw of the eccentric at the given angle of advance with the perpendicular OG. Produce OE to K. On OE and OK as diameters describe the two valve-circles. With centre and radii equal to the given laps describe the outside and inside lap-circles. Then the intersection of these circles with the two valve-circles give points through which the lines OH, OF, OQ, and OR can be drawn. These lines give the required positions of the crank.

Numerous other problems will be found in Holmes on the Steam-engine, including problems in valve-setting and the application of the Zeuner diagram to link motion and to the Meyer valve-gear.

Port Opening. The area of port opening should be such that the velocity of the steam in passing through it should not exceed 6000 ft. per min. The ratio of port area to piston area will then vary with the piston-speed as follows:

For speed of piston, 100 200 300 400 500 600 700 800 900 1000 1200 ft. per min.

Port area = piston

area X

.017 .033 .05 .067 .083 .1 .107 .133 .15 .167 .2

The length of the port opening may be equal to or something less than the diameter of the cylinder, and the width = area of port opening its length. The bridge between steam and exhaust ports should be wide enough to prevent a leak of steam into the exhaust due to overtravel of the valve.

Auchincloss gives: Width of exhaust port width of steam port + 1⁄2 travel of valve - width of bridge.

Lead. (From Peabody's Valve-gears.)-The lead, or the amount that the valve is open when the engine is on a dead point, varies, with the type and size of the engine, from a very small amount, or even nothing, up to 3% of an inch or more. Stationary-engines running at slow speed may have from 1/64 to 1/16 inch lead. The effect of compression is to fill the waste space at the end of the cylinder with steam; consequently, engines having much compression need less lead. Locomotive-engines having the valves controlled by the ordinary form of Stephenson link-motion may have a small lead when running slowly and with a long cut-off, but when at speed with a short cut-off the lead is at least 4 inch; and locomotives that have valve-gear which gives constant lead commonly have 4 inch lead. The lead-angle is the angle the crank makes with the line of dead points at admission. It may vary from 0° to 8°.

Inside Lead.-Weisbach (vol. ii. p. 296) says: Experiment shows that the earlier opening of the exhaust ports is especially of advantage, and in the best engines the lead of the valve upon the side of the exhaust, or the inside lead; is 1/25 to 1/15; i.e., the slide-valve at the lowest or highest position of the piston has made an opening whose height is 1/25 to 1/15 of the whole throw of the slide-valve. The outside lead of the slide-valve or the lead on the steam side, on the other hand, is much smaller, and is often only 1/100 of the whole throw of the valve.

Effect of Changing Outside Lap, Inside Lap, Travel and Angular Advance. (Thurston.)

Zeuner gives the following relations (Weisbach-Dubois, vol. ii. p. 307): If 8 travel of valve, p = maximum port opening;

L=

steam-lap, = exhaust-lap;

R = ratio of steam-lap to half travel =

L=

.5.S'

X S;

2p

S=2p+2L = 2p + R × S; S= 1 R

If a = angle HOF between positions of crank at admission and at cut-off, and B = angle QOR between positions of crank at release and at sin (180° - B compression, then R

sin (180° a).
sin a

r = 1⁄2·

sin 128

Ratio of Lap and of Port-opening to Valve-travel.-The table on page 831, giving the ratio of lap to travel of valve and ratio of travel to port opening, is abridged from one given by Buel in Weisbach-Dubois, vol. ii. It is calculated from the above formulæ. Intermediate values may be found by the formulæ, or with sufficient accuracy by interpolation from the figures in the table. By the table on page 830 the crank-angle may be found, that is, the angle between its position when the engine is on the centre and its position at cut-off, release, or compression, when these are known in fractions of the stroke. To illustrate the use of the tables the following example is given by Buel: width of port = 2.2 in.; width of port opening width of port +0.3 in.; overtravel = 2.5 in.; length of connecting-rod =2% times stroke; cut-off 0.75 of stroke; release = 0.95 of stroke; lead-angle, 10°. From the first table we find crank-angle = 114.6