508 APPLIED MECHANICS 
the pressure of the water are high. In the valves shown in Figs, 810: id 
811, the area through the seat is less than qo on account of the presence 
of the arrangements for guiding the valve.» These valves are therefore 
full open when the lift is something less than }d. 
An annular valve which has two seats is shown in Fig. 814. ay 
and d, are the diameters of the annular opening between the seats, a 
h the lift of the valve, then, neglecting the effect of the ribs between t the 
seats, the valve is full open when ai — d) = (dy +d,)h, that is, is : 
h=i(d,-d,). Tf d,=4d,, then h=}d,. 
The india-rubber disc valve is shown in Fig. 815. The thickness of 
the india-rubber is generally § inch to } inch for small valves, and m ay 
be as much as { inch in the largest sizes. The area of the seat or grat- 
ing in contact with the india-rubber should be sufficient to prevent. he 
pressure between them exceeding 40.lbs. per square inch. The peng 
forated guard limits the angular lift of the disc to about 30°. 
The Gutermuth valve, shown in Fig. 816, is an ingenious form of ff Ap 
valve. This valve is made from a sheet of special bronze of high” 
tenacity. Part of the sheet forms a spiral coil, the inner end of which i 
turned over and enters a slot in a spindle. The flat or uncoiled part 0 7 
the sheet forms the valve proper, and this is thicker than the rest. 
projecting ends of the spindle are rigidly held in bearings, so that 
flap is always in its correct position over the port. Before clampiagg 
down, the spindle is rotated until the spring of the coil is of the nece ary 
stiffness. The advantages claimed for this valve are, (1) the port 
entirely uncovered with a relatively small deflection of the metal of the : 
valve, (2) quite a small force exerted by the water is sufficient to 
deflect the valve, (3) the valve closes promptly when the flow ceases. 
440. Fluctuation of Delivery in Crank-driven Pumps.—In many 
cases the plunger or piston of a pump is driven through a crank and © 
connecting-rod, the crank being fixed to a shaft which has uniform — 
angular velocity. In other cases the plunger or piston is connected — 
directly to the piston-rod of a steam cylinder, and there is a crank shaft — 
whose crank is also connected to the piston-rod by a connecting-rod, On — 
the crank shaft is a heavy fly-wheel, which causes the angular velocity of | 
the shaft to be fairly uniform. 4 
In all such cases the velocity of the plunger or piston varies during” 
each stroke in a well-defined manner, and the curve which represents the — 
variation of the pent or piston velocity may be constructed as fully — 
explained in Art. 260, p. 300. The velocity of the water through the — 
delivery valve at any instant will evidently be proportional to the velocity ‘ 
of the plunger or piston at that instant. Hence a plunger- or piston- 
velocity diagram will also be a rate of delivery diagram. This disgraly 4 
may be drawn on a stroke base or, preferably, on a time base. 
First consider a simple plunger pump having one plunger. Due 
the suction stroke there is no discharge through the delivery valve, but — 
during. the delivery stroke the variation of the velocity of discharge is 
shown by the plunger-velocity diagram. The result for one revolution of 
the crank is shown at (a), Fig. 817, on a time base, 
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