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The quantity of water that flows through a pipe depends primarily on the head, pipe diameter, nature of interior surface and the number and shape of the bends.
(a) Consider figure 28.1. Indicate the water head.
(b) How is the water head calculated if it is mechanically established by pumping?
(c) A pipe line, 1/2 a mile long, 12 inches in diameter, discharges water under a head of 100 feet. Find the velocity and quantity of discharge .
(d) What length of straight pipe should compensate for the loss of head in pipe due to a right angle bend in pipe.
Pj Problem of Interest is of type force (force-push).
(a) Consider figure 28.1. Water head = h1 - h2. In essence, the water head is the difference in potential energy relative to ground. That is, (h1 - h3) - (h2 - h3). The pressure that forces water out of the pipe is directly related to the head and is zero when h1 = h2.
(b) Head = vertical distance corresponding to the mechanical pressure.
For example, 1lb/in2 = 2,309 ft head, and 1 foot-head = 0.433 lb/in2.
(c) Formula in focus is:
V = C(hD/(L +54D))1/2 -------------(1).
This formula is an approximation with 5% to 10% accuracy.
Where V = approximate mean velocity in feet per second
C = coefficient associated with pipe diameter. A table of pipe diameters and C is usually available.
For this problem, C = 48 for pipe diameter of 1 foot.
D = Diameter of pipe in feet.
h = total head in feet
L = total length of pipe line in feet.
So, substituting in equation (1), we have:
Velocity of discharge, V = 48[(100 x 1)/(2640 + 54(1))]1/2 = (0.037)1/2 = 9.233 ft/sec.
Area of cross section of pipe = πr2 = π(0.5)2 = 0.7854 sq-ft.
So, Discharge = 9.233 x 0.7854 = 7.252 cubic-ft/sec.
(d) Loss of head due to a bend in pipe = equivalent length (Le) of straight pipe that causes same loss in head as the bend.
Experiments indicate approximately, that a right angle bend has radius of 3 times the diameter of the pipe.
Formula for right angle bend:
Le = 4d/3. Where d is diameter of pipe in inches.
So, Le = (4 x 12)/3 = 16 feet of straight pipe.
Formula for valves:
Le = 4d/6.
The point . is a mathematical abstraction. It has negligible size and a great sense of position. Consequently, it is front and center in abstract existential reasoning.
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