Using the equation of continuity in the form of (nD /4)V = (nd /4)V2, where D is the diameter of the pipe and d is diameter of the throat, the above equation may be solved for V2 to produce
Let us now express Pj - P2 in terms of the indicator deflection, Ah. Apply the manometric equation in b in the sequence 1, 4, 3', 3, 2. Thus,
where Ah14, Ah3'3 (=Ah), Ah32, and ynd refer to the head difference between points 1 and 4, points 3' and 3, and points 3 and 2, respectively. ynd is the specific weight of the indicator fluid used to indicate the deflection of manometer levels (i.e., the two levels of the indicator fluid in the manometer tube). Equation (3.16) may be solved for P1 - P2 producing P1 - P2 = Ah(yind - y). However, in terms of an equivalent deflection of water, Ah^o, Pi - P2 = Aha ,oy. Thus,
If the tapping in d is used where the indicator fluid is lighter than water and the above derivation is repeated, ynd - y in Equation (3.18) would be replaced by y - ynd. Note that AhH O is not the manometer deflection; it is the water equivalent of the manometer deflection.
AhH Oy may be substituted for P1 - P2 in Equation (3.15) and both sides of the equation multiplied by the cross-sectional area at the throat, At, to obtain the discharge, Q. The equation obtained by this multiplication is simply theoretical, however; thus, a discharge coefficient, K, is again used to account for the nonideality of actual discharge flows and to acknowledge the fact that the head loss, h, was originally neglected in the derivation. The corrected equation follows:
where values of K may be obtained from c of Figure 3.5 and At = nd /4. Because P1 - P2 = yAhH o, Equation (3.19) may also be written in terms of P1 - P2 as follows q = KAtJ 2g(Py-P2- (3.20)
Equation (3.20) may be used if the venturi meter is not oriented horizontally. This is done by calculating the pressures at the points directly and substituting them into the equation.
When measuring sewage flows, debris may collect on the pressure sensing holes. Hence, these holes must be cleaned periodically to ensure accurate sensing of pressure at all times. In a of Figure 3.5, an automatic cleaning arrangement is designed using an external supply of water. Water from the supply is introduced into the piping system through flow indicator, pipes, valves, and fittings, and into the venturi meter. The design would be such that water jets at high pressure are directed to the pressure sensing holes. These jets can then be released at predetermined intervals of time to wash out any cloggings on the holes. of course, at the time that the jet is released, erratic readings of the pressure will occur and the corresponding Q should not be used. Line pressure of 70 kN/m in excess over source water supply pressure is satisfactory.
Example 3.6 The flow to a water treatment plant is 0.031 cubic meters per second. The engineer has decided to meter this flow using a venturi meter. Design the meter if the approach pipe to the meter is 150 mm in diameter.
Solution: The designer has the liberty to choose values for the design parameters, provided it can be shown that the design works. Provide a pressure differential of 26 kN/m between the approach to the tube and the throat. Initially assume a throat diameter of 75 mm.
From the appendix, p = 997 kg/m3 and ¡1 = 8.8(10-4) kg/ms (25°C); therefore,
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