The momentum equation tells us what happens to

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The momentum equation tells us what happens to pressure along the duct. For the case of steady- state, inviscid (no wall friction) flow along a continuous streamline in a constant density medium, the Bernoulli equation conserves momentum. constant = h g + 2 v + P = h g + 2 v + P 2 2 2 2 1 2 1 1 where P is the pressure, g is gravity and h is the fluid elevation at arbitrary points 1,2 along the flow streamline. The difference in pressure between the points is called the recoverable pressure difference because we can get the original pressure back by simply restoring the original velocity and elevation. Any viscous losses, like friction, cannot be predicted with the Bernoulli equation - these are unrecoverable , irreversible losses. Pitot-Static Tubes Recoverable pressure differences can be used to measure fluid velocity. The measurement of velocity by a Pitot-static probe is based on the stagnation of the momentum of fluid in the moving stream to a zero-velocity pressure force at the Pitot-static probeinlet, a relationship that can be derived from the Bernoulli equation when v 1 = v and v 2 (at the probe entrance) goes to zero: 2 2 v p p p fluid dynamic static stagnation w here P stagnation is the total pressure at the forward facing inlet to the Pitot-static probe where the velocity becomes zero, P static is the static pressure along the sides of the Pitot-static probe where the velocity is unchanged from the upstream duct velocity v . The pressure difference, P, is called the dynamic pressure because it is related to the change in fluid velocity. We can calculate the duct velocity from the dynamic pressure as, air P 2 = v Note that this expression is only accurate if the P-S tube points directly into v 1 such that all of v 1 is stagnated. If the P-S tube is misaligned, the measured velocity will be too low.
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ME 4600:483 Lab Notes Revised 11/16/2015 Flow Measurement Page 3 of 18 To obtain an estimate of the volumetric flow in the duct from a series of pitot-static tube velocity measurements, one must integrate the velocity over the duct area. dA v = A v = Q A AVG There are a number of different methods for approximating the above integral. The simplest method is to divide the duct cross-section into a number of equal area sectors, and measure the "average" velocity at the center of each sectors. We can then estimate the velocity by calculating the sum: avg pipe Numsectors i i pipe Numsectors i i i v A Numsectors v A A v = Q * * 1 1 The above method only works if the positions of the velocity measurements are carefully chosen. Figure 4 shows how to split the pipe into 6, 12 or 24 equal area sectors. The specific radial positions are given in the figure. The dynamic pressure, P, can be measured using capacitance-based differential pressure (DP) cells or manometers. A manometer relates the pressure difference to the difference in height of two columns of liquid supported by the respective pressures. The equations of hydrostatics tell us
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