Elbow Flow Meters
Clifford A. Pugh, Hydraulic Investigations and Laboratory Services Group
Elbow meters are based on the principle of "conservation of momentum." Momentum conservation requires that the momentum flux (momentum per unit time) remain unchanged as steady flow occurs through an isolated system of fluid. Since momentum is a vector quantity, a change in direction of the flow causes a reduction of momentum in the original direction which is offset by an increase in the new direction. In an elbow, such as the mitred elbow shown in figure 1, the momentum in the horizontal direction is changed by the pipe turning down. This change in direction causes the flow to exert a force on the pipe elbow.
F = Q (V2-V1) [Momentum Equation] (1)
F = Force
= the fluid density
Q = the discharge (flow)
V = the velocity vector
Figure 1 - Forty-five degree mitred bend with pressure taps on the inside and outside of the bend. The pressure differential is related to the square of the velocity.
This force results in an increased pressure on the outside of the bend and a decreased pressure on the inside. The pressure difference is proportional to the square of the velocity. The general form of the equation would be :
Figure 1 shows recommended pressure tap locations for a mitred bend according to ID Tech, inc. ID Tech sells an "Electronic Flow Calculator" based on an elbow meter. The coefficents of discharge (Cd) for mitred bends (determined empirically by ID Tech) are proprietary. Their toll free phone number is 888-782-0498.
Figure 2 shows a multiple level outlet at Beltzville Dam in Pennsylvania. Differential pressures across opposing pressure taps (P1 and P2) and stream gage measurements were used to develop the rating curve and equation shown in figure 3 (Hart and Pugh, 1975).
Figure 2 - Water Quality Control outlet at Beltzville Dam (Hart and Pugh, 1975). The pressure differential between P1 and P2 was empirically calibrated to obtain a discharge relationship. The elbow meter is used to set desired outlet flows for normal operations.
Figure 3 - Elbow Meter Calibration, Beltzville Dam.
Similar empirical relationships could be developed for pipe bends in the field by using a strap on acoustic flowmeter (or another flow measurement method) to obtain the data and develop the equation. One differential pressure transducer would be connected to the high and low pressure taps to measure the differential and obtain flow. This is a simple and relatively accurate device if the pressure taps are properly installed and the "burrs" are cleaned from the inside of the tap. A slight rounding of the edge of the taps helps to improve their performance.
As a practical matter, the lower limit of an elbow meter is about 2 ft/s. Pressure differences and discharge measurement accuracy are very low below this velocity. Flow Tech recommends an upper velocity limit of 10 ft/s, this is probably due to seperation at the sharp bend in the mitred elbow. Higher velocities may be allowable for elbows with a constant radius such as the example in figure 2 and 3.
(1) Hart, E. D., and Pugh, C. A. , "Outlet Works for Beltzville Dam, Pohopoco Creek, Pennsylvania," Technical Report H-75-10, U. S. Army Corps of Engineers, Waterways Experiment Station, May, 1975.