15. Equation Coefficients

The previous examples show that coefficients are used in water measurement to correct for factors which are not fully accounted for using simplifying assumptions during derivations of equations. For the convenience of using a measured water head, h1, instead of the more complex total head, H1, Cv is used because velocity head is often ignored in equations.

Orifices require an area correction to account for jet contraction in an orifice, the flow is forced to curve around and spring from the sharp edge, forming a contracted jet or vena contracta. Thus, the contracted area of flow, Ac, should be used in hydraulic relationships. Thus, the area, Ao, of the orifice must be corrected by a coefficient of contraction defined as:

eqn (2-50)

Properly designed venturi meters and nozzles have no contraction, which makes Cc unity because of the smooth transitions that allow the water to flow parallel to the meter boundary surfaces. Ultimately, the actual discharge must be measured experimentally by calibration tests, and the theoretical discharge must be corrected. A common misconception is that coefficients are constant. They may indeed be constant for a range of discharge, which is the case for many standard measuring devices. Complying with structural and operational limits for standard devices will prevent measurement error caused by using coefficients outside of the proper ranges. Some water measuring devices cover wider ranges using variable coefficients of discharge by means of plots and tables of values with respect to head and geometry parameters.

Coefficients also vary with measuring station head or pressure tap location. Therefore, users should make sure that the coefficients used match pressure or head measurement locations. Water measurement equations generally require use of some to all of these coefficients to produce accurate results.

Often, composite numerical coefficients are given that are product combinations of area or a dimension factored from the area, acceleration of gravity, integration constants, and the correction coefficients. However, geometry dimensions and physical constants, such as acceleration of gravity, are better kept separate from the nondimensional coefficients that account for the difference between theoretical and actual conditions. Otherwise, converting equations from English to metric units is more difficult.

Equation 2-49 also applies to orifices and nozzles. The coefficient of discharge for venturi meters ranges from 0.9 to about unity in the turbulent flow range and varies with the diameter ratio of throat to pipe. The coefficient of discharge for orifices in pipes varies from 0.60 to 0.80 and varies with the diameter ratio. For flow nozzles in pipelines, the coefficient varies from 0.96 to 1.2 for turbulent flow and varies with the diameter ratio. ASME (1983) and ISO (1991) have a detailed treatment of pipeline meter theory, coefficients, and instruction in their use.