Multiphase Flow Metering:An Evaluation of Discharge Coefficients
Authors: Ajienka J.A. , Enaibe O.E. , Owolabi O.O. 
Published: March 2010

Abstract
The orifice discharge coefficient (CD) is the constant required to correct theoretical flow rate to actual flow rate. It is known that single phase orifice models and methods of prediction of critical flow do not apply to multiphase flow.
Thus the questions that must be answered are how do we determine values of discharge coefficient for multiphase flow metering? Can the values of discharge coefficient for critical multiphase flow be used for subcritical flow?
Figures, tables and equations of CD are presented with which metered multiphase flow rates can be corrected to obtain actual multiphase flow rates for both critical and subcritical flows. It is shown that CD for multiphase critical flow cannot be the same for multiphase subcritical flow.

Introduction
Multiphase flow is a complex phenomenon: As a result the majority of published correlations are highly empirical. This affects the general validity of these correlations for all ranges of fluid properties as they are limited to the quality and scope of the data base from which they are developed. Therefore, correlation which performs well within the range of data used to develop it, may fail outside this range.
Multiphase flow through restrictions is usually evaluated under critical or subcritical flow conditions. As a standard oilfield practice, wellhead flow performance is evaluated under critical flow while flow performance through subsurface chokes and safety valves is done through subcritical flow. Critical or sonic flow is flow in which downstream pressure and temperature perturbations are not transmitted upstream to affect the flow rate unlike in subcritical flow.
Available critical multiphase orifice flow correlations can be categorized as follows(1):Analytical models, applying mathematical analysis based on fundamental principles, to a simplified physical model with the development of equations.Empirical correlations using dimensional analysis to select and group the most important variables.Empirical correlations from field or laboratory data.
Examples of category 1 correlations are those of Tangren et al.(2), Ros(3), Poettmann and Beck(4), Ashford(5), a generalized model by Ajienka(6) and Ajienka and Ikoku(7). The simplified form of the generalized model applicable to both continuous liquid phase flow and continuous gas phase flow is given by Equation (1):
Equation (1) (Available In Full Paper)
where
Equation (2) (Available In Full Paper)
Equation (3) (Available In Full Paper)
Equation (4) (Available In Full Paper)
Letting
Equation (Available In Full Paper)
To be equal to X, then:
Equation (5) (Available In Full Paper)
Equation (6) (Available In Full Paper)
Flow is critical if the pressure ratio (X = Xc) is equal to the critical pressure ratio. Otherwise, flow is subcritical.
Ashford's analytical correlation for critical flow is given by:
Equation (7) (Available In Full Paper)
where
Equation (8) (Available In Full Paper)
Equation (9) (Available In Full Paper)
Equation (10) (Available In Full Paper)
While the earlier analytical models assume that critical flow
TABLE 1: Empirical coefficients of category 3 correlations. (Available in full paper)
Occurs at a constant pressure ratio of 0.554 (for k = 1.04) as with single phase flow, the Ajienka and Ikoku model uses a predicted critical pressure ratio which is realist