Orifice Flow for Two-Phase Fluids – Part 1
We have continued our effort to verify a calculation model for use with determining the flow rates of two-phase fluid through orifices based on the data of Richardson, et al.1 In that work, they have indicated that the homogeneous equilibrium method (HEM) used for two-phase nozzle flow produces the most accurate results for measured flow rates for two-phase orifice flow involving highly volatile mixtures. These mixtures include well-mixed natural gas and propane; poorly-mixed natural gas and propane; and well-mixed natural gas, propane, and condensate (Tables 3 – 5, respectively).1
Mass flux integration. For integration of the mass flux of flow through a nozzle, Richardson, et al. used the enthalpy-based equation1,2; however, we typically use the more common density-based equation3,4. We have used the density-based mass flux integration to calculate the homogeneous equilibrium mass flux for two-phase flow based on the Richardson data, and obtain flow rates that are very close to the experimental results. A comparison of our calculation results to the experimental data is seen in the figure below.
The mass flux integration was evaluated numerically using pressure-density data from isentropic flashes generated using the NIST REFPROP program5. For the numerical integration, trapezoidal integration across constant pressure steps over the full pressure range from the reported inlet stagnation pressure to atmospheric pressure was used. The point at which the integration was maximized was taken as the estimated throat pressure, and the integration was re-evaluated from the inlet pressure to a pressure moderately below the throat pressure (in effect, increasing the precision of the numerical integration by decreasing the step size used). Not all runs could be modeled using REFPROP due to convergence failures for some isentropic flashes, particularly for cases near the thermodynamic critical point; nonetheless, over 100 runs were modeled. In some cases, a few isentropic flashes were performed, then fit to a property model as suggested by Simpson6.
Discharge coefficient. While we used the calculated discharge coefficient for each run in reporting the results above, we need to establish a generic discharge coefficient to use for estimating purposes. Excluding the runs with liquid fractions greater than 0.8, Richardson et al. show a slight dependency of the discharge coefficient with the liquid mass fraction within the throat of the orifice. Using the liquid mass fraction at the throat is a bit inconvenient as it requires additional computations after evaluating the mass flux. Looking at the data, the average discharge coefficient is 0.947, with a standard deviation of 0.026. For most cases, we would simply use the average discharge coefficient as greater precision is not warranted given our typical estimates.
For cases where a little more precision is warranted, a simple linear regression fit can be used; where the discharge coefficient, Kd, can be expressed as a function of the inlet quality, xinlet:
In Orifice Flow for Two-Phase Fluids – Part 2, we’ll take a look at how to deal with the cases having an inlet quality of 0.0 – 0.2 (or as Richardson, et al. report, liquid fractions of 0.8 – 1.0).
 Richardson SM, Saville G, Fisher A, Meredith AJ, Dix MJ. “Experimental Determination of Two-Phase Flow Rates of Hydrocarbons Through Restrictions”. Trans IChemE, Part B, Process Safety and Environmental Protection, January 2006; 84(B1): 40-53.  Institute of Petroleum. Guidelines for the Safe and Optimum Design of Hydrocarbon Pressure Relief and Blowdown Systems. 2001; London.  Darby R. Chemical Engineering Fluid Mechanics. 2001; Boca Raton: CRC Press.  API Standard 520 – Part I:2014. Sizing, Selection, and Installation of Pressure-Relieving Devices in Refineries. Part I – Sizing and Selection, 9th Edition. American Petroleum Institute, Washington, DC.  Lemmon, E.W., Huber, M.L., McLinden, M.O. NIST Standard Reference Database 23: Reference Fluid Thermodynamic and Transport Properties-REFPROP, Version 9.1, National Institute of Standards and Technology, Standard Reference Data Program, Gaithersburg, 2013  Simpson, L.L. “Navigating the Two-Phase Maze.” 394-417. In Proceedings of the International Symposium on Runaway Reactions and Pressure Relief Design, Boston, MA, August 2-4, 2005; AIChE/DIERS, New York, NY.