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 equation^1,2; however, we typically use the more common density-based equation^3,4. 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, K_d, can be expressed as a function of the inlet quality, x_inlet:

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).