In a previous Fireside Chat, Orifice Flow Calculation Basis, we provided information on the flow of compressible fluids through sharp-edged orifices.  While that topic was primarily geared toward the flow of gases through an orifice (for vapor blowdowns), we had briefly mentioned flow of a two-phase fluid through an orifice and provided a reference to Diener and Schmidt¹ (DS Method).  We’ve encountered problems implementing this method for flashing liquids (liquids having a saturation or bubblepoint pressure less than the inlet pressure yet greater than the outlet pressure) and wanted to provide an update.

With the flow of two-phase or flashing liquid through an orifice, there are significant non-equilibrium effects, mostly attributed to the time scale associated with vapor nucleation relative to the flow across the restriction.  For orifices, in which there isn’t much of a run up distance for the acceleration of the incoming fluid, this results in essentially frozen flow.  For us old-timers, we can recall the use of the frozen flow orifice calculations in the API RP 520 Part 1 5th Edition for two-phase flow or flashing liquid flow.  The reason those equations were used is that the results actually fit the orifice flow data well – it really wasn’t until DIERS came along and tested nozzles instead of orifices that any significant discrepancy was widely recognized.²  The data for nozzles showed a substantially lower flow rate than what was being predicted by frozen flow, which prompted the implementation of the homogeneous equilibrium method for nozzle flow.  This lower flow rate has been attributed to the occurrence of gas formation within the nozzle, especially for nozzles greater than 10 cm (4”) in length.³  It is interesting to note that the data for lowly subcooled or low quality two-phase flow show that there is still likely some thermal non-equilibrium (boiling delay) happening in these cases.^{3, 4}

While working on orifice flow calculations for flashing liquid flow, we ran into difficulties matching the calculation results from the DS Method to experimental results from Richardson, et al,⁵ prompting us to reconsider the use of the DS Method.  Various references state that subcooled flashing liquids flowing through orifices essentially behave like nonflashing liquids, but for us there has always been a question about the effective pressure differential.  Using the data of Richardson, et al, we have convinced ourselves that the effective pressure differential is based on the pressure downstream of the orifice (as opposed to the saturation pressure or some other effective pressure, which is what we might use for highly subcooled liquids flowing through a nozzle), regardless of the degree of subcooling of the liquid.  In the figure below, we plot the flow data of Richardson for subcooled flashing liquid cases versus the saturated pressure ratio.  To generate a useful two-axis plot, we use the dimensionless parameters of Leung⁶ for the normalized mass flux and saturation pressure ratio.  The data tables of Richardson provided fluid composition, inlet pressure (P₀), inlet temperatures, orifice diameters, and measured flow rates.  The NIST REFPROP program⁷ was used to generate the inlet densities (ρ₀) and saturation pressures (P_sat).

Normalized mass flux versus saturation pressure ratio

With this basis, we can obtain flow rates that are very close to the experimental results from Richardson, et al.  The average discharge coefficient for subcooled flashing liquid flow from Richardson is 0.594, with a standard deviation of 0.01, which corresponds to the average ASME PTC 19.5-2004 liquid orifice discharge coefficient of 0.596.  A comparison of our calculation results to the experimental data is seen in the figure below.

Calculated flow versus experimentally measured flow

Incidentally, the work of Richardson et al was performed for the IP (now Energy Institute) Guidelines for the Safe and Optimum Design of Hydrocarbon Pressure Relief and Blowdown Systems.⁸

In summary, for subcooled flashing liquids, in which the vapor pressure is greater than the back pressure but less than the incoming pressure, we use the liquid orifice flow calculation with the following parameters, regardless of the degree of subcooling:

• Flow basis = liquid orifice flow ^9,10
• Upstream pressure = incoming pressure
• Upstream fluid density = density at the saturation pressure
• Downstream pressure = actual back pressure
• Discharge coefficient = liquid discharge coefficient