A subcooled yet flashing liquid, in which the vapor pressure is less than the relief pressure yet greater than the superimposed backpressure, represents an occasional non-ideal sizing condition, as the flow is neither fully liquid (which the liquid sizing equations are geared towards), nor fully two-phase – there is a transition between liquid and two-phase flow as the pressure of the fluid is decreased along the relief path.  Dr. Leung makes a useful distinction based on the location of that transition – either within the nozzle of the relief valve itself (lowly subcooled) or at the nozzle exit plane or further downstream (highly subcooled).^1,2

A transition pressure ratio based on the saturation pressure and compressibility (omega parameter) of the fluid is used to quantify this transition, and two separate calculation methodologies are employed depending on the situation.  For cases in which the flashing occurs downstream of the nozzle, and the total backpressure is less than the saturation pressure, the flow through the nozzle is a liquid but is also choked, with the choking pressure at the saturation pressure; therefore, the sizing equation looks like the liquid sizing equation, but with the effective downstream pressure as the saturation pressure instead of the total backpressure.  Since the flow is choked, and downstream effects do not impact the flow through the nozzle, we may select the compressible discharge coefficient (see our blog, discharge coefficient for non-ideal cases), although API Standard 520 Part I §C.2.3.1 indicates a preliminary discharge coefficient for design purposes may be the incompressible discharge coefficient³ and Dr. Leung would advocate a discharge coefficient as a function of the omega parameter and absolute back pressure ratio⁴.

An interesting problem can occur in these cases with significant differences in the discharge coefficient selected.  A typical next step in the calculation after the sizing is to design or evaluate the discharge piping based on the capacity of the relief valve.  Based on the initial assumption outlined above, the flow in the discharge piping is two-phase.  Arguments about two-phase pipe flow calculation methodology aside, there can be a situation in which the pressure drop in the discharge piping results in the total backpressure exceeding the saturation pressure, which results in liquid flow downstream of the relief valve nozzle exit.  One would then go back to the sizing calculation to update the total backpressure, perhaps employ a lower discharge coefficient to reflect the all liquid flow through the relief valve, perhaps also employ a non-certified liquid flow correction factor, and recalculate the capacity of the relief valve.  The capacity of the relief valve is thus lower (lower correction factors and lower pressure differential available across the nozzle).  A new pressure drop calculation in the discharge piping at the lower capacity may then find that the total backpressure is less than the saturation pressure, setting up a never-ending iteration loop.

In many cases, the difference in the two conditions is trivial, and selection of the case with the lowest capacity is sufficient for the design basis; however, for those cases where the different conditions have a material impact on design, we have a decision to make.  We can either report both results as boundary conditions; or we can modify the evaluation heuristic to employ a technique, such as that outlined in [4], that eliminates the discontinuity.  This decision should balance the time required for implementing alternative techniques (and justifying them) against the criticality of the result.

To illustrate a non-trivial case, we present a situation we recently encountered.  A 1½-G-2½ relief valve certified for vapor flow (but not liquid flow) was sized for highly subcooled liquid flow, with a saturation pressure of 70.4 psig.  The iteration calculation yielded the following results, with the first iteration having elevated backpressure issues, the second iteration having capacity issues:

a:  Pressure specified at the throat (exit plane) of the nozzle
b:  Total backpressure calculated in the discharge piping based on device capacity

Employing the technique in [4], an effective K_d of 0.82 was determined and used in the final result: