Relief Device Discharge Coefficient for Non-ideal Flow
Manufacturers of pressure relief valves are required to publish capacity data for their valves, and quite often this is translated into an effective coefficient of discharge (Kd). These discharge coefficients are typically published for water (as a non-flashing incompressible liquid), steam and/or air (as an ideal gas). The choice of discharge coefficient to use in the case of non-ideal fluid flow, such as for two-phase or supercritical fluids is not obvious if the relief device manufacturer does not provide specific guidance (and most do not).
There is a debate about the discharge coefficient to use, and it has been a challenge in proving due to a lack of test data and the complexity of fluid flow in non-ideal regions. Two hypotheses were published in the Journal of Loss Prevention in the Process Industries:
- Leung, “A theory on the discharge coefficient for safety relief valve”, J. Loss Prev. in Proc. Ind., 17 (2004), pp 301-313.
- Darby, “On two-phase frozen and flashing flows in safety relief values – Recommended calculation method and the proper use of the discharge coefficient”, J. Loss Prev. in Proc. Ind., 17 (2004), pp 255-259.
Dr. Leung’s hypothesis is that the discharge coefficient is based on the compressibility of the fluid; the discharge coefficient of a fluid having an omega factor (used as a correlating factor that represents the compressibility of the fluid) of 0 would be that of a liquid, and would approach that of an ideal gas near an omega factor of 1. On the other hand, Dr. Darby’s hypothesis is that the discharge coefficient is based on whether the fluid actually chokes in the nozzle, irrespective of the compressibility of the fluid; however, noting that the more compressible the fluid, the more likely the application is to result in choking behavior.
The discharge coefficient was introduced as a correction for the actual non-isentropic behavior compared to the ideal isentropic nozzle flow. Other corrections may be needed (for example, the effect of elevated backpressure on the device capacity) and there are added complications for multi-phase fluids (for example, the degree of thermal and mechanical equilibrium of the phases is needed), but the intent of the discharge coefficient is the correction for frictional and velocity profile effects1.
For choked fluids, this deviation is limited to effects in the nozzle itself while for non-choked fluids the deviation includes all of the effects downstream of the nozzle. As a result, Darby’s hypothesis makes more sense to us, and in the absence of specific guidance we would use the manufacturer’s certified vapor Kd value for cases where the two-phase or supercritical fluid chokes in the nozzle, and the certified liquid Kd value for cases where it does not choke.
As a side note, this is substantially incorporated into API Std 520 Part I 9th Edition §B.1.4.2:
“Note that when using the direct numerical integration technique, the choice of the applicable correction factors should be made based on the nature of the relieving fluid. For fluids that behave as incompressible fluids or those that do not choke within the PRV itself, correction factors pertaining to liquid and/or subcooled flashing liquid service have been used. For fluids that behave as compressible fluids or those that do choke within the PRV itself, correction factors pertaining to vapor and/or two-phase service have been used   .”
 Huff JE, Pressure Relief System Flow: Results of the DIERS Phase II Projects. In Fisher HG, Forrest HS, Grossel SS, Huff JE, et. al. Emergency Relief System Design Using DIERS Technology – the Design Institute for Emergency Relief Systems (DIERS) Project Manual. AIChE 1992. New York; p. 75.