Discharge Coefficient Loss Model
Exit orifice subsonic losses for the Relief Valve and Orifice (when using the Cd (Exit) loss model) junctions can be defined with a discharge coefficient, Cd.
When determining the pressure loss across the junction, xStream calculates a subsonic discharge coefficient area (CdA) for the orifice and applies the following set of isentropic compressible flow equations to solve for the static pressure and Mach number at the vena contracta
Where ṁ̇ represents the mass flow rate, To is the stagnation temperature, Po denotes the stagnation pressure, M is the Mach number, γ is the specific heat ratio, R is the gas constant, Z is the gas compressibility factor, and P is the static pressure. Note that the specific gas constant, R, is defined as the universal gas constant divided by the molecular weight of the gas.
It is assumed that the stagnation pressure at the throat corresponds to the upstream stagnation pressure, while the static pressure at the restriction is taken as the downstream stagnation pressure. This assumption disregards static pressure recovery, which could be a significant factor.
The following equation converts this loss to an equivalent K factor.
Subsonic Loss - Perry's (Inline)
Subsonic losses are defined for the Venturi and Orifice (when using the Cd (Inline) loss model) junctions using the discharge coefficient, Cd. This model differs from the exit discharge coefficient model in that it includes the effect of dynamic pressure recovery that occurs in the downstream pipe, making it more appropriate for inline junctions. The mass flow through the junction is calculated via the following equation: (Perry, 10-14 (1997)Perry, R. H., Green, D. W., Perry's Chemical Engineers' Handbook, 7th Edition, McGraw-Hill, New York, NY, 1997.)
Where ṁ is the mass flow rate, Y is the expansibility factor, A is the throat area of the orifice, venturi, or nozzle, P1 is the inlet pressure, P2 is essentially the pressure at the vena contracta, gc is the unit conversion constant, ρ1 is the inlet density, and β is the diameter ratio of the throat relative to the upstream pipe. While this is equivalent to the flow calculation in ASME MFC-3M-2004, the two approaches differ in calculation of expansibility factors and in the conversion from the differential pressure, P1 - P2, to the irreversible pressure loss.
Note: The differential pressure loss P1 - P2 is not equivalent to the change in static pressure across the junction. The static pressure at the throat, P2, is not equal to the static pressure at the inlet of the downstream pipe, which assumes fully developed flow and depends on the pipe area.
Expansibility factors, which can be viewed as a junction Output Parameter, are calculated in terms of the pressure ratio r and specific heat ratio γ.
The first expansibility factor equation is applicable to orifices and the second is applicable to nozzles and venturis.
As previously noted, the differential pressure is the pressure difference between the upstream pressure and the vena contracta, and does not account for the recovery of the dynamic pressure that occurs in the downstream pipe. Because the flow is assumed to be fully developed at the inlet of the downstream pipe, this recovery must be incorporated at the junction itself by converting the differential pressure loss to the irreversible pressure loss, which is treated as the stagnation pressure loss. This conversion is as follow for orifices:
Venturis and flow nozzles use the following conversion to stagnation pressure loss:
Perry's Chemical Engineer's Handbook defines sonic choking based on the critical pressure ratio, rc.
Note: The sonic loss model described below is used regardless of the subsonic loss model selected, therefore sonic choking in xStream is not defined by this pressure ratio. However, this caution message will be displayed if the junction is not considered sonically choked by xStream but would be according to the equation above.
A sonic CdA value will be assumed from the subsonic Cd and throat area when using this subsonic discharge coefficient loss model if no sonic CdA value is specified. This is to avoid issues with expansibility factor convergence. For that loss model, the sonic CdA and subsonic CdA should generally be equal.
The above equation for the mass flowrate can be simplified when the orifice is sonically choked. When the Mach number at the restriction is 1, the equation can be simplified as shown below. The value of CdA used here will be the CdA for sonic choking, which is entered as a separate value from the discharge coefficient, Cd, as is discussed in the Subsonic vs. Sonic Pressure Losses topic.
Note: The CdA for sonic choking may be different from the subsonic CdA loss model option in xStream. The discharge coefficient can vary at different pressure ratios due to the vena contracta moving closer to or farther from the orifice restriction. For the highest accuracy the CdA used for subsonic and sonic losses should be tested and entered separately. See the "Modeling Choked Flow Through an Orifice" white paper on AFT's website for more information.
Note that output K factor values may not be meaningful for sonically choked junctions, but sonic choking will be reflected in pressure loss and flow results.
Transient Theory
The above equations for the mass flow rate through an orifice are applied in the transient simulation similar to the approach used for other loss models. Inlet conditions are guessed at, a mass and energy balance is applied to find the outlet conditions, and then those conditions are used to determine the mass flow through the orifice for that guess. Iteration proceeds until the guesses for inlet conditions are consistent with the solved mass flow equation.
