Discharge Coefficient Loss Model
Subsonic Loss
Orifice subsonic losses for the Orifice, Relief Valve and Venturi junctions can be defined with a discharge coefficient, Cd.
When determining the pressure loss across the junction, AFT 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.
Sonic Loss
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.
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.
