Orifice subsonic losses for the Orifice, Relief Valve and Venturi junctions can be defined with a discharge coefficient, C_d.

Subsonic Loss

When solving the system, AFT xStream creates a subsonic C_dA for the orifice using the C_d specified for the subsonic loss model, and the specified orifice area. xStream then solves the pressure drop by iterating the following equation for pressure drop for a given mass flow rate. The equation below for mass flow rate can be derived from isentropic relationships, replacing the static pressure at the restriction with the downstream stagnation pressure, Po,out.

The assumption that the static pressure at the restriction can be replaced with the downstream stagnation pressure, Po,out, is necessary to find a solution, as the velocity must be known to find the static pressure, and the mass flow must be found before the velocity can be a known quantity. It is important to note that this assumption neglects static pressure recovery, which could be a significant 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 C_dA used here will be the C_dA for sonic choking, which is entered as a separate value from the discharge coefficient, C_d, as is discussed in the Subsonic vs. Sonic Pressure Losses topic.

Note: The C_dA for sonic choking may be different from the subsonic C_dA 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 C_dA 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.