Tuesday, June 23, 2026

For excess inflow into a system, the flow rate entering the system from a higher pressure source can be determined based on the flow limiting element between the high pressure source and the system of interest. These flow limiting elements may be a specific piping element (e.g. orifice or control valve) between the high pressure reservoir and the container to be protected, a source term that can be modeled as a piping element (e.g. orifice), the limitations of a fluid driver, or the entire piping system. The discussion below is for the restriction orifice as the flow limiting element.

Flow through an orifice. There are some situations where the excess flow entering the container can be modeled using flow through an orifice. The most obvious example is when a restriction orifice is present in the line connecting a high pressure reservoir to the container. Another example is at the location of a heat exchanger tube failure, where an orifice can be used to model the flow through one or both sides of the broken tube. In these cases, the common estimation technique is to ignore the effects of the upstream and downstream piping on the pressures at the orifice as well as any potential energy effects within the orifice itself, and to determine the excess flow entering the container using the following equation:^{2 [Equation 10-20], 3 [Equation 3-1.1]}

Where W is the mass flow rate, Q is the volumetric flow rate, ρ is the density, C_d is the discharge coefficient, Y is the expansion factor, P is the pressure, β is the ratio of the orifice diameter to the inlet pipe diameter, and the subscripts HPS and x represent high pressure side (inlet) and throat (orifice outlet) locations, respectively.

The form shown is based on Bernoulli flow of an incompressible fluid⁴, with the discharge coefficient C_d and expansion factor Y correcting that basis to actual single-phase flow; see Orifice flow calculation basis. The work of Richardson et al⁵ indicates the homogeneous equilibrium method is a more appropriate basis for two-phase flow through an orifice (see also Part 2).

Coefficient of discharge. The coefficient of discharge (C_d) is a function of the geometry of the orifice, the diameter ratio (β), and the Reynolds Number. At the high Reynolds Numbers typically encountered, the coefficient of discharge is essentially constant between approximately 0.6 and 0.8. The ASME PTC 19.5-2004 provides the following equation for the coefficient of discharge for orifices designed for use in flow meters (corner taps) with incompressible fluids:^{3 [Equation 4-8.7]}

Where R_D is the Reynolds Number at the inlet pipe diameter.

For liquids, this coefficient of discharge is approximately 0.62 for highly turbulent flows typically encountered.

Expansion factor. For incompressible fluids, the expansion factor is equal to one, and the pressure at the throat is equal to the downstream pressure. For the simplification of ignoring the effects of the downstream piping, the throat pressure is equivalent to the downstream relief pressure. Note that flashing liquids may exhibit choking behavior due to flashing; however, the work of Richardson et al⁵ indicates the degree of subcooling has very little effect on the flow of a flashing liquid through an orifice.

For gases, choking effects through the orifice (at the vena contracta) should be accounted for. For cases where the choking pressure is greater than the downstream relief pressure, that choking pressure is used as the throat pressure. In addition, the expansion factor is generally less than one. For nonchoked (subcritical) flow of an ideal gas having an isentropic expansion coefficient, γ, Perry’s Eq. (10-29)² indicates the expansion factor for orifices is approximated by:

At choked (critical) conditions, the flow rate can increase further as the downstream pressure falls below the choking pressure, because the vena contracta moves downstream and enlarges⁶. The higher discharge coefficient used for compressible fluids gives a conservative estimate, though further evaluation may be warranted; see Orifice flow calculation basis for the supercritical-flow background and the detailed methods of Cunningham⁶ and Benedict^7,8.

Blog series information. This blog is part of a series on the proposed updates to the CCPS Guidelines 2nd edition §3.3 Venting Requirements for Nonreacting Cases that were removed during final editing. See the general CCPS Guidelines for Pressure Relief and Effluent Handling 2nd Edition review for more information.

 

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[1] AIChE Center for Chemical Process Safety. “CCPS Guidelines for Pressure Relief and Effluent Handling Systems”. 2nd Edition, 2017; New Jersey: John Wiley & Sons, Inc.

[2] Boyce MP. Transport and Storage of Fluids – Head Meters. In Perry RH and Green DW. Perry’s Chemical Engineers’ Handbook. 1997; New York, McGraw Hill: 10.11-10-18.

[3] ASME PTC 19.5-2004 “Flow Measurement, Performance Test Codes”

[4] Darby R. Chemical Engineering Fluid Mechanics. 2001; Boca Raton: CRC Press.

[5] Richardson SM, Saville G, Fisher A, Meredith AJ, Dix MJ. Experimental Determination of Two-Phase Flow Rates of Hydrocarbons Through Restrictions. Trans IChemE, Part B, Process Safety and Environmental Protection. January 2006; 84(B1): 40-53.

[6] Cunningham RG. Orifice Meters with Supercritical Compressible Flow. Transactions of the ASME, July 1951 pg 625-638.

[7] Benedict RP. Generalized Contraction Coefficient of an Orifice for Subsonic and Supercritical Flows. Journal of Basic Engineering, July 1971 pg 99-120.

[8] Benedict RP. Generalized Expansion Factor of an Orifice for Subsonic and Supercritical Flows. Journal of Basic Engineering, June 1971 pg 121-136.

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