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 control valve as the flow limiting element.

Control valve sizing. Similar to a restriction orifice, a control valve or pressure regulator may be present in the line that connects a high pressure reservoir to the container. Again, the common estimation technique is to ignore the effects of the upstream and downstream piping on the pressures at the control valve as well as any potential energy effects within the valve itself, and to determine the excess flow entering the system using the control valve sizing equations. The manufacturer’s literature should be referenced for the appropriate control valve sizing methodologies and sizing characteristics at the anticipated opening, noting that the maximum sizing flow coefficient for the full-open condition may not be the same as the flow coefficient at the maximum of the control range, which is typically reported. Many manufacturers use the ANSI/ISA Standard 75.01.01-2007 (IEC 60534-2-1 Mod) control valve sizing methodologies, making this a convenient basis for new designs as well.¹

The sizing equations relate the valve flow coefficient C (commonly reported as C_v in US customary units or K_v in SI units) to the flow rate and the service conditions. For an excess-inflow estimate the coefficient is known, or bounded, from the manufacturer’s data and the equations are rearranged to solve for the flow rate. The form of the applicable equation depends on whether the fluid is a liquid or a compressible vapor, and on whether the flow is choked.

Liquid. For a liquid, the flow is incompressible and the controlling distinction is whether the liquid’s degree of subcooling relative to the pressure differential is enough to flash the liquid at the vena contracta, that is, to choke the flow through valve. The standard provides equations for both cases.

Vapor. For a gas or vapor the fluid is compressible, and the potential to choke the flow through the valve is even more likely. Using the ideal gas assumptions, the standard provides equations for both cases. Because the choked mass flow scales with the square root of P₁ρ₁, the result is sensitive to the inlet density; the manner in which that density is evaluated is the subject of the caution below.

Non-ideal (high pressure) gas behavior. The compressible-flow equations in the standard infer the gas density from the inlet pressure, temperature, and molecular mass through the ideal gas law, and they characterize the isentropic expansion through the valve using the ideal-gas specific heat ratio γ (by way of F_γ and x_T, which are referenced to air at γ = 1.40). A gas at high pressure can exhibit non-ideal behavior such that its thermodynamic properties cannot be evaluated by assuming the ideal gas law without significant error.²

The first correction is to the density. The standard provides the compressibility factor Z for this purpose: where real-gas behavior is appreciable, the ideal density is divided by Z, evaluated as a function of the reduced pressure and reduced temperature. In the mass-flow forms above this enters through ρ₁. The compressibility factor corrects the inferred density but it does not correct the isentropic expansion process itself.²

The second correction is therefore to the expansion process. At elevated pressure the real-gas isentropic exponent departs from the ideal-gas specific heat ratio on which F_γ, x_T, and the expansion factor Y are based, so that even a density-corrected calculation can incorrectly estimate the expansion factor and the location of the choke point. An enhanced sizing method that replaces the specific heat ratio with the isentropic exponent appropriate to the real gas, applied together with the compressibility factor, addresses this effect.³ The error introduced by ignoring these corrections is negligible at low values of ΔP/P₁ and low reduced pressure, but it grows as the inlet pressure becomes a significant fraction of the critical pressure, and particular caution is warranted for gases near their critical point or near a phase boundary.

Two-Phase. The standard is sufficiently accurate only for single-phase flow, and is unsuitable for conditions involving gas/liquid mixtures at the inlet to the control valve; this standard may calculate highly underestimated mass flow rates in that event. Although there is currently no applicable international standard for two-phase flow through control valves, some manufacturers publish correlations for sizing of their control valves for two-phase flow.

Attached fittings and non-turbulent flow. The typical forms of the equations assume a line-sized control valve with no attached reducers or expanders and fully turbulent flow. Where inlet or outlet fittings are present, the standard introduces a piping geometry factor F_P that modestly reduces capacity. Where the pressure differential is very small, the viscosity high, or the flow coefficient very small, the flow may be laminar or transitional (valve Reynolds number below 10,000), in which case the Reynolds number factor F_R applies. Neither factor typically governs an excess-inflow estimate, but both are given in full in the standard.

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.

[1] ANSI/ISA-75.01.01 (IEC 60534-2-1 Mod)-2007. “Flow Equations for Sizing Control Valves”. 2007; Research Triangle Park, NC: ISA.
[2] Fagerlund AC and Winkler RJ. “The Effects of Non-Ideal Gases on Valve Sizing”. In ISBN 9780876647028 – Advances in Instrumentation, Proceedings of the ISA International Conference and Exhibit, Philadelphia, Pa., Oct. 1982: 1323-1333.
[3] Riveland ML. “Enhanced Valve Sizing Methods for Fluids Exhibiting Real Gas Behavior”. Paper #92-0053, Advances in Instrumentation and Control, Proceedings of the ISA International Conference, 1992: 111-126.
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