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### Sizing relief valves for real gas or vapour

An appraisal of relief valve discharge area sizing methods for gas or vapour.

**TEK SUTIKNO**

Fluor EnterprisesFluor Enterprises

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**Article Summary**

At pressures and temperatures close to the critical regions, most gases or vapours generally do not behave like an ideal gas. The compressibility factor, Z, is reportedly a typical indicator of deviation from the ideal gas behaviour. Relief valve discharge area sizing methods derived from ideal gas assumptions could result in significant under-sizing of relief valves in gas or vapour services close to the critical regions. API 520 Part I states that the gas or vapour may be in these regions if the associated Z is less than approximately 0.8 or more than approximately 1.1, and three relief valve discharge area sizing methods for gas or vapour in these regions are described – isentropic expansion coefficient, Omega Leung method, and direct integration. These methods require a set of thermodynamic property data and may not lead to identical discharge areas.

The required property data can be conveniently generated from commercial process simulation programs, but the generated data could vary depending on the selected property package available in the program. Moreover, the three sizing methods stated by API may not result in the same relief discharge areas for a given relief case. This article discusses three examples of relief valve discharge area sizing cases using the ideal and real gas isentropic expansion coefficients and compares the results with those calculated by the direct integration methods. Additionally, the variation in the areas calculated from using different property packages is also presented.

Three different gas compositions are used as examples in the relief cases – a mixed butane stream in a refinery, a hydrogen rich stream in a hydroprocessing unit, and a natural gas from a pipeline. All of these examples involve gas phase relief only. A relief valve needs to operate adequately in low back pressure or high accumulated back pressure situations. The high back pressure case generally relates to an overpressure scenario where more than one relief valves are simultaneously discharging to a common flare header. This article focuses on the sizing calculations only for a choked (or critical) flow case where the choked pressure is higher than the relief valve back pressure.

The discharge flow of gas or vapour through a relief valve is generally assumed to follow an isotropic path described as P0ν0k = P1ν1k = constant, where P is pressure and ν specific volume. k, for ideal or perfect gas, is the ideal gas specific heat ratio, where Cp/Cv equals Cp(Cp-R). R is the gas constant; Cp and Cv are respectively specific heat at constant pressure and specific heat at constant volume. For relief valves in gas or vapour service where the back pressure is lower than the choked or critical pressure, Equation 1 from API 520 Part I can be used to calculate the required discharge area of the relief valve:

(1)1

where:

A – required discharge area, in2

W – required relief load, lb/hr

T – relief inlet temperature, oR

P1 – inlet pressure, psia

Z – Compressibility factor

M – molecular weight, lbm/lbmol

Kd, Kb, and Kc are respectively effective coefficient of discharge, back-pressure correction factor, and correction factor for rupture disc installation. These k factors are hereafter referred to as Ktotal. The value of Ktotal is assumed the same in all calculation cases for a relief gas/vapour stream example presented in this article. For the mixed butane, as an example, Ktotal remains at the same values for all area sizing cases for this mixed butane. C is a function of the ideal gas specific heat ratio (k = Cp/Cv) of the gas or vapour at inlet relieving temperature and defined in API 520 Part I as Equation 2:

(2)2

With the value of ideal Cp/Cv derived from the property package PP A1 available in a commercial simulation program, the discharge area can be calculated from Equations 1 and 2.

As indicated in API 520 Part I, deviation from ideal gas behaviours becomes significant and needs to be evaluated when Z factors are approximately less than 0.8 or more than 1.1. For relief cases where the gas or vapour is in the non-ideal region with a Z factor less than 0.8 or more than 1.1, the k can be replaced with n, the isentropic expansion coefficient, and the reversible adiabatic expansion path is described to follow P0ν0n = P1ν1n = constant. Derived from this pressure and specific volume correlation, Equation 3 can be used to estimate the values of n. Equations 4 and 5 can be used to calculate the mass flux (G, lbm/sec/ft2) and the required discharge area A. Kv in Equation 5 is the viscosity correction factor. Again, the various K factors are assumed to have the same values in each area calculation for a specific relief example case in this article:

n = ν ( ∂P) T Cp (3)

P ∂ν Cν

(4)3

(5)3

To estimate the values of n’s using Equation 3 for the three relief examples, the values of ν and Cp/Cv are generated from the selected property package available in the commercial simulation program. The gas or vapour at the relief valve inlet condition is expanded at a constant temperature and at a differential pressure increment to obtain the values of ν and Cp/Cvat pressures lower than the relief valve inlet pressure (P1), and the (∂P/∂ν)T at a constant temperature is estimated by taking the derivatives of a correlation equation developed from the isothermal plot of P versus ν. The values of n generated at each low pressure point are used to calculate the mass flux, G’s from Equation 4, and the maximum G value calculated corresponds to the choked velocity at Pt, the throat pressure at the relief valve minimum discharge cross-sectional area. The minimum discharge area (A) required for the relief valve can be calculated using Equation 5 with the maximum calculated G value.

Table 1 summarises the calculated discharge areas required for the mixed butane relief example using the isentropic adiabatic expansion method based on ideal gas and non-ideal gas. Several alternative property packages are also used to generate the parameter values needed for the sizing calculations. These packages are commonly available in commercially available software programs and have been regularly used for process simulation of hydrocarbon systems. Property Package PPA1 (Peng-Robinson) is the same as PPA, except different estimating options for enthalpy and density are used in PPA1.

The required property data can be conveniently generated from commercial process simulation programs, but the generated data could vary depending on the selected property package available in the program. Moreover, the three sizing methods stated by API may not result in the same relief discharge areas for a given relief case. This article discusses three examples of relief valve discharge area sizing cases using the ideal and real gas isentropic expansion coefficients and compares the results with those calculated by the direct integration methods. Additionally, the variation in the areas calculated from using different property packages is also presented.

Three different gas compositions are used as examples in the relief cases – a mixed butane stream in a refinery, a hydrogen rich stream in a hydroprocessing unit, and a natural gas from a pipeline. All of these examples involve gas phase relief only. A relief valve needs to operate adequately in low back pressure or high accumulated back pressure situations. The high back pressure case generally relates to an overpressure scenario where more than one relief valves are simultaneously discharging to a common flare header. This article focuses on the sizing calculations only for a choked (or critical) flow case where the choked pressure is higher than the relief valve back pressure.

**Isentropic expansion**The discharge flow of gas or vapour through a relief valve is generally assumed to follow an isotropic path described as P0ν0k = P1ν1k = constant, where P is pressure and ν specific volume. k, for ideal or perfect gas, is the ideal gas specific heat ratio, where Cp/Cv equals Cp(Cp-R). R is the gas constant; Cp and Cv are respectively specific heat at constant pressure and specific heat at constant volume. For relief valves in gas or vapour service where the back pressure is lower than the choked or critical pressure, Equation 1 from API 520 Part I can be used to calculate the required discharge area of the relief valve:

(1)1

where:

A – required discharge area, in2

W – required relief load, lb/hr

T – relief inlet temperature, oR

P1 – inlet pressure, psia

Z – Compressibility factor

M – molecular weight, lbm/lbmol

Kd, Kb, and Kc are respectively effective coefficient of discharge, back-pressure correction factor, and correction factor for rupture disc installation. These k factors are hereafter referred to as Ktotal. The value of Ktotal is assumed the same in all calculation cases for a relief gas/vapour stream example presented in this article. For the mixed butane, as an example, Ktotal remains at the same values for all area sizing cases for this mixed butane. C is a function of the ideal gas specific heat ratio (k = Cp/Cv) of the gas or vapour at inlet relieving temperature and defined in API 520 Part I as Equation 2:

(2)2

With the value of ideal Cp/Cv derived from the property package PP A1 available in a commercial simulation program, the discharge area can be calculated from Equations 1 and 2.

As indicated in API 520 Part I, deviation from ideal gas behaviours becomes significant and needs to be evaluated when Z factors are approximately less than 0.8 or more than 1.1. For relief cases where the gas or vapour is in the non-ideal region with a Z factor less than 0.8 or more than 1.1, the k can be replaced with n, the isentropic expansion coefficient, and the reversible adiabatic expansion path is described to follow P0ν0n = P1ν1n = constant. Derived from this pressure and specific volume correlation, Equation 3 can be used to estimate the values of n. Equations 4 and 5 can be used to calculate the mass flux (G, lbm/sec/ft2) and the required discharge area A. Kv in Equation 5 is the viscosity correction factor. Again, the various K factors are assumed to have the same values in each area calculation for a specific relief example case in this article:

n = ν ( ∂P) T Cp (3)

P ∂ν Cν

(4)3

(5)3

To estimate the values of n’s using Equation 3 for the three relief examples, the values of ν and Cp/Cv are generated from the selected property package available in the commercial simulation program. The gas or vapour at the relief valve inlet condition is expanded at a constant temperature and at a differential pressure increment to obtain the values of ν and Cp/Cvat pressures lower than the relief valve inlet pressure (P1), and the (∂P/∂ν)T at a constant temperature is estimated by taking the derivatives of a correlation equation developed from the isothermal plot of P versus ν. The values of n generated at each low pressure point are used to calculate the mass flux, G’s from Equation 4, and the maximum G value calculated corresponds to the choked velocity at Pt, the throat pressure at the relief valve minimum discharge cross-sectional area. The minimum discharge area (A) required for the relief valve can be calculated using Equation 5 with the maximum calculated G value.

Table 1 summarises the calculated discharge areas required for the mixed butane relief example using the isentropic adiabatic expansion method based on ideal gas and non-ideal gas. Several alternative property packages are also used to generate the parameter values needed for the sizing calculations. These packages are commonly available in commercially available software programs and have been regularly used for process simulation of hydrocarbon systems. Property Package PPA1 (Peng-Robinson) is the same as PPA, except different estimating options for enthalpy and density are used in PPA1.

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