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### Is your relief valve sizing method truly rigorous?

Pressure relief valves are devices that protect equipment from excessive overpressure.

**Ugur Guner**

Bryan Research & EngineeringBryan Research & Engineering

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**Article Summary**

In case of emergency situations, they should ensure a sufficient discharge of mass to reduce the pressure below the recommended pressure limits. The purpose of relief sizing is to determine the required area that can hold the required mass discharge from the valve under different overpressure scenarios. The discharged mass can be vapour, liquid, supercritical fluids or two-phase fluids. This article focuses on relief valve sizing methods for vapour phase and supercritical fluids at choked flow. The American Petroleum Institute (API Standard 5201,

Pressure relief valves (PRV) are the primary means of excessive overpressure protection. A pressure relief device is designed to open, relieve excessive pressure, reclose, and prevent further flow of fluid after normal conditions have been restored. PRVs consist of an inlet nozzle connected to the vessel, a movable disc that controls the flow through the nozzle, and a spring that controls the position of the disc. The operation of conventional spring loaded PRVs is based on a force balance. The spring-load is preset to apply a force that is opposite in amount to the pressure force exerted by the fluid on the other side when it is at the set pressure. When the pressure at the inlet of the valve is below the set pressure of the valve, the disc is seated on the nozzle to prevent flow through the nozzle.

The purpose of relief valve sizing is to determine the proper discharge area of the relief device and diameter of the associated inlet and outlet piping. Although an orifice is commonly used to describe the minimum flow area constricted in the valve, the geometry resembles a nozzle and area is determined by applying the equation for flow in an isentropic nozzle.

(1)

where A is the area of the valve, m is the mass flow rate through the valve, Gn is the mass flux and KD is the discharge coefficient.

The value of mass flow rate, m is determined by energy and mass balances on the vessel under the conditions of a specific relief scenario: a run-away reaction, an external fire, loss of cooling, thermal expansion of a liquid, control valve failure, etc. Calculation of mass flow rate discharged through the relief system is not within the scope of this paper.

The mass flux, Gn is calculated from either an appropriate theoretical model or numerically. The discharge coefficient KD accounts for the difference between the predicted ideal nozzle and the actual mass flux in the valve. It is available from the valve vendors.

It is important that the relief area be neither too large nor too small. An undersized valve would not provide the required overpressure protection, whereas an oversized valve will result in excessive flow. This can adversely affect the opening and closing characteristics of the relief valve, resulting in possible damage to the valve. Unexpected high flow due to oversizing also results in undersized discharge piping and effluent handling systems downstream of the valve. In addition, the cost of an oversized relief valve will be higher. Over predicted mass flux leads to an undersized valve, while under predicted mass flux results in an oversized valve. Hence, it is crucial to calculate mass flux correctly. In the next section, we will investigate theoretical models for mass flux through relief valves.

Mass flux in PRVs is modelled using an isentropic nozzle equation. The expression for the mass flux (G) in an ideal (isentropic) nozzle is obtained directly from an energy balance in the nozzle

(2)

where P1 is the pressure at the valve entrance, P is the fluid pressure, Pn is the downstream pressure (pressure at the nozzle exit or nozzle throat), Pn is the density at the nozzle exit (throat), and un is the velocity at the exit.

When a compressible fluid moves from a high-pressure upstream condition to a low downstream pressure across a nozzle, orifice, or a pipe, it expands. As a result, its density decreases and velocity increases. For a given inlet condition and with decreasing downstream pressure, the mass flux in the nozzle increases due to the expansion and flow area reduction until a limiting velocity is reached in the nozzle. This is called choked or critical flow. The limiting velocity is the sonic velocity of the fluid at the throat condition. The mass flux that corresponds to the sonic velocity is known as the critical mass flux. The pressure at which critical mass flux occurs is called the critical flow pressure. When the downstream pressure is lower than the critical flow pressure, mass flux will remain constant at the maximum value.

In order to solve Eq. 2 analytically, a relationship between pressure and density (or specific volume) is needed. For vapours and gases with a constant isentropic expansion coefficient, the expression for the pressure and specific volume relationship along an isentropic path can be shown as

(3)

where P1 is the pressure at the inlet, v1 is the specific volume at the inlet, and n is the isentropic expansion coefficient. P and V are the pressure and specific volume within the isentropic path. The major assumptions in the derivation of Eq.3 are that the gas follows an isentropic path and the isentropic coefficient is constant along this path.

Combining Eq. 2 and 3 and the definition of sonic velocity, the mass flux relation for choked flow can be obtained as;

(4)

In this equation, P1 and p1 are the inlet gas pressure and density, n is the isentropic expansion coefficient. The expansion coefficient is assumed to be constant along the isentropic nozzle path.

For gases, the density at the valve inlet can be written as

###

^{2}) recommends basing vapor-phase sizing methods on an ideal gas flow assumption.^{1}This assumption has been addressed by several groups in the industry and may lead to high levels of deviation in cases of near critical and supercritical fluids^{3,4}. In the most recent version of API 520, a rigorous approach for calculating mass flux through the valves is introduced in addition to the existing, ideal gas based models.^{2}Furthermore, API Standard 520 suggests using a real gas isentropic coefficient calculation method as an alternative to the ideal gas specific heat ratio for sizing relief valves. This paper will compare different vapour and supercritical vapour vent sizing approaches and compare their performance against a rigorous model. The rigorous model performs many isentropic flashes using appropriate thermodynamic packages offered in ProMax^{7}. The performance of each method is assessed through both pure component and mixture examples. In ProMax, a rigorous sizing method can be conveniently applied along with other alternative methods for API 520 sizing calculations^{7}.**Introduction**Pressure relief valves (PRV) are the primary means of excessive overpressure protection. A pressure relief device is designed to open, relieve excessive pressure, reclose, and prevent further flow of fluid after normal conditions have been restored. PRVs consist of an inlet nozzle connected to the vessel, a movable disc that controls the flow through the nozzle, and a spring that controls the position of the disc. The operation of conventional spring loaded PRVs is based on a force balance. The spring-load is preset to apply a force that is opposite in amount to the pressure force exerted by the fluid on the other side when it is at the set pressure. When the pressure at the inlet of the valve is below the set pressure of the valve, the disc is seated on the nozzle to prevent flow through the nozzle.

The purpose of relief valve sizing is to determine the proper discharge area of the relief device and diameter of the associated inlet and outlet piping. Although an orifice is commonly used to describe the minimum flow area constricted in the valve, the geometry resembles a nozzle and area is determined by applying the equation for flow in an isentropic nozzle.

^{5,6}The required orifice area for a relief valve is;(1)

where A is the area of the valve, m is the mass flow rate through the valve, Gn is the mass flux and KD is the discharge coefficient.

The value of mass flow rate, m is determined by energy and mass balances on the vessel under the conditions of a specific relief scenario: a run-away reaction, an external fire, loss of cooling, thermal expansion of a liquid, control valve failure, etc. Calculation of mass flow rate discharged through the relief system is not within the scope of this paper.

The mass flux, Gn is calculated from either an appropriate theoretical model or numerically. The discharge coefficient KD accounts for the difference between the predicted ideal nozzle and the actual mass flux in the valve. It is available from the valve vendors.

It is important that the relief area be neither too large nor too small. An undersized valve would not provide the required overpressure protection, whereas an oversized valve will result in excessive flow. This can adversely affect the opening and closing characteristics of the relief valve, resulting in possible damage to the valve. Unexpected high flow due to oversizing also results in undersized discharge piping and effluent handling systems downstream of the valve. In addition, the cost of an oversized relief valve will be higher. Over predicted mass flux leads to an undersized valve, while under predicted mass flux results in an oversized valve. Hence, it is crucial to calculate mass flux correctly. In the next section, we will investigate theoretical models for mass flux through relief valves.

MethodsMethods

Mass flux in PRVs is modelled using an isentropic nozzle equation. The expression for the mass flux (G) in an ideal (isentropic) nozzle is obtained directly from an energy balance in the nozzle

^{2, 5, 6.}(2)

where P1 is the pressure at the valve entrance, P is the fluid pressure, Pn is the downstream pressure (pressure at the nozzle exit or nozzle throat), Pn is the density at the nozzle exit (throat), and un is the velocity at the exit.

When a compressible fluid moves from a high-pressure upstream condition to a low downstream pressure across a nozzle, orifice, or a pipe, it expands. As a result, its density decreases and velocity increases. For a given inlet condition and with decreasing downstream pressure, the mass flux in the nozzle increases due to the expansion and flow area reduction until a limiting velocity is reached in the nozzle. This is called choked or critical flow. The limiting velocity is the sonic velocity of the fluid at the throat condition. The mass flux that corresponds to the sonic velocity is known as the critical mass flux. The pressure at which critical mass flux occurs is called the critical flow pressure. When the downstream pressure is lower than the critical flow pressure, mass flux will remain constant at the maximum value.

In order to solve Eq. 2 analytically, a relationship between pressure and density (or specific volume) is needed. For vapours and gases with a constant isentropic expansion coefficient, the expression for the pressure and specific volume relationship along an isentropic path can be shown as

^{2, 8};(3)

where P1 is the pressure at the inlet, v1 is the specific volume at the inlet, and n is the isentropic expansion coefficient. P and V are the pressure and specific volume within the isentropic path. The major assumptions in the derivation of Eq.3 are that the gas follows an isentropic path and the isentropic coefficient is constant along this path.

Combining Eq. 2 and 3 and the definition of sonic velocity, the mass flux relation for choked flow can be obtained as;

(4)

In this equation, P1 and p1 are the inlet gas pressure and density, n is the isentropic expansion coefficient. The expansion coefficient is assumed to be constant along the isentropic nozzle path.

For gases, the density at the valve inlet can be written as

^{8};###
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